Probability, Mortality and Life AssuranceAuthor(s): Philip RobinsonSource: Mathematics in School, Vol. 26, No. 5 (Nov., 1997), pp. 42-45Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215334 .

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Probability, Mortality

Life Assurance

by Philip Robinson

The aim of this article is to show how some very important practical lessons in commercial arithmetic may be given using only basic mathematics. Most pupils will have some form of insurance policy in the future and it is hoped that some real understanding may be gained by examining the mathematics involved.

Expected Values The idea of an expected value is intuitive and not difficult for most pupils to understand. I find that it is enough to ask them how many times they might reasonably expect a five, say, to appear if they roll a die 60 times. The answer is given quite quickly but it takes a little longer for pupils to formulate that they have multiplied the probability of obtaining a five by the total number of throws.

i.e. E(X) = Np (X)

It should be made clear that this is not a golden rule but rather, the value of E(X) is approached as N increases.

Present Values and Discount Factors

Writing the well known formula for compound interest (which might have to be demonstrated) we have

Pn = P(1 + iOn

where Pn is the value of P units invested at i% per annum, for n years. Our problem is to find the present value of the known value of a sum n years from now. Arranging the compound interest formula we have

Pn P =

(1 + iOn

P is called the present value of the sum. The factor - is (l + z)n

called the discount factor. A simple example takes the sting out of this for pupils:

If I want a1000 in five years time and the building society offers me 6% per annum, how much should I invest now? (i.e. what is the present value of a1000 under these condi- tions?)

We have P(1.06)5 is the discount factor. Hence P = 1 1

where is the discount factor. Hence P = (1.06)' (1.06)'

a747.26. Such work involves pupils with their calculators and it is

instructive to have them construct discount tables. (See Table 1).

Table 1 Present Value Factors to determine the present value of single payment received

'n' years from the present (or vice versa)

Years 1% 2% 3% 4% 5% 6% 71 8% 9% 10%

I .9901 .9804 .9709 .9615 .9524 .9434 .9346 .9259 .9174 .9091

2 .9803 .9612 .9426 .9426 .8070 .8900 .8734 .8573 .8417 .8264 3 .9700 .9423 .9151 .8890 .8638 .8396 .8163 .7938 .1722 .7513

4 .9610 .8238 .8885 .8548 .8227 .7921 .7629 .7350 .7084 .8830 5 .9515 .9057 .8626 .8219 .7835 .7473 .7130 .6806 .8499 .8209

0 .9420 .8880 .8375 .7903 .7462 .7050 .8863 .6302 .5963 .5645 7 .9327 .8706 .8131 .7599 .7107 .6651 .8227 .5835 .5470 .5132 8 .9235 .8535 .7894 .7307 .6768 .8274 .5820 .5403 .6019 .4665 9 .9143 .8368 .7664 .7026 .6446 .5919 .5439 .5002 .4604 .4241

10 .9053 .8203 .7441 .6756 .6130 .5584 .5083 .4632 .4224 .3855

11 .8963 .8043 .7224 .6496 .5847 .5268 .4751 .4289 .3875 .3505 12 .8874 .7885 .7014 .6246 .5568 .4970 .4440 .3971 .3555 .3186 13 ,8787 .7730 .6810 .8006 .5303 .4688 .4150 .3677 .3262 .2897 14 .8700 .7579 .6611 .5775 .5051 .4423 .3878 .3405 .2992 .2633 IS .8613 .7430 .8419 .5553 .4810 .4173 .3624 .3152 .2745 .2394 16 .8528 .7284 .6232 .5339 .4581 .3936 .3387 .2919 .2519 .2176 17 .8444 .7142 .6050 .5134 .4363 .3714 .3160 .2703 .2311 .1978 18 .8360 .7002 .5874 .4936" .4155 .3503 .2959 .2502 .2120 .1799 19 .8277 .6864 .5703 .4746 .3957 .3305 .2765 .2317 .1945 .1635 20 .8185 .6730 .5537 .4584 .3769 .3118 .2584 .2145 .1784 .1486

The table gives the present value of a single payment received 'n' years in the future discounted at x% per year. For example, with a discount rate of 7% a single payment of a1 in six years time has a present value of a0.6663 or 66.63p

(Extract from Table 33, Statistical Tables, Murdoch & Barnes. Macmillan 1974)

42 Mathematics in School, November 1997

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Table 2a

ftlc Edit Saelec Formal .Qptons

Charts aindow 8clp MS Sa- S* i

e A1/1.04

0.9615385 0.9245552

S i.

i i

.i. , "::

" a ,,,a ,; '

a -

--a.:=~ ,-:: ,

i .. :i a

pg ROMM

I 1 ,.

