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Calculate actuarially fair price, expected utility and hedging strategy.
Esperanza has been an expected utility maximize ever since she was five years old. As a result of the strict education she received at an obscure British boarding school, her utility function over returns on assets u(R) is strictly increasing and strictly concave u(R) = R1/2. Now at the age of thirty something, Esperanza is evaluating an asset whose return (R) is a random variable that takes on one of two possible outcomes: good (G), or poor (P). The returns for each of the two outcomes are as follows: RG = $4900, RP = 2500. The probability of each outcome is: Pr(G) = .475, Pr(P) = .525.
a. Now suppose that the asset has three possible outcomes: good (G), medium (M) or poor (P). The return for each of the three outcomes is as follows: RG = $4900, RM = 3600, RP = 2500. The probability of each outcome is: Pr(G) = 1/5, Pr(M) = 3/5, Pr(P) = 1/5. What is the expected return and what is Esperanza\\\’s expected utility for this new asset?
b. Suppose that Esperanza can purchase insurance that guarantees her a return of $4900 regardless of the return on the asset. How much would this full insurance cost if its price is actuarially fair? How much of a premium over the price of actuarially fair insurance would Esperanza be willing to pay for this full insurance?
c. The asset described in part (a) and the asset described in part (c) is related in what manner? In particular, using the terminology we used in class, compare these two assets. Using this terminology, and the description of Esperanza\\\’s utility function, explain the difference in the premium over actuarially fair insurance that Esperanza is willing to pay when facing the asset described in part (a) and in part (c).
d. Suppose that full insurance is not available. Instead, Esperanza can reduce her risk from the asset described in part (a) by a hedging strategy. This hedging strategy effectively yields to Esperanza the returns and probabilities of the asset described in part (c). How much would Esperanza be willing to pay a hedge fund to perform this hedging strategy on her behalf?