Uncertain Annuity. Professor Jones has just turned 90 years old and is applying for a lifetime annuity that will pay $10,000 per year, starting 1 year from now, until he dies (there is only one payment per year at the end of each year). According to statistical summaries, the chance (probability) that Professor Jones will die at any particular age is as follows. age 90 91 92 93 probability .08.09 .09.09 94 95 96 .10 .10 .10 97 .10 98 99 .10 .07 100 .05 101 .03 What is the life expectancy of Professor Jones? What is the present value of an annuity at 8% interest (yearly compounding) that has a lifetime equal to Professor Jones's life expectancy? What is the expected present value of the annuity? Hint: compute the probabilities of survival to various ages i. For example, 990 = 1, 491 490 -0.08 = 0.92, 492 = 291 -0.09 = 0.83, and so forth. Denote by P(n) be the present value of an annuity over n periods, where n may be a fractional number, that is, n ER+. The formula for P(n) can be derived from the usual formula for annuities (for n E N) by using the relation ch - en in) for c>0. Prove that P(n) is a concave function by showing that its second derivative is always negative. Use the concavity of P(n) to show that the expected present value of an annuity with a random number of periods n is always smaller than the present value of the annuity evaluated at the expected value of n. Hint: use Jensen's inequality, which states that E(F(X)) 0. Prove that P(n) is a concave function by showing that its second derivative is always negative. Use the concavity of P(n) to show that the expected present value of an annuity with a random number of periods n is always smaller than the present value of the annuity evaluated at the expected value of n. Hint: use Jensen's inequality, which states that E(F(X))