What's new

CAPM Spreadsheet (SML) - Covariance calculation

isagasta

New Member
Subscriber
Thread starter #1
Hi David,

In the spreadsheet, under the SML tag, could you please explain the Covariance(port,market) formula you are applying here? Many thanks!


1564441660708.png

Many thanks.
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
#2
Hi @isagasta It applies the covariance property that COV(aX + bY, cW + dV) = ac*COV(X,W) + ad*COV(X,V) + bc*COV(Y,W) + bd*COV(Y,V). More specifically, please see https://www.bionicturtle.com/forum/threads/question-on-cml-and-market-portfolio.1278/post-4678 i.e., as emphasized below
Hi Martin, Thanks for asking about that: I made a note to clarify this non-obvious step on revision.
It applies property of covariance http://en.wikipedia.org/wiki/Covariance
... specifically: COV(aX + bY, cW + dV) = ac*COV(X,W) + ad*COV(X,V) + bc*COV(Y,W) + bd*COV(Y,V)
... not in Gujarati per se but generalizes from Gujarati's properties
... and we are here simplistically only using two assets where both the portfolio and the market are two-asset portfolios.

so given:
a = %weight Asset A in Market Portfolio M
b = %weight Asset B in Market Portfolio M; i.e, b = 1 - a
c = %weight Asset A in Portfolio P
d = %weight Asset B in Portfolio P; i.e., d = 1 -c
Ra = Return (Asset A)
Rb = Return (Asset B)

Unrealistically, both portfolios P & M are 2-asset portfolios, same assets, but different mixes:
Return (Market Portfolio M) = a*Ra + b*Rb
Return (Portfolio P) = c*Ra + d*Rb
Cov (M, P) = Cov(a*Ra + b*Rb, c*Ra + d*Rb), and per the above covariance property
= a*c*Cov(Ra,Ra) + a*d*COV(Ra,Rb) + b*c*COV(Rb, Ra) + b*d*COV(Rb, Rb)
...since COV(Ra,Ra) = variance (Ra) and COV(Rb,Rb) = variance(Rb)!
= ac*variance(Ra) + bd*variance(Rb) + covariance(Rb,Ra)*[ad + bc]
 
Top