13 Grapefrut hedge Faer D Jones has a erop o grapetruit that

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?

Solution

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 = S_T^g W + (F_T ? F₀)h,

Where, where S_T^g is the spot price of Grape and F_T 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, S_T^O

Var[y] = Var[S_T^gW + (F_T ? F₀)h] ----------------------1

We minimize this by solving the optimal h^* that set the derivative of variance to zero

=> h = ?? (S_T^g *?_g / S_T^o*?_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 (S_T^g ) = 0.50 and Spot price of orange (S_T^o ) = 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