University of Cape Town

School of Economics

ECO4053Z Financial Economics I

EXAM 2007

PART I: SHAKILL HASSANTime: 1 ½ hoursThis part consists of two questionsAlthough they may seem challenging, they are both bookwork, so stay calmThis section is closed book of courseAnswer both questionsExplain and motivate all steps very clearlyPurely mathematical answers with no explanation will be severely penalised (and will not exceed half the available marks)

Question 1: Mean-Variance Analysis

Consider a setting with N risky assets and no risk-free asset. Show that any portfolio on the mean-variance efficient frontier can be obtained as a combination of any pair of efficient portfolios. Comment with regard to optimal portfolio choice. Explain all steps clearly.Hints: recall the optimisation programme that identifies one portfolio on the frontier. Suppose this programme is solved for two portfolios, offering distinct rates of expected return. Now show that any other portfolio on the frontier, associated with any other rate of expected return, can be obtained as a combination of the two identified portfolios. (This is the two-fund theorem.)

Question 2: Equilibrium Asset Pricing

Derive the standard Capital Asset Pricing Model. State all assumptions; explain all steps clearly.

Good luck.

PART II: HAIM ABRAHAM

Time : 1 ½ hoursAnswer all questionsAll questions carry equal weight

Question 3

The current price of a stock is R30. The risk-free interest rate is 5%. Each month for the next 2 months the stock price is expected to increase by 8% or reduce by 10%.

(i) Use a 2-step tree to calculate the value of a derivative that pays off , where ST is the stock price in 2 months.

(ii) If the derivative is American, should it be exercised early?

Question 4

(a) The term structure of interest rates is flat in the US and in France. The American dollar interest rate is 11% per annum and the euro interest rate is 8% per annum. The current exchange rate is 2.1 euro = US$1. Under the terms of a swap agreement a financial institution pays 5% per annum in euros and receives 10% per annum in US$. Payments are exchanged annually with one exchange having just taken place. The remaining life of the swap is two years. The principals in the two currencies are US$10 million and 20 million euros. Assume all interest rates are continuously compounded.

Value the swap as a series of forward foreign exchange contracts.

(b) Consider two companies, X and Y. Company X is based in the UK and wants to borrow US$50 million at a fixed rate of interest for 5 years in US funds. Since the company is unknown in the US, this proves to be impossible. Company X, however, has been quoted 12% per annum on fixed rate 5-year sterling funds. Company Y is based in the US and wants to borrow the equivalent of US$50 million in sterling funds for 5 years at a fixed rate of interest. Unfortunately, the company has not been able to get a quote, but has been offered US$ funds at 10.5% per annum. Five-year government bonds currently yield 9.5% per annum in the US and 10.5% in the UK.

Suggest an appropriate currency swap that will net the financial intermediary 0.5% per annum.

Question 5

Consider the following table,

Bond principal(R)

Time to maturity (years)

Annual coupon (payable ½ yearly)

Bond price(R)

100 1/2 0 98100 1 0 95100 3/2 6.2 101100 2 8 104

(i) Calculate the zero rate for maturities of ½ year and 1 year.(ii) Compute the price of a 2-year bond providing a semiannual coupon of 7%

per annum.

Question 6

(a) When you buy a call with a strike price K2 and exercise date T2 and sell a call with a strike price K1 and exercise date T1 (T2 > T1), the outcome is a diagonal spread. Show diagrammatically the profit of the diagonal spread when (i) K2 > K1 and when (ii) K2 < K1.

(b) The two-month interest rates in the US and South Africa are 3% and 8% per annum (continuously compounded), respectively. The spot price of the US$ is R0.65. The futures price of a contract deliverable in 2 months is R0.66. what arbitrage opportunities does this create?

(c) Show on a diagram the variation of profit and loss with the terminal stock price for the two portfolios below,

(i) Two shares and a short position in one call option(ii) One share and a short position in two call options.