LIFE ANNUITIES

In this topic, you will begin to learn how life insurance companies make important calculations which combine the principles of compound interest and probability. In a life insurance company, annuity contracts represent payments being made only if a person is alive, while life insurance contracts represent payments being made only when a person dies. We will explain how to calculate the present value and the accumulated value of life annuity payments, that is, payments which are contingent on a designated person being alive.

Present and accumulated values for life annuities can be calculated in a manner quite similar to the method for annuities. We will begin by considering the present value and accumulated value of a single payment.

PAYMENT WITH BENEFIT OF SURVIVORSHIP:

Asking the question: “How much should a person now age 25 pay for the right to receive $100 at age 50 if that person is then alive to receive it?”

This is the same thing as asking:

“What is the present value to a person now age 25 of $100 payable at age 50, calculated with the benefit of survivorship?”

The phrase with benefit of survivorship is used to distinguish this situation from one where only rates of interest are involved. If only rates of interest were involved in finding the present value, the answer would be:

Basic equation for present value”

But now the element of survivorship is also involved because the person must survive in order to receive the payment. Therefore, with benefit of survivorship means that payment will be made only if the designated payer or recipient is alive at the time the payment is due.

To begin solving the problem posed above, it is necessary to consult a mortality table.

The money paid in will earn interest over the 25- year period. For this example, the rate will be assumed to be 5%. The basic equation for finding present value can be used to show that all the money paid in equals the present value of all the money to be paid out 25 years later. The amount each pays in can then be found:

$29.53 is the amount each person would pay in if all were to receive $100, 25 years later (dead or alive). The $28.88 payment will benefit of survivorship is smaller because in that case only those who survive are to receive their $100.