A 65-year old female client is considering either purchasing a single-premium insurance policy or investing in a tax-free municipal bond. The fundamental feature of this insurance 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 tax codes. In her case, each dollar of premium generates 3.6914 dollars of insurance to be paid at her death. On the other hand buy now and hold until death investment strategy in tax-free municipal bonds gives return at annual rate of 7.1% (and all of the dividends associated with these bonds can be reinvested at this rate). Thus, in two years the bonds would be worth (1.071)2 = 1.147 dollars per each dollar invested, and, e.g., in 12.5 years they would be worth (1.071)12.5 = 2.357 dollars per each dollar invested. The worksheet Residual Life describes the random variable R representing the residual life of 65-year old females (residual life is the number of years till her death), according to the official government statistical data. To simplify for the purposes of this question, all deaths are assumed to 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

A) What is the expected value of the random variable R?

B) If one million dollars is invested in the tax-free municipal bond with buy now and hold until death investment strategy, as described above, what is the probability that the value of investment at time of death will be less than two million dollars?

(Hint: You may want to consider a new random variable with the same outcomes and their probabilities as the random variable R, but with numerical values associated with each outcome transformed to represent the value of investment.)

C) If one million dollars is invested in the tax-free municipal bond with buy now and hold until death investment strategy, as described above, what is the expected amount of money on hand at the time of death? (You should ignore any potential tax effects and assume that the probability of default is zero.)

a) 2.31 million dollars

B) 3.55 million dollars

C) 3.81 million dollars

D) 4.17 million dollars

E) 8.1 million dollars