Calculating portfolio allocation beta with different asset classes?

I'd like to calculate portfolio allocation beta on a portfolio that has different asset classes. The portfolio may be made up of:

Short term bond fund (with beta tied to Barclays U.S. Aggregate Bond Index)
Sector fund 1 (with beta tied to DOW)
SP500 fund (with beta tied to SP500 index)

I understand how to calculate portfolio beta if all assets are benchmarked off the same index but not when beta is heterogenous. Can anyone provide a formula(s) of how it is calculate along with some simple examples?

• $\beta$ is always measured against a specific target. That is, you can't claim to have a $\beta$ of $x$ and not specify what benchmark you're measuring against. So, what index do you want to compare your portfolio to? That's really the only question you need to answer. Jul 7 '12 at 16:32

Let's first restate the formula of the beta of a portfolio $P$ relative to a benchmark $B$:

$$\beta_P=\frac{Cov(r_P,r_B)}{Var(r_B)}$$

As chrisaycock said in his comment, the key thing to understand is that the beta is a statistical measure computed relative to a benchmark. Hence, I believe that the real question you should be asking is:

Which benchmark should I choose to compute a $\beta$ for a global portfolio?

There is no real answer to that question; it depends who you want to present it to and what you are trying to demonstrate. The easy (and naive) answer would be to use a global equity index such as MSCI World. I believe that it is better to look into a factor analysis where you estimate the sensitivity of your portfolio to different risk factors, as I discussed in this post.

The different factors you will choose will then be several indices of the different asset classes (MSCI World, Bonds Global Aggregate, etc).

• Thanks. I plan to use this technique in Excel investopedia.com/articles/financial-theory/09/… to calculate beta for each asset. What is the suggested look back period? I will need to manually paste in newer closing prices if I want the most current beta. How often should that be done? Jul 7 '12 at 20:06
• @4thSpace You still seem to be missing the point. Don't worry about computing $\beta$ just yet. Instead worry about deriving one or more appropriate indices. SRKX is suggesting that you use factor analysis to determine what really drives your portfolio. Jul 7 '12 at 22:21
• @4thSpace I want to second SRKX's answer and say that what you propose is misguided at best. A multi-asset class portfolio will have multiple risk sources, and $\beta$ is an inappropriate measure. SRKX is right to call what you propose naive. Jul 9 '12 at 14:59