(i) Let U(W) be the utility function of an investor's wealth, W. The first and second derivatives of U with respect to Ware positive. Explain what these conditions imply about the investor's economic characteristics. (ii) You are given the following information on projects A and B: State of Probability Return on Return on nature project A project B 10% 3% 20% 4% 4.5% 50% 6% 7% 20% 12% 14% Explain with reasons which of the two projects the investor in (i) above should choose in order to maximise his expected utility of wealth.6 The Black-Scholes formula for the value of a European call option on a non- dividend paying stock at time t can be written as: c= So(d,) - Kerr-o @(d_) where In(S /K) + ("+0/2(T-D) d, = ONT-t In(S / K) + r - 0/(T - 1) d, = ONT-t K = strike price T = time of maturity S = price of stock at time t r = risk-free rate o = volatility and () = cumulative distribution function of the standard normal distribution. Using the Black-Scholes formula show that the call price, c, is the maximum of S - Ke "or zero, depending on the strike price, when o tends to zero. [9]7 An investor wishes to construct a portfolio consisting of a risk-free and a risky asset. His expected utility is given by E[U] = 1 - 12Op where r and o are the mean and standard deviation of the portfolio rates of return. The risk-free asset has an expected rate of return of 5% p.a. The risky asset has an expected rate of return of 8% p.a. and variance of 4%% p.a. Determine the portfolio that will maximise the investor's expected utility. [9] 8 (i) Define the delta, gamma and theta of an option. [3] (ii) Describe, using a numerical example, the concept of delta hedging. [6] [Total 9] 9 (i) Describe the forms of the Efficient Markets Hypothesis (EMH). [3] (ii) Outline the role that portfolio managers have even if the market is perfect and fully efficient. (3]