exercise 1 Stock BBB has a spot price equal to 80$ and a dividend equal to 10$ will be paid in 5 months. The on year interest rate is equal to 8% (c.c). 1. Calculate the 6 month forward price? 2. Calculate the value of the contract for the short position if after 3 months the spot price of the stock is equal to 75$, the interest rate is still at 8% SOLUTION1 1. I = 10* e 0,08*0,1667 + 10* e 0,08*0,4167 = 19,54 = ( S I) e r ( T t) = (80 19,54) e 0,08*0,5 = 62,93$ 2. I = 10* e 0,08*0,1667 = 9,87 = ( S I) e r( T t ) = (75 9,87) e 0,08*0,25 = 66,45$ f r ( T t) 0,08*0,25 = ( 0 ) e = (62,93 66,45) e = 3,45
EXERCISE 2 A stock trades at 10$ and will pay a 0.50$ dividend in 3 months. Risk-free rate is 6% cc for all maturities. 1) Determine the 2-month and 4-month forward prices of the stock 2) Now assume that you buy 100 stocks forward for the 4-month maturity at the price you just determined. the stock trades at 12 $, interest rates have dropped to 4.50% and the company has announced that the dividend will be of 1$ and not just 0.50. Determine the market value of your deal (i.e. the original forward purchase of 100 stocks) SOLUTION 1) The 2-month forward is unaffected by the dividend, and is equal to: 2 0.06 / 6 = 10 e = 10.1005 To calculate the 4-month forward we must take into consideration the dividend, the present value of which has to be subtracted from today's stock price 4 = 4 3 3 ( 10 0.50 e 0.06 / ) e 0.06 / = ( 10 0,4926) e 0.06 / = 9, 699 2) After 1 month the 3-month forward price is 3 = 6 4 4 ( 12 1 e 0.045 / ) e 0.045 / = ( 12 0,9926) e 0.045 / = 11, 132 The value of the purchase of 100 stock forward at t0 is: V = 100 (11,132 9,699) e 0,045 / 4 = 141,697
EXERCISE 3 It s December 1st 20xx. A stock trades at 400 $ and will pay a known dividend of 15$ on the 1st of May 20xx+1. The c.c. risk-free rate is 6% for all maturities. 1) Determine the fair price of a 1-year forward contract on the stock. 2) Now assume an intermediary quoted a 1-year forward price of 415 $. Explain how you would build an arbitrage transaction in order to exploit the differential, if any, between market and fair price. SOLUTION EXERCISE 3 1) To calculate the 6-month forward price we must take into consideration the dividend, the present value of which has to be subtracted from today's stock price = 0.06*0.5 0.06 0.06 ( 400 15 e ) e = ( 400 14,56) e = 409. 27 2) If the forward price is 415 there is an arbitrage opportunity. Since the forward price is higher then the spot prices the cash and carry should be applied. More specifically the strategy is : borrow 14.56 for 6 months and 385.44 for one year; buy the stock spot and sell forwards. T=0 T=6M T=12M dividend +15 Borrow 4 months +14.56-15 Borrow 12 385.44 409.27 Buy Spot Sell forward -400 + stock +415 stock 5.73
EXERCISE 4 The two-months c.c. interest rates in Switzerland and the United States are 3% and 8% respectively. The USD/SR rate (number of USD for 1 SR) is 0.65. The futures price for a contract deliverable in 2 months is 0.66. What arbitrage opportunities does this create (assume forward and futures prices are the same). Show how you would build the arbitrage transaction. 1) The forward equilibrium exchange rate is SOLUTION EXERCISE 4 = 0.65e (8% 3%)2 /12 = 0.6554 To exploit arbitrage opportunities we shoud T=0 T=2M Borrow USD +100 USD Repay -101.34 USD Exchange USD and CH -100 USD +153.84 CH Invest CH - 153.84 CH Receives + 154.61 CH BUY USD and sell Buy +101.34 USD Chf at the market rate Sell -153.54 profit 1.06
EXERCISE 5 A Swiss company is importing goods from the US and has to pay 500,000 USD in four months. The company s bank offers it the following exchange rates : Bid Ask Spot CH/USD (number of CH per one USD) 1.32 1.33 orward 4 months CH/USD (number of CH per one USD) 1.29 1.31 The 4 months CH interest rate is equal to 2.0% and the 4 months USD interest rate is equal to 5.4%. 1) Explain how the Swiss company could hedge its currency risk and how much (CH) will have to pay at the end of the 4 months; 2) Using the X bid rates only, determine whether the forward X rate is an equilibrium one or not; 3) If the forward market rate is not an equilibrium rate, state whether the Swiss company would be better off using this market rate or trying to get the equilibrium rate. SOLUTION 1) The swiss company can hedge the currency risk by buying forwards dollars. At the market forward rate of 1.31 the company will have to pay 500,000,000 USD * 1.33 chf/usd= 665,000,000CH 2) The equilibrium rate is 3) or the company it would be better try to use the equilibrium rate. The company should therefore borrow CH, exchange CH into dollars and invest in dollars
EXERCISE A one-year-long forward contract on a non-dividend-paying stock is entered into when the stock price is $ 40 and the risk-free rate of interest is 10% per annum with continuous compounding. a) What are the forward price and the initial value of the forward contract? b) Six months later, the price of the stock is $ 45 and the risk-free interest rate is still 10%. What are the forward price and the value of the forward contract? orward price = 12 = 40e 0. 1 = 44. 21 Initial value is zero, like for every forward entered at market conditions Six months later we need a six-month forward 6 = 45e 0.1 0.5 = 47.31 0.10 0.5 V = ( 47.31 44.21) e = 2. 95
EXERCISE Suppose that you enter into a short futures contract to sell July silver for $ 5.20 per ounce on the New York Commodity Exchange. The size of the contract is 5,000 ounces. The initial margin is 4,000$ and the maintenance margin is 3,000$. What change in the futures price will lead to a margin call? What happens if you do not meet the margin call? Value of contract = 5,000*5.20 = 26,000 Since this is a short position, in order for my balance to drop 1,000$ the value of the contract must increase to 27,000. This happens when the price rises to 5.40$ If the margin call is not met the position is closed out
EXERCISE The sd of monthly changes in the spot price of live cattle is 1.2 cents per pound. The sd of monthly changes in the futures price of live cattle for the closest contract is 1.4. The correlation between the futures price change and the spot price change is 0.7. It is now Oct. 15. A beef producer is committed to purchasing 200,000 pounds of live cattle on Nov. 15. The producer wants to use the December live cattle futures contracts to hedge the risks. Each contract is for the delivery of 40,000 pounds of cattle. What strategy should the beef producer follow? 1.2 Hedge ratio = 0.7 = 0. 6 1.4 200,000 0.6 40,000 Number of contracts = N = = 3
EXERCISE A company has a $ 10 million portfolio with a beta of 1.2. The S&P is currently 900 and one futures contract is on 250 times the index. How can the company use futures contracts on the S&P 500 to completely hedge its risk over the next 6 months? What position should it take to reduce the beta of the portfolio to 0.3? The company has a long equity position and should short the index 10,000,000 1.2 Number of contracts = N = 53 900 250 The company should short 53 contracts If the company wants to achieve a beta of 0.3 (3/4 reduction of the original exposure) it should sell 3/4 of 53 = 40 contracts