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:

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