Investments, tenth edition

 

  

KEY TERMS 

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  C H A P T E R  

2 3

  Futures, Swaps, and Risk Management 

827

 KEY EQUATIONS 

   Interest rate parity (covered interest arbitrage):    F

0

5 E

0

 

¢

1

1 r

US

1

1 r

UK

≤

T

   

  Hedging  with  futures:     Hedge  ratio

5

Change in portfolio value

Profit on one futures contract

   

  Parity for stored commodities:  F  

0

   5   P  

0

 (1  1   r  

 f 

   1   c )  

  Futures price versus expected spot price:    F

0

5 E(P

T

) 

¢

1

1 r

f

1

1 k

≤

T

    

  PROBLEM SETS 

    1.  A stock’s beta is a key input to hedging in the equity market. A bond’s duration is key in fixed-

income hedging. How are they used similarly? Are there any differences in the calculations 

necessary to formulate a hedge position in each market?    

    2.  A U.S. exporting firm may use foreign exchange futures to hedge its exposure to exchange 

rate risk. Its position in futures will depend in part on anticipated payments from its customers 

denominated in foreign currency. In general, however, should its position in futures be more or 

less than the number of contracts necessary to hedge these anticipated cash flows? What other 

considerations might enter into the hedging strategy?  

   3.  Both gold-mining firms and oil-producing firms might choose to use futures to hedge uncertainty 

in future revenues due to price fluctuations. But trading activity sharply tails off for maturities 

beyond 1 year. Suppose a firm wishes to use available (short maturity) contracts to hedge 

commodity prices at a more distant horizon, say, 4 years from now. Do you think the hedge will 

be more effective for the oil- or the gold-producing firm?  

 

 

 

4.  You believe that the spread between municipal bond yields and U.S. Treasury bond yields 

is going to narrow in the coming month. How can you profit from such a change using the 

municipal bond and T-bond futures contracts?     

    5.  Consider the futures contract written on the S&P 500 index and maturing in 6 months. The 

interest rate is 3% per 6-month period, and the future value of dividends expected to be paid 

over the next 6 months is $15. The current index level is 1,425. Assume that you can short sell 

the S&P index.

     a.   Suppose the expected rate of return on the market is 6% per 6-month period. What is the 

expected level of the index in 6 months?  

    b.   What is the theoretical no-arbitrage price for a 6-month futures contract on the S&P 500 stock 

index?  

    c.   Suppose the futures price is 1,422. Is there an arbitrage opportunity here? If so, how would 

you  exploit  it?     

   6.  Suppose that the value of the S&P 500 stock index is 1,600.

     a.   If each futures contract costs $25 to trade with a discount broker, how much is the transaction 

cost per dollar of stock controlled by the futures contract?  

    b.   If the average price of a share on the NYSE is about $40, how much is the transaction cost per 

“typical share” controlled by one futures contract?  

    c.   For small investors, a typical transaction cost per share in stocks directly is about 10 cents per 

share. How many times the transactions costs in futures markets is this?     

   7.  You manage a $16.5 million portfolio, currently all invested in equities, and believe that the mar-

ket is on the verge of a big but short-lived downturn. You would move your portfolio temporarily 

Basic

Intermediate

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828 

P A R T   V I

  Options, Futures, and Other Derivatives

    into T-bills, but you do not want to incur the transaction costs of liquidating and reestablishing 

your equity position. Instead, you decide to temporarily hedge your equity holdings with S&P 

500 index futures contracts.

     a.   Should you be long or short the contracts? Why?  

    b.   If your equity holdings are invested in a market-index fund, into how many contracts should 

you enter? The S&P 500 index is now at 1,650 and the contract multiplier is $250.  

    c.   How does your answer to ( b ) change if the beta of your portfolio is .6?     

    8.  A manager is holding a $1 million stock portfolio with a beta of 1.25. She would like to hedge 

the risk of the portfolio using the S&P 500 stock index futures contract. How many dollars’ 

worth of the index should she sell in the futures market to minimize the volatility of her position?  

    9.  Suppose that the relationship between the rate of return on IBM stock, the market index, and a 

computer industry index can be described by the following regression equation:  r  

IBM

   5   .5 r  

 M 

   1  

.75 r  

Industry

 . If a futures contract on the computer industry is traded, how would you hedge the 

exposure to the systematic and industry factors affecting the performance of IBM stock? How 

many dollars’ worth of the market and industry index contracts would you buy or sell for each 

dollar held in IBM?  

