| < + d 2( + d 3( + • • • ad inf.)
= F r^r<
Now let P be the annual payment to be found which will be
substituted for the periodical payments of F.
Then the present value of all the payments of P, assuming the
first to be due now, is
¥(l + v + v 2 + v 3 + ■ ■ ■ adinf.)
= P _L = pi+i
1 —J! I
Now the present values of these two series of payments must
be equal to one another, and we therefore have
P^- = F l
1 - v 1 - v l
whence P = F -= ;
1 -v'
-F-iL
chap, ii.] THEORY OF FINANCE 15
But we see that the annual sum payable in advance for t years
which a sum of F payable now will purchase is
^ ' F ' . l = F- d
(1-M>- l+»l-u« 1-d'
Our result is thus confirmed by general reasoning.
Suppose now the first payment of the duplicand be due t years
hence ; the present value of all the payments is then
F(v t + v°- t +v^ + ■ • ■ ad inf.)
= F
= F
1-D*
1
(i+O*
And the present value of all the annual payments of P, the first
being assumed to be payable a year hence, is
P( u + „2 + l ,s + . . . ad inf.)
- P-i
i
We therefore have in a similar way as before
PI=F, l
i (1 + if - 1
(i+iy-i
Now the annual payment which requires to be set aside to
accumulate to the sum F due at the end of t years is
1
F— =F-
s- (l+.-y-i
this result also being arrived at by general reasoning.
6. The schedule given in Article 39 illustrating the redemp-
tion of a sum by equal payments including principal and interest
is very instructive. It is shown how the capital contained in the
mth payment of the annuity is „»-m + i. We also know that
the capital contained in the first payment is — , and in the
16 ACTUARIAL THEORY [chap. ir.
K
second is — (1 +i), because the interest on the first repayment
Vj
of capital has been released and must be utilised to increase the
capital contained in the second payment. Similarly in the third
instalment the capital is — (1 + i) 2 , and generally in the mth
instalment the capital is — (1 +i) m - 1 .
K K
That — (1-M') m-1 = — v n-m+i j s easily proved.
s — - ct — .
n\ 7i I
For — (l+t)™- 1 = ±- — '-
By the first way of looking at the matter, the repayments of
capital in t years amount to
K K K
— («» + »»-i + • • • + v n ~ f + !) = — (a-. - a—,.) = K a—,
a _ \ ' a V rt| n-tV a n-t\
n| ft| i|
And by the second the total capital repaid in t years is
« I 71 1
These two expressions are identical, for
K K K
— * n = - — " re Vi = — «»{l+(l+i)+ •■• +(l + n ( - 1 }
7t | 71 J 71 |
= («« + t!« - 1 + ... + V n - ' + 1- )
a- K '
*l
K r \
»|
If K
= K - — a .
a— «-*l
7l|
Again, the capital returned in the first payment is Jj",in
a—
»
the second — a™- 1 , and so on, and in the last it is — v. Now
chap, n.] THEORY OF FINANCE 17
the present value of the capital in the first payment is
K K K K
v — v 71 = — v n + 1 , of that in the second v" — v 71 ' 1 = — v n + 1
a— a— a— a—, '
K K
of that in the last v n — v = — r u + l . Therefore the total
a—, a— :
n\ n\
value of all the capital repaid in the n instalments is n — v n + 1 .
This expression is of use in ascertaining the value to be paid for
an annuity-certain allowing for income-tax, when tax is deducted
from the whole annual payments without regard being had to the
proportions of capital contained therein. It is obvious that, for an
annuity of 1 for n years, a purchaser in these circumstances should
not pay a— , but should deduct the value of income-tax on capital
at t per unit, or tnv n+1 . Thus the net price paid for the annuity
will be a— — lnv n + l . This result is necessarily only approximate,
as an adjustment should now be made for the reduction of interest
following on the reduction of capital invested, and for the con-
sequent increase of capital returned in the successive payments of
the annuity.
In making up a schedule such as that given in Article 39, it
should be carefully noticed that it is only necessary to work out
the figures in column (3). The first value in this column is
The succeeding values are obtained by continued multipli-
'SI
cation by (1 + 1). The figures in all the other columns are obtained
from those in column (3). In forming the schedule in this way,
however, a periodical check should be applied, the figure in column
(3) opposite m being (1 +t) m_1 .
When the annuity is payable q times a year, it should be
assumed that interest is convertible at the periods of payment of
the annuity. The schedule should then be formed in respect of
an annuity for nq intervals at rate of interest — , bearing in mind
the formula previously found, namely
a— = — a — i at rate of interest —
n| q ml q
18 ACTUARIAL THEORY [chap. ii.
7. Jn the circumstances mentioned in Article 40, where an
investor has lent money, repayable by an annuity and yielding a
given rate of interest, say z, but where he is able to accumulate the
sinking fund returned to him annually at a lower rate only, say i' ,
it will be seen that for an advance of 1, the borrower must pay
interest amounting to i per annum, and also the sinking fund at
rate i' to replace the advance of 1 or - — In other words,
**l
1 is the value of an annuity of i + — — , and by proportion
s'—
^ — is the value of an annuity of 1 per annum.
1 + tY-.
n\
To find in such a case the amount of capital outstanding at the
end of t years.
If K was the original advance, the annual payment being
V 'si
the sinking fund will have accumulated at i' to
K ,
ii. i
in t 3'ears. Now, if the borrower be asked to repay the capital
outstanding for the convenience of the lender, he should pay only
the balance outstanding after deduction of the accumulation of
17"
sinking fund, that is, K - — — s'—.
s—, 'I
If, on the contrary, it be to the borrower's convenience that he
should repay the balance of capital, the lender must receive such
a sum as will enable him to purchase an annuity of K ( i + - —
' S ni
for the remainder of the term, that is, K| i a \a , the
value of the annuity being calculated at rate i', as that is the rate
returned by investments elsewhere.
The third case may, however, arise where both parties desire
to end the contract, and in such circumstances it will be sufficient
if the lender get such a sum as will enable him to set up a similar
chap, n.] THEORY OF FINANCE 19
contract for the remainder of the terra. That is, he should get
the value of an annuity of K f i + - — j for the unexpired period
on the same terms as the original annuity was calculated. We
saw that the value of an annuity of 1 for the whole n years was
1 — £-',— . Hence we get the value of an annuity of K ( i -\ — ; —
for (n-t) years as K ( i + _j_\ ' j^L_.
V s-Jl+ts —
8. The general rule given in Article 43« for finding the present
value of a series of payments of any amounts, to be made at any
times, the value to be so calculated as to yield the purchaser the
remunerative rate, i, on his whole investment throughout the
longest of the periods, n years, and to return him his capital at the
accumulative rate, i', should be most carefully noted, as it is in-
valuable in finding the present value of varying annuities of this
nature. It is sufficient to know the first part of the rule ; namely,
that the present value may be found by multiplying the amount
accumulated at rate i ' to the end of the n years of the series of
payments by — —
9. In Article 50 it is shown how to approximate to the rate of
interest by means of Finite Differences, given the value of the
annuity and the term. By the same means an approximation may
be made to the value of an annuity at a rate intermediate between
the rates in a given table of values.
The general formula may be stated in the form
1(4 -
h
I'nx+h = Unx + T ;1 «b + ^, &Unx+ ' '
where the values at intervals of x in the rate of interest are given.
For example, if tables of values at 3 per cent., 3| per cent.,
4 per cent., etc., be given, and it is desired to find the value at,
say, 3J per cent, we have
a (3j%) = a (3%) + S Aa (3%). + — 9 -- a \»:j
20 ACTUARIAL THEORY [chap. ii.
For a term of 20 years,
0(8jx) = 14-87748 + f(- -66508) + fx -04301
= 14-87748 --99762 + -01613
= 13-89599
10. With reference to Article 65, a, it should be noted that the
value of an annuity-certain for n years deferred t years, interest
at rate i during the first t years and at rate j thereafter, can be
conveniently expressed only in the form
„ .,, l-(l+i)" TO
(l+i)' 1 v . JJ — .
J
It should not be written in any modification of the formula
id!— = a- — a—
t] «| n+t\ t\
11. To find the annual premium payable in advance for t years
required to provide an annuity-certain for n years, the first payment
of which is to be made at the end of t years.
The value of the benefit to be obtained is v t ' l a— ,.
»|
The value of the payments to be made to secure this benefit
(P being the required annual premium) is
?(l+v + v 2 + ... -l-i'*- 1 ) = Pa-,
Now the value of the benefit must equal the value of the
payments made for it, whence we have
Pa— = « (-1 fl_
t\ n\
and P = ^
If the premium be payable half-yearly, we have as before the
benefit side = w t-1 a— , and the payment side
=iM i+ ir+(> + ir--+(>+ir"}
— a— where interest is at rate —
chap, ii.] THEORY OF FINANCE 21
p c' -1 a—
Therefore — = 'lL
2 1 + a .
2(-l|
u^zti being calculated at rate — and the other functions at
rate i,
12. Sinking-Fund Assurances are of importance, as they are more
frequently in use than formerly was the case. They are employed
to provide sums required for the redemption of debenture issues at
their due date, to return at the expiry of a lease the capital sum
paid for property held on leasehold, and, in short, to secure the
payment of a sum of whatever nature at the end of a term certain.
The present value of such a sum, that is, the single premium
to secure it, is v n .
Putting P— for the annua] premium to secure this benefit,
the present value of the premiums is
whence P-,(l+rt — ri ) = v n
and P-, =
I 1+a^n
If the premium be payable p times a year, we have the payment
side equal to
P [ f i \-i / i \-2 /, i \-(p-D
|[-(-r + ( i+ r-- + ( i+ i
+
p(p>
= JTjfl + a r .)
p
where a is calculated at rate — .
7'J) - 1 | p
From this we get
p(3>)
ml
V n
p 1 + a-
1 ?l
1ip~l\
We have so far made but a simple application of the formulas
already obtained, interest being assumed to remain constant
throughout the whole terra of n years. It is, however, the case
that the rate of interest has shown a tendency to decline for many
22
ACTUARIAL THEORY [chap. ii.
years, though of late this tendency appears to have received a
check, which however is probably of only temporary effect. It
will in any event be prudent to make allowance for such a fall,
and we must seek formulas to give effect to this consideration.
Suppose the rate of 3£ per cent, to hold for 10 years, there-
after falling £ per cent, every 10 years till a minimum of 2 per
cent, is reached. Then the value of 1 due at the end of (10 + m)
years (m < 10) is v (H) x l> (3J) .
At the end of (20 + m) years (m < 10), the value is
10 io
V) x v m x V
At the end of (60 + m) years the value is
in in 10 10 10 ' 10 m
"m x v w x v m x "«) x «'<»> x v w x V
where m has any value.
The present value of the annual premiums where the sum is
due at the end of (10 + m) years (m < 10), is
P ( a fi|M) + "(Si) a m|(3i))
10 m
and P = p w) x w v m
a i0| (Si) + "(Si) a m| (3i)
Similar values of P where the sum is due at the end of
(20 + m), (30 + m), (40 + m), and (50 + m) years, m in each case being
less than 10, may be found.
Finally, when the sum is due at the end of (60 + m) years, m
being of any value, we have for the value of the annual premiums
u / 10 J. 10 10 4.
1 t a ibl(34) + \h) a io|(3j) + "(s« "(3» a iol(3) + '
10 10 10 10 10 10 s
+ v m v w v m V}) Vi) Vd^h^)-
And, the value of the benefit being as found above, we may at
once determine the value of P.
13. To find the value of an annuity-certain of 1 for n years
paying the purchaser a desired rate of interest and securing by
a Sinking Fund policy the return of his capital with one year's
interest at the end of the year following the last payment of the
annuity.
A purchaser would pay 1 for an annuity-due for (n + 1) years
of (P + d), where P is the premium payable in advance
chap, n.] THEORY OF FINANCE 23
charged by an office for a Sinking Fund policy of (n + 1) years
term, and d is the interest in advance on 1 at the rate desired.
For an annuity for n years of (P^rj-r + d) he would therefore pay
1 — (P.-^ttt + *0> an< ^ f° r an annu ity of 1 for n years he would pay
1
P .-TT, +
1.
Again, for the annuity of (P + d) for n years, we saw that
the policy effected was for 1, and therefore for an annuity of 1 the
policy will be for — ,, and the annual premium will be
r ' P , + d
P
P ^n + d '
We have now to see
(1) What the total capital invested is ;
(2) How each annual payment is divided between interest and
premium ; and
(3) Whether the policy returns the capital invested with one
year's interest at the end of (re + 1) years.
(1) The value paid for the annuity is, as above, . — - 1
n+U
But in addition the purchaser must pay the first P ,T+T]
premium on the Sinking Fund policy, which is P + d
Therefore the total capital invested is . . =r ,
(2) Each annual payment is 1,
whereof there is
v d
Interest on ^ of capital . . . p ,
p
it+ii
• P__. + d
»+i|
And premium on policy
Together .
24 ACTUARIAL THEORY [chap. ii.
(3) The capital invested is, as before, . . . p —-,
One year's interest thereon
P ^ + <
Together, making up the amount payable 1
under the policy .... P^rr, + d
If P , be a net premium, and calculated at the same rate as
d, then the price paid for the annuity, -, - 1, is equal to
»+i|
a— , the value of an annuity-certain for n years.
For, since P — =-= = = . ( i )
»+i| (1+0* l+i\a— Tl )
1 -d
& — r^r
» + l|
P-rr.+d
«+i| a-^
1
= a-
P-^.+cf - "»+H
»+i|
And ~ , — 1 = a— ^-. - 1
EXAMPLES
1. If an annuity-certain is payable twice a year, interest con-
vertible four times a year, and the effective rate of interest is i,
what is the amount of the annuity in n years ?
The general formula to be applied is
. (> + iJ
p
(' ♦ i )'
l
CHA1> - "•] THEORY OF FINANCE 25
where j is the nominal rate of interest convertible q times a year.
But by the terms of the question U + J-Y = (1+i) where
q = 4. Also p = 2. Therefore we have
1 (1 +iy - i
* _ 2(1+0* -1
2. Find the amount per annum payable momently, interest
convertible momently, for n years, corresponding to a yearly
payment of a for n years, interest convertible yearly.
Let K be the amount per annum required. Then the value
1 — e ~~ n ^
of K for n years will be K , — , which must be equated to
the value of the payments of a, that is, to a ^ — : — i- — .
i
We therefore have
l_ e -m l-(l + i)-»
K 5 — (l :
6 i
whence K = a i— : — '- — x
1 -«-«»
3. An annuity-due of 1 per annum is to be allowed to accumulate
until the first payment has doubled itself. Assuming that this occurs
at the end of an integral number of years exactly, find what is then
the amount of the annuity. Prove the result by general reasoning.
If n be the number of years it takes the first payment to double
itself, we have the amount of the annuity-due at the end of that
time equal to
(l+i) + (l+«) 2 + • • • +(l+»)*
(1 + Q" - 1
i
2-1
= (1+0
= (1+0
since (1 + i) n = 2
1+i
This is the value of a perpetuity-due of 1 per annum, and our
result is easily proved to be correct. For, the first payment having
accumulated to 2, of this 1 may be paid away and the remaining
1 accumulated for n years further, while the second and succeeding
payments will accumulate to 2 in succession, yielding 1 per annum
26 ACTUARIAL THEORY [chap. ii.
to be paid away and 1 per annum to be re-invested and accumu-
lated, and so on, ad infinitum ; all which is obviously the value of a
perpetuity-due.
4. Find the value of an annuity-certain of 1 payable half-
yearly for 48 years and 48 days at £3, 3s. 2d. per cent, interest,
given log 1-01579 = -006804 and log -22132 = T-345027.
We must assume here that interest is convertible half-yearly,
and then remembering that where both annuity is payable and
interest convertible p times a year a_ = — a — | at rate of
p np\
interest — , we have
P
«+£
(2) _ J_ "(-01579)
lstJ _W, 2 -01579
a
96 +H
To evaluate v , we have
„-£ . (» + |L)
log « + 865 =-. (96+^) loffW
= _ ( 96 + S) lo S 1-01579
" -( 96 + 3l) x - 006804
Therefore a
= T-345027
= log -22132
( 2) _ 1- -22132
m ~ -03158
4S+ 365|
= 24-657.
5. Each payment of a perpetuity is divisible equally among five
funds. It is arranged that, instead of the perpetuity being shared
as at present, four of the funds should for a fixed number of years
each in succession, receive the annual payments in full, and that
the fifth fund should be entitled to the perpetuity in full
thereafter.
chap, ii.] THEOBY OF FINANCE 27
Find the number of years which must elapse before the fifth
fund comes into possession.
Here the benefit which the fifth fund is procuring is a perpetuity
deferred t years, and the benefit it is forgoing is a fifth part of a
perpetuity, and these two must be equal.
Hence v l — = — — r-
and t = -r^ —
log v
6. Find the value at 3 per cent, of an annuity-certain for 30
years, the annual payment to be reduced by one-half after the end
of each period of 10 years. Given v 10 at 3 per cent. = -74409.
Here the value of the annuity may be written
a = a iol + T' ,10 "iol + T t,20a iol
v l-" 10 1- '74409 8 _„ n
Now a— . = : — = ^-5 = 8-530
iol i -03
t-io _ = -74409x8-530 = 6-347
10;
and v™a^ = -74409x6-347 = 4-723
Therefore a = S-530 + -V x 6-347 + — x 4-723
2 4
= 12-884.
7. A shareholder in a life company holds £2000 of its paid-up
capital, the dividends on which are increased 10 per cent, every
quinquennial valuation. Supposing a valuation to have just taken
place and the dividends for the next five years to be fixed at £100
per annum, what is the value of his interest in the undertaking
upon a 5 per cent, basis ?
The annual payment for the first five years is 1 00, for the next
five 100x1-1, for the next five lOOx(l-l) 2 , and so on. The
present value of all these payments is
100a- + u 5 100(l-l)a F1 + i)i°100(l-l) 2 a rj + • • •
= 100« S] {l + r3(M) + «»(M) 2 + • • •}
= 100a-:
•II -^(M)
which at 5 per cent, is equal to £3134-547.
28 ACTUARIAL THEORY [chap. ii.
8. Calculate the price to be charged for an annuity-certain of
£25 for 30 years, assuming 4 per cent, interest for the first
10 years, decreasing thereafter by A per cent, per annum each
period of 10 years. (Use the Tables given at the end of the
Theory of Finance.}
We have
10 10 10
a - fl i0|(4%) + V (4%) a i0|(3i%) + V%)"(3K) a iO|(3%)
= 8-11090 + (-675564 x 8-31661)
+ (•675564 x -708919 x 8-53020)
= 8-11090 + 5-61840 + 4-08528
= 17-81458
and 25xa= 445-3645
= £445, 7s. 3d. nearly.
9. In connection with a feu-duty of £20 per annum, a duplicand
is payable every 21 years, the next being due 7 years hence.
Find the present value at 4 per cent, of all future duplicands ;
and the equivalent addition to the feu-duty if all duplicands be
dispensed with. Given v 3 = -88900, u 4 = -85480.
The present value of all future duplicands is
20(i> 7 + u 28 + d 49 + • • ■ ad inf.)
v 7
= 20—- —
1 - u 21
Now j' = i^xt* = -88900 x -85480 =-75992,
and d 21 = (v-f = -43883.
Therefore the value of the duplicands
- 20 x -75 " 2
" 20x I--43883
= 27-083.
This may be looked on as the benefit.
If, then, the annual addition to the feu-duty be P, we have the
payment side
= P^ = 25P.
Equating the two sides, we have
25 P = 27-083,
whence P = 1-083.
= £1, Is. 8d.
•]
THEORY OF FINANCE
29
10. The annual value of a certain property is £20, and this
increases each year by 1 per cent. The property is subject to a
feu-duty of £8 a .year, but the feu-duty payable at the end of
every 21st year from the present time is to be, not £8, but the
full annual value of the property at that time. Give a formula for
the present value of the feu-duty.
The value of the annual payment of £8 in perpetuitv is
obviously — r- .
But every 21st year the £8 is not receivable, and the deduction
on this account is therefore
8(»a + «* 2 + « 88 + • • • ad inf.)
~ 1 _ „21
Instead of the £8, there is receivable the full annual value of
the property, at the end of every 21st year, and the present value
of this is
20(l-01) 21 « 21 + 20(l-01) 4 V 2 +20(l-01) 63 t.63+ • • • ad inf.
20(1 -01)211)21
1 -(l-01)2ii,2i
Therefore the full value of the feu-duty is expressed by the
formula —
4- -8
(1-01)2^21
l~"pi ^ ""1 - (l-01)2»t> 2
^+20
11. Construct a schedule showing the repayment of a loan of
£1750 by means of an annuity-certain for 4 years payable half-
yearly at 5 per cent, interest.
Half-
Year.
Interest
contained in
each Payment.
Principal
contained in
each Payment.
Principal
Bepaid
to Date.
Principal
still
Outstanding.
l
43-750
200-318
200-318
1549-682
2
38-742
205-326
405-644
1344-356
3
33-609
210-459
616-103
1133-897
4
28-348
215-720
831-823
918-177
5
22-954
221-114
1052-937
697-063
6
17-427
226-641
1279-578
470-422
7
11-761
232-307
1511-885
238-115
8
5-953
238-115
1750-000
30
ACTUARIAL THEORY
[chap. II.
12. Construct a similar schedule of the repayment of a loan of
£1300 in 7 years at 3 J per cent.
Interest
Principal
Principal
Principal
Year.
in Annual
in Annual
Repaid
still
Payment.
Payment.
to Date.
Outstanding.
1
45-500
167-108
167-108
1132-892
2
39-651
172-957
340-065
959-935
3
33-598
179-010
519-075
780-925
4
27-333
185-275
704-350
595-650
5
20-848
191-760
896-11-0
403-890
6
14-136
198-472
1094-582
205-418
7
7-190
205-418
1300-000
13. A loan of £10,000, bearing interest at the rate of 4 per
cent, per annum, payable half-yearly, is to be repaid by 40 equal
half-yearly payments, including interest and instalment of principal.
Having given (1-02)- 20 = -67297, find
(a) The amount of the half-yearly payment.
(6) The amount of principal included in the first and in the
twenty-first half-yearly payments respectively,
(c) The total amount of principal repaid, after payment of the
twentieth half-yearly sum.
(«) The half-yearly payment —
10000 x -02
1
"(»»
40| <2%>
200
1- (-67297)2
= 365-557.
(b) The principal in the first payment
10000
x «40 = 365-557 x (-67297) 2
40 1
= 165-557.
That in the twenty-first payment
= 10000 x v20 = 365.557 x . 6729 7
= 246-009.
chap, ii.] THEORY OF FINANCE 31
(c) The total principal repaid after the twentieth
payment
lnnnn 10000
= 10000 - x a—,
a-, 20|
40]
= 10000 - 365-557 x l ~ " 67397
•02
= 10000 - 5977-405
= 4022-595.
14. Given a— at 4 per cent. = 15-6221, and a— at the same
rate = 8-1109, find the capital included in the 15th payment of
the former annuity.
The capital included in the 15th payment of a—.
_ j,25 -15+1 = v \l
the value of which is found as follows : —
l_„io
«S5I
i
whence
v io
=
1 ~ ia ro l
=
1- -04x8-1109
=
•675564
Therefore
„n
=
■675564
1-04
=
■649581.
15. A life office advances £1000, repayable in 10 years, by an
annuity to secure interest at the rate of 5 per cent., and provide
for the accumulation of the sinking fund at the rate of 3 per cent.
When the sixth annual payment becomes due, the borrower desires
to cancel the arrangement and repay the loan at once. Find the
amount of capital actually outstanding and state what sum you
would advise the office to accept in satisfaction of its claim.
The capital outstanding is the original loan less the accumula-
tion of sinking fund,
= 1000 - 100Q x (l-03>- nv . = 522-988
„_ v ' 5 1 (3%)
*10| (3%)
But the redemption money should be the value of an annuity-due,
32 ACTUARIAL THEORY [chap. h.
of the same amount as the office was receiving, for 5 years at
3 per cent.
= 1000(_L_ + ■05)(l+a r ) = 647-330
V 10| (3%) '
16. Given that the amount of an annuity-certain of 1 is
26-87037, and that the present value of the same annuity is
14-87748, find the rate of interest.
Using the formula
1 . _L
a—, s
we have
In the present example
1 1
a—, s—
n\ n\
~~ 14-87748 26-87037
= -067216 - -037216
= -03.
17. A perpetuity of £7 , 10s. payable yearly, and a composition of
£7 , 10s. payable at the end of the 10th and every 20th year there-
after, are to be redeemed by an annuity payable half-yearly for 30
years. Find the amount of the annuity, taking interest at 4 per cent.
Here it will be convenient to find separate expressions for the
value of the old benefit which is being given up and for the value
of the consideration which is taking its place, equating the two
thereafter to ascertain the amount of the annuity.
The Benefit Side = 7-5 j-L + (u 10 + u 30 + d 50 + . . a d inf.)\
,./l V™ \
= 7 ' 5 (t + t^o)
- 7-5(25 + :°™^
\ -543613/
= 196-8205.
Now, if P be the half-yearly payment, the Payment Side
= P x 34-7609
interest being assumed to be convertible at the periods of payment
of the annuity.
chap, ii.] THEORY OF FINANCE 33
Equating, we have
Px 34-7609 = 196-8205
, „ 196-8205
and P = 377609
= 5-662.
= £5, 13s. 3d. nearly.
18. If a sum of £1,000,000 be borrowed at 4 per cent, interest,
payable annually, and £60,000 be applied each year towards
paying the interest and reducing the principal, in what time will
the loan be finally discharged ?
Here we have 1,000,000 = 60,000 a—, where n is unknown.
n\
To find n we proceed as follows : —
6a-
»l
=
100
»l
=
16|
1 -v n
=
§
v n
=
i
IT
n log D
=
-log 3
n
=
-log 3
-log 1-04
- -4771213
- -0170333
=
28-01.
The time is therefore practically 28 years.
19. An annuity of £50, payable by half-yearly instalments for
20 years, is bought at 14 years' purchase. Find, approximately,
the rate of interest realised by the purchaser.
It should be explained that, when " r years' purchase " is
spoken of, it means that the price paid is r times the annual
rent of the annuity. In this case £700.
[Effective rate = 3f per cent, almost.
20. An annuity-certain for 35 years is bought at 20 years'
purchase. What rate of interest is made on the investment ?
[Rate of interest = 3| per cent, very nearly.
C
34 ACTUARIAL THEORY [chap. ii.
21. An annuity of £80, payable in half-yearly instalments for
25 years, is bought for £1400. Required the half-yearly rate of
interest which is made on the investment.
Here we have 1400 = 40 a—,
\ 50|
Hence a—, = 35.
50 1
Rate of interest = 1 J per cent, very nearly.
22. Find the rate of interest at which a is calculated when
it = 16-938, given annuity values for the same term as follows : —
3 per cent. 17-413
31 per cent. 16-482
4 per cent. 15-622
Using formula (43) of this chapter in the Theory of Finance,
we have
AnK - -931+ -0355
= -0025, approximately,
whence i = -03 + p = -0325, or 3 \ per cent.
23. From the tables given at the end of the Theory of Finance,
calculate the value at 3J per cent, of an annuity-certain for
20 years.
The formula to be used is
i/i-i
where A a and A 2 a represent the successive differences of a
\p/ ) \ 6 / ) (3%),
a (U7V an< ^ a (i°/) ^ or a P el 'i°d °f 20 years.
The true value of a—, at 3J per cent, is 14-539.
24. From the tables given at the end of the Theory of Finance,
calculate the amount at 7\ per cent, of an annuity-certain for
25 years payable half-yearly, interest convertible half-yearly.
chap, ii.] THEORY OF FINANCE 35
The formula to follow is
A(A-i)
where A ar^ and A** m) represent the successive differences
° f ,S (3J%)' V» and S (HX) for a P eriod of 50 years.
The true value of \ s at 3|- per cent, is 68-032.
Neither in this question nor in the previous one will the formula
quite give the true value of the function, as second differences
are assumed to be constant.
25. Assuming one rate of interest throughout, obtain pro-
spectively and retrospectively the value of a Capital-Redemption
Policy of 1 taken out n years ago for a period of t years at an
annual premium of P_, and prove the identity of the two
expressions.
Before attempting this question, the student should know
something of prospective and retrospective policy-values, though
he will come more in contact with them when discussing life-
policies at a later period.
When a capital-redemption policy is entered upon, the value of
the benefit to be ultimately received is exactly equal to the value
of the series of premiums to be paid therefor. As time goes on
and the date of payment approaches, the value of the capital sum
obviously increases, while on the other hand the premiums to be
paid are fewer and their value consequently decreases. Thus the
value of the benefit now exceeds the value of the premiums still to
be paid. For this difference the office must keep a sum in hand
which is called the " policy-value." The policy- value has here
been looked at from the " prospective " point of view.
But again, after the policy has been in force for a number
of years, the premiums which the office has received have been
invested and accumulated (at the rate of interest assumed in the
calculations). These accumulations constitute the value of the
policy, which has here been discussed from a " retrospective " point
of view.
In the case in the above question the value of the benefit at
36 ACTUARIAL THEORY [chap. ii.
the commencement of the contract is v l ; and the value of the
premiums is P-,(l +a ), whence P-, = -
M * - ^ I m 1 -f- a-
i - 1 1
After n years the value of the benefit is increased to »'-», while
the value of the premiums is reduced to P-;(l +a -). There-
" t\ y t-n-ly
fore prospectively for the value of the policy we have
Again, the n premiums already paid have accumulated to
P-pj(l +i)s—, and therefore retrospectively
These two expressions are identical, for
(| V (-11-1 ' 1 _l_ n V £-71-1/
l+a-
t-ii
= r< r (i+0"{q+o-^- i }-{(i+o-^- TC - i n
L (l + a ,)xi -I
«-i| y
x (1+Q{(1 + »)*-!}
= P-(l-M>_
26. Calculate the net level annual premium for a capital-
redemption assurance of £100 payable at the expiration of
50 years, assuming 3| per cent, interest for the first 10 years,
3 per cent, for the next 20 years, and 1\ per cent, thereafter.
The Benefit Side = 100 x ^ x ^ x „» s
The Payment Side
D/ 10 10 20 v
= PC^KSiZ) + "«» X a 20| CB%; + "«» x " X a 20|(2J%))
100 x -7089188 x -5536758 x -6102710
8-6076866 + -7089188(15-3237994 + -5536758 x 15-9788911)
23-954
25-743
= -931
= 18s. 8d. nearly.
chap, ii.] THEORY OF FINANCE 37
27. Find the annual premium per cent, for a Leasehold Assur-
ance to mature at the end of 30 years,
(a) assuming 3 per cent, interest throughout ;
(b) assuming 3 per cent, for the first 20 years and 2i per cent.
thereafter.
Answers : (a) £2, 0s. lOd. per cent.
(6) £2, 2s. 8d. per cent.
28. It is desired to have a policy providing £1000 at the end
of 30 years. The policy is to be by annual premiums under a
special system which provides for the premium being doubled at
the end of 5 years. Calculate at 3 per cent, interest the premium
payable during the first 5 years.
Here we may state the Benefit Side as 1000o 30 = 411-987,
and the Payment Side as
P{(l+^,) + « 5 (l+^,)} or P {2(1 +«-)-(!+*_)}
= Px 35-660.
Equating the Benefit Side to the Payment Side, we get
Px 35-660 = 411-987,
whence P = 11-553
= £11, lis. Id. nearly.
29. Express in the simplest form for applying to Interest
Tables the annual premium required to provide £1000 at the end
of Zn years, the premium to be reduced by one-half from the
beginning of each n years.
The simplest formula for this premium is
1000 1' 3 "
a— + a— + 2a -
Zn | "In I 11
^ H >&
Then the premium for the first n years is 4P, for the second
n years 2P, and for the remainder of the period P.
30. Calculate the reserve required at the end of 30 years
under a Leasehold Assurance policy with a premium of £10 —
(a) assuming 21 per cent, throughout ; (6) assuming 3^ per cent,
for 10 years, decreasing thereafter by \ per cent, per annum each
period of 10 years.
38 ACTUARIAL THEORY [chap. 11.
(a) The reserve required is (by the retrospective method)
1 °(%l( 2i %)- 1 ) = 450 '
(6) The reserve here is
' 10 K*ui«»- l)(l-° 3 ) 10 (1-0275)10 + ( ,_^ _ i) (i.0275yo
31. An insurance office calculates its Leasehold Assurance
premiums at 3 per cent, interest, and allows as the surrender value
of a yearly-premium policy the premiums paid, with the exception
of the first, accumulated at 3 per cent, interest, less a deduction of
10 per cent. Given u 20 at 3 per cent. = -55368, find —
(a) The annual premium required to provide a Leasehold Assur-
ance policy for £100 payable at the end of 20 years ;
(b) The surrender value allowed by the office for such a policy
at the end of 10 years.
(«)
p
r 20l
=
v™ -55368 x -03
(l+*> 2 -o, (L03)(l- -55368)
=
£3, 12s. 3d. per cent, nearly.
(*)
S.V.
=
•9 { 3-613 x (,_-!)}
Now
*ro.
=
and
v w
=
(„20)j = (-55368)*
=
■74410.
Therefore
*m
=
1- -74410
•03 x -74410
=
11-464,
and
S.V.
=
•9x3-613x10-464
=
34-026
=
£34, 0s. 6d. nearly.
32. If A represents the fund of a life assurance company at
the beginning of the year, B the fund at the end of the year, and
I the amount received for interest during the year, find the rate of
interest realised by the office during the year.
. „ represents the instantaneous rate or force of interest,
chap, ii.] THEORY OF FINANCE 39
being the annual rate per unit at which the funds are increasing
by interest at any moment of time, assuming that the increase or
decrease in the funds is uniform throughout the year.
— represents the effective rate of interest or the rate
of interest actually realised by the company during the year.
33. An estate, the clear annual value of which is £800, is let by
a college at a rent of £300 per annum on a lease for 20 years, which
may be renewed at the end of 7 years on payment of a sum of
money. Interest being reckoned at 6 per cent., what sum should
the tenant pay on renewing his lease ? Given log 106 = 2-0253059,
log 4-688385 = -6710233, and log 3-118042 = -4938820.
The lease has 13 years to run; the tenant wishes the term
extended to 20 years. Therefore, if he is to continue at the same
annual rent, he must pay the difference between the full annual
value and the rent for the period of extension, or
500 13 |a 7l = 500( %| -« lT| )
CAA /l - i> 20 1 - u 13 \
= 500
= 500
i
Now log(l-06)- 13 = -13 log 1-06
= - 13 x -0253059
= --3289767
= T -6710233
= log -4688385
And log(l-06)- 20 = - 20 x -0253059
= - -5061180
= T -4938820
= log -3118042
•4688385- -3118042
Therefore 500 13 |« = 500 x ^
= 1308-619.
The sum to be paid by the tenant is thus £1308, 12s. 5d. nearly.
CHAPTER III
Varying Annuities
1. In the scheme of figurate numbers it is to be noted that
the mth term of any order is equal to the sum of the first (to — 1)
terms of the preceding order.
Again, from the consideration that the terms of the (r-l)th
order are the first differences of the terms of the rth, those of the
(r— 2)th are the second differences, and so on, and those of the
first are the (r- l)th differences of the terms of the rth order, and
from the formula
u m = u 1 + (m-l)A« 1 + (w ~ 1 9 )(w " 2) A« Bl + • • -
+ («-!)(« -2) • • • (m-r+l) ^_ hi + . . .
I r — 1 1
we have
t-- = *__. + (m-l)At-- + ("'-!)(>*- 2) A2< __ . .
m| r | 1 [ r | *■ ' l|r| 2 1 1 «• I
(to -1) (to -2) ■ • • (to- r+l) Ar _i
+ P^T l| f|
(all higher differences vanishing)
_/ + Cm-1M- 4- ( m - 1 )( CT - 2 ) f a. . . .
~ l T\T\ + y™ i -) t T\^T\ + jjf h\^T\ +
(to-1)(to-2)- • • (»»-r+l)
|7^T Mil
But the first terms of all orders except the first are zero, and the
first term of the first order is 1 . Therefore
(TO-l)(m-2) • • ■ (m-r+l)
m| r| r- 1
chap, hi.] THEORY OF FINANCE 41
2. A proof by induction of the value of a-, - is as follows :—
We must premise that
. a in ^xi - v ' 1 f ^\ ~\ + (1 + 1) ^ H*r "ii ^ " (fl *i f±g : a *
-^ < s+Ti^)-( ff ^T|frr l +^^2| r -n| + ^^3|^+ • • +^" 2 "n-
= JL {«» <_ + o» / _ _ - »» < - 0« <
j |_ u| r-l| »| r| «+l| r| »+l| r+l|
= — (o-!-i-o"< i)
8 v »| r| »+l| r+lK
n| r| r
which follows the same law as the expression we assumed
for a—, —,, and which has been obtained therefrom by assuming
n | r |
nothing but the truth of the equation above premised. Therefore
if the expression holds for the rth order it also holds for the
(r+l)th. But we have seen that it holds for the third order,
therefore it holds for the fourth ; therefore for the fifth ; and so
on, until we reach the rth order, when we have the general
expression
a *»(«-!) • • • (n-r + 2) ^
»|r-l| |r — 1
n | r | i
3. The following may serve as an alternative explanation of the
value of an annuity of the rth order as found by general reasoning —
Suppose one is entitled to a perpetuity of the (r— l)th order,
a—. — , but prefers not to spend the payments as they fall due.
Instead, they are invested, and the interest on the investments
chap, in.] THEORY OP FINANCE 43
alone is spent. Now for the first (r - 2) years nothing is received
and nothing can be invested. At the end of the (r- l)th year a
payment of t— —^ is received, which is invested, and yields in a
year interest of it-—^—^, or it- This amount is spent at
the end of the rth year.
But, further, at the end of the rth year a payment of t is
J r\ r — 1|
received, which is invested along with the previous t ,, and
the interest received at the end of the (r+l)th year is
z (* Ti T + '-; ;,)j or '< — ti— >
v r-l\ r — 1| r\r-iy r+llrj'
since the with term of any order is equal to the sum of the first (m - 1 )
terms of the preceding order. This amount of i I- — _ is spent at
the end of the (r + l)th year.
The payment of t — , — , receivable at this time is also
r+l| t -\\
invested and the whole interest received at the end of the (r + 2)th
year, i(t——. -, + I — - — -,) or i I — , _ , is spent.
J v r+l| r | r+l| r-l|' r+2|r| r
This process goes on in perpetuity. But we notice that the
annuity being spent is i a—, —
We therefore have
a — i — = ia —i — .
qo I r- 1| col r j
and a-.-, = J^lzzll
If, however, the payments receivable cease at the end of
n years, we have an annuity for n years, and in this case we must
take account of the payments of the annuity of the (r - l)th order
which have been held back and not spent. The sum of these at
the end of n years is
(t + I + t — , — + • • ■ +t ")
^ r-l| r-l| i-|r-l| r+llr-l| ^ n\r-l/
which is of course equal to t — — -,
^ w+l|r|
We thus see that the value of an annuity for n years of the
(r-l)th order is equal to the value of i times an annuity for n
years of the rth order plus the value of a payment of t— — —. due n
years hence. In symbols
a = ia ■ + v n t — ,-, -;
»| r — 1 1 «| r\ »+l| r\
a v n( _
whence «-, -, = nlr ~ 11 . 5±liiJ
44 ACTUARIAL THEORY [chap. hi.
4. As explained in Article 25, Finite Differences are of great use
in finding the value of any varying annuity, and it is here also that
the use of the values of annuities according to the several orders
of figurate numbers comes in. For it will be observed that, in
stating u v u , u v etc., in terms of k x and its successive differences,
the coefficients of the differences follow precisely the same laws
as the scheme of orders shown in Theory of Finance, Article 3.
Thus we have
SKj + V 2 U 2 + V S U $ + • • • + V n U^
= a—,—,u. + a— —Am. + a- — A 2 k, + • • +a A r ~V
n 1 1 1 1 n 1 2 1 1 ' » | g | 1 ' n | r | 1
where the rth and higher differences of the series «j, u. p u v etc.,
vanish.
If the series be a perpetuity, we have, as is shown,
o Mj + u 2 m 2 + v s u s + ■ ■ -ad inf.
= a- — u, + a__A?<, + o-.-AV + • • • + «- -A r ~V
oo|l|i oo 1 2 1 i ml8| 1 °°M 1
where the rth and higher differences vanish
w. Am. A 2 m, A'- 1 ^
= "4- + — t 1 + -r^J- + • • • + -— i
From this we may get a formula for the value of the series of
payments u v u v u^ ■ ■ • u n , as follows : —
v M, + vhi 2 + d 3 m 3 + ■ • • + v n u n
, ft + £ + . . . + *£,)
-•('¥
+ n; — + • • + —
This will be found useful where the differences are not numerous,
and the number of terms unknown.
As an example, suppose it is required to find the value of the
annuity whose payments are 1, 5, 11, • ■ • 109.
Here u x = 1, Amj = 4, A-u 1 = 2, and
109 = u n = u 1 +(»-l)Aii 1 + ("zlKlZ^A**,
= l+4(«-l) + (/i-l)(»-2)
= n 2 + n - 1
whence ra=10.
chap, in.] THEORY OF FINANCE 45
Also w X i — 131, m 12 = 155, A« n = 24, Aht n = A\ = 2.
Therefore
w+5n 2 +lli.8+ • • • +109r 10
/l 4 2\ ../131 24 2\
EXAMPLES
1. A 20-year sinking-fund policy is effected on an increasing
scale of premiums, beginning at £100 per annum and rising by £3
each year till the end of the term. At the end of the fourth year
it is proposed to commute further payments. Determine their
value on a 3 per cent, basis.
The premium due at the beginning of the 5th year is £112,
which will increase by £3 per annum for each of the succeeding
15 years. Now, assuming for the moment that the premiums will
be payable at the end and not the beginning of each year, we have
their value equal to
112 ^I| + 3a Ia|2|
Adjusting this expression (since the premiums are actualty payable
at the beginning of each year) by multiplying it by (1 + i), we
have the commutation price of the future premiums equal to
(1+0(112^ + 3^)
a— - 16i> 16 \
112a-, +3^51
(1+0(1
16| i /
12-56110- 16 x-623167\
1-03(112 x 12-56110 + 3
1-03(1406-8432 + 3
-03
12-56110-9-97067
■03
= 1-03 (1406-8432 + 259-043)
= £1715-863.
2. Prove that if a denote the value of a varying annuity of
11 1 r I jo j
the rth order for n years then (1 + if a— — , is equal to the sum of
the first (n-r+l) terms of the expansion of (l-e)- r in powers
of v.
46 ACTUARIAL THEORY [chap. hi.
Remembering that
t _ _ (m-l)(m-2) • • ■ (m-r+l) _ c
lr-1
m-l r-1
and that where m < r, t—— = 0, we have (I +tYa— l - i
m\r\ ' K ' n\r\
^ T ' y l|r| T 2|r| T S|r| T ^ l n\r\>
= (i+O r C. 1 c r . 1 ^+ r c r _ 1 ^ + i + . . +.. 1 c r _ 1 «-)
= ( l + iy/,r + ri ,r + i + fc+l}v+2 + . . +(»-!) ("-2)- • 2-1 „.].
I, |2 |n-r|r-l J
= l +rj)+ K^±I)^ + . . + r(r+l)- •(»-!),,,-,
which is the sum of the first (rc-r+1) terms of the expansion
of (1 — w)"'' in powers of i>.
3. Find at 4 per cent, the value of the annuity of which the
first three payments are 40, 45, and 52 respectively, and the
last 325.
The first step necessary is to find the term of the annuity.
For this, we use the formula
u n = u x + (n-l)Ai h + 0-l)(rc-2) A2Mi + . . .
In this case
325 = 40 + 5(rc-l) + («-l)(n-2), since Aw x = 5 and A 2 Kj = 2
Hence rfi + 2» - 288 =
(»+18)(*-16) =
n=16.
Now, proceeding to find the value of the annuity, we have
a = 40 a_ + 5 a— , — + 2 a— , —
16| 16| 2 | 16| 3 |
= (40 x 11-652) + (5 x 77-744) + (2x341-879)
= 1538-558.
4. What is the value at 5 per cent, of the annuity whose pay-
ments are 16, 26, 58, 124, etc., the sum of all the payments being
1322480?
Here it is necessary to find n, the term, from the formula
v w(«-l) A , n(n- 1)(tz- 2) .„
chap, in.] THEORY OF FINANCE 47
Then proceed to apply the formula of varying annuities —
a = u,u—— + Au,a-~ + A'-mo-,-- + Ahr.a
The value required is 287998-936, as follows :—
a = 16 a + 10 «— , — + 22 a + 12 a
40|1 | n 40| 2 | T 40| 3 | T 40| 4 |
= (16x17-159) + (10x229-545)
+ (22 x 2374-991) + (12 x 19431-595)
= 274-544 + 2295-450 + 52249-802 + 233179-140
= 287998-936.
5. Find the present value of an annuity of the ?-th order, to yield
interest at the rate i per annum on the whole capital for the entire
term of the annuity, the capital to be replaced by means of a
sinking fund accumulating at the rate j per annum.
Here we must resort to the general rule given in Article 43a of
Chapter II. that the present value of any series of payments, n
remaining constant, may be found "by multiplying the amount,
accumulated at rate j to the end of the n years, of the series of
payments by - — — ."
v : J 1 + 1 s -,
n|
Now the amount of an annuity of the rth order accumulated at
rate i is s' and we therefore have the value of an annuity of
J « | r I •"
s
the rth order under the conditions laid down as 1 ".J where
n\
s and s'— are taken at rate /'.
n | r | ?i |
6. Find the value of an annuity-certain for 20 years whose
several payments are 1, 2, 3 ■ • ■ 20, the value to be so
calculated as to yield the purchaser 5 per cent, on his whole
investment throughout the whole of the 20 years and to return
him his capital at 3 per cent.
Applying the rule cited in the preceding example, we have
_ ,? 20| I] (3%) + A 20l 2] (3%)
1 - , - 05 %;(3X)
48 ACTUARIAL THEORY [chap. hi.
since the amount of the annuity to the end of 20 years is
(%|r] + %|Ji) at 3 per cent.
26-87037 + 229-01248
1 + 1-34352
= 109-187.
7. A loan of £42,000 has been made with the following
condition as to repayment : Annual instalments of principal for
20 years, the first being £4000, the second £3800, the third
£3600, and so on ; interest at the rate of 4 per cent, being paid
annually on the outstanding amounts. Immediately after pay-
ment of the fifth instalment, it is arranged to repay the balance of
the advance with a premium, the lenders being able to re-invest at
only 3 per cent. Show how the premium should be computed.
The value of the capital at 3 per cent, is
and of the interest on the outstanding amounts of capital
•04(24000*-, myj - 3000a lT| m%) + 200^- (3%) ).
Therefore the whole value of the outstanding loan is
3960a l5| T| (3%) - 320a l5| 21 (3%) + 8fl l5| 31 (.»'
And the premium required is
396 %H(3%)- 320 %|2](3%) + 8a 1 T|3 1 ( 3% )- 24000 -
CHAPTER IV
Loans Repayable by Instalments
1. In connection with the discussion of loans repayable by instal-
ments, it is important to remember that the symbol C stands for the
capital actually returnable by the borrower, that is, taking account
of any discount or premium on the par value of the loan, and
further, that j is dependent on this definition of C, being the ratio
which the annual payment of interest bears to C. These two
points must be clearly borne in mind in all questions of this
nature.
2. The following wording, differing slightly from that of Mr
King, may be useful to explain the general formula for the value
of a loan, repayable by instalments at stated periods of time, with
interest in the meantime at rate j, so as to yield the purchaser a
given rate of interest, i.
Had the borrower contracted to pay interest at rate i per
annum on the capital C, then the required value of the loan would
have necessarily been C, since, interest at rate z being payable at
the end of each year, with C repayable, the investor would have
realised rate i on the purchase price. The value of the capital at
rate i being K, the value at rate i of the interest, on the basis
assumed, would have been (C - K), which then is the value of the
annual payments of interest, if these were made at rate i. But, in
point of fact, the annual payments of interest are made at rate j,
and therefore by simple proportion their value is -4- (C - K).
Adding to the present value of the interest the present value of
the capital as already noted, namely K, we have the whole value
of the loan equal to K + -4- (C - K).
i
3. In the converse problem to find the rate of interest yielded
by a loan purchased at a given price, we have the equation
j _ Jv^~ — ) } whence i may be found as explained in Articles
11 and 12.
50
ACTUARIAL THEORY
[CHAP. TV.
Mr Ralph Todhunter has given (/ LA. xxxiii. 356) a very
useful formula for approximating to the rate of interest yielded by
a bond bought at a premium. In the explanation accompanying
the formula, he suggests that the premium might be dealt with
in practice bv writing down the book-value out of each dividend
by an equal proportionate part of the premium, the remainder of
the dividend being treated as interest. Taking the case of a bond,
repayable at par at the end of n years, interest meantime at rate j,
purchased at a premium of p per unit, Mr Todhunter gives us the
following schedule : —
Year.
Book- Value of Bond at
beginning of Year.
Interest.
1
2
3
n-l
n
1 + p
1 + p
n 1
n-2
1 + p
n 1
■i
1 + — p
n 1
1
1 + — p
n r
J n
J n
j - 1 -
J n
j -^
J ii
J n
giving an average rate of interest
• P
J n
1 / n-l n-2
1 + — [p + p + p +
n V n n
+
2 1 \
— P + — P )
n I n ' j
i + - V(" +1 )
re in
n+ 1
1 + -7T-P
chap. iv.J THEORY OF FINANCE 51
The result by this formula will as a rule be sufficiently close for
practical purposes ; but if greater accuracy be desired, the rate of
interest thus found may be used to obtain K in the formula
t = "£jj — ^r-s when a very close approximation would be obtained.
A — K.
Mr Todhunter, however, points out that his formula should only
be applied as a final result when n and j are not large. The
reason for this will be readily understood from the following : —
Using Makeham's formula, we have
_ j(C-K)
A-K
or in the example submitted by Mr Todhunter
1 +p - v n
where v n is calculated at rate i.
Hence i(l - v n )+pi = j'(l -v n )
and pi = [/-i)(l - v n )
j-i = ^{l-ci+o-"}- 1
_ pi ym j^— i + ^ i j
= TV ~-IT l + \3~— 1 i
H
n \
n+ 1 . n- — 1 .„
Neglecting higher powers of i than the first, we have
i + l.N
. . « /, ra + l.\
whence i — ■
. « + l
1 + ~s — i>
Zrc
which is Mr Todhunter's formula as given above.
52
ACTUARIAL THEORY
EXAMPLES
1. A bond for £775, 15s. 10d., repayable at par in five years,
and bearing interest at 3f per cent, payable half-yearly (first
payment six months hence), is bought at a price (£750) to yield
the investor 4J per cent, on his investment. Draw up a schedule
showing the amounts that must be added to capital each half-year
so as to gradually write up the sum invested to the redemption
value.
The schedule will be as follows : —
Half-
Tear.
Book-Value at
beginning
of Half-Year
Interest on
Book-Value
at 2i%.
Interest
received atp
13% on
£775,15s lOd
Amount to
be added to
Book-Value.
(3) -(4)
Book-Value at
end of
Half-Year.
(2)+(5)
Half-
Year.
p.)
(2.)
(3.)
(40
(50
(6.)
(70
1
750-000
16-875
14-546
2-329
752-329
1
2
752-329
16-927
14-546
2-381
754-710
2
3
754-710
16-981
14-546
2-435
757-145
3
4
757-145
17-035
14-546
2-489
759-634
4
5
759-634
17-092
14-546
2-546
762-180
5
6
762-180
17-149
14-546
2-603
764-783 -
6
i
764-783
17-207
14-546
2-661
767-444
7
8
767-444
17-267
14-546
2-721
770-165
8
9
770-165
17-328
14-546
2-782
772-947
9
10
772-947
17-391
14-546
2-845
775-792
10
2. The value to yield 4 per cent, of a 5 per cent, bond for
£1000 due after five years is £1044, 10s. 4d. Give a schedule
of the amounts to be carried each year to principal and interest,
with the amount of principal outstanding at the beginning of each
year.
Here the bond's book-value has to be written down (not up, as
in the former case), and the schedule will accordinglv be : —
Year.
Book- Value at
beginning
of Year.
Interest on
Book-Value
at 4%.
Interest
received at
5% on
£1000.
Amount
carried to
Principal.
(4) -(3)
Book -Value at
end
of Year.
(2) -(5)
Year.
(10
(2.)
(30
(40
(5.)
(6.)
(7.)
1
1044-517
41-781
50-000
8-219
1036-298
1
2
1036-298
41-452
50-000
8-548
1027-750
O
3
1027-750
41-110
50-000
8-890
1018-860
3
4
1018-860
40-755
50-000
9-245
1009-615
4
5
1009-615
40-385
50-000
9-615
1000-000
5
chap, rv.] THEORY OF FINANCE 53
3. The five per cent, stock of a colonial municipality is
redeemable at par in 20 years. What can a purchaser give for it
in order to make 4 per cent, on his investment ?
Using Makeham's formula,
A = K + 4- (C - K)
we have K = 100i> ( 2 4 ° %) , j = -05, i = -04, and C = 100.
Therefore
= 113-590.
4. Two loans were granted 10 years ago — (a) £20,000 at 4| per
cent, per annum nominal, repayable by 60 equal half-yearly
instalments which include both principal and interest ; (6) £20,000
at 4£ per cent, per annum nominal, repayable by 60 equal half-
yearly payments of principal, interest being also paid on the
balance from time to time remaining outstanding. Find in each
case the amount of the payment due to-day, and show the amounts
of principal and interest included in the payment. Find also the
sum for which each loan may be redeemed to-day, assuming
interest at 3^ per cent, per annum nominal.
First, as to the payments due to-day —
(a) In this case the periodical payment is always the same, and
, . 20000 , , 20000 4i ... , .,
is equal to whereof x «.„.„. is principal, and the
6C| (2|%) 60|
. , 20000., n > . . .
remainder, or (1 - v S „J, is interest.
20000
(U) The payment due now consists of (1) — — — of principal
and (2) interest on the balance of principal outstanding at the
beginning of the twentieth half-year, that is
(20000 -19™) -0225.
Thus the whole payment due is 20000 j-^ + (l - ^h -0225 j
20000 x — a-Tg. r/ y and of the future payments of interest
54 ACTUARIAL THEORY [chap. iv.
Secondly, as to the redemption price —
(a) As the amount paid to redeem the loan can only be
invested at 3J per cent., the future half-yearly payments must be
valued at that rate, that is, the price is
20000 ,, s
HZ — x ^ l+a maix)
60! (21%)
(6) The value of the future instalments of capital is
1
60 "wo-iX?
20000(— x-0225a-- r ,.,„.,-— x-0225ffl iffi - 7 .,.,A Therefore the
^60 •wi 1 10}%) go 4 <>i 2 1 (i|%)y
redemption price is
20000 {(i + S x -° 225 ) Vo,n«%)4 * •° 225 <^«>,
and A = 114.
1 - j) 60
SinCC ^75 - 36 ' 964
v™ = -35313
Therefore trying If per cent., we have
■02 110-38-8443
' ~ 1-1 X 114-38-8443
= '®1 x -94678
= -01721.
Hence the approximate yearly rate of interest is
(101721) 2 - 1 = -03472, or 3-472 per cent.
7. A bond for £1000, bearing interest at 5 per cent, payable
half-yearly, and repayable at par in 30 years, is purchased for
£1250. What rate of interest does the investment yield to the
purchaser ?
As before, we have
.C-K
A-K
where it will be well to treat i and j as interest for a half-j'ear.
We then have
1000- 1000 i>8°
■025
1250- 1000 a 80
Trying d 60 first at 2 per cent, we get
AOK 1000 -304-782
1250-304-782
= -0184.
56 ACTUARIAL THEORY [chap. iv.
Seeing that 2 per cent, is too large, we try again with If per cent,
and get
nnf . 1000-353-130
i — -025
1250-353-130
= -0180.
Now a reduction of \ per cent, makes a change of -04 per cent,
in the result. Therefore to get the true rate we have
2 - x = 1-84 - ^lr
•25
whence x = -19
and i = -0181.
This being the half-yearly rate, the effective yearly rate is
(1-0181) 2 - 1 = -0365, or 3-65 per cent.
8. Towards the close of 1905 the Japanese Government issued
a four per cent, loan at 90 per cent., repayable at par on 1st
January 1931 (or in certain circumstances earlier), coupons for a
half-year's interest payable 1st January and 1st July each year,
with first payment on 1st July 1906. Allowing for discount on
the instalments the issue-price may be taken as 89J on 1st January
1906. Find the rate of interest realised.
.C-K
. Q9 100- 100 v m
" 89-5-1 00 v™
= (taking u 50 at 2£ per cent. = -29094) -0235
= (taking u 60 at 2-1 per cent. = -32873) -0237
whence approximately the half-yearly rate is -0236 and the effective
rate is (1-0236) 2 - 1 = -04776 = about £4, 15s. 6d. per cent.
9. A debenture of £100, redeemable at £110 on 1st July 1915,
and bearing interest at the rate of 4| per cent, per annum, payable
half-yearly on 1st January and 1st July in each year, is purchased
on 1st April 1905 for £109. How would you calculate the yield to
the purchaser ?
