Minimum Variance Ratio

[frm_student_1]

Hi David & All,

I have a question I could use your help with. In some places in the curriculum the hedge ratio is given as:

corr(A,B) * stdev(A)/stdev(B)

in a few areas it is given as:

-corr(A,B) * stdev(A)/stdev(B)

when is it the latter case?

 

[David Harper CFA FRM]

Hi @frm_student_1

I agree, specifically Hull gives corr(A,B) * stdev(A)/stdev(B), while Miller uses -corr(A,B) * stdev(A)/stdev(B). They are not really different. I have likened this to how VaR is sometimes given as a negative (which is mathematically true: VaR is typically a loss!) but mostly used as a positive (because, by convention and for some convenience, the axis is switched such that losses are positives). Miller's negative is mathematically accurate because, assuming the correlation is positive, his negative is making explicit that a short position is going to be the most effective hedge against a long position. In the two-asset portfolio variance, a short position is signified with a negative weight.

Hull is not different, really. His context is a futures contract hedging a spot position. He just assumes out the negative. If the correlation is positive and the underlying position is a long spot postion, Hull's optimal hedge (minimum variance ratio) will be positive (as both volatilities MUST be positive). Mathematically that maybe isn't pure, but he's just assuming that we know to hedge with a short futures position.

In this way, you can't go wrong with Miller's [-corr(A,B) * stdev(A)/stdev(B)] but you are still better off to apply the judgement that Hull implicitly delegates with [corr(A,B) * stdev(A)/stdev(B)]. I hope that helps!