Return on Portfolio

[Aisha]

Let us consider a portfolio P with n assets with a proportion of xi invested in asset ( with i=1,2,3...n and sum(x)=1 ). Let the annual return Rp on this portfolio be assumed to conform to a single-index model of asset return . Then, can the Return on Portfolio (Rp) be expressed as :

Rp= sum(xi*ai) + Rm * sum(xi*bi) + sum(xi*ei) [ sum over i=1 to n ]

where ai and bi are constants. ai represents the expected return of security i , bi represents the sensitivity of security i wrt return on market and ei represent the error term of the security i which is independent of movements in market.

To find the variance of return on portfolio, can we expresses var(Rp)= var (sum(xi*ai) + Rm * sum(xi*bi) + sum(xi*ei))
var(Rp) = (sum(xi*bi)^2)*var(Rm) + var(sum(ei*xi))
Genreally, var(Rp) is expressed as : Var(Rp) = var(Rm)*bp^2 + var(ep) where ep is the component of the portfolio return that is independent of movements in market risk .
On comparing the two var(Rp) equations above, is it okay to conclude that in case of single factor model , bp ( sensitivity of the portfolio p wrt market return) , bp= sum(xi*bi) and ep= sum(xi*ei)?

Thanks in advance!

 

[Joe Hook]

Hi Aisha,

Your logic here is all correct although note that your Var(Rp) can be rewritten slightly as:

var(Rp) = (sum(xi*bi)^2)*var(Rm) + sum((xi^2)*var(ei)).

This is because the ei's are independent and so the variance of the sums is the sum of the variances.

Thanks
Joe