T9. R63. P2: Describe the challenges associated with VaR measurement as portfolio size increases.

[mkaabb96]

@David Harper CFA FRM Hey David. I am having difficultly in understanding the formula for calculating the Covariance of Assets with the Portfolio i.e. Cov(i,p). As per your example in the study notes, you have shown the formula for COV(euro, portfolio) which is confusing. What if the correlation is not 0 but any number between 0 and 1? how will we calculate the covariance in that scenario?

This concept is really important as we use covariance to calculate the beta of asset as well.

Looking forward to your reply.

 

[David Harper CFA FRM]

Hi @mkaabb96 This is elsewhere many times discussed but quicker for me to just re-explain it. Cov(i,P) is just applying basic covariance properties (see https://en.wikipedia.org/wiki/Covariance) where w_eur = weight of EUR in the portfolio and the portfolio is weighted between the two such that P = w_eur*eur + w_cad*cad. So:

Cov(eur,P) = Cov(eur, w_eur * eur + w_cad *cad ); i.e., the covar between eur and the portfolio that contains some weight (w_eur) of eur
Cov(eur,P) = Cov(eur, w_eur *eur ) + Cov(eur, w_cad*cad)
Cov(eur,P) = w_eur*Cov(eur, eur ) + w_cad*Cov(eur, cad); and since Cov(eur, eur) = variance(eur) we have:
Cov(eur,P) = w_eur*variance(eur) + w_cad*Cov(eur, cad) <-- that's what is shown, but Cov(eur, cad) = ρ(eur, cad)*σ(eur)*σ(cad), so of course we also have the equivalent:
Cov(eur,P) = w_eur*variance(eur) + w_cad*ρ(eur, cad)*σ(eur)*σ(cad)

In the screen you shared, the final "0.00" is a zero Cov(eur, cad) because the ρ(eur, cad) = 0 but this correlation can be non-zero and then the XLS will show a non-zero as the final number in that expression. I hope that's helpful,

 

[yLam4028]

Dear David,

do we have an intuition of why beta of individual asset would all equal 1 in optimal portfolio ? Thank you

 

[David Harper CFA FRM]

Hi @yLam4028 we do not expect individual assets to have 1.0 betas. As the theory here is very well-developed, I want to be careful about terms like "optimal." Here is a long note I wrote on the topic that you may find helpful https://forum.bionicturtle.com/threads/week-in-financial-education-2021-05-24.23840/post-88829