TEMPORARY LIFE ANNUITIES

Life annuities may be either temporary life annuities or whole life annuities. Both kinds have many practical uses in actuarial calculations. In a temporary life annuity, each payment is made only if a designated person is then alive, but the payments are limited to a fixed number of years. In a whole life annuity, the payments continue for the entire lifetime of a designated person.

There are also temporary life annuities in which the first payment is made at the beginning, that is, on the same date on which the present value is calculated. These are known as temporary life annuities due. The use of the word “due” is comparable to its use in annuities certain. This type of annuity is important in life insurance calculations, because premiums payable for a life insurance policy represent an annuity due.

The line diagram of two five payments life annuities, one immediate and one due, look like this:

To Illustrate: Using the given table and 5% interest, calculate the present value at age 30 of a three – year life annuity due of $100 per year.

Age (x)	lx
30	9480358
31	9460165
32	9439447
33	9418208

Values of at 5% interest are as follows:

n
1	0.952381
2	0.907029
3	0.863838

Solution:

The line diagram for this life annuity due appears as follows:

                                               $ 100                  $ 100                        $ 100

^{Temporary life annuity due}                *

                                              Age 30                  31                              32                           33

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. The first payment is due upon the evaluation date. Hence, its present value is simply $100 (l30); it is not multiplied by any discounting factor. The denominator is the number living at the evaluation date:

Basic equation:

Present Value = $ 285.35

The present value of annuity immediate of the same question must be less than the present value of the life annuity due ($285.35), because each payment in the annuity immediate is paid one year later than its counterpart in the annuity due. Hence, there is a smaller probability that it will have to be paid, and there are a greater number of years in which interest is earned.

To Illustrate: 

Calculate the present value at age 65 of a four year temporary life annuity due of $50 per year? (Use given table and 6% interest)

Age (x)	lx
65	9019487
66	8935696
67	8847340
68	8753363
69	8652385

Values of at 6% interest are as follows:

n
1	0.943396
2	0.889996
3	0.839619

Solution:

The line diagram for this life annuity due appears as follows:

$ 50                $ 50                      $ 50                        $ 50

_______________________________________________

Age 65                66                       67                            68                          69

Basic equation: