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1855.] 239
On the Methods pursued at the Present Day for estimating the Value
of Contingent Reversionary Interests. By Robert Tucker,
Esq., one of the Vice Presidents of the Institute of Actuaries,
and Actuary to the Pelican Life Insurance Company.
[Read before the Institute the 26th February, 1855, and ordered by the
Council to be printed.*]
J.HE subject proposed for discussion this evening is one of con-
siderable importance to all who engage in the pursuits of an
actuary. The old practice of valuing life contingencies by a pet
table of mortality and at one fixed rate of interest has long since
been found to work unprofitably ; and parties seeking investments
in securities of this description have been compelled, for their own
safety and advantage, to adjust the valuation of the actuary before
determining the price to be given for the particular property under
consideration. If such a state of things is unsatisfactory to the
public, it must be equally so to the actuary to find his calculations
used only as a partial guide, instead of being, as in fact they ought
to be, the index by which a proposed purchaser may know the rate
of interest and the other advantages to be derived by obtaining the
security in question at the price fixed by the actuary. In the
ordinary transactions of a Life Assurance Office, where the fluctua-
tion in the rate of mortality is protected by the admission of large
numbers, it is necessary also to protect its pecuniary interests by
limiting the amount to be assured on each life. Even with this
precaution, it not unfrequently happens that particular years have
proved less favourable than might have been expected — not so
much from an increase in the number of deaths, as from the
average amount of the sums insured by the policies so lapsed being
greater than the average of the whole number. When, however,
the average of a number of years is taken, and the amount insured
upon any one risk is properly limited, these fluctuations become
less apparent and generally disappear.
If it be necessary to protect an Office by paying due regard to
the limit of its risks, how much more necessary is it to do this in
dealing with life interests not involving the usual considerations of
average ! It should be the duty of the actuary, in valuing all life
interests and reversions, to get rid of these contingencies in each
case ; or at all events to point out how they may be insured so that,
after the premium for any particular risk is provided for, there may
* This paper was read by way of opening a discussion on the subject to which it refers.
240 On the Methods for estimating [April
remain to the capitalist — whether an individual or a Company it
matters not — a fixed rate of interest upon the investment, as secure
as in an ordinary mortgage.
The importance of these considerations has heen pointed out by
Mr. Jellicoe in more than one of the many papers he has read
before the members of this Institute. It is to Mr. Jellicoe the
profession is indebted for a clear exposition of the mode of dealing
with isolated cases of contingent interests, and to him it properly
belongs to open the discussion this evening. My apology for doing
so must be the reason assigned to me by Mr. Jelhcoe — that I am
the latest writer on the subject, having furnished a letter " On the
value of contingent reversionary interests," which appears in the
last Number of the Journal of this Institute.
With this introduction, I proceed to notice some of the cases
of contingent interests which occur most frequently in practice.
They are —
1. A simple life interest, or an annuity on a single life.
2. A simple reversion, determinable on the death of a single life.
3. A contingent life interest, or an annuity on a single life
commencing on the decease of another life.
4. A contingent reversion, determinable on one life surviving
another.
A party purchasing or making an advance on a life interest
should secure his capital by insuring the life of the annuitant.
The value of a life annuity so secured is, as given by M.t. Griffith
Davies, -=— 1 where d is the discount of £1 for one year ; and
p is the Office premium per pound per annum.
Let the annuity be £100, age 40 ; the rate of interest required
by a purchaser for the use of his money, 5 per cent, j and the pre-
mium charged for the insurance, the Northampton Three per Cent. :
then J ^i 1 J x 100=11-256 x 100=^1,125. 12s., the value
of the annuity.
In addition to the sum advanced, it is usual to insure one year's
annuity; therefore in this case 1125-6 +100= 1225-6 will be the
sum to be insured, the annual premium on which will be 1225-6
x -033975 =41-639.
Now 1125-6 being the sum paid for the annuity,
and 41-639 being the first year's premium,
1167*239 will be the sum actually advanced, upon which 5 per
cent, interest is to be allowed to the purchaser for
the use of his money.
1855.] the Value of Contingent Reversionary Interests. 241
1167*239 X •05=58-361 =the annual interest,
and 41 - 639=the annual premium;
and the sum of these = 100-000 = the annuity purchased.
