# Optimal Hedge Ratio

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What is Hedging, Hedge Funds, Details about Hedging, Hedge Funds, Foreign Exchange Risk in India, Global Currency Hedging
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O PTIMAL H  EDGE  R   ATIO (The attached PDF file has better formatting.) Study recommendation: Some exam problems test the optimal hedge ratio in varioushedges. Know the basic formula and its relation to CAPM betas and bond durations.Exercise 1.2: Cocoa Hedges A cocoa merchant with inventory of cocoa worth \$10 million at present prices of \$1,250 per metric ton is considering a risk-minimization hedge of the inventory using the cocoacontract of the Coffee, Cocoa, and Sugar Exchange. ! The volatility of returns for the cocoa inventory is 27%. ! The futures contract size is 10 metric tons. ! The volatility of the cocoa futures is 33%. ! For the grade of cocoa in the inventory, the correlation between the change in thefutures price and the change in the spot cocoa price is 85%. A.What is the risk-minimization hedge ratio?B.Should the cocoa merchant be long or short the futures contracts?C.How many contracts should be traded? Part A: The optimal hedge ratio equals SF the correlation between the hedged good and the hedging instrument ( D ) S ×the standard deviation of the hedged good ( F ) F ÷the standard deviation of the hedging instrument ( F ), or  SFSF hedge ratio = D × F / F The optimal hedge ratio equals 0.85 × 0.27 / 0.33 = 0.6955. Take heed: The volatility is the standard deviation times the square root of time. The ratioof volatilities equals the ratio of standard deviations, since the square root of time appearsin both numerator and denominator. Part B: The cocoa merchant now owns the cocoa inventory. The merchant’s risk is that ahigher supply of cocoa or a lower demand for cocoa will reduce the price of cocoa. Themerchant must be short the cocoa futures contract. Mnemonic: A party that owns an asset or commodity faces a risk that its price will decline. ! The long position pays a fixed price and gains if the price increases.  ! The short position receives a fixed price and gains if the price decreases.To hedge, the merchant must gain from the futures contract if the price declines, so themerchant takes a short position.The cocoa merchant has goods to sell. The merchant is long the goods, so he or she mustsell futures contracts. A trader who has goods to sell must sell the futures contracts, anda trader who wishes to buy goods should buy the futures contracts. Part C: If the hedge ratio were unity and the spot price and futures price had the samevolatility, one should trade futures contract just equal to the value of the hedged good, or \$1,250 × 10 × Z = \$10 million, or Z = 800 futures contracts. The optimal number of contracts to trade is 800 × 0.6955 = 556 contracts. Note that each contract is for 10 metrictons, and each metric ton is worth \$1,250.The optimal hedge ratio minimizes the variance of the total portfolio. If one buys futurescontract for each unit of the underlying good of the same size, the total portfolio isN goods + × N futures.The variance of this portfolio is SSFF N × F + 2 × N × × D × F × F + × N × F   222222 We set the partial derivative with respect to equal to zero: SFF N × [2 × D × F × F + 2 × F ] = 0 22SF = – D × F / F The negative sign for  means that if the investor is long the commodity, he or she mustbe short the futures (if the correlation is positive), and if the investor is short the commodity,he or she must be long the futures (if the correlation is positive). Exam Problems For exam problems, the optimal hedge ratio comes in several guises. A.Exam problems on commodity futures give standard deviations and correlations. Anexam problem might give the slope coefficient from a regression of the spot price onthe futures price. The slope coefficient is the optimal hedge ratio.B.Exam problem on stock portfolios hedged with strike price futures give the beta of theportfolio, which is the optimal hedge ratio.C.Exam problems on interest rate futures give the durations of the fixed-income securitiesportfolio and of the asset underlying the futures contract. The durations are like the  volatilities, and the correlation is one. The ratio of durations is the optimal hedge ratio.  Exercise 1.3: Cocoa and Poppy Hedges A poppy farmer owns half of Afghanistan, with poppy farms that produce \$150 million of poppy at present prices of \$3,000 per metric ton. The volatility of returns for the poppy is51%.Because of regime changes in Afghanistan, Iraq, and Turkey, the poppy farmer isconcerned about declines in the price of poppy. There is no poppy futures contract, butthe farmer knows that the price of poppies is inversely correlated with the price of cocoa.He is considering a risk-minimization hedge of the inventory using the cocoa contract of the Coffee, Cocoa, and Sugar Exchange. The contract size is 10 metric tons. The volatilityof the futures is 33%. The correlation between the cocoa futures and the poppy price is –45%. Compute the risk-minimization hedge ratio and determine the number of contractsto be traded.
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