Final Exam Spring 016 Econ 180-367 Closed Book. Formula Sheet Provided. Calculators OK. Time Allowed: 3 hours Please write your answers on the page below each question 1. (10 points) What is the duration of a bond that sells at par, has a coupon rate of % (paid annually; with annual compounding) and a remaining time to maturity of years?. (10 points) Suppose that I buy a ten-year zero-coupon bond with a yield to maturity of percent (with semiannual compounding). One year later, the yield to maturity has fallen to 1 percent. What is my return on holding this bond for one year? 3. (5 points) Suppose that the ten-year (par) government bond yield is percent, the ten-year swap rate is 1.8 percent, and the LIBOR rate is 30 basis points above the repo rate, and is certain to remain so for the next ten years. Suppose that there are no regulatory constraints, transactions costs or default risks. In this case, there is profitmaking arbitrage opportunity. In this arbitrage: (a) Do you go long or short the government bond? (b) Do you pay fixed or floating in a swap contract? (c) Do you borrow or lend in the repo market? (d) What collateral do you give or receive in the repo contract in part (c)? (e) In this arbitrage, what percentage return do you earn (as a percentage of the value of the government bond that you go long or short in part (a))? 4. (5 points) Gilman Financial Services has a CDS spread of 50 basis points per annum on a five year contract. In the event of default, the recovery rate will be 75 percent. What is the approximate probability of Gilman Financial Services defaulting at some point over the next five years? 5. (1 points) Short Questions. Parts (i)-(v) are multiple choice: in each case, only one answer is correct. (i) Which of the following are commonly referred to as NINA loans: A. Loans for which the new identity of the applicant was not applicable. B. Mortgages that did not require any income or asset verification. C. Industrial loans made by the New Industrial National Association. D. Loans made to finance solar power projects motivated by climate change. E. Mortgages that have not been reviewed by the insurance or accounting departments. (ii) Which of the following describes a risk reversal, as discussed in class: A. A strategy of reducing the Sharpe ratio on an investor s portfolio. B. A strategy of receiving fixed in a variance swap contract. C. A strategy of paying fixed in a variance swap contract. D. A CDO strategy that goes long the equity tranche and short the mezzanine tranche. E. An option strategy that goes long an out-of-the-money call and short an out-of-the money put. (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 were discussed in class as possible causes for the sharp decline in oil prices in 015: A. A decline in oil supply owing to turmoil in the Middle East. B. The slowing economy in China. C. Technological difficulties in the fracking industry. 1
D. The economic recovery in the US. E. Lack of financial credit to oil producers. (v) Which of the following is true for the price of a European call option on a dividend paying stock. A. It can be higher or lower than the price of an American call option on the same stock at the same strike price and expiration date, depending on the risk-free rate. B. It can be higher or lower than the price of an American call option on the same stock at the same strike price and expiration date, depending on the strike price. C. It cannot be higher than the price of an American call option at the same strike price and expiration date. D. It cannot be lower than the price of an American call option at the same strike price and expiration date. E. It must be the same as the price of an American call option at the same strike price and expiration date. (vi) A 90 day T-Bill with a face value of $10,000 costs $9,875. What is the interest rate, on a discount basis? (vii) What is the current approximate level of ten-year US government bond yields? 6. (5 points) The interest rate is %. An insurance company must pay $10,000 *1.0 7 =$11,487 in seven years. It wants to immunize this liability with a portfolio of a five year zero coupon bond and a nine year zero coupon bond. How much will the insurance company invest in each of these two bonds? 7. (10 points) Consider the following data for a one-factor economy. All portfolios are well-diversified. Portfolio Expected Return Beta A 10% B 3 F 6% 0 (a) According to Arbitrage Pricing Theory (APT), what is the expected return on portfolio B? (b) Suppose that another portfolio, portfolio E, is well-diversified with a beta of 1 and an expected return of 9%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy? 8. (9 points) Suppose that a stock is trading for $0. Its volatility is 0.4. The stock pays no dividends and the riskfree interest rate is zero. You buy a European call option on this security at a strike price of $0 with an expiration of 6 months hence. According to the Black-Scholes formula, what should this call option be worth? 9. (5 points) A call option to buy a barrel of oil at the end of the year for $50 costs $8.60. A call option to buy a barrel of oil at the end of the year for $51 costs $8.30. Assume that risk-free interest rates are zero, that investors are rational and risk neutral, and that there is no chance of oil being between $50 and $51 per barrel at the end of the year. What is the probability of the oil price being above $51 at the end of the year? 10. (10 points) Assume that risk-free interest rates are zero. Suppose that a non-dividend paying stock trades for $50 and that the price of a European call option to buy this stock for $40 in one years time is $8. Investors are able to borrow and lend freely at the zero interest rates, and can go long or short shares or options without constraints. Is there an arbitrage opportunity? If so, what exactly is it? Please say exactly what you would go long or short including the amounts. 11. (5 points) The one-year interest rate in the US is 0.5% and the one-year interest rate in Brazil is 15.5%. The exchange rate today is that one US dollar buys 3.5 Brazilian real. According to covered interest parity, what is the one-year forward dollar/real exchange rate?
