13. (Grapefrut hedge Faer D Jones has a erop o grapetruit that be ready fo harve
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Question
13. (Grapefrut hedge Faer D Jones has a erop o grapetruit that be ready fo harvest und sa?e as 150,000 nounds of grapefruit juice in 3 monts Jones is woied about possible price changes, so he is consideiing hedging There is no futures contract tor grapefruit juice, but there is a futurescontact for orange juice His som, Gavin, nevently studied minium- variance hedging and suggests it as a possible approach Currently the spoi pries are ? 20 per pound (or orange juice and $t 50 per pound for grapefruit juice The standud deviation ol the prices of orange juice and grapefruit juice is about 20% per year, and the correlation coefficienu between them is about 7 What is the ninimum-variance hedge for fariner Jones, and how effective is this hedge as compared lo no hedge?Explanation / Answer
Answer )
We find such a position 'h' in orange juice futures that minimizes the cash flow of the farmer expected after 3 months. The cash flow 'Y' of the grape fruits proceeds and payments from futures contract at time T = 3m is
y = STg W + (FT ? F0)h,
Where, where STg is the spot price of Grape and FT the forward price of Orange at time T,
and W is the size of the crop and F0 the forward price at time 0.
The spot price of Orange at time T, STO
Var[y] = Var[STgW + (FT ? F0)h] ----------------------1
We minimize this by solving the optimal h* that set the derivative of variance to zero
=> h = ?? (STg *?g / STo*?o) W ,
here ,? =correlation =0. 7 , ?g (standard deviation in grapes) = 0.2 , ?0(standard deviation in oranges) = 0.2 , W = 150,000 pounds, Spot price of grapes (STg ) = 0.50 and Spot price of orange (STo ) = 0.20
=> h = -0.7 * ( 0.50 / 0.20) * 150000 = -262500 Pounds.
Hence the ratio of variances and standard deviations are
(1- ?2) W2?g2 / W2?g2 = 0.51
with minimum variance
= (1 - ?2 ) ^ 0.5 = 0.71.
. Thus, the standard deviation of the cash ow is roughly 30% smaller with the minimum variance hedge
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