CHAPTER VI
ANNUAL PREMIUMS

1. We have found the value of a whole life assurance of 1 due at the end of the year in which (x) will die, representing the value by the symbol Ax. This is the single premium calculated at a given rate of interest i and on the basis of a given mortality table, which should be paid for the benefit; so that if we have a large homogeneous group of lives each aged exactly x and each paying Ax into a fund which will earn interest at rate i, the fund will be able to pay 1 at the end of the year of death of each member and the last such payment will exhaust the fund if the lives fail in accordance with the given mortality table.

Having deduced a proper Ax for this group of lives, we may find that some members cannot afford to pay the full premium in advance. Each such member might prefer to have his share of the cost spread uniformly over his lifetime.

In the case of the single premium we made each member of the group aged x pay the average cost of the amount assured. We now carry the same principle over to the case of annual premiums and say that the present value of what each member shall pay must be equal to the present value of what he shall receive. Accordingly if P. represents the annual premium payable by a man now aged x for a whole life assurance of 1 to be paid at the end of the year of his death, we have

Px.ax=A

assuming that the annual premiums are paid in advance each year. This principle that the value of the single premium is equal to the value of all the annual premiums is fundamental.

Ax is the cash value of the benefit to be received by the assured:

Px.ax is the cash value of the annual payments necessary to secure that benefit.