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Questions FRM Handbook Example 12.4


New Member
Dear David,

In the FRM handbook of Jorion, I think I have found a mistakes in the answers, but I am not quite sure. Hope you can verify.

A bronze producer will sell 1,000mt of bronze in three months at the prevailing market price at that time. The SD of the price of bronze over a three-month period is 2.6%. The company decides to use three-month futures on copper to hedge. The copper futures contract is for 25 mt of copper. The SD of the futures price is 3.2%. The correlation is 0.77%. To hedge its price exposure how many futures contract should the company buy/sell?

A) Sell 38 futures
B) Buy 25 futures
C) Buy 63 futures
D) Sell 25 futures.

The anwer, according to Jorion, is answer B.

However, I think it should be D. Hull states in chapter 3 of his book that a short hedge is appropriate when the hedger already owns an asset and expects to sell it at some time in the future. Which is the case here.

Am I interpreting this wronly?
So, who is right?

Thanks in advance,



New Member
Optimal Hedge Ratio = h* = rho*Sigma(S)/Sigma(F) hence h* = (0.77)*(2.6%)/(3.2%). I am sure the 0.77 can never be in %.
So h* = 0.6256

N* = - h* NA/Qf = - (0.6256) * (-1000mt)/ 25mt
He is selling 1000 mt in 3 months time and needs to hedge that with futures (contract size of which is 25mt).
So N* = 25 (positive sign hence buy)


New Member
Dear skcd and David,

I do not agree, in Hull, h* is defined as rho*sigma(S)/sigma(F). So, we agree on the h*.
However, N* is defined as h*NA/Qf, so, without the minus sign (page 59 of Hull 6th edition).
Leading to N* = -25, so selling.

In fact I don't see any minus sign in Hull, contrary to Jorion.

So, my conclusion is still selling...

David, can you please give a hint???


New Member
Come on, screw hull! See the common sense here. You are going to sell the underlying in 3 months. Are you going to sell futures as well to hedge it??????
Hedge of a call option is going short the stock. Hedge of a short call option is to buy and hold stock.
I guess you are just getting hung upon formulae.


New Member

I something in it, but it is not in line with Hull.

Looking example 12.5: FRM exam 2002- Question 35:

A company expects to buy 1 mln barrels of crude oil in one year. The annualized volatility is12% The company chooses to hedge by buying a futures contract on Brent crude. The annualized volatility of the Brent futures is 17% and the correlation coefficient is 0,68. Calculate the variance-minimizing hedge ratio.

There is stated buying 1mln barrels in one year and buying futures contracts... The opposite (selling and selling as I think it should be) to the other question...
So, can you explain this then.... (This is totally in line with Hull!!!) Or is this question in your opinion wrong? Please read also pages 48/49 of Hull.

David Harper CFA FRM

David Harper CFA FRM
Staff member
I agree with Johannes, example 12.4 is wrong.

In Hull, the optimal hedge ratio formula gives the number of contracts to short; i.e., +N means short, -N means go long the hedge.
(I would argue, in this case, Jorion has simply been more precise to include the negative sign)

But i agree too, that thinking this thru is superior to trusting the +/- sign.

The producer will be selling, say bronze is $1 today, but producer will be selling bronze in one year.
Producer (a.k.a., hedger) wants to be short the forward contracts (the hedging instruments), because:

say bronze drops to $0.80 (i.e., future spot price), then producer will "lose" $0.20 on the lower spot
But he/she will profit on the short forward. The short forward makes a profit, so it hedges loss on underlying.
A long forward would not hedge, it would be a double-down on his losses.
Answer is wrong and should be (D).

Hope that helps, David


New Member
1)I dont remember Hull and no time to read 48/49 pages
2)I think i was right and stand by it for the case of selling bronze.
3) There are conventions with contracts. I am not sure if that is relevant here.
But lets say buying a future contract in oil might mean delivering oil (brent crude or WTI or light sweed or whatever f##$in crude)

So, when you are expecting to buy oil in one year you will hedge it with a contract that requires you to deliver or sell oil and buy dollars.

I don't know but it could be that a)oil futures are defined as delivering oil while b)copper/bronze futures are defined as buying them in future.

I am not sure.

You have a very valid point.

Plus note that this isn't an apples to apples comparison as its only the h* (hedge ratio) not the offsetting hedge number that you are referring.