Question
Consider an investor who has initial wealth w to invest, and can choose between a safe and a risky asset. Specifically, he can put a
Consider an investor who has initial wealth w to invest, and can choose between a safe and a
risky asset. Specifically, he can put a share s of his initial wealth in the risky asset. The
investor has a strictly increasing and strictly concave utility function over final wealth y. The
safe asset has a return of zero. The risky asset has a random return R, which can take on
values {r, 0,r}, with r > 0. The probabilities of these returns are qp, 1 q, and q(1 p)
respectively, with 1 >= p, q > 0.
(a) Under what conditions on p, q will the investor invest a positive share s in the risky
asset? (5 marks)
(b) If the investor puts a share s of his wealth in the risky asset, write down expressions for
final wealth y for each of the three possible returns on the risky asset. (5 marks)
(c) If utility of final wealth u(y) = ln y, write down an expression for the expected utility of
final wealth, using your answer to (b). (5 marks)
(d) If q = 1, p > 0.5, w = 1, find the optimal value of s for the investor (Hint: it can be an
interior or a corner solution.) (5 marks)
(e) How does your answer to (d) change if 0 < q < 1? Explain. (5 marks)
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