## Friday, April 27, 2012

### PRESENT VALUE OF A LIFE ANNUITY

As stated earlier, an annuity is a series of payments and the present value of the annuity is the total of the present values of each of the individual payments. To find the present value of a series of payments where each payment is made only if the designated payer or recipient is alive to pay or receive it, we can now use the method for calculating the present value of a single payment.

For Example: The present value at age 35 of a life annuity of \$100 per year for three years (with the first payment due at age 36) can be represented by the following line diagram:

The present value of each of the three payments can be calculated individually, as follows:

The Payment Due at Age 36
Basic equation:

The Payment Due at Age 37
Basic equation:

The Payment Due at Age 36
Basic equation:

The present value at age 35 of this annuity is the total of these three expressions. The common multiplier (\$100) can be factored out. The fractions to be added together have a common denominator (l35). Hence the present value of the annuity can be expressed as:

The numerator of this expression represents the total to be paid out to the survivors at each age, with each such amount being discounted at interest to the evaluation date. The denominator represents the number of persons alive on the evaluation date, among whom this total present value to be paid in must be allocated.

If, for example, given table and 3% interest were used, the present value of the annuity would be calculated as follows:

 Age (x) lx 35 9373807 36 9350279 37 9325594 38 9299482

 n 1 0.970874 2 0.942596 3 0.915142 4 0.888487

From above:

To Illustrate: Using the given table and 6% interest, calculate the present value at age 45 of a life annuity of \$30 per year for four years, first payment due at age 46.

 Age (x) lx 45 9048999 46 9000587 47 8948114 48 8891204 49 8829410

 n 1 0.943396 2 0.889996 3 0.839619 4 0.792094

Solution:
The line diagram for this life annuity appears as follows: \$30               \$30                   \$30                       \$30    Age 45      *

46                   47                    48                       49

The expression for the present value will have a numerator representing the total to be paid out to the survivors at each age, with each such amount being discounted at interest to the evaluation date:

1. 2. 