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Investments

Zvi Bodie, Alex Kane, Alan J. Marcus

Chapter 23

Futures, Swaps, and Risk Management - all with Video Answers

Educators

Chapter Questions

Problem 1

A stock's beta is a key input to hedging in the equity market. A bond's duration is key in fixedincome hedging. How are they used similarly? Are there any differences in the calculations necessary to formulate a hedge position in each market?

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Problem 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.
a. In general, however, should its position in futures be more or less than the number of contracts necessary to hedge these anticipated cash flows? (Hint: Think about the firm's stream of cash flows extending out over many years.)
b. What other considerations might enter into the hedging strategy?

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Problem 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 one year. Suppose a firm wishes to use available (short maturity) contracts to hedge commodity prices at a more distant horizon, say, four years from now. Do you think the hedge will be more effective for the oil- or the gold-producing firm?

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Problem 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?

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Problem 5

Consider the futures contract written on the S\&P 500 index and maturing in one year. The interest rate is $3 \%$, and the future value of dividends expected to be paid over the next year is $$\$ 70$$. The current index level is 4.000 . Assume that you can short sell the S\&P index.
a. Suppose the expected rate of return on the market is $8 \%$. What is the expected level of the index in one year?
b. What is the theoretical no-arbitrage price for a 1-year futures contract on the S\&P 500 stock index?
c. Suppose the actual furures price is 4,024 . Is there an arbitrage opportunity here? If so, how would you exploit it?

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Problem 6

Suppose that the value of the S\&P 500 stock index is 4,000 .
a. If each E-mini futures contract (with a contract multiplier of $$\$ 50$$ ) 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 $$\$ 40$$, bow much is the transaction cost per "Typical share" controlled by one futures contract?
c. For small investons, a typical transaction cost (including the bid-ask spread) for direct trades in stocks might be about 5 cents per share. How many times the transactions costs in futures markets is this?

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05:04

Problem 7

You manage a $$\$ 42$$ million portfolio, currently all invested in equities, and believe that the market is on the werge of a big but short-lived downturn. You would move your portfolio temporarily into T-bills, but you do not want to incur the transaction costs of liquidating and reestablishing your equity position. Instead, you decide oo temporarily hedge your equity holdings with E-mini S\&P 500 index futures coatracts.
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 $$\$ 00$$ index is now at 4.200 and the contract multiplier is $$\$ 50$$.
c. How does your answer to part (b) change if the beta of your portfolio is . 6 ?

James Kiss
James Kiss
Numerade Educator
05:04

Problem 8

A manager is bolding a S1 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?

James Kiss
James Kiss
Numerade Educator

Problem 9

Suppose that the relationship between the rate of return on Digital Computer Corp. stock, the market index, and a computer industry index can be described by the following regression equation: $r_{\text {Degeal }}=.5 r_M+.75 r_{\text {lederery }}$. 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 Digital stock? Specifically, how many dollars' worth of the market and industry index contracts would you buy or sell for each dollar held in Digital?

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Problem 10

Suppose that the spot price of the euro is currently $$\$ 1.10$$. The 1-ycar futures price is $$\$ 1.15$$. Is the interest rate higher in the United States or the curo zone?

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Problem 11

a. The spot price of the British pound is currently $$\$ 1.30$$. If the risk-free interest rate on 1-year government bonds is $1 \%$ in the United States and $2 \%$ in the United Kingdom, what must be the forward price of the pound for delivery one year from now?
b. How could an investor make risk-free arbitrage profits if the forwand price were higher than the price you gave in answer to part (a)? Give a numerical example.

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Problem 12

Consider the following information:
$$
\begin{aligned}
r_{\text {Us }} & =2 \% ; r_{\text {Ux }}=3 \% \\
E_0 & =1.30 \text { dollars per pound } \\
F_0 & =1.29 \text { (1-year delivery) }
\end{aligned}
$$
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?

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Problem 13

Farmer Brown grows Number 1 red corn and would like to hedge the value of the coming harvest. However, the furures contract is traded on the Number 2 yellow grade of com. Suppose that yellow com 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 Bruwn buy or sell to hedge his position?

