1. Hedging Using Futures Contracts Finance (Derivative Securities) 312 Tuesday, 22 August 2006 Readings: Chapters 3 & 6

2. Long v Short  Long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the price  Short futures hedge is appropriate when you know you will sell an asset in the future & want to lock in the price

3. Arguments For Hedging  Companies should focus on the main business they are in and take steps to minimise risks arising from interest rates, exchange rates, and other market variables

4. Arguments Against Hedging  Shareholders are usually well diversified and can make their own hedging decisions  It may increase risk to hedge when competitors do not  Explaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult

5. Convergence Time Spot Price Futures Price t1t1 t2t2

6. Basis Risk  Basis is the difference between spot & futures  Basis risk arises because of uncertainty about the basis when the hedge is closed out

7. Long Hedge  Suppose that F 1 : Initial Futures Price F 2 : Final Futures Price S 2 : Final Asset Price  Hedge future purchase of an asset by entering into a long futures contract  Cost of Asset = S 2 – (F 2 – F 1 ) = F 1 + Basis

8. Short Hedge  Suppose that F 1 : Initial Futures Price F 2 : Final Futures Price S 2 : Final Asset Price  Hedge future sale of an asset by entering into a short futures contract  Price Realised = S 2 + (F 1 – F 2 ) = F 1 + Basis

9. Choice of Contract  Choose delivery month as close as possible to, but later than, the end of the life of the hedge  When no futures contract available on asset being hedged, choose contract whose futures price is most highly correlated with the asset price Creates two components to basis

10. Optimal Hedge Ratio  Proportion of exposure that should optimally be hedged is where  S is standard deviation of  S, the change in the spot price during the hedging period  F is standard deviation of  F, the change in the futures price during the hedging period  is correlation coefficient between  S and  F

11. Index Futures  To hedge risk in a portfolio, the number of contracts that should be shorted is where P is value of the portfolio, β  is its beta, and A is value of the assets underlying one futures contract

12. Reasons for Hedging Equity Portfolios  Desire to be out of the market for a short period of time Hedging may be cheaper than selling portfolio and buying it back  Desire to hedge systematic risk Appropriate when you feel that you have picked stocks that will outperform the market

13. Hedging a Portfolio  Suppose that: Value of S&P 500 is 1,000 Value of Portfolio is $5 million Beta of portfolio is 1.5 Risk-free rate is 4%, dividend yield is 1% p.a.  What position in 4-month futures contracts on the S&P 500 is necessary to hedge the portfolio? N* = 1.5 x 5,000,000/250,000 = 30

14. Hedging a Portfolio  Suppose index is 900 in three months’ time, futures price is 902 Futures gain is 30 x (1,010 – 902) x 250 = $810,000 Index loss 10%, dividend 0.25% per 3 months, overall loss –9.75% E(R P ) = R f + β[E(R M ) – R f ] = 1 + 1.5(–9.75 – 1) = –15.25

15. Hedging a Portfolio  Expected value of portfolio after 3 months 5,000,000 x (1 – 0.15125) = $4,243,750  Hedger’s position 4,243,750 + 810,000 = $5,053,750 Value has increased by the risk-free rate (1%)

16. Rolling Hedges Forward  Can use a series of futures contracts to increase the life of a hedge  Each switch from one futures contract to another incurs an element of basis risk

17. Interest Rate Futures  Treasury Bonds:Actual/Actual (in period)  Corporate Bonds:30/360  Money Market Instruments:Actual/360

18. Pricing  T-bond Cash price = Quoted price + Accrued Interest  T-bill Quoted price: where Y is the cash price of a Treasury bill that has n days to maturity

19. T-bond Futures  Cash price received (short position) = Quoted futures price × Conversion factor + Accrued interest  Conversion factor for a bond is approximately equal to the value of the bond on the assumption that the yield curve is flat at 6% with semi-annual compounding

20. T-bond Futures  Suppose that: Cheapest-to-deliver bond is 12% bond with conversion factor of 1.4000 Delivery in 270 days Last coupon 60 days ago, next in 122 days, then 305 days Yield curve flat at 10% Quoted price $120  What is the quoted futures price?

21. T-bond Futures  Cash price: 120 + (6 x 60/[60 + 122]) = 121.978  PV of next coupon: 6 e -0.1(0.3342) = 5.803  Cash futures price: (121.978 – 5.803) e 0.1(0.7397) = 125.094  Quoted futures price: 125.094 – 6 x (148/[148 + 35]) = 120.242 120.242/1.4000 = 85.887

22. T-bond Futures  Factors that affect the futures price: Delivery can be made any time during the delivery month Any of a range of eligible bonds can be delivered The wild card play

23. Eurodollar Futures  If Q is the quoted price of a Eurodollar futures contract, the value of one contract is 10,000[100-0.25(100- Q )]  A change of one basis point or 0.01 in a Eurodollar futures quote corresponds to a contract price change of $25  Settled in cash  Expires on third Wednesday of the delivery month and all contracts are closed out Z is set equal to 100 minus the 90 day Eurodollar interest rate (actual/360)

24. Eurodollar Forward Rates  Eurodollar futures contracts last as long as 10 years  For Eurodollar futures lasting beyond two years we cannot assume that the forward rate equals the futures rate

25. Eurodollar Forward Rates

26. Duration  Duration of a bond that provides cash flow c i at time t i is:  where B is its price and y is its yield (continuously compounded)

27. Duration  This leads to:  When the yield y is expressed with compounding m times per year  This expression is referred to as “modified duration”

28. Duration Matching  This involves hedging against interest rate risk by matching the durations of assets and liabilities  It provides protection against small parallel shifts in the zero curve

29. Duration Hedging  Suppose that: On 2 August, fund manager has $10m invested in govt bonds Hedges porftolio with December T-bond futures, priced at 93.0625 Duration of portfolio in 3 months is 6.8 CTD bond is 20yr 12% bond, yield = 8.8%, duration will be 9.2 at maturity of futures

30. Duration Hedging  No. of contracts: (10m x 6.8) / (93,062.50 x 9.20) = 79.42  If portfolio rises to 10.45m on 2 Nov, futures price = 98.50, loss on contracts: 79 x (98,500 – 93,062.50) = $429,562.50  Net change in manager’s position: $450,000 - $429,562.50 = $20,437.50

31. Duration-based Hedge Ratio F C = Contract price for Interest Rate Futures D F = Duration of asset underlying futures at maturity P = Value of portfolio being hedged D F = Duration of portfolio at hedge maturity