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Lab 3

# Lab 3 - Pricing an Annuity Central Indiana Life Insurance...

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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: 1-0.017049=.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 1-0.018668=.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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Lab 3 - Pricing an Annuity Central Indiana Life Insurance...

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