Pricing an Annuity
Central Indiana Life Insurance Company’s customers can use a portion of the funds accumulated in
their
401(k) retirement plan
to buy an
annuity
that pays $30,000 a year until death.
Part 1:
When Martin Dempster retired at age 61 on January 1, 2004, he opted to purchase the
above mentioned annuity. His first payment is made immediately. The actuaries at CILIC predicted
that interest rates would average 5% per year over the life of the annuity.
You have been assigned
to compute how much CILIC should charge Martin for the annuity. (For sake of simplicity, we will
assume that the payments are made once a year on January 1, although monthly payments would be
more typical.)
Discussion:
Suppose, for example, we knew that Martin would die between the 5th and 6th
payments. Then, CILIC would make the following payments:
Year
1
2
3
4
5
Payment
$30,000
$30,000
$30,000
$30,000
$30,000
At 5% interest, the present value of these payments is
30000+30000(1.05)
1
+30000(1.05)
2
+30000(1.05)
3
+30000(1.05)
4
=136,378.52
which is the least we would charge for the annuity.
(We would certainly charge more to cover
administrative costs and to allow for a modest profit.)
However, we have no way of knowing how long Martin will live. We solve this problem using
what is referred to as a
mortality table
. This table lists, for each age, the fraction of people at that
age who typically will die before their next birthday. The “Probability of Death” column on page 3
was copied from a mortality table for males.
Using this information, we can calculate the probability that Martin will be alive in the future.
Martin is 61 when he retires.
The entry in the “Probability of Death” column corresponding to age
61 is 0.017049.
Hence, Martin’s chance of surviving until age 62 is:
10.017049=.982951
According to the “Probability of Death” column, the fraction of 62 year olds dying before age 63 is
0.018668 so the fraction surviving is
10.018668=.981332
and we expect
.981332*.982951=.964601
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to still be alive after 2 years.
Continuing in this way, we can compute, for each year after 2004, the
probability that Martin will still be alive.
We multiply Martin’s $30,000 annual payment by the
probability that he is still alive and take the present value of the total.
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 Spring '08
 Staff
 Subroutine, Return statement, The Return, Martin Dempster

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