• We are pleased to announce that the winner of our Feedback Prize Draw for the Winter 2024-25 session and winning £150 of gift vouchers is Zhao Liang Tay. Congratulations to Zhao Liang. If you fancy winning £150 worth of gift vouchers (from a major UK store) for the Summer 2025 exam sitting for just a few minutes of your time throughout the session, please see our website at https://www.acted.co.uk/further-info.html?pat=feedback#feedback-prize for more information on how you can make sure your name is included in the draw at the end of the session.
  • Please be advised that the SP1, SP5 and SP7 X1 deadline is the 14th July and not the 17th June as first stated. Please accept out apologies for any confusion caused.

Return on Portfolio

A

Aisha

Member
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!
 
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
 
Back
Top