   10.  Suppose that the spot price of the euro is currently $1.30. The 1-year futures price is $1.35. 

Is the interest rate higher in the United States or the euro zone?  

   11.      a.    The spot price of the British pound is currently $2.00. If the risk-free interest rate on 1-year 

government bonds is 4% in the United States and 6% in the United Kingdom, what must be 

the forward price of the pound for delivery 1 year from now?  

    b.   How could an investor make risk-free arbitrage profits if the forward price were higher than 

the price you gave in answer to ( a )?  Give  a  numerical  example.     

   12.  Consider  the  following  information:   

r

US

 5 4%;    r

UK

 5 7%

E

0

 5 2.00 dollars per pound

F

0

 5 1.98(1-year delivery) 

    where the interest rates are annual yields on U.S. or U.K. bills. Given this information:

     a.   Where would you lend?  

    b.   Where would you borrow?  

    c.   How  could  you  arbitrage?     

   13.  Farmer Brown grows Number 1 red corn and would like to hedge the value of the coming har-

vest. However, the futures contract is traded on the Number 2 yellow grade of corn. Suppose 

that yellow corn typically sells for 90% of the price of red corn. If he grows 100,000 bushels, 

and each futures contract calls for delivery of 5,000 bushels, how many contracts should Farmer 

Brown buy or sell to hedge his position?  

   14.  Return to  Figure 23.7 . Suppose the LIBOR rate when the first listed Eurodollar contract matures 

in January is .40%. What will be the profit or loss to each side of the Eurodollar contract?  

   15.  Yields on short-term bonds tend to be more volatile than yields on long-term bonds. Suppose 

that you have estimated that the yield on 20-year bonds changes by 10 basis points for every 

15-basis-point move in the yield on 5-year bonds. You hold a $1 million portfolio of 5-year 

maturity bonds with modified duration 4 years and desire to hedge your interest rate exposure 

with T-bond futures, which currently have modified duration 9 years and sell at  F  

0

   5  $95.  How 

many futures contracts should you sell?  

   16.  A manager is holding a $1 million bond portfolio with a modified duration of 8 years. She 

would like to hedge the risk of the portfolio by short-selling Treasury bonds. The modified dura-

tion of T-bonds is 10 years. How many dollars’ worth of T-bonds should she sell to minimize the 

variance of her position?  

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  C H A P T E R  

2 3

  Futures, Swaps, and Risk Management 

829

   17.  A corporation plans to issue $10 million of 10-year bonds in 3 months. At current yields 

the bonds would have modified duration of 8 years. The T-note futures contract is selling at 

 F  

0

   5  100 and has modified duration of 6 years. How can the firm use this futures contract to 

hedge the risk surrounding the yield at which it will be able to sell its bonds? Both the bond and 

the contract are at par value.  

   18.  If the spot price of gold is $1,500 per troy ounce, the risk-free interest rate is 2%, and stor-

age and insurance costs are zero, what should be the forward price of gold for delivery in 

1 year? Use an arbitrage argument to prove your answer. Include a numerical example show-

ing how you could make risk-free arbitrage profits if the forward price exceeded its upper 

bound value.  

   19.  If the corn harvest today is poor, would you expect this fact to have any effect on today’s futures 

prices for corn to be delivered (postharvest) 2 years from today? Under what circumstances will 

there be no effect?  

   20.  Suppose that the price of corn is risky, with a beta of .5. The monthly storage cost is $.03, and 

the current spot price is $5.50, with an expected spot price in 3 months of $5.88. If the expected 

rate of return on the market is 0.9% per month, with a risk-free rate of 0.5% per month, would 

you store corn for 3 months?  

   21.  Suppose the U.S. yield curve is flat at 4% and the euro yield curve is flat at 3%. The current 

exchange rate is $1.50 per euro. What will be the swap rate on an agreement to exchange cur-

rency over a 3-year period? The swap will call for the exchange of 1 million euros for a given 

number of dollars in each year.  

   22.  Desert Trading Company has issued $100 million worth of long-term bonds at a fixed rate of 

7%. The firm then enters into an interest rate swap where it pays LIBOR and receives a fixed 

6% on notional principal of $100 million. What is the firm’s overall cost of funds?  

