1. Copyright © 2001 by Harcourt, Inc. All rights reserved.1 Chapter 10: Futures Hedging Strategies The law of the conservation of risk is like the law of the conservation of misery. You can only pass it around. You cannot get rid of it. Tanya Styblo Beder Risk, February 1999

2. Copyright © 2001 by Harcourt, Inc. All rights reserved.2 Important Concepts in Chapter 10 n Why firms hedge n Hedging concepts n Factors involved when constructing a hedge n Hedge ratios n Examples of short-term interest rate, intermediate- and long-term interest rate, and stock index futures hedges

3. Copyright © 2001 by Harcourt, Inc. All rights reserved.3 Why Hedge? n The value of the firm may not be independent of financial decisions because u Shareholders might be unaware of the firm’s risks. u Shareholders might not be able to identify the correct number of futures contracts necessary to hedge. u Shareholders might have higher transaction costs of hedging than the firm. u There may be tax advantages to a firm hedging. u Hedging reduces bankruptcy costs. n Managers may be reducing their own risk. n Hedging may send a positive signal to creditors. n Dealers hedge so as to make a market in derivatives.

4. Copyright © 2001 by Harcourt, Inc. All rights reserved.4 Why Hedge? (continued) n Reasons not to hedge u Hedging can give a misleading impression of the amount of risk reduced u Hedging eliminates the opportunity to take advantage of favorable market conditions u There is no such thing as a hedge. Any hedge is an act of taking a position that an adverse market movement will occur. This, itself, is a form of speculation.

5. Copyright © 2001 by Harcourt, Inc. All rights reserved.5 Hedging Concepts n Short Hedge and Long Hedge u Short (long) hedge implies a short (long) position in futures u Short hedges can occur because F The hedger owns an asset and plans to sell it later. F The hedger plans to issue a liability later u Long hedges can occur because F The hedger plans to purchase an asset later. F The hedger may be short an asset. u An anticipatory hedge is a hedge of a transaction that is expected to occur in the future. u See Table 10.1, p. 405 for hedging situations.

6. Copyright © 2001 by Harcourt, Inc. All rights reserved.6 Hedging Concepts (continued) n The Basis u Basis = spot price - futures price. u Hedging and the Basis    (short hedge) = S T - S 0 (from spot market) - (f T - f 0 ) (from futures market)    (long hedge) = -S T + S 0 (from spot market) + (f T - f 0 ) (from futures market) F F If hedge is closed prior to expiration,  (short hedge) = S t - S 0 - (f t - f 0 ) F If hedge is held to expiration, S t = S T = f T = f t.

7. Copyright © 2001 by Harcourt, Inc. All rights reserved.7 Hedging Concepts (continued) n The Basis (continued) u Hedging and the Basis (continued) F Example: Buy asset for $100, sell futures for $103. Hold until expiration. Sell asset for $97, close futures at $97. Or deliver asset and receive $103. Make $3 for sure. u Basis definition F initial basis: b 0 = S 0 - f 0 F basis at time t: b t = S t - f t F basis at expiration: b T = S T - f T = 0 u For a position closed at t:    (short hedge) = S t - f f - (S 0 - f 0 ) = -b 0 + b t

8. Copyright © 2001 by Harcourt, Inc. All rights reserved.8 Hedging Concepts (continued) n The Basis (continued) u illustrates the principle of basis risk. u This is the change in the basis and illustrates the principle of basis risk. u Hedging attempts to lock in the future price of an asset today, which will be f 0 + (S t - f t ). u A perfect hedge is practically non-existent. u Short hedges benefit from a strengthening basis. u Everything we have said here reverses for a long hedge. u See Table 10.2, p. 408 for hedging profitability and the basis.

9. Copyright © 2001 by Harcourt, Inc. All rights reserved.9 Hedging Concepts (continued) n The Basis (continued) u Example: March 30. Spot gold $387.15. June futures $388.60. Buy spot, sell futures. Note: b 0 = 387.15 - 388.60 = -1.45. If held to expiration, profit should be change in basis or 1.45. F At expiration, let S T = $408.50. Sell gold in spot for $408.50, a profit of 21.35. Buy back futures at $408.50, a profit of -19.90. Net gain =1.45 or $145 on 100 oz. of gold.

10. Copyright © 2001 by Harcourt, Inc. All rights reserved.10 Hedging Concepts (continued) n The Basis (continued) u Example: (continued) F Instead, close out prior to expiration when S t = $377.52 and f t = $378.63. Profit on spot = -9.63. Profit on futures = 9.97. Net gain =.34 or $34 on 100 oz. Note that change in basis was b t - b 0 or -1.11 - (-1.45) =.34. u Behavior of the Basis. See Figure 10.1, p. 409.

