Calculating a Holding’s Beta to a Weighted Multi-Asset Benchmark
Summary
The document asks how to calculate a portfolio holding’s beta against a benchmark made from several assets with stated target weights. One proposed method is to build a benchmark return series and estimate the holding’s beta against it. That raises a practical definition choice: whether to rebalance to the target weights each period or use a different treatment of benchmark weights through time.
A second proposal uses a covariance matrix covering the benchmark assets and the holding. It computes covariance between the weighted benchmark and the holding, then divides by the benchmark variance, following the standard beta relationship. This formulation corresponds to treating the benchmark weights as fixed in the return calculation, as in a continuously rebalanced benchmark. The document poses the method for discussion but provides no answer, empirical comparison, or implementation detail. Results depend on the benchmark’s rebalancing convention and the return data used, so the convention should match the benchmark being analyzed.
Key ideas
- A holding’s beta to a benchmark is its covariance with the benchmark divided by the benchmark variance.
- For a weighted benchmark, the benchmark return series depends on how its component weights are maintained through time.
- A covariance matrix can calculate the covariance between a weighted benchmark and an individual holding.
- Fixed target weights in each period represent a continuously rebalanced benchmark convention.
- The calculation should use a rebalancing convention consistent with the benchmark definition.
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# How to calculate beta against a multi-asset benchmark
# How to calculate beta against a multi-asset benchmark
Lets say that I have a benchmark, $BM$ that consists of 3 assets- 30% asset $A$, 30% asset $B$ and 40% asset $C$. Now, lets further assume I am trying to construct a portfolio that uses $BM$ as its benchmark. In order to calculate my residual risk, active risk, etc, I need to calculate $\beta_n$ for each holding $n$ in the portfolio. I am wondering what the most consistent way to do this is.
One thought is to create a return stream for $BM$ and use it to calculate $\beta_n$ for each holding as I would for a single index benchmark. But questions arise as to the weighting I should use- do I assume the benchmark is continuously rebalanced (i.e.- scale each daily return of $A$, $B$, and $C$ by 0.3, 0.3, and 0.4 respectively? Or do I solve for the original weights I would have had at some time in the past to get to my 30/30/40 today? Or something else entirely?
Another thought, is to use the covariance matrix, and some "tricky" weighting. e.g.:
We know that the variance of a portfolio $P$ can be calculated from its covariance matrix ($V$) and weight row vector ($x$) like:
$\sigma^2_{P} = x V x^T$
We also know that given 2 portfolios ($P_x$ and $P_y$) with the same holdings but different weight vectors ($x$ and $y$), we can calculate the covariance between them as:
$cov(P_x,P_y) = x V y^T$
So, could I solve for the covariance matrix for holdings $A$, $B$, $C$, and $n$, ($V^*$) and create two weight vectors:
$x = [0.3, 0.3, 0.4, 0.0]$ and $y = [0.0, 0.0, 0.0, 1.0]$ and solve:
$\beta_n = \frac{xV^*y^T}{xV*x^T}$
I believe this may be the same as the "continuously re-balanced benchmark" above, but could be a bit more straight forward programming-wise. Thoughts? Am I just making this more complicated than it needs to be?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.