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Unformatted text preview: Hedging with forwards and futures Example A gold mining firm anticipates to produce 500,000 ounces of gold during the next year, and wants to fully hedge its gold price risk. Three possible hedging alternatives Forward hedge Futures hedge Money market hedge Forward Hedge Suppose the entire gold production is sold at the end of the year. Short forwards on 500,000 ounces of gold that mature in one year. Payoff at maturity 500,000 S T + 500,000 (F t,T  S T ) = 500,000 F t,T Conditions for a perfect forward hedge: Asset to be hedged must exactly match the asset underlying the forward contract. No asset mismatch. Delivery date of the forward contract must exactly match the arrival date of the cashflow to be hedged. No maturity mismatch. Number of units of the asset to be hedged must be an exact multiple of the number of units of the forward contract. No rounding error. Futures Hedge Gains and losses on futures contracts are settled daily. Cash flow at T : (short futures) where F t =F t,T denotes the futures price at time t , and F T =S T . How can we turn this into a perfect hedge? Solution : Short futures for 500,000/(1+r) T ounces of gold at t =0 . At t =1 increase the position to 500,000/(1+r) T1 oz of gold. At t =T1 increase the position to 500,000 oz of gold. h ( F F 1 )(1 + r ) T + h 1 F 1 F 2 ( ) (1 + r ) T 1 + ... + h T 1 F T 1 F T ( ) T2 T1 T days 1 2 Conditions for a perfect futures hedge No asset mismatch. No maturity mismatch. No rounding error. Interest rates are known in advance. (And the hedge must be tailed. ) Tailing a Hedge Intuition : Markingtomarket increases the volatility of a futures hedge, because gains and losses are carried forward at the prevailing interest rate, i.e., they grow at the rate r . Therefore, the hedge ratio must be reduced. To tail a futures hedge you hedge only the present value of the exposure . Tailing factor = discount factor: Tailing requires continuous adjustments to the hedge position. The hedge ratio grows over time as the maturity of the exposure declines. 1 (1 + r ) T or e rT Example 2 Suppose a trading company entered into the commitment to deliver 150 million barrels of oil in 10 years at $26/barrel. What is the (varianceminimizing) futures hedge? Assume that the continuously compounded riskfree rate is 5% p.a. 150 million e0.05*10 = 82.3 million The varianceminimizing futures hedge is only 55% of the original exposure! Money Market Hedge Suppose derivatives markets are not available. How can you hedge a long exposure of 500,000 oz of gold without derivatives? 1. Borrow 500,000/(1+ g ) oz of gold from a bullion dealer....
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 Fall '09
 Bergman

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