Derivation of minimum variance portfolio

[Jorge.Beca]

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]

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]

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://forum.bionicturtle.com/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,

 

[Amierul]

Hi @David Harper CFA FRM , Good day.

I have a question regarding the R8-P1-T1 spreadsheet, on the PPC-MVP Section.
Appreciate if you can help to answer them.

1. How can asset be negative? What is the assumption there?

thank you!

 

[David Harper CFA FRM]

Hi @Amierul In the MPV/PPC/CAPM worksheet, any negative values are weights (e.g., volatilities must be positive) and the weight of an asset (in the PPC) or the weight allocated to the risk-free rate (in the CML) can be negative. For an asset, negative signifies that we are shorting the asset; i.e., we sell it (today) rather than buy it. It is no different for the risk-free rate because this is simply the rate earned on a risk-free asset: in the case of a negative allocation to the risk-free rate, it signifies that we are selling the Rf asset, otherwise known as borrowing rather than lending (in the CML, leverage enables an expected return above any asset simply because we borrow cash to buy more equity assets). In this way, negative weights are key to the model and signify shorting/borrowing rather than the "typical" buying/lending. I hope that's helpful!