Trying to figure out a portfolio beta question for work that has me stumped. If I take the weighted average beta of three assets vs the S&P, which is my understanding of how portfolio beta is calculated, why is it different than the total portfolio beta vs the S&P? I assume it’s because of the correlation of the three assets in the portfolio, so which is the correct measure of “portfolio beta” and why? Thanks in advance, nobody I work with can figure this one out. Matt
Beta of port. = Covariance of your portfolio and S&P (market)/ (std.mkt)^2) Correlation of the asset in the portfolio is the reason. To calculate the covariance between the market and the portfolio, you need to know the std. of your portfolio and std. of market. Remember, std. of your portfolio is not a linear calculation. I am too lazy to do the math, if you take the weighted average of your asset std. you will get the upper limite of your portolio std. (because you are ignoring the correlation factor). Therefore, I think if you take the weighted average of your asset beta to calculate the port. beta, the resulting number will be higher than the actual port. beta. I hope I made sense this morning, I am still in vaction mode.
Thanks for the quick response, I’m off to a series of meetings but will look at this shortly…
ws and i may be saying the same thing, but here goes. i think weighted averaging the beta’s is fine, but the beta model assumes that the residuals are uncorrelated. but if there’s a relationship between the stocks that isn’t captured by beta, it’ll affect the performance of the portfolio’s beta. the r-sqrd’s of beta’s are usually pretty low anyway. doesn’t explain that much. at very least you need to account for simple factors like oil price.
mullrich Wrote: ------------------------------------------------------- > Trying to figure out a portfolio beta question for > work that has me stumped. > > If I take the weighted average beta of three > assets vs the S&P, which is my understanding of > how portfolio beta is calculated, why is it > different than the total portfolio beta vs the > S&P? I assume it’s because of the correlation of > the three assets in the portfolio, so which is the > correct measure of “portfolio beta” and why? > > Thanks in advance, nobody I work with can figure > this one out. > > Matt Sorry to ask what may be obvious, Matt, but are you sure you’ve double checked your numbers? If I understand your question you’re saying that you’ve got betas for each asset individually and a beta for the portfolio, but your beta for the portfolio is not the same as the weighted average beta of the assets? If so, that certainly shouldn’t be the case, and I wonder if you’ve got a formula/math error. Here’s why: Let’s suppose you have three assets with returns rA, rB, and rC. Suppose you combine them in a portfolio with weights wA, wB, and wC. So, for any given period, your portfolio return, rP, is wArA+wBrB+wCrC. Now, if you were to calculate the beta of this return series relative to the S&P 500 (call it rM), we’d do: beta(port) = COVAR(rP, rM)/VAR(rM) = COVAR(wArA+wBrB+wCrC, rM)/VAR(rM) = [wA*COVAR(rA,rM)+wB*COVAR(rB,rM)+wC*COVAR(rC,rM)]/VAR(rM) = [wA*COVAR(rA,rM)]/VAR(rM) + [wB*COVAR(rB,rM)]/VAR(rM) + [wC*COVAR(rC,rM)]/VAR(rM) = wA*beta(rA,rM) + wB*beta(rB,rM) + wC*beta(rC,rM) So, mathematically, it must be that the beta of the weighted average returns is equal to the weighted average of the asset betas. I put together a small spreadsheet with three assets and 250 returns (using a random number generator) to make sure I’m not losing my mind and sure enough, the portfolio beta was equal to the weighted average beta. I can’t upload the file to the forum here, but if you want to see it, send me your email and I can forward. Hope that helps. If I missed something or misunderstood your question, please clarify.