Suppose the annuity, instead of being immediate, is deferred
until the extinction of another life. Let A, aged 25 years next
birthday, be entitled to an estate producing £1,000 per annum on
the death of B, aged 65 years last birthday. What amount of
annuity should he grant for what may remain of his life after the
death of B, in consideration of an immediate advance of £1,000,
the lender to receive 5 per cent, interest for the use of his money,
besides the premium on the amount necessary to secure his outlay,
according to the Northampton three per cent, rates? Here, in
addition to the advance of £1,000, an annuity during the joint lives
of A and B must be purchased or charged, equal in amount to the
deferred annuity, in order to fulfil the conditions stipulated for of a
clear 5 per cent, for the use of the money, freed from all contin-
gencies.
Suppose the joint life annuity can be purchased according to
the Carlisle Table, interest 3£ per cent. The value of the survivor-
ship annuity being -j— 1— AB, we have
-^ 1=12-958
d+p
AB= 8-331
4*627=value of A's interest per £.
The sum to be advanced to A being £1,000, the equivalent rever-
sionary annuity to be granted by him will be
4*627~ 216 123 -
Now 216-123 X 8-331=1800-521=price.of annuity during the joint lives;
1000- =sum paid to A;
216*123=one year's annuity;
making together 3016-644, the sum to be assured.
3016-644 x -02404= 72-52 the annual premium.
1000-
1800-521
2873*041 = sum advanced, or presumed so to be.
•05
143-65205 = interest on do.
72*52 = annual premium.
216-172 = annuity as above, very nearly.
242 On the Methods for estimating [April
It occasionally happens, that instead of an immediate advance
being required by the " tenant in reversion," it is only an annuity
during the joint lives of himself and the " tenant in possession " —
that is, he desires to increase his present at the expense of his future
income. This does not differ in fact from the example last given :
the cost of the joint life annuity is the only sum advanced to the
expectant, the rest of the calculation being the same in both
instances. But it may be urged that, if the Company making the
advance also grant annuities as a branch of their business, they are
charging on the one hand 5 per cent, to the borrower, and allow-
ing in return only 3 \ per cent. This is true; but it should be
borne in mind that the transactions are separate and distinct — that
the Company on the one hand, as a Loan Office, advances the price
of the annuity, and receives it on the other as an Annuity Office ;
and it seems but reasonable that a fair rate of interest should be
allowed for the use of the money, and a fair per centage of profit
charged on the annuity. If the negotiation were a mere family
arrangement, then the relative values of the two annuities — that
is, the survivorship and joint life — should alone enter into the
calculation.
This method of dealing with contingent interests is, to my
mind, much more satisfactory than charging a higher rate of
interest and leaving the contingencies unprotected. Taking 5 per
cent, as a fair and remunerative rate of interest, and charging a
more moderate premium than that indicated by the Northampton
Table, no borrower need complain of being overcharged ; and no
Office need be damaged by employing its capital so profitably,
and at the same time adding to the number and amount of its
insurances.
It is scarcely necessary to occupy the time of the meeting by
going through an example of the two cases of reversionary sums
before alluded to^l-rf(l + A), and \-{p + d) (1+AB). The
principle involved is the same in all; and they have been fully
treated in the transactions of the Institute and under the head of
" Correspondence " therein. The advantage of the plan recom-
mended is to render a contingent life interest or reversion a mar-
ketable security, and to make it as available for the purpose of a
loan as an ordinary mortgage, an immediate life annuity, or an
absolute reversion : and this is done, as Mr. Jellicoe observes, by
rendering certain that which is uncertain ; treating each security
as an investment not subject to any contingency whatever, and
putting the holder of it into the receipt of a given rate of interest
1855.] the Value of Contingent Reversionary Interests. 243
so long as he retains it, and reproducing his capital at the termina-
tion of the contract.
There is one other subject which I desire to bring under the
notice of the members ; and although not (strictly speaking) within
the limits proposed for discussion this evening, it is so immediately
connected with it that a notice of the one would be incomplete
without the other : it is, the mode of dealing with isolated cases of
annuities certain. Like the ordinary tables of life annuities, tables
of the values of annuities for terms of years should not be too
implicitly relied upon as a guide to parties making occasional
purchases in this description of security who have not the means of
reinvesting their surplus annuity in the same way : the condition
of the ordinary tables being, that the reinvestments are made at
the same rate of interest as the purchase is supposed to yield.
Now as this is obviously a matter of much uncertainty, and in
isolated cases almost of impossibility, no correct idea can be formed
of the aqtual rate of interest realized until the amount of surplus
annuity is determined; that is to say, until it is known how
much will be annually required to reproduce the capital at the
end of the term, at some rate which may be safely assumed for the
purpose.
In a paper read before the Institute in November, 1850, Mr.
Hardy has given a table of the values of £1 per annum for a
certain number of years, so as to pay a fixed rate of interest on the
original purchase money, and to replace that purchase money at
the expiration of the term, at a different rate of interest.
