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Question 1

1.1 Fisherâ€™s Separation Theorem is vital to finance theory today. With reference to the concepts of wealth maximisation and net present value, detail and discuss. Additionally, comment on the roles played by capital markets and shareholders in Fisherâ€™s analysis.Â  Â (10 marks)

(a)A second-order stochastic dominance is consistent with utility functions that have positive marginal utility and risk aversion.

(b)If asset â€œAlphaâ€ has a higher mean and higher variance than asset â€œBetaâ€, it is stochastically dominant, in accordance with the first-order criterion.

(c)If asset â€œAlphaâ€ is stochastically dominant over asset â€œBetaâ€ according to the second-order criterion, it is also dominant according to the first-order criterion.

(d)A risk-neutral investor will use second-order stochastic dominance as a decision criterion only if the return of the underlying assets are normally distributed.Â  (8 marks)

1.3State the minimum set of necessary conditions needed to obtain mean-variance indifference curves.Â  (7 marks)

2.1An investor whose initial wealth is Â£2,000 is offered an opportunity to play a fair game with 2 possible outcomes: winning Â£400 with a probability of 50% or losing Â£400 with a probability of 50%. The investorâ€™s utility function is the natural logarithm of wealth, U(W) = lnW. What are the certainty equivalent and risk premium of this risky game?Â  (8 marks)

2.2Assume the same facts as above in Q2.1 except for the payoffs: the relevant payoffs are now winning Â£900 with a probability of 50% and losing Â£900 with a probability of 50%. What is the risk premium? Compare the risk premium with that in Q2.1 and interpret.Â  (5 marks)

2.3You have a logarithmic utility function U(W) = In W, and your current level of wealth is Â£5,000. Suppose you are exposed to a situation that results in a 50/50 chance of winning or losing Â£1,000. If you can buy insurance that completely removes the risk for a fee of Â£125, would you buy it or take the gamble?Â  (6 marks)

2.4Using the same facts as in Q2.3, suppose you accept the gamble and lose, and consequently your wealth is reduced to Â£4,000. Imagine now that you are faced with the same gamble and have the same offer of insurance as in Q2.3, will you buy the insurance this time?Â  (6 marks)

3.1 Johnâ€™s initial wealth stands at Â£720, and his investment options consist of two securities, j and k, the current prices of which are Â£8 and Â£9, respectively. The uncertain future is partitioned into two states, S1 and S2. In state S1, the prices of securities j and k are Â£10 and Â£30, respectively while in state S2, the security prices are Â£20 and Â£10, respectively.

(a)If John wanted to buy a completely risk-free portfolio (i.e., one that has the same payoff in both states of nature), how many shares of j and k would he buy? (You may buy fractions of shares.)Â  Â (8 marks)

(b)What is the one-period risk-free rate of interest?Â  (5 marks)

(c)Assume a third state is available. In state S3, the prices of securities j and k are Â£18 and Â£6, respectively. Would you be able to find a completely risk-free portfolio? Why not?Â  (5 marks)

3.2 Suppose that there are two equally likely states of nature and that an investor's utility is equal to the log of consumption, so that

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Explain the meaning of the state prices and find the market prices of the two securities. Also determine the risk-free rate implied by the state prices.

4.1 For all his clients, a portfolio manager manages portfolios that lie on the Markowitz frontier. Two of the portfolio managerâ€™s portfolios, A and B, are under review. The expected returns of the two portfolios are substantially different. The review concludes that Portfolio A is virtually identical to the market portfolio and should be superior to portfolio B. Do you agree or disagree with this conclusion? Explain with reference to the CAPM.Â  (12 marks)

4.2 The portfolio manager believes that portfolio A has a higher expected return because it has greater non-systematic risk than portfolio B. define non-systematic risk and explain why you agree or disagree with the portfolio manager.Â  (13 marks)

The hypothesis of capital market efficiency has attracted a great deal of interest and critical comment. Summarise the concept of the efficient market hypothesis and critically appraise the same.