11. Copyright © 2001 by Harcourt, Inc. All rights reserved.11 Hedging Concepts (continued) n Some Risks of Hedging u cross hedging u spot and futures prices occasionally move opposite u quantity risk

12. Copyright © 2001 by Harcourt, Inc. All rights reserved.12 Hedging Concepts (continued) n Contract Choice u Which futures commodity? F One that is most highly correlated with spot F A contract that is favorably priced u Which expiration? F The futures whose maturity is closest to but after the hedge termination date subject to the suggestion not to be in the contract in its expiration month F See Table 10.3, p. 412 for example of recommended contracts for T-bond hedge F Concept of rolling the hedge forward

13. Copyright © 2001 by Harcourt, Inc. All rights reserved.13 Hedging Concepts (continued) n Contract Choice (continued) u Long or short? F A critical decision! No room for mistakes. F Three methods to answer the question. See Table 10.4, p. 413 worst case scenario methodworst case scenario method current spot position methodcurrent spot position method anticipated future spot transaction methodanticipated future spot transaction method

14. Copyright © 2001 by Harcourt, Inc. All rights reserved.14 Hedging Concepts (continued) n Margin Requirements and Marking to Market u low margin requirements on futures, but u cash will be required for margin calls

15. Copyright © 2001 by Harcourt, Inc. All rights reserved.15 Hedging Concepts (continued) n Determination of the Hedge Ratio u Hedge ratio: The number of futures contracts to hedge a particular exposure u Naïve hedge ratio u Appropriate hedge ratio should be  N f = -  N f = -  S/  f F F Note that this ratio must be estimated.

16. Copyright © 2001 by Harcourt, Inc. All rights reserved.16 Hedging Concepts (continued) n Minimum Variance Hedge Ratio u Profit from short hedge:    =  S +  fN f u Variance of profit from short hedge:        S 2 +   f 2 N f 2 + 2   S  f N f u The optimal (variance minimizing) hedge ratio is (see Appendix 10A)   N f = -   S  f /   f 2 F F This is the beta from a regression of spot price change on futures price change.

17. Copyright © 2001 by Harcourt, Inc. All rights reserved.17 Hedging Concepts (continued) n Minimum Variance Hedge Ratio (continued) F Hedging effectiveness is e * = (risk of unhedged position - risk of hedged position)/risk of unhedged positione * = (risk of unhedged position - risk of hedged position)/risk of unhedged position This is coefficient of determination from regression.This is coefficient of determination from regression.

18. Copyright © 2001 by Harcourt, Inc. All rights reserved.18 Hedging Concepts (continued) n Price Sensitivity Hedge Ratio u This applies to hedges of interest sensitive securities. u First we introduce the concept of duration. We start with a bond priced at B: F where CP t is the cash payment at time t and y is the yield, or discount rate.

19. Copyright © 2001 by Harcourt, Inc. All rights reserved.19 Hedging Concepts (continued) n Price Sensitivity Hedge Ratio u An approximation to the change in price for a yield change is u with DUR B being the bond’s duration, which is a weighted-average of the times to each cash payment date on the bond, and  represents the change in the bond price or yield. u Duration has many weaknesses but is widely used as a measure of the sensitivity of a bond’s price to its yield.

20. Copyright © 2001 by Harcourt, Inc. All rights reserved.20 Hedging Concepts (continued) n Price Sensitivity Hedge Ratio u The hedge ratio is as follows (See Appendix 10A for derivation.):  Note that DUR S  Note that DUR S  -(  S/S)(1 + y S )/  y S and DUR f  -(  f/f)(1 + y f )/  y f u u Note the concepts of implied yield and implied duration of a futures. Also, technically, the hedge ratio will change continuously like an option’s delta and, like delta, it will not capture the risk of large moves.

21. Copyright © 2001 by Harcourt, Inc. All rights reserved.21 Hedging Concepts (continued) n Price Sensitivity Hedge Ratio (continued) u Alternatively, F N f = -(Yield beta)PVBP S /PVBP f where Yield beta is the beta from a regression of spot yields on futures yields andwhere Yield beta is the beta from a regression of spot yields on futures yields and PVBP S, PVBP f is the present value of a basis point change in the spot and futures prices.PVBP S, PVBP f is the present value of a basis point change in the spot and futures prices.

22. Copyright © 2001 by Harcourt, Inc. All rights reserved.22 Hedging Concepts (continued) n Stock Index Futures Hedging u Appropriate hedge ratio is  N f = -  N f = -  (S/f) F F This is the beta from the CAPM, provided the futures contract is on the market index proxy. u Tailing the Hedge F With marking to market, the hedge is not precise unless tailing is done. This shortens the hedge ratio. F See Table 10.5, p. 422.