.. a ,a, ]a]aPa- .......... I

a:..'.a r ca iii~

Table 2b

ple Edlt Seled Format ,ptlons

(harts 4indow Uelp

a 0.96153851

0.952381

a:: 0.92455621"bl(1-05

i ,

-a

i.:._ a

0.8548042

"... !i;) ."

N] 0,8219271 l I : ' I:

i!!aii

~b~0.7 so3i45 ~`:;~ 0.7599178

0.7306902

'Uii 0.7025867 ~ 0,6755642

0.6495809 aaaa u!a 0,624597

-aai~ 0.6005741 a'aa

.-. 0.5774751 ,, iii~~a~ 0,5552645 ' 0.53390:2

0.5133732 _

I I

il a.aE a..a

0 61 R I aaau illaillinalia i Ii i:.i..iaa:. a a a `a..````a`..````L`a`aa``:``.a2;a;aa

a aaa?!aTa :-\a

?/i ::; :::i?!:ii:!i::~ "i ii~ :i:!iii~i.i::ii ::::iil i:iii.~~iiii:iii:iiii:ii:a:::ii.i: :.a;i:i :.:/ a:a.a:ia ai/a:;.ia? .i]ai./ i~a:ii~a:a~i~~ :i:ii,:ia... i-.a ~

.:::i. . .\::: :; :.;:i::i.!::i- ::ai:i : : / :;!.:.:,i :;: ais] :i i!!i

aa,a.""; aa-a"'-'a' a ....' . '"aa a-'"" ..... " " a --a aa-- , aa-! a ,", aN.:,x .a-

- aa a '. aN-g % aaa

a,a a L -.

a-:--'aa. -aaa- aaaaa aa-a-a- a-aaa-

a aa:<sa -N :aa a aa, , ,':,a..a , a

Some pupils might write a simple program to generate these values or simpler still, a spreadsheet will do the task very quickly.

Discount factors are very important in the commercial world. Debts can be sold at a discount, i.e. if a company is owed

a sum of money due to be paid in three months time, it can sell the debt on to a third party at a reduced value, usually less than its present value. i.e. at a discount.

Mathematics in School, November 1997 43

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Mortality Tables

Table 3

Age Males Females Age Males Females

x Ix qx Ix qx x Ix qx Ix qx

0 100000 .0 1271 100000 .00984 5 89152 .0 1098 93014 .00624 1 98729 .00085 99016 .00072 56 88173 .01224 92453 .00686 2 98645 .00051 98945 .00045 57 87094 .01361 91819 .00752 3 98594 .00038 98900 .00031 58 85909 .01509 91129 .00824 4 98557 .00035 98869 .00025 59 84612 .01670 90379 .00901

5 98522 .00032 98844 .00022 60 83199 .01843 89564 .00986 6 98490 .00030 98822 .00020 61 81666 .02028 88681 .01077 7 98461 .00027 98802 .00019 62 80010 .02229 87726 .01176 8 98434 .00025 98783 .00019 63 78226 .02448 86695 .01284 9 98409 .00024 98764 .00018 64 76312 .02687 85582 .01400

10 98385 .00024 98746 .00018 65 74261 .02949 84384 .01528 1 1 98362 .00024 98728 .00018 66 72071 .03238 83095 .01669 12 98338 .00026 98710 .00018

6. 69738 .03555 81708 .01828 13 98312 .00029 98693 .00019 68 67259 .03903 80214 .02008 14 98283 .00034 98675 .00022 69 64634 .04285 78603 .02212

15 98250 .00041 98653 .00026 70 6186-1 .04703 76864 .02443 16 98210 .00053 98628 .00030 71 58955 .05160 74987 .02704 17 98158 .00102 98598 .00033 72 55913 .05658 -2959 .02998 18 98057 .00111 98566 .00035 73 52749 .06198 70772 .03329 19 97948 .00102 98531 .00035 74 49480 .06783 68416 .03698