Here i must be considered as the rate yielded to the purchaser
per half-year. At 1st July 1905 the value of the capital will be
chap, iv.] THEORY OF FINANCE 57
110 x v™, and of the interest 2-25(1 +a- 7 -,.\ As that date is half
a period forward we must discount these values for that time, and
we get the equation
109(l+i)i=110^ + 2-25(l + «- |w ).
i must now be approximated to, and this result being the half-
yearly rate we obtain the effective yearly rate from (1 +if- 1.
It is probably better, however, to proceed in the same way as
in our other examples. Then, i being the half-yearly rate as
before, we have
■0225 110(1 +t>- 110 v 2 °i
1 ~ "FT* 109-110j> 2 °*
As before, i may be approximated to, and the yearly rate found
from the result.
The formulas produce the same result.
10. A bond for £100, bearing interest at 6 per cent, per
annum payable half-yearly, and redeemable at par at the end
of 40 years from the date of issue, was issued at par 30 years ago,
the present market value being £115.
(a) Find the rate of interest yielded to a purchaser now buying
at the market price.
(b) Find the rate of interest obtained by the original holder
when his profit on sale is taken into account.
(c) If it were proposed to convert the bond into one for £125
bearing 3£ per cent, interest, redeemable at par in 30 years, what
gain or loss would there be to the holder of the bond on
conversion ?
, . . . C-K nQ 100-100t>»
(«) * = J* JTR = "° 3 x 115-100,20
Trying 2 per cent.,
100-67-297
1 ~ ' 06x 115-67-297
•03 x 32-703
47-703
= 2-0567 per cent.
58 ACTUARIAL THEORY
Also trying 2£ per cent.,
100-64-082
1 ~ x 115-64-082
•03x35-9 18
50-918
= 2-1162 per cent.
2-25 gives 2-1162
and 2 gives 2-0567.
Thus a difference of -25 gives a difference of -0595.
Therefore 2 + x = 2-0567
+ -
1595
whence
x = -0744
and
i = 2-0744
per
cent.
The yearly rate then = (1-020744) 2 -
-l = -i
(b) i =
•03 115-
1-15 * 100-
■IK
-in
11)60
Trying 3 per cent..
04192 = 4-192 percent.
•03 115-115x 16973
' ~~ 1-15 X 100- 115 x -16973
•03(100-16-973)
100-19-519
= 3-0949 per cent.
Also trying 3\ per cent.,
•03(100-14-676)
1 ~ 100- 115 x -14676
-03 x 85-324
83-123
= 3-0794 per cent.
A difference of -25 in the rate per cent, gives a difference
of --0155.
Therefore 3 + x = 3-0949 - '^^x
•25
whence x = -0894
and z = 3-0894 per cent.
and the yearly rate = (1-030894) 2 - 1 = -06274 = 6-274 per cent,
(c) A = K + 4(C-K)
= 125i)3o+ -^r(125-125u 3 <>)
chap, iv.] THEORY OF FINANCE 59
Where i = -04192, as found in (a)
.aqf;
A = 36 - 465+ OT2^ 125 - 36 ' 465 )
= 36-465 + 73-920
= 110-385.
Thus there is a loss of 4-615, or, say, £4, 12s. 4d. to the holder
of the bond on conversion on the terms given.
11. A foreign corporation issues a loan of £390,000 4 per cent,
bonds, repayable by annual drawings as follows : — £10,000 at the
end of 5 years, £11,000 at the end of 6 years, £12,000 at the end
of 7 years, and so on, till the whole is repaid. The issue-price
being 94| per cent., what rate of interest is paid by the
corporation ?
Here it is necessary to find the term when the last instalment
is paid. We have
390 = 10 + 11 + 12+ • • • +{10 + («-l)}
20 + fra-l)
whence n — 20.
Therefore the value of the instalments of capital is
iooo^io^+W-
C — K
Using now the formula z = j —- — - and trying 4£ per cent.,
A — lv
we have K = 196943-6, as found belov
Also
C = 390000
sol 1|
-=-•045
^20121
10 -an
x lOOOti*
K
=
13-00794
8-29286
A = 390000 x -945 = 368550
4-71508
and j = -04.
Therefore
104-7796
130-0794
nt 390000-196943-6
368550-196943-6
234-8590
838-561
= -045.
196943-6
Thus the rate is 4J per cent.
60 ACTUARIAL THEORY [chap. iv.
12. Having given the value of a— at 4£ per cent. = 13-0079
and at 5 per cent. = 12-4622, find approximately the rate of
interest yielded by an annuity for 20 years, in which the payments
are successively 20, 19, 18, etc., when purchased for £150.
The successive terms of this annuity involve first differences
only, and consequently its value may be stated in the symbols
20a nr\ ~ %i2i
Now at 4| per cent. a—-- = 13-0079,
v™ = \-ia- = 1- -585355 = -414645
and ,__ - ^- W ° - 13-0079 8-2929 _
2o| 2 1 -045 -045
Therefore 20a_ _-«-,- = 260-158-104-778 = 155-380
20| 1 1 20| 2 |
Again, at 5 per cent. a—. — = 12-4622.
„so = i _ ; a _ | = 1_ -623110 = -37689
Therefore IQa^^-a-,- = 249-244-98-488 = 150-756
We see that a rise of -005 in the rate means a fall of 4-624 in
the price, and to bring the price from 155-380 to 150 (a fall of
5-380) the rate of interest must be increased by
•005x5-380 _ . 005g2
4-624
Therefore approximately at the price of 150 the rate of interest
realised is 5-082 per cent.
13. Apply Todhunter's formula to determine the rate of interest
yielded by a terminable 6 per cent, debenture, repayable at par at
the end of 20 years, purchased for £119, 10s.
chap. iv.J THEORY OF FINANCE 61
Here we use the formula —
i =
' n
1 + -s— P
■Ofi ' 195
06 -~2Q-
01
l + gx-195
= -04558
= £4, lis. 2d. per cent.
14. Twenty years ago a Local Board borrowed £100,000 at
5 per cent, from an Assurance Company, such loan being repayable
by 30 equal annual payments, including principal and interest.
The Board now offers 3f per cent, debentures, repayable at par
60 years hence, but now issued at 90, in equitable fulfilment of the
contract with the Assurance Company. What amount in deben-
tures should the Company accept ?
The annual payment made under the contract of the original
loan is , and there are 10 such payments still to be made.
°30l (5%)
If the Company is to forgo the receipt of these they should be
commuted at the rate of interest presently ruling in the market,
and the proceeds should be employed in buying the new debentures
at 90.
To find the rate of interest now obtainable on investments we
may take the rate yielded by these debentures as fair. Thus we
have
.C-K
where j =-0375, C = 100, A = 90, and K = 100i> 60 at, say, 4£ per
cent. = 8231
•0375(100-8-231)
90-8-231
= -0421 approximately.
62 ACTUARIAL THEORY [chap. iv.
The value of the 10 annual payments outstanding is therefore
100000 , ,_ '
' x °«ii,™«) and tne amount or debentures this sum will
a — , lu i \ % zl /o)
30| (5%)
purchase at the price of 90 is
100000
X d
"mIto 10|(4 ' 21%) _ 100000 x 8-02677
•9 15-3724o x -9
= 58016-994 = £58,016, 19s. lid. nearly.
15. A Company has an issue of 6 per cent, debentures
maturing after 5 years, which are quoted at a price which yields
4 per cent., and it proposes to redeem them by issuing 5 per
cent, debentures for the same nominal amount in lieu. Show how
to find the number of years for which the 5 per cent, debentures
should run so that the holders would still realise 4 per cent, on
their investment.
In the general formula
A = K + i-(C-K)
j
when C = 1, we have
A= 1-(1-K)(l -£)
Under the present arrangement of 6 per cent, debentures
■06\
A = Wi-ix^-S)
- 1 + -°Hi^
On the proposed altered basis of 5 per cent, debentures
A = i -( i -w( i -S)
Equating these two
I
•oi(i-; %) )= -02(1-4,)
= l + -01o-, „
l + -01a-„„ o/ , = l + -02a_ „
»l(4#) 6 | (4%)
CHAP. IV.]
THEORY OF FINANCE
Therefore
v l = 2v 5 -1
(4%) (4%)
and
log(2v 5 -l)
n = , *■ '•'
logu
1-80879
1-98297
= 11-23.
63
The period may therefore be put at nearly \Y\ years.
CHAPTER V
Interest Tables
1. In Articles 16 and 17 is discussed the formation of a table of
log (1 + i) n . The column of values thus formed may be used further
to get the values of (1 + i) n by taking the natural numbers corre-
sponding to the logs. As this is not done by a Continued Process,
a periodical check is not sufficient. Each value must be separately
checked. This may best be done by taking independently the
logs of the table last found and comparing the results with the
original logs.
2. If a column of (1 + i) n has been formed, a column of v n may
be obtained from it by the use of a table of reciprocals. By taking
the reciprocals of the values thus found the original table of
(1 + i) n should be reproduced, and a check put upon the work.
Again, a column of log v n might be formed in the same way as
a table of log (1 +i) n (in this case there is no need to start at the
end of the table as for forming a table of v n directly) and the
natural numbers corresponding would be the values of v n , each of
which requires to be checked as before.
3. In view of what has been said about Leasehold Assurances it
will be well to discuss methods of forming a table of annual
premiums for such. It is necessary to note that as the premiums
are due at the beginning of the year, we have to deal with
annuities-due throughout.
First, we have P_ =
and hence log P__ =
v % v %
a—, 1 + a — -
n\ n-l\
= log v n — log (1 H
■IK
Therefore, being supplied with values of v n and a— , we proceed
to form columns of log v n and log (!+«_) and, deducting the
P. v.]
THEORY OF FINANCE
65
value in the latter of these from the value in the former, we obtain
log: P , from which P — . may be obtained at once.
The following schedule shows the process : —
Again, we have
Term
71.
I0g1)"
(2) -(8)
l °S0-+ a n-lp
(1)
(2)
(3)
(4)
(5)
1
2
3
4
etc.
n \ a—
1
(l+i)"a
(1+0*-
m| x " y " n\ "n+l|
Therefore knowing- the values of s—. we form therefrom a column
of * r - 1 and take the reciprocals of these, the results being
the values of P-
n
method : —
The following schedule indicates the
Term
71.
"^i- 1
»l (2)
(i)
(2)
(S)
1
2
3
4
etc.
Further, P— =
3 71.
n|
!-(!-«") _ 1 '^
1+a — n l + « — r , (l + z)a-i
1+a
r»-l|
E
66 ACTUARIAL THEORY [chap. v.
It is on this relationship that Orchard's Conversion Tables for
annual premiums are founded. By them if one enters with the
given value of a , the result is P — They are fully discussed in
Chapter VIII. of the Text Book, Part II., and are only mentioned
here to show their use in forming a table of P —
In all the above methods it has been assumed that one rate of
interest holds throughout the term of the assurance^ but it is not
implied that this assumption always holds good.
INSTITUTE OF ACTUARIES'
TEXT BOOK — PART II.
CHAPTER I
The Mortality Table
1. A Mortality Table is defined in this Chapter as an instrument
by means of which are measured the probabilities of life and the
probabilities of death. In its final form a mortality table sets
forth the history of the experience by means of the number living
and the number dying columns. If we refer to page 494 of the
Text Book, we find the following figures : —
Age.
Number LiviDg.
Number Dying.
127,283
14,358
1
112,925
3,962
2
108,963
2,375
3
106,588
1,646
4
104,942
1,325
etc.
etc.
etc.
These figures tell us that, according to this experience, out of
every 127,283 persons born, 112,925 on the average survive to age
one, 108,963 on the average survive to age two, and so on. Or
again, they tell us that out of the same number of births,
14,358 on the average die before attaining age one, 3,962 on
the average attain age one, but die before attaining age two, and
so on.
It cannot be too carefully impressed upon the student that a
68 ACTUARIAL THEORY [chap. i.
mortality table does not give absolute but only relative ov average
results ; in other words, it is not intended to be inferred from
these figures that 127,283 children were in reality found who
were all born at the same moment of time, that 14,358 died before
attaining age one, that 3,962 actually attained age one but died
before attaining age two, and so on. An arbitrary figure called
the radix is selected to represent the number of entrants at the
initial age, and the figures submitted are only on the average, and
relative to one another.
Were we asked to form a mortality table representing the
experience of Edinburgh during the calendar year 1906, it would
not be sufficient to give us merely the deaths that occurred
in Edinburgh during that calendar year; arranged according to
year of age. The summation of these deaths would have no
relation whatever to the /„ persons out of which the d deaths
actually occurred, nor again would the (/„ — d ) persons have any
relation to the /, persons out of which the d x deaths occurred,
and so on.
It is possible that the deaths column so supplied us might adopt
a quite irregular form, for it naturally depends on the number
living at each age out of which the deaths occurred. For example,
the deaths between twenty and twenty-one might be twice as
numerous as those between twenty-one and twenty-two, owing to
the fact that the number living between twenty and twenty-one
happened to be fully twice as numerous as those living between
twenty-one and twenty-two.
Again, it would not be sufficient that it be added to our data
that the number born in each calendar year for many years past
had been equal to the annual deaths. The migration element
would require to be kept before us, since people might be
emigrating and immigrating in different numbers and at entirely
different ages.
Before a mortality table can be formed in the way here dis-
cussed, it is essential that the population be proved to be in
every way stationary ; that is, that the annual births be equal
to the deaths, that the births all take place on the same day of
the year, say 1st January, and that there be no emigration or
immigration.
chap, i.] TEXT BOOK— PART II.
EXAMPLES
(a ) 2 (a ^ 3
1. Prove that m x = ry + lifL + l!lL +
We have by Text Boole formula (9)
whence
%
=
X
2 + m
m
X
■ —
' 2 %
2-o
=
?,(l-iO
-l
=
(<7 T ) 2
+
(O 8
■A
+
2. Prove that » = 1 - ?n + -H™ ") 2 - \(m ) 3 + • - •
and that 9j .= w^ -J(«) 3 + lOJ 3 - iK) 4 + • ■ ■
From Text Book formula (8), we have
1 -\m
* z
* } "> " l+lm
- X
= (i-H)( 1+ H) _1
= l-»«,+TW ! -iW 3 + • • •
Also from Text Book formula (9)
in
X
X
= lllx {l-hn x + l(mf-^ mj Y+ . . }
= »» -ifm ) 2 + i(m ) 3 -v(»« ) 4 + • ■
The latter result might have been derived directly from the
former, since q = 1 — p .
70 ACTUARIAL THEORY [chap. i.
3. Given the following particulars
x Q%
20 -00572
21 -00608
22 -00643
23 -00668
24 -00691
find how many of 10,000 people living at age twenty die during
each year of age up to twenty-five.
Here we are given / 20 and q 20 , and hence we may find d 20 , since
from Text Book formula (5), l 2Q x q 20 = d 20 . Also from Text Book
formula (1), / 2 i = ^20 ~~ ^n> anc ^ being given q„ v we may similarly
find d 21 , and so on for d. 2V d 23 , and d., v In the example, rf 20 = 57,
d 21 = 60, (/.,o = 64, rf 23 = 66, and 36 = -99115, p 37 = -99090, p 3S = -99062, ? j 39 = -99030.
Now, from Text Book formula (4), we have I = / », and
therefore
^36
=
'35 Pss
30000 x
•99139
=
29742
^37
:
^36 Ps6
29742 x
•99115
=
29479
'38
=
29211
'30
=
28937
'«
=
28656
CHAPTER II
Probabilities of Life
1. It is very important that the student should have at his
finger-ends the values of all probabilities in which two lives may
be involved, and for that purpose he should practise, till he attains
complete proficiency, writing down the values of the following,
giving in addition the symbols, where these are possible. The
answers should be carefully compared with those given in the
Text Book.
The probability that : —
1. (x) will survive n years.
2. (x) and (^) will both survive n years.
3. Neither (x) nor {y) will survive n years.
4. At least one of the lives (,r) and (j/) will survive n years.
5. (x) will survive n years and (j/) die within n years.
6. Exactly one of the lives (x) and ( if) will survive n years.
7. At least one of the lives (x) and ( 7/) will fail within n years.
8. Both (x) and (^) will die in the rath year from the present
time.
9. The first death will happen in the rath year from the present
time.
10. The second death will happen in the rath year from the
present time.
11. One only of the two lives will fail in the nth year.
12. Neither of the two lives will fail in the nth year.
13. One at least of the two lives will fail in the nth year.
14. (.i) will survive n years and Q/) will survive (n — 1) years.
With regard to the last of these, it is useful to note that,
besides the form given in the Text Book, this probability may be
written : —
p x ,» = p x -p ,. x .p = p x .b , _
g'l n-1* y *x n-l'x+1 »-l« y 1 x n-1' x+l:y
72 ACTUARIAL THEORY [chap. ii.
2. To find the probability that r at least of m lives will survive
n j-ears.
An alternative proof of the formula
Z r
p L =
% l xyz ■ ■ ■ (m) n +Z) r
is as follows : —
This probability is equal to the sum of the probabilities that
exactly r, exactly (r+l), exactly (r + 2), and so on ad inf., will
survive n years (though, when r + k > m, the individual prob-
abilities will have no value).
But by Text Book formula (14), we have
P
n xyz • • • (m)
Hence
)
xys • • • (m)
(i+zy+i
7s Z'+i Z r + 2
(1+zy+i ^ (1+Z) ! '+ 2 (l+Z)'-+3 T
Z>- j- Z Z 2 |
(i + zy+H + (i + z) + (l + z) 2 + " j
z>- / Z \-i
(i + z)'-+iv i + z
Z''
~ (l+zy
3. In expansion of Text Booh, Articles 34 and 35, it may be
asked what are the expected deaths and expected claims respec-
tively amongst m joint-life policies on (x) and (j/) for £K each.
The expected deaths are m(cj +(/), and the expected claims
£Km(l -p ). In the former of these expressions, however, it
might fairly be argued that it is incompetent to take account of a
second death on any one policy in the year, as the lives are not
traced be3'ond the first death.
But in last-survivor policies this does not hold, and the
expected deaths are as before m (q + q), while the expected
claims are £¥Lm(q x q ).
In contingent insurances payable if (x) die before (j/), the
expected deaths may be considered to be mq , and the expected
claims are £Kmq (1 — ^q ).
4. It is most necessary that a clear perception should be
obtained of the nature of the force of mortality, and the following
chap, ii.] TEXT BOOK— PART II. 73
explanation is offered in the belief that it will assist towards the
attainment of that object.
On page 495 of the Text Book will be found a table of a , the
' x'
rate of mortality. This is the probability that a person aged x
will die within a year, and it is deduced from the elementary
equation q = __?.
X
It must, however, be evident on consideration that the rate
of mortality is not constant between ages x and (.r+1), then
suddenly rising to q x+l ; nor constant between ages (tf+1) and
(.r+2), then suddenly rising to g , and so on. The probability
that a person of any age will die within a year obviously depends
upon the number who are alive at that age (whether the age may
be expressed by an integer or not), and the deaths within one year
after that particular age.
Now it is frequently necessary in actuarial work to have the
probability that a person aged x will die at a particular moment.
The function, however, that is tabulated is the force of mortality
at age .r, which is clearly defined in Text Book, Article 38, as
" the proportion of persons of that age who would die in a year, if
the intensity of mortality remained constant for a year, and if the
number of persons under observation also remained constant, the
places of those who die being constantly occupied by fresh lives."
It may be useful at this point to introduce an illustration to
assist in making the idea clear. Let us consider the speed or the
" force " of a railway train. This is generally measured by the
distance covered in the course of an hour, e.g., the speed is 40 miles
an hour. Any other function if tabulated would convey but little
meaning. For the same reason the force of mortality is always
measured as within one year.
Suppose now we wish to measure the rate at which a train
is travelling at any particular point. We might ascertain precisely
the distance covered during the following minute, when simple
proportion would give us the distance covered in an hour. It is
obvious, however, that a minute is too long a period within which to
measure ; that, in fact, the rate at which the train was travelling
may have varied considerably within that interval. A better
result would be obtained were we to measure the distance covered
during the following second, and resort as before to simple pro-
portion. In other words, the smaller the interval of time within
74 ACTUARIAL THEORY [chap. ii.
which we measure, the more accurately shall we be able to gauge
the rate at which the train is travelling at any particular point.
It will be noticed that our answer gives us the distance that the
train would cover during one hour, were the speed at which
the train was travelling during the infinitely small interval of
measurement to remain constant for an hour.
When now we come to measure the force of mortality at any
age x, we might work out the probability that a person of that
age will die within one day. Multiplying the result by 365, we
should get an approximation to the force of mortality. In symbols
i-r i
X X + —
385 . . n
fx, = 365 -, approximately.
X
A day, however, is too long a period within which to measure.
A better result would be obtained were we to reduce the interval
to one hour. This would give us
I -I . i
X X + ■
24x365
fj, x = 24 x 365 j approximately.
X
The smaller the interval within which we measure, the more
accurate will be our result. Hence we say that
1 X X+t
X
where t approaches the limit 0.
When t approaches the limit 0, we have dl =1 -I , and
F x X X ~\~ Z X
t, the infinitely small increase in x, is written dx. We therefore have
i dl
J- X
P*~ T~dx~
X
dl
where — - is the first differential coefficient of I with respect to x.
dx x
It may be pointed out here that the value of the force of
mortality among lives assured varies between zero and infinity.
The value is nearly zero in the case of lives of age at entry .r who
have just passed the medical examination for life assurance. It is
infinitely great when we come towards the end of the mortality
table, say when there is only one person alive, and that one about
to die. In the last year it rises from a fraction less than unity to
chap, ii.] TEXT BOOK— PART II. To
infinity. It may be noted that the rate of mortality can never
exceed unity.
If the column I followed a mathematical law, it would be a
dl
simple matter to evaluate ~, and hence /j. . The several formulas
that have been suggested for / will be discussed later in Chapter
VI. Meantime we must take it that the column I does not follow
X
a mathematical law, and be content to obtain an approximate
formula for a .
At this point we find it useful to resort to the method of Central
Differences. The ordinary formula of Finite Differences for
interpolation between two of a number of given values of a
function is
,, l(t-l) ^o , t(t-l)(t-2) A3 ,
u , = u + t\u + -A 1 A-u + -A ^A 1 A 3 « + • ■ •
x+t x x 2 ' T 3
where all the known values but one are on the same side of the
unknown value, for t is supposed to lie between and 1. In a
scheme for Central Differences we choose values of the function
which are distributed more nearly equally on each side of the
required value. We have the following
Also,
x-1
1 -,
x-\
u
X
oo
b
X
"l+l
C x+1
U x+l
%+i
b x+l
U x+2
where
a =
X
a x-l + a x+l
2
=
a ,+a . + b
x-l x-1 X
2
=
a , + M
X-l - x
and a
X
-1
a -lb
X - X
1, u
-I
X x-l
=
u -a + lb .
X X - X
76 ACTUARIAL THEORY [chap. it.
Then since
x+t x-1 *■ J x-1 9 x lo x+1
neglecting differences beyond the second, and substituting for
u , and a , their values as found above, we have
X-1 X-1 '
u j_ = (u -a +U) + (t+l)(a -U) + (±±Dlb
X+t V X X - X' v / V x 2 x' o x
= u + ta + — b
x x 2 x
Adopting now the notation of the mortality table, and
writing I for u s we have
I ,= I +ta + b
X+t X x 9 X
whence -5±^ = a + — i
t x 2 *
from which we get, in the limit when t approaches 0,
dl
X
dx •<
a , + a , ,
x-1 x+1
2
Wl ~ x-1 1
2 01 " V
i dl
x
^ / rf.r
X
,, _ 1 s+l~ as-1
e-1 a; |
, 2 /
2/
X
\ d x-l + d x
/** / x V 2
! 2/
But
Therefore, /j.
We might otherwise arrive at the same result by a process
slightly different.
du 71 - u r
- — — - i when t approaches the limit 0,
dx
I x x 2 * 13 x J
t
when t approaches the limit 0,
chap, ii.] TEXT BOOK— PART II. 77
*J± _ Al , + (blk* + (^- 1 )^- 2 ) a3„ + . . .
c/a- * + 2 * 6
when £ approaches the limit 0,
= An -^A 2 n + M s w - . . .
a 1 a: ^ a:
= Au - TrA' 2 ^ approximately.
Hence, since in this approximation third and higher differences
are held to vanish, and therefore second differences are constant,
du
—^ = An - iA 2 ?( approximately,
dx x - x i
= Au - I (An -Au ,)
x - ^ a: ai-l'
= l(Au x + Au x _J
u —u
1 approximately.
2
From this we get as before
dl 1^-1,
X _ g + 1 X~l
dx~ ~~ ~ 2
, *-l x+1
and ^ = 7 - {
X
This formula provides a good working approximation to the
value of the force of mortality in almost all cases.
Again arguing from the same formula as the preceding result
was obtained from, we may obtain a general formula for /x^.
We have successively
u = -4- -7-*= -4-( A/ --^ 2/ +IAH - . . . )
r * I dx I v x T * 7 x
X X
= — — ( — d) (stopping at first differences)
X
= %
fi = - _L { - d --£-(• Fy)
die
live
live
=
*yz ^ "x'
live
die
die
=
P x (. l -Py)( l -Pz)
die
live
die
=
Py( l -P x )(. l -P s )
die
die
live
=
p z ( l -p x )0-'P y )
die
die
die
=
(1 -/0(1 -*,)(!-
-p)
If these probabilities be summed, the result will be found to be
unity, and thus proof is obtained that all possible contingencies
have been noted.
2. The probability that two persons aged respectively twenty
and forty will not both be alive at the end of 20 years is -38823.
Out of 96,223 persons alive at age twenty, 6,358 die before they
attain age thirty. Find 1 30 5 30 .
I 30930 = * _ 30P30
- 1 60
l 30
_ 1 '60 ^40 '20
~ I I I
40 '20 '30
= -34495
3. Given that the probability that two persons aged twenty-
five and fifty respectively will both live 25 years is -27516, and
that, by the same mortality table, out of 93,044 persons alive at
age twenty-five, 82,277 attain age forty, what is the probability
80 ACTUARIAL THEORY [chap. u.
that a person aged forty will survive till the attainment of age
seventy-five ?
„, _ hs. _ hb '50 ^25
( 40 '50 '25 '40
— SO ; ?5 hd _ „ v '26
~ 7~" / ~ 25^25: 50*/
'25 : 50 '40 '40
-„ c1 - 93044
= -27516 x
82277
= -31117
4. An annuity society is formed in which members may secure
an annuity of m at age x + n by payment of a single sum at age x.
If k members aged x start the society and / new members of the
same age join each subsequent year, find how many members will
be entitled to rank for annuities at the end of n + t years and the
corresponding amount payable.
Of I entering now / will survive at the end of n+t years,
/
and therefore of k entering now k JE+'izi w jH survive at the end
X
of n + 1 years. Again, of / entering one year hence / . , will
survive at the end ef n + 1 years from now, and of I entering one year
hence / x+n+ will survive ; and so on for succeeding years.
X
Thus the total number surviving at the end oi n + t years and
entitled to rank for annuities will be
, h+n+t , j( x+n+t-l , x+n+t-2 , :c+?»l
k — j— + I \ ? + — + . . . + -j-J
xxx X
Each of these gets an annuity of m, and therefore the total
amount of annuities in force will be
m> K+tPx + l L+t-lPx + n+t-lP. + ■ +.P,)}
5. Obtain from the Text Book mortality table the numerical
values of the probability that out of three lives 30, 35, and 40
(1) One, at least, will die in the 10th year.
(2) Not more than two will fail in the 10th year.
(3) All will die within 20 years.
chap, ii.] TEXT BOOK— PART II. 81
(1) This probability in symbols is
1 -(l-9l930)(l- 9 l?3 5 )(l- 9 l940)
V 89685A 86137A 82277^
_ 88879 85213 81176
89685 X 86137 * 82277
•03274.
(2) This is the probability that all three will not die in the
10th year, and is equal to
*■ ~ 9l?30 X !>l?35 X 9 1940
- 1 _ 39 v 44 v *49
'30 '85 '40
, 806 924 1101
= 1 - „»„„- x „„,„„ x
89685 86137 82277
= -9999987.
\y) 1 20530 :35 :40 = (* ~" 20^3o) V- ~ 20P35) V- ~ 20P40)
_ ^30 ~ 4o v ^35 ~ ^55 v MO ~ ^60
~ I I ~~l
'30 '35 '40
16890 19571 23435
X ■ „„,„,. X
89685 86137 82277
= -01219.
6. There are X persons living aged x, and the number of
combinations of them taken 3 together is 35. What is the
probability that, at the end of n years, the number of combinations
of the survivors taken 3 together, will be at least 10?
By inspection one may see that the number of combinations of
seven persons taken 3 together is 35, and of five taken 3 together
is 10. Thus the question is to find the probability that at least
five persons out of seven of age x will survive n years.
Zr
Now
Therefore, p.
IS) " (1+Z)*-
Z 5
71 XXXXXXX
(1+Z) 5
= Z5-5ZS + 15Z''
F
82 ACTUARIAL THEORY [chap. n.
7. Find the probability that, out of five lives all aged x, one
designated life, A, will die in the year and be the first to die.
This may happen in several different ways,
(1) A alone may die in the year, the other four surviving to
the end of the year. The probability of this event happening is
(2) A and one other may die in the year (A first), and the
other three survive to the end of the year. This probability is
(3) A and two others (A first), the probability being
(4) A and three others (A first),
(5) All (A first), K<0 6 -
The total probability therefore is
i x (p x Y + AKi x )Kp x ) s + AK%) s (p x ?
+&Wp x +W
+ K%)%+(.i x f}-(p x f]
= M(p*+<0 5 -(pJ 5 }
= w-(p x y}
8. Find expressions for the following probabilities : — ■
That out of 25 persons aged x,
(a) Exactly 5 will die in a year.
(b) Not more than 5 will die in a year.
(c) 5 designated individuals and no more will die in a year.
( = <%
approximately as
x-l X
d x
21 >
x
=
X-l X
= < u
3
that is, according as
d , >
X-l
= < d *
1 n
12. Show that q = -j- 1 I ,,u ,,dt
^x I J „ x+t^x+t
X "
--if 1 *.**
1 d? ,,
x+t ^dt
«+* dt
o dt
X
I
X
13. Prove that I = \ l t u. At
x J x+t nr+t
-•+» rfi
x+t dt
o di
= -(',+. -0
= i
[ cha p- i'. TEXT BOOK—PART II. 85
14. Show that, approximately colog p = fj, +^(q ) 3 .
c°l°g e P x =-I°g e P x
(O 2 (O s
= ^ + ^- + ^-+ etc - ■ •(!)
Also /» ,, = m approximately
= 9, + — + — + etc. . .(2)
Stopping at the term involving (9 ) 3 , and deducting (2) from (1)
colog A -^ = ^_|_
andcolog A = n x+i + jstiy
15. Show how to obtain an approximation to //. .
Here the ordinary approximate formula //, = J ~ 1 x+1 fails
~* X
us ; for if we wrote fn = . — 5f2±_ we could assign no mean-
la;]
ing to /,,,,. The / persons aged x are select, and come under
observation at that age for the first time, and consequently we
know nothing of the persons of age x — 1, of whom they are the
survivors. We must accordingly seek another approximation.
L d Jm
= ~ T ( A/ m-^ 2/ m + -J A8/ W - ■ • ■) approximately.
[*]
If then we take the column L ,, I, , ,, /. , „, etc., and difference
fcc]> [x]+l> H + 2' '
it, we obtain successively A/ r ,, A : / r ,, A 3 / r ,, etc. If, further, these
' J [xy [ry [xy ' '
be divided by 1, 2, 3, etc., respectively, and the sum of the odd
terms deducted from the sum of the even, the result, divided
by / , will give us an approximation to /x .
86 ACTUARIAL THEORY [chap. ii.
Or we may proceed thus —
1 dl..
dlos I, ,
dx
\ rflogj / (M being the modulus of common log-
= ~ ~M dx arithms and equal to -4342945).
= - ^(Alog 10 / w -|Anog lo ; M + l-A31og 10 / M -. . . )approx.
Following a method similar to that indicated above we obtain
another approximation to the value of y. .
CHAPTER III
Expectations of Life
1. The definitions of the following functions and the dis-
tinctions between them should be carefully noted.
The Complete Expectation of Life at any age is the average
future lifetime of each person of that age.
The Curtate Expectation of Life at any age is the average
number of complete years which will be lived by each person of
that age.
The expectation of life, or more properly the complete expecta-
tion of life, is also sometimes called the " mean after-lifetime " ; the
"average after-lifetime"; the "mean duration of life"; or the
" average duration of life."
The most probable after-lifetime at any age is the difference
between that age and the year of age in which the life will
most probably fail, that is, the year in which most deaths
occur.
The Vie Probable at any age is the difference between that age
and the year of age to which there is an even chance of living,
that is, the year in which the number living is reduced to
one half the original.
2. In Text Book, Article 24, Lubbock's formula is applied bo
find a more exact expression for e than e + i. The deduction of
r XX"
the formula itself may be presented as follows : —
To find the sum of the series
« + « 1 +« 2 + • • • +'h + u 1 + L + u i + l+ • ' *
T T t t
to the end of the mortality table, the values ultimately disappear-
ing whatever function u may represent.
88 ACTUARIAL THEORY [chap. hi.
We have then
« i -« 0+ 1a„ + iii_ — ^« + Ai — .pi — L* Uo+ . .
1(1 _ l) 1(1 -l)(l- 2)
u^u.+ lAu, + 1A1_ /AX+ -iAJ ^Vj Z A 3 Mo+ . .
etc. etc.
And summing these
t-i t - 1 2 2 - 1 < 2 - 1
2 « « = S + -g" A "o - T2T A2a ° + HT A3 "» - • • •
since coefficient of A?<
= 1(1+2+. . • +73I ) = i lzi) = td
coefficient of A 2 w
= i< 12 + 22+ " • ' + ('- 1 ) 2 }-^(l + 2+. ■ • +TTT)
(t-l)*(2f-l) _ /-J. _ * 2 - 1
12i! 2 4 "T2T
and so on.
Similarly
2( " 1 £-1 / 2 -l / 2 -l
2— « = tel + — A % - - I 2 F A 2 Kl + f—^
2Jl-« = <« 2 + -^ A« 2 - ■ JW A\+ ^A3„ 2 ....
and so on.
lp. in.] TEXT BOOK— PART II. 89
Now summing these summations, we get
"o + "i + w 2+- • • + u i + u i+l + "i+l + • • •
t
t-l,
fi-l,
= <(»/„ + MJ + M2+ • • • )+ t -—-(Au + An 1 + Au,+ • • •)
12<
(A 2 « + A 2 m 1 + A 2 k 2 + • • • )
+ 'i-^ 1 (A3 Wo + A3« 1 + A3« 2 + . . . )- etc.
< — 1 l" — 1 £ 2 - 1
= '(«o + «i + "2+ • • • )--2-"o+ l2T A? 'o - -35- A2 "o+ etc.
Subtracting u from both sides we have
u 2_ + n 2 + • • ■ +w i + u i+l + u i+l+ ■ • ■
t t t t
t-l t 2 -1 t 2 — 1
= t( Ul + u 2 + ■ ■ • ) + _« + -j2fAu - -^-A 2 « + etc.
For the application of this formula to the case of the complete
expectation of life, we may proceed as follows : —
If we were to say
i = t (/ *+iH+ 2+/ *+3 + • • ' )
X
we should be wrong in that we take account of no more than
complete years lived. We should therefore obtain a somewhat
better result from
K = 2 V( / *+jH +1 H + hH + 2+- • •)
x
The same error in principle appears, however, for we take account
only of complete half-years lived ; and we shall obtain a correct
result only from
where — is smaller than any assignable value.
90 ACTUARIAL THEORY [chap. in.
We now use the summation formula, and say
i =L(t(l +t +. . .) + — * + — A/ - tzlm r + etc.l
* ir\ l »+i w ' 2 » 121 * 24/ x I
X K
= e +* : + i — * : ? + ■ • «
* 2i 12/ 2 /
= (where — is infinitely small) e + h + j\r — ^— r^ +
A/. - -i 7 A2/_
T
But in discussing the force of mortality in Chapter II. we
showed that
dl
= A/ - -|A 2 Z approximately.
Therefore
1 dl
K = e * + ^ + TW rff approximately
I , - ' ,.,
. i _ a! ~ 1 j:+1
- ^+2 24/
X
The process by which e was obtained was perfectly general,
and we may similarly write
U +U,
e — e + a — — -
ij/ xy 12
and generally
u, + u + pi + • ■ -to to terms
,8 =e ...1 . r % r v r ~
xyz . . . (m) xyz . . . (m) « 12
EXAMPLES
1. Given the following mortality table, deduce, in respect of a
life aged eighty-two, (a) the curtate expectation of life, (6) the vie
probable, (c) the age at which it is most probable that he will
die. Find also the average age at death of the 129 lives, aged
ninety-five.
TEXT BOOK— PART II.
91
X
I
d
X
I
d
X
X
X
X
82
10096
1712
92
575
209
83
8384
1540
93
366
144
84
6844
1361
94
222
93
85
54S3
1180
95
129
58
86
4303
1002
96
71
34
87
3301
830
97
37
18
88
2471
671
98
19
10
89
1800
527
99
9
5
90
1273
402
100
4
3
91
871
296
101
1
1
Answers: (a) 3-582.
(6) 3-5 approximately,
(c) 82.
The average age at death of those aged ninety-five will be
96593.
2. From the Text Book mortality table ascertain the values of
lio^so and slis^o-
I io%> = 9-599, 6 \ 1& e i0 = 12-643.
3. Deduce a formula for \ 8 without making the assumption
of a uniform distribution of deaths.
By Text Book formula (27) we have approximately
e = e + i — iVh.
NoW \J X = K~nPxK+n
= («, + i - T5/0 - »/>„ .+„ + *- TS^+J
CHAPTER IV
Probabilities of Survivorship
1. In Text Book, Article 3, Q 1 is derived from formula (1) by
giving to n successively every integral value from unity upwards.
It may be derived by a similar process from formula (2)
7 1
n-l\^xy
Here then Q*
xy
d
:+n
-i x
y+n
-*
i
n
x+n
r
-1 X
I
y+n-
-4
i ,
a; +7i
I X
X
-1 ~~
I
V
x+n
X
1 ^ 1 + l ^
/ 2Z
■ j/
1 // / 11,
y 1 I x+n-1 y+n-1 _ x+n y+n-l
2 \ 11 11
x y x y
1 ., , ^ ^ I +
. x+n~\ y+n _ x+n y+n
i i ' i i
x y "% i
)
10
& 6
x : i/-l , x - 1 : i y
slnC e ^ i n i - i
2. The formula Q 1 = -Ul - "^-^ + ^=Ll2) j s , by the intro-
xv 2 V P»-i P„_i '
duction of ages x - 1 and y - 1, in a form unsuitable for application
to select tables. To render it in a suitable form we have
Ql = J_ (l _ g » : "- 1 + e *- I: A
Xy 2 V Py-l Px-l '
= J_ (l _ Px:y-li l+e x+l:y) + W^j+l))
chap. iv.J TEXT BOOK— PART II. 93
since e , = p ,(l+e ,, ) and e , = p , fl+e ,,)
x : y-1 J x : y-l> x+1 : y' x -1 : y * x— 1 : y*- x : y+l- /
Q 1 =4(l-p(l+c^ ) + p(l+e _,,)\
mi 2 I ^ x+ln/ ' ' y*- x:y+Uj
3. By the use of Tcri Sooi formula (2) we may also very easily
arrive at the probability that (x~) will die before (?/) or within t
years after the death of Q/), i.e , formula (14), as follows : —
We have Q 1 -= . = (1 - ,p ) + * «+t+«-i y+»-i
x y
/ v d Z
— n _ t) 1 4- *+' ~ x+t+ro-l y+n-j
{ t P * J + / /J
X X + ( J/
= (!-A) + AQiTts,
= l-A( 1 -Qxi:P
4. To find the more restricted probability that (x) will die within
t years after the death of (?/), we have the required probability
2d (7 -I )
y+tt-lA x+n-h x+n+t-y
Tt
y a:
""' t/+7t-l x+n-k _ x+t y-\-n-l x+n-j-t-%
ii " i ii,
y x x y x+t
^x-y tt x^~x+t : y
= (1-Qy-A(l-Q_L )
x+t : y
x+f.y
— Ql — _ _Q1
^•x : y ( 1 1) ^-xy
which is obviously correct ; for, if we take the probability that (x)
will die before (?/) from the probability that (x) will die before (j/)
or within t years aftei - , we are left with the probability that (x)
will die within t years after the death of (jf).
5. The allied probability that (.i) will be alive t years after the
death of (jy) is found as follows : —
Taking the «th year, the probability of (j/) dying therein
d
is y , and the probability of (x) being alive at the end of
l y
94 ACTUARIAL THEORY [chap. iv.
t yeavs after the middle of the rath year (since (y)'s death will
occur at the middle of the year on the average) is *+"+*-* Hence
x
the required probability
2d ^ J , , , ,
i i
y x
= ,pQ , , * as above.
This probability is the same as the probability that (x) will not
die before (?/) or within t years after Q/)'s death, since
x+f.y
= i-{i- t p.(i-QjL )}
x+t :y
= l ~ ™x : yjt))
6. The preceding probability must be clearly distinguished from
the probability that (x) will be alive at the end of the tth year
succeeding that in which (jj) dies, which may be found thus —
Taking the nth year, the probability that (^) will die therein
is v +n ~ } an d the probability that (x) will live to the end of the
y
I
tth after the rath year is x+n+ . The total probability required
X
must therefore be
" "+V 1 , ^^ = 2 u.,P ( iP ~ P)
y %
— *P (2 7? , . x n p — 2 p , , x p)
(/ aA ■«.-' x+t n-1* y n 1 x+t n 1 y J
= „ ( e x+t:y-l _
v x\ n x+t :
v F y-1
0r = «PjP.+|( 1+e . + t+i :»)-«,+«:,}
7. We may find an alternative formula for e I as follows : —
J y\x
Taking the rath year, the probability of (_!/) dying therein is
{ ,P - /J,), and (x)'s expectation of living to the end of that
year and of each year after amounts to
LPx + n+ lPx + n +2Px + • * • ) = nPj~ l+e x+J-
chap. iv.J TEXT BOOK— PART II. 95
The expectation of (,r) after (y), should (y) die in the nth year, is
therefore n P x (n-iP y - nP y X l +e x+J> and the total expectation
ei = 2 r> ( , p - p)(l+e , ).
y \x nrx\.n-l* y ti L yJ^~ x+n J
That this expression is identical with that in the formula
ei = S p (1 - p)
y]x n L aA n 1 y*
may be shown as follows : —
\Px(. l - n Py) = lPx(. l -lPy) + 2Px( l -2Py) + 3Px(. l -3Py)+ ' '
^lPx% + 2Px(ly + l\%) + i Px( C ly + l\% + 2\1y') + - ' '
=%(A + A + «P.+ • • ' )
+ 2 |? !/ X 3?'x( 1+e :C+ 3)+- • •
^%' x\n-\" y %*■ y'^~ x+n'
8. In finding e —=- we have, as shown in the Text Book,to divide
x:y(t\)
the expectation. Then the expectation for t years depends on (x)
alone, and is equal to i e . Thereafter, in order that (x)'s living
to the end of the (t+ l)th year may count, (t/) must live to the end
of one year ; that (x)'s living to the end of the (t + 2)th year may
count, ( y) must live to the end of two years ; and so on. We have
therefore
„ __ | . , x+t+l y+l + x+t + 2 j/+2 + '
e x:y(f\) " \Cx ~ r / I
x y
1 1 4-1 I 4- . . .
I . , „ x+t+l y+1 ^ x+t + 2 g+2 T
_ \t e x + tPx e x+t:y
= e - , p (e , , - c , , )
9. Coming to the probability that (x) will die first of the three
96 ACTUARIAL THEORY [chap. iv.
lives (x), (y), and (z), we may proceed in the same manner as
in the case of two lives.
O 1 = S s+tt-1 y y+n-h z+n-j
^xyz I L I
X y z
_ y x+n-1 x+n y+n - 1 y-J-7i z+w-1 z +n
" ~7 x 2/ 12
a? 2/ *
_ "S\_ ( x+n-1 y+n-l z+n-1 x+n y+n-l z+n-1
~tv / / 1 m~
x y z x y z
. x+n-1 y+n-l z+n __ x+n y+n-l z+n
x y z x y z
I , J I , I I I
, a+w-l y+w. g+w,-l x+n y+n z+n-1
/ / / m
x y s x y z
i _>. J , i i , * L i , \
/ z z i i i J
x y z x y z
40 - „
Py-\:z-l Px-liy-1 Py-l P*-l:»-l
6 C C ■ xz xyz-'
= (l-Q 1 )^ ~e )~(e -e )
\ ^yz J ^ x xyz J ^ xz xyz'
= (e - e ) - Q He - e )
^ x xz J ^~yz^ x xyz J
Or we may proceed thus : —
e 2 \ =2px|o 2
vz\x n r x In* vz
n"x\\n 1 z \n ' yz'
= e \ -e 1 ]
z\x yz\x
= (e - e ) - Q \e - e )
*» x xz J ^~yz^ x xyz J
12. The following should be carefully noted.
ITiere are two formulas for q— : —
3 xy
T-xy *x *y *xy
and q— = q x a
*xy 7 x ?y
We may express in similar forms the probability that two
G
98 ACTUARIAL THEORY [chap. iv.
children, ten and fifteen years old respectively, will both die
before attaining age twenty-one, viz. : —
and |u9io x |o? 1 5
The probability that one at least will so die is
1 -( 1 -|n9ioX 1 -| 6 '?i5)-
Now to obtain the probability that both the children will die
before twenty-one in the lifetime of their mother aged fifty, we
have analogously
1 11 9 10 : 60 + 1 6 9 15 : 50 — M 69ii0:15' : 50 "*" 0^10 : 15 : 60 X 1 5^16 : 56-'
and lll9 1 1 0: 5 o X | 6 9i5:50
The probability that one at least will so die is
1 ~ - 111?™ : 5oX X -Ie
since the former is taken from actual experience, while the latter
is taken from a table which represents suitably the mortality of
the lives concerned.
2. If the probability that (x) will die before (z) is -1996 ;
(x) „ both (J/) and (s) is -1610;
O) „ 0) is "2990 ;
(,y) „ both 0) and (s) is -2602;
find the values of the following probabilities : —
(1) That the survivor of (x) and (y) will die before (s).
(2) That (x) will die before (z), (y) having died first.
chap, iv.] TEXT BOOK— PART II. 99
(1) Ql. = Q2 +Q2 = Ql _Q1 ,Q1 _Q1
*• ' **xy:z ^xyz^^xyz %* ^j + V W
11 ■ »
= -1996- -1610 + -2990- -2602
= -0386 + -0388
= -0774
(2) Q^ = -0386
1
3. Find the probabilities that, in the tth year from the present
time,
(a) A life now aged x will die, having survived a life now aged y
by at least m years, and a life aged s by at least n years.
(6) The last survivor of three lives (x), (y), and (s) will die.
(c) A life (z) will die, leaving (x) and (y) surviving.
(a) To fulfil the conditions, (x) must die in the fth year, (y) on the
average before the middle of the (t - m)ih year, and («) on the
average before the middle of the (t - re)th year respectively.
The probability is
'x+t-l y y+t-m-j z~ z+t-n-j
I X / X I
X y z
-j-i|9, + l -i|9 v + .-i|9,-«-i|?, ir -t-i|?»,-t-i|? H
+ t-ll^xyz
(c) t^ x lz±J x l J+!zl
x y
4. Find expressions for the probabilities that of three lives
(x) (_y) and (z), (x) will die
(1) In the same year as (s), whether first, second, or third of
the lives.
(2) At least t years after (j/), and at least t years before (s).
100 ACTUARIAL THEORY [chap. iv.
(1) This probability is equal to the sura for all values of n of the
probability that both (x) and (s) will die in the nth. year
^n-1* as n* x'^-n-l' z n' Z-*
— «( ,P + P — - p X p - p X n p)
\n-\±xz n l xz n-1 1 x n* z n l x n-1* z'
€ e
— 1 _l O c _ x~l : z _ x : g-l
' " *"" P x -X '" P 2 _j
(2) If (,r) die in the (ra + / + l)th year, then to fulfil the conditions
(#) must on the average have died before the middle of
the (?i + l)th year, and (z) must live on the average till the
middle of the (n + 2t+ l)th year. The probability required
is therefore
2 x+n+t x y y+n+i z+n+it+j
I I I
X y z
V x+t+n y~ y+n+$ z+it+n+j
1 I /
x+t y z+2t
tPx X 2(Pz^x+( : g : z+2t
5. Required (.r)'s expectation of life ten years after the death
of a life presently aged y.
From the whole expectation of life of (.r), we must deduct the
part during the life of ( y) and for ten years after. The expectation
required is therefore
e x~ e x : y(W\) = e x~ < e x~ wPx{ e x+10~ e x+10:yJ>
loPx^x+lO ~ e x+10 : y)
Or we may proceed otherwise. The (ra+10)th year will count
only if (,r) survive it, and if (y) die within n years. The prob-
ability of its being reckoned is therefore (1 - p ) +10 f>j and the
sum of this expression for all values of n is the expectation
required
V n-^yJn+lO-^x
~ V)Px^{ ~ nPy'nPx+10
V>PJ< e x+10 ~ e a;+10 : y)
6. Write down in respect of the (t+ l)th year the probability
indicated by each of the following symbols: — (a) Q 2 , (6) Q 2 ,
(c) Q 3 , (d) Q 1 -, (e) QI , (/) QJ_
V > **-xyz' V J ^x-.yz' V ■> ^xy.z KJ J and the
average future lifetime of each of these L persons is therefore
KT. + T. +1 )
L years.
X
Now the average future lifetime of L persons living between
and 1 will be £(T + Tj), and of Lj „
1 „ 2 „ M T i + T 2)' and of L 2 «
2 „ 3 ,, 2(^2 + ^3), and so on.
CHAP. V.]
TEXT BOOK— PART II.
105
Therefore the average future lifetime of each of the whole
existing population will be
i(T + T ] ) + KT 1 +T 9 ) + KT 9 + T,)+. . .
L + L, + L 2 + • ■ •
. |T + T 1 + T 2 +. ■ ■
T n
Y»
Similarly the average future lifetime of the existing population
Y
who are aged x and upwards may be shown to be -=?.
X
4. Now if the average present age of the existing population is
Y Y
?p-j and if their average future lifetime is ttt, it is obvious that
the average age at death of the existing population will be
2Y„
-rp- 2 ; and for those of the existing population who are aged x and
2Y
upwards the average age at death will be x + ttt^-
X
5 The distinction drawn in Text Book, Article 21, may be
shown thus : —
In the case of the stationary population which is recruited each
year by births, we have for the average age at death
%d + %d 1 + %d 2 + ■ • ■
d + d 1 + d 2 + ■ ■ •
T
= —2
/„
In the case of the existing population we have as follows : —
(i)
0-1
1-2
2-3
etc.
Number
living.
(2)
etc.
Average age at death
of each.
(3)
2 + L n
* +
KT\ + T 2 )
KT 2 + T 3 )
etc.
106 ACTUARIAL THEORY [chap. v.
Multiplying the number living at each age by the average age
at death of each of the group, and dividing the sum of these pro-
ducts by the total population, we have for the average age at death
of the existing population
£L + fL 1 + |L 2 +-- - KT + T 1 ) + HT 1 + T 2 ) + HT 2 + T 8 )+...
L + L 1 + L 2 +... L + L 1 + L 2 + ■■ ■
= £T n + T 1+ T 2 +. • • + jT + T 1 + T g +
T T
- Xo , Y
To + T
2Y_o
T
In the case of e we have the average age at death of the
population and, assuming that there are /„ annual births, this
2Y
average age is the same every year. In the case of -=-^ we
A o
multiply the number at present living in each age group by the
average age at death of the group, and by this process obtain
the average age at death of the present members of the
community.
Applying the same process to those of the present population
aged x and upwards we have for their average age at death
(* + i)L.+ (* + §)L^ 1+ (* + f)L, +g+ ... + •yr, + T, +1 +T +1 + .. ;
L ,+ L . + i +L . +2 +- h+h + i + h + 2 +
Y
Y
—
X
+
— +
T
X
T
S
—
X
+
2Y
X
EXAMPLES
1. Having given a complete table of p x , accurately representing
the probabilities of life at all ages, show how, from the deaths
taking place in one year, to calculate approximately the total
numbers living in a stationary population where there is no
disturbance from immigration or emigration.
chap, v.] TEXT BOOK— PART II. 107
Since the population is stationary, the deaths in any one year
must equal the births, that is
l = d + d 1 + d. 2 + d s + ■
We are also given p^ p v p 2 , p s , etc., and therefore since
h = h x Pi
l 2 = l 2 X P-2
etc.
we can successively obtain ly, /.„ l v etc.
Now making the assumption of a uniform distribution of
births and deaths, the total population numbers
= u + i i+h+ i 3+ ■ ■
2. A military power desires to maintain a standing army of
1,000,000 men. Five years' service is compulsory on all males
attaining the age of twenty. How would you apply a table,
showing the mortality amongst males, to ascertain the annual
number of recruits required to maintain the army at its proper
strength ?
By the table, an entry at age twenty of /„ males will support
a population between twenty and twenty-five of T 90 - T„-. By
simple proportion we find the number of recruits of age twenty
necessary to support an army of 1,000,000 men of ages twenty to
. . , . , 1000000 ,
twenty-nve to be = =— ( 20 .
*■ 20 _ * 25
This formula naturally only takes account of the numbers in
time of peace. It further does not allow for the necessary
selection by medical examination of the recruits, nor for the effect
thereof on mortality.
3. In a stationary community supported by 5000 annual births,
each member, on attaining the age of twenty, makes a payment of
£20, and contributes £1 at the end of each succeeding year until,
and inclusive of, the sixtieth birthday ; receiving thereafter an
annuity of £15, payable at the end of each year. In respect of
each contributor who dies before receiving the first payment of
£15, a payment of £5 is made. Find expressions for (a) the
number of contributors, (6) the annual receipts, (c) the total
yearly annuity-payment, and (d) the annual death claims.
108 ACTUARIAL THEORY [chap. v.
(a) The number of contributors is — — (T 20 - T 60 ).
k
(6) The annual receipts are
^0°_(20/ 20 + l n + Z 22 + . • . +l m ) = ^(20Z 20 + N' 20 - N' 60 )
(c) The total yearly annuity-payment is
5000 , 5000 ,_. T ,
—r- xl5(l 61 + l e2 + l 6a + ■ ■ ■ +L-i) = -r-xl5N 60
l o o
(d) The annual death claims are
5000 K ,l J 1 J N 5000 S/I 7 A
-7— x5(d 20 + d 21 + d 22 + • • ■ +L 20 + r2iL 21+ ^L 22 + . • + r^L 54 ).
'0
110 ACTUARIAL THEORY [chap. v.
7. A system of free education is introduced into a community,
embracing all children from the age of five to thirteen inclusive.
The present number of such children is A, but the birth-rate in
the community, previously stationary, has been increasing during
the last four years at 1 per cent, per annum. What would you
estimate to be the number of children under education at the end
of twenty years, and how many children will have passed out
during the period ?
Assuming that, during the period the population was stationary,
a table of mortality was formed showing the column of L and the
column of T, we may proceed as follows ; —
In the first year of the system the number under education is
A / L 5 + L 6+ L 7+ • • ■ + L ]3 \
In the second year A f^C 1 " 01 ) + L 6 + L T + ; ' ; + L us\
\ T - T J
In the third year A { ML2D! + ^(W+ L r+ • " • ±hA
*■ ^-5 ■'■14 I
and so on, till at the end of the twentieth year the number
will be
A | L B (l-01)i» + L B (l-0iy» + L T (l-01)"+. ■ ■ +L,„(l-0iyi 1
l T 6 - T w J
The number that will have passed out during the twenty years,
having attained the age of fourteen, will be
A?4i-{l0 + (l-01) + (l-01)» + . . . +(l-01)i"|
8. In a pension society it is a condition that if a member dies
after m years from entry his widow is entitled to an annuity. If
on the average there enter the society in the course of the year
k new members aged x, each with a wife aged y, how many
widows entitled to draw annuities will there be at the end of the
rath year of the society's existence ?
The widows between (y + m) and (y + m + 1) years of age are
on the average each (y + m + J), and that they may be entitled to
annuities their husbands must have died between (x + m) and
(x + m + J). Similarly the husbands of the widows whose average
age is (y + m + -f ) must have died between (x + m) and (x + m + f )
chap, v.] TEXT BOOK— PART II. Ill
if their widows are to get annuities, and so on for (n - m) terms.