The difference between this table and the ordinary tables of
annuities consists in the employment of two rates of interest instead
of one. That which a purchaser realizes, or is supposed to realize,
Mr. Hardy calls the remunerative rate; and that which is to repro-
duce the capital, the accumulative rate. In the ordinary table the
remunerative and accumulative rates are the same.
The connection between an annuity for a term certain and a
life annuity, and how one expression may be deduced from the
other, has been clearly shown by Mr. Hardy and other writers.
The same result may be arrived at thus : —
The value of an isolated life annuity is, as before explained,
s= j— 1. Here ft, the annual premium, is payable in advance,
and the sum insured includes this premium and the first year's
interest. Now suppose the premium to be paid at the end of the
year, and the annuity up to the day of death : then the sum to be
244 On the Methods for estimating [April
insured need only be the actual sum advanced ; we shall have,
therefore, denoting this annual premium by it,
7rs+j's=a, or s.(ir+i)=a
s= ; = ; when a=l.
ir + i ir + i
This expression is equivalent to the value of an annuity for a
term certain where ir is the surplus annuity at the accumulative
rate, and i is the annual interest at the remunerative rate.
By the aid of Corbaux's Tables of Compound Interest, and
Barlow's Table of Reciprocals, the value of : is readily ob-
tained ; it being given by Corbaux for every year, half year, or
quarter, according as the ratio of interest is annually, half yearly,
or quarterly. For example : Let three annuities be purchased, for
the respective periods of 10, 20, and 30 years, to pay 5 per cent.,
with the surplus to be reinvested at 3 per cent. The several values
of ir are, according to Corbaux, -08723, '03721, and- '02102,
*=-05 ; the successive values of -^ are ^^, ^^, ^j^,
respectively equal to 7'287, 11 '466, and 14'081, which will be
found to coincide with the values given in Mr. Hardy's table.
One of the most useful applications of this table is to test the
comparative advantages of an investment in perpetual or in ter-
minable annuities ; and I propose to close these remarks by one
example, taken from the prices of the Government securities as
given in Wetenhall's list.
On the 14th of October last, immediately after the payment of
the half yearly dividend, the Long Annuities had just 5 J years to
run; their market price was 4f, and the corresponding price of
Reduced 3 per Cents, due at the same time was 94.
Per annum.
£. s. d.
Now £100, invested in Long Annuities at 4§, will 1 oo 17 9
produce J
Deduct property tax at 14c?. in the pound .
leaves net
Required for sinking fund, at 3 per cent.
giving a net interest of
A similar sum, invested in Reduced, at 94, gives
Deduct property tax
leaves a net interest of
1
6
8
21
17
10
15
G
6
£3
15
3
3
3
10
8
£3
2
1855.] the Value of Contingent Reversionary Interests. 245
On comparing the price of Long Annuities with the ordinary
tables of annuities certain, it appears that such an investment
would be equivalent to a 6 per cent, purchase, and that a gain
of more than 2§ per cent, interest would be derived from selling
Reduced and buying Long Annuities ; but when it is found that
the property tax is levied on the whole annuity, and is equivalent
to more than 40 per cent, on the interest yielded by Reduced
instead of 6 per cent., and when the reproduction of capital has to
be provided for at a rate of interest which may be safely relied
upon, about 2 per cent, of the supposed advantage disappears, and
this would be further reduced by the payment of income tax on the
interest of the reinvestments.
Supposing there were no income tax, and the surplus annuity
to be still reinvested at 3 per cent., then if
*'=' 06 ^ = Wik™ = ^77= 4 - 207 '
'•=-° 51 ^7 = IttJ^I = ^87 = 4 ' 3735
From which it appears, that an annuity of £1 for 5 \ years certain,
which can be bought for the sum of £4<. 7s. 6d., including com-
nnssion, would pay the purchaser (upon the supposition above
stated) £5. 2s. per cent, for the use of his money, and enable him
to reproduce his capital in a 3 per cent, stock at par.
On the Sickness and Mortality amongst the European and Native
Troops of the Madras Army.
WE owe to the courtesy of W. H. Scales, Esq., a medical
officer in the East India Company's service, Madras Presidency,
the following series of. tables on the sickness and mortality
amongst the European and Native troops in the Madras army;
and as, from the extension of life assurance in our Indian posses-
sions, every information of this kind is of value, we present a
summary of them to our readers. In the original tables the num-
bers " treated " and " died " to the " strength " are given for each
year, as well as the sickness and mortality according to length of
service ; but for life assurance statistics the facts here recorded are
the most important. We would especially draw attention to the
returns classified according to ages.