Final Exam. 5. (24 points) Multiple choice questions: in each case, only one answer is correct.

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1 Final Exam Fall 06 Econ Closed Book. Formula Sheet Provided. Calculators OK. Time Allowed: 3 hours Please write your answers on the page below each question. (0 points) A stock trades for $50. After six months, its price will either rise to $60 or fall to $40. If it rises to $60, then after a further six months, its price will either rise to $70 or fall back to $50. If it falls to $40, then after a further six months, its price will either fall to $30 or rise back to $50. Riskfree rates are 0% per six months. Find the price of a European call option today with a strike price of $60 and an expiration of one year hence.. (5 points) The five-year zero coupon bond yield is.5% and the six-year zero coupon bond yield is.5% (both with annual compounding). Find the forward rate from five to six years hence. 3. (5 points) Consider a stock that pays no dividends. It trades for $30. European at-the-money call and put options with an expiration of one year hence both trade for exactly $5. The risk-free interest rate is %. Assume that there are no transactions costs or borrowing constraints. In this case, there is profit-making arbitrage opportunity. In this arbitrage: (a) Do you go long or short the call option? (b) Do you go long or short the stock? (c) Do you go long or short the put option? (d) Do you invest or borrow at the riskfree rate? (e) In this arbitrage, if I go long or short one call option in part (a), how much should I borrow or invest at the riskfree rate? 4. (5 points) The one-year interest rate in the US is % and the one-year interest rate in Brazil is 4.75%. The exchange rate today is that one US dollar buys 4 Brazilian real. The one-year forward dollar.real exchange rate is that one US dollar buys 4.59 Brazilian real. Assume that there are no transactions costs or borrowing constraints. In this case, there is profit-making arbitrage opportunity. In this arbitrage: (a) Do you buy or sell Brazilian reals in the spot market? (b) Do you buy or sell Brazilian reals in the forward market? (c) Do you borrow or lend in dollars? (d) Do you borrow or lend in Brazilian reals? (e) If, in part (a), I bought or sold 4 reals (for $), how much profit (in reals) would I make on this arbitrage? 5. (4 points) Multiple choice questions: in each case, only one answer is correct. (i) Which of the following best describes the OIS rate: A. The fixed rate on an interest rate swap contract where the floating rate is the average federal funds rate. B. An international oil price. C. Interest rates charged on offshore loans. D. An information system used to warn regulators of problems after the financial crisis. E. An index of the foreign exchange value of the US dollar against a basket of foreign currencies.

2 (ii) A company can invest funds for five years at LIBOR minus 30 basis points. The five year swap rate is 3 percent. Which of the following is the fixed rate that the company can earn investing floating and converting to fixed using this swap: A..4%. B..7%. C. 3.0%. D. 3.3%. E. 4.0%. (iii) Which of the following best describes the Supplementary Leverage Ratio: A. An alternative leverage ratio that takes account of how liquid banks assets are. B. A measure of the amount of additional leverage that banks are allowed to take on in the Dodd-Frank act. C. An alternative leverage ratio that takes account of banks derivatives and repo exposure. D. An alternative leverage ratio that applies only to foreign banks operating in the US. E. An alternative leverage ratio that omits preferred shares from the definition of equity. (iv) Which of the following was the immediate trigger of the financial crisis of 907? A. A collapse in house prices leading to losses on mortgages. B. An unsuccessful attempt to create a short squeeze in a copper mining company. C. A slowdown in the agricultural sector. D. The Federal Reserve raising interest rates sharply to support the value of the dollar. E. The collapse of the railroad industry which had been marked by overbuilding. (v) Suppose that in March, I take the long side in a June Eurodollar futures contract for $ Suppose that in June, the effective federal rate is 38 basis points, the one-month LIBOR rate is 55 basis points and the three-month LIBOR rate is 65 basis points. Which of the following describes the payment that I will make or receive: A. I pay $550. B. I receive $550 C. I pay $55. D. I pay $5. E. I receive $5. (vi) Which of the following best describes the US Treasury yield curve over the last 50 years. A. The yield curve has typically been upward sloping. The occasions when it was downward sloping came just before bursts of inflation. B. The yield curve has typically been upward sloping. The occasions when it was downward sloping came just before recessions. C. The yield curve has typically been downward sloping. The occasions when it was upward sloping were associated with poor bank profitability. D. The yield curve has typically been downward sloping. The occasions when it was upward sloping came just before recessions. E. The yield curve has typically been downward sloping. The occasions when it was upward sloping came when the federal funds rate was unusually high. (vii) Which of the following is the standard deviation of annual US stock returns (S&P500) over the past 90 years. A. 4% B. 7% C. % D. 6% E. 40% (viii) According to the dividend discount model, if a stock will pay dividends of $5 next year, dividends will grow at % per annum for ever, and the stock has a required return of 5%, then its price should be: A. $5 B. $00 C. $83 D. $66 E. $500

