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Market portfolio and derivative of weight?

David Harper CFA FRM

David Harper CFA FRM
Staff member
@agattik Miller's is a strictly correct solution to the derivative (i.e., solving for a local minimum in the net portfolio's variance): by assuming the portfolio variance is given by σ(A + B) it is assuming the that (B) is added to (A) such that, when the correlation is positive, we need to short the hedge instrument (hence the negative sign). Hull's h* = ρ*σ(s)/σ(f) is not really different, at all. In solving the slope of the regression line, it is just solving (strictly) for the relationship, which is the β(ΔS, ΔF), and not the actual hedge position; however, Hull's can dispense with the sign (+/-) because we don't really need the sign to tell us (and I do recommend against such a mechanical use of the formula's output), and also, as suggested, I think it's better to grok that the hedge ratio is the regression beta. I hope that helps. @emilioalzamora1 recently assisted on the same/similar question here https://www.bionicturtle.com/forum/...and-correlation-miller-ch-3.11966/#post-61807 ie..,
Question: Why do i have to multiply the correlation (A,B) in negative on the formula despite on the text stated the correlation is +0,5?
1. The minimum variance hedge ratio, h* = -ρ(A,B)*σ(A)/σ(B).

Hi @jlahuerta

the min. variance hedge formula includes the "minus sign" and this minus sign remains in case the correl coefficient is positive (in this case +0.5).

If the correl coefficient would be - 0.5, then the formula would get you get a positive sign at the right-hand side of the equation.

Having a positive correlation between Security A and B can be regarded as an unfavourable property (from a diversification perspective).
  • Given a positive correlation (0.5) between Security A and B, you will have to short Security B in order to guarantee a minimum variance.
  • Given a negative correlation (e.g. -0.7) between Security A and B, you would rather go LONG Security B (because of the diversification benefit).