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Derivation of minimum variance portfolio

Jorge.Beca

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Thread starter #1
Hi, I am going through the excel sheet (R8-P1-T1-Elton-CAPM-v3) in section 9, and I saw the formula to find out Wa for the Minimum variance portfolio. It is not very intuitive to see that that formula is the 1st derivative of the portfolio variance with respect Wa. I know the formula is not part of the exam, but I wonder, for the person that is curious or with forgotten math concepts, if it is possible to add a little description next to the formula (or what is the alternative, because not everything is explained in the forum). Thanks.
 

Jorge.Beca

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Thread starter #2
Sorry, me again. Another example of what I mean is that in the same excel sheet, tab T1-SML, where is the formula for Covariance (Port, Market) in line 31 coming from? Thanks.
 

David Harper CFA FRM

David Harper CFA FRM
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#3
Hi @Jorge.Beca

Re: MVP, Yes, I have tagged our edit project (the master list of edit tasks) with adding that annotation. The derivation has been much discussed and elaborated already in the forum (if you search "minimum variance portfolio"); e.g., https://www.bionicturtle.com/forum/threads/p1-t2-305-minimum-variance-hedge-miller.6800/

Re: Cov(Port, Market)
  • Let Portiolo allocations (weights) = w(aP) + w(bP) and let Market allocations (weights) = w(aM)) + w(bM)
  • Per https://en.wikipedia.org/wiki/Covariance, this is an application of basic cov property: cov(aX + bY, cW +dV) but both the portfolio and the market are allocations between the two assets:
  • Cov(Port, Market) = Cov[ w(aP)*A + w(bP)*B, w(aM)*A + w(bM)*B ] = w(aP)*w(aM)*cov(A,A) + w(aP)*w(bM)*cov(A,B) + w(bP)*w(aM)*cov(A,B) + w(bP)*w(bM)*cov(B,B) ... and Cov(A,A) = σ^2(A), cov(B,B) = σ^2(B). I hope that's helpful,
 
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