3 6. (5 points) Let the effective annual rate be 5% and consider a project that pays $75 in one year, $50 in two years and another $00 in three years. What is the present value of this project? 7. (5 points) Suppose that the one-year interest rate is % this year, and is expected to be.5% next year, % the following year,.5% the year after that and 3% the year after that. According to the expectations hypothesis of the term structure, what should the five-year yield be? All interest rates are expressed with annual compounding. 8. (5 points) Suppose that the CAPM holds, that risk-free rates are zero, and that the market portfolio has an expected return of 6% and a standard deviation of 0%. Hopkins asset management fund is an asset with no idiosyncratic volatility, and a standard deviation of 0%. What is the expected return on Hopkins asset management fund? 9. (9 points) You invest $00 in a bond that sells at par and has a coupon rate of 3% (paid annually; with annual compounding) and a remaining time to maturity of 3 years. What is the duration of this bond? 0. (9 points) You have a portfolio that is invested 70% in stocks and 30% in bonds. Last year, your stock portfolio earned 0% and the bond portfolio earned 5%. The overall portfolio earned 8.5% returns. The benchmark stock portfolio earned 8% and the benchmark bond portfolio earned 6%. A benchmark overall portfolio puts half the weight on the benchmark stock portfolio and half the weight on the benchmark bond portfolio, and thus had an overall return of 7%. Your portfolio outperformed the benchmark by.5 (8.5-7) percent. Using the formulas discussed in class, how much of this is attributable to picking between asset classes (stocks and bonds)? How much is attributable to selecting within stocks or bonds?. (9 points) Alexandra has the utility function uw ( ) = where w denotes her wealth, in dollars. She is w / offered a gamble that has a 4 7 chance of giving her wealth of $00 and a 3 7 chance of giving wealth of $9. What is the certainty equivalent of this gamble?. (9 points) Suppose that interest rates are zero and a stock trades for $00 and has a volatility of Suppose that the assumptions of the Black-Scholes model are satisfied. What is the price of an at-the-money European call option with an expiration of one year hence? 3

4 Solutions. If in 6 months, the stock price is $60, to replicate the option I buy one share and invest 50 = At the end,. this gives me $0 on the up branch and 0 on the down branch. The cost is $4.55. The option is worth $0 on the up branch and $0 on the down branch. So the price of one option is $7.7. If in 6 months, the stock price is $40, the option is worthless. So today, I buy one share and invest 40 = and this gives me $0 on the up branch and $0 on the down. branch. The net cost is $3.64. So the value of an option is $ points for incorrectly assuming that the probabilities of each branch are 50 percent and working out the riskneutral value correctly. points off for algebra error which is.76%. An acceptable answer is the approximation (6*.5) (5*.5).75% 5.05 = point off for algebra error. 3. (a) I go long the call option. (b) I go short the stock. (c) I go short the put option. (d) I invest at the risk-free rate. 30 (e) I should invest =. All parts get point for right, 0 for wrong. 4. (a) I sell reals in the spot market. (b) I buy reals in the forward market. (c) I lend in dollars. (d) I borrow in reals. (e) I borrow 4 reals, convert to $, invest the $ to get $.0. Then, I convert back at the forward rate to get 4.68 reals. To repay my loan, I have to spend 4*.475=4.59 reals. This gives me a profit of 0.09 reals. All parts get point for right, 0 for wrong. 5. (i) A (ii) B (iii) C (iv) B (v) D (vi) B (vii) D (viii) A All parts get 3 points for right, 0 for wrong PV = + + = point off for an algebra mistake. No credit if it isn t set up as the present value equation. 3 points if you do something weird to the interest rate. /5 7. (.0*.05*.0*.05*.03) = %. Simply averaging the five rates gives % and is an acceptable answer. point off for an algebra mistake. points for correctly computing the total return over five years, which is however not what is meant by yield. 4

5 8. The beta must be and so the expected return is %. 9. Duration is (0.083*) + (0.946*3) = Between asset classes: ( )* ( )*0.06 = are 0.4%. Within asset classes 0.7*( ) + 0.3*( ) = 0.0 are.% 5 points for getting one of the two correct, or transposing between and within / / / CE = = = 70 = / So CE = / 0. = 5 The certainty equivalent is $5. points off for algebra mistake 3 points for correctly working out expected utility, but none for working out expected wealth (rationale: expected utility is part of the certainty-equivalent calculation, expected wealth is not) d = [ln() + ] / 0.5 = 0.5 d = 0.5 C 0 = 00* *0.403 = 9.74 points off for algebra mistake 3 points off for not looking up normal tables correctly 5