Therefore the number of widows entitled to draw annuities is
I I \ s+m+i*- x+m x+m+i' "*" 'y+m+V-'x+m~ .e+m + y
x y "-
y+n-i^ x+m x+n-y J
= k( p x I e-i e")
M?U x m\n-m y m\n-m xy'
An alternative method of arriving at the same result is to
deduct from the number of females who are alive and who have
become entitled to annuities by the survivance of their husbands
for the necessary m years the number of females who are alive and
whose husbands are still alive. The result is obviously the number
of widows who are entitled to draw annuities. Following out this
plan we get
I I \ y+m+h x+m y+m+% x+m y+n-b x+m'
x y <*
~ *■ J+m+J x+m+i + y+m+f x+m+? + ' " ' + 'y+n-$ x+n-y J"
= if p x I e — I I) as above.
Mrc-i a: m\n-m y m I n - m xy'
9. A railway staff in a stationary condition is recruited
annually by 500 entrants at age twenty, who are required to
contribute to a pension fund. At age sixty they have the option
of retiring on pension, and retirement is compulsory at age
sixty-five. Assuming that there are no secessions other than by
death, and that one-half of those who reach age sixty retire at
that age, the others remaining till age sixty-five, give expressions
for : —
(1) The total number of present contributors.
(2) The total number of present pensioners.
(3) The total number of future years' service with which
existing contributors will be credited.
(1) The total number of present contributors is
•^{(L 20 + L 21+ . • • +L 64 )-K^o + L 01 +- • • + L M )}
500,
= ~{(T S0 -T 66 )-i(T 60 -T e6 )}
= ^{T 20 -KT 60 + T 65 )}
112 ACTUARIAL THEORY [chap. v.
(2) The total number of present pensioners is
^{K L C0+ L 61+- ' • +L 64 ) + ( L 6 6 + L 00+ • • ■ )}
'20 «• I
l 20 ^ }
= ^xKT 60 + T 65 )
(The total of present contributors and present pensioners makes
up the whole population of age twenty and upwards. This is
obviously correct, and goes to prove our results.)
(3) J(T 20 + T 21 ) represents the total future lifetime of the L 20
persons living between twenty and twenty-one. Of this number
of years, however, \ (T 60 + T 65 ) is lived after retirement. Hence
iCTao + T^) - i(T 60 + T 65 ) represents the number of future years'
service in respect of the L 20 persons living between twenty and
twenty-one.
Similarly |(T 21 + T 22 ) - \ (T 60 + T 66 ) represents the number of
future years' service in respect of the L 21 persons living between
twenty-one and twenty-two.
We shall obtain similar expressions for L 22 , L 23 , etc., to L 59 ,
that for the last being \ (T 69 + T 60 ) - \ (T 60 + T w ).
Further, for the JL 60 persons between sixty and sixty-one
who have not retired the number of future years' service is
il^C^Mjo + IV) - T 65 }. Also for the £L 61 persons between sixty-one
and sixty -two years of age the expression is \ {J(T 61 + T 62 ) - T 65 } ;
and so on, till finally for £L 64 it is \ {J(T 64 + T 6S ) - T 65 }.
The total number of future years' service with which existing
contributors will be credited is therefore
™ {(Y 20 " \o) - 20(T 6 o + T 05 ) + i(Y«, - Y 65 - 5T 66 )}.
CHAPTER VI
Formulas of De Moivre, Gompertz, and Makeham,
for the Law of Mortality
1. Under the supposition named at the beginning of Text
Book, Article 7, viz., that the numbers living at successive ages
are in geometrical progression, it may be shown as follows that
" there would be no assignable limit to the duration of human
life, and the values of annuities would be equal at all ages."
Let k be the radix of the table, and r the rate at which the
population is decreasing, and therefore fractional. Then the
numbers living at successive ages will be k, kt, kv 2 , nr 3 , etc., to
infinity.
At any age x we have the force of mortality
1 dl
— __ X
F *~ i die
x
1 dKr*
Kf dx
= xr* log r
Kr x »e
= - log r
which is independent of the value of x, and therefore constant at
all ages.
Also the complete expectation of life at any age x
1 r
-.1
Kr x J
J
1
log/
Kr^+tdt
o
r*dt
H
114 ACTUARIAL THEORY [chap. vi.
e x is therefore also constant at all ages ; and thus the first part
of the statement is proved.
Anticipating by a little the theory of annuities, we have for
the annuity value under the supposition
a
X
=
xJ o
vH , t dt
x+t
=
1 r
J
v t Kr x + t dt
=
I?
r f dt
==
1
log
v + logr
1
8 + ?x
which as above is independent of x, and is therefore constant at
all ages. Thus we find the second part of the statement correct.
2. Under Gompertz's law the formula for I is expressed in
terms of three constants. The constant k does not vary with
the age, and is therefore of importance only in fixing the radix
of the mortality table.
From any three values of the number living, the three con-
stants may be deduced. For example, taking the values of log I
at ages x, (x + i), and (x + 2f), we proceed as follows : —
log / = logk + c x logg
lo S l x+t = log & + <;*+' log g
lo S l x+zt = logi + C+^logg
Taking the first differences of both sides of the equation
log t P x = ^O'-^logg
l °S t P x +t = c*-H(c*-l)logg.
Now dividing each side of the second equation by the
corresponding side of the first, and taking the logarithm of the
result, we have
log (log t p x+t )- log (log t pJ = tlogc,
from which c may be found, and by substituting its value in one or
other of the second set of equations, logg may also be found.
Again by substituting these two values in the first equation of all,
chap, vi.] TEXT BOOK— PART II. 1 15
log k may be found, and from the three constants now known the
table may be wholly determined.
3. In the case of a mortality table following Gompertz's law it
may be shown that
I,? 1 = Q 1 x 1,7
\t *%y ^xy \ t 'xy
a relation which, it was suggested in the notes on Chapter IV.,
might be used as an approximation in any mortality table.
x y J o
1 /""
By Gompertz's law = -5—=- / I , .1 , , Bc x +'dt
J r I I / x+t y+t
x y
rX 1 r 00
c x + cy I l x+t y +t v '
x y J
c* If"
= &T&TT\ l x+t l y+ i (.f , 'x+t + ' J 'y+t') dt
* y
c*
Q.
c x 4. c y ^-xy
c x
C x + cV
1 f*
Als ° \t?ly - IjJ^ 'M+th+t^+t*
nj ' x+t i y+t & x+t + r y+t yt
x y'
c*+~cy , .
* y
Q 1 > + Gx e ,
A, B, C, etc., being obtained from seven equations depending on
the unadjusted data.
One curious point connected with Makeham's law is the fact
that the value of logc as deduced from different mortality
experiences is very nearly identical. The following examples
may be given . —
Seventeen Offices
. -03956
H MF . . . .
„ -0400008
Thirty American Offices
. -041280
Gotha Life Office
„ -039625
o [M] .....
. -039
Commenting on this, Makeham has said : " The practically
identical agreement in the value of log c in these several instances
could only result from the rate of deterioration of the vital force
being the same for each individual. Thus extended, Gompertz's
law may be stated as follows : The vital force or recuperative
power of each individual loses equal proportions in equal times ;
and the proportion of vital force so lost by each is universally the
same, being approximately represented by log c = -04."
Makeham's first modification of Gompertz's law was never
intended to be applied to every mortality table. In some cases
it was shown to fail to represent the experience. Where
geometrical second differences of log I were not found to exist,
he suggested that geometrical third differences might exist, "not
as in any way superseding the method of second geometrical
differences, but merely as a substitute which may be available in
certain cases where the other is found to be unsuitable." This
118 ACTUARIAL THEORY [chap. vi.
method lacks the neatness of application to joint-life contingencies
possessed by the well-known first modification, and has never
proved to be of more than theoretical interest. It may simply
be stated that under the second modification I is of the
X
form ks^g^'uf 2 , whence we find that p = A + FLz + Bc*.
EXAMPLES
1. Find, on De Moivre's hypothesis, the most probable number
of deaths among 1000 persons aged thirty.
The most probable number of deaths may be found by deter-
mining the greatest term in the series (p 30 + <7 30 ) 100CI -
But by De Moivre's hypothesis p 30 = ^1 = ^ ~ ^ =
and 9so = 5Q-
Therefore we require the greatest term in the expansion of
1000-ra + l 1
^55 J_y°oo.
\56 + 56/
Now the (n + 1 )th term = the rath term x
n 55
And therefore the (re + l)th term > the rath term
so long as 1000 - n + 1 > 55»
or 56re < 1001
or » < 17ft.
That is, the 18th term is the greatest.
But the 18th term involves 17 deaths ; therefore the most
probable number of deaths is 17.
2. Prove that upon De Moivre's hypothesis the force of
mortality is equal to the rate of mortality at all ages.
1 dl
** ~ ~T ~dlc
x
1 d(8&-x) .. . , , . .
= ~ 86^ di — ( J h yp° thesis )
1
86 -a:
= %
chap. vi.J TEXT BOOK— PART II. 119
3. If the force of mortality varies inversely as the complement
of life, deduce the forms of I and e .
XX
1
Let [l =
(o - x
k
0) — X
d log I
But a = - r_£
r * dx
Therefore log I = - \ u, dx
a X J ' X
— dx
iu — x
= k log (oj - a:)
whence l x = (a> - «)*
Also ■ / -2±-'<&
"-/.
x I
.'
But I V -t ^- ^ =
fr a-.r-Q * 1 -( M -a- <)''+!
(o-.r)* " (co-.r)* A + l
1 (a; - ayn- 1
Therefore e = , .
* ( which is the loss to the granter of
the annuity for each survivor, and therefore the total loss in
respect of the survivors of l x entrants at age x is I q (1+a ).
Again, if the annuitant dies during the first year the whole of
the accumulation, that is (l+z')a, will be set free. The total
amount set free by the deaths among I entrants at age x will be
d (1 +i)a . It is assumed that the mortality actually experienced
will be that shown by the table adopted, and it may easily be
proved that
I ..a (1+a ,,) --= d (1 +i)a
Expressing Text Book formula (3) in slightly altered form, we have
l(l+i)a = / ,,(l+a ,,)
x^ ' x a+iv X+V
From this it may be seen
(1) That, unless d die during the year, the accumulations of
chap, vii.] TEXT BOOK— PART II. 121
the purchase-monies of l x entrants will not be sufficient to meet the
annuities payable to the survivors. This confirms Mr King's
remarks in Article 4 that "we must always presuppose a suffi-
cient number of such benefits to form an average; so that
the contributions for those that never mature may be available
to meet the deficiency in the contributions for those that actually
become payable."
(2) That, when an annuitant dies, the reserve released is not
to be considered profit to the office, since the amount released has
to be applied to make good the deficiency under the contracts of
those who survive.
2. The following simple proof shows the identity in result of
Barrett's and Davies's forms of commutation columns. According
to Barrett, who used the initial form,
„ _ B - + i
. A *+l +A s + 2 +
A
X
= ( 1+0 B — ^ +1 + (!+»•)" — V,+ - • •
(1 + •)»-',
(l+i)<»v x l x
X
D
X
N
_ X
D
X
according to Davies, who used the terminal form.
3. In Text Booh, Article 25, it is shown that the value of a
life annuity is less than the value of an annuity-certain for the
term of the curtate expectation, and it may similarly be shown
that the value of an assurance of 1 payable at the moment of death
122 ACTUARIAL THEORY [chap. vii.
is greater than the value of 1 due at the end of the term of the
complete expectation.
A. = l-rf(l+«J
> 1 -d(l +a— i), since a < a—,
x e x\
,.1+e
> i\ e x +K where 1 - k is the time lived by (x)
in the year of death.
Therefore A (l+i)* > v^
or A > v e x since A = A(l+i) K , k being the
X X X^ / °
time before the end of the year, at which death occurs.
Or we may prove it thus : — Let there be d quantities each
eaual to v, d , each « 2 , d , „ each v 3 , and so on. The arithmetic
mean of these d + a" , , + a" L _ + • • • quantities is
X x+1 x+2 *
d x+ <*.+! + d x+2 +
= A
X
and their geometric mean is
i x +l+ ad x+^
**+** „+!+** „
V d x+ d x+l +d x+2+- ■ ■
= „ 1+ «*
But the arithmetic mean of any number of positive quantities
which are not all equal is always greater than their geometric
mean, as proved on page 6.
Therefore A. > v 1+ '»
and
A > v ex as before,
Again
A
foolp _ x
X
since A >
v 6 " and a < a—.
From this we argue that the annual premium required to provide
a payment of 1 at the moment of death is greater than the annual
CHAP. VII
] TEXT BOOK— PART II. 123
premium payable in advance required to provide 1 at the end
of the term-certain of the complete expectation of life.
4. If we are given any two of the three functions a . A , and P x
we can find i, the rate at which they are calculated.
(1) Given a and A .
V / a; x
1 — ia
By Text Book formula (22) A^ = — ?
whence (l + i)A = 1 - ia
1-A.
and i
a + A
X X
(2) Given A^ and P x .
dA
By Text Book formula (37) P^ = j—jr
X
whence (l+i)T x (l-AJ = i\
P (1 - A )
x V X'
and
A - P (1-A)
(3) Given P x and a x .
By Text Book formula (38) P„ = — d
J v ' x 1 +a
X
whence (1 + i) ? x (1 + aj = 1 - ia s
1 ~ *.+u i +°J
and
5. The form of the equations given in Text Book, Article 39,
can be adapted to the case of endowment assurances in every
instance.
A xn\ = V (. l+a x:^- a x:^T\
1 - ia x:^Tl
1+i
x:n-l\
ii
o„ - a
All through ^.^ni is substituted for a,, and it will be found a
124 ACTUARIAL THEORY [chap. vii.
useful exercise to reason these equations out for endowment
assurances as is done in Text Booh, Articles 30, and 41 to 44, for
whole-life assurances.
6. In connection with Text Book formula (41), it should be
noted that
D +D + • • • +D ,
x+1 T X+2J '- L -"
xn\
=
D
x+l +
d x
D x+2+- ■ ■ + B x+n-
-1 . x+n
D
X
' D
X
=
a
x :
+An
7t - 1 1 xn\
Hence
A n
=
a — — a — -r
xn \ x : n - \\
and
pi
xn\
=
A 1 ,
xn |
1 +a — -,
x :n-\\
=
a —.- a — rr
xn\ x'.n-\\
1 + a — TT
x:n-l\
This formula is frequently useful and exhibits the method of
finding the annual premium for a pure endowment by means of
tables of temporary annuities alone.
7. By application of Text Book formula (57) we have
a— = a + a —a
xy x y xy
And also by Text Book formula (63)
A— = v(l +a—)-a-
xy v xy J xy
= {"(1 +«,)-",} + HI + «,) - «,} - HI + a xy ) - aj
= A +A -A
x y xy
But it must be observed that we cannot write
P-=P +P -P
xy x y xy
which will be at once apparent when it is remembered that
A-
P_ = xy
*» " 1+a-
xy
A +A -A
_ x y xy
1 + fl-
aw
A A
y xy
1 + a— 1 + a— \+a—
xy xy xy
chap, vii.] TEXT BOOK— PART II. 125
8. In Text Book, Article 88, A -, is described as an
' ' wxyz . . . (m)
assurance on the last r survivors of in lives, by which is meant
that it is an assurance payable at the end of the year in which
at least r lives cease to survive, i.e., the year in which occurs
the death of the (m-r+l)th person. Now the probability that
the (in — r + l)th death will occur in the reth year is
i(7 L = ( p L- p L).
7i- 1[ wxyz . . . (m) rt-1 wxyz . . . (m) n wxyz . . . (my
Therefore A 2L = 2«"( p L - p
(m) Ti-1 wxyz . . . (m) *i irayz . . . (m)
= Tjfl+Sy" 73 M-2u ra p
iuei/z . . . im) 71 wxyz . . . (m)
= v(\+a t)-a.
But affain
A 1 = 2r»
imi/z . . . (m) wxyz . . . (m)
1-
, (m) Ti-1 J wxyz . . (m)
= 2tc{Z'-rZ''+i+ ! ^ wx icy ir; xy xz yz>
+ (A +A +A + A ) - A
V toti/ was ii'>/- xj/2' »™>/z
A ? = Z 2 - 2Z 3 + 3Z 4
-( A .» + A .y+ A «.+ A «, + A » + V
" KKxy + Kxz + Kyz + KJ + 3 Kxyz
A _s = z 3 -3Z 4
= f A +A +A + A ) - 3A .
V it-J-i/ if« »»!/z J; !/- y
A — * = Z 4 = A
126 ACTUARIAL THEORY [chap. vii.
9. The Text Book skeleton formula (57) can be applied to
temporary benefits and deferred benefits, by altering the meaning
ofZ.
Thus we may say
r(r+ 1")
I a * , = Zr-rZr+i + -^-^ Z'+S- . . .
I n wxyz . . . (m) J,
where Z r signifies the sum of the values of the temporary
annuities on r joint lives for n years, for all the combinations of
r lives that can be made out of m lives.
Or again |A A = Z r -rZ r + l + ^t ~ Z'+»- . - .
where Z r signifies the sum of the values of the assurances deferred
n years on r joint lives for all the combinations of r lives that can
be made out of m lives.
10. To find the present value at rate i of the amount at rate j
of an annuity-due of 1 per annum to accumulate during the lifetime
of 0).
The following is perhaps a clearer demonstration than that
given in Text Book, Article 98.
If (x) die in the first year the amount of accumulated annuity
payable is (1 +j) = (1 + 1 /> IT
If (x) die in the second year the amount payable is
{(i+i) 2 +(i+i)} = (i+»v
And so on, and generally if (x) die in the fth year the amount
payable is (1 +J)s-^.
The present value is to be taken at rate i, and the probability
d
of (x)'s death in the tth year is *+' - . Therefore the present
value of the amount to be paid in the event of (x)'s death in the
tih year is
(l+jj-l d^ f ,
vHl+jY- — — x -±t±
K JJ .1 I
chap, vii.] TEXT BOOK— PART II. 127
Hence A = 2 <=U -V(1 + j) Q±jlzl fs+izl
J x K K '
where A' is calculated at rate J, which is such that J = *
* l + i 1 + J'
For an alternative solution we have
+ « 3 {(i +j) + (i +i) 2 + (i +i) 3 }^ +
= (i +j) (-5 — j+1 z * +2 — ) + (i +i) 2 ( ° +1 ^ +2
a; '
_ (i+j)M x+ (i +J y-M x+1 + (i + j?M x+i +. ■ .
D
X
11. To find the present value at rate i of the amount at
rate j of a temporary annuity-due of 1 per annum for n years to
accumulate during the lifetime of {£), the accumulations to be
payable only in the event of (x)'s death within that period.
As in the previous problem, the present value of the amount
to be paid in the event of (x)'s death in the tth year is
v\\+j)
(l+i) 1 -!^
Hence A = 2 (=1 v f (l + j)
, (1 +/?-! d t+l _,
1 +,; f rf -- — - d
j
1+ J,
. '-(A*- A 1 -)
; ^ xn\ xn\
i+y i
where A — , is calculated at rate J, which is such that •=— f~- = -. T -
arn | 1 + Z 1 + J
128 ACTUARIAL THEORY [chap. vii.
According to the alternative method
A = ,(i+i)f +4(i+i) + (i+i) 2 }%+- • ■
X X
+«"{(i +i) + (i +i) 2 +•••+(! +M %^
a;
= (1 +i) (^ ^— ? ^±^ij + (1 +./)*(— =±i ^'j
x a;
+ . . . +(1+jQ« '+- 1
_ (1 +j)(M x - M^J + (1 +j) 2 (M* +1 - M, + J +.■■+(! jjjy, - M^ + J
D
12. A different problem from the last is to find the present
value at rate i of the amount at rate^' of an annuity-due of 1 per
annum which is to be allowed to accumulate until all of / persons
are dead.
This means that each payment is to be allowed to accumulate
at rate,/ up to the limit of life, and the whole must be discounted
from that time at rate i. Therefore
A = „„_. x w+jy-'+it+iV+jr-'- 1 + ■■■ + L-iQ+A
VI + i) x /
i±iV (1+0
01-x
J + i)
where a' is calculated at rate j.
X J
For the value of a temporary annuity-due which is to be
allowed to accumulate to the end of the term under similar
conditions, we have
A = v « w+jy+^+jy-^- ■ • +*, +n -i( 1+ .a
i x
-\i + i) x - r
= U — •) + a 'x:n^i\) w ^ ere a ' x -^zi\ * s calculated at rate j.
chap, vii.] TEXT BOOK— PART II. 129
13. To find Lhe value of a temporary assurance for n years
on the life of (x), commencing at 1 and increasing by 1 per
annum so long as (x) and (tj) are jointly alive.
According to the terms of the contract it will be noticed that
1 is payable in any case provided (x) dies between ages x and
M - 1*1
x + n. lhe value of this portion is therefore — x x+n . This
X
sum assured of 1 falls to be increased by 1 provided (y) lives one
year and (x) dies between ages x + 1 and ,t + n. The value of
M , - M M
this second portion is therefore p — x+ ^ — 5±^. And so on
r r v D
X
for every year until the rath, the last increase taking place if
(if) lives n - 1 years and (x) dies between - ages x + n-1 and x + n,
]VI -M
the value thereof being , p *+*-* £±2.
x
Therefore the whole value of the assurance is
A = ^- /(M - ]VI )+p (]VI - ]VI )+ • . .
Y) I ^ x x+n' "y\ x+1 x+n'
+ , p (M _, , - 1VI ) 1
n-l 1 y\ x+n-1 x+n J j
14. The annuity of Text Book, Article 99, is payable so long
as (x) lives, but not more than / years after the death of (jf).
Now as to the first t years, it is obvious that (j/) does not come into
consideration at all ; but, in order that a payment may be made on
(.c)'s surviving the (< + l)th year, it is necessary that (jj) should
have lived at least one year, and similarly for following years.
The value of the whole annuity is therefore
vl .+l + r ~ l x+2 + - •• +VH *+t Vt+ll x +t+ l l y + l+ vt+2l x +W l y + 2 + ---
I + 11
x x y
„ » H x+t vl x +t+ l l V+l + v2l *+t + °Jy+2+ ■ —
x x+t y
= a *ri+ vt tPx a x+f.y
= a __|±f(a fl )
x D *+' x+f.y'
x
15. To find the value of an annuity to (.t), the first payment
I
130 ACTUARIAL THEORY [chap. vii.
to be made at the end of the tth 3 r ear succeeding the year in
which (jy) dies.
The value of this annuity could be obtained by deducting
from a the foregoing annuity, for this would give the value of
a life annuity to (x) less that of an annuity payable so long as (x)
lives jointly with (?/) and for t years after the death of (y), should
(x) live so long. The difference is obviously the value of the
desired annuity.
Thus a = a —a — =;
x x:y(t\)
= a -{a - -£+< ( aa ) I
x-\-t / \
- -^T^x+t x+f.y)
X
Or we may proceed as follows : —
a = 2 -"-VJ±t- 1 V'< ( |»,
n = l Z Z M *+»
» x
Z -I Z
- y,„n+t K-fn-1 !/+"■ J+n+t /i , •>
Z Z ^ z+i+ 2 (ra-2) + • • • +«»(»-»)
u
(jl - 1)0—!=-;
-"nm
11
(n - l>-i -
a—, — nv 11
n\
n
n\T\
n i.n/
7l| r-l| n+llr|
134
ACTUARIAL THEORY
[«=
CHAP. VII.
(»-!)(!- u TC ) - (a^ - nv n )
n
-0+W
n — (T + z")a— j
?H
Again, from the reasoning in Text Boole, Article 108, we may
form a mortality table for joint lives as follows : —
Ages.
Couples
remaining.
Couples broken.
x+\ :)/ + l
x+2 :y + 2
x + o :y+3
etc.
nm
(«-l)(m-l)
(«-2)(m-2)
(n - 3) (m - 3)
etc.
ra + m- 1
n + m- 3
» + »n-5
m + m - 7
etc.
Hence
33/
i>(re + m - l) + y 2 (» + m - 3) + u 3 (re + ?« — 5)+ • . • to ra terms
— i f» + m-l)a— lT1 - '2a—. —. I
nm \ y J "HI »|2||
f a—. - rai)"-\
\(»+m-l)a S |-
Also
*!/ rem
— i) {rem --(n + m) + l] + ii 2 {«?« - 2 (ii + ?«) + 4}
+ 1> 3 [nm -3(n + ?ii) + 9\+ to re terms j
-L [{„*» _ (re + m) + 1 { ^ - - ( ra + ,„ _ 3>- | r| + ^ j, ]
and a— — = a—
1L j 1 | 7t
a—. — nv n
n(n - 1")
%2\ 2
"I 3 I
chap, vii.] TEXT BOOK— PART II. 135
22. In Text Book, Article 115, we have the value of a temporary
annuity on two lives, the term varying with each life. Similarly
we may have a deferred annuity on two lives, the period for
which it is deferred varying with each life. For example, the
annuity payable till the survivor of two lives, aged six and eleven
respectively, attains majority is
|l5 a 6+|l0 a il-|l0%:ll
And by analogy the annuity payable to the survivor of two lives,
aged six and eleven respectively, but deferred till each or the
survivor has attained majority, is
isl^ + ioCn-islVii
The general formula for such an annuity is
\a + \n — l« if in < n
n\ a m\"x n\ ux
or I a + I a - i a if m > re.
n\ a m\ x m| ax
23. Again, the assurance payable if either of two lives, aged
six and eleven respectively, dies before attaining majority, is
A i - ! - = A 1 - + -J^- 1 A l -
(6:15|)(ll:l0|) ■"''67TT 1 : 10| T D 16:5|
6 :11
and for the annual premium we shall have
D
P* -m *
A 1 _ . 16:21 A l
'eill'ilOp F) 16 :6|
6 : 11
(<3:15|)(11:10|) _ D io-21
a : 11 : i6I + T) a i6 • 5J
6:11
Generally
*_1 , a+m : x+m a 1
l ax' m| £) a+m : n-m\
P. 1 -i., 1 -;. = ^r — if m < n
, a+m:x+m
axm\ ]} a+m : n-m\
ax
A^ -. + "+»=»+? A-4 .
if m > n
D
a+»:»+»
axn\ Y) '•
ax
136 ACTUARIAL THEORY [chap. vii.
24. Also the assurance payable if both lives, (six) and (eleven),
die before attaining majority is
Ai _ i ~ = A 1 -+A 1 A 1 - J -
(6:16|)(11 :10|) 6:15| T 11:10| (6 :15|)(11 :10|)
_ Al I 41 A 1 _ 16 :21 n
6: Ml" 1 " ll:10| '6 : ll 1 : 10| i-» 16:51
6 : 11
The annual premium for this benefit is
Ai _ i _
P l _ 1 _ = (6:16|)(11:10|)
«S;15|)(11:10]) D 16:21
a (0 : 16|) (11 : i0|) ~D 16: M
6 : 11
Al , ll A ! 16 : 21 a i
6 :15| "^ 11 :10| '6:li:10| n 16 : 5 1
6 : 11
, _ _ 16 :21
a 6:15| + a il :10| a e : 11 : 10] H 16 : 5 1
6 : 11
Generally
A i + Ai- -A'- «+»:«+,» A4 ,
Pi_ i_ = — ^ 2 if m < n
(ax'n\ J) *+«:«-»|
or = Pj it m > n
_ a+n : x+n
ara~| xni\ axn\ J) x+n :m-n\
ax
25. The following are problems connected with national
insurance. Given a stationary population where the numbers
living at each age correspond with the figures in some known
table of mortality, find
(a) The amount that will require to be subscribed to provide
1 at the death of each member of the community.
The deaths in the first year are (d +d +d +. ■ . +d u _ )=--l
„ second „ (d 1 +d + d^ + . . ■ +d u _ 1 )^ ^
» third » (d 2 + d s + d 4 + .. . + d u _ , ) = / 2
and so on.
chap, vii.] TEXT BOOK— PART II. 137
The present value of the benefit is therefore
C 1 Z A = vl + W 1 + v%+- ■ ■
= v k( l + a o)
= ; oOo + A o)-
If each member of the community is to contribute the same
single jjremium irrespective of age, we must divide the value of
the benefit by the population. The total population is
/„ + *, + /,+ • ■ • = / (l+e ).
Therefore the single premium that each must contribute is
W 1 + c o) 1 + e o
(6) The fund that will require to be subscribed to pay an
annuity-due of 1 per annum to every member of the community.
K' ll x(. 1+ ( = l o( l + a o) + ho- + «i) + hv + a d + ■ • •
- I 1 ~ A ° 4- I 1 ~ A l + / l ~ A 2 + . . .
= \{Qo+k + h+- ■ ■)-K' 1 iA x }
= ^ (l+e )-^o(l+«o)}
(l+z)/ (l+c )-/ (l+fl )
i
(c) The fund that will require to be subscribed to pay an
annuity-due of 1 per annum to every member of the community ;
no payments to be made at and after age y.
The expression for this may be stated thus : —
y,"- 1 Id+a)-^' 1 l(l+a)
that is, we deduct from the fund required in respect of the whole
community that portion of it which is not required in respect of
those who are at present aged y and upwards, and from the result
138 ACTUARIAL THEORY [chap. vii.
we deduct the value of the deferred annuities payable after
attainment of age y to those who at present are of younger age.
The expression may be reduced as follows : —
_ (i + o { v 1 + e o) - ^ + d m+2'-
■ d,
, d
y v.
[a:]+>l-l> x+n' x+n+1''
For the annuity commutation columns we have
\x] [xY [x]+l [x]+l' x+n x+n'
T W= D M + D M+ l + D M + * + '
' [x]+n - 1 + x+n + i+n+1 +
Also the assurance commutation columns are
(x] [xf M+l M+l x+n x+n'
M [x] ~ C M + C M+1 + C M+2 + '
+ [x]+n-l """ x+n + x+n+1 +
28. It will be useful to discuss here the formation of select
mortality tables. If we refer to the O tables we find a
mortality table in the following form : —
Age
at
Entry
[*]
Years elapsed since Date of Assurance.
Age
attained
x + n
i
2
etc.
n or more
hx]
hx]+i
fa+2
etc.
''x+n
142 ACTUARIAL THEORY [chap. vii.
The most obvious way to form such a mortality table is to
assume the same radix at each age at entry. Successive multipli-
cation by the probabilities of living will then give us a complete
table of mortality for each age at entry, since
{x] x P[x\ = Wi' Wi x p m+i = Wa'
and generally where k > n
l m X P W X P M+ i X ■ • ' X P M+ -i X /».+. X P*+n+l X ■ ■ ■ X P.+fc-l
But by this method we should have an independent mortality
table for each age at entry, and the extent of the monetary
tables following thereon would be prohibitive. A better plan is,
after forming the table for the first age at entry as above
by working along the first line and down the column I to the
limit of life, to form the remaining tables for the succeeding ages
at entry by working backwards along their respective lines.
Thus since
P
I
X+71 + 1
[*+l] + 9l-l 1
I
Therefore I „ , = — x+n+l
■r[a;+l]+«-l
Also I ,„, „ = ta;+1]+ ' t - 1 and so on.
[l+l]+» -2
[ai+l]+n-2
But I , is already calculated in the ultimate column,
x+n+l J
therefore ^ +I]+B _i may be found by dividing by p [x+1]+ „_ ll and
successively l, x+1]+n _ 2 , ■ ■ ■ \ x+1 y ^ n tne same way, commencing
with I „, we may work back to I, , „,, and so on for the
x+n+2' J [e+2/
other ages at entry.
The advantage of this method of formation is that only one set
of " ultimate " monetary values is necessary.
29. Now out of I select lives at age at entry x, I will be
alive at the end of n years. But according to the method by which
we have constructed our table I , is also the number alive at
x+n
the end of n years out of / mixed lives who were alive at age
attained x. It follows therefore that the difference between I, ,
■]
TEXT BOOK— PART II.
143
and / is the number of damaged lives included in the / . Again,
if from the deaths in each year of age amongst the / mixed lives
we deduct the deaths in the corresponding years amongst
the I , select lives we have the deaths in each year of age
[x] J b
amongst the (/ - / ) damaged lives. The mortality experience
is as follows : —
Age.
Survivors of
Number of
Damaged Lives
Dying in a Year.
Mixed Lives.
Select Lives.
Damaged Lives.
,r + l
x + 2
x + n
h+l
',+ •2
'x+n
kx]
l m+i
kx]+2
''x+n
l x ! [x]
h+l ~ kx]+l
lx+2 ~ kx)+2
lx+n~h+n = ®
d x - <\x]
dx+1 - d [x]+l
d x+2 d \x]+2
6
It will be noticed that the effect of assuming that selection
becomes unimportant after n years is that all the damaged lives
die before the end of the n years.
30. To find the single premium required to permit of (x)
effecting, n years hence without fresh medical examination, a
whole-life assurance by annual premiums.
The annual premium to be paid by (,r) is fixed at P , but
considering that his medical examination takes place at the
present time, the premium he should pay is ~P [x]+n - The single
premium to be paid now is therefore the present value of
an annuity-due, of the difference between these two premiums,
deferred n years, or
i| a [XP- [x]+n
D.
' [x+n]
)
[x]+n
D
M
( P ,
[x]+n
( [x]+n
P
P )a
[x+nV I
[x+ny [x]+n
nU
[x+n]-' [x]+n
D
[x]
(■ [x]+n [x+ny V a [s] \x]n
which is in the form most suitable for calculation.
)
144
ACTUARIAL THEORY
chap. vir.
For the annual premium for
should have
years for such a benefit we
rp - p in
^ [2I+™ [x+n']-' [x
)+n
M
(P - P
[x)+n
M
[1] n I
ml/
It is to be noted that this annual premium can only be accepted
along with the annual premium for an ordinary policy running
during the n years ; otherwise by withdrawing at any time the
assured would exercise against the office issuing the policy an
option for which allowance has not been made in the calculation.
31. To find the annual premium for a short-term insurance
upon the assumption that all the healthy lives withdraw at the
end of the first year.
The necessity for taking such an option into consideration
arises from the fact that the annual premium for a short-term
insurance frequently diminishes with an increase in the age, the
original date of termination of the contract remaining unchanged.
Thus, in symbols, it may happen that
p!_ •> P J_ > P J_ > etc > P 1 —
xn\ ^ x+1 : Ti-1 1 x+2:n-2\ ^ c ^ j+«-1 :1|
The reason for this is that the decrease in the term of the
insurance has a greater effect in reducing the premium than the
increase in the age has in raising it. The following table (based
on the O tNM] Table at 3^ per cent.) illustrates the point.
Short-Term Insurance Premiums per unit assured.
Age.
Term.
Age.
1.
2.
3.
4.
5.
6.
Y.
30
31
32
33
34
35
36
37
38
39
40
■00470
•00479
•00491
•00504
•00517
•00533
■00548
•00565
•00587
•00605
•00630
■00552
•00562
•00575
•00588
•00602
•00619
•00636
•00657
•00678
•00699
•00726
•00615
•00627
•00640
•00655
•00671
•00690
■00709
•00731
■00754
•00779
■00808
•00663
•00677
•00692
■00708
■00726
■00747
•00768
■00792
•00818
•00845
•00878
•00700
•00715
•00732
•00750
•00770
■00792
•00815
•00842
■00870
■00901
•00936
■00729
•00745
•00763
•00782
•00804
•00828
•00853
•00882
•00913
•00946
•00983
•00752
•00769
■00788
■00809
•00832
■00858
•00886
■00916
■00949
•00984
•01025
30
31
32
33
34
35
36
37
38
39
40
chap, vii.] TEXT BOOK— PART II. 145
Here it will be seen that P^ > P^- > etc. > P^-, and
so on.
It follows, therefore, that an office issuing policies, say for
seven years, runs an appreciable risk, in that all the lives which
are still select at the end of the first year may drop their policies
and effect new ones for the remaining six years at a lower rate.
The value of this risk is ascertained in the problem before us.
If I persons enter at age x, d die within the first year,
and i I+11 withdraw at the end of the year. Therefore the number
to enter on the second year is I, , , , - 1, , ,,. Out of these,
J [X] + 1 [3+1] '
d w+i- d ix+n die within the second y ear » leavm S W«~k+u+i
alive ; d 2 - d die within the third year, leaving
Wi"Wn+J alive > and so on -
Therefore the benefit side
^M + ^M+l -^+l]) + ^M+2- rf [x+l] +1 ) + - • • + p "( d M+ .-i- rf [ . + i ]+ .- 8 )
i]
vd r , + v 2 d r , , , + v s d r , „ + •■•+«"£,,
[x] [»]+! [g]+2 [x]+n-l
[x]
_ ^ + i] + ^+ 1] + 1 +- • • +*'«Wi ]+ ._,
hx]
M M- M M + » ^,+11-^+1]+.-!
And the payment side
P'l _
[*]»l
>, f / w +
v x (vd + v-d , n + v 3 d , a + •
^ x x + 1 sr + 2
since v < 1
>
c ,+ c , +1 + c , +2 +- • ■
>
M
X
154 ACTUARIAL THEORY [chap. vii.
Or again, since the present value of 1 payable at the end
of the year of death of (x) is clearly less than 1, we have
M
A =
Hence
A * - : d* < x
D > M
X X
2. Express in terms of the D and N columns and the rate of
discount, the annual premiums for
(1) An endowment assurance to mature in n years.
(2) A whole-life assurance, premiums limited to n payments.
Subtract the second from the first and give a verbal inter-
pretation of the result,
D_
The premium for (1) is
^-V.-i
d - ( m ,
and for (2)
N ,-N ^ ,
x~l x+n-1
X
The difference is ^ x+ ™ * — , which is the annual premium
x+n-l
required to provide an annuity-due during the lifetime of (x) after
n years, consisting of d, the interest in advance on 1, payment of
which in the case of (2) is deferred from the end of the rath year
■ — as would happen under (1) — to the end of the year of death of
(x). During the ra years the benefits are identical.
3. Find the rate of interest, given
(a) a =13-257 and A =-19304
(b) a =13-164 and P =-04147
^ ^ X X
(c) A =-19414 and P =-00927
v ' X X
Approximately (a) 6 per cent. ; (b) 3 per cent. ; (c) 4 per cent.
4. Find the value of A , having given P = -01662, a = 17-155,
and;; =-99229.
chap, vii.] TEXT BOOK— PART II. 155
We have A ,, = l-d(l+a ,)
a
= l-d~Z
V P*
ia
P x
The only unknown quantity in this expression is i, which from
the given values of P and a we ascertain to be practically equal
to '04. Substituting: this value for i we get A , , =-30847,
5. Required the cost of a deferred annuity, of which the first
payment is to be made at the end of four years, and which is
then to continue for twenty years certain and thereafter for so
long as a life presently aged x may live.
The first part of the annuity is an annuity-due for twenty
years certain deferred four years, and the second is an annuity-
due on (x) deferred twenty-four years. The cost is therefore
Al+asp + wl*.
N
, \ , z+23
= («S§1 ~ a w) + -Q-
6. Give an algebraical proof that
1-A
" - 1
a
d
1 + » ^ « 1 + * i
156 ACTUARIAL THEORY [chap. vii.
/. Give the formula for a whole-of-life assurance on (x) by
three payments, the first to be made immediately, the second to
be half the amount of the first and to be made at the end of three
years, and the third to be half the amount of the second and to
be made at the end of seven years.
M
The benefit side = ^=-^
The payment side = P
whence P =
D
X
M
D , + *D„ + , + *D, + 7
8. Investigate a formula for the annual premium payable during
life for an assurance on the life of (.r), the sum assured not to be
paid in any event for twenty years from the date of the policy.
The benefit divides itself into two parts. If (x) should die
within twenty years the sum assured is payable at the end of that
period and its value is i> 20 (l — p ). The other part is an assurance
on (a:) deferred twenty years, i A . Therefore the benefit side is
equal to
The payment side = P(l + a ).
■tfl°(l - p)+ I A
Hence
1 +a
9. X has an income of J per annum ; he can insure his life
at P per unit ; and investments will yield i per unit after his
decease. How much of his income must he spend in premiums,
in order that his representatives after his death may enjoy a
perpetual income derived from the policy, exactly equal to the
balance ? Assume that the income is payable at the beginning of
each year.
Let S be the sum for which X must insure his life. Then
SP is the amount of his income which he spends in premiums,
the balance of his income being (J - SP ). The income (payable
at the beginning of the year) which will be derived from the
chap, vii.] TEXT BOOK— PART II. 157
proceeds of the policy will be Sd, d being the interest in advance
corresponding to i. We now have the equation
J-SP = Sd
X
J
whence S =
and S P =
J p *
,,20
10. In consideration of a yearly premium of , an assurance
a z:20|
company offers a life aged x a policy securing a sum of 1 payable
at the expiration of twenty years, if (x) be then alive, and a sum
of S payable fifteen years after the end of the year of death of (.r) if
this event take place during the twenty years. Find the value of S.
The value of the premiums to be received is
n
20
x a
1)20
\ : 20|
And the value of the benefit granted is
D * + 20+ S » 1!i (M- M , + 20)
D
X
Hence Si' 15 (M - M L0 .) = « 20 D -D _„
and S =
x ^ ' a;+20
M - M ^
X x+20
11. Investigate the change in the value of q produced by
assuming an increase in the rate of interest to represent an
increase in the rate of mortality. Illustrate from the case where a x
is extracted from the 4 per cent, table and assumed to represent
the 3 per cent, value of a table showing higher rates of mortality.
We must first obtain q in terms of values of a^ and accordingly
we have
Q =1-0 =1 =
158
ACTUARIAL THEORY
[chap. VII.
Here, then, we get increased rates of mortality by taking a and
a at a higher rate of interest while i remains the lower rate,
and we may write q' x
rate j (> i).
To obtain the old rate of mortality from a similar formula we
have
(i+iK
(l+i)a'
1 - -t; ; — - where a' and a , , are at
1 + a ' , * x + 1
x+l
1
1 +d
x+1
and q > x - % = fl_^*\ A_^A
1 +o'
3+1
Taking the example of 3 per cent, and 4 per cent, we have
■ow r
the increase in q , i.e.. q - q = -= ^— where a' and a , , are
1 +«
a+1
at 4 per cent.
12. Calculate from the values given below the net annual
premiums at age thirty for the following policies : —
(a) Whole-life assurance, premiums payable throughout life.
(6) Whole-life assurance, premiums limited to ten payments,
(c) Ten years' temporary assurance.
(d^) Ten years' pure endowment,
(e) Ten years' endowment assurance.
(/*) Ten years' double-endowment assurance.
X
D,
N.
M,
30
37879
805450
14419
31
36557
767571
14201
32
35272
731014
13980
33
34023
695742
13758
34
32808
661719
13535
35
31627
628911
13310
36
30480
597284
13083
37
29364
566804
12855
38
28279
537440
12626
39
27225
509161
12395
40
26201
481936
12164
chap, vii.] TEXT BOOK— PART II. 159
a) ^o = 1^1 = -017902
' N 30 805450
b) M 3Q 14419 14419 _
N 30- N 40 "" 805450-481936 ~ 323514 ~ u * 40 ' u
c) M 80 -M 4 J , 14419-12164 _ ^255_ _
N so-N 40 805450-481936 "~ 323514 ~ UUDa ' u
D 40 26201
ho ~ N 40 323514
= -080989
■) M 3° M4 ° + D40 = -006970 + -080989 = -087959
"40
CO ^sji-MiO+l^io = -087959 + -080989 - -168948
13. What is the annual premium at 3 per cent, for a temporary
insurance for three years on a life aged thirty ? Given Z = 92529,
/„.= 92079, l Zi == 91472, Z M = 90763.
pi „ _ ^X+^+^BB
30:31 " ' 'ao + Si + ^w
= -97087 --96469
= -00618
14. Given P and A , show how to find at rate of interest i the
x aar
annual premiums for
(a) Joint-life Assurance on two lives aged x.
(6) Last-survivor Assurance on the same.
O) p ■=
>■ ' XX
(b) P- =
V / XX
1+rt
XX
(LA
XX
X
I- A
X
d
A-
1+fl-
2A -A
X X
1 + "2a -a
160 ACTUARIAL THEORY [chap. vii.
and
ACTUARIAL '
THEORY
A
P
X
X
P +d
X
a
X
1
p +d
X
- 1
a
1-A
X3
; -1
as before.
15. Find without using commutation columns an expression for
the annual premium for an assurance payable only in the event
of (,r) and (y) both dying within n years.
A-i-
pj. xyn)
•wl " \+a „
xy : n-l\
a ;
_ „ _ xyn\
xy:n-l\
where the values of a r and a —, are found from the formula
xyn\ xy:n-l\
%T\ = a xt\ + Vl ~%T\
vl , , + vH , „ + • • • +v t l ,,
__ x+l x+2 x+t
I
x
vl , , + vH , „ + • • • + v* I , ,
■ y+l y+2 y+t
I
V
p *, + l* y+ i+« ! W y+8 + - • • +Vtl x +t l y+t
I I
x y
16. Deduce the single and annual premiums for an assurance
for ten years, payable as to one half at the first death and as to
the other half at the second death of three lives (x), (v), and (s).
Is there any practical objection to making the quotation, and if
so, how would you propose to meet it ?
The single premium is
K|io A ^ s +|io A ^)
whei ' e a 4 :ij = %n + «„n + Vi - -Vh
chap, vii.] TEXT BOOK— PART II. 161
In deducing the annual premium the above is the benefit side.
Payment side = P(l +„_?-)
if it is desired to make the premium level throughout the whole
status. A certain risk, however, attaches to the issue of the policy
on such a footing. If one of the lives, say (x), die early, the
remainder of the benefit could be obtained by the survivors, (y)
and (s), provided they are in good health, at a smaller premium
than P as found from the above. In such a case the office would
not receive the stipulated premium throughout the whole of the
status assumed in the calculation. To get over the difficulty we
may
(1) Make the premium payable only during the joint existence
of all three lives, whence payment side = P(l+a _^-\
(2) A premium may be accepted which is to be reduced by
half on the first death.
Payment side = P{l+-? r (a Ki+a— ^"H.
■> t - v xyz : 9 1 xyz : 9 y >
(3) Probably the best way is to issue two policies, each for \,
one of which will be payable on the first death, the premium
being ^ Pf^T • ioi ' an( * *^ e ot ^ er payable at the second death, with
premium -J Pi,. A— .
1 | 10 xyz
17. Find the annual premium for an assurance of £100 payable
as follows :—
(a) £50 at the death of the first of two lives and £50 at the
death of the second, the premium to be reduced by one half from
the date of first renewal after the death of the first life.
(b) £33, 6s. 8d. at the death of the first of three lives, the
premium then to be reduced by one-third ; £33, 6s. 8d. at the
second death, with a similar reduction in the premium ; and the
remaining £33, 6s. 8d. on the death of the last survivor.
(a) The benefit is obviously 50(A + A ).
The premium to be paid depends as to one -half on the
life (x) irrespective of the life (y) and as to the other half on the
life (^) irrespective of the life (x). The value of the payment
side is therefore
P{Ki+}-
162 ACTUARIAL THEORY [chap. vii.
Or again, the payment side depends as to one-half on the joint
lives of (,r) and (if) and as to one-half on the life of the last
survivor. That is,
Payment side = PQa^ + fa-) = P{1 + l( % + a y )}.
A +A
Hence P = 50, ^ v —,
(b) The premium required for this benefit, found in a manner
similar to the above, is
A +A +A
33-3 : *- — y - ?—
1 + l(a +a +a)
The payment side may be expressed in either of the forms
18. Find a formula for the annual premium, payable till the
benefit is entered upon, for a deferred annuity of 1 to begin to
run on either of two lives, presently aged twenty-five and thirty
respectively, attaining age sixty. If (30) attain age sixty and
(25) be also then alive the annuity will be payable thereafter
during the joint lives and the life of the survivor.
Benefit side = « 30 30 P25 : 30O55 + a 60 ~ « 65:60 )
+ " 3 VsoC 1 - so^Ko + ^as^st 1 - 30P30K0
Payment side = P { 1 + 1 ffl a^-^ + v*\ p 9 Jl - 30 pJ 1 5 a 65 }
And p _ ^ 3oP25:3o( g 55 + a 60- a 55:60) + l ' S0 30P30( ] -SO^^O + ^SS^C 1 gaj^Kj
1 + |29 a 26 + U a 80-|s9 fl S6:8O + B ' ,O 80P26( 1 -«lP«>)( 1+ |4 a iiB)
19. Find the annual premium for an assurance payable at the
second death of the four lives (w), (x), (3/), and (s).
Benefit side = A — ?
wxyz
Payment side = P — 8 (l+o — ?)
chap. vu.J TEXT BOOK— PART II. 163
Therefore equating
A —
p S _ wxyz
wxyz
wxyz
1 -d
1 +a — -
wxyz
where a — = = a +a +a +a -3a
wxyz wxy ivxz wyz xyz wxyz
20. Find the single premium to assure a perpetuity of £100 in
the event of A, who is aged thirty, dying within ten years, the
first payment of the perpetuity to be due at the end of the year
in which A dies. Interest is to be taken at 4 per cent, and the
mortality is to be assumed to follow De Moivre's hypothesis.
Given v w at 4 per cent. = -675564.
By De Moivre's hypothesis the number living at age thirty is
86 — 30 = 56, and one dies every year. Therefore
V + D 2 + • • • + V 10
so:io] = gg
l-i>io
56;
•14484
The single premium for the perpetuity is
Ai - x 100 x ii? = -14484 x 100 x 26
30 : lOf i
= 376-584
= £376, lis. 8d. nearly.
21. On De Moivre's hypothesis as to the law of mortality, find
the annual premium at age x to provide an endowment assurance
payable at age x + 1 or previous death.
p _ _ s'l
*" "" 1+a *:^|
But on De Moivre's hypothesis where n represents the comple-
ment of life at age x, we have —
164 ACTUARIAL THEORY [ CHAP - vn -
. _ v + d 2 + • • • + v l + (re - f)vt
■« = n '
- _Ji
«71 + («-<>'
Also a — = "(»-l) + A"-2)+- • ■ +f i - 1 (^-< + l)
x:t-l\ n
(*
-iH-rn-'
l t^\\2\
n
B f
^-(t-iy
-1
i
a r\ + ( n -
■(>'
where a— ^
Hence P^ =
"' ra+ ("- 1 ) ffl ?3T|- a FrT|2T
22. Find the value of an annuity to be payable until the
survivor of three children, aged five, eleven, and thirteen respec-
tively, attains majority.
°(5 :iij) (11 :10|) (18:81) == °5 : fef + "ll : ioj + a is :W]
~ a 5 : 11 : ioj _ a 5 : 13 : 5] _ °ll : 13 : t\
+ a —
^ 5 : 11 : 13 : 8 1
23. Find the value of an assurance payable should three
children, aged ten, twelve, and sixteen respectively, all die before
attaining majority.
A ~T~I i _ J _ = A 1 - + w s d - v ) v A * -
(10 : 11|) (12 : 9 1) (16 : 5 1) I0:12:16:5j ^ SOB'S' 10 : 12 lb : 17 : 4|
+ A1 "jPl^C 1 -6^^10^6:41 + A 1 -jPldX 1 -SO0)sO2 A 17:4|
+ A 1 - s P lfl )( 1 - fl Pu),Pio A l , .:jI
24. How many damaged lives are there among the under-
mentioned 740,925 lives aged forty-five who have been insured for
three years t and how many of them will die in each of the
following five years ?
CHAP. VII.]
TEXT BOOK— PART II.
165
Give a short sketch of the reasoning which leads to your
figures.
Age
X
Years elapsed since
Date of Insurance.
Age
X
1
2
3
4
5 or more.
h*)
{x-n+i
'[a:-2]+2
V-3]+S
\x-i\+i
'.
45
46
47
48
49
50
730692
720100
709190
698088
686620
674923
736363
726015
715320
704292
693058
681445
739403
729402
718906
708042
696823
685377
740925
731220
721033
710325
699220
687728
741538
731895
721908
711423
700413
688992
741700
732100
722100
711700
700800
689400
45
46
47
48
49
50
The number alive at age forty-five of those who have been
insured for three years is 740,925, but the number who are select
at age forty-five is 730,692. Now both these numbers are reduced
by mortality to the same figure at age fifty, namely 689,400. The
surplus of 740,925 over 730,692 must therefore represent damaged
lives who all die off in five years. The number of damaged lives
among the 740,925 is thus 10,233 and the deaths amongst these in
each year are shown as follows : —
Age.
(i)
Number of
Mixed Lives
Surviving.
(2)
Number of Lives
Surviving out
of those select
at Age 45.
(3)
Number of
Damaged Lives
Surviving.
(2) -(3)
00
Number of
Damaged
Lives Dying.
(5)
45
46
47
48
49
50
740925
731895
722100
711700
700800
689400
730692
726015
718906
710325
700413
689400
10233
5880
3194
1375
387
4353
2686
1819
988
387
The difference at each age between the number surviving of
those select at forty-five and the number surviving of those who
at age forty-five had been insured three years shows the number
of damaged lives surviving at each age, and the first differences
of this column show the number dying in each of the five years.
25. Given a select mortality table showing /
[xf
[*-l]+l> [s-2]+2>
166 ACTUARIAL THEORY [chap. vii.
\x-s]+z' ^[1-41+4' an( ^ ^> express the probability of a select life
aged x at entry being at the end of five years,
(a) In existence, irrespective of the state of his health then,
(6) In existence, and still a select life,
(c) In existence, and an unhealthy life.
In the form of tables described, selection is assumed to wear
off in five years, therefore at the end of that time the number
of / persons select at age x who are still alive is merged in the
ultimate table and is expressed by I , their health not being in
consideration. But the number of select persons of age a: + 5 is
by notation I. +s p therefore the number of unhealthy is the
remainder of the total / . Accordingly the probabilities
required are
(a)
(6)
(<0
x+S
l lx]
'[x+S ]
M
X + 5 ~ [»+»]
26. A life office secures every year K new assurers all aged
x at entry. At the end of a quinquennium how many of the
entrants during that time may be expected to be unhealthy.
Of I, , persons who enter at age x there are alive at the end of
five years I , of whom some are select and the others unhealthy.
But the number of select lives of age x + 5 is L . „. Therefore the
number of unhealthy lives of that age is I — I . Similar
expressions give the number of unhealthy lives at the end of four,
three, two, and one years. Thus, if the number of entrants each
year is K, and if we assume them all to enter at the beginning of
the year, we get the number of unhealthy lives at the end of five
years as
J~ \ v-Vj+6 ~ l*+W + ( [*]+4 ~~ [x+t) + Cv]+s ~ {x+sy
+ (Wa " h*+i? + (Wi ~ kx+i])}
chap, vii.] TEXT BOOK— PART II. 167
27. A person aged x wishes to be allowed to effect ten policies
each for £1000 as follows :—
(1) At age x at the normal annual premium for that age.
( 2 ) » (*+l)
( 3 ) » (* + 2)
etc. etc. etc.
(10) „ (* + 9)
It is required to find the single premium payable to provide
for this option.
For the second of these policies the premium to be paid in
absence of any arrangement would be P , whereas the premium
arranged for is P [a . +ir The value of the option on this one policy
N
is therefore (P. „, - P r _) -g±l.
\ M+i [x+iy n
M
Similarly for the third policy the difference in premium
to be allowed for is (P. , , - P r , „,) and the value of this is
N
( P M+2 — P i>+2i) n" 1 " 3 ' ^ n( ^ so on ^ 01 t ^ le otner seven policies.
M
The single premium to be paid for the option is therefore
100 °{( p ,m-V 11 )% 1±1 + ( p M+2 - p , +2] )% 1±2+ - • •
N i
+ fp _p ) M+9 L
w
28. Calculate the following option premiums : —
(a) The single premium per cent, required to permit of (30)
effecting at the end of five years a whole-life policy at the normal
annual premium for his then age, without fresh medical examina-
tion. Use the O tM1 Table at 3i per cent, interest.
(6) The yearly addition per cent, to the short-term insurance
premium for seven years required to permit of (40) effecting a
whole-life policy at the end of that period at the normal annual
premium for his then age, without fresh medical examination.
Use the [NM] Table at 3 per cent, interest.
168
ACTUARIAL THEORY
[chap. VII.
(a) Using the formula given on page 143, we have
100(P
[30]+5 [ssy (- a rf
l)
[80] [30] : 5 |-
= 100(-02024--01959)(19-793- 4-633)
= -985, say 19s. 8d.
(6) Similarly for this option premium we have
100 ( P 4r- p M71 )(a,
lay v [40] "[40] : 7
:T)
[40] : 7 |
(3-445 - 3-377) (18-102 - 6-250)
6-250
= -129, say 2s. 7d.
29. Find the annual office premium for a whole-life assurance
to (x), the expenses being 8 per cent, of the first and subsequent
gross premiums with further initial expenses of 2 per cent, on the
sum assured and 5 per cent, on the first gross premium.
To get the value to the office of the payment side we must
deduct from the value of the gross premium the value of all
expenses. Thus, if P be the gross premium, the value of the
payment side is
P(l + a.) - -08P(1 + a.) - -02 - -05P
and this is equal to the value of the benefit, A .
Hence P(l + aj - -08P(1 + a J - -02 - -05P = A^
P{-92(l + o c )--05}=A :c + -02
A +-02
•92(l+a.) --05
30. Given the following office rates for immediate annuities on
female single lives, aged fifty and sixty respectively, find at these
ages the annuity which £500 will purchase, it being a condition
that the annuity is to be payable during the joint lives and the
lifetime of the survivor, but is to be reduced by one-half after
the first death.
Age
last Birthday.
Annuity which
£100 will purchase.
Price of
Annuity of £10.
