Suppose a farmer is expecting that her crop of grapefruit will be ready for harv

Question

Suppose a farmer is expecting that her crop of grapefruit will be ready for harvest and sale as 150,000 pounds of grapefruit juice in 3 months time. She would like to use futures to hedge her risk but unfortunately there are no futures contracts on grapefruit juice. Instead she will use orange juice futures. Suppose each orange juice futures contract is for 15,000 pounds of orange juice and the current futures price is F0=118.65 cents-per-pound. The volatility, i.e. the standard deviation, of the prices of orange juice and grape fruit juice is 20% and 25%, respectively, and the correlation coefficient is 0.7. What is the approximate number of contracts she should purchase to minimize the variance of her payoff? Please submit your answer rounded to the nearest integer. So for example, if your calculations result in 10.78 contracts you should submit an answer of 11.
If answer is (without rounding) 8.75, then why not 0.875 (y*=Cov(Ftorange,Ptgrapefruit)/Var(Ftorange)=0.875 or (#contractsorange/#contracts grapefruit) = 15000/17142,65 = 0.875?

Explanation / Answer

Optimal contrack Size = (Size of Desired Underlying / contract size of 1 future contract)* Correlation * Risk of Future contract / Risk of Future contract

Optimal no. of contract = (150,000 /15,000) * 0.7 * 0.2 /0.25

Optimal no. of contract = 5.6

there no. of contract = 6 nos

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