Tn7 Hedge Ratios And Futures Contracts

Document Date: November 2, 2006 An Introduction To Derivatives And Risk Management, 7th Edition Don Chance and Robert Brooks Technical Note: Hedge Ratios and Futures Contracts, Ch. 11, p. 366 This technical note supports the material in the Determination of the Hedge Ratio section of Chapter 11 Forward and Futures Hedging, Spread, and Target Strategies. We derive the minimum variance hedge ratio and the price sensitivity hedge ratio. Derivation of Minimum Variance Hedge Ratio The variance of the profit from the hedge is 2 2 σ 2Π = σ ∆S + σ ∆f N f2 + 2Cov ∆S,∆f N f .

The value of Nf that minimizes σ 2Π is found by differentiating σ 2Π with respect to Nf. ∂σ 2Π 2 = 2σ ∆f N f + 2Cov ∆S,∆f . ∂N f Setting this equal to zero and solving for the minimum variance hedge ratio, N *f , gives N *f = −

Cov ∆S,∆f 2 σ ∆f

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The second derivative (2 σ 2∆f > 0) verifies that this is a minimum. Derivation of Price Sensitivity Hedge Ratio The value of the bond position can be specified as V = B + VfNf, where Vf is the value of the futures contract. Now we wish to find the effect of a change in r on V. Since ∂Vf / ∂r = ∂f / ∂r,

∂V ∂B ∂f = + Nf . ∂r ∂r ∂r The optimal value of Nf is the one that makes this derivative equal to zero. We do not know the derivatives ∂B / ∂r and ∂f / ∂r, but we can use the chain rule to express the equation as ∂V ∂B ∂y B ∂f ∂y f = + Nf = 0 . ∂r ∂y B ∂r ∂y f ∂r

This procedure introduces the yield changes, ∂yB and ∂yf, into the problem. Usually it is assumed that ∂yB / ∂r = ∂yf / ∂r. Substituting and solving for price sensitivity hedge ratio, N *f , gives

N *f = −

(∂B/∂y B ) . (∂f/∂y f )

N *f = −

(∆B/∆y B ) (∆f/∆y f )

This value is approximated as

 ∆B  ∆y f  = −   ∆f  ∆y B 

IDRM7e, © Don M. Chance and Robert-Brooks

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