HEC Paris MBA Program Name:... Financial Markets Prof. Laurent E. Calvet Fall 2010 MIDTERM EXAM 90 minutes Open book The exam will be graded out of 100 points. Points for each question are shown in brackets. There are 5 questions carrying equal weight. Answer all five questions. You are allowed to use a calculator. All other electronic devices are strictly prohibited. Good luck! Q1 Q2 Q3 Q4 Q5 Total 1
Problem 1. Decide whether the following statements are true or false. No explanations are necessary. (a) Forwards and futures are examples of derivative contracts. (b) You short a stock if you believe that it will go up. (c) You are financially better off if you receive $1 in five years than if you receive $1 today. (d) The IRR rule always gives the same answer as the NPV rule. (e) A futures contract is typically settled daily. (f) A forward contract is typically settled daily. (g) Inordertotradeafuturescontract, youneedtodepositfundsinamargin account. (h) Consider an interest rate that is quoted as 10% per annum with semiannual compounding. The equivalent rate with continuous compounding is 9.758% per annum. (i) According to the liquidity premium hypothesis (also called liquidity preference theory), long-term interest rates are typically higher than short-term interest rates. (j) A zero-coupon bond typically trades above its face value prior to maturity. 2
Problem 2. You want to buy an apartment in Versailles, which costs 700,000 euros. You can put 300,000 euros down, and for the rest you get a 20-year fixed rate mortgage from your bank. The annual percentage rate is 5% per year, compounded monthly. How big is your monthly payment? You assume that there are no other taxes and fees involved. Solution: Denote the unknown monthly payment amount by C. You are liable an annuity with monthly cash flows C, and you know that the fair value is 400,000 euros. There are T = 240 monthly cash flows and the monthly interest rate is r = 5%/12. The annuity formula implies that: 400,000 = C ] [1 r 1 (1+r) T or equivalently: C = 400,000 r 1 (1+r) T. So the amount you have to pay each month is C = 400,000 0.05/12 = 2,639.82 euros. 1 (1+0.05/12) 240 3
Problem 3. A riskless coupon bond is offered in the market at a price of $124.73. It has coupon payments of $10 in one year, $10 in two years, $10 in three years, and a coupon and principal payment of $110 in four years. In this problem, we assume that all yields and interest rates are compounded annually. (a) Compute the YTM based on the market offer price. Solution: The YTM satisfies the equation: $124.73 = $10 1+YTM + $10 (1+YTM) + $10 2 (1+YTM) + $110 3 (1+YTM) 4. We check by trial and error (or with a financial calculator) that YTM = 3.30%. (b) Usingtheyieldcurveofzerocouponbonds, pricethisbondanddetermine if it the offer in the market is a fair price. The annualized yields on zerocoupon bonds are given below. Solution: The bond is worth: Maturity Annualized Yield 1 year 1.00% 2 years 2.00% 3 years 3.00% 4 years 3.50% P 0 = $10 1.01 + $10 (1.02) 2 + $10 (1.03) 3 + $110 (1.035) 4 = $124.52. The offer price is slightly higher than the price implied by zero yields. (c) The zero yields are now 5% per annum for all maturities. What is the bond worth? Solution: The bond is worth: P = $10 1.05 + $10 (1.05) 2 + $10 (1.05) 3 + $110 (1.05) 4 = $117.73. The higher interest rates negatively impact the bond price. 4
Problem 4. You are asked to compute the price of the following foward contracts. (a) Suppose that you enter into a 6-month forward contract on a non-dividend paying stock when the stock price is $30 and the risk-free interest rate (with continuous compounding) is 12% per annum. What is the forward price? Solution: The price of the 3-month futures contract is F 0 = $30e 0.12 0.5 = $31.86. (b) A stock index currently stands at $350. The risk-free interest is 8% per annum (with continuous compounding) and the dividend yield on the index is 4% per annum. What should the futures price for a 4-month contract be? Solution: The price of the 3-month futures contract is: F 0 = $350e (0.08 0.04) 4/12 = $354.70. (c) The spot price of silver is $9 per ounce. The storage costs are $0.06 per quarter payable in advance. Assuming that interest rates are 10% per annum for all maturities, calculate the forward price of silver for delivery in 9 months. Solution: The present value of the storage costs is: $0.06+$0.06e 0.1 0.25 +$0.06e 0.1 0.5 = $0.1756. The forward price is therefore: F 0 = ($9+$0.1756)e 0.1 9/12 = $9.89. The forward contract can be replicated as follows. At date t = 0, we borrow $9.06, purchase 1 ounce of silver on the spot market, pay the storage cost, and store our purchase. In 3 months, we borrow $0.06 and pay the storage cost. In 6 months, we borrow $0.06 and pay the storage cost. In 9 months, we pay back our debt, which now amounts to: $9.06e 0.1 9/12 +$0.06e 0.1 6/12 +$0.06e 0.1 3/12 = $9.89, and take the silver out of storage. 5
Problem 5. A stock is expected to pay a dividend of $2 per share in 2 months and in 5 months. The stock price is $50, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a 6-month forward contract on the stock. (a) What are the forward price and the initial value of the forward contract? Solution: The present value of the dividends is The forward price is therefore I = $2e 0.08 2/12 +$2e 0.08 5/12 = $3.9079. F 0 = (50 3.9079)e 0.08 6/12 = $47.97. The value at origination of a forward contract is zero. (b) Three months later, the price of the stock is $48 and the risk-free rate of interest is still 8% per annum. What is the value of the short position in the forward contract? Solution: The present value of the future dividend is now The forward price is now I 1 = $2e 0.08 2/12 = $1.9735. F 1 = ($48 $1.9735)e 0.08 3/12 = $46.96. The value of the short position is therefore (F 0 F 1 )e 0.08 3/12 = $1.00 6