P1.T3. Financial Markets & Products - Hull

[CarlosB]

Hi, regarding this question:

4. A wheat farmer hedged her future sale of 100,000 bushels of wheat by selling forward 10
contracts (each for 5,000 bushels). The standard deviation of monthly changes in the spot and
futures price of wheat is, respectively, $0.60 and $0.90. What was her correlation assumption?
a) 0.67
b) 0.75
c) 0.80
d) 0.90

The answer is:

The optimal hedge ratio = correlation * spot standard deviation / futures standard deviation.
The optimal # of contracts = optimal hedge ratio * quantity being hedged / quantity of
contract.
Correlation = (optimal # of contracts * quantity of contract / quantity being hedged) * futures
standard deviation / spot standard deviation.
In this case, correlation = (10 * 5,000 / 100,000) * 0.90 / 0.60 = 0.75

Should it not be: spot standard deviation/ futures standard deviation?? 0.6/0.9??

 

[ShaktiRathore]

Hi
It should be .75 only. Optimal hedge ratio=Cov(spot,futures)/Var(futures)=correlation*stdDev of spot*stdDev of futures/stdDev of futures^2=correlation*stdDev of spot/stdDev of futures
Note above correlation is Cov/std Dev of spot*stdDev of futures its not stdDev of spot/stdDev of futures.
Thanks

 

[CarlosB]

Hi Shakti,

Thanks for the reply, but the the issue is that the answer from the notes has: futures
standard deviation / spot standard deviation (0.90 / 0.60) and not "spot standard deviation/ futures standard deviation" (0.6/0.9). If I do it like the latter, I will have 0.9 and not 0.75. Could you please help out?

Thanks,
Carlos

 

[ShaktiRathore]

Carlos
What r u referring to? Hedge ratio =correlation*spot stdDev/futures stdDev but correlation is calculated from hedge ratio by manipulating above equation as correlation=hedge ratio*futures stdDev/spot stdDev so hedge ratio has spot stdDev/futures stdDev is not correlation perse we obtained correlation from hedge ratio which has futures stdDev/spot stdDev not spot stdDev/futures stdDev which is contained in hedge ratio formula not correlation formula.Please dont get confused..
Thanks

 

[David Harper CFA FRM]

@CarlosB

As @ShaktiRathore shows, the fraction gets reversed because we are solving for the correlation. In other words, if we are given the correlation of 0.75, then

• solving for the optimal hedge ratio = rho*sigma(spot)/sigma(futures) = 0.75*0.60/0.90 = 0.50 optimal hedge ratio. However, we are essentially given the optimal hedge ratio of 0.50 because, as Number of contracts, N = h*Q(A)/Q(F) ---> h = N*Q(F)/Q(A) = 10*5,000/100,000= 0.50. As we are given the (implied) optimal hedge rate of 0.5:

• solving for correlation: as h = rho*sigma(spot)/sigma(futures) --> rho = h*sigma(f)/sigma(s) and, in this case, rho = 0.5*0.90/0.60 = 0.75.
  
• And we can "verify" with the original: optimal hedge = rho * sigma(s)/sigma(f) --> 0.50 = 0.75*0.60/0.90. I hope that explains, thanks.

 

[CarlosB]

Thanks for the help!