INSURANCE

A contract for the payment of a certain sum on the occurrence of a certain event. The term a urance is generally confined to those contracts under which a certain sum is to be paid on the death of an individual or individuals now living; while insurance is applied to those which provide for the payment of a sum on the occurrence of events not depending on the duration of human life, and which may never happen, such as the lo of ships at sea, the destruction of houses by fire, . A urances on Lives, are contracts which stipulate for the payment of a certain sum of money on the death of one or more individuals, in consideration of an immediate payment, or, more frequently, of an annuity or annual contribution, to be continued during the existence of the lives a ured. Contracts of this kind are of i.Timense importance to society. Every man whose income depends on his own life or exertions, and on whom others are dependent for support, must be sensible of the advantages of arrangements by means qf which, at a small sacrifice of immediate comfort, he is enabled effectually to provide against the casualties of life. They are of a totally different nature from gambling. Though nothing can be more uncertain than the continuance of an individual life, yet nothing is more invariable than the duration of life in the ma ; consequently, the exact value of life a urances can be calculated without any uncertainty whatever, and a man, by effecting an a urance, secures to his representatives, against the risk of accident, the advantages they would have from his enjoying his exact proportion of the average duration of life. Such transactions provide against destitution, and tend directly to the accumulation of capital; they will, therefore, be encouraged and protected in all well-governed communities. Method of computing the Value of A urances. The value of a urances on lives are computed in nearly" the same manner as those of annuities, the principles being the same in both cases. A tcfcle of mortality must first be selected, from which we deduce the probabilities of living over the different years of life. Having obtained these, and a umed a rate of interest, we proceed as follows: let the probabilities that an individual of a given age will hve over 1, 2, 3. 4, 5, . years be pi, p2, p3, 7>4, p., . respectively; also, let r be the rate of interest, and?.'=-,—:"f:^, and sup ASSURANCE. pose that the sum a ured is to be paid at the end of the year in which the life fails. Now, the value of 1/. to be received at the end of the first year is v, but it will not bo received if the life continues to the end of the year; and, as the probability that the individual will live over the year is />i, the probability that he will not live over it is 1—/71, therefore the value of 11. to be received at the end of the year, subject to the contingency of the life failing in the first year, is (1— pi,) v. The probability that the life will continue to the end of the second year is p^, and that it will continue one year only, pi, therefore pi — pi is the chance it will drop in the second year; and tho value of 1/. to be received at the end of the second year is t»-, therefore the present value of 1/. to be received if the life fails in the second year is (pi — p2) v"^. In like manner, the probability that the life will fail in the third year is p2 — pa; and the value of 1/. to be received at the end of three years is v^, therefore the present value of 1/. to be received at the end of three years, if the given life fails in the third year, is (p2 — Pi) i>^- The same proce is continued from year to year, till the probability of living over a year becomes nothing. Now, the whole value of the a urance is manifestly equal to the sum of all its partial values for the different years; therefore, denoting the value by I, we get I = (1 _p,) V -t- (pi —P2) V^ -I- {Pi—Pi) 1/3 + . or, separating this into two series, I = V {\ + pi V -If;j2 V- -^ p. 1/3 -i- .) — iP\ f + y>2 V"^ ■\- p^ V^ + .) But it is shown in the article Annuity that the series Pi V -\- p^v^ + ps v^ + . denotes the value of an annuity of 1/. on a life, the probabilities of the continuance of which are represented as above; therefore, calling this annuity A, we have I = « (1 -f- A) — A. Since v = j, ^., this formula becomes by substitution i = r^.(i-»-A). The sum now found js what ought to be paid down, in order to receive 1/. on the failure of the given life: but by far the most usual practice is to pay for the a urance by means of an annual premium, the first payment being immediate, and the others at the end of each succe ive year. Let tr be the annual premium; then the value oi all the premiums after the first is obviously the same thing as the value of an annuity of the same amount, and is, consequently, equal to!r A. Hence, the value of all the premiums is t + a- A, or a- (1 + A), which is nece arily equivalent to the a urance. We have therefore the equation sr (1 + A) = w (1 -|- A) — A, A whence!r = v — j, ^ This formula is very easily computed when we are in po e ion of a table of annuities, and it shows at once the annual sum which an individual of any age ought to pay, in order to secure to his representatives 1/. (and, consequently, any other sum) at his death. Temporary A urances. The values of temporary a urances, or engagements to pay a certain sum in case a given individual dies within a given number of years, are easily found from those on the whole of life. For example, let it be required to fiijd what sum ought to be paid for I/., to be received if an individual now aged 40 shall die within seven years. Let I be the value of 1/. to be paid on the death of a person aged 40, and I7 the present value of the same sum, to be paid on the death of a person aged 47. Seven years after this the value of an a urance of \l. on the death of the person now aged 40 will be 17; but the present value of IZ., to be received certainly at the end of seven years is v'^, and the probability that the life v/ill continue seven years is pn; therefore the present value of I7, on the contingency that the life will not fail within seven years, is p^ z»7 I7; subtracting this from I, the value of \L to be received certainly at his death, there remains I — p-j v'^ l7, to denote the value of the temporary a urance. This may be expre ed by the following rule. Multiply the a urance on a life seven years older than the given life by the present value of 1/., payable seven years hence, and also by the probability that the given life will survive seven years; subtract the product from the a urance on the given life, and the remainder is the value of a temporary a urance for seven years in a single payment. In order to find the equivalent annual payment, it must be recollected that the first payment is made immediately, and that seven payments are to be made in all; consequently, all the payments after the first are equal to a temporary annuity of the same amount for six years, or one year le than the given term; consequently, if -r represent the annual premium, and A', a temporary annuity of 1/. for one year le than the given term, the value of all the premiums to be received is sr -|- sr A' or jr (1 + A'), which by hypothesis is equal to the a urance; consequently, to find the annual premium, we have to divide tho value of the temporary a urance iu a single payment by 1 -f- A'. H [s. 110]