QUESTION ONE

1. A risk neutral expected utility maximiser is facing an investment opportunity, which is a risky project. Her utility, u (y) is a function of the net monetary return of the project, where y is the net return. When she makes an investment of I (>0), the project yields a gross return of 200ln (I) on the investment with probability 0.2 and a gross return of 1000ln (I) with probability 0.8.

How much should she invest to maximize her expected utility?

2. Suppose there are two risky projects A and B. A random variable X_Adenotes risky project As return, while a random variable X_B denotes risky project Bs return. Both risky projects have the same expected return, but the variance of the return on project A is larger than that of project B. Can you conclude that a risk-averse agent always prefers project B over project A? Explain.

3. Suppose you are facing an investment opportunity that has a 1% chance of making 10y and 99% chance that it pays nothing. As an expected utility maximizer, whose utility function, u (z) = exp (cz) where z is a monetary payoff and c > 0, what is the risk premium for this investment? Describe the risk premium as a function of c and y.

4. How does the risk premium in (c) change as y changes.

QUESTION TWO

There are distinct portfolios, A, B and C.

Portfolio	Expected Returns	Standard Deviation
A	0.2	0.2
B	0.3	0.4
C	0.27	missing

_AB = 0.5

1. Determine the variance of portfolio C.

2. Using portfolios A and B above and given that the risk-free rate is 3%, compute the maximum feasible Sharpe ratio for the tangent portfolio, T.

3. Compute the correlation, _{A, T}, between the tangent portfolio, T and portfolio A.

4. If T is the tangent portfolio, evaluate the beta of portfolio A when the risk free rate is 3%.

QUESTION THREE

1. Can you be certain that a rational investor with the utility function u (x) = -3 x² + 45x will prefer mean-variance efficient portfolios? Explain.

2. If you did NOT know an investors utility function, what other conditions will be required so that we are certain that a rational investor will prefer mean-variance efficient portfolios?

3. The correlation between the returns on risky assets A and B is zero. Can we combine A and B to create a risk-free portfolio? Explain.

4. Can two distinct risky assets A and B be combined to create a risk-free portfolio that comprises A and B only? Explain.

5. Define the efficient frontier of risky assets, mathematically, where there is no risk-free asset but short sales on risky assets are allowed.

QUESTION FOUR

a) The cash flows, CF of a firm are distributed as follows:

State	CF	State Probability
1	0.65	0.2
2	0.75	0.4
3	1.65	0.4
4	2.25	0.2

1. A cash inflow is expected to be realised, following the distribution above, at the end of each year, for ten consecutive years. Calculate the expected cash flow and its standard deviation.

2. The expected return of the stock market is 20% per annum with an annual standard deviation of 1.5 while the correlation of cash flows of the firm and market returns is 0.3. Calculate the beta of the firm.

3. Calculate the risk-adjusted discount rate if the risk-free rate is 4% per annum.

4. Evaluate the present value of the cash flows.

2. Discuss the value and difficulty associated with estimating the fundamental value of risky investments. Explain the importance of valuing the flexibility that a firm has to strategically change its operations after a project has commenced operations. Compare the value of this flexibility with the value of the cash flows.