23. Copyright © 2001 by Harcourt, Inc. All rights reserved.23 Hedging Strategies n Short-Term Interest Rate Hedges u First we need to familiarize ourselves with the basics of T-bill and Eurodollar futures. F The T-bill futures is priced using the IMM index method. Let discount rate be 8.25Let discount rate be 8.25 Futures price is quoted as 100 - 8.25 = 91.75. This is the IMM Index.Futures price is quoted as 100 - 8.25 = 91.75. This is the IMM Index. The actual futures price is 100 - 8.25(90/360) = 97.9375 or $979,375 per $1 million contract.The actual futures price is 100 - 8.25(90/360) = 97.9375 or $979,375 per $1 million contract. Each basis point move amounts to $25.Each basis point move amounts to $25.

24. Copyright © 2001 by Harcourt, Inc. All rights reserved.24 Hedging Strategies n Short-Term Interest Rate Hedges (continued) F The Eurodollar futures is also priced using the IMM index method. Note that the actual spot Eurodollar pays interest added on to the principal. For example, if the rate is 10%, then $100 deposited for 90 days grows to $100(1 +.10(90/360)) = $102.50.Note that the actual spot Eurodollar pays interest added on to the principal. For example, if the rate is 10%, then $100 deposited for 90 days grows to $100(1 +.10(90/360)) = $102.50. The futures, however, uses the IMM method, as previously illustrated with T-bills, and treats it as though the underlying Eurodollar is a discount instrument.The futures, however, uses the IMM method, as previously illustrated with T-bills, and treats it as though the underlying Eurodollar is a discount instrument.

25. Copyright © 2001 by Harcourt, Inc. All rights reserved.25 Hedging Strategies n Short-Term Interest Rate Hedges (continued) u Hedging the Future Purchase of a Treasury Bill F See Table 10.6, p. 426 for example. u Hedging a Future Commercial Paper Issue F See Table 10.7, p. 429 for example. u Hedging a Floating-Rate Loan F See Table 10.8, p. 431 for example. F This is called a strip hedge. Note also the rolling strip hedge and the stack hedge.

26. Copyright © 2001 by Harcourt, Inc. All rights reserved.26 Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Futures Hedges u First let us look at the T-note and bond contracts F T-bonds: must be a T-bond with at least 15 years to maturity or first call date F T-note: three contracts (2-, 5-, and 10-year) F A bond of any coupon can be delivered but the standard is a 6% coupon. Adjustments, explained in Chapter 11, are made to reflect other coupons. F Price is quoted in units and 32nds, relative to $100 par, e.g., 93 14/32 is 93.4375. F Contract size is $100,000 face value so price is $93,437.50

27. Copyright © 2001 by Harcourt, Inc. All rights reserved.27 Hedging Strategies (continued) n Intermediate and Long-Term Interest Rate Futures Hedges (continued) u Hedging a Long Position in a Government Bond F See Table 10.9, p. 434 for example. u Anticipatory Hedge of a Future Purchase of a Treasury Bill F See Table 10.10, p. 436 for example. F Note the use of a regression estimate of the hedge ratio. u Hedging a Corporate Bond Issue F See Table 10.11, p. 437 for example.

28. Copyright © 2001 by Harcourt, Inc. All rights reserved.28 Hedging Strategies (continued) n Stock Index Futures Hedge u First look at the contracts F We primarily shall use the S&P 500 futures. Its price is determined by multiplying the quoted price by $250, e.g., if the futures is at 1300, the price is 1300($250) = $325,000 u Stock Portfolio Hedge F See Table 10.12, p. 440 for example. u Anticipatory Hedge of a Takeover F See Table 10.13, p. 442 for example.

29. Copyright © 2001 by Harcourt, Inc. All rights reserved.29 Summary n Table 10.14, p. 444 recaps the types of hedge situations, the nature of the risk and how to hedge that risk

30. Copyright © 2001 by Harcourt, Inc. All rights reserved.30 Appendix 10A: Derivation of the Hedge Ratio n Minimum Variance Hedge Ratio u The variance of the profit from a hedge is       S 2 +   f 2 N f 2 + 2   S  f N f F Differentiating with respect to N f, setting to zero and solving for N f gives N f = -   S  f /   f 2N f = -   S  f /   f 2 F A check of the second derivative verifies that this is a minimum.

31. Copyright © 2001 by Harcourt, Inc. All rights reserved.31 Appendix 10A: Derivation of the Hedge Ratio (continued) n Price Sensitivity Hedge Ratio u The value of the position is F V = S + V f N f u Use the following results:   V f /  r =  f/  r   y s /  r =  y f /  r u Differentiate with respect to r, use the above results, set to zero, apply the chain rule and solve for N f. The approximation is

32. Copyright © 2001 by Harcourt, Inc. All rights reserved.32 Appendix 10B: Taxation of Hedging n Hedges used by businesses to protect inventory and in standard business transactions are taxed as ordinary income. n Transactions must be shown to be legitimate hedges and not just speculation outside of the norm of ordinary business activities. This is called the business motive test.