50
60
£5 13 8
7 5 10
£175 19 1
137 2 10
chap. vii.J TEXT BOOK— PART II. 169
The annuity required is the sum of two annuities of equal
amount on the lives, and if 2P be the amount to be received
during the joint lives we have
500 = P(a' 60 + a ' 60 )
= P(17-5954 + 13-7142)
Hence P =
17-5954 + 13-7142
= 15-970, or say £15, 19s. 5d.
£500 will therefore purchase an annuity of £31, 18s. lOd. during
the joint lives, to be reduced to £15, 19s. 5d. on the first death.
31. Given tables of office premiums for endowment assurances
and double-endowment assurances, show how to employ them to
obtain the office premiums for the following benefits : —
(a) £100 payable on attaining a given age or at previous
death, together with a guaranteed bonus of £33, 6s. 8d. payable
only if the given age is attained.
(6) A similar benefit, but with a guaranteed bonus of £50.
(a) This is equivalent to a term assurance of £100 coupled
with a pure endowment of £133, 6s. 8d., which may be split into
an endowment assurance of £66, 13s. 4d. payable on attaining the
given age or at previous death and a double-endowment assurance
of £33, 6s. 8d. payable on death within the term and £66, 13s. 4d.
on attaining the given age. Therefore if P'^— , and (DP)'^ be the
office premiums per £100 assured for endowment assurances and
double-endowment assurances respectively, we obtain the required
premium from the formula
l p '^+KDP) Mn .
(b) By a similar process of analysis we find the premium for
the second benefit to be
32. From tables of office "Whole of Life" and "Limited
Payment" premiums for each age at entry, show how to find the
sum assured that could be given at age x for a single payment
of S and a future annual payment of P.
170 ACTUARIAL THEORY [chap. vh.
x an( ^ ^'x ' 3e ^e omce single and annual premiums at
age x per unit assured.
We must split the single payment to be made now into P and
(S - P) in order to put the payments of P on the basis of an
annual premium.
Then since a single premium of A' insures 1, a single premium
of (S - P) insures —rr .
X
Also since an annual premium of P' insures 1, an annual
P
premium of P insures pr.
X
Therefore the whole amount insured by a single payment of
S - P P
S and a future annual payment of P is . , + -rp- _
33. A man aged forty next birthday desires to effect a policy
payable at death for £5000. He proposes to make a first payment
of £1000. Find the future premium to be charged annually, given
tables as in the preceding question.
Let P be the future premium. Then by our formula above
Hence
A' 40
P' "
r 40
P(±.
^40
"ATo) =
= 5000-
1000
A< 40
and
P =
5000-
1
1000
A '40
1
pTd"
A' 4 o
34. Express in commutation form the annual premium for an
endowment assurance to (x) payable in n years or at previous
death, the premium for the first five years being only one-half
of the premium for the remainder of the term. Find therefrom
the premium for the first 5 years, and for the remaining 25 years,
for a 30 years' endowment assurance on a life of 30. Interest 4
per cent. Given N 29 = 502353-5 ; N s0 = 474646-5 ; N 34 = 376007-4 ;
N 59 = 58526-9; and N 60 = 52931-0.
chap, vii.] TEXT BOOK— PART II. 171
M - M , + D
Here the Benefit side = — ^ ^ *±»
X
/N - N N - N \
And the Payment side = Pf^^i r -±^ + _5±i *J^z2)
M - M L + D
Whence P = XT * XT x+n * +n
N , + N , -2N , ,
x-1 x+i x+n-1
Using now the figures given we have
p __ M30 - M 60 + D 60
N 29 + N 34 -2N 69
But M 30 = t-X 29 -N 30 = 50 ^f 53 ' 5 ~ 474646-5 = 8385-7
58526-9
M fl0 = «N 69 -N 60 = ^Jf -52931-0 = 3344-9
and D 60 = N 69 - X 60 = 58526-9-52931-0-5595-9
Therefore P 8385-7-3344-9 + 5595-9
502353-5 + 376007-4-117053-8
= -013972.
The premium for the first five years is accordingly -013972 and
thereafter -027944.
35. Find by the [NM] Table, with interest at 3| per cent,
throughout, the annual premium per cent, required at age thirty
to provide a debenture policy under which 5 per cent, interest
payable half-yearly is to be provided on the sum assured for
15 years from the end of the year of death, at the end of
which period the sum assured is to be payable.
As explained on page 148, the benefit at the end of the year of
death is 1 plus an annuity-certain, for the period stipulated, of the
excess of the guaranteed rate of interest over the rate assumed by
the office. In this case the premium will therefore be
lnnn L -05- -035 )
100P [M] { 1+ 2— fl so, (1 U)}
= 1-788(1 + -0075x23-18585)
= 2-099, say £2, 2s.
172 ACTUARIAL THEORY [chap. vii.
36. Find by the [NM] Table, with interest at 3 per cent,
throughout, the annual premium required at age thirty-five to
provide an annuity-certain of £100 payable half-yearly for 20
years, the first payment to be made at the end of the year of
death.
Modifying the formula given on page 147 to suit the case of a
half-yearly annuity we have for this premium
P [35] x 50xa. WmX)
= -02212x50x30-36458
= 33-583, say £33, lis. 8d,
37. Find by Mr Lidstone's formula the annual premiums per
cent, required for the following joint-life endowment assurances.
(a) Lives (30) and (35) for a term of 20 years on the basis
of the [M] Table with interest at 3i per cent.
(6) Lives (20) and (40) for a term of 30 years on the same
basis with 3 per cent, interest.
(a) Following Mr Lidstone's formula we have
(. i [30] : 20] + [35] : 20] ~~ 20| /
= 3-841 + 3-923-3-416
= 4-348, say £4, 7s.
W 1O0 (W30i +I V3O|- P 30-,)
= 2-460 + 2-947-2-041
= 3-366, say £3, 7s. 4d.
38. Write down formulas, with and without commutation
symbols, for the annual premium for a joint-life term policy.
Calculate with the help of tables of logarithms the annual
premium for a three-year term policy on the joint lives of A and B,
each aged thirty-six next birthday, at 3 per cent, interest, having
given
log l x = 5-9097, log l s7 = 5-9076, log l 3S = 5-9042, and log l m = 5-9002.
How would you approximate to such a premium in practice ?
The annual premium in commutation symbols is
M -M
xy x+n : y+n
N -N
x-\:y-\ x+n-l:y+n-\
chap, vii.] TEXT BOOK— PART II 173
from which we may pass to a formula without commutation symbols
as follows : —
M - M
_xy x+n : y+n
N -N
a;-l:j/-I x+n-1 : y+n-1
ON , - N ) - («N L , , , - N , ^ )
^ x - 1 : ?/ - 1 wys v x+n-1 : y+n-1 x+n ; y+n J
N * - N
x-l :y~~\ x+n-\ : y+n~\
N - N ^
gy x+n : y+n.
N , ,-N ^ , ^ ,
_ p/ «+l ',+1 + "Vs l y+2+ ■■■+ "" l x+n K+n
l Jy +vl X+ l l y + l + ■••+»"- 1 '.. H »-iVh>-1
Substituting the ages, etc., required for the second part of the
question we have the premium desired as follows : —
_ V hl hi + V ks ^38 + V ^39 ^39
^36 ^36 + V ''S7 '37 + V 's8 ^38
At 3 per cent. « = -97087, also log v = T-9872, log u 2 =T-9743,
and log v 3 = 1-9615.
In dealing with the logarithms of the numbers living the
characteristics may be ignored as they are the same in every case
and only determine the position of the decimal place in the corre-
sponding natural number.
Then log l S6 l 36 = 2 log Z 36 = 2 x -9097 = 1-8194 = log 65-978
log vl s1 l s7 = logv +21ogZ sr =T-9872 + l-8152 = l-8024 = log 63-445
log «% s / 38 = log« 2 + 21og/ 38 = r-9743 + 1-8084 = 1-7827 = log 60-632
log » 3 V 39 = log« 3 + 21ogZ 39 =T-9615 + 1-8004 = 1-7619 = log 57-796
Therefore
v _ "*87*37+ ,,i!< 38 / 8S + 1 ' 8/ 89*39
__ „ ? 63-445 + 60-632 + 57-796
_ 65-978 + 63-445 + 60-632
= -97087 - -95695
= -01392.
In practice, as explained on page 152, we would take it that
p 1 _ pi _+pi -
r |86:S61 : 3 1 " ' 36 : 3 | T 36 : 3 |
174 ACTUARIAL THEORY [chap. vii.
Using the figures given, we find P' —='00699, and hence the
approximate joint-life short-term premium is "01398, which com-
pares very favourably with the true value.
39. Find by the [NM] Tables, using 3i per cent, interest, the
annual premium for a joint-life short-term assurance for four years,
the lives being aged thirty and forty.
Using the same formula as before
P 1 _ = „ _ [30]+l [40]+l + ' ' ' +V WhT[40]+4
I30:40l:4| " 7 7 i. . . , ,,3/ 7
[30] [40) T T " [30]+3 [40]+3
= -96618- -95086
= -01532.
The practical formula is in this case
pi -+pi _
130:401 : 4| 30 :4| r 40 :4|
= -00663 + -00878
= -01541.
40. Given >][44] = 13-791, %M = 13-463, ^^ 13-006,
aIld "[44][44] =12 ' 391, find the ValUe ° f VH44] C 1 ) ^ fimte diffel ' _
ences, (2) by central differences, stopping at second differences.
(1) For a finite-difference formula we have
T
1 d
d d) - d U)
*<» - * W
etc.
1 + 1-01 rf w
i + i-oi a o>
1 d
1 + 1-02 d W
etc.
1 + 1-02 d U)
etc.
where column (4) is constant.
3. The following is an alternative method of forming the
conversion table from single to annual premiums.
dA
Therefore -=-
A
A
dA
T&-0
We must therefore make up a preliminary table of the
reciprocals of the single - premium values, less unity. Then
putting the value of — on the fixed plate of the arithmometer and
d
multiplying by each of the values in this preliminary table, we
obtain a series of which we must again take the reciprocals to get
the required values of P. This method does not fulfil the condi-
tions of a continuous method. But good checks may easily be
applied either by working backwards from the values of P to the
values of A and comparing with the original values of A, or
otherwise.
CHAP. VIII.]
TEXT BOOK— PART II.
181
4. Conversion tables are of great value in the working out of
premiums, whether single or annual, where it is frequently easier
to obtain the annuity value than the premium. For example,
If we enter
the Conversion
Table with
We obtain in the
Single-Premium
Conversion Table
Annual-Premium
Conversion Table
•*st:»-l|
xy
**»:»-! |
a w
i-d(i++ ,. 1 )
184
ACTUARIAL THEORY
[chap. viri.
Entering the annual-premium conversion table with the value
of this expression we obtain P^.
8. To form conversion tables for continuous functions we first
sketch out the following scheme for single premiums : —
Annuity-
Value.
(l)
Corresponding
Assurance Value.
(2)
A
of Co). (2).
(3)
1
1-01
1-02
etc.
1-5
1-5(1-01)
1-5(1-02)
etc.
-S(-01)
-5(-01)
-a(-oi)
etc.
Therefore starting with the initial value of A = 1 — 8, and by
the continuous addition of — 8(-01), we form the table at the rate
of interest involved in 8.
Again, for annual premiums we have
Annuity
Value.
Corresponding
Premium Value.
A
of Col. (2).
(i)
(2)
(3)
1
1-5
ife-
1-01
4-
1 1
1-02 1-01
1-02
J--5
1-02
1 1
1-03 1-02
etc.
etc.
etc.
We must first then form a preliminary table of the reciprocals
of the successive values of the annuity and find the differences of
these reciprocals. Thereafter the successive addition of the
differences to the initial value 1—8 gives the table. As with
ordinary tables, one series of differences is sufficient for the
formation of tables at all rates of interest. That is to say, the
constant addition of 8 ..-8 to the values at rate i will also yield
Vis the table at rate j.
CHAP. VIII.]
TEXT BOOK— PART II.
185
EXAMPLES
1. Verify by actual calculation the following values of A,
which correspond to values of a advancing from 10 to 11 by
differences of -1, at 3J per cent, interest; and insert correct values
in place of those which are incorrect.
a.
A.
10 -0
•60241
■1
•59905
•2
•59508
•3
•59157
•4
•58775
•5
•58434
•6
•58072
•7
•57741
•8
•57349
•9
■56938
11-0
■56627
It will be found that the 2nd, 3rd, 5th, 8th, and 10th values
are incorrect, the true values being -59880, -59518, -58795, -57711,
and -56988 respectively.
2. Check all the figures in the following table, using Rothery
and Ryan's Conversion Tables to obtain the single and annual
premiums.
Rate.
Basis.
a.
A.
P.
Whole - Life Assurance,
age forty .
Ot««21%
18-280
•52976
•02748
Endowment Assurance,
age thirty, payable at
sixty ....
OW3i%
16-163
•41961
•02445
Leasehold Assurance,
term 20 years .
3%
14-324
•55367
•03613
Joint -Life Assurance,
ages twenty-five and
thirty-five .
[M] 3%
16-725
•48373
•02729
Absolute Reversion, i.e..
the present value of
1 payable at the death
of (x), otherwise A x ,
age seventy
O tq/] 4%
7-882
•65838
...
186 ACTUARIAL THEORY [chap. viti.
3. Given tables of joint-life annuities, the columns D and /
for single lives, and conversion tables, show how, by means of
these, you would arrive at a premium for a joint-life endowment
assurance on (x) and (_y) payable at the end of 20 years if both be
alive, or at the first death before then, half premiums only to be
payable for the first five years.
The benefit side = A ^-..
m/:20|
To obtain the value of this we must enter conversion tables
with a ,-n which is equal to
% ~ 19 | % = a xy~ V 1 A X lgPy %+W : y+19
= n g+ 19 «+i9
xy D I *+19 : y+V»
x y
all the parts of which we obtain from the tables given.
The payment side = P(l + ^ : jgj + 6 | a^.^)
a —, we have found above and
xy : 19|
x+b y+b n v s+19 1+19
S I ay : 15| D I K x+b:y+b-> J) / .e+19 : y+19
x y x y
Hence the value of P may be found.
4. Use the tables at the end of the Institute Text Book and
tables of logarithms to find at 4 per cent, the single and annual
premiums for a joint-life endowment assurance on two lives each
aged thirty-seven, the sum assured to be payable at the end of
23 years or first death preceding.
Here we have
A S7:37: 23| = 1 ~ d ^ + fl 87 :37 : 22])
and P 37:37:2l = H^ Z ~ d
T 37:37:22|
N
N0W a 37:37:2^ = " S 7 : 37 " iT^
37 : 37
= 13-054-1-524
= 11-530
lo g N 59:59 = 9 ' 40776
log D 37:37 = 9-22491
•18285
= log 1-524
Entering the 4 per cent, single-premium conversion table with
this value we get
CHAP. VIII.]
TEXT BOOK— PART II.
187
Value corresponding to 1 1
Do. -5 = -01923
Do. -03 = -00115
Deduct
■53846
■02038
37 : 37 : 23|
■51808
Entering the 4 per cent, annual-premium conversion table with
11-530 we get
Value corresponding to 11 = -04487
Do. -5 = -00333
Do. -03 = -00020
Deduct
■00353
• 37 : 37 : 2S|
■04134
5. Given P, find the corresponding A by the use of the ordinary
single- and annual-premium conversion tables.
Enter inversely with P the annual-premium conversion table
and with the result thereby obtained enter directly the single-
premium conversion table, which will give us the A required.
6. Show how to construct a conversion table from which A
may be found directly by inspection, P being given.
P
Since A =
P + d
we may proceed to form the table on the
following system :-
Annual
Premium
Value.
(i)
log (1).
(2)
(3)
(2) -(3)
= li
ogA.
log- X (4)
= A.
(5)
P
P + AP
P + 2AP
etc.
logP
log(P + AP)
log(P + 2AP)
etc.
log(P + d)
log(P + AP + <2)
log(P + 2AP + d)
etc.
log
P + d
. P + AP
l0S P + AP + (
log
P + 2AP
P + 2AP + d
etc.
P
f + d
P + AP
P+AP+d
P + 2AP
P + 2AP + d
etc.
188 ACTUARIAL THEORY [chap. viii.
The work must be done in duplicate to ensure accuracy, as the
method is not continuous.
Again since — = -^ — = 1 + — , we may proceed by first
drawing up a table of reciprocals of P, P + AP, P -f 2AP, etc. ; then
multiplying each of these by d on the arithmometer and adding
1 to each result we obtain a table of reciprocals of A from which
the successive values of A may be found.
As this also is not a continuous method, the work must be
done in duplicate ; or checked by doing all the calculations in
reverse order when we should obtain P, P + AP, P + 2AP, etc.
7. If a single-premium continuous conversion table be entered
inversely with e~ nS , what does the result obtained represent?
The equation upon which such a table is founded is
A = 1-Sd
And as we are to enter the table inversely we have
1-A
a = —
In the particular case before us, A = e~ nS .
1 — e~ n ^
Therefore a = 5
o
which is the value of an annuity of 1 for n years payable
momently with interest convertible momently. (See Theory of
Finance, Chapter II., Formula (15).)
8. What would be the result of entering single- and annual-
premium conversion tables, calculated for continuous functions, with
a— ; and what does — S represent ?
Entering with a— we have
A = l-8a-
= 1 - S
1
1-e-
= e"
chap, viii.] TEXT BOOK— PART II. 189
that is, the present value of 1 payable at the end of n years,
interest being at i per annum convertible momently.
Also P = 4- -8
= S -8
1 -e~ nS
-ttM 1 -^ 1 -■-"'>}
which is a year's premium payable by momently instalments,
interest at rate i convertible momently, for a sinking-fund assurance
due in n years.
1 1 - 8a
-i- -8 = _^»
which is the premium payable per annum by momently instalments
for a whole-life assurance payable at the moment of death. (See
Chapters IX. and X.)
9. Find by means of Rothery and Ryan's Conversion Tables the
single premium corresponding to annuity -983, interest 4 per cent.
The practical difficulty here is that the tables start from unity
for the annuity values, while the given annuity is less than unity.
But A = l-d{\+a)
= i - d(i + rTo) + a
Therefore enter the table with the value 1*983 and add d to
the result.
Then the single premium corresponding to annuity
1-983 at 4 per cent. . . . . = -88526
d at 4 per cent. . . . . . = "03846
Single premium corresponding to annuity -983 . = -92372
The value of d might be obtained by taking the difference
between the single premiums corresponding to annuities 1 and 2
respectively.
CHAPTER IX
Annuities and Premiums Payable Fractionally
throughout the Year
1. Formula (1) of this chapter may be arrived at by the
following method which is somewhat similar to that of Text
Book, Article 3.
o| a . = \ - °
i|
Therefore interpolating
k
, la = a - —
tC | X X yyi
m
But o = — ( , I a + „| a + ••+ la}
* m \1| x _2[ x ml xl
mm m
— /(a ) + ( a -— )+ • • • +
m + \
m — 1
= « +
. , (m'i (m) 1
Also a = a + —
* x m
m+\
= a + -k —
* 2m
m — 1
= a - — n —
* 2m
2. Formula (5) which applies to the case where m — 2 may
be made general as follows : —
chap, ix.j TEXT BOOK— PART II. 191
For each year which (x) completes, the amount to the end of
the year of the m payments of — each is
1 m-l m-2
■^{(i + O* +(i + m +• • ■ +1}
and the value of these payments for the whole life of (x) is
■j m-l m-2
".-^{P+O^ + P+O* +•••+!}
Now in respect of the year of death, if (x) dies in the rth of
the m periods (r > 1) the amount to the end of the year of the
payments which he has received is
1 m-l i m-2 1 ro-r+1
—(i + m +— (i + m +• ■ • +—(l+«) m
m K J m K ' in
Taking the summation of this expression for every value of r
from 2 to m, and dividing the result by — since, on the assumption
of a uniform distribution of deaths, death is equally probable in
each of the m parts of the year, we have as the amount,
at the end of the year of death, of the payments made during
that year
1 1 m ~l _ 9 TO ~ 2 1 1
_L/!!LlL(i +,•)"» + ^_^(i +z -)™ +. . +_Ln +,■)«}
m { in v ' m m K J )
The value for the whole of life of the payments made during
the year of death is therefore
A
* m
/Ezi(i + i)» +^_^(i + m +• ■ • + — (i + O m l
and we have
, m-l m-2
% = «.s-{Ci + 0- +(i + 0- +•••+!}
+ A I{!Li(i + ;)ir + !^(i + i)» + . . + _L(i+i)»j
* m \ m K J m K J »i J
192 ACTUARIAL THEORV [chap. ix.
Expanding the successive powers of (1 + i), but stopping at first
powers of i in the expansions, we have
«, = a — \ (1 + i) + ( 1+ i)+ ■ ■ ■ + H
* 'm l\ m J \ m j )
. 1 fm-1/, m-1 A m-2/-, m-1 \
+ &—{ 1 + i)+ 1 + i + ...
x m L in \ in J m \ m J
+ l(i + i-M
m \ m J )
1 / m-1 A 1 -*'«„. 1 r»« - 1 (m-l)(2m-l)
= a —( m + ——i)+——J!—\—-+± ^ i,l
* m \ 2/ 1+2 ml 2 6m J
«i - 1 . m-1 (m - lV2m - 1) . m-1 . m-1 .
= a + — = za + — % 1- tts l ^ — ta s — i
x 2m * 2m 6m 2 2m * 2m
(taking : = (l+z)~ = 1-i approximately)
1 + &
m-1 m 2 — 1 .
= ^ + ~2~ik 6m 2 " %
If then in this formula we give to m the value 2 we have
(2)
a = a +
which is formula (5) of the Text Book.
This general formula can be rapidly applied in practice and
on the whole gives very good results. If we calculate by it the
annuities of Text Book, Article 28, we get the following values.
/? = 20-14127 a™ = 10-46974
a® = 20-26533 a^ = 10-59380
a 30 = 20-39002 d 60 = 10-71849
3. The argument of Text Book, Article 12, is very involved,
and the following is an alternative method of obtaining formula
(7), which is founded on the assumption of a uniform distribution
of deaths.
chap, ix.] TEXT BOOK— PART II. 193
1 1 / -L — ro-l
(ml 1 1 / - - ^±
t. N m in. ....
m m /
vl , ,
z+1
1 1 f i./ , ?«-l ,\ - . m-2 ,\
ra '* <- \ x+1 m V \ x + 1 ra V
ro-l / J \ 2./ »» — 1 \
+ «»(/,, + — rf )+^ ,,+W 1+ mU ,. + d ,, I
^ x+1 m x ) x+1 \ x + 2 m x + 1 )
%_/ m — 1 \ ~\
\ x + 2 m X +^J >
= _^+i — x+2 x jLf(i + r) m +(i+o » +... +ij
a:
+ _5 «±i x^|(«- ixi+0" +0-2)(l+i)» +...
+ (l+i)™|
2^ i_
«(l+i){(l+i)»-l}-i(l + i)»
- a s X T + a: T
«8{(1 + i) m - 1} ™ 2 {(1 + i) m - l} 2
The sum of the expression which is the coefficient of A may
be found as follows : —
ro-l m-2 1_
Let 2= (OT-l)(l+2') m + (m-2)(l+i) m +...+(l+i)»
1 m-l 2^
(l+i)™2 = (m-l)(l+i) + (m-2)(l+i) m + ... +(l+i) m
1 m-l J_
2{(i+i)™-i}= »«(i + «)-{(i+0+(i+0 m + ••• +(i+*') m }
= ».
set |
Am) _ Am) x+t (m)
a=t I x J_) s+J
»ra — 1 *+; / m — 1>
a * 2ro ET
a:
a _ "x+t a W-'/i
D *+' 2m
a n _^ri- D ^'
V^' 2™ /
^1 2m V D
5. To find i la . that is, the yearly annuity whose payments are
T
made at the end of — , 1 + — , 2 + — , etc., years respectively.
It is quite distinct from i|a (f> , the annuity payable t times
T
1 12 3
a year in instalments of — each at the end of — , — , — -, etc.,
of a year respectively, which is the same as a[
«)
1/i l+i 2+1- \
t
. (m) _ (™) _ i_ _ I Ji±2?(V™ ) - — ^ approximately
(m) if ^.L5±H^
~ a ™i t \ D '
X
t I ' ml J
t \ m \ \m »:»-J ' J
= T {('-»KfT+ fl "wl J
9. To find P (m) .
( m) _ m - 1
1 1
a * ~~ a * 2w
* V 2m a /
A
„(m) z
Now P^, = -p
a < 1
2m y x ' j
2m
P
1 _^li(P +d)
2m y * J
Again, multiplying each side by J 1 - ^— (P^ + d) V> we
P f_!^p>^ = P,
have
chap, ix.] TEXT BOOK— PART II. 197
from which we see that the addition to P to obtain P (m) mentioned
X x
in Text Book, Article 42, is for loss of premium " ; P (mJ P and
2m *> *
for loss of interest —= — P d.
2m *
The addition for loss of premium is found as follows : — The
chance of the second instalment of the mthly premium not being
received is — , of the third — , etc., of the rath m ~ , on the
m m m
assumption of uniform distribution of deaths. Therefore the whole
premium lost in the year of death is
1 /I 2 m-\\ P, ot-1
/ 1 2 m - 1\
— + — + • ■ ■ + )
\m m m J
m \m m
and the annual premium required to insure against this loss is
m - 1 (m) t, ,
— s — r r as above.
2m * x
The loss of interest on the second instalment being postponed
1 P d 2
— of a year is — - ; on the third being postponed — of
m mm m
p(™)
a year — - , etc ; on the rath being postponed of a
J m m " r r m
p(m) _
year — — — d. Therefore the loss of interest in accepting
mm
an rathly premium is
pC) , , . p(™> ,
** d f, . „ . . , ,0 r x m - 1 d
|l+2+ ■ • ■ +(m-l)| =
m - 1 „(m) , .
P d as above.
2m x
10. To find P^.
xt\
m - 1 /, D ,
(m) m - I f , ". r+ t
a n = a ; — ( 1
xt\ xt\ 2m V U
H 1 2« (a. P *)}
a «ti ,
x xt\
198 ACTUARIAL THEORY [chap. ix.
= *«i{ 1 ~ ^T ( p -i + *>} since P ^ = p il + p ii
[ml d
EXAMPLES
1 . Show that a x = a + ^ - ^ approximately.
By Text Book formula (2)
(l+i)*+l
{— - + Ki+»r T
■ ul -/a,
+ ^
= -J{2 + 2(1 + 0* -1(1 + 0'*} + Ki+0
Neglecting higher powers of ?' than the first
-i
„< 2 >
< =x( 2 + 2 + i-0+Ki -*0
a „+i
chap. ix.J TEXT BOOK— PART II. 201
2. Given the following formulas : —
(a) A^ = v(\+a x )-a x
(b) 1 = fa, + (1+0 A.
(c) A — v — iva
09 a] = i-«*(i+)
See Text Book, Chapter VII., Article 43, for explanation.
. , (m) (ro) / 1 (m)\ (m)
(e) a = a -\ ha JA
v J % «° \77J «° / *
This is explained in Te.i;< Z?oo&, Articles 13 and 14 of this chapter.
3. Find the present value of an annuity to (x) (payable half-
yearly) commencing at £1, the payments to be doubled every 10
years.
This annuity is equivalent to an annuity of 1 commencing
now ; plus an annuity of 1 commencing on attainment of age
a: +10; plus an annuity of 2 commencing on attainment of age
x + 20 ; plus an annuity of 4 commencing on attainment of age
x + 30 ; and so on. Therefore, taking cP = a + i-, we have the
' 3 to X X &'
whole value
X X
chap, ix.] TEXT BOOK— PART II. 203
4. Use the Text Book table at 3^ per cent, to find the half-
yearly premium to secure at age thirty-five a double-endowment
assurance payable at the end of 1 years or previous death.
Here we may write
P(2) i A^+A.1
2 2 a?!
a ■ —,
M - M , +2D ,
*- x-1 x+n-l' ^ x x+n'
J_ M 3 ,-M 45 + 2D 45
2 (N S4 -N M )-'2o(D 3i -DJ
1 9806-7769 + 2x16570
T (474131 - 260247) - -25(25839 - 16570)
1 35177
2 211567
■08313.
5. Prove that the value of a temporary annuity-due to (x) for n
years, payable half-yearly, is approximately ^a.— { + a^)-
a ( - = a - - m ~ V l - D - t+,t \
a »l *»l 2m V D )
Ir D , -v
2m\ ™l ^ y ^ ' D J
X
= _L/( OT+ l) a +( OT _l)Ca- i -l + -^±5U
2m \ v 7 m| ^ >\ xn\ D x /J
= 5— {(m+l)a. -i + (»»-l)«r -}
2jw ' »| \ / xn\'
Hence in the particular case
"raTf "^ " j ,7]
(2 > = i( 3a +„ )
6. Prove that the value of a temporary annuity to (z) for n
years, payable half-yearly, is approximately ^(3a^ + a m 7])-
204 ACTUARIAL THEORY [chap. ix.
a V± = d*>--Z±?:cP
D
X
D "x+n
D ^
x+n ,
= ( a x + ¥)--jr(. a x+n + ¥) approximately
X
4 V. a;«. | ' xny
7. Using the ordinary approximate addition to a yearly
annuity to make it payable m times a year, prove that for an
assurance of 1 on (x) the premium payable m times a year
m—\ ,
W \ TO + 1 /
OT+1
a +-7i
The premium payable at the beginning of each ?«th part of a
year for a whole-of-life assurance on (x) is
p W
m W
1 1 - da,
1 X
a
X
(m) »l - 1
1 - aa ;t —
1 * 2m
m
(m)
a
(m) ra - 1
a. = s
2m
(m) m + 1
since a = a + -= — .
1 * 2m
m—1 ,
—^ — (I
m + l
a , +
•i)
2m
chap. ix.J TEXT BOOK— PART II. 205
8. Given P^ at 3 per cent. = -01930, find the corresponding
half-yearly and quarterly premiums.
Since P (m) = -
, "- 1 (p.+«o
Therefore P (2)
2m
•01930
l-i(-01930 + -02913)
•01930
1- -01211
1930
98789
= -01954
p(2)
and the half-yearly premium — ^- = -00977.
Also P W =
P
•01930
1- -01816
1930
98184
= -01966
p W
and the quarterly premium — jj— = "00492.
9. An industrial assurance office grants whole-life policies by
weekly premiums of 2d. Find what sum assured can be granted
at age x, if the agent introducing the business is allowed as com-
mission the whole of the office premiums for the first 13 weeks
and 20 per cent, on them thereafter, and the office expenses
amount to 20 per cent, on the premiums from the commencement.
Assume that premiums, commission, expenses, and claims are on
the average paid in the middle of the year, and that the year
contains 52 weeks.
206 ACTUARIAL THEORY [chap. ix.
The premium of 2d. a week for 52 weeks = 8s. 8d. = - 43 of £1.
It is assumed to be payable in the middle of the year on the
average, and therefore the value of the premiums is
= -43xi(a +a) approximately
N +^D
= -43
D
Of the premium 20 per cent, for commission and 20 per cent,
for expenses has to be allowed for the whole period of the assur-
ance. This being 40 per cent, of the premium, its value is
•173 x — -. In addition to this 20 per cent, for commission,
X
a further 80 per cent, (making together the whole premium) has
to be allowed for the first thirteen weeks, and this being payable
on the average in the middle of the year, according to assumption,
its value is -086 -|±i
X
Thus the value of the net premium received by the office is
•43 * ' x - -173 ' - ' - -086 -gfci
X XX
If S be the sum assured to be allowed, its value is
M^l+t)*
X
And, payment and benefit being equal in present value 3 we get
•26(N. + jD,)-086D, +t
M.(l+0*
10. Given a table of temporary life annuities, show how you
would find approximately the value of an annuity on (x) payable
yearly for 21A years, the first payment of 1 being due 2 J years
hence.
In commutation symbols
X
since an immediate annuity-due for 21^ years means a payment
chap. ix.J TEXT BOOK— PART II. 207
of 1 at the beginning of each year for 21 years, with a payment
of i at the end of 20| years.
Now D x+n + B x+H + D, +4i + . . . + T> x+Wh + |D B+B
D , + 2 + D x + 3 , D , +8 + D , + 4 , , D , + ffl+ D , + g 8 , D , +M
= g + 2 + • ■ • + — :t — 2 + ~2~ a PP rox -
= (D, +1 + D x+2 + D i+3+ • • • +D a:+23 )-(D^ +1+ iD i+2 )
" N ,- N „ 2 3-^ D , +1 + ( D , + l + D , +2 )}
Converting the expression into annuities, we have
2ir*:2iJT ~ a x:~23\ ^( a x:T\ + "*:¥])
I a..
This shows us to be correct, for the annuity under consideration
is one for 23 years but with no payment for a year and a half.
11. If the force of mortality be constant from age x to age
c m — 1
(x + n), show that a — , may be expressed in the form -; where
v ' ym\ J r logc
y <£ x and y-\-m ^> x + n.
Let u, = u, = etc. = « , = - log r.
y j/+i y+in °
d log I
Then — -, — - = — log r
dy S
l °g l y = J^logr + log*
Z = fa*
y
t P y = r 'G > ™)
and d' » = «'r* = c* where ur = c.
But a -: = I «',» d/
_ = | v* ,p ,
ml J *' »
fm
y o
= u m _ „ w m {(1 + i )m _ 1 }/">
We may reason this out as follows. If 1 were payable certainly
at the end of the first mth part of a year its value would
210 ACTUARIAL THEORY [chap. x.
i
be v m . But as it is not payable till the end of that rath part in
which (x) dies we must deduct the value of an annuity of the
j_
interest on v m for every mth part which (x) completes. Now
i_
{(1 +i) m - 1} is the interest on 1 for the mth part of &. year, and
JL JL JL
therefore v m {(l+i) m -1} is the interest on v m for that period.
Again a is the value of the annuity which pays — at the
' m
end of every ?nth part. Therefore the value of the annuity to
provide t! m {(l +i) m - 1} at the end of every such period is
mv m {(l +i) m - \}a™\ Hence A^' = v m -mv m {(l + j)» - \}a^\
When m is infinite v m = 1, and m{(l +i) m - 1} = 8 ; therefore
A = 1 - 8a" .
4. Care must be taken in adapting these formulas to the case
of endowment assurances. We have
A -, = A^ + A 1
and on the assumption of a uniform distribution of deaths
A ( ^ = A%x J-+ A-i
art) X7i | «, . Eft I
and A ( ^ = A^fl + nd-il + A I
rc?a | xn\ ^ ' X7Z- 1
Also A -. = A -, x -=■ + A -.
xn\ xn\ g am |
and A -, = A 1 -, (1 + i)i + A I
xn\ xn | *- ' xn\
Again, without any such assumption
— — — (
A V, = v m - mv m { (1 + i) m - 1 } a — n
**l iv. ' / ) x:n-—\
ml
And A -i = 1 - So" — ;.
chap. x.J TEXT BOOK— PART II. 211
6. The proof of Text Book, Article 16, may be shown more
clearly.
^dt
x+t
«—*
A = —T v*l ^a
X
~ Uo * +i V i x+t dt J
1 f° dl , ,
--ty.*-**
Now by the method of integration by parts
\p d £ dx - j d S dx -ht dx
and n#* = r d -pd X -r q p*
J r dx ] dx J o dx
r<° dl . r ,0 dv t l ,, f dv*
Hence j Q ^dt = £-£** - j Q W
/■CO
= —I — I I , ..V 1 log t)
1 in ^
And A, = -\(-l x + z(yi x+t dl\
dt
dt
x+t
= l-Sd.
X
We may also proceed thus : —
dD , . +t + s )
212 ACTUARIAL THEORY [chap. x .
We have
I /-to I /-co
= TT"/ D L ,u rft + S— / D /ft
D / X+t r X+t D /n *+'
that is 1 = A + So"
s a:
and therefore A = 1 - Sa .
X X
6. Formula (14) may be deduced as follows, probably more
easily than by the method of the Text Book.
/oo
o ,Pt
ng both sic
x Jo \ ax J
dt
Therefore, differentiating both sides with respect to x
da. /•<» /d/
dx
I
But
d _x±l
d.p I
t*X _ X
dx dx
dl ^, dl
1 X+t _ j X
x dx x +* dx
~ l x l x+tl J 'x+t + l x+t l xl J - x
da /•«
Therefore -pS = | «'(« -« , ,Vp
/■GO fOO
"Jo ' * Jo ' * C
dl
dt
x+t
= a a — A
, , -r da
And A = a a - 5
X 'XX j
dx
chap, x.] TEXT BOOK— PART II. 213
fn
7. Similarly a — = I v f p dt
X7l\ I C\ X
Therefore -^ = P v'(-^)dt
dx J \ dx /
fn
= I v* (u, — a , \ p dt
J Q Vas r x+l ) t L x
fn fn
= ^l V \P X dt -j vt tP^.
X+t dt
a-.-A*-,
And A 1 -, = u a — , -
xn ' x am.
xn
xn\ r x m \ g x
8. The following method of deducing formula (20) is perhaps
preferable to that indicated in the Text Book.
r
e = I n dt
Jo' x
Therefore, differentiating both sides,
' .i/r.,/.)
dx dx\J (
>d
- s:m°
/-oo /-co
Hence = l> x ] o #,* ~ ] tP.^+t*
)dt
'dt
= u, e —1
X X
de '
And a = i-(l + -P°)
r * e \ dx)
X
= -|-{l-i(^_a-^ +1 )} approximately.
X
9. The following three annual premiums should be noted : —
("JP is the annual premium payable once a year required to
214 ACTUARIAL THEORY [chap. x.
provide an assurance of 1 payable at the moment of death and
A
accordingly is equal to - — — .
X
P is the annual premium payable by momently instalments
to provide an assurance of 1 at the end of the year of death and
A
is equal to — ,
a
X
0°'P is the annual premium payable by momently instalments
to provide an assurance of 1 at the moment of death and is equal to
A
EXAMPLES
1. Find the weekly premium required to provide a double-
endowment assurance to {£) payable at the end of n years.
Provision is to be made for the immediate payment of claims.
The benefit side
(M -Rl )(l+0*+2D _,_
>■ x x-\-n'^ ' x-\-n
D
X
As to the premiums, it is usual in the case of weekly con-
tributions to assume that they are best represented by a continuous
annuity, since the interval is so short. The payment side will
thus be
Pxa-,
P(a -v n p a , )
D
where P is the total contribution per annum
Hence P
(M - M _, 5(1+0* + 2D ^
(N -N ^ ) + i(D -D ^ )
The contribution to be paid at the beginning of each week is
p
therefore — .
52
x.J TEXT BOOK— PART II.
215
2. Use the Text Book table at 3 per cent, to find the value
of A gQ . — by three methods.
0) A «0:fiii
50
SO : 20| " f\
SO
(14462-9409)1-015 + 16605/ , , i\
36949 -(taking (1+,-)*= 1 + _J
= -58821. /
( M so- M 5o){ + D 60
W A 30:20T = n
SO
= 0^62^9X^+^6605
36949
= -58819.
0) A so :2 -o] = l ~ S %
We may put a H =
N ao- N 5o + KD 30 -D 60 )
30 : 20( n
30
(735104 - 230450) + ^(36949 - 16605)
36949
= 13-933.
Therefore A^.^ = 1 - -02956 x 13-933
= -58814.
3. Find the necessary formulas for the single premiums for
the following benefits : —
(a) An assurance payable 1 2 years after the death of (x).
(b) An assurance payable at the end of 12 years from entry,
provided (x) has died at any time within the 12 years.
(a) The required single premium is v 12 A .
For A , one or other of the following formulas may be taken,
\ = A.(l+0*
— i
A = A x -s-
X x 6
or A = 1 - Sd
X X
216 ACTUARIAL THEORY [chap. x.
(6) For this assurance we have
4. Show that, by a mortality table which follows Makeham's
first modification of Gompertz's law, A = —logs a +(/x +log*)n'
where a' is calculated at a rate of interest j, such that
1 c
~x+t dt
\dt
— If"
x J
1 /•<*>
= T / v * l x+t {- l ogs-(\ogg log^^+'jf
x J
■■ — T logs r vtl *+t dt ~T ( io gg i °g c y( c K +( *
x J a; •/
= -logs^+^ + log^a'^
since fj, = - log * — (log g log c)^
and therefore - (log g log c) c 1 = fjt, + log *.
a'^, being calculated at the required rate j which is such that
1 c
i+j ~~ i+r
5. Prove that
(6) -f(/a)=-ZA
(a) — a = — —
dN _ dT>
D -J-2-N -=-?
(D) 2
chap, x.] TEXT BOOK— PART II. 217
Now — N = JLTb dt
Tb
SJ
o \dx
\dt
rr±D
J o \cte
X+t ,
dt
x+t
= -D
X
J
and — D x = -D^ + S), (see page 211).
a -(D)2 + ND(u +8)
Therefore "a = " ,^'v,' '
d* * (DJ 2
This may be simply found also from the relation
_ da
A = u. a - ~.
r * i dx
(b) —(la) = l—a+a—l
= /{("«. +S)a -ll-iua
a: l^a; J x ' x r x x
= -1(1 -8a)
X^ X'
6. Prove that
=
-I A.
X X
(a)
dM
X
dx
' X X
(b)
dA
X _
dx
A>, + 8)-
"ft
__
l-8a
(a) A
Therefore M f = D^-sTd *
J o
And differentiating both sides
dM dD ,«, /riD
X _ X _ g / ( 5
rf^ <£e J \ dx
dD /-co /«/D
J _ » / ( ;
dx J V rfi
= -«,D
™ * a:
+(
x+t
rf<
rf/
218 ACTUARIAL THEORY [chap. x.
M
J X
dA d D
rj-\ % _ X
*• ' dx dx
dU. - dD
D _J1-M -j-*
* dx x dx
' (D) 2
-#;+MD> <+ 8)
7. If the force of mortality be constant ( = c), prove by direct
._ c
integration that A x = ~T~S'
Let c = -logs = i* = p.
x+t'
d log l x
Since « = 3—
r " dx
- d log I
then j * = -log*,
dx
and integrating, log l % = x log s + log k
I = ks x
.. dt
x+t
Now A,= x/ )) «'^«/»:
X
_, _LJ «*&*+'( -log*)
= I t)'i'(-log*)
Jo
A 1
dt
-logs
— log vs
- logs
- log s- log u
c
c + S
CHAPTER XI
Complete Annuities
1. On the assumption of a uniform distribution of deaths, the
value, at the beginning of the year of death, of the correction
to the yearly annuity to make it complete is as shown in Text
Book, Article 5,
1 / 1 1 1 Ja
—lv r +2v r +3v r + ■ ■ ■ + rV r )
From this is seen clearly the point discussed in Text Boole,
Article 3, viz., that the correction given by formula (1) is too
large. The average capital payment is
.1(1 + 2 + 3 + ■ . . +r)
1 r(r+l)
r+1
= i when r is infinite.
But to arrive at the value at the commencement of the year
of death, the earlier and smaller payments are multiplied by
JL JL -1
values (»«•, v T , v T , etc.) which are greater than the values
r r-1 r-2
(u 7 , v r , v T , etc.) by which the later and larger payments are
multiplied. Thus the true correction is less than |i)i.
1 i. A —
Now let * = v T + 2v T +3v r + ■ ■ • +rv T
\ 2 3_ Z. it!
v^s = v T + 2v T + ■ ■ ■ +(r-l> r +rv r
220 ACTUARIAL THEORY
s(l-v T ) = v r +v r +v r + • • • +fl r -ru r
= v r j--r« r
1 -v v
*{(l+i)''-l} = j--n;
1 -W T ,
iv(l +i) r
— I rv
{(1 +«')'-!}
iv(l +i) r
{(1+0' -I} 8 {(1 +0'-l}
Therefore the value of the correction at the beginning of the
year of death is
iv(l+i) r v
T T
And at the end of the year of death
i(l+i) T 1
r*{(l+0' -1}2 r {(l+iy-l\
Making r infinite we have as the correction
j_ l_
S 2 S"
Multiplying this by A and adding the result to a we have
a /* 1\
* a; a;^g2 g J
A Z '- 8
7 a ,+ A ,i2-
= a +A -7s-
— I
since A = A x -~- approximately.
chap, xi.] TEXT BOOK— PART II. 221
2. In the case of annuities payable m times a year we have only
to alter the interval of time from a year to the mXh part of a year,
and the payment from 1 to — If then we break up the mih part
of a year into r parts as before, we have as the correction
1 If —
2_ _3_
,mr + 3^mr + . . . +rv w,
which is precisely analogous to the correction in the case of the
yearly annuity. We have merely to alter the rate of interest
from i to {(1 + i) m - 1}. Accordingly, at the end of the rath part
of the year in which death occurs, the value of the correction is
2_ i
1 r {(l+i)™-l}(l+Q«- _ 1 -|
ml JL L J
r 2 {(l+i) mr -l} 2 r{(l+i) mr ~l}
l
But when r is made infinitely great, (1 + i) mr is unity, and
1 -Is / -i-
r {(l + j)mr _ 1 } is log (1 + i)™ = — . And writing iw for { (1 + » ) m - 1 }
we have as the correction
Jo
1 | w
1_
'(«)
S 2
Therefore
a
since
(m) (m) A (m) ^(m) - *
= a x +A * 32
(m) -r ^(m) - 8
= a „ i A „ — ; — 5~~
A = A (m) x ^P approximately.
3. Making no assumption as to distribution of deaths, and
taking r very great and approaching infinity, we have as the
value of the correction in respect of the (re+l)th year for a
yearly annuity
222 ACTUARIAL THEORY [chap. xi.
+f (M. + .+!^-M c+1l+1 )}
Now
M ^ = M _,_
M ^ A i = M _,_ + — AM x +— ^s ^ A2M ,+.+ ' " •
x+n-\ — x+n r s+ti 2 I+
r
etc. etc. etc.
r — 1 /r — 1
c-^-o
Therefore 2 = rM^ + ^AM^ — A'M +(i+ . . .
where AM i+ , = M, + , +1 - M +?l , etc.
Hence if we stop at second differences the correction is
^-{— (rM ^ + ^AM _, -^p=^A2M ± )-M xl ]
1 / - r - 1 - r 2 -l- - \
= rr( M , +— -AM , -- TS - r A 2 M ^ -M.J
D V + —A^l + i)h, and that
(m) m - 1
a x = a x + ~^~> we have the value of the annuity as at present
constituted
v * x+nJ
If K be the annual payment in future under the conditions
required, the value of the future payments
Hence, equating
and K = P G + ",+,)
2. Given that £100 will purchase an annuity of £5-026 payable
yearly, find the corresponding annuity also payable yearly but
with a proportion to the date of death, using 3 per cent, interest.
Here a = ™L
x 5-026
= 19-897.
And d x = a, + ^(1+0*
= + A- A,(l + i)i by formula (2)
Therefore 3$ = a ?0 + 1 + J{ 1 - d(l + aj } (1 + i)i
= 7-299 + -625 + -125(1- -02913 x 8-299)1-015
((1 +i)i being taken as 1 + - s -— 1-015)
= 7-924 + -096
= 8-020.
Adding the loading of 10 per cent, we get 8-822 as the price
of an annuity of 1. Therefore £250 will purchase an annuity of
250 =28-338.
8-822
The annuity, payable quarterly in advance and with a propor-
tion to date of death, which can be purchased is therefore
£28, 6s. 9d.
4. Show that A =1 -id approximately.
1 - ia„
We have A =
l+i
whence (1+?')A = l-ia
But (l+i) = (l+i)i(l+i)i
= (1 + i)i( 1 + _ j approximately.
Therefore (1 + i)U/l + ±\ = 1_«
and (1 + i)iA x = 1 - ia x - -1(1 + i)U
*" a * i s A^ = 1 - i& x approximately.
TEXT BOOK— PART II. 227
1
5. Show that a = v 2m « . approximately.
.(m) TO - 1 1 —
«» = « + -^ — + 7T- A
* x 2m 2m x
K x 2J 2m y x
= a — - — 8a
x 2m x
= all - jr — ) since 8 = d approximately.
v im a approximately,
since v im = (1 - dy
= 1 - - — approximately.
CHAPTER XII
Joint-Life Annuities
1. In a table in which Gompertz's Law holds we have funda-
mentally
u = Be*
and / = kg*
whence p = g« (« -*)
and a = 2s' p
(putting we first find w such that
3« =a„„ + a.„ + m„ and then a at the given rate of interest is
r w "80 "40 "50 ««» °
the value required.
3. If in the relation 2c w = (?+ & we assume that x < y
we have
2c™ = c x (l + cv- x )
1 +cv- x
r W-X — __■___
log(l+ci'- :l 9-log2
and w - x = ° v = '- 2—
logc
230
ACTUARIAL THEORY
[chap. xir.
from which we see that the value of w-x is the same for all
values of x and y where the difference between x and y is constant.
In other words, the addition to be made to the younger age to
find the equivalent equal age is constant where {y - .r) is constant.
We might therefore form such a table as the following : —
y-x
w-x
log (1 + cV- x )- log 2
logc
1
2
3
4
5
etc.
Entering this table with the difference between the two ages,
we find, in the second column opposite, the addition to be made
to the younger age to obtain the equivalent equal ages.
If there be three lives we have
3c u = if + cv + cf.
where x < y < z.
From the above table find w such that 2c™ = d^ + cn. Then
Sc u = •2c w + c z
= c w {2+c z - w )
2 + c z ~ w
-.it. ~iit *j n ^
log(2 + c 2 - w )-loff3
u — w = — 5_^ — I 1 5
log c
We might then form a second table of which the first column
should be integral values of s - w and the second should be the
corresponding values of the above expression which is equal to
u — w. Then for annuity values, etc., involving three lives we
should find from the first table the value of »- x corresponding to
y-x and hence find id ; and thereafter find from the second table
the value of u - w corresponding to s - w and hence find u. The
value of the required function is then that of a similar function
on three lives all aged u.
CHAP. XII
.] TEXT BOOK— PART II. 231
4. Still considering a table subject to Makeham's first modi-
fication of Gompertz's law we have as before
(putting cf + cv = c w )
a = 2u ( , p
xy t' xy
= 2t) t s ! W cI + c, )( t '- 1 )
2uV «V c ™( c _1 )
t*w
= a'
w
1 .1
where a' is calculated at such a rate /, that = — - — ..
«° '' 1+j l+i
Similarly a g = 2u«s«s*g°V-i>
where e w = c^ + c» + c 2 and the annuity is calculated at a rate J
such that ; = = . .
1+j 1 + 1
Generally a , . = a'
J xyz- • • (m) w
where c w = c" + cv + c s + ■ ■ ■ to m terms, and the annuity is
I gm - 1
calculated at a rate ;' such that ; = r -
J 1+7 1+z
The problem may be stated still more generally.
a _ ~2 v t s mt a -(c x +c ll +c z + ■ - ■ to to terms)(c'-l)
xyz- • ■ (m) b
putting c* + & + c z + • • • to m terms = rc w
2»' ' p
(r)
i-f www- • • (r)
a'
} s m ~ T
calculated at a rate of interest / such that : = ^ ; -
J \+j 1 + 1
In practice it would be most convenient to make r=l, as,
under this second method, tables of annuity values have to be
calculated at special rates depending on the number of joint lives,
and these special tables will of course be most easily prepared for
single-life annuities. A table of c x for all values of x must also
be prepared, as a table of /* in this connection is inconvenient.
232 ACTUARIAL THEORY [chap. xh.
5. A constant increase in the force of mortality under
Makeham's law has the effect of an increase in the rate of
interest. For if in the expression
/*» = - log*- (log g log c)c*
we add a constant - log r, where r is a positive fraction and con-
sequently — log r is also positive, we have
p' z = - (log s + logr)- (log g log cy?
whence simply
/' = h x r x g cX
= r x l
X
tP'z = r \P X
Also the value of an annuity on (r) in the new table is
a' _,, s = ~2v* p'
x(i) u x
tPx
t*x
xQ)
where a is calculated at a special rate j which is such that
1 r 11
= - — .. From this we see that : < . since r is a
1 +j 1 + 1 \+j \+i
positive fraction ; and consequently j > i. It may be mentioned
that an increase of •01 in the force of mortality is very nearly
equivalent to a rise of 1 per cent, in the rate of interest.
Further, in any table, as indicated on page 211,
1 dD dlosD
^ D dx dx
X
Now in a table where jj, is increased to jx -logr
rflo S D'_
dx
Also in a table where S is increased to 8 - logr
dlogD"
dx~^ = ^ + ( S - lo g0
dlogW dlosD"
Therefore £__■ = £ — ■
dx dx
D' = D" for all values of x
X X
and a = a" .
chap, xii.] TEXT BOOK— PART II 233
Thus a constant addition to the force of mortality is equivalent
to the same constant addition to the force of interest. From this
fact the practical assumption is made that a constant addition,
say "01, to the rate of mortality is equivalent to the same constant
addition, 1 per cent., to the rate of interest.
Though a! by the extra mortality table is equal to a . by the
normal, it does not follow that the corresponding single and
annual premiums are also equal.
For
A'
(0
=
i-^O+V
=
1 - d...(\ +a, ..) since d...=a...
(») v OX (0 0)
whereas
A o>
=
1 - d. ..(1 + a, .,)
0) v OX
Again
F «)
=
1 d
l+> w
=
1 d
But
P 0)
=
6. An increase in the constant B has the same effect as
increasing the age. For if p x = A + Be*, let
uf = A + B'c*
' X
where B' > B.
Find h such that B' = Be",
Then p x = A + Bc*c*
= A + Bc?+ h
= Px+h
7. In a table which follows Makeham's first modification, when
it is required to find the value of the annuity a^—^y it is not
correct to put it equal to ^ _ ^ where «/*„ = /*„ + /*„ + /*,+ • • •
to m terms.
We must proceed more slowly.
, . Z ''
a x^rr. +1 ; y ) }
Hence A MW " HA WM +^ w (l+a M:M+1 )-^ M (l + « w+1:M )}
Again, since A = 1 - d(\ +a ), we may put the formula in a
form for finding the value of A*, given tables of joint -life
annuities.
Ky = i U - «*(* + %) + %(! + a , : s+ l) - *Pj} + «,+! : „)>
2. In calculating the necessary joint-life annuities or the
suitable commutation columns as described in Text Book, Article
17, it is not at all essential that both (,r) and («/) be taken from
the same mortality experience. The two lives are of quite
different classes, and the risk will be most accurately calculated
by taking their mortality from different tables. The standard
basis at the present time is to take the O table for (if) who
is in the position of an annuitant or life-tenant, and the O
table for (x) who comes into property on the death of (y),
and desires to insure against his dying before that event.*
Tables of A 1 and P 1 on this basis have been calculated by
mj xy J
* See, however, the word of caution on pp. vi. and vii. of British Offices'
Life Tables, 1898; "Select Tables, Whole-Life Assurances— Males. "
chap, xm.] TEXT BOOK— PART II. 241
Messrs Baker and Raisin. Further, it is wrong to calculate the
values from an aggregate table, for (x) is usually the younger
life, and in such a case his mortality would be underestimated,
while that of (y), the older, would be overestimated, both
errors operating against the office.
3. Text Book formula (7) may be shown simply thus : —
A 2 = 2tj™ ,|q 2
xy Ti-1 I *xy
= 2»»( jo - ,|oM
Ml-1 | ?x ti-1 I ?xy'
= 1v n , I q - 2u™ , I o 1
7i-l I l x Ti-1 *x
xy
= A - Ai
x xy
Again A 2 = 2jj m in 2
° *y 71-1 | JX1/
= 2«»( 1(7 - lfl-1)
*>7i-l | '*/ 71-1 I *:ry'
= Stj^C ,1(7- ,1(7 + Jff 1 )
^Tt-l J 1 y 71-1 | ^an/ 7t-l | ixy f
A - A + Ai
!/ xy xy
All the 7e.rf Book formulas (9) may be similarly deduced, but
by summing from 1 to n only.
Thus I A' = 2?u» ,i(»i
| ft a^/ A ft-1 | J xy
= 2%»r jo - jo 1 )
= I A -I A 1
\n xy \n xy
4. An alternative formula for i n A^ may be found as follows :—
I A 1 = A 1 -v n p A~ — —
\n xy xy n l xy a:+7t:y+7i
1 „ ( . ^Wftiy+ft-l, s+ ft-l:y+ft \
2 ti^A •+»:»+• p j+ii _ 1 p^.j J
= 1(, A -.l^tl+ii^iil)
-tf \ ft XJ/ « » /
*y-l *x-l
Q
242 ACTUARIAL THEORY [chap. xin.
5. The Text Book formulas (8) may best be obtained by
deducting the corresponding values in (9) from the whole benefits.
