As an example of the manner of computing the net single premium for whole life insurance, let us take 1,319 insurances of $1,000 each, age 9o, and employ the Actu aries' Table and 4 per cent. interest.
Dividing this sum by 1,319 the net single premium is found to be $904.41.
A whole life insurance is thus seen to be a term insur ance for the term of possible life; that is, for a term of 1 years equal to the highest age of the table, plus 1, less the age at entry.
If one desires to pay premiums otherwise than once for all in advance, he must, of course, pay sums that will be equivalent in value to the net single premium. Thus, if he wishes to pay by means of an annual premium, it must be an annual premium equal in present value to the net single premium for the insurance.
In the case of a whole life insurance from age 9o, for instance, the following is the way to find the net annual premium, equivalent to the net single premium of $904.41, the first premium payable in advance, and subsequent pre miuTs at the beginning of each year if the insured has survived. Such is precisely the character of a life annuity due; that is : a sum payable at the beginning of each year in case one survives. Therefore, the net annual premium is a life annuity due; and if we know the present value of such an annuity of $1 per annum, it will be the present value of such a net annual premium of $1. But since the net single premium and the net annual premium are to be equivalent in present value, the number of dollars and cents in the net annual premium may be found by dividing the net single premium for the insurance by the present value of the annuity of $1. The present value of a life annuity due of $1 from age 90 has been found to be $2.485; the single net premium at age 90 is $904.41. The net annual premium then becomes $904.41 4- 2.485 = In like manner the net annual premium for a term insurance may be found by dividing the single premium for the same by the present value of a term annuity due of $1 for the same period. In computing the present value of the term annuity due of $1, it is to be remembered that it is a life annuity for one year less than the term, with $1 added for the advance payment.
The net annual premium, payable for a limited number of years, such as five, ten, fifteen or twenty payment life, may be found by dividing the net single premium for whole life insurance by the present value of a term an nuity dile of $1. The principle is the same in all cases; the net annual premium, whether for the same term of years as the insurance, or for a shorter term, must be equivalent in present value to the net single premium; that is, the net annual premium must be as many dollars as the present value of a life annuity due of $1 for the payment term, is contained in the net single premium.
So far as the mathematics of it goes, there could be a premium for a term insurance, payable during a longer term than that covered by the insurance; but two things make this impracticable, except under extraordinary con ditions : During the term of the insurance not enough would in such case be collected to pay claims; and, if the deficiency were advanced from other funds, security would need to be taken that the premiums falling due after the insurance expired would be paid, since there would no longer be any incentive to pay them.
The same reasoning applies, almost without change, to the case of furnishing insurance from birth at a net level premium for life, since such premium will be smaller the first year than the current cost, which would occasion a deficiency. Therefore, insurance at the earliest ages is sold on the natural premium plan, the premium being level but the amount of the insurance increasing each year.
Whole life insurances, with both amounts and pre miums definitely fixed in the contracts whether at single or at annual premiums, were first offered by the Equitable Society for the Assurance of LiVes and Survivorships, of London, which was organized for that purpose on the mutual plan in 1762, through the efforts of Thomas Simp son and James Dodson. Limited premiums were a later development. The plan was at first deemed so dubious, notwithstanding its mathematical demonstration, that by the advice of a parliamentary committee a charter was re fused the Society, which, in consequence, became a mere voluntary association.