Need help with a simple calculation...

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wawa

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I would appreciate some help with this simple calculation; however, simple as it might be I'm stuck...

"It is now April 1st. We want to buy 3'750 pounds of coffee in June, and want to hedge that. Since futures on coffee are for delivery in May and July, we use a combination of these two futures for the hedge. The standard deviation of the spot price on coffee in May and June is assumed to be 4.3 and 4.6 cents per pound and the futures price in June for July coffee has the standard deviation 5.0 cents per pound. The correlation coefficients are assumed to be (May,June) = 0.96, (June,July) = 0.92 and (May,July) = 0.90. Determine the best hedge, i.e., how many pounds of coffee the May and July futures should pertain to. Answer with numbers rounded to whole pounds. Answer: The May future should be A pounds of coffee and the July future should be B pounds of coffee."

How do you reason to get to the numbers A and B, and what are they.... :(
 
This is kind of a messed-up problem, because the combination needs to be changed in May when the May contracts expire. That means you can hedge with the May and July for part of the time and then switch to all July. That means we need some slightly different data than what we have, but we'll go with it.

w1 = weight of May coffee
w2 = weight of June coffee
p1 = price of May coffee
p2 = price of July coffee
P = price of June coffee

Eqn 1: w1 + w2 = 3750
Eqn 2: Minimize Var(w1*p1 + w2*p2 - 3750*P) in w1 and w2

So now you expand out that Var thing into

Var(w1*p1 + w2*p2 - 3750*P) = w1^2*Var(p1) + w2^2*Var(p2) + 2750^2*Var(P1) +2*w1*w2*Cov(p1, p2) - 2*w1*3750*Cov(p1, P) - 2*2*3750*Cov(p2, P)

Since they've given you all the numbers you need you plug those in and get an eqn in w1 and w2. Take derivs and set = 0. Solve.

This problem is beyond the scope of the CFA exam. Is it Book 7 or PRM exam?
 
Thanks, Joey. Book 7 or PRM? It's actually neither of those two; it's a university summer term course they call 5B4020---Web based course in Financial Mathematics. I signed up for it when I had all this idle time no longer stuying for the Level I exam as I felt I had too much idle time on my hands.
 
Well, it will prepare you excellently for easier problems than this on L II and (especially) L III.
 
wawa Wrote:
-------------------------------------------------------
> Thanks, Joey. Book 7 or PRM? It's actually neither
> of those two; it's a university summer term course
> they call 5B4020---Web based course in Financial
> Mathematics. I signed up for it when I had all
> this idle time no longer stuying for the Level I
> exam as I felt I had too much idle time on my
> hands.


What university?
 
Set up in matrix form and solve:

(I) : Cov(Ft(1), Ft(1))*beta1 + Cov(Ft(1), Ft(2))*beta2 = Cov(Ft(1), S)
(II): Cov(Ft(1), Ft(2))*beta1 + Cov(Ft(2), Ft(2))*beta2 = Cov(Ft(2), S)

(I): Var(Ft(1))*beta1 = Cov(Ft(1), S) - Cov(Ft(1), Ft(2))*beta2

--> beta1 = ro(Ft(1), S)*SD(S)/SD(Ft(1)) - ro(Ft(1), Ft(2))*beta2*SD(Ft(2))/SD(Ft(1))

Plug into (II):

(II): ro(Ft(1),Ft(2))*SD(Ft(2) * ( ro(Ft(1),S)*SD(S) - beta2*ro(Ft(1), Ft(2))*SD(Ft(2)) + Var(Ft(2))*beta2 =
= ro(S, Ft(2))*SD(Ft(2))*SD(S)

Solve for beta2

0.90 * 5 ( 0.96 * 4.6 - beta2 * 0.90 * 5) + (5)^2 * beta2 = 0.92 * 5 * 4.6

beta2 * 25 * (1 - 0.90^2) = 5 * 4.6 * (0.92 - 0.9 * 0.96)

==> beta2 = 4.6/5 * (0.92 - 0.9*0.96)/(1 - 0.90^2) = 0.2711

Solve for beta1 (plug and chug)

= ro(Ft(1),S)*SD(S)/SD(Ft(1)) - ro(Ft(1), Ft(2))�SD(Ft(2))/SD(Ft(1)) =

= 0.96 * 4.6 / 4.3 - 0.90 * 0.2711 * 5 / 4.3 = 0.7432

Answer: The May future should be 2787 pounds of coffee and the July future should be 1017 pounds of coffee

Phwew! It seems so simple now but it's so easy to make a mistake on the way...
 
ChadD Wrote:
> What university?

This course (link): http://www.math.se/5b4006.html

I'm afraid there might be lingustic problems for everyone but to Scandinavians. The course material itself consists of Hull's book "Options and Futures" plus some additional material for the mathematics itself, in English, the math is not included in Hull's material. The lectures are rolled out using Windows Media Player.
 
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