Thus : —
|A 2 = A 2 - I A 2
n I xy xy | n xy
= (A -Ai )-(l A - | AM
V x %y' v I % x \n xy J
= I A - I A 1
»| x n\ xy
6. The application of Davies's and De Morgan's types of joint-
life commutation symbols respectively to the case of A^ may be
shown more clearly than in Text Book, Articles 14 and 15, in the
following manner : —
We have A 1 = -r— T 'Zv t d , , , I ,, ,
xy I I x+t-l y+t-i
x y
Now under Davies's form, where x > y, we may write this
x+t-l y+t-i __ _ v x+t-l x+u^ y+t-l y+V
D 2D
xy xy
YiiX+tfl l 4-7 7 - 7 7 -7 7 \
v x+t-l y+t-l x+t-l y+t x+t y+t-l x+t y+P
2D
xy
t ,(N , ,+N , )-(N +N )
>• x - 1 : y - 1 x-1 :y' ^ x : y-l xys
2D
xy
Also where y > x we have
' S/oV+ X l x+ t-l l q +t-l + l x+t-l l v +t~ l x+t l y +t-l~ l x+t l y+^
2D
xy
«(N , -N J + (N , -N )
>• x-1 : y-l x '.y-l' ^ x-1 : y xy/
2D
xy
Under De Morgan's form we shall write
x+y
St) 2 (. l x +t-lh)+t-l + l x+t-l i y+t~ l x+t l y+t-l~ l x+t l y+t)
2D
xy
(«N , -N ) + w4(N , -N ,)
^ x-1: y-l xy' ^ se-1 :i/ %;y-l'
2D
chap, xni.] TEXT BOOK— PART II.
7. To find Ai -=:.
x:y(t\)
vd + v 2 d , , + • . . +v l d
Al — ^ -. x x + l x+ t-1
4. »+' V+i x+t+1 y+lj
243
I I,
x y
u ( I , , vd , ,1 , + « 2 :f+1 )
- "P^C 1 + Vl : y+lJ - ^J 1 + Vl : v : .+l) + Vj} + % : ,+i : .+i)}
Since fl ,:,-i :,-i = ^:»-.,-,( 1 +V: 1 :) etC -
In this form the expression may be applied to select tables.
11. To prove Text Book formulas (22) to (27) inclusive we have
d ^ A - I I
j\2 _ 2d™ x + n ~ 1 y y+n-j z+n-j
*»* / I I
1 x y z
d A d I A
_ y„7i »+»-! z+rc- 1 _ y j, ic+n-1 jz+ti- 1 z+7t-$
" 11 III
% z x y z
= Ai -Ai
xz xyz
^2 _ g t ,7t «+«-! ( y y+n-j z+n-j . z z+n-j y+n-j \
V ' I \ I II I J
x y z z y
d , , / / , , I , , I , .1, A
\»,n x+n -I t y+n-j , z+n-j _ q y+n-j z+n- j \
I \ I + I 11 J
x y z y z
« »
x y 7
A 1 +A 1 -2Ai
a^ az aa/3
246 ACTUARIAL THEORY [chap. xiii.
d ± i l -I + x ' -I , ,
A.8 _ Vj,ti x+n-1 y y+n-j z z+n-j
xyz ~ ~ \ I I
x y z
_. 2i,n »+"-! _ 2j)» *+"-! V+n-h - Yv n x+n-1 z+n-j
x x y xi
d 1,1,
4- 2»i» x+n-1 V+n- l z+n-j
+ I I I
x y z
= A - Ai - Ai + Ai
x xy xz xyz
Al _ V„;t x+n-l l y+n-* z z+m-$ z+?t-$ y y+ft- 1
* : »» 4 V / / / I
z z
v
I . , I
, y+n-j z+
+ I
't
l
— Y,M x+n-\ ( y+n-j , z+n-j y+n-j z+m-A
- lv —r~\~r- + —i n — )
= A 1 +A* -A 1
xy xz xyz
A-
xy:z
(It should be noted that therefore A 1 — = A - A 3 )
v x : yz x xyz'
A 1 - Y-,71 z+n-j f x+n-1 y+n-j , y+n-1 x+n-j\
z i j/ y a;
= A 1 +A 1
xyz xyz
1 ^ Jd ^ , I -I , , d , , / -/ A
%,)! z+n-j l x+n-1 _y__y+n :: j y+rc-1 a; x+n-j )
I \ I I + I 7 /
z x y y x
I , ,/d , , d , , d , ,/, , d ,/
y„M »+"-« ( »+«-! , y+n-1 _ x+n- 1 y+n- j _ y+n-1 x+n -j
i \ i + i u n
z x y x y y x
= A 1 + Ai - Ai - A i
xz yz xyz xyz
12. To find P 2
xy
A -A 1
p2 _ x xy
x
In granting such an assurance by annual premiums, (y) must
be medically examined as well as (x). For were (z/) in bad health
and about to die, the office would be granting a whole-life
assurance to (x) for P 2 , which is less than P .
v / xy' x
chap, xiii.] TEXT BOOK— PART II.
247
13. To find the annual premiums corresponding to the assur-
ances in Text Book formulas (22) to (27) inclusive.
In obtaining the statuses for annual premiums, care must be
exercised that premiums are not taken into account beyond the
period when they certainly cease to be payable by reason either of
the benefit being paid or of the chance of its payment having passed.
A2
xyz
p2 _
f 1+a
1 xz
Here both (x) and {tj) must be medically examined.
A2
P 2 = "U"
^ 2 "" 1+a -
x:yz
A3
P 3 = X1IZ
*n* " 1+a
X
In these two cases all three lives must be examined.
Ai -
pi _ _ x:yz
x:yz ~ \ +a _
x:yz
The same difficulty arises as in the case of P 1 -^, since, e "
* x:y(t\y ' '*=>''
if (z) die early, say during the rth year, then possibly
^xTf.y+t < ^\:y~z> anc * an °P tion ma y be exercised against the
Ai -
office. The alternative is to make P 1 — = — ivH. w ith a corre-
*:»* 1+a
xyz
sponding risk of granting the benefit at an insufficient single
premium, if (z), say, be on deathbed, (x) alone is medically
examined.
J 1 _
ixy'-.z
P- =
xy:z
3
A —
1+a
xyz
A-
xy:z
3
l+a-
xy:z
In the last two cases (x) and (j/) must be examined.
14. Another and probably better method than that suggested
in the Text Book of applying Simpson's rule to the calculation of
A 1 — and A— may be pointed out.
x'.yz xy:z J *■
3
Ai - = A 1 +Ai -Ai
X : %z xy xz xyz
248 ACTUARIAL THEORY [chap. xnr.
Then finding w such that a = a we may write
° w yz J
Ai - = Ai + Ai - Ai
x'.yz xy xz xw
Also A- = Ai +Ai -Ai -Ai
xy'.z xz yz xyz xyz
S
Find w such that a = a , and w' such that a , = a ; and we
have A- = A 1 +Ai - Ai - Ai ,.
xy : z xz yz xw yw 1
3
15. To find the single and annual premiums for an assurance
payable on the death of the survivor of two children, ten and
fifteen years old respectively, provided both die before attaining
age twenty-one during the lifetime of their mother, aged fifty.
The single premium is
I A 1 +|Ai -(i A-i- + 16:21:S6 , Ai )
11 10:50 T 6 15:50 \| 6 '10:151 :50 T F) 5 16:56/
10 : 15 : 50
To obtain the annual premium divide this expression by
|6 a i0 ISO - *" |6 a iB :60 — |6 a i0:15:60 + ^ 6^10:50^ — e^ie) |s a i6 : 56
16. To find Ai _
ao : xy
A \.~ = Ai - + AJ --AAr -
ao'.xy a;xy b'.xy 'aW'.xy
= (Ai +Ai -Ai ) + (A} +AJ -AJ )
** ax ay axy' ^ bx by bxy'
( A 1 i A _L _ A— 1
V- aW :xy>
= (Ai + Ai + AJ + Ai )
v ax ay bx hy'
-(A* +Ai +Ai +AJ +Ai+Ai)
v axy abx aby bxy abx aby'
^ abxy dbxy'
From observing the method of arriving at this result, any
similar complicated benefit of the form A-, * , , ^ may
*■ abc • • • (tti) : xyz • ••(«) j
be worked out.
17. To find A 2
wxyz
A 2 = A 2 +A 2 +A 2
wxyz wxyz wxyz wxyz
1 1 1
= (Ai -Ai ) + (Ai -Ai ) + (A> -A 1 )
^ wyz wxyz' \ wxz wxyz' ' V wxy wxyz'
= (Ai +Ai +Ai )-3A!
> wyz wxz wxy/ wxyz
p. xm.] TEXT BOOK— PART II. 249
18. To find A 3
wxyz
A 3 = A 2 +A ! n + A» A
wxyz w(xy)z wx(yz) w(xzyy
1 1 1
= (Ai -Ai -) + (Ai -A* -) + (Ai -Ai -)
^ wz wz: xy' \ wx wx : yz' v wy wy: xz'
= {A 1 -(A 1 +A 1 -Ai )\
i wz ^ wzx wzy wzxy' '
+ {Ai -(A* + A 1 -A 1 )}
( wx ^ wxy wxz wxyz-' >
+ {A 1 -(A 1 +A 1 -A 1 )}
i wy v wj/a; -uji/s wyxz' >
= (Ai +A 1 +AM-2(Ai +A 1 +A' ) + 3A»
v wx wy wz' v. was/ wxz wys' wa^/z
19. To find A 4
wxyz
A 4 = A -Ai -A* -A 8
wxyz w wxyz wxyz wxyz
= A -A} -{(Ai +Ai +Ai )-3Ai }
w wxyz l >• wxy wxz. wyz' wxyz'
-{(A* +A 1 +Ai)-2(A! +Ai + A 1 U3A 1 }
* V wx wy wz' v wxy wxz wyz' wxyz>
= A -(A 1 +Ai +A 1 ) + (A 1 +A 1 +Ai )-Ai
w ^ itfffi icy wz 7 ^ wxy wxz wyz' wxyz
Or A 4 = A -A 1 —
wxyz w w : xyz
= A -(A 1 +A 1 +A 1 ) + (A 1 +A 1 +A 1 )-A J
w ^ wx wy wz'' ^ wxy tvzz wyz-' wxyz
20. Text Book formula (29) may be easily obtained in a manner
similar to that already shown for formula (14) of Chapter X.
a
d_
dx
I v*. p dt
Jo tlx «
a = | v*l — ,p ),» dt
But as shown on page 212 ^jj, ■= 0*. - f i+ ,)A
f A^-^+t) t Pxy dt
Therefore -r- a
u a -A 1
r x xy xy
And since ^ = £(3. +1 : , ~ Vi:») approximately
250 ACTUARIAL THEORY [chap. xiii.
In the same way, since
a = I v'p dt
xyz J t^xyz
d = I v t ( — 4 p \p dt
dx *v Jo \dx* 7' v °
= I vHu, -u ,,vp eft
I ^.r"a; ~x+t)t*xyz
d .
im a -A 1
r a; aa/z aa/z
and since — a = ¥a , , -a , ) approximately
fl r xyz 2 ^ a-f 1 : j/2 z-lia/z' rr j
we have A 1 = u, a +h(a , -a ,,. )
ici/2 r s ki/3 ^ v x-1 :yz aj+l : ys /
which is Te.ri 2Joo£ formula (31).
21. We might obtain the expression for A 1 in Text Book
formula (40) as follows : —
s i» = rr/o '*■+''*+«'*«
~a;+S
= TT f ' ' ; ' iV ' //
C' + d'T .
z y
s+e v+i
\-jyi x+t i y+t ^ +t +^ +t ) dt
= ^it/„ rt -+« / »+>-+* + / V< )( ^
"" A
c^ + c^ ^
22. Following, in the case of Gompertz's law, the method
adopted by Mr Colenso with regard to Makeham's formula in
/. I. A., xxxi. 342, we have this expression: —
*y I I Jo *+t y+t
x y
b^ r
= t-7 v l c f l ,J , t dt
I I J0 x+t y+t
1 C
= u, a' (a! being calculated at rate j where : = z — :)
r « ») v i ° ' 1+1 \+i
= /j, a' {a being at the same rate j, and w being such that
u, = a +/x ).
*w n x r y J
chap, xiii.] TEXT BOOK— PART II.
251
23. Under Makeham's rule for ^ Text Book formula (38)
may also be obtained as follows : —
X y J u
- Aa ~*y + -fi{? H * + t l y + t Bctdt
= A % + ^ n7 "°V« W Bc * +t+Bc!,+ ^
= -r^Aa +_£_(A -2Afl )
c* + cv "y d* + &> *y *r
-A_ Acr
c x + cv "v c? + cy *y
ti* — c x — c v
- A + log s a , since A = - log s
(f + cv *v c* + cy ° w'
Also A 1
xyz . . . (ro)
( X ' vH _,, ^ ^, , ,(A + Bc*+ ( ) ' ' 'x r y >w'
= — log s d + (u, + log s)d ' since A = - log s.
O ww \l x O / WW O
Similarly
A 1 = -logs a + C «, +log*W
:cyz ° www v x o y www
And generally
A 1 . . = -log* a . , + (« +log.s")o' , ,
xyz . . . (m) ° www . . . (m) ^ r :c o / www . . . (m)
Mr Colenso gives tables of -log s g , log (/j +log s), and
l°g,„a' from which values of A 1 may be easily calculated.
°10 www xyz J J
Basis : Carlisle Table of Mortality, rate of interest 3 per cent.
25. Text Book formula (41) may be obtained thus : —
chap, xhi.] TEXT BOOK— PART II. 253
'xyz . . . (m)l :abc . . . (»)
1
- r vh ^
J *+' : !/•
.1
I J n x+t : y+t : z+t . . . (m) a+t : b+t :c+t.. . tn)
xyz ... (m) . ate ...(«.) J ° v
x {mA + B(c*+* + c»+' + c*+* + . . . to m terms)}*
_ m \n , c x + c v + c z -\ to m terms
xyz . . . (m) . abc . . . (*) c* + c v + c* +... to ™ terms + c° + c 6 + ... to ra terms
1 /""
X ■ / ■y'Z /
/ , / n x+f.y+f.z+t . . . (m) a+t :b+t:c+t . . (n)
'xyz . . . (m) . ahc . . . (ny u
x {/*,+, + /*„+, + /*, + , + ■ • • to m terms
+ Pa+t + Pb+t + P c +t + ■ • ■ t0 n terms - ( w + re ) A W
c x + c y + c s + • • • to m terms _
c" + cv + c? + ■ ■ ■ +c a + c b + ■ ■ • to (?n + n) terms xyzabc . . . (m+n)
n(c x + c y + c z + ---tom terms) - ra(c a + c 6 + . . . to n terms)
+ . . . to (in + n) terms
x log (8 . , , .
& xyzabc . . . (m+n)
since A = — log s.
26. Text Book formula (42) may be obtained directly in a
manner which throws light on the ordinary assurance of the
same kind proved in Text Book, Article 21.
The benefit may be divided up as follows : — (1) An assurance
of 1 payable at the moment of death of (x) provided (y) be then
alive, and (2) a temporary assurance for t years to be entered on
by (x) at the death of (y). Thus
I /-co
x y
] /-co —
A 1 + —— I v n l I ix (A , - v\p , A , Ll )dn
zy I I In x+n y+n^y+vS x+ii t<- x+n x+n+U
/■CO _
I v n l I ix . A , . , dn
J x+t+n y+n r y+n x+t+n
x y
_ v'p
= Ai + A 2 - -r^ f
xy xy I I
x+t y
= \- vt t Px X iTt:y
254 ACTUARIAL THEORY [chap. xm.
EXAMPLES
1. Find an expression for A 1 on the assumption that the
chance of (x) and (y) dying in the same year may be neglected.
For the complete value of A 1 we have the formula
But as the chance of both deaths occurring in the same year may
be neglected, we omit the second term in the expression, and we
have for the value of A 1 under the conditions specified
xy r
2v n ( .p — p) p
\n-\ r x n> ayn 1 y
= W » p *-i=y - p )
V , n
J x-1
This benefit is of use when studying formula (11) of Text Book,
Chapter XIV., and its modification in formula (14).
2. Given the values of single- and joint-life annuities, find the
annual premium payable during the joint lives of (x) and (z/) for
an assurance payable on the death of the last survivor of (,r) and
(y), but half the sum assured to be payable on each death if (x)
dies before (y).
Do you see any objection to making the premium payable
during the life of the last survivor ?
The benefit splits into two parts : where (x) is the survivor the
assurance is payable on his death, and where (y) is the survivor
half is payable on each death. Therefore the whole value is
A2 +|Ai + |A 2
xy 2 xy * xy
= (A -Ai) + iAi +i(A -A')
\ x xy' * xy * v y xy'
= 1-^(1+^ + ^-1^)
chap, xni.] TEXT BOOK— PART II. 255
As the premium is to be payable throughout the joint lives the
payment side
= P(l+o )
1 -d(l +a +la -la )
and P = - ^ x - y ? xyJ
1+a
mi
The objection to making the premium payable throughout the
life of the last survivor is that, if (x) were to die in the early years
of the contract, then (?/) might be able to secure the benefit of
■| payable on his own death at a smaller premium, provided he
were in good health.
3. Deduce a formula for the annual premium for an assurance
payable if a life aged x dies within the next five years, or if he
lives five years and dies after another life now aged y.
The first part of the benefit is A*. —. ; for the second part, if (jj)
also lives the five years, we have v 5 ^p A-^r — -=•, and if (y) dies
J ' b r xy x+b:y+b' w '
within five years, v 6 p (1 — p )A . Therefore the whole benefit
is
A 1 Ti + ^.P A-^- -nc + » B e P C 1 - S P ) A , s
x:5\ 5' xy x+b : y+b b^x^- b r y' x+b
= A -« 5 t BA i( + » 5 ,p (A , . - A-^r . -th)
x 5 J x x+b b*xy\ x+b x+b .y+b'
+ JJ 5 c O A , . -D 5 ,p A , .
5 J " x x+b b 1 xy x+b
= A -« 5 c p A-? -^
x 5 1 xy x+b :y+b
Similarly the payment side is
P { % : 61 + V \Pxy\+b + V %Px( l - bPy) V J
= Pa
a;
A — v b V A 1
Hence P = -* & xy x+b:y+5
4. Express the value of an annuity-certain for n years, payable
quarterly, to begin to run at the death of (x) if he die after (y).
256 ACTUARIAL THEORY [chap. xiii.
On the required contingency happening the value of the
annuity is Ja^-r calculated at rate of interest — Therefore the
value of the annuity at the present time is
5. Deduce a formula for the annual premium for an assurance
payable on the death of (,r) if (^) has died five years or more
before him, the premium to be payable during the currency of
the assurance.
The benefit here is equivalent to an assurance on (x), less an
assurance payable if he dies before (y) or within five years after
(y), that is
A -Al -=r
x x:y(b\)
The premium will be paid throughout the whole of (x)'s life,
and the payment side is equal to
A - A 1 -~.
whence P = x _ '-"^
6. State a formula for the annual premium for an insurance
payable t years after the death of (x), if (?/) has survived him and
died before the end of the t years.
Benefit side
= tt'fetl"-* x + n ~ l V+n-l Vpn-} x+n-1 v+n+t-j\
\ It It/
N v. ii r. it *
v xy t r y x:y+t'
The premium will be payable so long as (?/) survives jointly
with (x) and for / years after (x)'s death, if (xf) lives so long.
Therefore payment side = P(l+a .zsjO
Equating the two sides we have
p __ v xy t 1 y x:y+t'
chap, xrn.] TEXT BOOK— PART II.
257
7. Give the formula for the single premium for an assurance
payable at the death of the last survivor of (x) and ( y), if that
occur in the lifetime of (z), or within t years after the death
of (4
c D i }
= {^""TT 1 ( A x+t~ A £rt:z)f
+ { A .-%f(V- A ^:,)}
_ ( j^ x+t : y+t ,j^ -A— -A— -L. ^l
1 xy D x+t:y+t x+t:y+t:z x+t :y+t :z) f
"- xy I
8. Find the annual premium for an assurance payable at the
death of (x), unless (^) die within the first n years and in the
lifetime of (.r).
To get the benefit we must deduct from the ordinary assurance
on (x) the value (1) of an assurance payable should (x) die after (y)
within n years, and (2) of an assurance payable should (x) die after
n years, {jj) having died within n years.
The benefit side is
A -{I A 2 +(1- p) \A\
x l \n xy v n*y'n\ x>
= A -| A+l A 1 - | A +v n p A ,
x \n x \n xy n\ x n* xy x+n
= \ A 1 +V n p A ,
\n xy n±xy x+n
which is correct, being the assurance payable should (x) die before
(y) within n years, or should (x) die after n years, {jj) having
survived that period.
The payment side is
Pfl +a — r,+v n pa.,)
<• xy:n-l\ n* xy x+n'
And P =
I A 1 +t" p A ,
\n xy n*xy x+n
\+a — rr-. + v n p a ,
xy:n~l\ n*xy x+n
9. Find the annual premium for an assurance on the life of
(x), the policy money to be payable at death or on the expiry of
n years, provided that in either case two other lives (y) and (z)
are then in existence.
R
258 ACTUARIAL THEORY [chap. xiii.
The benefit side is
I A 1 +v n p
\n xyz n^xyz
And the payment side is
*• xyz:n-l\'
I A 1 +V n p
Hence P = |ro , g ^?
10. Give the formula for the annual premium for a temporary
assurance of 1 payable in the event of (x) dying before (y) within
n years, (s) having died previously.
The benefit side is
A^ = I A 1 -I Ai
n xyz \n xy \n xyz
1
The assurance will not cease on the death of (z), and therefore
that life does not come into account in settling the currency of
the payment side, which accordingly is
pn+o — )
And P
Ai -I Ai
n xy \ n xyz
11. Deduce the annual premium for an assurance payable on
the death of (x) if he attains age x + n and dies before (jf) and
after (s), the premium to be payable throughout the whole period
of the status.
The benefit side = i A 1 - I A 1
?i I xy n\ xyz
As in the previous question the death of (z) will not disturb
the continuance of the assurance, and therefore the premium will
be payable so long as (x) and (^) jointly survive. The payment
side is accordingly
P(l+fl )
|A1 - |A1
and P = "1 "ti n \ x v z
\+a
xy
12. How would you arrive at the annual premium for an
assurance payable at the death of the last survivor of three
chap, xm.] TEXT BOOK—PART II. 259
lives aged 40, 50, and 60 respectively, provided a life aged «>0
is dead before the happening of the death of the survivor?
The benefit side = A 2 = a a i
20:40:50:60 ^40:50:60 40:50:6 0:20
The payment side = P(l+a ^
^ 40 : 50 : 60'
A A I
Hence P = 40 : 60 : 60 40 : so : go : 20
+ % : 50 : 60
40:60:60:20 == A 40 : 20 + A 50 : 20 + A 60 : 20 ~ A 40 : 50 : 20
40 : 50 : 20 ~~ 40 : 60 : 20 _ 40 : 60 : 20 ~ 50 : 60 : 20
50 : 60 : 20 + 40 : 50 : 60 : 20 + 40 : 60 : 60 : 20 + A 40 : 60 : 60 : 20
13. A sum of 1 is to be divided among such of the existing
children of a widow aged w as may be alive at her death. What
is the share of (,r), (a) assuming that there are two children aged
x and y respectively now alive, (6) assuming that there are three
now alive aged x, y, and z respectively ?
(a) If both (,r) and (y) are alive at (w)'s death then (x) receives
A, but if (y) has died previously (x) receives 1. Therefore the
value of (a-)'s share is
AA 1 +A2
* wxy wxy
1
= AAi +(Ai -A* )
A wxy V wx wxy-'
= Ai -AA 1
wx -> wxy
(JJ) If all three are alive at death of (to), (x) receives A ; if (x)
and one other only are then alive, he receives A ; and if he alone
is alive at death of (w), he receives 1. Therefore his share is
AAi +"i(A2 + A 2 ) + (A» +A» )
o wxyz - *■ wxyz wxyz' V wxyz wxyz-'
1 1 21 12
= AA 1 +i(Ai -Ai +Ai -Ai )
■* wxyz * ^ wxz wxyz wxy wxyz'
+ (Ai -A 1 -A 1 +Ai )
^ wx wxy wxz wxyz'
= Ai -AfA 1 +Ai ) + AAi
wx " ^ wxy wxz' •=> wxyz
14. Given four lives (x), (a), (6), and (c), find the value of an
assurance to yield at (x)'s death £1000 if one and only one of the
lives (a), (6), and (c) shall have predeceased him, and £3000 if
260 ACTUARIAL THEORY [chap. xnr.
two and only two shall have predeceased him. The expression is
to be reduced to assurances which determine on the first death.
The value of the assurance is
1000A 2 „ + 3000A3 .
xaoc xabc
= 1000(Ai +A 1 +A 1 , -3Ai )
^ xao xac xbc xaoc-'
+ 3000( A 1 + A\ + Ai - 2Ai - 2A 1 - 2A\ + 3Ai )
V xa xb xc xab xac xbc xabc J
= 3000 (A^ + Al + A L ) - 5000 (A^ + A^ + A\J + 6000 A^
15. Three partners, A, P>, and C, aged respectively 30, 35,
and 40, possess a capital of £10,000, and their proportionate
interests in the business are 2, 3, and 5 tenths. How would you
calculate the premium for an assurance to cover the risk of
having to pay out the representatives of the partners who may
happen to die first and second ?
The value of the assurance required is
2000A i : sFTlo + ™ 0OA L -. aoTIo + 5000A i, : ^
It would be advisable that three separate policies should be
effected, either by single or annual premiums, one for each part
of the above benefit. If, however, one annual-premium policy is
essential, the payment side will be
P ( 1+a 807l57lj) = P ( 1+fl 30:S5 + a 30:40 + a 35:40- 2fl 30: 8 5:4o)
oil I q a i 1-5A 1
Hence P = 1000 30:35:40 36:30:40 40:30:35
1+a s0:35 + a 30:40 + fl! 35:40~ 2o 30:35:40
It is possible that under such a policy an option may be
exercised against the office in the event of one of the lives dying
early, say C in the first year. The premium for the remainder of
2000A1 +3000A1
the benefit, viz. 2~ 2^i, will probably work out
+ a 31 : 36
at less than P as found above.
16. Give a formula for P * , — . , the annual premium for an
assurance on (30) payable in the event of his dying within 10
years or before (60).
chap, xm.] TEXT BOOK— PART II. 261
The benefit side is
A 1 = = A 1 _4-Al -Al
30:10|:60 ^30 :10| ^ ^30 : CO A 30:60:ioj
The premium will be payable so long as (30) survives jointly
with the survivor of 10 years certain and of (60), and for the
payment side we have
^on . in, . en * «V
SO :10| :60 SO : 10| :60
30 : lofTio ' a 30 : ioj + a 30 : 60 ~ a 30 : 60 : ioj)
A 1 - + A 1 -A 1 -
Therefore P 1 = = so:io|^ 30:eo 30:60:io|
30 : 10| : 60 o i „ _ „ _
30 :10| T 30:60 30:60:10]
With the premium payable during a status such as this, it is
possible for an option to be exercised against the insuring office
in the event of (60) dying within 10 years, say at the end of the
tth year. For at that date, provided (30) is in good health, he
may obtain an equivalent benefit for Pg^rr . 10 _,, which might be
less than the premium found as above. On the other hand, it
would not do to make the premium payable so long as (30), (60),
and 10 years certain survive jointly, unless evidence as to the
health of (60) were produced. For if (60) were dying, then (30)
would secure a short-term assurance for 10 years for a very
inadequate single premium.
17. Use Mr Colenso's tables (J. I. A., xxxi. 354-6) to find the
value of A 3 i 5:72:79 and <->P£ :T1 , :W
5 » : 72 : 79 = " lo § S fl ~35 : 72 : 79 + 0*116 + lo S *) «" 8 5 : 72 : 79
= -log .9 V73 : 71-73: 71-73 + <>35 + l0 S*Kn-73: 71-73: 71-73
since /" 3 5 + / u 72 + /"-9 = -01020 + -06558 + -11692
= -19270
= 3 /*71-73
Now -log,«- 71 . 7S:71 . 73:n . 73 = -02930
and log 10 (,u s6 + log,) = 3-29072
loo- a' „„..„ = -68205
iu »10 71-73: 71-73: 71-73
3-97277 = log ]0 -00939
262 ACTUARIAL THEORY [chap. xiii.
Therefore A 8 i 5 . 72 . 79 = -02930 + -00939
= -03869, or £3, 17s. 5d. % very nearly.
( " J)P B 1 B :72:79 = r^f^ approximately.
2^ K 3B : 72 : 79
■03869
4-053
= -00955, or 19s. Id. % very nearly.
18. Investigate an expression for — a —, and show what approxi-
mate conclusion this leads to on the assumption that Makeham's
law holds.
, -a —, = -r- I v\p dt
dx xn \
)dt
dt
d f n
dx) Q tlx
- r/ufy t
1 l x
fn fn
= u I v* t p dt - I «' p a , dt
'xj U x J t*x r x+t
= a a —. - A 1 -,
•x xn\ xn\
On Makeham's hypothesis p — p. = 'Bc x (c t - 1), and therefore
- 1 -a—=-'Bc x l « ( (c ( -llp
dx *»l Jo H x
dt
Considering the definite integral, we see that it represents
either
(a) The value of a temporary annuity with increasing pay-
ments ; or
(6) The difference between the values of two uniform annuities,
one calculated at the ordinary rate of interest, say i, the other
1 c
calculated at rate j such that =
J l+j 1+2
chap, xin.] TEXT BOOK— PART II. 263
19. Show that on Makeham's hypothesis A 1 = — — - — =
J r "v* 31og s-S
for all values of x, y, and z, which satisfy the equation
O >x 'y ''z
l°g* " M x + lo g*
Since
^r*% i^y >z
iog* /\+ lc g*
B(2c c -c»-c')
Be®
2c* - c" - c~
Therefore Sc* = logs(2c* -c* -c»)
= log *{ 3c 1 -(cx + cu + c*)}
c*f31ogjj-S)
C- + CV + C-- = ^-T^—
But by Text Book formula (41)
c A _ 1
v ij/z = ' c x + cy + c z xy* cf + cv + c*
rX — C" -t- V — it .
Ti _ A ' logtftf
substituting 1 - Sa^ for A^
«■ C ° te _ c !±^^log,a-
substituting for Sc* its value as found above
= c x + &> + C
logs
= c*
c*(31og s - S)
logy
31og J - 8
264 ACTUARIAL THEORY [chap, xiii
20. Find an expression for -^ a .
^ di *»
d - dv d -
di *u di dv *v
— P— *
dij dv tPxv
/■CO
Jo tlxy
= - v I tv*.p dt
Jo tlx »
21. Give a formula for the fine, to be paid as a single premium
at the outset, for the option to increase at the end of n years,
without further inquiry as to health, the sum assured by a survivor-
ship policy payable only in the event of (x) dying before (j/).
If, on the option being exercised at the end of n years, the
premium to be paid is to remain at the rate of P * per unit for
the future, the difference between that premium and the premium,
which, looking to the effect of selection, should be charged,
(P—L. Y is (P—L- r -rr- - P f \r ,)■ The whole value of this
difference for the period after n years is then
[«]+» : [y}+n (V l pi yi , \
D M+» :[»]+» M[i/]A T lx]+n:[y]+nJ-
MM
If, however, when the option is to be exercised, the premium
to be paid is that for a similar benefit at the then ages, we must
substitute for P* , in the above formula the premium P— i .
mlV\ r [x+n] [y+n]
22. A select life, (x), desires a contingent insurance against (y)
with the proviso that, if he be alive at the death of (?/), he shall
have the option of converting his policy, as at the next renewal
date, into a whole-life assurance at the ordinary annual premium
applicable to his then age without medical examination. Obtain
an expression for the net annual charge required for this option.
If the option is assumed to be exercised at the end of the tth
year its value is
(• [x]+t~ ix+ty\x]+t
chap, xiii.] TEXT BOOK— PART II. 265
and the probability of its having to be exercised then is
t-l\%ttP[x]
Therefore the value at present of the option in respect of the
tfth year is
" t - 1 1 %]tP{xV{x]+t ~ P [s+<]) a M+<
Summing this for every value of t and dividing by a we
obtain the addition to the ordinary premium to cover the option
t-l\%ltP[xp-"[x]+t~ lx + ty\x]+t
a MM
= 2 <-i|9[i/]( P M+< ~ P [s+<]X a M 3 a M -Fy)
a M»]
23. Show how to find approximately the net annual premium,
on the basis of the Makeham graduation of the Carlisle Table,
at 3 per cent, interest, for an assurance payable in the event
of (x) predeceasing (y), (1) (x) only, and (2) both (x) and (y),
being resident in a foreign country. It may be assumed that the
extra risk is represented in the case of a single life by a constant
addition of -01 to the force of mortality at all ages.
A' 1
(1) <»>P'i = , l v ,
K) xy * + «„
where A^ = j ^\p xt p/ x+t dt
J o
o
= A' 1 + -01a'
xy xy
both A' 1 and a' being calculated at a special rate of interest j
xy xy °
such that - ; = — 7T, where -logr =-01.
1 +j 1-03'
/"OO
Also a' = I v ( .p' ,P dt
" xy j t" xt' y
. r
J
'V' P .P dt
v'r
P ,P <
I xt 1 y
a found as above.
xy
266 ACTUARIAL THEORY [chap. xiii.
A" 1
(2) (»)p"i =
*» i + a
xy
,.dt
_ TOO
where A"^ =. J^\p' xt P y ^ x
= A"i +-015"
xy xy
both A" 1 and a" being calculated at rate ; which is such that
xy xy ° v
1
r 2
: 1-03'
where
■log r— '01.
Also
a xy
=
f o V \P'xtP'y dt
=
I v'r 2t ,p (It
Jo Uxv
=
a" as above,
xy
CHAPTER XIV
Reversionary Annuities
1. The temporary reversionary annuity
I a I = I a — I a ,
\n y\x I to x 1 7i xy'
where (x)'s chance of receiving payments is confined to the first
n years, must not be confused with the annuity to (x) after (y) for
life which is to be entered upon only in the event of (y) dying
within the first n years. To obtain the latter we must deduct
from the reversionary annuity, a y the reversionary annuity after
n years should both lives survive that period, v n p a l .Its
value will therefore be
( a - a ) - v n p (a —a )
*• x xy' m-rm/V x+n x+niy+n- 1
This again is different from the deferred reversionary annuity
n\ y\x n\ x n\ xy
= v n p (1 - p )a , + v n p a , I ,
n r x x n l y' x+n n r xy y+n\x+ll
under which, as pointed out in Text Book, Article 5, it is not
necessary that both lives should survive the period of deferment.
Another benefit to be distinguished from the foregoing is the
annuity to commence on the death of (j/) and continue during the
subsequent lifetime of (.r), but in any event to the end of n years
certain from the present time. This is a reversionary annuity
on n\ after (^), a \—, together with a deferred reversionary annuity
on (x) after Q/), n \a y \ x ; then
a I -i + I a I = C°-i ~ a -i) + ( \a - \a )
y\n\ %\ y\x ^ n\ yn\-' ^n\ x n\ x\y
Finally, this is different from the annuity-due to run for » years
268 ACTUARIAL THEORY [chap. xiv.
certain after the death of (^), and for so much longer as (x) may
live, the value of which is
Afl+a — n) + «"7=^l = Afl+a — n")+C« -a 1=^^)
y< n-1]' £) V x+n x+n\y'
2. To find the annual premium for an endowment assurance to
(x) payable at age (x + n) or previous death, the premium to be
doubled in the event of the death of (j/) before (x) during the
n years.
We have benefit side = A -,
xn\
Paymentside = P(l + «„— ) + P x )„_,«,).
= P(l + 2a — T .-a —^)
^ xin-l\ zy:n-l\'
Whence P = , — ^
1 + 2a — ^ — a — r,
a;:m-l| xy:n-l\
The difficulty arises, however, that if (#) die early, say in the
tih year, (x) may then obtain his benefit at a premium of
P . . — r, which might be less than 2P as found above, and the
office would not in this case obtain the premium on which it
reckoned in making the contract.
3. To find the proportions in which the purchase price of a
last-survivor annuity on (x) and (?/) should be paid by them.
Each is entitled to half the annuity during the whole of his
life, and to the remaining half for the period succeeding the death
of the other life. That is, (x) is entitled to
la + la I = a — la
A x ^ y\x x " xy
and (?/) is entitled to
la +la I = a — la
J y z %\y y £ xy
Now a —la +a — la = a +a - a
x * xy y * xy x y :
xy
= a—
xy
which is the whole purchase price. Therefore (.r) and (y) must
pay (a - la ) and (a -la ) respectively.
x j \ x " xy* y x y
4. To find the value of a last-survivor annuity on (x) and (y)
which is to be reduced by half at the first death.
chap. xiv.J TEXT BOOK— PART II.
269
The whole annuity is payable during the joint lives, but half
only during (,r)'s life after (y), or (y)S life after .v. Therefore
the value is
a + la I + la I = Ma + a )
xy 2 j,| j, t 2 x \y !l", T ",j
which is obviously correct.
5. The identity proved in Text Book, Article 18, may be very
easily demonstrated by working from expression (1) to expres-
sion (10).
= V P X %+v\P X (% + l \%)+V\P z (% + 1 \ «
D *
xy
ay
D
(ro)C ! + (™>C — + • . .
or = •» — * +1 ! » +1 if x < y
(ra)M 1
x y
chap, xiv.] TEXT BOOK— PART II.
271
Here, again, the method of forming («0C 1 and (™)M l is suffi-
ciently clear.
If the annuity is to be entered on at the death of {y) with a
proportionate payment to the date of death of (x), we have
a =
The denominator will take the form D or (l+i>-*D
xy K . ' xy
according as x > or < y ; and the numerator in either case will
take the form
(ra)C 1 + (ro)C -JL + . . . = (ro)M 1
11/ i+l:|+l T " x xy
where ( ro )C J = v*+U , ,d /'a , + — A ,\
Tables must therefore be formed of this function for every value
of x with every value of y, and the summation of these values
from x upwards where x -y remains constant will give ( ro )M 1
xy
as above.
8. On consideration of the argument in Text Book, Articles
21, 23, and 26, we have clearly on the analogy of formula (10)
d ft = ^tPxit^Py- tPy^flt
= ^Pxit-lPy- tPy)(\+t~^2~)
x * y-\
= a I — •
which is Text Book formula (14).
9. The method of finding the value of d(™) given in Text Book,
Article 30, may be made more clear by a simple graphic
illustration.
p Q E s
L J ! ! etc.
I i I l I
A B C D E
272 ACTUARIAL THEORY [chap. xiv.
Let A, B, C, etc., represent the ends of rathly intervals as
from the date of effecting the contract ; and let P, Q, R, etc., be
at the middle of these intervals.
Now on the assumption of the deaths being equally distributed
throughout each interval, if (^) die, say, in the interval AB, his
death will on the average occur at P. Therefore in the case of
the formula,
y\x x xy
the first payment, of — will be made at B, and the succeeding
payments at C, D, etc.
But in the case of the benefit represented by aW it is desired
that the first payment of — should be made at Q, and the
m
succeeding payments at R, S, etc. If therefore the formula affl is
1 — 1
to be employed, it is clear that not — but v 2m — should be paid
at B, C, D, etc., for this will amount with interest to — at the
m
correct dates of payment Q, R, S, etc.
If the transaction be effected thus, (x) will receive a payment
of v 2m — , to which, on the conditions of the benefit oW, he is
m vW
not entitled, in the event of his dying between B and Q or
between C and R, etc. Making a deduction for this over-
payment and keeping to our previous assumption as to equal
distribution of deaths, we have the value of the correction as
shown in Text Book, Article 30, = — A 2 approximately.
1m w ^ J
A further correction is necessary to make the annuity complete,
and the value of this is also approximately — A 2 as shown.
These two corrections are equal in value but opposite in sign,
and we therefore have finally,
i
i(m) = j;2mf a (m) _ a {m)\
y\x K x xy '
10. To find | i«.
\% y\x
J_]
As before, let payments of v 2m — be made under the annuity
m
chap, xiv.] TEXT BOOK— PART II. 273
i a<™) - i a(™) at B, C, D, etc., which will accumulate to at
Q, R, S, etc. : the value of this annuity being,
1
v 2m(, a (m)_ I a (m)\
^ [ n x I ft xy J
Now let E be the end of the n years. At E a payment
of z> 2m — is due, but is not to be made, because the annuity is to
cease at the end of n years and the payment of — to which this
m
would accumulate by, say, T falls outside that period, and
accordingly is not payable. A deduction similar to that for
the whole-of-life reversionary annuity must also be made. We
then have,
-i- 1 -i- 1
I gMi = „2m(| a (m)_| ffi (m)) v 2m v n p (I - p ) - — x | A 2
|»^|a; \-\n x \n xy ' m % c x^ n r y' 2m \ n ^
Besides an addition to make the annuity complete in the same
manner as for the whole reversionary annuity we must add a
proportion for the period between S and E. Therefore
-J- n — 11
I aW = u 2m d m )-\ d m f)-i — v Sm v n p (1 - p) + -^—x\ A 2 V
|jli/|x ^\n x l» >»' \m n r x^ n^y' 2m \ n X V \
+ /JL XI A 2 +J_t)»ffl(l- n)\
— /I — 1 \
= v 2m ( I a( m > - I iiW) - ( — v 2m - 7t- ) v n p (1 - p ")
11. To find |.
n\ y]x
We have ia<™> = ia( m >- iaW
?i| y\x n\ x n\ xy
= v n v a< m ) —v n p a< m )
n r x as+ft n r xy x+n:y-\-n
= v n p (I - p )d m ) +v n p aW , ,
n r x^ n c y' x+n n r xy y+n\x+7i
With regard to the first portion of this formula, if (x) survive and
(^) die within the n years, the first payment of the annuity ajW will
be made at the end of — of a year after n years, and therefore
m
S
274 ACTUARIAL THEORY [chap. xiv.
d m ) — df- m ) . Also in regard to the second part, d< m > , is of
x+n x+n => r ' j/+n|x+n
the same nature as d(™>, which has been already discussed.
y\ xr J
Therefore,
|dV) = v n p(\- p)d< m ) +v n p k m ) , ,
n\ y\x n 1 x^ n 1 y' x+n n r xy y+n\x+n
= v» p (1 - p )/flW '+ s-A , ^
w-^aA 71-^3/' I a:+w 2?72 x+n J
1
+j»p DSwfaW -ffl( m ) )
n r xy K x+n x+my+n'
In the above formula - — A , would be more exact than
2m x + n
- — A , , but all throush this chapter correction for payment of
claims at moment of death is ignored.
12. By Text Book formula (22) the annual premium for a
reversionary annuity is found to be
a —a
Pa
y\ x 1+fl
xy
Now the value of a reversionary annuity occasionally decreases
as the lives grow older, and therefore the annual premium for an
annuity to (.r) after (if) may be greater than that for a similar
annuity to (ai+1) after (^ + 1). This is a state of matters the
reverse of what is usually found in assurance contracts, with the
consequence that a level premium to be charged throughout may
be too small at first and afterwards too great ; and were the
assured to realise this they might drop the policy and get the
same benefit for a lighter premium after a few years. The first
year's risk is vp q (1 + a ) = q a , and Pa i must never be less
than this.
13. To find the reversionary annuity to two children aged 10
and 15 respectively, which is to commence on the death of their
mother aged 50 and to continue till both children have attained
majority.
This is a temporary annuity to (10) after (50) for 11 years,
together with a temporary annuity to (15) after (50) for 6 years',
chap, xiv.] TEXT BOOK— PART II. 275
less a temporary annuity during the joint lives (10) and (15) after
the death of (50) for 6 years. That is,
[ll ffl S0[l0 + lo a 5o|l5~ U^ollOUS
= |li a iO~|li a 50:10 + |6 a i5 _ lG a 50:15 _ lo a iO:15 + lo a 50:10:16
■ 1,
(10 : ll|X15 : 6|) 60 :(10 : ii|X15 : 6|)
14. To find a~\-r
xy\ab
(a) «— 1-7 = a-r-a-r —
*■ ' xy\ab ab ab'.xy
abxy xy
= (a+a 1 +a+a)-(a.+a +a + a^ + a, + a )
*• a b x y J v ab ax ay bx by xy'
+ (a, +a . +a +a, )-a .
^ abx aby axy bxy' abxy
— (a +a) + a
\ x y J xy
= (a +a.)-(a .+a +a + a, + a, )
*■ a b' ^ ab ax ay bx by'
+ (a.+a.+a +a, )-a ,
^ abx aby axy bxy' abxy
(b) fl— !-=■ = a— I +a— \.—a— I .
v ' xy\ ab xy\a xy\o xy\ ab
= (a I +a | -a \ ) + (a ],+a \, - a I.)
*• i|o T y\a xy\a' ' v x\b y\b xylb'
-{a\,+a\,-a I , )
v x j ab y\ab xy\ ab'
= (a — a + a - a — a +a )
^ a ax a ay a axy'
+ (a, - a. + a, - a, - a, + a, )
*■ b bx b by b bxy'
— (a.-a,+a,—a.—a,+a, )
*- ab abx ab aby ab abxy'
= (a +a,)-(a +a + a. + a, + a , )
v a b' V. ax ay bx by ab'
+ (a +a, +o.+fl,)-fl,
^ axy bxy abx aby' abxy
15. The problem of Endowment Assurance Instalment Policies
discussed at page 148 is sometimes further complicated by the
introduction of a beneficiary. If the life assured die before the
date of maturity, the beneficiary is to receive an annuity for life
with the guarantee of n payments certain ; but if, on the other
hand, the life assured survive the endowment period, then the
annuity guaranteed at that date is for n years certain, and con-
tinued beyond the n years to the last survivor of the life assured
276 ACTUARIAL THEORY [chap. xiv.
and the beneficiary. This extra benefit for the first (m + n - 1 ) years
is the value of an annuity of — to {y), which however is not
payable so long as (x) is alive, nor for n years after, or, in symbols,
~V|m+i-l y~\m+n-\ y:x(n\y
And for the period after (m + n—V) years, the benefit is a
reversionary annuity to (3/) after (x) deferred (m + n- 1) years ; in
symbols,
1
n m+n-\\ x\y
The whole extra benefit is therefore
re Mm+m-l y \m+n-l y:x(n\) m+n-l\ x\y'
1 D
n ^ y:m+n-l\ yn\ J) y+n :x :m-l| m+w-l| x\y'
(See value of 1 a —=. deduced on page 131.)
To get the extra annual premium, we must divide this function
by a — j in order to guard against an option as already discussed.
This portion of the premium will cease to be payable on the death
of (^) before Qc).
EXAMPLES
1. An annuity of £100 per annum, payable until the death of
the last survivor of three lives, A, B, and C, aged respectively
20, 30, and 40, is to be divided equally between A and B during
their joint lives, afterwards between the survivor and C, if living,
and ultimately is payable to the last survivor. Find the value of
A's interest.
Given
«20
=
20'2246
ffl 20 : 30
= 16-1739
"so
=
18-4156
°30 : 40
= 13-9872
%
=
16-1026
fl 20:40
= 14-5274
%
=
14-1097
fl 20:4Y
= 12-9502
a m
=
13-8064
a 20:48
= 12-7015
chap. xiv.J TEXT BOOK— PART II. 277
A's interest = K:30 + Ho!20:40 + a 3oTTo| 2 o
= £ a 20 : 30 + (i a 20 : 40 ~ ^ a 20 : SO : 4o) + (°20 ~ fl 20 : 30 ~ °20 : 40 + °20 : 30 : 40^
= a 20 - £ a 20:30 - i a 20:40 + i a 20:30:40
To find the value of a„„ ...... we have
«S0:40 = 13 - 9872 =
S
= £k{l(a +a ) + (Ai - A* +Ai -AMfl + fl— Tl )\
i*V xz yz' *- xz xyz yz xyz'^ »-l|-"
10. There are two formulas for a | , viz. (a) 2w* p (1 - p ) and
(6) '2v t t _ 1 \q x t p(l + a ). Give the corresponding formulas for
fl— L j and prove their identity.
0) \z\x = 2v \Px( l -tPy)( l -tPz)
( 6 ) %\x = 2 ^-l|^ X A( 1 +«.+*)
chap, xiv.] TEXT BOOK— PART II. 281
^W* - tP) 0- - t P z ) = ^ t P x x \ t q-
= 2?, W?,-i + l|^ + ---+ ( -ll < 7ji)
= •y.?iiC 1 +'!P, + i+« , A + i+-")
+• • •
= 2/u ( .p x. ,\q—(\+a ,,)
11. A father aged 50 wishes to secure to his two children
aged 8 and 10 respectively, an annuity of £n, to commence at his
death and to continue until the younger child, or the elder if he
be the survivor, attains the age of 21. Find a formula for the
value of the annuity. Would it be safe to grant such an annuity
to be secured by annual premiums ?
The value of the annuity is
£ra (|l3 a 50|s + |li a 50|l0-|li a 50| 8 :10)
One difficulty in connection with accepting payment by annual
premiums is in determining the period during which the premiums
should be made payable.
If they are made payable throughout the whole status, the
risk is run of the contract being dropped, and of a new one being
effected at a cheaper premium in the event of the early failure
of one of the children's lives. In this case to obtain the annual
premium we should divide the single premium by
(■ + a 50 : 8 : i2| + ffl 50 : 10 : 10| ~~ a 60 : S : 10 : ioj )
If, on the other hand, the premium is accepted only duiing
the joint status, there is a risk of either (8) or (10) being a bad
life, the premium payable being considerably underestimated as a
consequence. In this case we should divide by (1 + a 50 ■ s ■ jo • ib])'
It is possible, too, that even if all survive, the annual premium
for a similar benefit for the remainder of the term may diminish,
in spite of the increase in the ages.
A contract by annual premiums should therefore, if possible, be
avoided.
12. Find, according to the Text Book Table, the value, at 3 per
cent, interest, of a contingent annuity for the remainder of
30 years certain from the present time, the annuity to commence
282
ACTUARIAL THEORY
[chap. XIV.
on the failure of the joint existence of two lives both now aged 30,
but only in the event of such failure taking place after the
expiration of 5 years and before the completion of 10 years.
The formula to be used is
V \P.
. x a„
5' 30:30 35 : 35 1 25|
-, - V W ,
10^30: 30 X ^40 : 40 1 20|
_35j_35/ , 60:60 „ \
_ D l a 25l~ a 35:35 + f) 60:60/
" ?>« ■■<•>■ 35:35 '
-(%l" a 40:
30:30
60 : 60 ,
D„
5/" \
-( a 25|-°35:35)-
logD g
logD„
30 : 30
= 1-807- -726
= 1-081
= 9-42108
= 9-52032
40 ^ T) 60:60i
40 : 40
40 : 40 f
^"201 "'40:40-'
■^30 : 30
~( a 20l °40:4o)
"251
log 2-271 =
1-90076
17-413
15-142
2-271
•35622
r-90076
■25698
log 1-807
lo S D 40:40 = 9>31707
lo sD 30:30 - 9-52032
1-79675
a 20j
°40 : 40
= 14-877
= 13-718
1-159
log 1-159
= -06408
1-79675
1-86083
log -726
13. Required the annual premium for an annuity payable to
the last survivor of (.r) and (#), to commence at the end of n years
if both be living, or at the first death if that occur within n years.
The benefit side
i) n p a , I ,
n' xy y+n \x+n
= v n p a
n^xy"x+n:y+n y\x
+ a I - v n p a
xly n* xy :
= v u p (a , + a
nl j;lA x+n V
xy x+n\y+n
a
x+n x+n:y+n
ui'xy v "x+n ' y+n x+n:y+n
- a , + a , ) + a - a + a - a
y+n x+n : y+n' x xy y :
= a + a - la + u n p a ,
x y xy n 1 xy x+n : y+n
chap, xiv.] TEXT BOOK— PART II. 283
This might be written a-- \ a and in this form it means an
annuity to the last survivor, not payable, however, so long as they
both survive within the first n years ; which is the required annuity
expressed in other terms. Our reasoning is thus proved correct.
The payment side = Pa —
* ' xyn |
a + a - 2a + v n p a
And P = v xy n °>y x +K'-v+n
a -,
14. Ascertain the annual premium for a reversionary annuity
which (x) desires to provide for his wife (y) and child (z) after
his death. The annuity is to be £100 so long as (y) survives, but
to be reduced to £50 at (y)'s death.
The benefit side consists of two parts : (a) an annuity to (y)
after (x)'s death ; and (6) an annuity to (z) after the death of the
survivor of (x) and (y) ; that is
100a I + 50o-i
x\y xy\z
A difficulty arises in determining the status for payment of
the premium. If the annuity were not subject to reduction on
Q/)'s death the premium might be made payable during the joint
life of (.r) and the survivor of (y) and (z). But in this case, should
(y) die early, the reversionary annuity to (z) might be obtained
at a smaller premium than that so found. Again, if the premium
were made payable during the joint existence of (x) and (y), and
(y) were to die early (she not being subject to medical examina-
tion), the contract is practically a reversionary annuity to (z) after
(a-) at an insufficient premium. The best plan is, if possible, to
have separate contracts for the two portions of the benefit, and
have the premium for the former payable during the joint lives of
(.r) and (y), and for the latter during the joint lives of (x) and (z).
15. Under a deed of separation, A covenants to pay an annuity
of £K per annum to his wife B so long as she lives, and the terms
of the deed make his estate liable for the annuity after his death.
He wishes to free his estate from this liability so long as his
daughter C survives him, and he applies to an insurance office for
a quotation for the annual premium for such a contingent annuity.
Find the net annual premium.
The company will have to pay the annuity so long as C
284 ACTUARIAL THEORY [chap. xiv.
survives jointly with B after the death of A, and the premium
will cease on the first death of A, B, and C.
Therefore if the ages of A, B, and C are x, y, and z respectively,
the value of the payments is
Kxai = K(a -a )
x\yz ^ yz xyz'
and of the premiums P(l+a )
K(a -a )
whence equating and solving P = — -2? ^-
16. Determine the annual premium for an annuity of s to
continue during the lifetime of B aged y, after the death of A,
aged x; with the proviso that should A survive the age of
x + n a sum of t is to be at once paid to B, if then alive, instead
of the annuity.
The value of the benefit is
s\(a - a )-v n v (a , - a , , )} + tv n p
l\ y yx J n"xy^ y+n y+n : x+n J > n r xy
The value of the premium which will run during the joint
existence for n years is
P(l+„ )
Hence P
,{(« - a )- >+ » : '+» (a^ -a ^ )| +< «+»:»+«
(> y yx J D y +n y+n:x+n->) J)
1+a — r .
xy:n-l\
17. Give the formula, reduced to its simplest form, for the
annual premium for an annuity of 1 to a female aged y to be
entered on at the death of her husband aged x if that occur
within the next 20 years ; but to be entered on at the end of 20
years if (#) be then alive, whether her husband survive that time
or not. The annuity is to be payable by half-yearly instalments,
and with a proportion to date of death of the annuitant.
The value of the benefit is
4(2) _ ^20 „ (,J(2) _ M) 1
"x\y " 20-fW- a;+20|!,+20 y+W
= »K"f - *») - ''V^OfU - *?U V+ J - K 2 j 20 + i Vj}
= ^y ~ %) + V2 °20^{%+20( 1 - «*) + V \+20 : ,+20 + i + i^+SoC 1 + J }
if a(2) = a + \ for both single and joint lives.
chap. xiv.J TEXT BOOK— PART II. 285
The value of the payments is
whence P may be found by equating the two sides.
18. Write down an expression for the net premium payable by
a husband aged 40, to provide an annuity of 1 to his wife aged 30,
should she survive him ; the premium to be payable quarterly in
advance for a period not exceeding 20 years, and the annuity to
be entered upon at the death of the husband, and to be payable
quarterly with proportion to the death of the widow.
Using Sprague's formula (17)
l
fi(m) _ „2m(- a (m) _ a (m)\
y\x *> x xy '
— m-\
= v 2m (a „ ) if we assume sW = a-\ —
we have iW |so = vi(a s(j - a 30 . 40 ) for the value of the benefit.
And for the payment side
0: 20j
o^x^tU^
- pm/a -Wi - 50:6o N ll
_ 20 r i a 80:40:20| + 8l ^ D )t
„(ml _ a (m) _ x+n:y+n (m )
*y»| " xy J) x+n:y+n
xy
TO + 1 I) x+n:y+n ( a m + l \
% 1m D \ *+»:»+» 2m J
xy
m + 1 /.. _ x+n:y+n \
\ xv
%n\ „,
Hence «« - — «-»-«,:«>
D,
_ , 6.A _ 50:60 \
a 30 : 40 : 20| + 8 1 T) I
\ ^30 : 40 7
19. A, aged 45, wishes his son, aged 15, to receive an annuity-
due of £20 on A's attaining 60 years or previous death. Find the
286 ACTUARIAL THEORY [chap. xiv.
yearly premium at 3 per cent, interest, using the O Table for
the father and the O for the son.
Benefit side = 20a,. ,-nl,.
45 :14| 115
= 20 ( a i6- a i5 : 45:i4|)
Payment side = P(l +« 15 . u . -^
Hence equating the two sides
p ^_ 15 15 : 45 : liy
1+a i5:45:i4]
An approximation to a . . — may be found as follows : —
1 _ ,
° 15:45:U| ^5:45:151 + ^
But by Lidstone's formula P 15;46:lT| = P 16:1 ^ + P 46:1 ^- P lT|
Now [M] a [16];1 ^ = 10-899, [NM \ 45];1 ^ =10-192, and flfij = 11-296
Therefore entering conversion tables, we get
P [16] :i5] =- 05492 > P W]!K | =- 06022 ' ^ %l =- 05220
whence P [15] . [46] . lTj =-06294 approximately.
