Question

1. a. Two investors, A and B, are evaluating the same investment opportunity, which has an expected value of £100. The utility functions of A and B are ln(x) and ​​​​​x​², respectively. Which investor has a certainty equivalent higher than 100? Which investor requires the higher risk premium?

b. (i) Describe suitable measures of risk for ‘loss-aversion’ and ‘risk aversion’.

(ii) Concisely define the term ‘risk neutral’ with respect to a utility function u (w), where w is the realisation of a random variable W.

(c) Suppose you are facing a lottery that has a payoff of 10b pounds with probability 0.01 and that of 0 with probability 0.99. You are an expected utility maximiser with a utility function, u(x) = −exp(−ax) where x is the payoff in money terms and a > 0 is a parameter. What is the risk premium for this lottery - describe the risk premium as a function of ‘a’ and ‘b’.

(d) How does the risk premium in (c) change as ‘b’ changes.

2.

​​​​​​​There are two distinct portfolios, A and B.

Portfolio	Expected Returns	Standard Deviation
A	0.2	0.1
B	0.3	0.2

1. A portfolio D assigns weight w to portfolio A and weight (1-w) to portfolio B. What should w be so as to minimise the variance of portfolio D, if ρ_AB = -0.5? The variance of a portfolio is given by:   
2. If ρ_AB = 0.5, and A and B are frontier portfolios, what is the expected return and standard deviation of the minimum variance portfolio?
3. Given that A and B are frontier portfolios and that another frontier portfolio C, has an expected rate of return of 25% and variance of 2 %, evaluate the new ρ_AB.

3.

1. Project A has a payoff of 1 with probability 0.8 and 100 with probability 0.2. Project B has a payoff of 5 with probability 0.99 and 1585 with probability 0.01 Which project is preferred by an investor who maximises expected utility and has an utility function u(x) = ln 2x? Comment on the preferences.
2. Is it true that if an investor with utility function u (x) = x ^1/2 prefers risky project A to risky project B, another investor with utility function v(x) = 5x^1/2 + 5 also prefers A to B? Explain.
3. A risk neutral expected utility maximiser is facing a risky investment opportunity. Her utility is a function of the net payoff of the project in terms of money u(y) where £ y is the net payoff. When she makes an investment of £ x (> 0 ), the project yields a gross payoff of £ 30*x ^1/2on the investment with probability 0.4 and a gross payoff of £ 10* x^1/2with probability 0.6. The net payoff is the difference between gross payoff and the cost of the investment. How much should she invest to maximize her expected utility?
4. State the conditions needed to ensure that the choices made to maximise expected utility will coincide with the choices in a mean-variance framework.

4.

1. The cash flows of a firm that has just conducted an Initial Public Offering (IPO) are expected to be either £10 million per annum for 10 years and zero afterwards with probability ‘p’ or £5 million forever with probability (1-p). The risk-adjusted discount rate for this firm is 10% per annum. I. Express the current fundamental value of the firm in terms of ‘p’. II. If the current stock market value of the firm is £55 million, what is the value of ‘p’ implied by the market?

(b) The cash flows of a firm are expected to be £1 million per year starting next year for the first ten years and are then expected to start declining forever at the rate of 5% per year. The risk-adjusted discount rate is 10% per annum. What is the present value of the cash flows?

c) Investment analysts regularly prepare forecasts and reports for their clients on the valuation of the firms they follow as analysts. Briefly discuss the factors that should be taken into account in arriving at such valuations.

Answer 1

1)a)

The utility function of A and B are x and x2 respectively meaning that the utility derived from B is greater than that of A. Hence B has a certainty equivalent more than 100 and A will require higher risk premium to move forward with the investment opportunity.

b)i)

Loss aversion means that a gain of 1 dollar is likely to cause less addition to happiness(utility) than the subtraction to utility caused by a loss of 1 dollar.

Risk aversion means the hesitation of a person to agree to a situation of unknown payoff than another situation with a predictable payoff but with a lower quantum of payoff.

Measures of loss aversion:

• only entertain those opportunities with no chances of loss
• invest smaller amounts across various sectors

Measures of risk aversion:

• obtain insurance
• regularly review existing systems in place

ii) A risk neutral person is one whose marginal utility of money remains the same with an increase in money and the total utility is thus moving upward constantly with a positive slope. Thus, he is indifferent between a certain income and an uncertain income even if both of them offer the same net value to him.