Choosing a Benchmark for Portfolio Beta Across Asset Classes
Summary
Portfolio beta is defined relative to a chosen benchmark as the covariance of portfolio returns with benchmark returns divided by the benchmark’s return variance. When holdings span bonds, sectors, and broad equities, their individual betas may refer to different indices, so those values cannot simply be combined as though they shared one common reference.
The document emphasizes that there is no universally correct single benchmark for a global or multi-asset portfolio; the choice depends on the purpose of the analysis and its audience. A broad global equity index is offered as a simple, potentially naive option. As an alternative, the portfolio can be analyzed against multiple risk factors, using representative indices for asset classes such as global equities and aggregate bonds. This approach describes sensitivities to several sources of risk rather than collapsing them into one beta. The discussion offers the conceptual framework but no worked numerical example or rules for selecting factors, estimating them, or handling changing exposures.
Key ideas
- Portfolio beta measures covariance with a specified benchmark relative to that benchmark’s variance.
- Beta values tied to different benchmarks are not directly interchangeable.
- The appropriate benchmark depends on the purpose of the measurement.
- A multi-factor analysis can describe portfolio sensitivity to indices representing different asset classes.
Tags
Full text
# Calculating portfolio allocation beta with different asset classes?
# 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?
## Answer by SRKX (score 5, accepted)
https://quant.stackexchange.com/a/3758
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.