Entering conversion tables inversely with this value we obtain
a n K , .r,.!!-i.-nn = 9-861
[15] : [45] : 14|
and
% 3 = 23 ' 223
Therefore
P =
20(23-223-9-861)
10-861
= 24-605, say £24, 12s. Id.
20. There are S persons at present entitled to an annuity of £K
per annum each, their ages being respectively a, b, c ■ ■ ■ I, m, n.
Upon the death of an annuitant, the next on a waiting list steps
in. The waiting list consists of T persons, aged respectively
p, q, r ■ ■ ■ x,y, z. It is required to find the value of the interest
of (z) who is Tth on the list.
chap, xiv.] TEXT BOOK— PART II. 287
There are (S + T) persons involved altogether, and (s) will
come in only when the (S + T - 1) persons, apart from him, are
reduced in number below S. Therefore his chance of getting a
payment in any year is the probability that he is alive multiplied
by the probability that less than S persons out of (S + T - 1 ) are
alive.
Now the latter probability is equal to the sum of the prob-
abilities that none, that exactly 1, that exactly 2, etc., that
exactly (S - 1) persons out of (S + T - 1) are alive
= p B+ p M
tr abc . . . Imnpqr . . . xy t r abc . . . Imnpqr . . . xy
+ p m+ . . .
t^abc . . . Imnpqr . . . xy
+ p Szll
t*abc . . . Imnpqr . . , xy
z z 2
7 s-i
;+;r-^+ • • • +.
s
1+Z (1+Z) 2 (1+Z) 3 (1+Z)
- 1--*-
(1+Z) S
■■ s
t"abc . . . Imnpqr . . . xy
where abc- ■ • Imnpqr- - - xy represents (S + T - 1) persons.
The value of (z)'s interest will therefore be
£K 2t>* A (l - t P aic ... lmnpiT ...?y )
CHAPTER XV
Compound Survivorship Annuities and Assurances
1. It has been already pointed out that there are two formulas
for the reversionary annuity
a i = 2d' p x 1,0
y\x trx \t*y
and a I = 2d' 2p- -S±izl (l + a „)
y]x I I K x + tJ
x y
and these two are identical in value.
Similarly we may express compound survivorship annuities in
either of two ways. Thus
a 1 I = 2d' p x i.o 1
yz\x t^x \t"yz
for a payment will be made at the end of, say, the | xyz -2 x\yz -5 xy\z - xz\y'
and P(a +ia \ + Aa 2 i +a 2 i )
\ x & x\yz * xz | y xz\y'
whence P may be found.
2. Give a formula for the present value of £1 receivable on
the death of a person aged 50, provided another person now aged
20 has then either died or attained age 40.
This assurance, being payable on the death of (50), provided
either (20) has previously died or 20 years certain have elapsed,
may be represented by
A 2:S - = A - A 1 -
50:20:20| 50 50:20:20|
3. Determine A 3:4
w : xyz
21
This is the value of an assurance on the death of (to) provided
he die either third or fourth of the four lives (to), (x), (y), and (s) ;
and provided (s) and (y) have died first and second of the four
respectively.
Now A S:4 = A 2 -rfa 2 i
w : xyz ivxyz xyz \ w
21 11
= A 1 - A 1 - da 2 I
wxy wxyz xyz [ to
1
But a 2 | = 2d ( p x |,<7 2
xyz\w t r w | t*xyz
1 1
2« ( ,p (L<7 1 -|,<7 1 )
t r w\\t*xy \t l xyz'
1| -,
Q 1 a I - Q l a I approximately.
^-xy xy\w ^-xyz xyz\w rr J
Hence A S:4 = A '-A '-(/(Q'o I -Q'o i )
w :xyss wxy u-xyz ^^-xy xy\w ^"xyz xyz\w'
21
292 ACTUARIAL THEORY [chap. xv.
4. Find Ihe value of
(a)
a?4
1 xy ' :z
(*)
A 2 —
x:y(,t\)
(<0
A-
xy: z
1:3
(d)
A l:3
x:yz
«
A 1-
zlxv
(a) This is an assurance payable (1) if (x) die first, (2) if (x) die
third, (z) having died first, (3) if (^) die first, or (4) if (?/) die
third, (z) having died first. In symbols
A 1 + A s + A ! + A s
xyz xyz xyz xyz
1 1
(4) This is an assurance payable on the death of (x) if he die
more than / years after (jj). In symbols
x:y(i |)
(c) This is an assurance payable (1) on the death of (x) after
(jf), if (z) has died before (y) or if (z) is still alive ; (2) on the
death of (y) after (x), if (z) has died before (x), or if (z) is still
alive. In symbols
A s + A 2 + A 3 + A 2
xyz xyz xyz xyz
11 11
(d) This is an assurance payable on the death of (x) if he die
first or last of the three lives. In symbols
A -A 2 = A -A 1 -A 1 +2A 1
x xyz x xy xz xyz
(e) This is an assurance payable on the death of the survivor
of (x) and (^) if that should happen before the death of (z). The
alternative symbol is A—
J xy:z
8
5. Express A 2 '- 3 * and A 8;4 in formulas for summation
r w : xyz w : xyz
1 21
not involving the use of the integral calculus.
chap xv.] TEXT BOOK— PART II. 293
dill
^2:3:4 _ ^ t = °° tl t-* z +'-I v+t-j x+t-j ,,+t-^ j-
w.xyz " ^( = 1 / III w+t-i
1 z vj x y
A~8:4 _ V t = 0O v t-l * z+t-h y+t-1 w+'t-j i-fl-tT
w.xyz t = l I I 11 w+t-i
21 I y w x
6. The present holder of a title of nobility is aged w. It is
desired to effect an assurance payable on the death of his wife
aged x, provided that during her lifetime, the heir aged y having
died, the next heir aged z shall have succeeded to the title. Give
a formula for the single premium.
To fulfil the conditions both (w) and (y) must die before both
(x) and (z), but it is immaterial whether (x) dies before or after (s).
The single premium therefore is
A- 2:3 = A 3:4 + A 3:4
wy :x:z w:x:y:z w:x :y:z
"Y 12 2 1
It may be most easily expressed as an integral, as follows : —
re© —
A =J o V '- t Pj( 1 - t Py)tPJ l v+tH 1 - t Pj t Pf ,„}**+&
7. Calculate by the Text Book Mortality Table the value of
the following formula, using 4 per cent, interest : —
500A 7 i 0:70 -40ai 0:70 | 50
500A 7 i 0:70 -40< ;70 | 50
= 500x£A 70;70 -40x!« 70:70 | 50
(since by formula (4) of this chapter of the Text Book ffl^ . T0 | 50
= Q70:70 a 70:7o|50 = 2 a 70:7o|50^
= 250A 70:70- 20 Ko-«50:70:7o)
We have A n . n = l-d(l+ « 70 . 70 )
= 1- -03846(1 +4-054)
= 1- -19438
= -80562
294 ACTUARIAL THEORY
[<
And a tn = 12-522
50
a 50:70:70 = "*:*:* where ^50 + ^70 + ^70 =
3/*
By the Table of Uniform Seniority
C 50 + C 70 + C 70 = 3 C 66'4
a ,v*. m .± ***. = 3-850- -4 x -228
06*4 :66'4 : bo 4
= 3-759.
[chap. XV.
Therefore 500 A^- 40< :70 | 50
= 250 x -80562 -20(12-522 -3-759)
= 201-405-175-260
= 26-H5.
CHAPTER XVI
Commutation Columns, Varying Benefits, and Returns
of Premiums
1. In addition to the expressions derived in Text Book, Articles
9 to 1 4, the following should be carefully examined. It will be found
that these or similar expressions are very frequently required in
Chapter XVIII. in connection with valuation by the retrospective
method, and it is essential that the principles upon which they
are founded should be thoroughly understood.
N -i - N . + .-i '.(i+o-H+ia+Q"-^- • • +U-i( 1 + o
D ^ ' I
x+n x+n
This represents the accumulations to the end of n years of an
annuity-due on (x) for that period. It should be noticed that it
is greater than (l+i)s—; for each value of /in the numerator is
greater than I in the denominator, and the whole expression
is accordingly greater than (1 + i)* + (l + i) 71 ' 1 + • ■ ■ + (1 + i),
which is the value of (1 + i)s— r
N ,-i- N . +t -i '.(i+'r+^q+o-^ ••• +*, +t -i(i+o- ,+i
x+n x+n
x x+n _ aA 'J s+1'- J x+n-l
D ^ l j.
X+n 33+71
x+t x+n _ x+t^ '_ a+f+l'- '_ x+n -l
D /
x+n x+n
x x+n x+n x^- J a+1'- J x+n-l
D ^ " l M
x+n x+n
R-R -<]VI d(l+iy- 1 + 2d (l+i)»-s+ ... +td (1 +»)»-<
x x+t x+t x^ '_ x+V* ^ a;+E-l v '
S i ^
x+n x+n
R-R ^,-tM
X x + t
g — = ^-[ d x( 1+i T' 1 + 2d x+li l + i ) n - 2 +---
1 - 'n x+n
+ <{d. + ,_ 1 (l+0"-'+«Wl + 0"-'- 1 + ••■ +d *+n-l}]
596 ACTUARIAL THEORY [chap. xvi.
2. The following is probably a simpler method of obtaining
the values of varying and increasing annuities and assurances.
iN+P i , + N i , + N x ,+ . • • )
(va\ = x X+J x+2 + -
x
AN ± hS ^
D
X
When A = h = 1
CH = &
X
AM +*(M .. + M .,+ M r+ ,+ . • •)
v ^ _
x
AM +^R ,,
D
x
When A = A = 1
R
( IA X - w
AN +£(N ,, + N LO + • • • +N ^ .)
( V _a) = —^ i-«±i 5+"
<■ n[ 'x J)
AN ± h(S _,_. - S ^ )
D
X
When A = h = 1
S - s
(la) = « *+"
AM ±A(M ^.H-M ^„4- • • • +M .„ .)
Also (v^A), = g
X
AM ±A(R . -R .J
x ^ a;+l x+n'
D
a;
When A = h = 1
R -R ,
(I A) = * * + "
x x+n x-\-n
15
chap xvi.] TEXT BOOK— PART II. 297
^ 'xn\
D
X
£(N - N.J ± k{S „ - S ^ - (m - 1)N ^ \
D
When k = h = I
S -S . -«N
( Ifl )-i
v 'xn\
Also (vAV-:
A(M - M ^ ) + A{(M - M , ) + (M -M , ) + ...+(M , ,-M _,_ )}
v s x+ro-' — tv x+1 x+n' ^ x+2 x+n' *■ x+tt-1 x+n' >
D
X
MM - M ^ )±h{R ^ - R ^ -(n-l)M ^ J
^ a: x+n J *■ x+1 x+n ^ J x+n*
D
X
When A = h = 1
R -R , -»M ^
(IA)l-; = -= *±£ 2±*
*• 'xn\ J)
x
3. So long as the S and R columns are supplied, the working
out of increasing benefits by these formulas is therefore an easy
matter. But in cases where these commutation columns are not
available, a method which has been described by Mr Lidstone
(J. /. A., xxxi. 68) may be used. The proof upon which the
method rests is as follows : —
Let B be a benefit of any nature dependent on the life (.r),
and expressed by vp 1 + v 2 p 2 + v 3 p B + ■ ■ ■ , where p v p v p v etc.,
are the probabilities of a payment being made at the end of the
first, second, third, etc., years.
™ d -r, d r. dv
Then -^ B = — R-. x -p
di x dv x di
= ( ft + 2e ft + 3«» ft +. • • )(-« 2 )
= -(v' 2 p 1 + 2v 3 p 2 + 3v i p B + ■ ■ ■ )
Therefore _(1 +i) * B, = ip l + 2v% + 3v%+ ■ . ■
= ( IB X
298 ACTUARIAL THEORY [chap. xvi.
where (IB) is a benefit dependent on the same probabilities
as B , but increasing by 1 per annum throughout.
d AB -JA*B
But -7T B = — ^__ " approximately.
AB -|A2B
Hence (IB) = -(1+i) ^— —. - approximately.
This formula is perfectly general and applicable to any type
of benefit which increases uniformly.
For example, let it be required to find the value of (I A)
by the Text Book Table at 3£ per cent.
C lA ) 4 5 = ~ 1()A5 ^05
= - 207 {(-42692 - -46889) - | (-39003 - -85384 + -46889)}
= 207(-04197 + -00254)
= 9-21357
4. The difficulties and dangers attending the practice (recom-
mended in Text Book, Article 27) of omitting the denominator in
writing benefit and payment sides are such as to outweigh any
slight saving of trouble.
Theoretically, such expressions as(M - M ), (N , -N ),
etc., have no meaning as they stand (vide Text Book, Chapter VII.,
Article 9), and in practical use they will have different senses
according to the particular denominator used with them. There-
fore until the proper denominator is fixed the proper sense cannot
be ascertained. It is only after supplying denominators to both
that benefit and payment sides can be examined and compared to
check their accuracy. Further, where a second life comes into
the problem, and the denominator is omitted everywhere, the fact
may be overlooked that, e.g., D is the denominator for part of the
problem and D for the remainder, and thereby serious error may
be introduced.
6. To find the value of the temporary benefits mentioned in
Text Book, Article 46, we must stop at the nth value of u, that
is, u , which is equal to
(n- l)(»-2) .„
chap, xvi.] TEXT BOOK— PART II. 299
Then we shall have
(™U = ^[( N ,- N ,+>o+{^ +1 -S I+7l -(»-l)N i+TO }A Mo
+ { ». + , - *.+. ~ (• " ^ - ^^ N + „} A %
C vA )^ = f[( M «- M ^ + >o + { R , + i- R x + .-(»- 1 ) M , + 4 A "o
+ {SR ,-2R -(»-2)R x -^"^""^M U»«
To obtain (y—,a) x and (v^jA)^ we need only omit from these
formulas the terms which cut off the whole benefit at the end of
n years, but retain the terms which cut off the increase merely.
Thus
(V). = ^[ N . B o+( S , + i- S . + J A "o
X
+ {2S ^ - 2S . - (n - 2)S ^ } A*w„
l e+2 x+n v. y a:+«<
♦ ...]
(v-^A) = irMi«„ + (R , -R , )A«„
\ »| •'» D L x ° ^H -1 as+n'
+ {2R ^„ - 2R ^ - ( H - 2)R , } AV
I x+2 ai+m >. J x+n 1
*■■•]
6. The warning contained in Text Book, Article 98, should be
carefully noted, as the error presents itself in different forms. For
example, if it is desired to have a table giving the annual premiums
for pure endowments, one-half of the premiums to be returnable in
the event of death before the expiry of the term, it is incorrect to
take the arithmetic mean of the premiums for pure endowments
with and without return respectively. The correct office premium
for the new benefit is P(l + k) + c where
D ^ + Ac(R - R ^ -»M _,_ )
N
x+n * ^ x x+n x+n*
x-1 x+n-l *>• ' J\ x x+n x+n J
300 ACTUARIAL THEORY [chap. xvi.
while the proposed office premium is ir(l +k) + c where
1 c D ^ D ^ +c(R - R L - nM , ) -i
__ _±_ I X+n , x+n V- SC x+n X+n' I
2 IN ,-N _, , N ,-N _,. -(1+k)(R -E -»M,)
"- x-1 x+n-1 x-1 x+n-1 ^ ' /\ x x+n x+n'J
and these two are not equal.
The explanation is that if (x) were to die within the n years,
having taken out a policy at this proposed premium, the office will
return only one-half of the premiums paid ; but if, on the other
hand, he had effected two policies, one with and the other without
return, each for one-half of the sum assured, the office would have
to return the whole premiums under the former policy which
obviously are greater than the mean of the premiums under both
policies. In accepting the contract at the proposed premium the
assured is therefore allowing himself to be overcharged.
7. We proceed to discuss some practical problems not dealt
with in the Text Book.
It sometimes happens that (x) for some reason will not be
accepted by an office at the normal premium for his then age.
He, however, refuses to pay the premium for an older age which
they wish to charge him, but consents to his policy at the normal
premium bearing the condition that the sum assured will be paid
under deduction of a certain sum in the event of his dying within
t years and in full on death thereafter, t being usually fixed at the
expectation of life of (x). It is required to obtain a formula to
determine the amount of this " Contingent Debt."
First, let us assume the debt to be constant during the t years,
and equal to X.
Taking the life at his assumed or rated-up age we see that the
value of the premiums which the office should receive is
x+n^- x+n'
But they are to receive only P (1 + a ). Therefore they
lose premiums to the value of
( P x - p )( 1 + «x )
v x+n x'^ x+n'
Now the present value of the debt is
XA-L -
x + n : t\
Therefore equating and solving for X we have
X =
( P x -P)(l+« , )
^ x+n x/\- x+n'
x+n: 1 1
CHAP. XVI.J
TEXT BOOK— PART II.
301
Again, assume the debt to commence at fX. and decrease by
X each year till it disappears at the end of t years.
As before, the value of the premiums which the office forgoes is
(P ^ -P)N
(P _, -P)(l + a , ) = - x+n - *' x+n ~ 1
D
x+n
The present value of the debt is now
1 ^
x+n
fSM . -X(R ^ ^ -R ^ ±(i1 )
D
2+71
Equating and solving we have
(P _, - P )N ^ ,
v x+n x J x+n -1
x
tM
x+n
( R ;
x+n+l
R
x+n+t+l
)
In this investigation the damaged life is assumed to be a
normal life aged (x + n), and the extra rates of mortality for
successive years are accordingly as follows : —
Year of
Duration.
Extra rate of Mortality.
1
2
3
etc.
*x+n "x
Vx+n+\~1x+\
1x+n+i ~ 1x+1
etc.
It will be found that this extra mortality is at first small and
slowly increasing, but becomes great in the later years of duration ;
and this is a comparatively uncommon form of extra risk. Also
from the nature of the contract, the contingent-debt scheme should
be specially applicable to the case where the extra risk is at first
large and afterwards decreasing. Accordingly, the method of fixing
the amount of the debt is open to criticism in these respects.
8. We here give the methods of ascertaining the premiums
when they are loaded for bonuses in addition to the sum assured
of 1.
(a). Uniform Reversionary Bonus. — The problem is to find
the annual premium for an assurance of 1, to increase by 56 every
5 years, with an interim bonus of 6' in respect of each premium
302 ACTUARIAL THEORY [chap. xvi.
paid since the commencement or since the last increase, in the
event of death within any quinquennial period.
M IVf +M ,,„+ • • •
Benefit side = _5 + 56 x+i ?±i°
/ X V flj + 5 £ + 10
Payment side = ir
+ h D
X
N i
3J-1
D
X
whence we may obtain ir.
If no interim bonus is to be given for the first quinquennium,
the benefit side becomes
M * , „ M , + 5+ M , + 10+--- . ,, R, + 5- 5 ( M , + 10+M, +16 +---)
— + 56 -^ + 6 —
XX X
while the payment side remains
77 -D-
X
and the new value of 7r may be found.
If V = 6 the benefit side in the first case becomes
M R
— - + 6— i
D + D
and the payment side being
N
D
X
x-l
While if 6'= b, and no interim bonus be given for the first
quinquennium, the benefit side becomes
M 5M ^ + R ,.
x , i x+b x+&
D + D
X X
N
Payment side = ir
x-\
D
M+6(5M + R ^)
and in this case w = — 5 5±* !±5l
X-l
chap, xvi.] TEXT BOOK— PART II. 303
For an endowment assurance with a similar bonus, we have
M - M , +D /M - M + D
Benefit side = — - £±» £±? + 56 ( *+'■> *+ n *+"
, M * + 10- M s + ,+ D * + , t ,
+ g +
M , -M , +D M -M +D v
■ x+n-b x+n x+n , x+n x+n x+n\
X X '
(R -R ^)-5(M ,+M ,,„+... +M )
, iA x x+n' \ x+b z+10 x+n'
D
M . - M,^, + D ,^ 5 ( M ^,+ M „„+ ••• +M L ) + h(D , -M.)
_ % x+n x+n . i >■ x+5 x+W x+n' \ x+n x+n'
D "*" D
X X
(R -R _,_ )-5(M LB +M ,,„+... +M )
, 1 1 v x x+n' V »+5 s+10 x+n'
x
Payment side = tt x " 1 x+m ' 1
and 7r may at once be obtained.
If no interim bonus is to be given for the first five years, the
last term on the benefit side becomes
(R ^ K -R ^ )-5(M ^,„+M , + ... +M _,_ )
ii ^ x+b x+n' v x+10 x+lb x+n'
x
If b' = b, the benefit side becomes in the first case
M - M , +D ^ R -R ^ -bM ^ +«D ^
x x-\-n x+n . r x x+n x+n x+n
d + ■ D r
X x
and in the second case
M-M +D , 5M JK + R ^ S -R ^ -rM ^ +nT> _,_
x x+n x+n , i x+b x+b x+n x+n x+n
D + D
X x
Also, the form of the payment side remaining throughout
x-l x+n-1
* D
x
the various values of it may be deduced by equating and solving.
(b). Compound Reversionary Bonus. — To find the annual
premium for an assurance of 1, to increase by 5b per unit every
5 years, calculated on the sum assured and existing increases,
with an interim bonus of b' per unit in respect of each premium
paid since the commencement or since the last increase, also
304 ACTUARIAL THEORY [chap. xvi.
calculated on the sum assured and existing increases, in the
event of death within any quinquennial period.
Benefit side = ^-{(M,- M x+5 ) + (l + 5i)(M^ +5 - M, +w )
X
+ (1 + 56)»(M, +10 -M, +18 ) + . ■ • }
+ w{^'~ R , + 5- 8M , + 5) + ( 1 + B6 )C R « +B - R , +1 0" 5M , +1 0)
x v
+ (l + 5^(R 2+10 -R i+16 -5M x+16 ) + . . .}
N
Payment side = it *
X
and it may be obtained by equating and solving.
If no interim bonus is to be given for the first five years, the
first term of the second part of the benefit side will be omitted,
and it will then read —
W H - M - + 5> + P + B *XM, +6 " M, +10 )
+ (l + 5^(M i+10 -M x+1B )+. . . }
+ £{(l + M)(R„ + ,-R. +10 -BM B+10 )
+ (1 + 56M J _,.-R_ L1 -5M X J+. • • 1
lib' = b the benefit side in the first case becomes
2 r {M x + b(R x -R x+6 ) + b(l + 5b)(R +& -R +1Q ) + . . . }
X
and in the second case
i-{M^56M x+6 + 6(l + 56)(R^ 6 -R^ )
+ *(1 + 5^R +10 -R +1B ) + . • • }
The form of the payment side is constant, and therefore the
several values of tt for these benefits may be obtained.
To find the annual premium for an endowment assurance with
a similar bonus, we have
chap, xvi.] TEXT BOOK— PART II. 305
Benefit side
= lM( M s- M * +6 ) + ( 1 + 56 X M , +5 - M , +10 ) + (l + 56)s(M i+10 - M i+1B )+ ..
X
+ (1+56) ^ (M, +n _ 6 - M i+n ) + (l+56)T D>+i| }
+ £ {( R ,- R , +B - 5M J(+ a) + ( 1 + 5ft)(R +5 - V!o- 5M , +1 o)+ •••
a; '
+ (1+56) 6 (R^ ,-E -5MJ
v ' *- x+?l-5 .c+ji x+tt' J
N -N
Payment side = tt
x-l X+ll-1
D
whence we may obtain w.
If no interim bonus is to be given for the first five years the
benefit side becomes
]^{( M ,- M x + 5) + ( 1 + 56 )( M x + 5- M x +1 o)+- • •
n-5 n v
+ (1 + 56)~ (M, +n _ 6 - M x+ J + (1 + 5by Dx+n \
+ £{(l+56)(R i+6 -R +10 -5M i+10 )+. . .
X v
7t-5 -.
+ (1+56) 6 (R^ t -E -5MJ
1 V ' J \ x+n-5 x+n aj+tt'J
If b' = 6, the benefit side in the first case becomes
i-{M i + 6(R i -R, +6 ) + 6(l + 56)(R +6 -R +10 )+. . .
+ 6(l+56)^(R i+7i _ 5 - R + J + (l + 5iAD +h - M i+/[) }
and in the second case
^\M x+ 5bM x+ . j + b(l + 5b)(R x+5 -R x+l0 ) + . • .
x *■
+ 6(1 + 56)^(R^. 5 - R + „) + (l + 56)T(D, +|i - M„ +s) }
Then, the form of the payment side remaining unchanged, the
several values of ic may be obtained.
Where b' = b — - 01 it may be shown that the single premium
at 4 per cent, for an assurance of 1 with that compound rever-
U
306 ACTUARIAL THEORY [chap. xvi.
sionary bonus is approximately equal to the single premium at
3 per cent, for an assurance of 1. For ease in working, let it be
assumed that the bonus is compounded yearly. This will have the
effect of increasing the value of the benefit, which may then be
expressed
1 fl-01 , /1-01\ 2 , /1-OlX 3 , I
-rtroi d .HiW d »+ i+ wK+* + ' ' ' J
Now — .— = — --- approximately, and if we substitute this
1-04 1-03 ™ J
value for it we shall decrease the value of the benefit which will
now be
T|T03 ^ + (T03y 2 ^+i + (T03)3^+2 + ' ' ' )
= d"( C « + C ,+i + C :c+ 2 + - • • ) at 3 per cent.
X
The two approximations given effect to above act in opposite
directions, and will to some extent neutralise each other.
9. Under a scheme of Discounted-Bonus or Minimum-Premium
Policies the annual premiums are obtained by deducting from the
full profit premium the value of a certain rate of bonus.
(a). Cash Bonus. — At the several investigations cash bonuses
are usually declared as a percentage of the premiums received
since last investigation. On the assumption that investigations
are quinquennial, and that it is desired to apply a cash bonus of
100k per cent, of the premiums received in reduction of the full
profit premiums, the yearly reduction will be
D +D +D + • • .
£ x 5p- *+!> s+10^ z+15^
x n
x-l
Now, assuming that 5D „ = D , a + D ,, + D lt + D , + D
' & x+i x+3 x+i x+i 3+6 X+7)
etc., we have the yearly reduction equal to
ftP' D *+ 3 +D *+4+- • • +D *+7 +D s+8+- • •
N
x N ,
x-l
chap, xvi.] TEXT BOOK— PART II. 307
But again 5D *+* + 5D * +1 Q + 5D *+u + ' ' '
= a(i) = a -2
X X
D +D +D + . . .
Therefore Ax5F _*±$ x -±^ £±15
x-i
a -2
X a x +l
(6). Uniform Reversionary Bonus. — On the assumption that
it is desired to apply the value of a uniform reversionary bonus
of 5b to be declared every five years, the reduction in the annual
with-profit premium will be
^ x+5 3+10 rc+15 I
x~l
If an interim bonus at the same rate is also to be assumed the
reduction will be increased to
R
x-i
(c). Compound Reversionary Bonus. — -If we apply the value
of a similar compound bonus, the yearly reduction will be
5bM x+5 + 5b(l + 5b)M x+w + 5b(l + 5byM x+15+ . ■ ■
N *-r
Or including an interim bonus at the same rate, we have
KK- a + .)+frfl + 56XR +6 - R, +1 o) + &(i + 56) a (R +10 - R + J+ •
N ,
x-l
In any such system, if the bonus declared is greater than that
applied in reduction of the premium in accordance with any of the
above formulas, the excess is added to the sum assured. But if
the rate declared be less than the assumed rate, the difference
must be deducted from the sum assured, or else an increase must
be made in the premium payable, care being taken to ensure that
no option is permitted to the disadvantage of the office.
308 ACTUARIAL THEORY [chap. xvi.
10. To find the annual premium for a pure endowment payable
at age (x + n), the premiums received to be returned with simple
interest at ratej in the event of previous death, but the premium
to be calculated at rate i.
The difficulty here is in the return of the interest on the
premiums. In respect of the first premium paid, this return is of
the nature of an increasing assurance commencing at one year's
interest, and increasing by the same amount per annum. The
value of this therefore is
R - R , -«M ,
- ' x x+n x+n
J D
X
In respect of the second premium the return is of the nature of
a similar assurance deferred one year, and its value is
R - R^ -(ra-l)M _,_
/ x+1 x+n V- J x+n
J D
X
and so on for each premium, the value of the return of interest in
respect of the last being
R - R - M _,
• / x+n-l x+n x+n
J D
X
The value of the return of interest in respect of all the premiums
is the summation of these n expressions, and is equal to
2R-2R A -,R - ^- } M x
x x+n x+n 2 x+n
j - D
Therefore the benefit side
D ___ R - R , - «M
+ j[* (!+«) + <-'}
SR-SR -nil _^+l) M +
x x+n x+n 2 x + n
Payment side = w
D
X
N -N
X-\ X + 1h-\
D
chap, xvi.] TEXT BOOK— PART II. 309
Equating and solving, we have
= fD . +c(R -R-nM _,_ )+;c(SR -2R -wR _ w ( w + 2 ) M \~|
|_ z+« >■ a; x+n x+n' J \ x x+n x+n 2 *+» f I
-5-[ N .-i- N , + .-i-( 1 +0(H,-R, + .-»M, +)i )
-i(l+ K )J2R -211 -»R X _^±1) M \1
and •"•' = 7t(1+k) + c.
11. To find the annual premium for a similar benefit, with the
exception that the premiums are to be returned with compound
interest at rate j.
Here we have the benefit side
D (1 + ;)' - 1 d , , ,
= ^+Mi+K)+ C }2r^(i+/)^^ — ?f±
= _±Hi + {^(i +K ) + c }i+i(A'i--Ai n )
]) ' I V ' / J V V. sen I ami- 7
where A' 1 -; is calculated at rate J, which is such that
xn\ 3
l + J " l + j
t» i. *j„ »-l sc+m-1
Payment side — 7r — — '
D + c-^D(A' 1 -;-AU'
Hence •"■ =
N _N -(\ + k) ^ D(A'L-AL)
x+n-1 ^ 1
and tt' = 7r(l + k) + c.
Alternatively, we have the benefit side
D M - M M , -M ,
a; * ^
M , - M , •>
+ ( l +i7 -)»_-HLg ^}
a;
- %- + M 1 + K ) + c i w K 1 +i) M , + (1 +i) 2 M, +1 + • • •
I x
+(W) slM * +M - 1 -( 1 +i>si W ) M * + J
310 ACTUARIAL THEORY [chap. xvi.
The payment side being as above, we have
* = [D.+.+ c {( 1 +i) M .+( 1 +i) SM . + i+- • • +( 1+ i) ,lM , + ,-i
-CI +i)^ 0) M a+ J]-[N !r _ 1 - N^^- (1 + k){(1 +i)M ;c + (l +i)^ +1
and t — ir(l + k) + c.
12. The annual premium for a deferred annuity with a similar
condition as to return of premiums may be found by substituting
N for D in the above formulas.
x+n x+n
If in this last problem we assume that the net premiums are
returnable, we shall obtain
- = N, + , + [N._ 1 -N ji+|i . 1 -{(l +i ,-)M. + (l+i)»M^ 1
+ . . . + (1 +jy>M a+% _ 1 - (1 +j)s- m M x+n }]
If, further, we put j = i, the portion of the denominator within
brackets becomes
(l+0M a + (l+0*M s+1 + ■•■ +(l+^M x+a _ 1 -(l+0^ ) M, +n
= (l+OC^ 1 ^ +«•+«.+! +«*+«.+,+ ■ • • +«" +,+y .+.+- • ■ )
+(i +ty(v*+*d x+l +v*+*d x+2 + . . . +v*+-+id x+n +. . . )
+ • • • +(1 +i) n (?> X+n
N ,
D _,_ (l+0*-i
chap, xvi.] TEXT BOOK— PART II. 311
which is the annual premium under a leasehold assurance to
provide a at the end of n years certain.
This result is correct ; for, since the office has to return to those
who die within the n years their contributions along with com-
pound interest at the rate assumed in their calculations, it will
derive no benefit from those who so die, and therefore mortality
must be left out of account so far as these years are concerned.
13. To find the annual premium limited to t years and
returnable with simple interest at rate j for a pure endowment
with return.
Here there will be t expressions for the return of interest to
sum.
D , R -R -tM
Benefit side = -j±Z + {,r(l + K ) + c} Jl it* 5±?
X 3!
2R-2R -*R J( 2w -' + 1 ) M
x x+t x+n 2 *+*■
+^{^(l + «) + 4 g
Payment side = it x ~ 1 x+t ~ 1
X
■k may be found by equating the two sides and solving, and
hence also x'.
14. To find the annual premium for a whole-life assurance of 1
deferred n years, premiums to be returnable in the event of death
within the n years.
M , R -R , -»M ,
Benefit side = -Jtt* + {^(1 + «) + c} -• ?+^ *±*
X X
N
Payment side = tt *~
X
M , +c(R -R ^ -»M ^ )
Whence ir = 5±» — — — —
Wnence ir _ . _ M )
and it' = ir(l +k) + c.
Mr Stirling gives (/. /. v4., xxxi. 259) a simple practical
formula for obtaining this annual premium from a table of annual
premiums for pure endowments with return.
312 ACTUARIAL THEORY [chap. xvi.
The argument is as follows : — At the end of the n years the
premium for the assurance at the then age would be P , but
r rt x+n'
the office is to continue receiving only the premium -k ; therefore
at that time it must have in hand to meet the shortage in future
premiums a sum of
(P ^ -7r)a^
^ x+n ' x+n
■k must therefore also be the premium for a pure endowment, with
return, of this amount, or (P , -ir")a , xRP^ where RP —. is
v x+% ' x+n xn\ xn\
the annual premium for a pure endowment of 1, with return.
That is
7T = (P ^ -,r)a ^ xRPi
*- x+n ' x+n xn\
^ xn\ '
" x+n x+n
' xn\
p ^ a _,
x+n x+n
+ a„
xn\
Taking net premiums throughout and substituting for RP —
its value as found in Text Book formula (31), we get
M ^
_ x+n
N -(R -R , -«M )
a:-l v x x+n x+n'
which is the annual premium for a deferred assurance with return
of the net premium, agreeing with our first formula above if k and
c are zero.
Again, taking the loading as a percentage on the premium
only, that is 7t' = it(1 + k) and c = 0, and making the necessary
modifications on the value of RP —. as found in Text Book formula
xn\
(47), we have by Stirling's formula
M ^
- _ x + n
N , -(1+k)(R -R , -nM , )
x-1 ^ ' ^ x x+n x+n,'
which is the annual premium for a deferred assurance with return
of the office premium where the office premium is loaded with a per-
centage on the net premium only, agreeing with our first formula
ifc=0.
chap. xvi.J TEXT BOOK— PART II. 313
Now, taking ir = 7r(l +k) + c, and giving to RP - 1 - its value as
found in Text Book formula (47), we have by Stirling's formula
A L
v _ *+»
N x . t - N I+ , _ ! - (1 + k)(R x - R x+n - nM x+n )
V x+1l +c(R x -R x+n -nM x+n ) ' + %+ n
But by our first formula
M ^ +c(R -R , -nM , )
N -(1+ K )(R -R , -wM , )
*-l v J\ x x+n x+n'
and these two formulas are not identical.
The reason for the divergency will be found on examining the
formula
» = (P, -7r)a , xRPi,
v x+n J x+n xn\
Under the circumstances now being considered RP — , is loaded
to provide for the return of RP-^(1+k) + c. According to the
argument by which this formula was derived the office premium
which should therefore be returned is
^ x+n ' x+ 71 l xny- ' >
But the office premium which actually is to be returned is
(P , -7r)a , RPi(l+«) + c
^ x+n J x+n amp
and these are obviously not equal.
Mr Stirling, however, put forward the formula merely as a
useful method of obtaining the office premium for the deferred
assurance, the premium P also having to be considered a
premium with some loading. Its usefulness is considerable, for
the numerator is constant for assurances commencing at age (x + n).
The process is to add to a the reciprocal of the office premium
for a pure endowment with return, and divide P„,_a„.„ by the
result.
1 he formula is easily modified to apply to endowment assurance
and limited-payment policies.
For the endowment assurance payable at age (x + n + m), or at
death between age (x + ri) and that age, with return of premiums
if death occurs before age (x + ri), we have
P — a —
s+7i:m| x+n'.m\
V = - 1
+ a
RP
1 x+n:m\
314 ACTUARIAL THEORY [chap. xvi.
For a policy under which premiums are to cease to be payable
after age (x + n + in - 1), i.e. after (n 4- m) payments, we have
p ^ a^ -,
i
T>p 1 x+n:m\
xn\
15. To find the purchase-price of a life annuity of 1 to (x),
subject to the condition that should (x) die before he has received
in annuity payments the whole of the purchase-price the balance is
to be paid to his estate.
Let W be the purchase-price. Then we have
Benefit side = ^ + * '£ * +W+1
X X
Payment side = W
Equating and solving, we have
W =
N tt ( R 3+l R g + W + l)
D - M
Since W is still involved in the right-hand side of the equation
it will be necessary to make an approximation to its value in the
first place. The right-hand side on being worked out should then
agree with the assumed value of W. After two such approxima-
tions the true value might be found by interpolation.
This method of obtaining W is not quite correct, inasmuch as
W is usually an integer plus a fraction. But as Mr Manly, the
author of the formula, points out, the correction for the value of
the assurance of this fraction of the annuity is so insignificant
that it might be ignored.
These remarks apply also to the two following problems : —
To find the single premium for an annuity with a similar
condition but deferred n years, the net premium also being
returnable in the event of death within n years.
Let W be the purchase-price of the annuity.
N W M - (R - R 1
Benefit side = * +n + * V x+n + 1 «+»+w+i'
X X
Payment side = W
Hence we have W = *+" ^ x + m+1 x+*.+v+i>
D - M
chap, xvi.] TEXT BOOK— PART II. 315
To find the annual premium for a similar annuity to the last,
with the condition that all net premiums paid are to be returnable
in the event of death within the n years.
N tt(R - R )-(R - R \
Benefit side = '+» + — - x+n x+n+1 *+"+""• +i j
X X
N -N
Payment side = ■* !
Hence "
D
X
N — ( R — R ^
x+n \ x+n+i 3+«.+tt7r+l/
N -N , _(R - R )
£-1 x+n-1 *- x x+n'
16. To find the annual premium for a pure endowment payable
at age (x + n); the premiums to be payable only so long as another
life aged y is alive jointly with (x), and to be returnable if (.t)
should die within the n years.
The value of the return in question was discussed on page 129,
and making use of the result there given we have here
Benefit side
D ^, /M - M , M _,_, - M ^
= p + - + Mi + ")+c}( * D x+n + * +1 D x+n Py +---
X XX
M , ,- M
i- M _, \
*ri 5±? )
N -N
Payment side = tt s- 1 ^- 1 »+»-i = »+»-i
Hence we may obtain ir. Also it' = ir(l+ k) + c.
17. To find the annual premium for a similar benefit, but the
return of premiums to be with simple interest at ratej.
Following the method of the solution on page 308, we have
Benefit side
D L /M -M , M - M ,
= -^ + Ml +*) + *}(— '-g^'- + X+1 D x+n Py +'~
X XX
M . -M
, x+n-1 x+n \
" r Q n-lfyj
x
/R-R, -«M L R^-R^-^THlVI
+J>(l+«) + 4( J ^ — + ~ £± B —Py +
X
-if)
R - R - M ,
+ D
316 ACTUARIAL THEORY [chap. xvr.
N -N
Payment side = it *-i:?/-i *+»-! :.,+.»- 1
xij
Equating and solving, it, and hence it', may be found.
18. To find the annual premium for a similar benefit, but the
premiums to be returnable with compound interest at rate j in
the event of {x)'s death within n years.
Benefit side
= J ^iL +K l +K) + c} [{ (1+i) ^ +(1+i)2 ^i + ...
X XX
X *
X X X
+{(l+»^=- 1 }.-./'J
D f M' - M' , 1 M' - M'
a; s «/ x
i M' . - M' ■>
4. . X s+n- 1 x+n \
where D' . M' etc., are calculated at rate J, which is such that
1+J " 1+*
Payment side •= ?r «-i:»-i~ «+»-i:»+. ^
xy
whence tt and tt' may be found.
19. To find the annual premium for an assurance on the life of
(x) deferred n years ; the premium during that period to be payable
only if {jj) also is alive, and thereafter throughout (x)'s life, and
to be returnable should (x) die within the n years.
Benefit side
M _,_ /M - M , M-M
X xx
M , - M
, S+71 -I _x+n \
D n-l'y^
chap, xvi.] TEXT BOOK— PART II. 317
Payment side = iz C x ~ Uy ~ 1 ~ ' x + n - l: «+ n ~ 1 + T +"-»)
Equating these two sides we may solve for ir, and 7r' = 7r(l + K) + c.
20. To find the single premium for an annuity to the last survivor
of (x) and (j/) deferred n years, the premium to be returned in the
event of the annuity not being entered upon.
Let a' be the purchase-price. Then we have
Benefit side = i«- + «'xi A—
7i | xy \n xy
= ( \a + \a - \a )
\«| i w | y n] xy'
+ {o(1+k) + c}(i A +1 A -I A )
Payment side = a.
Hence equating and solving for a, we have
( \a + \u - \a ) + c(\ A +i A _i A )
Vti- | o: n \ y n \ xy-' ^]n x \ n y \n xy J
1-(1 + k)(i A +| A -I A )
\ J \\n x \n y \n xy'
and a! = a{\ + k) + c.
To find the annual premium for a similar benefit, all premiums
paid to be returned on the death of the survivor should that occur
within n years.
It has already been pointed out (page 151) that some difficulty
attends the fixing of the status during which the premium shall be
payable. We may consider both cases.
(a) If the joint lives be taken as the status, the benefit side
/•/M - M , JVI - M _,
= ^+Mi + «)+ C }{pV^ + " + d V----
*- x X X
M -M \ /M-M, M , ,- M ,
+ D~" " »-i'V V D D lx
"* ' N y v
1 " !/+ , l -l y+n \ xy x+%:y+v, __x+nly+n\
+ D »- i; V D J
y ' xy ■>
Payment side = tt(1 + a^._ i )
whence we may obtain tt and also it'.
318 ACTUARIAL THEORY [chap. xvi.
(b) If the premium be payable to the death of the last survivor,
we have
/R - R , -nM ,
Benefit side = i a- + {*■(! + k) + c}(-JL. *+ n *+«■
n\ xy I *• V D
, - nM , v
■ V+n x+n:y+n \
R _ R _ „M L R - R , -nM
, y y+n y+% _ xii x+% - ■■
D
v *y
Payment side = tt(1 +«^ :S rj[)
from which other values of 7r and ir' will be found.
21. In the problems connected with pure endowments with return
of premiums, the element of mortality is in practice frequently
ignored. This is in effect taking for granted that the life will
survive the term ; and if it does not, the office receives for its trouble
interest on the premiums which it has received and now has to repay.
Thus in the case of the annual premium for a child's pure
endowment to (,r) payable at the end of n years, with return of
premiums in the event of previous death, the net premium is
simply found from
v n
■K =
a— i
When the question is complicated by making the premium
payable only so long as the father (jy) shall survive (see page 315),
the net premium is taken as
v n
a —.
yn\
EXAMPLES
1. The sum of £s is deposited by each of / persons in a fund,
and accumulated at compound interest. £a is paid on the death
of each member, at the end of the year in which he dies, and at
the end of n years the amount remaining in the fund is applied to
the purchase of an annuity upon the life of each of the surviving
members. Find the amount of the annuity.
Let the amount of the annuity be p. Then the value of an
annuity of p to each of the survivors of / persons alive n years
ago is pi , a , .
CHAP. XVI
] TEXT BOOK— PART II. 319
But the accumulations of the fund are
Hence
*( t (l+0»-a{d > (l+0- 1 +d. +1 (l+»-)- g +---+rf, + .. 1 }
P
sD -a(M - M _,_ )
x ^ x x+n*
la,
x+n x+n
x+n
Alternatively, p being the amount of the annuity as before,
a(M - M , ) pN j_
Benefit side = V x x+nJ + C^±2
X X
Payment side = *
t? .• u ^ a; x+n' , * x+n
Equating^ we have =- - h — p- 3 — = s
X X
r x+n x *• x x+n'
~D D
X X
*D - «(M - M ^ )
= _^ V, x_+nJ as before
N ,
x+n
2. If / persons each secure by annual premium an endowment,
show that the amount which will be payable at maturity to the
survivors consists of the accumulated premiums paid by the
survivors and by those who die.
The annual premium is ~P x ±- = = _ ft*
s-l x+n-1
and its accumulations to the end of n years amount to
xn\\ v x+n J
N -N
P _L !S-1 x+n-1
~ xn\ v x+n
D N ,-N ^ ,
x+n x-1 x+n-1
N -N, . . U*+»
a;-l ai+n-l
x+n
which is the amount payable at maturity, being 1 for each of
the / , survivors.
x+n
320 ACTUARIAL THEORY [chap. xvi.
P_
3. For what benefit is — ™J the single premium? Explain
the formula verbally.
This is the single premium for an endowment assurance of 1
with the net premium returnable, since the value of such a
benefit is
A ~. + A xA n
xn \ xn\
the payment for it being A.
Y-,
Hence A = - — ^J-
1-A-r
xn\
dA-,
for P - = p-
xn \ 1 - A _,
xn\
Now -^p is the value of a perpetuity-due of P -,. But P -, will
insure the payment of 1 at the failure of the joint status of (.r)
and n | ; and after that, a fresh status of a similar kind being set
up and the payments of premium continuing under the perpetuity,
payment of 1 will be made on the failure of the second status ;
and so on indefinitely. And this is the benefit asked for, since on
payment of the endowment assurance 1 may be taken, and there
remains A to set up a second similar contract, and so on indefinitely.
4. " Suppose the annual premiums to increase or decrease a
certain sum every t years, and at the end of v intervals of / years
each the premium to continue constant during the remainder of
life, what annual premium should be required during the first t
years " ? Jones gives as the answer to this question
p M E + g( N E +( - 1 + N tt + 2 ( -l+- • • + N S + , ( -l)
X-l
while Chisholm, correcting him, gives
M
N s-l±'?( N * +J -l + N W 1 + - ■ • + N x + vt-l)
State the different conditions under which both answers are
correct.
chap, xvi.] TEXT BOOK— PART II. 321
This problem is discussed in Text Book, Articles 28, 29, and
35 of this chapter, and from what is shown there it will be
observed that Jones's solution proceeds on the assumption that
the premium increases or decreases by q per unit of the sum
assured, while Chisholm assumes the increase or decrease to be
q per unit of the premium. Each answer in its own case is correct,
the question being stated ambiguously.
5. Find the annual premium for an annuity to (x) after death
of (jy), all premiums paid except the first to be returned in the
event of (x) dying before (^).
Benefit side = a y | x + {it (1 + *) + c) «+ l! »+i
Payment side = ir (1 + a )
a _„ +c ^HlSI
x xy £)
AndTr = 2
i +a -n+K) x +y- y+i
xy v ' D
xy
6. Deduce a formula for the annual premium for an assurance
on the life of (x) against (jf) for n years, with return of all
premiums paid should (%/) predecease (x).
Benefit side
M 1 - M-4 R 1 - R-.-i--»M L.
xy
x+n:y+n + < /j , \ , £ i xy x+n:y+n x+n:y+n
xy
Payment side = ir
xy xy
N -N
x - 1 : y - 1 x+n~l:y+n-l
D
xy
whence
(Mi - M-L- _ -) + c(U l - R-r- i-»M- JL)
V. xy x+n. y-\-n J >• xy x+n:y+n x+niy +n'
(N n ,-NT , ^ )_(l +K )(Ri-R i _„M 1-)
^ x-l:y-\ x+n-\:y+n-l J <■ ' K xy x+n:y+n x+w.y+n-l
7. Find the annual premium required to secure to (x) a pure
endowment of 1 payable at the end of 20 years, with return of two-
thirds of the premiums in the event of death within the second
half of the period.
X
322 ACTUARIAL THEORY [chap. xvi.
Benefit side
= D g+20 + 2I n , ,A , c ] 10M ^+10 + R s+10~ R a;+20~ 20M a; +20
X X
N -N
Payment side = ir
x~l J+19
D
X
whence
N ,-!- N +19 -|(1 + k)(10M i+10+ R^ +10 - R, +20 -20M i+20 )
8. Find the annual premium limited to t payments for a whole-
life assurance to (x) subject to the condition that interest for each
year on the net premiums, up to and including the year of death,
is to be allowed by the office at rate i, which is the rate realised
by the office on its investments.
Benefit side
M
= — - +
D
X
i^ X
D
X
R +( )
N
,-N
Payment side
- v "
-1 X
D
+(-1
M
X
(N
-iS
C-l X
-ay-
- (N _,, . - jS
n X+t-1 3.
+ ( X
+s )
M
X
R
X
- R x+t
since N . -
x-l
iS -
X
-iR =
X
vS ,
x-1
R
X
S - iS - w'S
-S
X
-1 + «' S .
and similarly
N
X-
H-l~ iS :
-iR = R
c+( x+t x+t
A proof of this formula by general reasoning similar to that of
Text Book, Article 66, may be given.
The office can gain nothing from accumulation of interest on
the premiums, and therefore the payment side is the value of a
benefit of tr payable at the end of the first year if death occur in
the first year, 2ir payable at the end of the second year if death
occur in the second year, and so on, increasing up to the tt\x year,
after which the increase ceases, and the benefit remains at tir till
chap, xvi.] TEXT BOOK— PART II. 323
the year of death. The benefit side is simply the present value
of 1 payable at the end of the year of death.
9. Find the annual premium for an endowment assurance to
(x) payable at age Qc + t) or previous death, subject to the condition
that interest for each year on the net premiums, up to and
including the 2 th year or the year of previous death, is to be
allowed by the office at rate i, which is the rate realised by the
office on its investments.
Benefit side
_ M s~ M r +t + D +t , ■ f S,-S, +t -*N. +f R-R -M. +t )
D +t7r \ D + D 1
N ! - N _,, ■,
Payment side = *■- x ~^ x+t ~^
D
whence
M - M ,, + D ,,
X x+t x+t
( N ,-i " iS x ~ iR J - ( N , +( -! " iS x+t ~ iR x+t) + ^x+t + M x+t~)
M - M + D _,_,
X X+t X+t
R -R ^ t -tM ^, + tD ^
X x+t x+t x+t
since as before N . - iS - iR = R
a:-l a; a; a;
and N , , , - iS , t -iR , , = R
x+t-l x+t x+t x+t
and since i(N ,,+ M , ,) = iN ,, + viN , , ,-JN , .
*• x+t x+t' x+t x+t-1 x+t
= (1-»)N ^ n
= D ^ + N L , - «N _,_, ,
a+t a;+( a;+(-l
= D -M ,,
i+( a:+(
A proof of this by general reasoning may also be given. The
value of the payment side is as before that of an increasing
assurance of ir, 2tt, etc., up to tir in the 2th year. But in this
case the benefit ceases then entirely, tir being receivable also if
the life completes that year. The benefit side is simply the
present value of 1, payable at the end of t years, or at the end of
the year of previous death.
10. Find the annual premium for a pure endowment payable at
age (x + 1) ; the premiums to be limited to n payments, and to be
returnable in the event of death before age (x + i).
324 ACTUARIAL THEORY [chap. xvi.
D _,_, R - R -jiM ,,
Benefit side = -2±± + {tt(1 + k) + c} -i- ! '
Payment side = ir
D
X
N -N
x-1 sc+n-l
D
x
D ,, + cfR - R , -»M ,,)
^ x-1 x+n-1' v ' /V a ^4.71 3;+^
11. Obtain a formula for the office annual premium, P, required
for a policy on (x) for a term of n years, the assurance to cover
(1) an advance of £p made out of a trust fund at the beginning of
each year, (2) the premiums actually paid under the policy, and
(3) the legal expenses of the arrangement, say £a.
Benefit side
R - R , -«M R - E , -jiM
= p_J5 *+» !±S +{*(!+ «) + «.}_« E±5 !
x x
M - M
+ a - ' x+n
Payment side = it
D
X
x-I x+n-1
whence tt =
t
D
X
fp + cYR - R -«M , ) + a(M - M u )
(N - N )_(1 + K )(R _ R -»M )
and P = tt(1 + k) + c.
12. Obtain, in terms of commutation symbols and the rate of
interest, an expression for the annual premium for a deferred
annuity to be entered on at age 65 on a life now aged 40, the
premium to be returnable in case of death before 65, and the
annuity to be payable half-yearly, and to be complete.
Benefit side = ±- [ N(i5 + * D M + ±(1 + .>M B
40
+ {P(1 +K ) + C }(R 40 -R 65 -25M 65 )]
N„„-N
Payment side = P 89 64
40
chap xvi.] TEXT BOOK— PART II. 325
whence P =
N e5+i D 65 + Kl+O 4 M 66 + C (R 40 -R 66 -25M 66 )
N, 9 -N M -(1+K)(R 40 -R a6 -25M M )
and P' = P(1+k) + c.
13. Required the net single premium for an insurance upon the
life of (x) of £1000, increasing at compound interest during the
first 5 years at the rate of 4 per cent, per annum.
The single premium = 1^ ) {(1-04)C i + (1-04)2C ;c+1 + (1-04)3C i+2
14. An office proposes to grant endowment assurances under a
table of premiums reduced by anticipation of future bonus.
Investigate a formula for the annual reduction when the dis-
counted bonus is
(a) A quinquennial cash bonus of 1 per cent, per annum of the
premium.
(F) A simple reversionary bonus of £1 per cent, per annum
declared quinquennially and vesting when declared. It may be
assumed that the office does not allow interim bonuses.
(a) For a cash bonus the reduction is
■01 P' _ 5 ( D *+5 + D * + 10 +D * +1 5 + - • • +D X + J
xn\ N , -N , ,
as-1 x+n-1
(6) And for a simple reversionary bonus
5(M +M ,„+ M +. . . + M J - fra - 5)M , +raD
V. 1+5 x+W 2+15 x+a-6' \ ' x+n x+n
•01 X
N , - N , ,
x-l x+n~\
15. Obtain a formula for the office annual premium for an
endowment assurance policy on {x) to mature in 20 years, the
premium to be based on select tables, and to provide for a com-
pound reversionary bonus of p per cent, per annum declared
quinquennially, with interim bonuses at the same rate after the
first 5 years, and the loading to provide for an initial commis-
sion of k per cent, on the sum assured spread over the whole term,
a constant of / per cent, on the sum assured, and a percentage of
m on the gross premium.
326 ACTUARIAL THEORY [chap. xvi.
The problem to find the net premium for such an assurance
has been discussed in general terms on page 305. Here the
particular case is treated, and the office premium also deduced.
First, to find the net premium, we have benefit side
= IT Hi - M M+6 ) + ( l + Too") ( m W+6 " M M+M )
+ ( 1+ ^) 2 ( M M + 10- M M+ 15)
+ ( i+ iS>)H 1+1 5- M W+ 20)+( 1 +lS))X ]+2
+ Too \ 1 + Tooj ( r m+5 ~ r m+io ~~ 5 M w+io)
+ Too V 1 + Too; ( r m+k> - r m+i5 ~ 5 M [x]+is)
+ TOO V 1 + T00J ( R M+16 _ R M+20 - 5M M+2o)|
N - N
M M+20
Payment side — tr
whence
m T 100 [*i+6 T iooV 100; ^ Ws "w+io^
" [x] M+20 v. s
p / 5p\ 2 p ( 5p \ s
+ T00V 1 + TOoJ ( R M+io _ ^l+ie^ + TOOV 1 + TOoJ ( R M+i6" R w+2o)
+ ( 1+ T%)^ D M+20- M M + 20)}
Further, the office premium
_1 / & J_\
* ~ l_J?L^ + 100a W : 2 / 10( V
100
16. Find the annual premium for a deferred annuity-due to
(£), the first payment of which is to be made at age (x + n) with
premiums returnable in the event of death before that age. If
(x) survive the n years, the annuity is guaranteed for at least
t years whether he live so long or not.
chap, xvi.] TEXT BOOK— PART II. 327
Benefit side = -^(a- + ( ja, + J + {<1 + k) + c} R *~ *'+"" * M °+»
X u x
Payment side = wa. -,
■* xn\
Equating, , - °*+^ + .lVJ + c C R J - R .+. -* M ,+J
and ir' = ir(l + k) + c.
17. Find by the [M1 table at 3i per cent, the annual premium
at age at entry 30 for an endowment assurance of 1 for a term of
30 years with a uniform reversionary bonus of 1 per cent, on the
sum assured in respect of each year entered upon after the first.
Benefit side
. M [»] - M 6Q + D 60 , . Q1 R [3 q ]+1 ~ R eo - 2CjM eo + 29D 6 o
D D
[80] "[80]
N -N
Payment side = P [30 1 «?
[30]
M r 301 - M eo + D ao + - 01 ( R rsom " R eo " 29M eo + 29D J
N -N
[80] 60
(10267-77 - 4671-51 + 7432-6)+ -01(278665-44 - 58079-89 -
29x4671-51+29x7432-6)
614584-81650-3
13028-86 + 3006-5716
532933-7
16035-4316
532933-7
= -03009
18. Find by the O tM1 table at 3| per cent., for age at entry
40 with a term of 15 years, the annual premium for an endowment
assurance of 1, together with a compound reversionary bonus of
1 per cent, per annum, which is to compound every 5 years, with
an interim bonus at the same rate in the event of death during
the 5 years calculated on the sum assured and bonuses in force
at the beginning of the quinquennium.