20 97849 .00093 98497 .00035 T5 46123 .07416 65886 .04110 21 97757 .00087 98462 .00036 76 42703 .08096 63178 .04566 22 97672 .00083 98427 .00036 77 39246" .08827 60294 .05072 23 97591 .00081 98392 .00037 78 35781 .09610 57236 .05637 24 97511 .00081 98356 .00038 79 32343 .10445 54010 .06271

25 97432 .00081 98318 .00039 80 28965 .11334 50623 .06982 26 97353 .00082 98280 .00041 81 25682 .12278 47089 .07779 27 97273 .00083 98239 .00043 82 22528 .13278 43426 .08669

28 97192 .00084 98197 .00045 83 19537 .14333 39661 .09661 29 97110 .00086 98153 .00048 84 16737 .15440 35829 .10750

30 97027 .00088 98105 .00052 85 14153 .16591 31978 .11922 31 96941 .00091 98054 .00056 86 11805 .17776 28165 .13160 32 96853 .00094 98000 .00060 87 9706 .18986 24459 .14448 33 96762 .00099 97941 .00065 88 7863

.20215 20925 .15772 34 96666 .00105 97877 .00071 89 6274

.21453 17625 .17116

35 96564 .00113 97807 .00078 90 4928 .22693 14608 .18468 36 96455 .00123 97132 .00085 91 3810 .23929 11910 .19814 37 96337 .00134 97649 .00093 92 2898 25153 9550 .21143 38 96208 .00148 97557 .00103 93 2169 .26374 7531 .22442 39 96065 .00165 97457 .00114 94 1597 .27632 5841 .23703

40 95907 .00184 97346 .00127 95 1156 .28971 4457 .24914 41 95731 .00206 97223 .00141 96 821 .30430 3346 .26096 42 95534 .00231 97086 .00157 97 571 .32044 2473 .27331 43 95313 .00260 96933 .00176 98 388

.33844 1797 .28715 44 95066 .00293 96763 .00196 99 257 .35853 1281 .30330

45 94787 .00332 96573 .00219 100 165 .38087 893 .32252 46 94472 .00376 96361 .00245 101 102 .40551 605 .34538 47 94117 .00425 96125 .00274 102 61 .43241 396 .37231 48 93717 .00481 95862 .00305 103 34 .46140 248 .40349 49 93266 .00545 95569 .00340 104 19 .49214 148 .43881

50 92758 .00615 95244 .00378 t05 9 .52414 83 .47780 51 92187 .00694 94884 .00419 106 4 .55667 43 .51960 52 91548 .00781 94486 .00465 1C7 2 .58874 21 .56277 53 90833 .00877 94047 .00514 108 I .61896 9 .60521 54 90037 .00982 93564 .00567 109 4 .64382

110 1 .67391

Definitions

I, are the number of survivors to age x of 100,000 live births of the same sex who subsequently experience mortality similar to that of the population of that sex in England and Wales in 1980-82.

d, are the number dying between age x and (x+l), described similarly to 1,, that is l,-i...

q, is the initial mortality rate between age x and (x+l), that is, d,/l,.

(Extractfriom English Life T'ables No 1,4 1980-82. tHMSO 1987)

44 Mathematics in School, November 1997

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Mortality tables are tables of values which give the prob- abilities of individuals living to different ages.

One of the most important factors for life assurance com- panies to consider is the probability of individuals living to a certain age, given their present age (and health). In particular, the probability that they will live to collect a sum of money from the company. Insurance companies must be sure that they have enough funds laid by or invested to pay off policy holders at a certain point in the future. Mortality tables help to clarify this situation (see Table 2). These tables show the number of males or females (they have different expected life spans and therefore different tables apply), out of one hun- dred thousand born in a certain year, are expected to be alive at any time in the future. We see from the table that lx is the number of males living at age x. dx is the number that have died and qx is the mortality factor. i.e. it gives the probability that a person aged x years will die before reaching the age of x + 1 years.

lxl The probability of surviving the year is given by

x.

lx+ Consequently the mortality factor is qx = 1

-x. lx The given table is for males and females whose daily

routines and jobs do not incur particular risks. Soldiers, lion-tamers, divers and so forth have a greater risk of dying and so special values are needed; these need to be calculated in each individual case.

