Interview Question: Goldman Sachs: Imperfect Hedge

[John Eaft]

A farmer expects to have 150 thousand kilos of grapefruit juice for sale in 3 months.
However, the farmer is afraid that the spot market price may be very low.
To hedge her risk she’ll use futures contracts.
However, there are no futures contracts for grapefruit juice.
So she will use orange juice futures instead : each contract is for 15k pounds of orange juice & current futures price=118.56 cents per pound.
Standard Deviation of the prices of orange juice & grapefruit juice is 20% & 25% respectively & the correlation coefficient is 0.7.
What is the approximate number of contracts she should buy to minimize the variance of her payoff?

The correct answer is approximately 8. I get '1'. Where am I wrong?

The farmer goes short on y orange juice Futures, so at T she receives y*(F_0 - F_T).
IF she could sell grapefruit juice at time T she would receive a profit of G_T for example.
We want to minimize the difference between these two cash flows.
So we will minimize the variance of y*(F_0 - F_T) -G_T.
The variance of this is y^2 * Var (F_0 - F_T) + Var (G_T) - 2y Cov( F_0 - F_T, -G_T).
This is equal to y^2 * Var(F_T) + Var(G_T) - 2y Cov ( F_T , G_T).
We want to minimize so we take the derivative (with respect to y) to be equal to zero.
So 2y Var(F_T) - 2 Cov (F_T, G_T) =0.
So y Var(F_T) - Cov (F_T, G_T) =0.
So y = Cov (F_T, G_T) / Var(F_T) (1)
We are told the standard deviation of the prices of orange juice &grape fruit juice is 20% and 25%, respectively.
At time T, the price of orange juice is equal to its futures price. Ie S_T = F_T. So we have Var(S_T) = Var(F_T). And we know Var(S_T) = 0.2^2= 0.04. And so Var(F_T) = 0.04.
We also know that the correlation coefficient is 0.7.
The correlation coefficient is equal to Covariance(F_T,S_T) divided by the product of the standard deviations. So Cov(F_T , G_T) / (0.2*0.25) = 0.7. So Cov(F_T,G_T)= 0.7* 0.2 *0.25 = 0.035.
So (1) should give y = 0.035 / 0.04 = 0.875, And this is rounded to 1.
What is my mistake?

 

[financeguy]

Without checking your math on the portfolio variance minimization, you seem to have ignored the question of number of contracts. It looks like what you've done is figure out how many contracts of orange juice you should sell per one contract of grapefruit juice, i.e. the hedge ratio. I'm assuming you meant 150k pounds of grapefruit juice, or 15k kilos of orange juice in a contract, i.e. it's 150k pounds of grapefruit juice and 15k pounds of orange juice per contract, or it's 150k kilos of grapefruit juice and 15k kilos of orange juice per contract, so that there's no issue with converting between kilos and pounds. If that's the case, you'd have needed 10 hypothetical contracts of grapefruit juice to hedge, meaning you need to multiply your result by 10, making 8.75. That would get us to 9 as the answer with rounding to the closest full unit, but closer to that 8 number.

That aside, intuitively I'd have thought since orange juice has a lower volatility you'd need more than 1 as the hedge ratio but I haven't really thought it through properly.

 

[John Eaft]

Thanks for your reply. Observations:
1. It is 150,000 pounds and not kilos. Thanks for the correction.
2. You write in your reply that I have ignored the number of futures contracts on orange juice. Please observe that in my solution, I call 'y' as the number of futures contracts on orange juice, which the farmer goes short. Why do you call 'y' the hedge ratio , when in reality it is the number of contracts ? (this is how i define it).
-Please explain as I may have understood wrong.

 

[financeguy]

The price per pound of each of those commodities is what varies with the standard deviations given and with that correlation. You have ignored the 150k pounds of grapefruit juice and the 15k lot size. Write down the exact dollar pnl formula.

 

[gver]

i think the pound and kilos is not a mistake .. infact you might have written it the other way round .. grapefruit should be in pounds and orange juice in kilos ..

assuming that is true, then ,

minimising the variance is just another way to say how many contracts to hedge it in the best manner .. therefore, PnL from change in price of grapefruit = PnL from change in price of orange juice

= 150,000* vol_g = 15,000 * N * 2.2 (this is kilos to pounds) * vol_o * correl

N = 10 * 25%/(2.2*20%*0.7) = 8.11 ~ 8 contracts

 

[financeguy]

Orange juice futures are quoted in pounds in real life so it's hard to imagine they made up a hypothetical world in which they are quoted in kilos.

I don't think you can do what you've done there, gver. You need to write down the pnl function first and then take the variance of it as John has tried to do (just with a function that hasn't been scaled). The correlation, for instance, isn't just there to scale down the volatility of orange juice. In any case, here's how I think it's done:

PnL = 150k * G + N * 15k * 1.1856 - N * 15k * F, where G is the spot price of grapefruit juice in 3 months and F is the spot price of orange juice in 3 months

Var(PnL) = 150k^2 * Var(G) + N^2 * 15k ^2 * Var(F) - 2 * 150k * N * 15k * Vol(G) * Vol(F) * Corr(G,F), noting that the term with the current futures price of orange juice is invariant so doesn't matter, and also using Cov(G,F) = Vol(G)*Vol(F)*Corr(G,F)

Var(PnL) = 150k^2 * .25^2 + N^2 *15k^2 * .2^2 - 2 * 150k * N * 15k * .25 * .2 * .7, plugging in the known terms

dVar(PNL)/dN = 2 * N * 15k^2 * .2^2 - 2 * 150k * 15k * .25 * .2 * .7 = 0

2 * N * 15k^2 * .2^2 = 2 * 150k * 15k * .25 * .2 * .7

N * 15k * .2 = 150k * .25 * .7, cancelling out matching terms on each side

N = 10 * .25 * .7 / .2 = 8.75

This is very similar to what John was doing, but it incorporates the amounts of everything we need.

 

[rajanS]

n= .25/.20 * .70 = 0.875

This is just OLS regression of Y on X with Y being the type of security you want to hedge and X being the futures position needed to cross hedge. You want to minimize the variance in Y. Slope in a two variable regression is given by relative volatility of Y over vol of X times their correlation.
Optimal contract required is n*Q_A units of assets needed. Where QA is size of position to be hedged. Hence N = n*QA/QF where QF is the size of 1 futures position
We get 0.875*10 ~8.75