328 ACTUARIAL THEORY [chap. xvi.
Benefit side = j^[(M M - M B + DJ + -01(R [40] - R [4()]+5 )
[40] *-
+ -01(l-05)(R [40]+5 - R 50 ) + -01(1-05)^(R 60 - RJ
-M 65 { (1-05)3 -1} + D M { (1-05)3-1}]
N -N
Payment side = P
whence
D [40]
P =
[( M W ] - M 5 B + D 65 )+- 01 (V r \ o:+5 )
'[40] ""55 u
+ -01(l-05)(R [40]+5 - R 50 ) + -01(l-05)s(R 60 - RJ
+ {(1-05)3 _1 } (D 56 -M 66 )]
= I / (8202-99- 5689-39 + 9958-2)
" 352204-126234-5 V" J
+ -01(191831-83 - 151953-13)
+ -01(1-05)(151953-13 - 115901-12)
+ -01(1-1025)(115901-12- 84508-96)
+ -157625(9958-2 - 5689-39)1
14268-1029
" "225969-5
= -06314
19. Given that the office single premiums for pure endowments
of £100 at a certain age, with and without return of premium in
event of previous death, are £70 and £65 respectively, find the
office single premium for the pure endowment with return of half
the premium in event of previous death.
As pointed out on page 299, it would be wrong to charge
£67, 10s., as that would suffice for the return of £35 (i.e., half of
£70) and not of £33, 15s. in event of previous death.
Instead, let S be the sum assured under a policy without
return, and (100 — S) the sum assured under a policy with return,
so that the premiums on the two policies shall be equal. That is
(100-S)x-7 = Sx-65
S = 51-852
and 100 -S = 48-148
chap, xvi.] TEXT BOOK— PART II. 329
The premium under each policy will be 33-704, and the premium
required under the joint policy will therefore be £67, 8s. 2d.
20. Find, by the use of the following office rates, viz. : —
P '4o:i^ = -° 9479 > P'i:i5i = -01475, and A' 60 = -55396,
the annual premiums for an assurance on a male life, aged 40
next birthday, of £1000, payable in the event of death within
10 years, with a return of all the premiums paid if he survive
that term : —
(a) If the return of premium is to be made at the end of 10
years, and (6) if it is not to be made till death.
Let P be the premium required.
Then(«) P = lOOOP'i^ + lOPxP'^1,
. „ 10°<>P'i:iol
1_10P' i.
40:]0|
Now P'. n .i| is not given, but may be taken
= P'«:i5r P 'i:»| = -09479- -01475 = -08004.
Though by this method there is insufficient loading on the pure
endowment portion of the premium, yet the fact that there is a
term assurance for a very much larger figure makes up for this
loss.
1000 x -01475
Jl LlLlLU'lt -»
1 - 10 x -08004
14-75
•1996
=
73-898
Again (6) P
=
lOOOP'^ + lOPxP'.
P
=
1000P 'A : i*
^OPWso
=
14-75
1 - -8004 x -55396
14-75
•55661
—
26-5
A'
40:10| 50
330 ACTUARIAL THEORY [chap. xvi.
If in these formulas we assume net premiums throughout, we
1000 A* - 1000 A i ,-,
^ W = - - 10A V ^ (6) - a -10^ ' Which
a 40:l0| *
X X
the expression " until death " being taken to imply a complete
annuity.
N , - N _,_„
Payment side = ir x ~ x x+9
X
And 7r may be found by equating the two sides.
25. Given values of a as follows : at 3 per cent., 19*895 ; at
3i per cent., 18*441 ; at 4 per cent., 17*155, find at 3 per cent,
the annual premium for an assurance on (x) of £100, the sum
assured to increase by £1 per annum for each year entered upon.
100A + (IA)
Here P = x — — ^
l + «.
To find (I A) we may make use of the general formula
established on page 297.
(W).- -(1 + V
CHAP. XVI.]
We have
TEXT BOOK— PART II.
333
Rate of
Interest.
"■x
A,
= l-d(l + o x )
A
A-
3
H
4
19-895
18-441
17-155
■39141
■34258
•30173
- -04883
- -04085
•00798
Then (IA^
dA
A A -AA2A
-(1+0-
p =
Ai
-.. O o -04883 + -00399
•005
10-881
39-141 + 10-881
20-895
= 2-394
26. If the office premium is 15 per cent, greater than the net
premium, find at 3^ per cent, the approximate annual office
premium for an endowment assurance on a life aged 40 to be
payable at age 60 or previous death, all the office premiums to be
returned without interest in the event of death before age 60.
Given A^.^ at 3 per cent. = -20413, at 3£ per cent. = -19385,
A 40-20i at 3 i P er cent. = -55338,
13-207.
at 4 per cent. = -18428 ; and
and (1 + | 19 %) at 3| per cent. =
The benefit side = A
40:20|
+ P(l-15)(IA) 4 i 0:2 ^.
By Lidstone's general formula for any benefit
(IB) = _(l+,-) dB
di
Hence (IA) 1
40 : 20|
dA 1 —
.(1 + 40:20 '
= -(1-035)
= -(1-035)
= 2-054.
di
VA2A 1 -
•J 40 : 20|
•005
■00957 - \ x -00071
•005
334 ACTUARIAL THEORY [chap xvi.
Therefore the benefit = -55338 + P(W5) 2054
= -55338 + Px 2-362.
The payment side = ?(} + a iQ .m)
= Px 13-207
Equating the two sides we get
Px 13-207 = -55338 + Px 2-362
-55338
10-845
= -05103
The office premium therefore = -05103 x 1-15 = -05868.
27. Use Lidstone's two approximate formulas to find by the
[M] table at 3^ per cent, the annual premium for a joint-life
endowment assurance on two lives aged 44 and 45 respectively,
which shall increase by 1 per annum, i.e. 1 to be payable in the
event of the first death taking place in the first year, 2 if it
happen in the second, and so on, 20 if it happen in the twentieth,
also 20 if both survive the twentieth year.
The two formulas are
and
p -i =
xyn\
P n + P-i-
xn\ yn\
P »l
( IA ) -r =
AA
-(1 + —
. -;-iA2A -f
xyn\ - xyn\
Ai
3%
*i%
4%
P _
44:20|
•04391
•04206
•04030
P _
45 : 20|
■04438
•08829
•04254
•08460
•04078
•08108
P_
20|
•03613
■03416
•03229
•05216 -05044 -04879
Enter annual-premium conversion tables inversely to find
«.„ „* ,-sr H-301 10-869 10-461
44 : 45 : 19 1
Enter single-premium conversion tables to find
A ^-.n " 64172 -59864 55919
A - -04308 - -03945
A2 -00363
chap, xvi.] TEXT BOOK— PART II. 335
inual premium
-(1-035)-
Hence approximately the annual premium
•03945 - -00182
•005
11-869
= -720
28. A father, aged 35 next birthday, has a child, aged 1 next
birthday. An assurance of £150 is to be paid on the child attain-
ing the age of 21, provided the father be then alive. In the event
of the father's death an annuity of £5 is to be paid annually until
the child attains age 21, with a payment of £100 at that date.
The premiums are to cease on the father's death, and to be
returned on that of the child before reaching age 21. Deduce
the formula for the annual premium.
Benefitside - 150 ^ + 5 x ^a^ + 100^(1 - w pJ
/M-M M-M M_M\
X K -r JT !\ £) T ^35 D "t- +19F35 J5 J
N -N
Payment side = -k -^ 5ii»
D 85 :1
By equating the two sides, jt may be found, and -k = ir(l + «) + c.
29. A debt is to be discharged by n equal annual payments
to include (a) principal, (b) interest, and (c) the premium for an
assurance to provide for the cessation of the annual payments and
the extinction of the debt, if the debtor aged x die before the
expiration of the n years. What should be the amount of the
annual payment ?
If the annual payments are made at the beginning of each of the
n years, we have the annual equal payment of principal and interest
XT
to repay a loan of K in that time, — ; and the annual premium to
insure the balance of the loan outstanding is
^-{(M - M , ,) + "- 2 (M - M ,)}
a _'^ x x+n-V *■ x x+n-v V x x+l-> >
*l
X-l X+71-1
Adding this to — we get the whole annual payment required.
336 ACTUARIAL THEORY [chap. xvi.
Alternatively, we may proceed thus. If the loan be repayable
by n equal instalments of — the amount payable under the
»l
policy if (x) die in the first year is — a ^TTi > ^ * n tne secon d
a — ;
"I
a , and so on. Therefore the whole benefit under the
a— »- 2 r
policy is
K ( c , a srn +C «+i a «^l + - ' • +C «+.-i"n \
" d ;
and the premium to be added to the equal annual instalment
ot — is
K
a
»|
(. C **^T\ +C x+I a ^2\ + - • • +C x+n-2 a T^)
X-l X+71-1
The two results are identical, for
= C.(l+t> + «) 2 + • • • + «"- 2 ) + C, +1 (l+i> + i> 2 + • ■ . +«"-»)
+ • • • + C V f , l -2
+ . . . + V n-2Q
<■ as ic+m-1' *- a; x+n-2- 1 ' V a *+l'
30. Find the annual premium required at age x to secure £100
per annum for six years commencing on a child's sixteenth birth-
day ; the premiums are to be limited to (16 - x), and in the event
of death before age 16, the whole premiums paid are to be
returned, while if it occur between ages 16 and 21 a proportion
of the premiums paid, decreasing in arithmetical progression from
five-sixths in the seventeenth year to one-sixth in the twenty-first,
is to be returned.
chap, xvi.] TEXT BOOK— PART II. 337
Benefit side
= ioo^^ + {< i + K ) +c} { R --^-^ 6 -^ M 18
X x
X
Payment side = rr - i — j=r
whence 7r may be found, and 7r' = 7r(l +k) + c.
A better method of carrying through the transaction would be
to have six policies each for £100 payable at ages 16, 17, 18, 19,
20, and 21 respectively, with return of premiums paid in the event
of previous death.
31. Find the single payment required to redeem future
premiums under a child's endowment with returnable premiums,
effected at age x, and payable at age (x + 1), which has been n years
in force.
If ir and it' are the net and office premiums for the original
benefit, and if we write A and A' for the net and office single
payments now required, we may take it that A must be equal to
the value of the future premiums now to be forgone less the
return in respect of these premiums, and plus the return in
respect of the single payment. That is
N ,-N ,, , R^ -R -(t- n )M ,,
• _ x+n-1 x+t-l _ i x+n x+t v ' x+t
D~ " D ^
x+n x+n
M -L. ~ M _L,
+ {A(l+/c') + c'}-^ *-+!
x+n
Hence
. V ,i:+»t-l x+t-V ' x+n x+t V ' r+t> V x+n x+t'
D , -(1+ K ')(M ^ - M _,_,)
x+n v 'V. x+n x+t'
and the single premium required
A' = A(l+/c') + c'.
338
ACTUARIAL THEORY
[chap. XVI.
32. If you were furnished with the office annual premiums
for temporary assurances of £100 for from one to fifteen years at
age x, and also with full tables of annuities, how would you arrive
at the annual premium for an assurance of £1000 decreasing by
one-fifteenth each year, and at a premium also decreasing by
one-fifteenth annually ?
Let the premium required commence at 15P and decrease by
P per annum.
Benefit side
= 1000x T V(P'i :ri + P'i^a K ^ + P^ n a^ + ... +P'^a^)
where P' 1 .^ etc., are the office annual premiums per unit.
Payment side = P(%._r\ + \:T\ + a ,:T\ + ' " ' +a *:l¥|)
and P may be obtained therefrom.
A more practical method of carrying through the assurance,
which, however, would not fulfil the condition that the premium
should decrease by an equal amount each year, would be to get
(x) to effect fifteen policies, as follows : —
Term.
Sum Assured.
Premium payable at
beginning of each
year during term.
]
I
1
I
2
5
5
10
i
10
i
10
1
10
1
00
5
oo
6
00
6
00
5
1000
15
1000
1 5
1000
15
1000 1
15 J
P'l _
a;:l|
P'l _
x:i\
P'l _
s:3|
P'l _
as:15|
The sum assured decreases in the required manner, but the
premium does not, though it too decreases in total, since at the
end of each year there is always one premium less to pay out of
the fifteen with which we started.
CHAPTER XVII
Successive Lives
EXAMPLES
1. Give an explanation of the expressions
(a) A A A
(6)(l+0*A s A v A f
(a) This represents the present value of 1 to be paid at the
end of the year of death of (z), who is to be nominated at the end
of the year of death of (if), who is to be nominated at the end of
the year of death of (x).
(b) This represents the present value of 1 to be paid at the
moment of death of (s), who is to be nominated at the moment
of death of (y), who is to be nominated at the moment of
death of (x).
2. Show that the single premium for an assurance, with return
of the premium along with the sum assured, is equal to the value
of all the future fines on successive lives, where the lives are to
be nominated all of the same age as that at present of the life in
possession. Explain verbally why this should be so.
The single premium for an assurance with return of premium
is found as follows : —
Benefit side = A + B x A
X X
Payment side = B
A
and B = ,_*-
1 - A
X
Again, the value of all future fines on successive lives of equal
ages is
A i + (A i )= + (A i )3 + . . . ad inf.
A
1-A.
340 ACTUARIAL THEORY [chap, xvii,
Now, under the assurance, we obtain 1 + B at the death of (x),
whereof 1 will pay the fine falling due at his death, and B will
set up a new policy of like nature on a life then aged x, which in
turn will pay the fine at the second death, and provide B for a
third policy, and so on, ad infinitum.
3. Find the value of an annuity during four successive lives —
the second, third, and fourth lives being nominated on the deaths
of the first, second, and third lives respectively.
a , ,. „ ■ ■ = a + A (l+a) + A A (l+a)+ A A A(l+a)
1-A 1-A 1-A 1-A
= _— !?-l + A -__?+A A —-JI+A A A , '
d ™ d w x d w x v d
1-A AAA
W X y Z _ I
d
4. A copyhold estate is held on two lives, each renewable at
the end of the year in which it drops, by a life aged 10, on payment
of a fine of £8. Assuming the two lives to be now aged 30 and 35
respectively, find the present value of all the fines in perpetuity at
3 per cent, interest.
The value of the fines payable on the succession of lives
starting from the life now aged 30 is
8 ( A 80+ A 30 X A 10+ A 30 XA 10 > t
N = A , prospectively
n'.t x x+n r r J
,P(N , -N ,, ,)-(M -M ^ )
= LA_*rl *+'- 1J V * £±±1 retrospectively
z+n
Joint-life assurance
V = A , , -P fl+ffl , ,) prospectively
n xy x+n:y-\-n xy^ x+n'.y+n' r r J
P CN - N ) - (M - M )
= *iA »-! = »-! x+n-l:y+n-lJ V sy x+n:y+n-> retrospectively
x+n:y+n
Leasehold assurance
, V ij = °'""- P M( 1+8 rnrm) Prospectively
= P-(l +«')*-. retrospectively
Further problems similar to these will occiir later, and will
then be disposed of,
chap, xvni.] TEXT BOOK— PART II. 345
2. The notation for policy-values in regard to select tables
follows the rules already laid down (see page 140).
If / < n, the period during which selection is assumed to have
effect,
V = A -Pa
* [x] [x]+t lxflxl+t
= 1 a M+'
[X]
If t > n
t [x] ~' x+t [xf'x+t
%+t
= 1-
%]
If the life presently aged Qv + 1) is assumed to be still select the
reserve value is
[x+t] ~ [x]%+t] = ~ ~g
[X]
3. A very simple proof of Text Book formula (3) is as follows : —
V +P = A , -Pa ^
n x x x+n x x+n
= (n x+n + VP x+n A x+n+1 ) - P x VPx+n( l + %+n+O
= *?,+. + v P x +n{ A x+n+i - V x( l + V«+l)}
= <%+n + Px+n X n+l V x)
Or, retrospectively,
P (N -N , ,)-(M -M , )
n x + x = g *
iC+7l
p ,P,-rU-( M ,- M , W i) , , ,+» d + 9 *+"
KV + V x «+i v ^
346 ACTUARIAL THEORY [chap. xvnr.
Similarly, we have for pure endowments
y 1_ I p ]_ _ J^ 1 _ p 1 a
n xt\ xt\ x+n:t-n\ xt\ x+n : i - a- 1[
V Px+n x+n+1 : i-»-l| ~~ xT\ V Px+n\ + fl ar+m+l :t-n-i\'
= v Px+-nS x+n+1 : t - n - 1 1 ~~ xT\ ^ + tt x+n+l : t-n-ifi
= V Px+n X m+1 rrtj
P i CN - N ")
s\ „ - tt 1 , t> 1 si P- a:— 1 x+n-1' . t> 1
Or, again, \^ + P^ = —I g + P lf|
x+n
pi(N -N )
, xt P- z-1 x+»'
= «P,+. x — L "d
x+n+1
x+n '* Tl+1 ' xt|
= «p r . . _ x _ . , V
4. By Te^ .Boo£ formula (3),
n x x \-*x+n -lx+ii n+\ x J
= via , +(1-0 ^ ) ,,V}
'■"rt+n V Jx+tiAj+1 x>
= via (I- V)+ V \
<-*x+n \ ra+1 x' n+1 x>
Therefore ( V + P )(1 + i) = q(l - V )+ V„,
vfl i ax^ / ^x+nr- n+1 ay n+1 x
From this we see that the reserve value at the beginning of the
year, and the premium then paid, both accumulated to the end of
the year, are equal to the reserve value at the end of the year,
together' with a contribution towards the claims payable.
Now, if the mortality actually experienced agree with that
assumed, the account will work out as follows for I , policies,
' x+n * ;
each for 1, existing at the commencement of the year : —
Claims payable . . . . . . = d, +n
Contribution towards claims payable
= I xq (1- ,,V) . = d , (1 - ,,V)
x+n *x±wS n+1 x' x+fiS n+1 x J
Reserve released . . = d , x t ,V
x+n n+1 a;
Together d x+n
so that the claims payable are exactly met.
Profit or loss from mortality in an Insurance Office therefore
depends on a comparison of the claims payable less the reserve
chap, xvm ] TEXT BOOK— PART II. 347
released, with the expected contribution towards claims payable.
If the claims, less the reserve, exceed the contribution, there is a
loss to the office, and vice versa.
Therefore mere comparison of the claims payable with the
accumulations of premiums received is no test of profit or loss from
mortality. It is impossible to view the contracts in this light,
since at the date of the claim the accumulations of premiums
received are not available, having been applied to pay current
claims and to increase the reserve (apart from payment of
expenses and distribution of profit). Further, however long the
life live, even beyond his expectation at date of entry, there is a
loss to the office in respect of his claim (unless indeed he live
beyond the limiting age of the mortality table used), since V is
the average reserve in hand at any time while the claim to be paid
is 1. Every life assured on the books has to contribute towards
the deficit, and the profit or loss on mortality, as already stated,
depends on how this deficit compares with the contribution.
In the case of an assurance under which no further premiums
are payable, we have
Here the reserve value accumulated for a year is equal to the
reserve value at the end of the year, together with a contribution
towards the claims payable.
In the case of an annuity, we have
= < V
according as
Now, f V + P ) is the reserve value immediately after payment
of the (n+ l)th premium and B+1 V. the reserve value immediately
before payment of the (n + 2)th premium. Therefore the former
is > = < the latter, according as interest on the reserve is
< = > the current mortality risk.
348 ACTUARIAL THEORY [chap. xvm.
It will be found that, unless the policy has been a considerable
number of years in force, ^V^ + PJi < q x+n (l -„ +1 V.), and there-
fore V + P > ,V . On the other hand, where the policy has
n x x n+1 x > r j
been a long period in force („V,+ PJf may exceed q x+ Jl - n+1 VJ,
and therefore V +P < ,,V , that is the reserve value of the
n x x n+1 a;
policy will increase in the course of the year.
In the case of a temporary assurance
Vi- +pi_ > = < V 1 -
according as
And it will be found (where t is not very great) that interest on
the reserve is insufficient to provide for the current mortality risk,
and that V 1 ^ + PV. is greater than „,,V\ r .. Indeed V l T alone
n xt\ xt\ r 7i+l xt\ n xt\
frequently exceeds V 1 - as may be seen from Text Book, page
319, Table C.
In the case of a pure endowment
according as
Vl + PI > = < VL
n xt\ xt\ ^n+1 xt\
C V xF] + p K r|) z < - > 9 x+n x « + i V xT|
But obviously
Ulfj + xf\ )* > %+n X n+1 a*]
since the latter is negative.
Therefore V i + P i, < ,,Vi showing that the reserve
n xt\ xt\ n+1 xty &
value in this class increases in the course of the year.
For the endowment assurance (the summation of the last two)
we have
V-+P-> = < V -
» a* | T xt\ n+1 xt\
according as
C V rfl +P -l)* < = > Ix+nV-n+lVxTi)
that is, according as the interest on the reserve is < = > the
current mortality risk.
CHAP. XVIII.]
TEXT BOOK— PART II.
349
6. As an illustration of how to ascertain profit or loss from
mortality, let us assume in connection with the example worked
out on page 316 of the Text Book that the actual mortality experi-
enced for the first five years was as shown below, and not according
to the Text Book table, and that the annual premium was -01873.
Age.
h
X
30
89685
655
31
89030
740
32
88290
700
33
87590
720
34
86870
750
35
86120
765
etc.
etc.
etc.
We have first to construct a table, thus :-
Year.
Premiums
received.
V
» 30
as per Text
Book, p. 316.
Survivors
at
end of year.
Fund required
at
end of year.
1
2
3
4
5
etc.
1679-800
1667-532
1653-672
1640-561
1627-075
etc.
•01168
•02364
•03590
•04847
•06133
etc.
89030
88290
87590
86870
86120
etc.
1039-870
2087-176
3144-481
4210-589
5281-740
etc.
Then we may find the profit or loss, as
follows :
Year.
1.
2.
3.
4.
5.
Fund at beginning of year .
Premiums received
Interest ....
Claims .....
Fund in hand at end of year
Fund required do.
Profit ( + ) or Loss (-).
1679-800
1039-870
1667-532
2087-176
1653-672
3144-481
1640-561
4210-589
1627-075
1679-800
50-394
2707-402
81-222
3740-848
112-225
4785-042
143-551
5837-664
175-130
1730-194
655
2788-624
740
3853-073
700
4928-593
720
6012-794
750
1075-194
1039-870
2048-624
2087-176
3153-073
3144-481
4208-593
4210-589
5262-794
5281-740
+ 35-324
-38-552
+ 8-592
-1-996
-18-946
350 ACTUARIAL THEORY [chap. xvih.
7. By Text Book formula (10)
A ^ -A
» x i_ a
X
This result may be proved by general reasoning.
Suppose (x) to enter into such a contract as that described in
Article 60 of Text Book, Chapter VII., under which the amount
payable after deduction of the single premium shall be 1. That
is, if B be the single premium the total sum assured is 1 + B, the
loan on the policy is B, and the interest payable in advance on
this loan is dB. Then B = A^l + B)
A
whence B = - — ~
1 -A
X
1
The sum assured is therefore
1-A
The single premium, which is also the amount of ^
the loan, is . . . • . = — ~
1 -A
X
The yearly interest payable in advance, which is
also the annual premium for the net amount pay- ja
able, is ....
1-A.
The net amount payable at death is the sum
1 A
assured, less the loan, that is = -r — - — ^-, or 1.
X X
Now, after n years the reserve required under
1 A ,
1 ;„ x+n
r^x s • rrs:
the above single-premium policy for
from which deduct the policy loan which the office A
must take credit for .....
1-A
A , -A
The difference is ..... -^± x -
1 -A
X
which is the reserve required for an annual-premium policy under
which the net amount payable is 1, and which has been n years in
force. That is,
A _,_ -A
n *
1-A
chap, xvih.] TEXT BOOK— PART II. 351
8. The proof of Teat Book, Articles 41 to 43, seems rather
laboured. The matter may be put more shortly.
P' -P'
y x+n x
» » P' +d
x+n
V
n x
. C- Kd
1 +
(P, + »+«0(i + O
Now, according as c > = < Kd, the expression
x+n
C — Kd
(P, + .+d)Ci +
is positive, zero, or negative ; and V < = > V .
r ' ' ° ' n x n x
Or, again, V = 5±1
P . -P
X
c + d
+ n ' l+K
Therefore V > = < V
7i x n x
p -p p , -p
x+n x ^ _ , x+n x
c + d " P.„+d
p _i x+n
"+ n l+K
, i c + d
that is, as d > = <
l+K
, d c
or as a - > = <
l+K l+K
or, finally, as Kd > — < c
9. Text Book, Article 48, is very important, and has a wide
bearing in a consideration of the effect that an increase in the
mortality has on policy-values.
One is apt to assume that merely because one mortality table
exhibits higher rates ot mortality than another, the former requires
352 ACTUARIAL THEORY [chap. xvm.
larger reserves than the latter. But it is impossible to argue so
fast ; for we see that, if in the expression
P + d
1-
P , +d
we increase both P and P , , we cannot tell whether the
X X+IV
whole expression is increased or diminished without more minute
examination.
The same point is seen on examination of the value of V
found retrospectively,
r-fc{ U 1 +«)"+W 1+ ^ 1+ • • • + WiP +'')}
x+n ^
-{d,(i+0- 1 +^ +1 (i+f)«-»+...+d, +1l . 1 }]
If the rate of mortality as a whole is increased, it is quite true
that P is increased and I . decreased, both tending to increase
x x+n ' °
V. but at the same time I ,,, I , „, etc., are decreased, and
n if x+1' x+V ' '
d. d , etc., proportionally increased, all tending to decrease
V , and the final result may quite well be a lower value for V .
Now, as pointed out in Text Book, Article 48, " If the increase
be proportionally greater at the younger ages, the policy-value
will be diminished, and if the increase be proportionally greater
at the older ages, the policy- value will be augmented." Also,
Dr T. B. Sprague, in his paper, " How does an increased mortality
affect policy-values " (/. I. A., xxi. 109), says : "It seems that we
may fairly draw the following conclusions : —
(1) If two tables show the same mortality at young ages and
at higher ages an increasing difference in the rate of mortality,
then the one which shows the higher rate of mortality will require
larger policy- values.
(2) If two tables show the same mortality at high ages, but an
increasing divergence as we proceed to younger ages, then the
table which shows the lower mortality at younger ages will require
larger policy-values.
(3) If two tables, A and B, show the same rate of mortality at
the middle ages, say about 50, but at younger ages the table A
shows the higher mortality and at higher ages the lower mortality,
then table A will require the lower policy-values."
Finally, Dr Sprague tells us that "policy- values do not at all
depend upon the absolute rate of mortality exhibited, but only
CHAP. XVIII.]
TEXT BOOK— PART II.
353
upon the progression that the rate of mortality exhibits. It is the
table in which the mortality increases the more rapidly, that
requires the larger policy-values."
10. As shown in Text Book, Article 49.
V > = < V
n x n x
according as
1 +a
X
l+o.
1+a'
> = <
1 +a
x+n
If, then, it be desired to compare the reserves at 3 per cent,
of the H M experience with those of the M , a table should be
worked out for each age as follows : —
O 1
policy-values.
Ratio of annuities-due, — g , at 3 per cent, for comparison of
H M a T
Age.
Ratio __!
0%
20
■9720
25
■9778
30
•9813
35
•9850
40
•9881
45
•9888
50
•9895
55
•9890
60
•9870
Here then
H M V
1A 20 20
<
o M V
20 20
H M a„.
because
M a
u a 20
<
40
M a
U a 40
Also
H M V
lX 15 40
<
°V«
because
H M a
W a 40
() M a
U a 40
<
°\ 5
But
H M V
11 20 40
>
o M v
W 20 40
because
H M a
H a 40
>
H M a
a a 60
40
M a
oo
354 ACTUARIAL THEORY [chap. xvih.
The matter becomes rather more complicated when it is desired
to compare the H M and H M(6) reserves with the O m .
In this case when n < 5
H M V > = < M V
n 21 n x
H M a H M a ^
according as — ^-^ > = < — n^i?
M a M a A
x x+n
But when n = or > 5,
H M and H M(6) V > = < O m V
n X 71 X
H M(5) a n M
according as 1 „ x+n > = < 1 —
H M a M a
H M a H M < 6 >
a
that is, as -^ > = < — = — x -±?
M a M a ^
Having drawn up a table of the two ratios ^ M x and — = — ^±™
H M a H M < 5 Y
O^a OV
a+rt
we can tell, by inspection of the ratios, at what ages at entry and
for what terms the H and H M ^ reserves will be greater than,
equal to, or less than the O m reserves.
It is wrong to assume that, because one table or combination of
tables shows larger reserves than another for whole-life policies,
the same relation will hold for endowment assurances. Comparison
must be instituted between an entirely different set of functions,
viz., term annuities. For
V -
n xt\
> = < J x r\
a 'x7\
a '^
-^ _ ^ x+n:
a ,
x+tii
t-n\
t-n\
according as
This is not a merely theoretical point ; for as a matter of fact,
while the H M and H M(6) reserves are greater on the whole than
the O for whole-life assurances, the reverse is the case for
endowment assurances.
If two mortality tables yield equal policy-values, that is, if
.] TEXT BOOK— PART II. 355
CHAP. XVIII
a'
V = V , then the ratio — will be constant for all values of
x, and we may write
1+a' i
X - 1
But again
1 + a. 1+k
whence a' = — - 1
x 1+K
1 + a'
Then V = 1 -
1 +a'
X
x+n
= 1- 1+K
= 1-
1+a
~T+k
1+a
x-\-n
1+a
X
= V as required.
n « *■
d x
1 + fl i-i
i + «
1+K
a - k
X
< 1 +%+l)
a — k
X
a
X
X
which is Text Book formula (29).
The conclusions of Text Book, Articles 59 and 60, may be
proved directly for the rate of mortality, q' x .
356 ACTUARIAL THEORY [chap. xvm.
If 'W-O-t)
N x'
a' = i_p + _^
The addition to be made to the rate of mortality at age x if
equal policy-values are to be produced is therefore —,
which increases with an increase in x.
Now, if instead of this increasing function we make a constant
addition of r to the rate of mortality, the increase will not be so
rapid as is required to give equal policy-values. Therefore, on
the principles of Text Book, Article 48, the effect of adding a
constant to q is to diminish policy-values.
Again, if for — - ^ we substitute - r, the rate of mortality
will be increasing more rapidly in proportion than under the
formula which produces equal policy-values ; and therefore on the
principles of Text Book, Article 48, the effect of deducting a con-
stant from q is to increase policy-values.
11. In this connection it will be useful to discuss the reserves
required for policies upon lives subject to extra mortality. Extra
mortality will probably occur in one or other of three well-defined
ways.
(1) The extra mortality may be higher than the normal
throughout, the difference being small and slowly increasing at
first, but becoming great in the later years of insurance. We
should expect reserves under such a table to be greater than
those under the normal, as the increase in the mortality is pro-
portionately greater at the older ages.
(2) If the extra mortality is greater than the normal, but by
a constant difference throughout, we should expect the extra
mortality reserves to be less than the normal, since this constant
addition does not allow for the increase in the rates at older ages
being sufficiently great to give equal policy-values.
(3) The difference between the extra rates of mortality and the
normal may be great at first, afterwards diminishing, and finally
disappearing, until the two tables coincide. Here also the reserves
CHAP. XVIII.]
TEXT BOOK— PART II.
357
under the table of extra mortality will be smaller than the normal,
as the increase is proportionately greater at the younger ages.
The effect upon policy-values of an increase in the mortality
under a table graduated by Makeham's formula, /j. = A + Be*, may
be considered briefly.
If in this formula the value of A be increased, the result, as
shown on page 232, is equivalent to increasing the rate of interest,
and that, as proved in Text Book, Articles 69 and 70, results in a
lower policy-value. A lower policy-value is therefore the effect of
a constant addition to the force of mortality. If, on the other
hand, B be increased, the effect, as shown on page 233, is to
increase the age, and therefore a higher policy-value will be given.
If A and B be both increased, the ultimate effect cannot be
ascertained without further investigation, as the two increases
operate in opposite directions in their effect on policy-values.
There are two well-known methods of dealing with extra-rated
cases in valuation.
(1) Policies on such lives may be valued at the increased ages
which correspond to the higher rates of premium charged ; that is,
they are treated throughout as normal policies effected at such
increased ages.
(2) The policies may be valued at the true age, precisely like
normal policies of that age, each year's extra premiums being
assumed to meet that year's extra claims.
It is interesting to examine the extra rates of mortality which
are assumed to underlie each of these methods.
Suppose (x) to be a life charged the premium as at age x + r.
Under the first method we then have
Years
Normal Rate of
Assumed Rate of
Extra Rate of
Elapsed.
Mortality.
Mortality.
Mortality.
%
'x+r
*x+r "x
1
%+i
^x+r+X
"a+r+l _ ?x+l
I
%
+2
Qx+
T+2
1x+r+
2 Qx+2
%
+(
?*+
r+t
C lx+r+
t ~ 1x+t
358
ACTUARIAL THEORY
[chap, xviii.
In examining the second method, we know that for a normal
life
CV. + W+O = n + i V * + %U l -n + i V x )
while in the ease of the extra-rated life under this method
C.v,+p.+R)(i+0 = , +1 v.+9', + .(i-, +1 v;
where R is the extra premium, and q' represents the actual
rate of mortality. That is, the normal reserve plus the ordinary
net premium and extra premium accumulated to the end of the year
are assumed to be sufficient to meet the normal reserve at the end
of the year, and make the necessary contribution towards payment
of the claims actually experienced. Hence
R(l+i) «('. +1 + • • ■
= «p.(i - + V W ~ r )P.+i( 1 " r ) + • • •
= *>* + («') 2 a + - • •
= a" calculated from the normal mortality table
at rate of interest J, which is such that
1 1-r , 1 1 , . .
whence ■= : < -z ■. and i > i.
l+a' ,
Now V' = 1- -
= 1-
1+a'
1 +a
x+n
l+a"
X
= V" calculated from the normal
n x
mortality table at rate of interest j.
But since i > i, V" < V calculated at rate i, and therefore
«/ ^ ' n x n x
V < V .
n x n x
Hence it is seen that the effect of diminishing p x at each age
by a constant percentage is equivalent to using a higher rate of
interest ; that is, policy-values are diminished.
By a similar process it may be shown that the effect of
increasing p by a constant percentage is to increase policy-values.
13. The propositions of Text Book, Articles 71 and 72, may be
proved as follows : —
If a < a , ,
X x+1
then i>P x V+ a x+ i)< a x + i
• a *+i
and vp <
1 + ( Ka.i
X+1
°x+l
V - w > o - i— —
*' 1+a x+l
that is, P 1 T , > I
x:i\ ^ x+1
360 ACTUARIAL THEORY [chap, xviii.
Again, if a x < a x+l
<%
a
tx l+a
X
a
that is, Pi ;T1 > P^
Further, if a < a
3 X
then
1
l+a 1 + a ,,
that is P > P , ,
a - a , ,
Finally, ,V = -^ £±1
-' i * l+a
Therefore, if a < a ,,, ,V is negative.
14. Besides the two cases mentioned in Te.rf Z?oo£, Article 73,
in which negative policy-values occur, the following may also be
noticed : —
(1) Reversionary Annuity Contracts.
Here we have
Va I = a , I , - Pa | a ,
n y\x y+n\x+n y\x x+n:y+n
/a , —a ,
I x+n x+n:y+n
\ a ,
a - a \
_£ ^A a
a y it»:j+n
It has already been pointed out (page 274) that the annual
premium for such a policy may decrease with an increase in the
ages. If this were to happen, the above formula would give a
negative result.
(2) Contingent Insurance, - (x) against the survivor of (j/) and
(s), (z) having died.
Under such conditions
Vi - = A-i- -T--P 1 -a
n x.yz x-\-n:y-\-n x\yz x-\-n'.y+n
= fp_J ,_ pi _^ a
^ x+n'.y+n x'.yz' x+n:
y+n
chap xviii ] TEXT BOOK— PART II. 361
As pointed out on page 247, if n be small, P_!_ ____ may be less
than P 1 — , in which case this value will be negative.
x:yz' °
15.- "" «/,.«-!,
To prove V<™> = V (l + ^J P(W\
r » * » *\ 2m x )
V(™> = A -P(™l a (™)
7i a: z-fft a: e+ti
= A x -P<«>(V -Htfl)
x+n x y x+n 2m )
= A ^ -/p +^-lip(m)(P +d)\a^ +^PW
x+n y x 2m x x I x+n 2m *
(since PW = P. + ^PO-^ + d), see page 196)
2m
1
a J
2m
A j. -Pa^ + ^J.pW{l-(P+^a,J
z+71 x x+m o™ a; I *• a < m >
n xt | x-\-n:r-n\ xr \ %+n:r-n\
a?+m:r-»| xr\{ x+n:r-n\ 2m \ T> J)
. a _/p_ + w ^p(™>fpi_ + ■ '
and , V = A , , - P a , , , + P (1 - i)
n+t x x+n+t x x+n+t x^ '
which agrees with Text Book formula (31).
17. To find V«, ( t = -\
n+t x > \ l - m )
V(™> = A _F m >a< m )
n+t x x+n+t x x+n+t
= K+n+t ~ { P * + ^ W + «*)} «**M + P ^
* 2m
= A . + ^.«- p ,v. + .+ !! kr p S* ) { 1 -( p .+'0v^
= ( A x+ n +t ~ F x a x+%+t) \ l + ~2m~ P *7
chap, xvni.] TEXT BOOK— PART II. 363
which agrees with Text Book formula (36), since
V+<(,,V-V) = A -Pa +t(A -Pa )
n x Mi+l x n x' x+n x x+n^ ^ x+n+l x x+n+\>
-t(A _,_ -P a , )
v x+n x x+n'
= A ^ + l(A ^ ^ -A , )
x+n *• 2+71+1 x+n'
-P {a ^ +t(& ^ ^ -a , )}
x ' »+« v x+n+1 x+n'>
= A -Pa
x+n+t x x+n+t
if first differences are taken to be constant.
k
To find , ,V(»)
n+t x
■■("H
V(™) = A - P( m > x i I a( m >
»+( a; z+n+S x —-si x+n+t
m
= A -P(m/ a O0 - - +s\
x+n+t x \ x+n + t m J
- a -PW a (™) + P(™)( s\
* -..../ x \m j
x+n+t x x+n+t
k _ P(m)( a
x+n+t x \ x+n+t 2m
= A - P< w )fa , - ''^—^\ + F™/- - s)
x+n+t x \ x+n+t 2m / * \m J
- A, M+1 -{i',+=£ i W.+}{(i-<>-(i-^)}
Hence ^VN = A x+%+t - P A+m+| + P^l - - P,(l - *±1)
+ !!llpw{l-(P + i)a +P(l-0-Pfl-— )]■
2»» * \ x *+»»+* ^ ' *\ m J)
* \m J
2m
»+( v s r *^ m / 2ot * \ n + f x x \ m J)
p(m/I -s)
" \?n J
,m — \
+ d -2m-
364 ACTUARIAL THEORY [chap, xviii
The third and fourth terms of this expression are very small,
and in practice are usually ignored, it being assumed that
V(™> = V - P (l - ^-^)
n+t so n-\-t x x\ ml
= A , ,,-P a u L , + Pf- -s)
x+n+t x x+n+t x \m J
It may be noted that where the rathly premiums are instalment
premiums, the third term will not appear in the expression for
the exact value of y lm \ and therefore the value used in practice
n+t x x
will in that case be so much nearer the true value.
To obtain V( m )(i = — + s), we may also proceed by
interpolating between
/ pc™\
( nV(») + -i-] and t+iVW
p(m) p(m)
Thus VW = j,V(™) + -i- + m( fr+iV< m >- uV«— £-|
x m j
n+t x n-\ — x m \ n ^ — ZT x "+
= fcV(™> + sm( J;+iV( m )- , h Vf™M + F™> ( - - S )
Now, substituting for iVW and , i+iVW their values as
' o TiJ — a: n-{ x
m m
found by Text Book formula (36), we have
vw = fi + ^P^pc'Ar V +-f V - V)
«+t a; \ 2wz * / L 71 a Itl ?t + 1 re «. re'
+ «J v +— ( ^V - V)- V --( v - v)\l + pwf- -*
1 n * Tfl V71 +! x n x' n x 7^ «+■* a; n x' j J re \w
= f 1 + ^— ^ p(m)N ) -f v +f- + * N )( ^v - vU+pwf--^
The quantity — ^ — PjW is very small, and may be ignored;
and P may be substituted for P(™> in the second term. If
X J X
these alterations be made the expression will agree with Text Book
formula (38).
18. To find , V-:
n+t xr |
V_ = a P— x la
n+t xr\ " x+n+f.r-n-t \ xr\ l-t\ x+n+t : r - n - 1 1
chap, xvni.] TEXT BOOK— PART II. 355
Now i a _ = a
01 i/m| y:m+0\
,1a—; = a , — 1
Hence interpolating by first differences, where k is any fraction
of a year
„i a. — ; = a 1 — k
t| ym\ y:m+K\
T.-t\ a 'x+n+t:r-n-l\ ~ ^x+n+f.r-n-t] ~\ "
n+t xf\ "' x+n+t:r-n-t\ ~ xr~\ a x + n+t:r-n-t\ + a:?]^ ~0
Or following the method of Text Book, Article 78, we have
n+t xr] " n xr] Mi+1 xr] ~ n xTy xry< )
The two formulas are identical, since
V ~ + t( „V_.- V-.)
n xr | ^n-f X xr\ n xr y
~ x+n:r-n\ xr\ x+n:r-n\ <\ *+n+l:r-«-ll _ x7] x+ri+l:r-n-l y
-fA , i-P-ia , ,)}
^ x+fl.:r-m| a;r | x-\-n:r-n\ y *
= A , i + /fA , ,, ri-A , r")
x+m:r-n] V- a,'+n+l:r-TO-l | x-j-7i:r-7i|-'
_p_( a + /(a — a "H
= A , ., r , -P-ra
a-+ti+(:r-»-(| xr\ x+n+t:r-n-t\
19. To find V^, (t = - + »)'
V^ _ ^ _^ p(m) x j |a (m) fc+J|
»+' a:'' | i+«l+t:r-K-«| xr\ s\ x+n+t :r-n-—-r-\
- P ( "^( aW
x+ji+Cr-ji-*! zr|\ x+n+t:r-n-t\
m I
_A P^la a (P * I J) m ~ * \
x+n+t:r-n-t\ xr\\ x+n+f.r-n -t\ x+n+f.r-n -t\^ x+n+t:r-n-t\ ' 2m J
xr\\m J
366 ACTUARIAL THEORY [chap. xvni.
yW _ A /p _ + ^-^pWCPl_-i-(fl\a
n+t xr\ ' x+n+t:r-n-t\ |w| T 2m «f|^ w| ; J *+»+t:i'-»-l|
«-l p( ro )/-p ! _
+ { p fl +! sr I SiCPfc I+ -)}{(i-0-(i-^)}
,. w ~ 1 p(w)ffp.._i pi-^a + pi-n-rt-pi-/i-^±lYl
2to (W I \»- *+»+*: r-*-*| ct| ; rc+m+(:r-»-i| T ax|V ' xr^ m Jj
+ d m zlm(L- s )
2m XT \\m J
= V - - P -(\ - *±1\ + ^zlpW-f Vi- - pi-6 - £±^\
n+t xt\ xr\\ m J 2m iw|\»+« «r| xr|\^ nl jj
2m ™\\m J
The third and fourth terms of this expression are very small,
and in practice it may be taken that
yW = V P —(\ - iii\
n+t xr\ n+t xr\ xr{\ m J
m )
— a P — a + P —
x+n+t:r-n-t\ xr\ x+n+t:r-n-t\ xr\
As was the case for whole-life assurances, the third term in the
exact expression entirely disappears when the premiums receivable
are instalment premiums, and accordingly the approximate expres-
sion is in these circumstances so much nearer exactitude.
20. To find ,, V
V = A . , - P x.
n+t-.r x x+n+t r x 1-(| x+n+t:r-n-l\
x+n+t~r x 8, x+n+t :r-n-t\ r A v
chap, xvin.] TEXT BOOK— PART II. 367
21. To find ^ V(«), (, _ k <\
n+f.r x ' \ f <~S }
\ m /
, VW = A - F«/ a W - +.A
»+*:»• s x+n+t r x \x+n+f.r-n-t\ m I
= A -1 P + TO-1 F™YPi- + d'»la
*+»+« |r z T £ot r * V w| ' J *+«+*:r-»-t|
W-l p( m) ,-p_l , jx
2ot r * v x+n+f.r-n-t\ T ' x+n+*:r-»-t|
+ {A+ ! K 1 ^P(ftl^}{a-0-(i-'-±i)}
= x+n+t ~ *x\+n+t : r-n-t\ + r "*( ~ V ~ r *( ^~ )
OT-1 p(m) f/p_l pi_x pl (1 _ A _ pi_/l _ ^ +1 \\
2ot r * \ V *+»+t:r-»-«| 1 j!r|^V»+l:f-n-»| »r ' ' D
Together, as above,
. , ^ M , + , + "\+n + 2 V. " 2 V« - < R * +( - ^ M x +t
J* s ■
The reserve found retrospectively is
D^ [>( N «-i - N , + ,-i) - Ml + *) + '}(*« - R x+m - nM x+n )
-j{*(l + K ) + c}teR -2R^ -„R _<^+l) M \"|
"^ v ' ' y x x+n x+n 2 x+nf J
The two formulas may easily be proved to be identical if the
value of 7T as found on page 309 be remembered.
25. To find the reserve under a similar policy except that the
premiums are to be returned with compound interest at rate j in
the event of death before age (x + 1).
chap, xviii.] TEXT BOOK— PART II. 371
Prospectively, the reserve is
iVi + W i + K) + c} _L {( i + i>-,(M x+K - M x+t ) + (1 +jY+\M x+n - M x+j )
x+n x+n
+ V+jY+XM x+n+1 -M x+t ) + . . • +(l+j)KM x+t _ 1 -M x+t )}
^ x+n:t-n-\y
And retrospectively
f kil. W l + , )+4 ^ {[1+i)(Mr M i+t)
X+» X+tl
+ (l+i) 2 (M x+1 - M * + J+- • • +(l+i)»(M i+ ,. r Mj}
These two should be proved equal, given the value of ir as
found on page 310.
26. To find the value after n years of an assurance deferred t
years, premiums payable throughout life but returnable in the
event of death within t years.
Three cases arise, viz., n < = > t.
(1) n < t.
Prospectively, the reserve is
M ,, nM , +R , -R , -tM . ttN
_S±! +{„.(! + K ) + C } £±2 *g £±^ £±* _ - '+*- 1
a+7i a;+?& x+n
Retrospectively
jt(N -N , ,) R-R, -»M,
x+n x+n
Now these two expressions are identical, for
-J— IWN -N 0-{t(1 + k) + c}(R -R . -»M^)1
J) l >■ 3-1 x+n-V * ^ ' > v a; a-j-Ti x+n'-l
x+n
* d 1 - ["■{N.-i " C 1 + K )( R , " R * +( - M * + $ ~ M * + < ~< R * ~ R -+. ~ M * + }
x+n
+ M x+ t + 'C 1 + K )( nM x+n + \+n ~ K+t - tM x+t)
+ t.
M , N , ,
Prospectively -^±? - tt -^^
x+n a:+7?.
Retrospectively
N ,-N^ . R-R^.-iM,, M -M
v s-1 x+n-1 _ ^q + K ) +c }_? !+< £d± *±« *±»
These two expressions also may easily be proved identical.
We have seen that when n = t, the reserve which the office
must have in hand is A -ira. . This has to be provided out
of the premiums received, and the mortality up to age (x + 1) may
be ignored, as the premiums are returned in the event of death
previous to that age. We may therefore find at what rate of
interest the premium calculated by the exact formula will amount
in t years certain to this amount. That is, find j such that
tt(1 +j)s r ,,.. = A , — jra , ,
^ •> ' 8 10) X+t X+t
The reserve, when n
x+
x+n
(b) If (y) died after m payments had been made, occurring
in ( » - p)l , cases out of / .
Mb-1' y m' y' x+n x+n
Prospectively
M ,, M - M N
^+ ( +m{B-(l + K) + c}— £±^ *+'- T «+'-*
Retrospectively
a . N -l- N »+-l- M l +)c)+c} R «- R «- l ->-'" M x + .
x+n "x+n
(2) When w = t.
(a) If (3/) lived Q-l) years, occurring in p I cases out
01 / , ,.
x+t
M N
Prospectively -j^±± - tt ~^ ±l1
x+t x+t
Retrospectively
N ,-N ,, , R -R , -*M .
ff «-i ^+t-i _ {T ( 1 + K ) + c |_^ «gt £±e
x+t x+t
(6) If (^) died after m ( < i) payments, occurring in
( ,P - P )i . . cases out of I , ,.
M ^ N ^, ,
Prospectively "f^ 1 *"-
s+t £+(
Retrospectively
N-N R-R.-mlVI,
v x-l «+»-! _ {a .( 1 + K ) +c} _£ l±p E±»
(3) When » ><.
(a) If (#) lived (< - 1) years, occurring in t _ 1 P y i sc+n cases out
of / .
x+n
M N
Prospectively _^+n_ 7r ^ r l
x-\-n x+n
374 ACTUARIAL THEORY [chap, xvfii.
Retrospectively
N-3ST, R-R^,-/]VI M-M
x-l *'x+n-l l~(\ I ,A I c ) x x + t £±£ _ x + t x + n
T - — D
x+n x+n 3,+n
(6) If (y~) died after m (< i) payments, occurring in
( v - v ~)l . cases out of / , .
W-l' 3/ m"y J x+n x+n
M N j
Prospectively n ~ ^ T) "
Retrospectively
i ~
D
'N -N N -N \
i x-l a+m-1 , x+*-l a;+m-l\
71-1 ~ { D /
x+n x+n
R -R ^ -mM ,, M , - M ,
x+n x+n
28. The reserves for limited-payment policies, endowment
assurances, and temporary assurances have already been given,
both by the retrospective and prospective methods, and the values
of each by the two methods are equal.
The three classes may be looked at together in the following
manner : —
y _ a —Pa
n:t x x+n t x x+n:t-n\
N
= A , -,Pa x + JP .f-'- 1
x+n t x x+n t x Y)
x+n
y _ = A — P -a
n xt\ ' x+n:t-n\ xt\ x+n:t-n\
^N ^ n N , , ,
= A i »+'-i p a +p_ x +t-i
x+n J) xt\ x+n xt\ T\
x+n x+n
= A + -P r| a ++ (P < )N ^
k+ji o:(| x+n J)
a:+7i
yi- = A— P 1 - a
M N
= A E±i_pi_ a +pi_ x +'- 1
x+n J) xt\ x+n^ xt\ T)
x+n ' x+n
f Pl_ _ P W
= A - P 1 - a + xt \ x + tJ x+t-i
x+n xt\ x+n J)
x+n
chap, xvin.] TEXT BOOK— PART II. 375
In each of these expressions the first two terms are of the
general form A x+n - Pa a , +B , P alone varying with the nature of the
N
benefit. The third term has -=£i — constant with a varying
x+n
coefficient. This coefficient consists of two parts, the first of
which shows the correction to be made on the second term for the
value of the premium, and the second the correction to be made
on the first term for the value of the sum assured. Thus in the
case of temporary assurances, no premium will be received after
age x + t, and therefore the value of all premiums after that age
must be added to the liability ; further, no claims will be paid
after that age, and therefore the value of a premium starting
then which shall be sufficient to meet all these must be deducted.
29. These formulas may also be worked into the following
forms : —
N
,V = A , - P a , + P_£±ti
n;t x x+n t x x+n t x J)
x+n
f,P -P)N ,-T(N -N , ,) + PN
■ A — P A- x x x+t-\' x x
x+n t x x+n J)
x+n
(,P -P)N ,
= A-,Pa+ ' ' ^ x '- 1
x+n t x x+n J)
x+n
since
( a:^ x-1 — t-^x+t-l' x x x-l
V-
n xt\
= A+ _ P - + ^+fW>
x+n xt\ x+n J)
x+n
— A - P - a
x+n xt\ x+n
( p ,r p ,) N »-r^( N »-r N , + .-i)irfN I+l - ] + F A. ]
+ D
x+n
(P - - P )N .
v xt I x> x-\
~ A x+n xt | %+n + J)
x+n
since p *n( N *-i- N * +( -i)- rfN * + *-i = M x-™x +t + Vx +t - dN * +t -i
= ^x-^x + t-l^x +t ^x +t -^x + t- dN x +t -l
= M, = P A-!
376 ACTUARIAL THEORY [chap. xvm.
CP 1 - - P W
VL = A - - PV, a ^ + K "' * +tJ '+ 1 - 1
n xt | a+w xt \ x+n jy
x+n
= A, -PLa ±
x+n xl | x+n
(PL-P)N ^P^fN -N , ,) + PN -P N
, >• Xt] X> X-l Xt\^- X-l X + t-l' X x-l x+t x+t-l
x+n
= A , -P-a +
(^n-PJ^-!
x+n xt\ x+n jy
x+n
since PV,(N -N , ,)=M-M,=PN -P , N ,, ,
Xt\^ X-l X+t-l' X X+t r. X-l X+t X+t-l
Here each expression is of a perfectly general form indepen-
dent of the value of t, except in so far as t determines the value of
(P - P )N
the premium for the benefit, A , - Pa , H =-^ — - — , where
r ' x+n x+n J)
x+n
P alone varies, being the premium for the particular benefit under
consideration.
30. The reserves for policies under many special schemes may
be simply found by remembering the method by which the
annual premium was calculated. For example :
(a) To find the value after n years of a whole-life policy to
(r), under which interest at rate j is to be guaranteed on the sum
assured for I years after the death of (x), and thereafter the sum
assured is to be payable, the office assuming rate of interest i in
its calculations. It will be remembered (see page 148) that the
annual premium for this benefit is
p^+O'-'K-k,)}
Therefore the reserve will be
n x • ' w / ( | (t) I
(6) To find the value after n years of a whole-life policy, under
which the sum assured is to be payable in t equal annual instal-
ments, the first at the end of the year of death of (x). The
annual premium for this benefit (see page 147) is
P x^
chap, xviii.] TEXT BOOK— PART II. 377
and therefore the reserve will be
V x^l
» x t
31. To find the value after n years of a policy for the whole of
life under which the premium is P for the first t years, and there-
after is 2 P.
(1) When n< t, the value is
\+n ~ P (\+n + 1 - . | a *+ J Prospectively
P ( N . ! - N ^ J - (M - M )
^^ *+y *" * ^ Retrospectively
and these two are equal.
(2) When n = or > t, the value is
A x+ n ~ 2Pa x+« Prospectively
P(N + N, -2N , ,)-(M-M )
- ■ -?Z1 5±£zi »+»-^ I « jrW Retrospectively
which are also equal,
32. To find the value after n years of an endowment assurance
policy payable at the end of t years or previous death under which
the premium is to be P for the first r years and thereafter 2P.
(1) n < r.
Prospectively
A p ( a |- i a )
x+n\t-n\ ^ x+n:t-n\ r-n\ x+n;t-r\'
Retrospectively
IYN -N ,)-(M - M ^ )
>■ a;-l x+n-l' ^ x x+iw
(2) n = or > r.
Prospectively
D ,
x+n
A , ; — : - 2Pa , ; — ,
Retrospectively
P(N n +N , ,-2N , ,)-(M -M rJ _J
*■ x-l x+r-1 x+n-1' v x x+ws
D
x+n
378 ACTUARIAL THEORY [chap, xviii.
Again in each case the reserves by the two methods may be
proved equal.
33. To find the value after n years of a policy subject to a
contingent debt should (.r) die within t years.
(a) Where the debt is X for the whole period of t years.
Assuming that (x) is rated up r years, we have
(1) When n < t.
M, +r+ro (l-X) + XM„ +r+t
— 77 a , ,
T) x x+r+n
x+r+n
(2) When n = or > t.
A , , — 7T a , ,
x+r+n x x+r+n
(6) Where the debt is <.
A , , - 7r a , ,
34. To find the value of a double-endowment assurance policy,
2 being payable if (.r) live t years, or 1 if he die before that.
V = V-+ V-
% xt\ n xt\
= V-+A — Pia
n xt\ x+n:t-n\ xt\ x+n:t-n\
_ y ■ j^ 1 A - x + n:t ~ n \
n xt | x+n : t - n \ xt\ a .
art I
_ V - 4- A — A Id - V - "^
n xt\ T x+n:t-n] xt\^ n xt\J
= n V xt\( 1+A xt\) + ( A x+n:-nr\~ A xT\)
Similarly, to find the value of a half-endowment assurance
chap, win.] TEXT BOOK— PART II. 379
policy, 1 being payable if (x) live t years, or 2 if he die before
that.
V = 2 V -, - V I
n art I n xt\
= » V ^( 2 - A i|)-(V TO: ^-A^)
35. To find the value after n years of an annuity deferred
years which was purchased by single payment.
(1) n < I.
Prospectively 1 a
Retrospectively
I x a a (l+{) n isD
x 1 1 zV J _ 1 1 XX
I ^ ~ ~D
x+n x+n
The two expressions are equal, for
,|(iD N ,, D N ,
t\ X X x+t X _ x+t .
