Question

Farmer D Jones has a crop of grapefruit that will be ready for harvest and sale as 150,000 pounds of grapefruit juice in 3 months. Jones is worried about possible price changes, so he is considering hedging. There is no futures contract for grapefruit juice, but there is a futures contract for orange juice. His son, Gavin, recently studied minimum-variance hedging and suggests it as a possible approach.

Currently the spot prices are $1.20/lb for orange juice and $1.50/lb for grapefruit juice. The standard deviation of the prices of orange juice and grapefruit juice is about 20% per year, and the correlation coefficient between them is about 7.

What is the minimum-variance hedge for farmer Jones, and how effective is this hedge compared to no hedge?

Answer 1

Answer :-

. Short 7 four-month futures contracts and close out the position in 3 months when grapefruit is ready to be sold.

Since

Jones is going to sell fixed amount of grapefruit juice, wherew= 150,000 and the spotpriceSGfor grapefruit juice is $1.50,

we haveβ=cov(SG,SO)/var(SO)=ρGOσGσO/σ^2O=ρGOσG/σO

=0.7×20%×1.50/20%×1.20= 0.875.

So,h=-βw=-0.875×150,000 =-131,250

Hence, the farmer should short 131,250 pounds of orange juice.

By minimum-variance hedge formula, we haveσy=p1-ρ^2σx=⇒σy= 0.714σx.

Therefore, the risk reduce has reduced to 71.4% of the no hedge case.