   23.  Firm ABC enters a 5-year swap with firm XYZ to pay LIBOR in return for a fixed 6% rate 

on notional principal of $10 million. Two years from now, the market rate on 3-year swaps is 

LIBOR for 5%; at this time, firm XYZ goes bankrupt and defaults on its swap obligation.

     a.   Why is firm ABC harmed by the default?  

    b.   What is the market value of the loss incurred by ABC as a result of the default?  

    c.   Suppose instead that ABC had gone bankrupt. How do you think the swap would be treated 

in the reorganization of the firm?     

   24.  Suppose that at the present time, one can enter 5-year swaps that exchange LIBOR for 8%. 

An  off-market swap  would then be defined as a swap of LIBOR for a fixed rate other than 

8%. For example, a firm with 10% coupon debt outstanding might like to convert to synthetic 

floating-rate debt by entering a swap in which it pays LIBOR and receives a fixed rate of 10%. 

What up-front payment will be required to induce a counterparty to take the other side of this 

swap? Assume notional principal is $10 million.     

    25.  Suppose the 1-year futures price on a stock-index portfolio is 1,624, the stock index currently is 

1,600, the 1-year risk-free interest rate is 3%, and the year-end dividend that will be paid on a 

$1,600 investment in the market index portfolio is $20.

     a.   By how much is the contract mispriced?  

    b.   Formulate a zero-net-investment arbitrage portfolio and show that you can lock in riskless 

profits equal to the futures mispricing.  

    c.   Now assume (as is true for small investors) that if you short sell the stocks in the market 

index, the proceeds of the short sale are kept with the broker, and you do not receive any 

interest income on the funds. Is there still an arbitrage opportunity (assuming that you don’t 

already own the shares in the index)? Explain.  

    d.   Given the short-sale rules, what is the no-arbitrage  band  for the stock-futures price relation-

ship? That is, given a stock index of 1,600, how high and how low can the futures price be 

without giving rise to arbitrage opportunities?     

Challenge

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830 

P A R T   V I

  Options, Futures, and Other Derivatives

   26.  Consider these futures market data for the June delivery S&P 500 contract, exactly 6 months 

hence. The S&P 500 index is at 1,350, and the June maturity contract is at  F  

0

   5  1,351.

     a.   If the current interest rate is 2.2% semiannually, and the average dividend rate of the stocks 

in the index is 1.2% semiannually, what fraction of the proceeds of stock short sales would 

need to be available to you to earn arbitrage profits?  

    b.   Suppose that you in fact have access to 90% of the proceeds from a short sale. What is the 

lower bound on the futures price that rules out arbitrage opportunities? By how much does 

the actual futures price fall below the no-arbitrage bound? Formulate the appropriate arbi-

trage strategy, and calculate the profits to that strategy.        

    1.  Donna Doni, CFA, wants to explore potential inefficiencies in the futures market. The TOBEC 

stock index has a spot value of 185. TOBEC futures contracts are settled in cash and underlying 

contract values are determined by multiplying $100 times the index value. The current annual 

risk-free interest rate is 6.0%.

     a.   Calculate the theoretical price of the futures contract expiring 6 months from now, using the 

cost-of-carry model. The index pays no dividends. 

     The total (round-trip) transaction cost for trading a futures contract is $15.  

    b.   Calculate the lower bound for the price of the futures contract expiring 6 months from now.     

   2.  Suppose your client says, “I am invested in Japanese stocks but want to eliminate my exposure 

to this market for a period of time. Can I accomplish this without the cost and inconvenience of 

selling out and buying back in again if my expectations change?”

     a.   Briefly describe a strategy to hedge both the local market risk and the currency risk of 

investing in Japanese stocks.  

    b.   Briefly explain why the hedge strategy you described in part ( a ) might not be fully effective.     

   3.  René Michaels, CFA, plans to invest $1 million in U.S. government cash equivalents for the next 

90 days. Michaels’s client has authorized her to use non–U.S. government cash equivalents, but 

only if the currency risk is hedged to U.S. dollars by using forward currency contracts.

     a.   Calculate the U.S. dollar value of the hedged investment at the end of 90 days for each of the 

two cash equivalents in the table below. Show all calculations.  

    b.   Briefly explain the theory that best accounts for your results.  

    c.   On the basis of this theory, estimate the implied interest rate for a 90-day U.S. government 

cash  equivalent.