Evaluating Single-Premium Insurance Policies
You are an investment adviser and your client, a white American woman who just today reached the age
of 65, is considering purchase of a single-premium insurance policy. The fundamental feature of this
policy is that the premium is paid all at once when the policy is written, and this one-payment feature
does not run afoul of any IRS tax codes. In her case, each dollar of premium generates 3.6914 dollars of
insurance to be paid at her death. She seeks to evaluate this insurance policy as an investment by
comparing it with an alternative strategy of investment (and reinvestment) in tax-free municipal bonds.
Her objective is to maximize the expected amount of money for her estate at the time of her death.
Some of the data you need in order to help her make her decision can be found in mortality tables issued
by the National Center for Health Statistics. Table 6 of Vital Statistics of the United States -- Mortality
is included in amended form as Exhibit 1. It shows, for example, that of 100,000 new-born white
females, 87,633 reached their 65th birthday. Of these, 999 died prior to their 66th birthday. Similar
calculations can be carried out to age 120, at which point you may assume that all of the women are
deceased. The data in the table is available on the network.
Let R be the random variable representing the residual life of your client, where residual life is the
number of years till her death. To simplify calculations and the use of the table, assume that all deaths
occur half way through the year. That is, if she dies between today and her 66th birthday, R= 0.5. If she
dies between her 68th and 69th birthday, R= 3.5. The probability distribution for R can be computed
from Exhibit 1.
Assume that the annual rate on tax-free municipal bonds is 6.50% and that all of the dividends
associated with these bonds can be reinvested at this rate. Assume also that the bonds mature at the
moment of death, that is, half way into each year. Thus, in two and a half years they would be worth
(1.065)2.5 = 1.1705 dollars per dollar invested.
The insurance policy is the superior investment if she lives only a short time whereas the alternative
strategy of investing in municipal bonds is preferable if she lives a very long time -- say 20 years or
longer. Today, you must help her decide which investment strategy to pursue, prior to knowing the
exact date of her demise. To aid in the comparison, ignore any potential tax effects and assume that the
probability of default is zero for both investments.
Answer the following questions. We will discuss the solutions in class on the date listed on the syllabus.
This assignment is to be completed individually -- you may discuss it within your group, but each
student should solve and submit their own solution.
1. Compute the probability distribution for R from the data in Exhibit 1. What are the mean and
standard deviation of R?
2. Assuming she invests one dollar in the municipal bonds today, what is the expected amount of
money on hand at the time of her death? What is its standard deviation?
3. What is the probability that the payout on the bonds exceeds the payout on the insurance policy?
4. Based on her criteria of maximizing the expected amount of money on hand at the time of her death,
which investment is preferable, the bonds or the insurance policy?
5. Defining rates of return for random cash flows can be problematic, but let's try anyway. The
insurance company pays 3.6914 at the time of death. Define the implicit rate of return for the
insurance policy as that rate i you would have to earn on the tax-free municipal bonds to match this
payout in expectation. What is the implicit rate of return on the insurance policy?
This case was written by Professor Steven Lippman, UCLA.
Table 6. Life table for white females: United States, 2003
Amended with mortality statistics for 2003, ages 100+