A farmer has a crop of grapefruit juice that will be ready for harvest and sale as 150000 pounds of grapefruit juice in 3 months. He is worried about possible price changes, so he is considering hedging - a financial engineering technique that minimizes future uncertainties in the cash flow. Typically, hedging is carried out using futures contract. However, unfortunately, there is no futures contract for grapefruit juice, but there is a futures contract for orange juice. Still, the farmer might consider using the futures contract for orange juice as a replacement for futures contract for grapefruit juice, in the hope that these two contracts are highly correlated due to the similarity of the underlying products. Currently, the spot prices are $1.20 per pound for orange juice and $1.50 per pound 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 their prices is about 0.7 (highly correlated). What is the minimum variance hedge for farmer, i.e. how many orange juice futures contracts does the farmer need to purchase in total?
A farmer has a crop of grapefruit juice that will be ready for harvest and sale as 150000 pounds of grapefruit juice in 3 months. He is worried about possible price changes, so he is considering hedging - a financial engineering technique that minimizes future uncertainties in the cash flow. Typically, hedging is carried out using futures contract. However, unfortunately, there is no futures contract for grapefruit juice, but there is a futures contract for orange juice. Still, the farmer might consider using the futures contract for orange juice as a replacement for futures contract for grapefruit juice, in the hope that these two contracts are highly correlated due to the similarity of the underlying products. Currently, the spot prices are $1.20 per pound for orange juice and $1.50 per pound 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 their prices is about 0.7 (highly correlated). What is the minimum variance hedge for farmer, i.e. how many orange juice futures contracts does the farmer need to purchase in total?
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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Transcribed Image Text:3. (Minimum-variance Hedging)
A farmer has a crop of grapefruit juice that will be ready for harvest and sale as 150000 pounds of grapefruit juice
in 3 months. He is worried about possible price changes, so he is considering hedging - a financial engineering
technique that minimizes future uncertainties in the cash flow. Typically, hedging is carried out using futures
contract. However, unfortunately, there is no futures contract for grapefruit juice, but there is a futures contract for
orange juice. Still, the farmer might consider using the futures contract for orange juice as a replacement for futures
contract for grapefruit juice, in the hope that these two contracts are highly correlated due to the similarity of the
underlying products.
Currently, the spot prices are $1.20 per pound for orange juice and $1.50 per pound 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 their prices is about 0.7 (highly correlated). What is the minimum variance hedge for farmer, i.e.
how many orange juice futures contracts does the farmer need to purchase in total?
Hint: let h be the number of orange juice futures contracts that the farmer purchases. Let T be the time when the
grapefruit juice will be sold (in 3 months). The cash flow at time T will be two parts: (1) profits from selling the juice
(2) cash flow generated by futures contract. Can you write down the explicit form of this cash flow? Notice that this
cash flow is a random variable, then we can compute h by minimizing the variance of the cash flow (hence
minimum-variance hedging).
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