1. (5 points) Joe and Kate are both maximizing mean-variance utility in a model with two risky assets (stock A and stock B) and one risk-free asset. Joe has a coefficient of risk aversion of and invests 5% of his wealth in stock A, 5% in stock B, and 50% in the risk-free asset. Kate has a coefficient of risk aversion of 1. What share of her wealth does she invest in stock A? 3
Solutions 1. The duration is 1 10 [1* + * ] = 1.98 years 100 1.0 1.0 points off for algebra mistake. 100. The price of the bond today is 0 1.01 100 $91.41 = $81.95. After one year, the price of the bond is 81.95 =. 18 1.005 =. So the holding period return is 91.41 81.95 11.54% If interpret the question as saying that the interest rate is per 6 month period (rather than annualized with semiannual compounding), the answer would be 4.%. 8 points for this. points off for algebra mistake. 3. No partial credit. 1 point per part. (a) Go long the government bond. (b) Pay fixed in the swap contract. (c) Borrow in the repo market. (d) The collateral is the government bond I bought in part (a). (e) 0.5 percent (0. percent negative swap spread plus 0.3 percent LIBOR-repo spread). 4. Expected Cost=Expected Benefit 0.05=P(Default)*0.5 The probability of default is 10 percent. points for saying that the probability is percent (which is the per annum probability of default). No credit for mixing up recovery rate with loss given default. 5 No partial credit. 3 points per part. (i) B (ii) E (iii) C (iv) B (v) C 10,000 9,875 360 (vi) The interest rate is * = 5% 10, 000 90 (vii) 1.9%. Any answer in the range 1.5-.3 percent is deemed correct. Any answer that is a range is interpreted as the midpoint of the range. 6. The duration of the asset and liability must be the same. Let w be the fraction of immunizing portfolio invested in the five-year zero; 1-w is invested in the nine-year zero. Then 7=5w+9(1-w). Solving this gives w=0.5. So $5,000 is invested in the five-year bond and $5,000 is invested in the nine-year bond. Full credit for just saying 50% in the five-year and 50% in the nine-year. 7. (a) The key equation is Er ( i ) = r f + βλ i. Because the expected return on F is 6%, we know that the risk-free rate must be 6%. Because the expected return on A is 10%, we know that λ = 0.0. Hence the expected return on B must be 1%. 5 points for part (a). points off for algebra mistake. (b) With a beta of 1, the expected return on E ought to be 8%. So here is the strategy. Invest $100 in E. Payoff is $100*(1.09+F). Invest -$50 in A. Payoff is $-50*(1.1+F) Invest -$50 at the risk-free rate. Payoff is -$50*1.06. 4
Adding the pieces up, there is no cost today, but my payoff is $1. This is an arbitrage. I could similarly have formed an arbitrage by investing $100 in E, investing -$ 100 00 in B and borrowing $ at the risk-free rate. 3 3 0 0.4 8. In the Black-Scholes formula, d 1 = (ln( ) + 0.5) / 0.4 0.5 = 0.1414 and d = 0.1414. From the 0 normal distribution tables, N (0.14) = 0.5557 and N( 0.14) = 0.4443. Hence the call price should be C = (0* 0.5557) (0* 0.4443) = $.3. 3 points off for error resulting from failure to look up the normal tables correctly. points off for algebra mistake. 9. The vertical spread of going long the option at $50 and short the option at $51 costs 30 cents, and pays off $1 if the oil price is $51 or higher. Thus, the probability of oil being $51 or higher is 30%. 10. Buy 1 call option for $8. Short the stock (receive $50) and invest $40. All told, I receive $ on net. On expiration, if the stock price is greater than $40, my portfolio is worth zero. If the stock price is less than $40, then my portfolio is worth $40 less the stock price. This is an arbitrage. For full credit, all you need to do is state that you buy 1 call option and short the stock. No need to work through the implications (or even explicitly state that you invest $40, since it is at the riskfree rate). No credit for creating an imaginary put option that was not in the question since there is a mispricing, we do not know how the put would be priced. 1 1 11. 1.005 = 1.155. Solving this gives the forward exchange rate, F, of $1=4.0 reals. points off for algebra 3.5 F mistake. No credit for doing it backwards (i.e. having the forward rate show dollar depreciation). Full credit for doing it right, but expressing the answer as 1 real=$0.49. Er ( p) rf 1. Because Joe invests half his wealth in the risky portfolio with a risk aversion of, it must be that = 1 σ P. So Kate invests all of her wealth in the risky portfolio, which means that she invests 50% of her wealth in stock A. 5