The present value, at the beginning of the year, of £1, (or unity,) to be received on the 2nd day of that year, should the individual be, alive, and be sick, is

The present value of the like sum to be received:

The sum of these values for each of the 365 days in the

year

3

1+ 365

the sum of which series differs only by a very minute fraction from

So that, at the beginning of the year, the present value

of £1, to be received on each day of the year following the age A, should an individual be alive and be sick, will be

365

or
r
2

1+

ازار

GOE. which is the same as dis

counting, ting, for half a year, so many pounds as are equivalent to the units contained in the sum of the probabilities, that an individual will be alive and sick on each of the 365 days in the year mentioned.q on nydw iadł.aottersbienos odi moît mosqqo cals lliw end? * # to 5787 Venom sit to tasayeq ort 99ted 291971st 900m to fasrotai of seiz * This, it will readily suggest itself, must be near the truth, from the consideration that the present value spoken of will be equal, or nearly equal, to the sum of the daily

At page 45 it was demonstrated that represented the mean

quantity of sickness experienced by an individual in one year, as well as the sum of the probabilities, at the beginning of the year, of his being alive and sick on each day in that year; and we have just seen that the present value of the probabilities of receiving £1 on each day of sickness in the year is the same, or nearly the same, as the units contained in the mean quantity of sickness experienced by an individual, discounted for half a year. If, therefore, the daily allowance in sickness were made £, in

1

7

equal to the

present value of the several daily probabilities of sickness, for the year in question, and that would be the present value of a daily allowance in sickness after the rate of £1 per week. If therefore

ea

in place of representing the mean quantity of sickness for the

la

year, in the denomination of a day, it were made to express the mean quantity in the denomination of a week, then £ dis

counted for half a year, would be the present value of a daily allowance in sickness, after the rate of £1 for a week; and as it was before stated to be universal in Friendly Societies, to regulate the allowances in sickness by the sum allowed for one week, it will be convenient to adopt a week as the unit by which to measure the mean quantity of sickness, as well as the corresponding allowance in sickness; although the claim to such allow→ ance does in fact arise daily.

Let us put e, for the value, at age A, of a daily allowance in sickness, during the following year, after the rate of £1 per week, ea+1 for the like value, at the beginning of the year corresponding with the age one year older than A; e.+, for the like value at the beginning of the year corresponding with the age two years older than A; and e.+ for the like value, at the beginning of the year corresponding with the age n years older than A.

a

2

probabilities (£) discounted for half the year, over the whole of which it is as

la

Then if it be required to find the value, at age A, of a daily allowance to be made during sickness, for a given number of years, we must consider, first, what is the value of the allowance to be made during the year immediately following the age in question; which, we have already shown, is the quantity expressed by e.. But the value, at age A, of e.+1, that is the allowance to be made during the year following the age one year older than A, is lessened by two circumstances; in the first place e,+1 is the value one year after the age A, consequently the quantity e.+1 should, if the value of it be paid at age A, be discounted for one year, and its present value, at age A, would, from that cause, be reduced to ea+1. (1+r)1. Again it is not certain that the individual aged A will live to enter on the age one year older; so that the lastmentioned value must be further reduced in the ratio that the probability of the life A living one year bears to certainty; which it will be borne in mind is represented by 4+1; by this fraction, therefore,

la

If we represent the sum of this series for n + 1 terms by E, we

obtain this equation

will be

la

Ca + 3 • (1 + r)−.

la

Ca + n ⋅ (1+r) →n

la. Ca +la+1 · Ca +1 • (1 + r)~' + la+2 • Ca+2 · (1+r)2

+la+3 · Ca+3 · (1 + r)−3

+la+n · Ca+n • (1 + ) ̃”.
~]

Then will E, represent the value, at age A, of a daily allowance to be made in sickness, until the age (A+1) n + 1 years older

a

than A, after the rate of £1 per week. The value at the age (A) one year older than A, of a like allowance up to the age n+1 years older than A, will be expressed by E; the like value at the age (A,) two years older than A, will be E+2 and the same value at the age (A) n years older than A, will be E This last value will consist of but one term, and be

"

identical with eat with y

1a (1)

Let E represent the sum of the first t terms in the series indicated by E, so that E will be the value, at age A, of the daily allowance in sickness for t years only; t being understood, in this case, never to be greater than n.

』

And let E represent the sum of all the terms, in the series E., after the first t terms. So that Eat will be the value, at age A, of the daily allowance in sickness after the first t years. is evident, from the definition, that E+Eag Ea

(1)

Then it

The value, at age A, of all the terms after the first terms

91015191

or Ea

Ca+(+1)

(t+r)~(+1) +la+ (+2) • Ca + (+2) · (1 + r)−(+3) + . . . . .

lata · Ca+■ ' (1 + r) ̄" ], which is equal to [(1 +)~' ×

[laté• Ca+i+la+ ( + 1) • Ca + ( + 1) • (1 + r)−1 +la+«+2) • € a + (t + 2) • (b) ... &c.

If we multiply this whole expression by

ing unity, and which will not therefore alter the value of the quantity multiplied, we obtain the following equation:

€a+ ̧‹ +\) · (1+r)~' + ' + « +9 • Ca+ (+9) · (1+r)2 + · · . &c. )];

but the series in this expression, that between the last brackets is identical with the series indicated by E., and is the value, at the age t years older than A, of a daily allowance in sickness to the to the age n+1 years older than A, and the first factor

multiplied by the probability that a life aged A will live t years, From the foregoing theorem we deduce this rule il gomollot

To find the value, at age A, of a daily allowance in sickness until the age n41 years older than A, to be entered on at the end of t years, multiply the value, at the age t years older than A, of a daily allowance in sickness from that age until the age n41 years older than A, that is Eat, by the product of unity discounted for tyears into the probability of the life A living t years. These conclusions will be true whether the age t+1 years older than A be the oldest age in the mortality table used or not.

If therefore we have computed a table showing the values of E a daily allowance in sickness up to age n + 1 years older than A, for the several years of age to the age n years older than A, we can deduce therefrom the value, at age A, of the like allowance in sickness to commence from the age t years older than A, up to the age n + 1 years older than A; and t may be taken for any number not greater than n.

a

a

a [t]

By the definition it was seen that Ea)+ Ea =E; therefore, Ea()= E. — E. m; therefore the value, at age A, of a daily allow, ance in sickness up to the age t years older than A, is equal to the total value, at age A, of a like allowance up to the age n + 1 years) older than A less the value, at age A, of a like allowance from the age t years older than A, up to the age n+1 years older than A. If we make t, in the last given theorems, equal to one year we shall have this equation E[] = (1+7). E for the

a[1]

la

value, at age A, of a daily allowance in sickness to the age n+1 years older than A, to be entered on at the end of one year. The value at age A of E, is the same as ea, which is the value of the first term in the series represented by E., as will be seen at page 87.

Still making t equal to one, year, we shall have E

E.; and substituting e, for E) and (1+r)-1.

for Eau we shall have, »: »

la

Eati

J

M

[1] + Ea (1) lat!

la

So that if we know the

popit t69067

value, at one year older than a given age, of a daily allowance in sickness from that older age up to any other age, we can computethe value of a like daily allowance, from the given age first mentioned, without the trouble of computing the value of each term in the series represented by E ghidsdog od bol The following Rule corresponds with the last given the T

theorem.