1. Options and Speculative Markets 2005-2006 Hedging with Futures Professor André Farber Solvay Business School Université Libre de Bruxelles

2. OMS 03 Hedging with Futures |2 2.Hedging with futures Objectives for this session: –1. –2. –3.

3. OMS 03 Hedging with Futures |3 Identifying the exposure Exposure: position to be hedged Cash flow(s) –Future incomeEx: oil/gold producer –Future expenseEx: user of commodity Value –AssetEx: asset manager –LiabilityEx: financial intermediary General formulation: Exposure = M  S with: M = quantity, size (M > 0 asset, income M < 0 liability, expense) S = market price

4. OMS 03 Hedging with Futures |4 Setting up the hedge Futures position: Number of contracts n (n>0 long hedge – n<0 short hedge)  Size of one contract N  Futures price F Hedge = n  N  F Perfect hedge: choose n so that value of hedged position does not change if S changes

5. OMS 03 Hedging with Futures |5 Hedge ratio To achieve ∆V = 0 Hedge ratio: To achieve ∆V = 0 If M >0 : n <0 short hedge If M 0 long hedge

6. OMS 03 Hedging with Futures |6 Perfect hedge Assume F and S are perfectly correlated: then: h = - β and

7. OMS 03 Hedging with Futures |7 Basis risk Basis = Spot price of asset – Futures prices (S-F) Spot priceS 1 S 2 Futures priceF 1 F 2 Basisb 1 = S 1 –F 1 b 2 = S 2 – F 2 Cash flow at time t 2 : Long hedge: -S 2 + (F 2 – F 1 ) = – F 1 – b 2 Short hedge: +S 2 + (F 1 – F 2 ) = + F 1 + b 2 t1t1 t2t2 known at time t 1

8. OMS 03 Hedging with Futures |8 Minimum variance hedge Real life more complex: –1. asset to be hedged might differ from underlying the futures contract –2. basis (S –F) might vary randomly More general specification: Choose n to minimize the variance of ∆V

9. OMS 03 Hedging with Futures |9 Some math Take derivative and set it equal to 0: Solve for n:

10. |10 Hedging Using Index Futures Stock index futures: futures on hypothetical portfolio tracked by index. Size = Index × Value of 1 index point Example: S&P 500 (CME) - $250 × index To hedge the risk in a portfolio the number of contracts that should be shorted is where P is the value of the portfolio,  is its beta, and A is the value of the assets underlying one futures contract

11. OMS 03 Hedging with Futures |11 Reasons for Hedging an Equity Portfolio Desire to be out of the market for a short period of time. (Hedging may be cheaper than selling the portfolio and buying it back.) Desire to hedge systematic risk (Appropriate when you feel that you have picked stocks that will outperform the market.)

12. OMS 03 Hedging with Futures |12 Example Value of S&P 500 is 1,000 Value of Portfolio is $5 million Beta of portfolio is 1.5 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?

13. OMS 03 Hedging with Futures |13 Changing Beta What position is necessary to reduce the beta of the portfolio to 0.75? What position is necessary to increase the beta of the portfolio to 2.0?

14. OMS 03 Hedging with Futures |14 Rolling The Hedge Forward We can use a series of futures contracts to increase the life of a hedge Each time we switch from 1 futures contract to another we incur a type of basis risk

15. OMS 03 Hedging with Futures |15 Example: Stock index futures