Use of the table We see from the table that the probability of any male living to be 30 is 0.97027. If we wish to know the probability of any 30 year old male living to age 60 then we have

160 83199 0.85748

130 97027

This factor tells the company what proportion of males aged 30 who took out policies with maturation at aged 60, are likely to survive to collect their sum assured (the pay out). We see that for every 100 such policies the company needs provision for at least 86 policy holders. It is then required to know the present value of this sum per policy holder. So we have

Expected value = Sum Assured x Discount Factors x Mortality Factor = Present Value x Mortality Factor

Here we have Expected Value = a100 x 0.85748 x (1.06)30

= a14.93 Hence the company could not afford to sell such a policy

for less than a14.93 (plus administrative costs). The same idea applies to the calculation of annuities. An

annuity is bought for a single payment on the understanding that it will provide an income for life or a fixed period to be agreed. The company needs to calculate whether or not it will take more money than it pays out for a large number of policy holders. We can draw up a schedule for such an arrangement.

Problem: A man aged 65 pays aE75000 for an annuity which pays him a10000 per annum for 10 years or until his death, whichever is the sooner. If the working rate of interest is 6 a, decide whether this annuity is value for money in each case.

In order to make such a judgement we construct a table to find out the value of the payments now.

i.e. the expected value of the payments made by the company is a60711. The cost of this annuity, at a75000, seems rather high and at 6a, could not be considered a good investment even with additional administrative costs.

Mathematics in School, November 1997

Mortality Discount Expected payment Age Year Sum factor factor (to nearest a) 66 1 10000 0.97051 0.94340 9156 67 2 10000 0.93910 0.89000 8358 68 3 10000 0.90571 0.83962 7605 69 4 10000 0.87036 0.79209 6894 70 5 10000 0.83306 0.74726 6225 71 6 10000 0.79389 0.70496 5597 72 7 10000 7.75293 0.66506 5007 73 8 10000 0.71032 0.62741 4457 74 9 10000 0.66630 0.59190 3944 75 10 10000 0.62109 0.55840 3468

Total 60711

History

In 1706 Sherwin published his Mathematical Tables in which Halley (of comet fame) gave the investigation of the principal theorems dealing with compound interest and annuities. Halley included formulae for finding the rates of interest of annuities, something he had dealt with without explanation in his Discourse on Compound Interest. The details were pro- vided by De Moivre in a Letter to Dr Halley. [2] (Source: The Doctrine oflnterest and Annuities by Francis Baily of the Stock Exchange, London 1808)

Careers

Those interested in pursuing a career in this line of work should write to the leading professional body which concerns itself with problems of interest, annuities and pensions, the Institute of Actuaries, at Staple Inn, High Holbom, London. The profession is an all graduate profession and a degree in mathematics or statistics, or a degree with a high mathemati- cal content is required. To become a Fellow of the Institute of Actuaries it is necessary to pass a number of difficult examinations.

I have included a starter exercise for teachers interested in this work.

Exercise

1. Find the present value of a1 five years from now, at a 6a rate of interest. Under the same conditions, what is the present value of

a1000? 2. Calculate the probability that a 35 year old man will

live to be (i) 45 (ii) 60 (iii) 75 3. A widow aged 72 buys an annuity of a500 per

annum for 8 years, at a low cost of a2400. If the working rate of interest is 4a, use the mortality table to construct a table to help you decide the value of the annuity now. Is the annuity sold at a fair price?

4. Company A sells a debt of a5000 to company B, 2 years before the debt is due. The market rate is 5a and company B offers a price calculated at 8a. Find the difference in the offered price and the actual present value.

5. A saver pays a1000 per annum into a savings account on January 1, 1995 and on the same day each year thereafter for years. if the account pays 5a interest per annum, find the present value of the payments on January 1, 1995.

Answers:

1. 74.73p; a747.30 2. (i) 0.9816, 0.9874 (ii)

Author: Philip Robinson, Bedales School.

45

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