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05:25

Problem 14

Return to Figure 23.7, Suppose the LIBOR rate when the first-Listed Eurodollar contract matures in October is $1.0 \%$. What will be the profit or loss to each side of the Eurodollar contract?

Manasvee Singh
Manasvee Singh
Numerade Educator

Problem 15

Yields on short-term bonds tend to be mose 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 portiolio of 5 -year maturity bonds with modified duration four years and desire to hedge your interest rate exposure with T-bond futures, which currently have modified duration nine years and sell at $$F_0=\$ 95$$. How many futures contracts should you sell?

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Problem 16

A manager is holding a $$\$ 1$$ million bond portfolio with a modified duration of eight years. She would like wo bedge the risk of the portfolio by short-selling Treasury bonds. The modified duration of T-bonds is 10 years. How many dollars' worth of T-bonds should she sell to minimize the variance of ber position?

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Problem 17

A corporation plans to issue $$\$ 10$$ million of 10 -year bonds in three months. At current yields the bonds would have modified duration of eight years. The T-mote futures contract is selling at $F_0=100$ and has modified duration of six 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.

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Problem 18

$\alpha$. If the spot price of gold is $$\$ 1,500$$ per troy ounce, the risk-free interest rate is $2 \%$, and storage and insurance costs are zero, what should be the forward price of gold for delivery in one year? Use an arbitrage argument to prove your answer.
b. Show bow you could make risk-free arbitrage profits if the forward price is $$\$ 1,550$$.

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03:53

Problem 19

If the com harvest today is poor, would you expect this fact to have any effect on today's futures prices for corn to be delivered (post-harvest) two years from today? Under what circumstances will there be no effect?

Samit Deshmukh
Samit Deshmukh
Numerade Educator

Problem 20

Suppose that the price of corn is risky, with a beta of .5 . The monthly storage cost is $$\$ .03$$ per bushel, and the current spot price is $$\$ 5.50$$, with an expected spot price in three 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 three months?

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Problem 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 $$\$ 1.10$$ per euro. What cash flows will be exchanged on a 4 -year foreign exchange swap with notional principal of 100 million curos (or equivalently, at current exchange rates, $$\$ 110$$ million)?

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Problem 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 SOFR and receives a fixed $6 \%$ on notional principal of $$\$ 100$$ million. What is the firm's effective interest rate on its bocrowing?

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Problem 23

Firm ABC enter a 5-year swap with firm XYZ to pay SOFR in retum for a fixed $6 \%$ rate on notional principal of $$\$ 10$$ million. Two years from now, the market rate on 3 -year swaps is SOFR 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?

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Problem 24

Suppose that at the present time, one can enter 5 -year swaps that exchange SOFR for $5 \%$. An off-marker swap would then be delined as a swap of SOFR for a fixed rate other than $5 \%$. For example, a firm with $7 \%$ coupon debr outstanding might like to convert to synthetic floating-rate debt by entering a swap in which it pays SOFR and receives a fixed rate of $7 \%$. What up-front payment will be required to induce a counterparty to take the ocher side of this swap? Assume notional principat is $$\$ 10$$ million.

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Problem 25

Suppose the 1-year futures price on a stock-index portfolio is 3.828 , the stock index currently is 3,800 , the 1 -year risk-free interest rate is $3 \%$, and the year-end dividend that will be paid on a $$\$ 3,800$$ investment in the market index portfolio is $$\$ 80$$.
a. By how much is the contract mispriced?
b. Formulate a zero-net-investment arbitrage portfolio and sbow that you can lock in riskless peofits 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 doa'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 relationship? That is, given a stock index of 3,800 , how high and how low can the futures price be without giving rise to arbitrage opportunities?

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Problem 26

Consider these data for the S\&P 500 future's contract maturing in, exactly one year. The S\&P 500 index is at 4,290 , and the contract price is $F_0=4,292$.
a. If the current interest rate is $25 \%$, and the average dividend rate of the stocks in the index is $1.9 \%$, what fraction of the proceeds of stock short sales would need to be available to you to earn arbitrage profits?
b. Suppose now 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?
c. By how much does the actual futures price fall below the no-arbitrage bound?
d. Formulate the appropriate arbitrage strategy, and calculate the profits to that strategy.

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