D ~ ~W~ D = D = !-»'">+»
x+n x x+n x+n
(2) n = or > t.
Prospectively
Retrospectively
Prospectively a
,\a D N -N ,
< I x x x+t x+n
D _, D~
,|«D -(N -N , ) N
Again tl x x K x+t xW -
D D x + n
x+n x+n
36. To find the value of a similar annuity with the condition
that the premium is to be returned if Qc) die within the t years.
(1) n < t.
Prospectively
, I a , +{A(1+K)+C}A— ; :
t-n\ x+n ' l \ ' / ' I x+n:t-n\
Retrospectively
AD M -M
>-{A(l + *) + c}-S 1
x+n
D , ' v y ' D ^
x+n x+n
380 ACTUABIAL THEORY [chap, xviii.
But the latter is equal to
gi- [A{D - (1 + K )(M x - M x+ J} - c(M x - M x+n )]
x+n
+ V x+t + {A(l + K) + c}(M x+n -M x+t )]
N J _, JVT -M,, N ( + c(M-M ,)
+ WHk) + c) since A _ _ (1 + k)(m _ m <
x+n , x+n x ^ ' J\ x x+t>
= , Iff , +{Afl + K) + C}A-5- ; ,
t-n\ x+n I >. ' ' > x+n:t-n\
(2) n = or > <.
Prospectively «
Retrospectively
D M -M „ N , -N ,
A 5 -g--{A(l + >) + c} * p * +t --^ ^
o:+?t a+» a;+?i
But the latter is equal to
^[A{D^-(l +K XM,-M^)}- C (M,-M^ ( )-(N +i -N + J]
X-\-%
= IF" [ A{ D - (1 + «)(M x - M_)} - {N + c(M, - M x+t )} + N, + J
N
= a ,
37. To find the value at the end of n years of an assurance
with a uniform reversionary bonus of b per annum declared every
five years, an interim bonus of b' being granted in respect of
each premium paid since the date of last investigation should the
life die within a quinquennium, assuming re to be a multiple of 5.
Whole-Life Policy.
1 _[(l+re6)M J + hb(M , , +M , „„ + ...)
J) LV J x+n V x+n+5 x+n+10 '
x+n
+ b '{\+ n -m x + n +, + ^+n+,0 + -- • )}]-".
x+n
chap, xvin.] TEXT BOOK— PART II. 381
77 having the value found for the premium for this benefit on
page 302.
When b' = b, we have
(1+«6)M , + bR
x+n
ir having its appropriate value.
Endowment Assurance, maturing at age (x + i).
D l-V J\ x+n x+t t x+t j -r v \_ * x +n+b ^ x+n+10 T ^ l x+t>
x+n
+ (t-n)b(D x+t -M x+t ) + b'{(R x+n -R x+t )- H M x+n+&
+ M x +n+w + •■■+ M x+ t)n - -v»=^
When V = 6, we have
~ [(1+»6)(M ^ - M ^, + D ^,) + 6{R , -R , + (/-»YD ,-M,,)}!
x+n
— ira. , ; ;
"■ in these two formulas will have different values as found for the
two benefits on page 303.
38. To find the value of an assurance with a compound rever-
sionary bonus on similar conditions.
Whole-Life Policy.
d 1_(1 + 56)T { (Mx+n - M x+n+5 ) + (1 + 5b)(M x+n+& - M x+n+1Q ) + . . .
+ 6 '( R . + . " R , + , + 5 " 5M , + u +5 ) + *'d + 5b X R x+n+, ' K+n+ W ~ 5M * + « + I0)
If 6' = b,
J-(l + 56)T{ M x+ „ + 6(K s+n - R^ +5 ) + 6(1 + 56)(R, +n+B - R^ J
+ 6(1+5^(R^ +10 -R +n+u )+- ■ -}-™ +n
r will have a value appropriate to the benefit as indicated on
page 304.
382 ACTUARIAL THEORY [chap. xvm.
Endowment Assurance.
D^l + 56) * {(M x+n - M x+M+5 ) + (1 + 5bXM x+n+& - M x+n+ J + . . .
+ (1+56) » (M^-M^-Kl + S&^D^
+ 6 '(R +n - R I+ , +5 - 5M , + , +5 ) + 6 'd + ^)(R x+n+6 - R x+n+10 - 5M xm )
+ ... +6'(l + 5^(R i+( . 5 -R :c+i -5M^)} -™ +7[; ^
If 6' = 6,
_i-(l + 56)5 { M x+k + 6(R i+k - R x+n+b ) + 6(1 + 5&)( V* + 5 " V» + io) + ' • •
t-?t-5 t-n
+ 6(l+56)-T-(R^„ 6 -R, +t ) + (l+56) » (D, +( - M, + ,)}- TO<+ . !P ^
The values of 7r are indicated on page 305.
39. To find the reserve under a Discounted-Bonus or Minimum-
Premium polic}'.
(a) Cash Bonus.
It was found on page 306 that the deduction from the ordinary
premium for a discounted bonus of k per annum of the premium
N
was kV * + ~ . Therefore in finding the reserve of this class of
x-l
policy, we must add to the ordinary reserve for a full profit policy,
to allow for this decrease in future premiums ; but we must also
make a deduction from the liability, in respect of future bonuses
at this rate which will not be payable. That is
V = V +AF -^±- 2 a -5/fcP' *+"+ 5+ *+rc+io + ""
% x % x x~$ x+n x J)
x-l x+n
N N
= V +AF ^±- 2 a ^ -AF ■ XT " +1t+8 a ^
n x x M x+n x J^ a+n
x-l x+n-1
= V + H>' ( ^±J - ^ +TC+2 V ^
x a:-l m+7S-r
V being the reserve for a full profit policy.
chap, xviil] TEXT BOOK— PART II 383
(6) Uniform Reversionary Bonus.
The deduction from the ordinary premium was found to
R x
be b.^-?-. An addition and deduction must be made as above
x-l
explained, and we shall have
It R
V = V +6_L a -b x+n
■a x n x N x + w D
a; - 1 x+n
(R , R \
N , , N J *+»
X+ll-1 x-l'
(c) Compound Reversionary Bonus.
The deduction from the ordinary premium is
K* x - R . +5 ) + 6(1 + 5b)(R x+5 - R x+1( ) + 6(1 + 5b)%R x+w - R, +lg ) + ...
The reserve accordingly will be
V V + 6 ^-Va) + 6 (l + 56)(R, +5 -R, +I0 ) + -- a
% x n x J^" x+n
x-l
b ( R x + n ~ R , +m+5 ) + *q + B *X V, + 5 ~ R s + tt + 10) + -
D ^
x+n
. , _ r 6 (R, +ro - R, +m+ a) + Hi + 56 X R . + „ +6 - r ^ +1 q) + -
"• x+n-1
H R , - R , +5 ) + Kl + 56XR +5 -R, +10 )+ ,
x-l
-)•
40. The subject of surrender-values does not fall to be
discussed, but it may be remarked that a surrender-value is as a
rule granted only where a benefit is certainly payable, as under
whole-life assurances, endowment assurances, joint-life assur-
ances, etc. It is customary to allow no surrender-value where the
benefit is only contingent, e.g., temporary insurances, where the
sum assured is payable only should the life die within the term,
a contingency which may or may not happen ; contingent insurance,
(x) against (y), where the sum assured is payable only should (x)
die before (j/), which may or may not happen.
384 ACTUARIAL THEORY [chap, xviii.
In the case of a pure endowment with return of the premiums
in the event of previous deathj a surrender-value is given,
since a benefit is paid whatever happens ; but where it is a pure
endowment without return, usually no surrender-value is paid, as
the benefit is then contingent on survivance.
In the case of the double-endowment assurance, the method
by which the surrender-value is calculated requires special con-
sideration, since it must not be overlooked that only half the
benefit is certainly payable, the other half being contingent on
survivance.
41. The principles of Text Book, Articles 122 and 123, on which
formulas (56) and (57) are founded, may be stated generally so as
to apply to any kind of benefit.
First, let W be the amount of paid-up policy to be granted.
Then the value of a benefit of W must equal the value of the policy.
Second, the paid-up policy must equal the sum originally
secured less that proportion of it which the future premiums
will cover.
In the case of a whole-life assurance, we have
(1) WA , = V
(2) W = 1
P ,
x+n
Similarly for a deferred annuity-due, where P =
we have
x+t-l
N -N
x-1 x+t-l
N N PC N - N ")
/in w x+t-l _ s+i-1 \ x+n-1 x+t-V
^ D D
x+n x+n
(2) W = 1
^x+t-1
N -N
x+n-1 x+t-l
a — a
For a reversionary annuity, where P = -y — we have
xy
C 1 ) W K+* _ a x+n: y +J = ( a x+n ~ a x+n:y+J ~ P ( ! + a X +n:y+J
(2) W = 1
« , -« ,
g+w x+n:y+n
1 + a ,
x+n:y+n
chap. xvm.J TEXT BOOK— PART II.
385
W
42. To find the paid-up policy to be issued after n years in lieu
of a pure endowment policy payable at the end of t years with
return of premiums in event of previous death.
The value of the present contract is
< N «-i- N . + .-i)-^(R,-R^.-».M. | . )
x+n
Now, the n premiums paid are to be returned in event of death
before age x + t under the paid-up policy as under the original
contract, and therefore we shall write
x+n x+n **~
x+n
.hence W = " (N *-' " N « + .-i)-*'( R * -»■+.-» M.u J
D
x+t
43. Under a last-survivor assurance three cases arise in finding
the paid-up policy.
(1) (x) and (y) both alive.
WA-— - = A P.
a-
x+n:y+n x+n:y+n xy x+n:y+n
P-
or w = i-_ m
' x+n:y+n
(2) (x) dead.
(3) (y) dead.
WA ± = A - P- a
y+n y+n xy y+n
P-
or W = 1 - —3-
y+n
WA ± = A ^ - P-a ^
x+n x+n xy x+n
p-
or W = 1 *»
P ,
44. It is a common practice for offices to guarantee paid-up
policies under limited-payment whole-life assurances and endow-
ment assurances, the amount of each paid-up policy bearing the
same proportion to the original sum assured as the number of
premiums paid bears to the whole number payable.
2 B
386
ACTUARIAL THEORY
[chap. XVIII.
It sometimes happens however that the value of the guaranteed
benefit exceeds the value of the original policy, and the conditions
may be investigated.
Under the whole-life assurance by limited payments, where t
premiums were originally payable and n have been paid, the
amount of paid-up policy is — . Assuming the life to be still
select, we have
n
A -5, _ ^ A _ P a
2 [a;+m] t*+»] * [*] [i+n]:*-» |
according as
«[*]*[*+«]:«-» I > "" < [} t) [*+»]
or according as
P > = < 1 - — 1 P
t [as] ^ ^ I t Jt-nlx+n]
The following figures based on the O Table at 3 per cent,
illustrate the point : —
Age at
Entry.
Original
Number
of
Payments
w
< P M
(* - 7)«-n P l»+.] when » =
5
10
15
30
40
50
60
20
15
15
10
2-639
3-953
4-963
8-271
2-670
3-975
4-920
7-859
2-693
3-965
4-781
2-702
Similarly, in the case of the endowment assurance
n
1A_ >
t [x+»]:t-»|
< A
[x+n]:t-n\
P _ a
[x]t\ [x+n]:t-n\
according as
p _ > = < (i _ JL\p
[x]t\ ^ ^ ^ 2 y r [*+n]:«-»|
In the endowment assurance of practice the right-hand side is
always the greater, and therefore the value of the guaranteed
CHAP. XVIII. J
TEXT BOOK— PART II.
387
benefit is always less than the value of the original contract. The
following figures are on the same basis as before.
Age at
Entry.
(«0
Endow-
ment
Term.
(0
P M (1 +i)s—„ as shown
x+n
on page 295.
The following case will illustrate the absurdity of the sugges-
tion. Suppose an office
(1) Permits its whole-life policy-holders to alter to endowment
assurance on paying up merely the difference in premiums
accumulated at interest, and
(2) Permits its endowment assurance policy-holders to pay
only the whole-life premiums, the difference between the
premiums being allowed to accumulate as a debt against the
policy, which will be deducted from the sum assured on payment
either at maturity or at previous death.
No policies will be taken out under the second scheme, for
under the former just as good a benefit is secured for the same
premium, with the additional advantage that the accumulated
difference in the premiums will be deducted only at maturity and
not at previous death. To put the two classes on an equality,
the amount to be charged at the end of m years in the first
scheme should be altered to
/p p \ x-l x+tn-1
^ xm\ x' J)
x+m
Another illustration of the same absurdity may be given.
Suppose the assumed conditions to apply to all classes of assurance.
The life assured would then be well advised to take out a
temporary assurance which will give him the sum assured in the
event of death during the term at the lowest premium. Then at
the end of the term, if he survives, he alters to endowment
assurance, and pays the accumulated difference
(P-i-Pi )fl+i>-, = P±(1-m>-
and he should immediately receive 1, which is clearly wrong.
chap, xviii.] TEXT BOOK— PART II. 389
The correct amount to be paid by him is
N -N , , , N -N
/•p PI—") x ~ as+ro-l _ p _1_ x-1 x+m-\
>■ xm\ xm\' J) " xm\ £)
x+m x+m
= 1
Thus he provides the payment of 1, which has to be made to him.
From the formula given on page 387, we have
V V
p p , n x:n+t\ n x
~ x:n+t\ a _
x+n:t 1
fp p ^ a _ _ V
p , ^ x+n:t\ x:n+t\' 1 x+n:t\ % %
x:n+t\ """ a -
x+n:t\
V
= P
x+n:t\ a
x+n:t\
From this expression we observe that the future premium to be
paid is the endowment assurance premium at the present age,
less the reserve value of the existing policy spread over the future
premiums.
Again, we may argue as follows : — As no premiums are to be
received after age (x + n + t), and as the office will lose the interest
in advance after that age on the assurance of 1 which will then be
payable, the total loss to the office is
(P x + rf)%±^
X+lb
= (.V x + d )(*x+n-\+n:T\)
a x+n ~ a x+n:T\
Spreading this loss over the future premiums and adding the
result to the present premium, we have
p p a x+n~ x+n:T\
X a -x Xa -x+n:T\
Now let the premium to be paid remain at ? x ; to find the
altered sum assured, S.
390
ACTUARIAL THEO
We have
SA ^ -, - P a , -. =
V+Pa x -,
C n x x x+n\t\
x+n:t\
V p
n x , x
A — P -
x+n:t\ x+n:t\
[chap, xviii.
which may be explained by general reasoning. A -t is the
single premium at the present age for a sum assured of 1 payable
at age x + n + t or previous death. Therefore V is the single
V
premium for a sum assured of . n x . Also P , r , is the
* A — x+n:t\
x+n : 1 1
annual premium for a similar assurance of 1, and therefore P is
P
the annual premium for a sum assured of = — - — . Together
x+n:t\
these two sums make the total sum assured under the policy as
altered.
46. It is sometimes desired to apply the bonus on a whole-life
policy so as to limit the future premiums or to alter the policy into
an endowment assurance.
Mr Manly has put forward the following formulas in answer
to this problem, and has also supplied tables to facilitate their
application in practice.
If the future premiums are to be limited, let x v x v x 3 , etc.,
be the ages of the life assured at the successive periods of division
of profits ; y v y v y & , etc., the ages, at and after which the pre-
miums are to cease ; and Bj, B 2 , B 3 , etc., the amount of the
reversionary bonuses.
Then at the first investigation, we have
PN ,
X J/, -1
B,A =
i *i D
x 1
from which y 1 may be obtained.
chap, xvm.] TEXT BOOK— PART II. 391
At the succeeding investigation, we have
P(N -N n )
* v "2 - 1 Vl~ lJ
B *\ = D
from which y 2 may be obtained.
And we may proceed similarly at the following investigations.
If the policy is to be altered into an endowment assurance,
let y v y v y s , etc., be the ages at which the sum assured will be
payable, and let the other symbols have the same meaning as
before.
We have at the first investigation
N j
B,A = (P +d)-^—
la;, v x ' J)
from which we may obtain y v
At the next investigation
N -N ,
B A = (P +d) "-^ ^
2 x„ V- X ■> J)
H
from which y 2 may be found.
And we may proceed similarly at the succeeding investigations.
EXAMPLES
1. Given P 25 = -01521, P 42 = -02654, a 42 = 15-5679, find the
value of a whole-life policy for £1500 effected at age 25, which
has been 17 years in force, and to which a reversionary bonus of
£383 is attached.
Value of policy = 1883 x A i2 - 1500 xP 25 (l + a i2 )
= (1883 x P 42 - 1500 x P 25 )(l + a i2 )
= (1883 x -02654 - 1500 x -01521)(1 + 15-5679)
= (49-975 -22-815)16-5679
= 449-984, say £449, 19s. 8d.
V -
% xt\
392 ACTUARIAL THEORY
2. Give as many formulas as you know for V -
x+n:t-n\ xt\^ x+n:t~n-l y
p (N -N , ,)-(M -M _,_ )
xt\^ x-1 x+n-1* ^ x x+ns
D ^
x+n
*■ xt\ '>■ x+n:t-n-\\ J
{ x+n : t-n\ ~~ xT\ A + a 'x+n : t-n-iy
P
x+
1+a
CHAP. XVIII,
A (l ^U ^
K x-\-n:t-n\'
1
x+n:t-
1 +a
a; : £ - 1 1
x:t-l\ x+n:t-n-l \
l+a x:t—l
1-A
= 1-
x+n:t-n\
1-A
xt\
■A. : — A. — ,
x+n:t-n\ xt\
= 1
P
P , ; — i+d
z-\-n:t-n\
x+n:t-n\
' xt\
P , 7 i+<*
x-t-n\t-n\
= * C 1 l'irtlX 1 l V a:+l:t-i|)' '•( 1 1 *+»-l:<-»+l|'
3. Find formulas for V , V -,. and ,V , when the rate
of interest is zero.
V
n x
1+a
1 +a
— (when interest is zero) 1 -
1+e
x+n
l+e
chap. xviii.J TEXT BOOK— PART II. 393
1 +a
y _ __ i x+n:t-n-l \
1 +e 1
= (when interest is zero) 1
l+e x:t=T\
,V = A , - P(l+a , ; -,)
n:t x x+w t x^ x+n\t-n~~\\'
l+«.
= (when interest is zero) 1 x+n-.t-n-i\
+ e x:T=T\
A 1
since ,P = •= = -= when interest is zero.
' " l+(l x:t^\ 1+e *:*—
It will be observed that the formula for limited-payment whole-
life assurance is the same as for endowment assurance, so long as
»<(in the limited-payment assurance.
4. Under a policy taken out at age x which has been n years
in force, the sum assured and bonuses amount to S. Prove that
the value of the policy is equal to (s + ? x + ~ J A x+n - ( P x + -A
Value of policy = SA, + . - P,(l + %+ J
P(l-A _,_ )
C A, xV x+n'
~ x+n 1
= SA ^ - P (1 - A rMn ) (l + V)
x+n x^ x+n' \ i J
P P
= fs + P + -J)a ^ -(? + -J)
5. There are 44 policies of 1 each, all effected at age 40, which
have been in force 1, 2, 3, etc., up to 44 years respectively.
The sum of their values is 17-52789. Find the values separately
of the aggregate sums assured, and of the future net premiums,
having given A 40 = -379434.
By Text Book, formula 18, we have
2A = 2V(1-A ) + rh
x+n aA x' x
394 ACTUARIAL THEORY [chap. xvih.
Therefore
2A,„^ = 2V„„(1-A„„) + 44A -1 „
40+71 40^ 40' 40
= (17-52789 x -620566) + (44 x -379434)
= 10-87721 + 16-69510
= 27-57231
which is the value of the aggregate sums assured. And the value
of the future net premiums
= 2A„, -2V,„
40+ra 40
= 27-57231-17-52789
= 10-04442
6. Explain under what circumstances n V x < „_jV j, and give
an example from some known table of mortality where the
anomaly occurs.
V < V
"IX 71-1 £+1
if 1 _ x + n <; 1 — 3+"
if
a a , ,
X x+l
a a , ,
X ^ x+1
a
x+n x+n
if % < %+!
which is very unusual, but does occur at the infantile ages of all
mortality tables, and at very advanced ages of some mortality
tables which have been badly graduated, e.g. the Carlisle Table.
7. Find the reserve value after n years of a pure endowment
policy effected at age x to be payable at age (x + f), with premiums
limited to r and to be returned in event of death before age (x + 1).
The net premium for such a policy was found on page 324.
The reserve value after n years is
(1) n < r.
Prospectively
D , , nM ^ + R , - R ^ - rM J _,
*+« + {„-(! + K ) + c } _.*+» *+* *+<-._ f 'd
D ' ' v ' ' D ^
N , ,-N , ,
_ %+n-l s-fr-l
x+n
chap, xvm.] TEXT BOOK— PART II. 395
Retrospectively
N - N R - R - nM
v x-1^ *+*- 1 -{ ff (l +){ ) + c }_* *+" *+"
(2) n = or > r.
Prospectively
D J _, M - M
_£+' + r ^(l + K ) + c J_^±^ *f«
Retrospectively
N , 1 - N r4- 1 R - R - rM _,.
t - 1 «+ r - 1 -{ a -(i + K ) + c }_; **: £+?
In each case the values found by the two methods may be
proved equal, and it will form a useful exercise to do this as shown
for other similar problems.
8. Find the annual premium to secure an endowment assurance
to {£), maturing at the end of t years, with a guaranteed bonus of
£2 per cent, for each year completed ; and also the reserve value
at the end of n years.
Benefit side
M -M , + D ,, R R (f-l)M t +ID L1
x x+t x+t , .no x+1 x+t V J x+t x+t
D + ° 2 D
X X
Payment side
N , - N ^ ,
p J-l X+t-1
D
X
whence equating and solving
p M - M, + , + D x+ , + -02 {R +1 - R, +i - (I - l)M f+t + /D, +( }
N ,-i- N .+«-i
The reserve value after n years is
Prospectively
M s + ,- M s +( + D * +( . .no " M , + .+ R « + . + l- R . + «-( < - 1 ) M x + « + z.
1 0(8) N ^ +1D J
Here tt = ^A-l = «+' ± * +s
l + a x:7=T\ N ,-i- N «+,-i
and reserve value of policy
(1) n z.
N +4-D
(o) Prospectively « +n * *+*
10. What annual premium should be charged for an assurance
of £1000 to (.r), the premium being successively reduced by ^ of
the first premium and ceasing altogether after the tenth payment ?
What is the reserve value of the policy at the end of 6 years ?
CHAP. XVIII.]
TEXT BOOK— PART II.
Benefit side
1000M
X
D
a;
Payment side
D
1000M
^-i^toC^-Sx+io)
The reserve value at the end of 6 years
4 N — 2 (S S ^
S+6
397
11. Find the single and annual premiums for an assurance
upon a life aged x, the sum assured increasing in amount at
compound interest at rate j. What would be the policy-value
after t years?
The single premium is
(1 +j)M j +/(l +j)M x+1 +j(l +j)>M x+2 + ■ ■ .
D
X
And the value of this policy after t years is
(1 +jJ (1 +J)M, +( +j(1 +j)M, +t+1+ j(l +j)^M x+(+2+ ■ . .
x+t
To find the annual premium, we have the benefit side the
same as the single premium above, and
N
X
Payment side = P ■ *
Hence P
The value of this policy after t years is
n ^ - v (1 +j)M ar+t+ j(l +i)M, +t+1+ j(l +j) 2 M, +(+2+ ■ ■
_ p a+t-i
398 ACTUARIAL THEORY [chap. xvrn.
Or again, the single premium is
i
X
Find J, such that = ^ = , — 4. Then the single premium
1+ J l+i & r
= A'^ calculated at rate J, and the value of the policy after t
years is (1 +jJA' x+t .
A
In the case of the annual-premium policy we have P = - — — .
l+a
X
The value of this policy after t years is (1 +j)*A J — Pa .
12. A policy by annual premium was issued n years ago for
a reversionary annuity to (x) after (y). Find the reserve value at
the present time.
(1) (a) still alive Va i = a , i , - Pa i (1 + a )
v / w n y\x y+n\x+n y\x^ x+n:y+n'
(2) (y) dead Va i = a
\ / v-7 J n y\x x+n
13. Find an expression for the value at the end of n years of a
contingent insurance payable if the survivor of (x) and (if) die
before (z).
It must be ascertained whether (x) and (y) are both alive.
Then if both are alive
Vl = A - Pi (l+a 1
n xy\z x+n :y+n:z+n xy.z^- x+n:y+n:z+ns
If (x) is dead,
Vi = A-J Pi (l+a )
n xy'.z y+n:z+n xy:z^ y+n:z+n'
And, similarly, if (y) is dead,
Vi- = A J Pi (I + a )
n xy'.z x+n:z+n xy:z K x+n:z+n'
14. Obtain by the retrospective method a formula in terms of
annual premiums for V 1 - , and prove by general reasoning and
algebraically that
P -Pi-,
y a: xn\
% * P —
r xn\
chap, xvm.] TEXT BOOK— PART II.
399
PV,(N ,-N )-(M - M 1
n xt\ jj —
x+n
pi
M -
X
M
IC+71
xt
N ,-
x-1
N
N
3
D _,_
P 1 -
xt\
,-N
;-l x
P1 -i
xn\
+»-l
p n
Similarly, V = xK "- 1 5±
P (N - N ,)-(M - M )
p -pi-
x xn\
xn\
These results are correct, for the risk already undergone is
that of a temporary assurance for n years, which would be covered
by an annual premium of F^— ; the surplus premiums paid are
therefore (P^ - P^) and (P^- P^) respectively, which obviously
must have sufficed to purchase certain amounts of endowment at
age (.r + n). Now, P i will secure an endowment of 1 ; therefore
( P ir| _P kl) and ( P x _P ^) wiU secure endowments of
pi _ pi p _ pi
*' 1 "' and -JL-jSil respectively.
xn\ xn\
These, then, must be the accumulated reserves under the two
policies.
Likewise, in the case of the endowment assurance we should have
and when n = t,
n xt\
V -
t xt\ p _1
xt\
xt\
£71 1
pi l
xn\
xt\ '
-P 1 -
Xt\
= 1
showing that the accumulation of overpayments will exactly
amount to the sum required at maturity.
400 ACTUARIAL THEORY [chap, xviii.
15. Given at 31 per cent. V = -50084, and P = -05917,
* r n x 7 x+n '
find the premium for the age at entry (P ), the value of the sum
assured (A ), and the amount of paid-up policy equivalent
to V .
Since
V
n X
=
p , -p
x+n x
P , +d
x+n
therefore
P
X
=
p (i _ V)-dV
&+«\ n xJ n x
=
•05917 x -49916 - -03148 x -50084
=
•01377
A ,
x+n
=
P^
x+n
x+n
=
•05917
•05917 + -03148
=
•65273
w
=
V
n x
A ,
x+n
•50084
•65273
—
•76730
16. Having given that at 4 per cent, interest, P 20 -^=-01615,
and \o:^ = - 51078 > find 20 V 2 :40J and 2o( FP )20:40|
V _=A P -a —
20 20:40| 40:20| 20 :40| 40:20)
= -51078- -01615 x 1 ~ '^J 8 since a = ^
= -51078- -20543
= -30535
V -
(T?T>\ _ _ 20 20:40|
20^- '20:40| 4 —
40:201
•30535
■51078
= -59781
chap, xviii.] TEXT BOOK— PART II. 401
17. A whole-life policy was effected n years ago at age x, and
the sum assured is now to be reduced by half, the value of the
rest of the policy being applied in reduction of the annual premium.
Find the future annual premium.
(a) We may take half the premium formerly payable and
deduct therefrom the value of half the policy divided by the
annuity-due for the rest of life.
Thus, iP a -^ = R-KP.+.-PJ
x+n
= p -!?_,
x * x+n
(6) Or we may take the premium at the present age for a
policy of \) and deduct therefrom the whole value of the old policy
divided by the annuity-due at the present age.
V
Thus, «> --2-i = «> -(P^ -P)
3 * x+n a * x+n v x+n x*
x+n
= p -I? ^
x * x+n
(c) Or we may reason that the value of the present policy
must equal the value of a sum assured of \ less the value of the
future premiums. Thus, if P be the future annual premium,
V = 1A ^ - Pa ^
n x * x+n x+n
and Pa , = U - V
x+n * x+n n x
* 2 x+n >• x+n x* > x+n
hence P = P - AP ,
x * x+n
18. (x) and (y), who are insured under a joint-life policy for
£1000, desire at the end of n years to have it converted into two
single-life assurances for £500 each. What premiums will be
payable by (x) and (#) respectively ?
The joint-life policy has acquired a value of ^V^ of which
(x)'s share is V 1 and ( y)'s V 1 , these two parts together
*• J n xy \J / n xy A
making up the whole value of the policy. Now, if from the
premium payable by (x) at his present age for a £500 policy we
deduct his interest in the old policy-value spread over the whole
2c
402 ACTUARIAL THEORY [chap. xvih.
of his life, we shall arrive at an equitable premium to be paid by
him. Thus we have for (z)'s new policy for £500 a premium of
1000 V i
500P 2-J2
x+n a
x+n
And similarly for («/)'s new policy the premium is
1000 Vi
500P , !LJ*
y+« a
v+n
19. A whole-life policy for £1000, effected at age 20, has been
thirty years in force, and has accumulated bonus additions of £450.
It is proposed to devote part of the bonus to convert the policy
(including the remainder of the declared bonus) into an endowment
assurance maturing at age 65. Using the H table at 3 per cent.,
and ignoring the question of loading, find how much bonus will
remain attached to the policy after the alteration has been made.
The office has in hand at present 1000 30 V 20 + 450A 5Q . If X be
the amount of bonus to be surrendered this will be reduced to
1000 V +(450- X) A , and it will require to have in hand for
the new contract
1000(A 50:1 ^-P 20 a 60:1 ^) + (450-X)A 60: ^
The difference between these two reserves must be the present
value of the bonus to be surrendered. That is
XA 60 = 1000(A 50:1 ^-P 20 a 60:1 ^)-1000 30 V 20 + (450-X)(A 60: ^-A 60 )
x .. "MOV,-;,- 1000P 20 a 50: ^- 1000 80 V 20 + 450(A 60:llq - AJ
A —
50:15|
685-47 - 154-10 - 353-53 + 53-70
•68547
= 337-783
= £337, 15s. 8d. approximately.
Bonus amounting to £112, 4s. 4d. will therefore remain attached
to the policy.
At future divisions of surplus the whole-life bonus only will be
applicable to the policy, and it will have to be converted into
endowment assurance bonus by simple proportion.
chap, xvin.] TEXT BOOK— PART II. 403
20. Give the prospective and retrospective values after n years
of a child's endowment policy effected at age x, payable at age
(.r + 1), with premiums to be payable only so long as the father (y)
lived along with (x) and to be returnable in event of death of (x)
before age (x + f). Prove the formulas identical.
Prospectively —
If the father is alive to-day, which will happen in I x jp
cases of the survivors,
D^ Mi+^o + cjr M )+ ..
T\ ' ry I \ /V x +n x+u •'y+TiV x+n+\ x+t'
- • - 1+71
i+» x+n
N -N
i+n-l:;/+ro-l i+t-l:y+t-l
x+n:y+n
If the father died after n payments : ^ +m ( 7l _ 1 P !/ - n P y ) cases,
x+rt x+n
If the father died after {n - 1) payments : l x+n { n _ 2 P y ~ n . x P y )
cases,
£±1 + { ,t(1 + k) + C ^ ^p ^
etc. etc.
If the father died after 1 payment : Z j+b (1 -p y ) cases,
n M - M
£±i + {,r(l + ,c) + C }-*g '-±*
z+71 z+7t
The sum of all these cases is
M...-M... M.,„-M M -M , +t
" + 2Py D~
x+n x+n x+n
M...-M.,, M x+ , +1 -M j+ M,+ t -x- M ,+A
x+n *+» I+ "
ULd^ + {,r(1 + K) + c} I — S +p y^-jT + » p » — d~
i+7i-l:a+7i-l _ s+ t-liy+t-l |
~ ^ D~ ~ J
404 ACTUARIAL THEORY [chap, xviii.
Retrospectively —
*{j x (i +iy+i x+1 x p v (i +iy- i +i x+2 x 2 p y {\ +o-2 + .. .
/M - M _,_„ M _-M ,
+U-i x .-iP,( 1+ ')}-Mi + ")+^( ' B , + . + + y» %-h>' + '
i"y v x+n n-l"y v x+n J
N , ,-N , , /M -M ^ M ,,-M ,
/ ^z+m i <■ ' > \ v x+n ^y v x+n
V N
2"y yX+n 1 »-l"j/ v x+n J
To prove the two final expressions identical, we must first
recall the expression from which the value of the premium was
obtained, page 315. Adapting that expression to the present case,
we have
D _■_, / M - M ^ M^.-Ikf, M, ,,-M,,
-rr+Ki + «) + . •/ I IV 1-1-11 x+u r y \. z+tc £+«''
+ p (M , - M , ,) + , n p (M L _,_, - M ,,)+•• •
n"j/\ x+n x+tJ n+lt'y^ x+n+\ x+i'
D I
+ v(M -M "M-irfN -N ^ £i± »±?~|
x+n:y+n y -*
= 8 i-K« + W 1 + K ) + c H(M I - M i+J ) + ^(M >+1 - M, +( )
+ 2^( M ,+2- M , +i )+ ■•• + t - 1 Py( M x+t-l- M x+t)\
-'"'V^x-l-.y-l' x+t-l:y+t-U J) - + 7r v. W x -l:»-l _ x+m-l:!/+«.-l) 7"
xy
= IT
- {< l +K) + c}{(M x ~ M x+n )+P y {M x+l -M x+ J
+ 2Py(. M x+2- M x+J+ ••• +n-lPy( M x+n- 1 - M x+J}]
N -N , /M - M , M ,,- M ,
* i-l:»-l jc+»-l:y+n-l f^Q , ^ , c \ / » »+» , „ «+l «+»
^ „x+ts I v T > T J I yx+n T -fy u a:+™
M , ,- M ^ .
4- . . . + V X+% X+ 1
T ^n-l-fj/ w a!+» /
which is the reserve by the retrospective method.
21. Mr G. H. Ryan formed for a particular purpose a hypo-
thetical mortality table by adding -01 to the H 3 per cent.
P at all ages. Show that the policy-values by such a mortality
table are necessarily less than the H 3 per cent, values.
406 ACTUARIAL THEORY [chap. xvnr.
By this hypothetical table
F F
■y/ x+n X
' 2; F +d
x+n
( p , + ,+-oi)-(P,+-oi)
p , -p
x+n x
But the true value by H M table at 3 per cent.
P , -P
n x P , +d
x-\-n
Therefore V < V for every value of x and w.
22. Each of I. persons effected n years ago at age x a whole-
life policy for 1 at an annual premium of P . Find the reserve
required to be held to-day in respect of the survivors. How much
of this is required for the lives that are still " select," and how
much for the now " damaged " lives ?
The total reserve required is
[x]+n\ [x]+n ~ \x]\x\+v)
Now of these I, , , survivors, I, , , are still select. The reserve
[aj+w * [x+n\
required for them will accordingly be
[a:+«]V lx+n] [x] [x+ny
The remainder of the lives are the "damaged," and they
number /, , - /. . ,. The reserve required for them will be
{< d lxl+n - d [x+n}) + V K d [x ]+ n+l ~ d [x+n)+l) + • • • }
~* [»]«(■ [*]+»"" lx+ny + V { [x]+n+l~ [x+n]+lJ~*~ ' ' ' I
(M r , - M r , 0-P ,(N r ,, -N. , ,)
_ (j _ ; \ v [x]+n [x+ny [x]^ [x]+n [x+ny
~ y\x]+n Ix+nV D - D
[x]+n [x+n]
chap, xviii.] TEXT BOOK— PART II. 407
The reserve for the "select" lives, together with the reserve
for the " damaged," will equal the total reserve required
/ (A -Pa ~) + (l -I V M+ n ~ 1 [*+»]) [»i( N r»i+n ~ N ri+«i)
[*]+» [£+>i]
= M [« + .]- p M N [ , + , 1 CM M+ ,-M [it+|t] )-P M (N M+)( -N ri|+Bl )
M -P N
__ M+» [s] [afl+n
23. A policy for a term of w years is granted at an annual
premium to a life aged x, with the option of continuance at the
end of the term as an endowment assurance, maturing at age
(x + n + 1), on payment of the normal yearly premium at age (x + n\
Find expressions for the value of the policy on a select basis,
(1) at the end of the (n - l)th year, and (2) at the end of the
(« + l)th year, assuming the option to have been exercised.
The premium payable during the first n years is
M -M CP P -VN -N \
p [x] [x]+n , V* [»]+»:» I [!+»]:« | A [x]+n x [x]+n+t ->
N r , - N r , L N r , - N r ^
[x] [x]+n [x] [*]+»
while the premium thereafter for t years if the option be exercised
is P
[x+n]:t\
(1) The reserve at the end of the (n - l)th year is
P C N M - N M+ .-i)-( M M - M W + -i)
(2) The reserve at the end of the (n + l)th year, the option
having been exercised, is
A p _ „
[*]+»+! :*-l I [x+n]:t\ [a;]+»+l:t-l |
24. If in the formula a = A + Be* the value of c be 1, what
effect is produced upon (1) /^ (2) ("^ and (3) ("H^?
408 ACTUARIAL THEORY [chap, xviii.
(1) If c = 1, then yu. = A + B, which is constant for all values
of x.
(2) It has already been proved that under such conditions as to
mortality
a = ~ (see page 114)
jx -\- o
and A = — ^— = (see page 218)
Therefore <»>P = £±i
p + d
= /*
(3) (»>V = A -W'Pa ,
^ ' n x x+n x x+n
IX. + 8 ^^ + 8
=
which is obviously correct, as the premium for each year will
meet the year's risk, and there will be no reserve to accumulate.
25. Prove that the expected death-strain under a whole-life
policy, subject to an annual premium payable throughout life,
increases with the duration of the assurance if, at all ages on the
basis of the valuation mortality table and rate of interest, A 2 a is
algebraically > iAa.
The expected death-strain in the (n + l)th year of a policy
effected at age x is q (1 - V ), and in the following year
a (1 - „V ). The strain will therefore be increasing, if
"x+n*- ~n+l x' °x+n+V* m+2 x>
that is, if a ^ ^±2+J < q J^+l
x x
a x+ji+l — Px+n a x+n+l *"■ a x+n+2~Px+n+i a 'x+n+2
0r if %+n+l - V + l >x+n < "x+n+1 ~ ( l +')V+1
that is, if A 2 a , > iAa ^
which is the condition desired.
chap, xvni.] TEXT BOOK— PART II. 409
26. At the commencement of a certain year a company has on
its books l x persons who have been assured for I each for n years,
and who will be subject throughout the year to a special rate of
mortality q x + w^ and in respect of the claims which occur amongst
them, the company undertakes to pay only the reserve values at
the end of the year. On the assumption that the extra mortality
will cease at the end of the year, and that it will not prejudicially
affect the lives remaining assured, state under what conditions the
company will make a profit from the arrangement, and find an
expression for the amount of such profit. (Assume the premiums
due at the beginning of the year.)
The office will have in hand at the end of the year
^ v .- + p ,-J( 1 +«) = / * H X »+l V «-. + d .
which under normal conditions is sufficient to meet its claims and
the reserve values required for the survivors.
Under the conditions stated the number of deaths is increased
to I (q +w ) = d +w I. but the amount paid to each is only
,V . Further, the survivors will be/ — (d +w l) = l ,—w I
n+l x-n ' x *• x x x' x+l x x-
Therefore to meet its actual claims and the reserves for the sur-
vivors the office should have in hand at the end of the year
Vj+l x x/n+1 x-n ^ x $ %Jn+\ x-n
= l n x ,,v +d x ,,v
x+l Ti+1 x-n x ?i+l x-n
The office will therefore make a profit, if
/ .+l x .+l V .-n + d . > l x+l X n+l V x-n + d x X n+l V X - n
that is, if d x (l- n+1 V x _J>0
which must always happen, as n+1 ^ x _ n is fractional.
Further, the amount of the profit must be d.(l - B+1 V j ._J.
27. The actual claims for the year in an office exceed the
expected amount. Does the difference represent the loss from
mortality during the year ? Give the reasons for your answer.
This difference does not represent loss from mortality. The
office has in hand for each policy on its books a reserve value of
certain amount, greater or smaller. A comparison merely between
410 ACTUARIAL THEORY [chap. xvih.
actual and expected claims cannot therefore indicate the profit or
loss from mortality. The loss to the office through the death of
any single life assured is not the sum assured, as would be implied
in such a comparison as that suggested, but the difference
between the sum assured and the above-mentioned reserve. It
is this difference, then, that must be taken account of, if we are to
make a true investigation into the profit or loss from mortality.
The reserve held by the office at the beginning of the year and
the net premium then paid, both accumulated to the end of the
year, provide two things : —
(1) The reserve value required at the end of the year ; and
(2) A contribution towards mortality risk.
The total of the contributions to mortality during the year
would require to be compared with the net loss to the office in
respect of the claims, i.e., the difference between the sums assured
and the reserve values as described above, the balance between
these totals representing profit or loss from mortality.
It is indeed conceivable that the actual claims in an office
might exceed the expected, and nevertheless a profit from
mortality result. For the claims might have occurred chiefly
among old assured lives where the reserve values were con-
siderable, and the actual loss to the office consequently small.
On the other hand, the actual claims might be well within the
expected, and yet there might be a loss from mortality, due to the
fact that the claims occurred chiefly amongst recently assured
lives where the reserves in hand were small.
In this connection it may be pointed out that in the annual
reports issued by insurance companies, one may sometimes observe
a table giving the distribution of the claims experienced according
to the age attained at death. Such a table, however, conveys but
little information regarding the mortality experience of the com-
pany. It is obvious that much would depend on a variety of
considerations, such as the average age at entry and the class of
insurance effected. A company transacting little whole-life
busineso, and that chiefly at the younger ages at entry, but doing
a large endowment assurance business, would tend to compare
unfavourably in such a comparison with another office doing very
little endowment assurance business, and a considerable amount
of whole-life business chiefly at the older ages at entry. We
could not conclude merely for this reason that the profit from
mortality in the first company is less than in the second.
chap, xviii.] TEXT BOOK— PART II. 411
28. Assuming that an office had on its books at the commence-
ment of a year a group of 1000 lives aged 40, each of whom was
insured under a policy for £100 (without profits) payable at age 55
or death, and effected exactly 10 years previously at an annual
premium of £3, 14s. ; also assuming that 10 of these become claims
(payable at the end of the year of death) during the year, the
remainder being still in force at the end of the year ; that the
office earns 4 per cent, interest on its funds, spends 10 per cent, of
its premiums, and makes an H 3 per cent net valuation ; find
the total profit to the office earned by the group during the year.
How much of this is (1) profit from mortality ; (2) profit from
interest ; (3) profit from loading ?
From Hardy's " Valuation Tables " we find
100 ,o V SO:25J = 100A « :1 lH- 100P 30: 2 -^0:i5i
= 66-847 -3-244x11-383
= 29-921
100 ll V M:»i = 100A «:iii- 100P »:»i a .l:iii
= 68-528 - 3-244 x 10-805
= 33-477
The accumulation of the fund is as follows : —
Fund at beginning of year, 1000x29-921 . . . 29921
Add-Offiee premiums paid, 1000 x 3-7 . . 3700
Less — 10 per cent for expenses . . 370
3330
33251
Add also one year's interest thereon at 4 per cent., the
rate realised ....■••• 1330
34581
Deduct the claims payable 1000
Actual fund at end of year 33581
From which deduct the office's liability, 990 x 33477 . 33142
Difference, being total profit earned by the group
during the year
439
412 ACTUARIAL THEORY
Total profit, as before . , ,
Made up thus : —
(1) The liability at beginning of year was
H M 3 per cent, net premiums, 1000 x 3-244
Interest for year at 3 per cent., the valua-
tion rate ......
[chap.
XVIII
■
439
29921
3244
33165
995
34160
Less — Claims payable as before . . . 1000
33160
Deduct — Liability at end of year as before . 33142
Difference, being profit from mortality . 18
(2) The office premiums were as before . 3700
The net premiums .... 3244
Difference, being loading , . 456
Less — Expenses as before . . 370
86
Add — Interest for year at valuation rate 3
Profit from loading . . . . . 89
(3) The interest realised was as before . 1330
Deduct — Interest already taken credit
for in heading (1) . . 995
Do. heading (2) . . 3
998
Difference, being profit from interest . 332
439
29. What is meant by an H and H valuation ? How do
the reserves by this basis compare with those required by Dr
Sprague's Select Tables ?
For the purposes of an H and H valuation, in the
fundamental formula
V = A .-P(l+a , )
n x x-\-n x^ x+n'
chap, xvin.] TEXT BOOK— PART II. 413
we insert values for the functions on the right-hand side of the
equation, as follows: P^— the H M net premiums are used
throughout; A^ and a. +B ,— when the policy is of less than
five years' duration, the values from the H M table are used,
but when five or more years have elapsed, the H M(6) values are
taken.
In the case of policies, which have been less than five years
in force, the H reserve is less than that required by Dr Sprague's
Select Tables at all ages at entry. For those policies of five or
more than five years' duration we may compare the formulas,
as follows : —
y(H M and H M ( 6 )) _ .(H M ( 6 >) _ p (H M ) (H M ( B ))
n. x ~ x+n x & x+n
and V = A -i
F+d
X
•
i
F
1+ T
F
F +d
Total outlay ....
v 2y
One year's interest thereon .
F x + d
/ p '\
4 + t)
F +d
X
F
1 + —
F +d
X
2. Where the life interest to (.r) is limited to n years, the formula
is on the same lines as Text Book formula (1). The policy to be
effected is an endowment assurance which will return the total
outlay with a year's interest at the end of the first year in which
no payment of life interest is made, that is, at the end of the
(n + l)th year or the year of previous death ; and we shall have
1
a xn\ ~ p
x:n+\
+ d
1
416 ACTUARIAL THEORY [chap. xix.
3. The problem as stated in Text Book, Article 13, is much more
frequent in practice than the problem to find the value of the
life interest, and therefore it will be well to state the matter from
that point of view.
Since =; , - 1 purchases a life interest of 1, a sum of 1 will
P + d r
F +d
purchase a life interest of ■= — -^ —., and the policy to be
\ a; J
effected is for = — ^ K . Then we have
Amount paid to vendor ..... 1
F
First premium on policy
Total outlay
The annual income is .
which provides for
One year's interest on total outlay
1
-(F
V X
+ d)
V
1
-( P 's
+ d)
p>
d
i-(p;+o
d
i-(p;+<0
d
i-(P'.+ <0
l-(F+d)
chap, xix.] TEXT BOOK— PART II. 417
4. In connection with the Reversionary Life Interest, formula
(2), as mentioned in Text Book, Article 51, was given by Mr Charles
Jellicoe. It is assumed under it that an annuity for the joint
lives is actually purchased or set up in the books of the office.
Dr Sprague, arguing that if the number of contracts entered into
is sufficiently large, no such procedure is required, or, as a matter of
fact, carried out, suggested formula (5), where one rate of interest
is assumed throughout, and which, without the correction for £
payable if (jj) die first, reads
1-(P' +d,.,)(l+a ,..)
y\ x F +d,_
* w
Or, if we are to assume a higher rate of interest till the life
interest comes into possession
1_(F +d ,.,0(1+ a ,.„)
y\ x P' +d...
X (t)
For a complete reversionary life interest we might use the
formula
&' \ = a —a
v\z % %y
P'
P' +d a *u
A ^ l-(P>^)(l+%)
- V 2) F x + d
l -(P +t Q(l + « )
= «* p-^
X
which agree with formulas (16) and (17) of Text Book, Chapter XIV.
If now we give „ ^ - (1 + a^) for an annual reversionary
charge of 1, we shall give 1 for a similar charge of
P' +d
In this case a policy must be effected
r-(F, + «0(i + O
f m . I Proceeding as before, we have
2d
418 ACTUARIAL THEORY [chap. xix.
Amount paid to vendor ..... 1
P..
First premium on policy
i-(p; + <0(i+v>
F +d
Price of joint-life annuity . . . — - — x „ , a
1 -cp.+^ci+o •
Total outlay
l-(F. + d)(l+0
The annual income, from the annuity
during the joint lives and from the P' + rf
life interest after (j/)'s death, is
which provides for
One year's interest »
on total outlay . 1 -(F. + ^l+a^)
Annual premium on p'
policy
l-(P, + d)(l+«^)
F +d
1 - (P's+^+y)
xy'
At (x)'s death the sum assured is received ^ j^r, — , » , . , r
which repays
The total outlay
i-(F.+«9(i+0
And one year's in- ^
terest thereon
i-(P>rfXi+v>
l-(F+d)(l+0
«»
i-(p;+rf)(i +%)
5. For the Absolute Reversion, Jellicoe's formula (8) assumes
as before that an annuity is actually or constructively purchased,
while Sprague's formula (9) rejects this as not in accordance with
customary practice, and adopts one rate of interest throughout,
assuming that reversions will be purchased in sufficient numbers to
warrant this procedure. It is true that a specially large reversion
chap, xix.] TEXT BOOK— PART II. 419
might throw out the average on which the latter argument rests,
but this merely indicates that the office roust avoid contracts
of such size, just as they have a, limit for the amount of assurance
on any one life.
As before, let us consider the reversion which can be purchased
for a given sum, as that is the more usual problem. A reversion
of 1 is purchased for 1 - d (1 + a ) ; therefore 1 will purchase a
reversion of -= r^ T . And an annuity of -, ^ R must
1-dQ+aJ l- x = Qogl x + logl x+1 +... + log/ a ,_ 1 ) + log*/(.r + ^Tl+... + JJT~l)
Therefore
2£ ~ ' log D x at rate j = 2£ " 1 log D x at rate i + (log v - log v) ^ ~ ^ % + '* ~ ^
2. A table of A may be formed with the help of Gauss's
logarithms in a way similar to that in which A 1 is formed as
shown in Text Book, Article 99.
A x = v % + v P A x+1
= ^-i) + ^A + i
where II = 1
X P x
Hence logA^ = logvp x + ]ogU x + [t](log A x+1 -logU x )
Starting then at the end of the table we have A u = v and
log A = logv. From logi; deduct logII M as tabulated (for
the Text Book table at pages 499 and 501); enter Gauss's table
with the difference as argument, and to the result add
426
ACTUARIAL THEORY
[chap. XXI.
lo g n W -2 and lo S v Pa>-2> and We haVe l0 S A o>-2- Fr ° m
log A w deduct logII w ; enter Gauss's table, and to the result
add log II U and logvp u3 , and we have logA u , and so on, to
age 0. Then take the antilogs, and the table of A is formed.
The table of A when formed may be simply checked with
the table of a . Fol-
ic
A X + A 3+1 + --- +K-l = ( V - da X ) + ( V - da X +l)+--- +(«-*»«-!)
3. Besides the method of tabulating P given in Text Book,
Article 56, we might enter annual-premium conversion tables
with a^ and so obtain P . as described in Chapter VIII. Or
again, we might make use of a table of reciprocals, which we should
enter with 1 + a, and from the result deduct d. Thus —
Age
(1)
a
(2)
1
1 + a
(3)
1 + a
(4)
Neither of these methods, however, is a continued method.
4. The arithmometer may be employed to form a table of A -j
in the same way as described in Text Book, Article 61, for A .
A preliminary table of the differences between the temporary
annuities must first be drawn up, thus : —
1 ■ a — rr — a . ■, — ^i
x:n-l\ x+1 : 7i — 2|
U x+l:n-2\ ~ a x+2:n^S\
a x+2:n-3\ ~ a x+a:n-4\
etc.
chap, xxi.] TEXT BOOK— PART II. 427
where the age at which the annuity ceases is always the same,
viz., .r + n - 1 .
Putting 1 on the slide and d on the fixed plate with the
regulator at subtraction, multiply d by (l+a — -.) and the
value of A -j will result. On changing the regulator to addition,
continued multiplication of d by the series of differences found as
above, will give the values of A n . — -, A , „ — K ., etc. For
' ° aj+l:»-l|' x+2:»-2|'
A , , — r , = 1 - d(l + a , , — -;) = A — + d(a — -, - a , , — -,)
etc. etc.
5. Instead of using the values of - Aa to help in forming the
table of policy-values as described in Text Book, Article 78, we
may use the annuity-due values themselves.
For V = l-f*±*
Therefore, putting — on the fixed plate, multiplying succes-
sively by a , a , etc., and using the "effacer" between each
operation, we get the values -^ 3 -l±l y etc., the complements
a a
X X
of which are the required policy-values.
6. The values of endowment assurance policies may be similarly
arrived at, since
V-r = 1-
% xr\ a
a , 1
xr\
Or they may be formed on the principles of Text Book, Article
78, since
x+n : r -n- 1| x+n+1 : r - n - 2 1
V — = V — +
lt+1 xr\ n xr\ l+a
x:r-l
428
ACTUARIAL THEORY
[chap. XXI.
A preliminary table, as for the tabulating of A^, must there-
fore be formed, consisting of
x:r-l\
-,;+l:r-2|
ir+l:r
— a
x+2:r-S\
a x+2:r-a\ a x+Z:r-i\
etc.
Then with the regulator at addition, and = on the
1+a — n
fixed plate, the successive multiplication by these differences will
give us ,V —„ „V -,, „V -;,
° 1 iff J ir' 3 xr\'
etc.
The results may be checked by addition for
V _ + V -+ . . . + V -
1 xt I 2 xr | r-1 rr|
1+a
a x+\:r-2\ a x:r-l \
■ a
»+2:r-3|
1+a
+ ... +
"x:r-l|
1+a
x:r-l\ ' a;:r-l| ' ~x:r-H
(. r - 1 >,:¥=l\-(. a .+l:7=»\ +a ,+i:7=t\ + ■••+ a x+ ,-,-.r)
1 + a *:^
7. The construction of tables of policy-values for limited-
payment policies is a slower process, as the premiums have to be
valued separately from the sums assured and the difference taken.
As a preliminary, a table of differences of annuity-values should
be formed, as in the case of endowment assurances.
Years in force.
Annuity A
n -
n-
n-
e
]
-2
-3
-4
j
x+n-1:T\
a x+n-Z:T\ '
x+m-4:3|
'x+2:»-8|
(I -
*+l:«-8|
a:+™-2:l|
~ a x+n-S:2~\
' a x+S:n-i\
' a x+2:n-3\
chap. xxi.J TEXT BOOK— PART II.
429
Putting b P t on the fixed plate, and multiplying by 1, we get
the value of the premiums outstanding at the beginning of the last
year of premium payment, i.e. ^ Then the successive multi-
plication of ? x by the quantities found above and their continued
addition will give the value of the premiums outstanding at the
beginning of each year down to the second. For
Pa _ = p -i- P n
n x x+k-2:2| m, i » i j+»-2:l|
» x a i+vi-3:sl "" » x*x+n-2:T\ + n x^ a x+n~3:2\~ "x+n-i-.T^
etc. etc.
The results may be checked by addition, since the total
The value of the premiums must be deducted from the corre-
sponding assurance value to get the value of the policy. The
total may be checked by addition, for it should be equal to
(A„ , , + A + . . . + A„ , „ , ), less the above summation of the
^ 2.-1-1, X-\-t. X-j-ib — 1'
values of the premiums.
The value of the policy after the premiums are paid up is, of
course, just the assurance value.
EXAMPLES
1. Show in detail how to obtain a table of annual premiums for
whole-life assurances from the values of q without constructing
the life table. Assuming a rate of mortality represented by a
constant addition of -01 to q according to a standard table, explain
how the required premiums could be approximately obtained
without special tables.
Write down in a column in reverse order the values of p x from
age a) - 2 downwards. From these values prepare a column of
log vp .
430
ACTUARIAL THEORY
[chap, xx r.
Then logvp a2 = log« w2
lo g«P B -J + W lo g fl a -2 = l °% a 0>-$
l0 g^ W -4 + W l0 g a a,-3 = l0 S a W -4
etc. etc.
From this last column pass to the values of a u
Enter annual-premium conversion tables with these values, and
obtain P
a a% , etc.
10 - 2' to - 8 J
etc. The following schedule exhibits the
process :-
Age
»«
l-(2)
= P,
log (3)
+ log»
log (4)
= log« x
log- 1 ^)
= %
1 ,
l + («)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
to-2
o) — 3
CO — 4
etc.
With reference to the second part of the question it was shown
on page 232 that it may be reasonably assumed that the addition of
a constant "01 to the rate of mortality will have the same effect as
an increase of - 01 in the rate of interest per unit. We may
examine this assumption with reference to an increase from 3 per
cent, to 4 per cent, in the rate of interest employed in annuity
values. The assumption is that in any table
w'.
1 ,
1-03 P *
1-03
W-
1-04
P
P x
1-04' *
1-
<7* =
9, =
1-03
1-04
1-03
01
\-0i% + 1-04
= q +-01 approximately.
CHAP. XXI.]
TEXT BOOK— PART II.
431
If; then, the premiums required are 3 per cent, premiums,
we shall obtain good approximations if we enter a 3 per cent
annual-premium conversion table with 4 per cent, annuities as
found by the method in the first half of the question, thus
following the formula
P' =
1+ | |