It is common practice to have portfolio constraints like:
wi ≤ 0.05
That is, the weight of each asset can be no more than 5%.
Proxy for risk
We think that is what we want to do because we are so used to doing it. But why should we care about the weight of assets?
I don’t think we do. I think that that constraint is a proxy for constraining risk.
The reason we’ve used weight as a proxy for risk is a lack of technology. We didn’t have the ability to constrain risk on an asset-by-asset basis so we used a constraint that we could do.
There are weight constraints that are about liquidity risk — these are sensible as weight constraints. But they are different than blanket constraints over all assets.
Attributing variance to assets
We can partition the variance of a portfolio into pieces attributed to each asset.
Let’s start by noting that the portfolio variance is:
where w is the vector of portfolio weights and V is the variance matrix of the assets. In R notation this is:
w %*% V %*% w
We can change this slightly to get our partition of the variance:
w * Vw
where the * operator is element-by-element multiplication. In R this is:
w * V %*% w
Just as weights sum to 1, we are interested in the risk fractions summing to 1. We get this with:
f = (w * Vw) / w’Vw
We started with:
wi ≤ 0.05
I’m claiming that what is really wanted there is:
fi ≤ 0.05
This is a type of constraint that is now available in the Portfolio Probe software.
If there is a benchmark, then the benchmark weight vector b needs to be subtracted from the portfolio weight vector. So we have:
f = ((w – b) * V(w – b)) / (w – b)’V(w – b)
Here f is the risk fraction of each asset deviating from the benchmark.
Without a benchmark only assets that are in the portfolio contribute to the variance — f is zero for assets not in the portfolio. With a benchmark it is only assets that have the same weight in the portfolio as in the benchmark that are guaranteed to contribute zero to the variance. Assets not in the portfolio can be quite risky relative to the benchmark.
The application that led to the original user request was an asset allocation with a benchmark.
While risk fraction constraints can be used more generally, one application of them is in the creation of risk parity portfolios. These are portfolios where each asset class contributes the same amount of variance to the portfolio.
I quite like the idea of controlling how much risk comes from various pieces of the portfolio. However, I’m not convinced that equality across a semi-arbitrary categorization of assets is going to be the best thing to do. I would prefer more thought going into the balance.