## 2013/08/10

### Volatility weighting primer

Volatility weighting is one common means of attempting to improve the risk profiles of strategies, that is to give them smoother returns. It consists of taking some asset (or the returns from a strategy) and dividing the investment you make by the (estimated) future volatility of the strategy. The goal of this is in fact not reduce the volatility of returns, per sé, but rather the volatility of volatility. In practice it seems that volatility weighting does seem to work and gives higher Sharpe ratios (a measure of the amount of return for every unit of risk taken).

Some simple maths (skip this bit if you fear maths)

Consider for instance an asset with returns
$r_t = \sigma_t (\gamma + \epsilon_t)$
with (for simplicity) sigma σ and epsilon ε independent and epsilon ε is mean zero with variance 1. Here σ represents the volatility of the process. It is predictable (the value for the next period is known today) but it is random – it changes from period to period. The Sharpe ratio of this strategy is
$\frac{\gamma}{\sqrt{(\gamma^2+1)\frac{\mathrm{Var}{\sigma_t}}{\mathrm{E}[\sigma_t]^2}}+1}$
Notice that this ratio is maximised if volatility is deterministic, that is if the coefficient of variation of volatility
$\frac{\sqrt{\mathrm{Var}{\sigma_t}}}{\mathrm{E}[\sigma_t]}$
is zero. Making volatility deterministic is, in this case, exactly what volatility weighting does.

What we need for volatility weighting to work

1.  To be able to forecast volatility. It is typical to model volatility as a predictable process, but in practice we cannot even observe volatility directly. There is, however, some evidence that volatility is sticky, so that one can predict it to some degree.
2. We need the portion of returns not depending on volatility not be too large. If this is not the case, then we do not have the multiplicative nature of the return series. It does, however, seem that the effect of this portion of returns is not so large as to completely invalidate the use of volatility weighting.

Why volatility weighting works

1.  I have already mentioned that there is an effect of stabilising the volatility and that this creates a more stable returns series. In particular we saw earlier that volatility weighting seems to work by reducing the coefficient of variation of volatility.
2. There is, however, another possible effect, volatility timing. If returns are negatively related to volatility then volatility weighting will mean investing less when volatility is high and more when it is low, which intuitively seems to be a good strategy. This won’t work if the conditions in the previous section are not met, though.
It is not clear which of the above two effects is more important in practice, or if they can even be separated in any meaningful manner. It is clear, though, that volatility weighting can work even where the relationship with volatility is positive (so the second effect is not then the most important). It is also true that for some strategies, for instance momentum, the relationship with volatility is often negative and so we can expect the timing effect to play a role as well.

Own volatility or underlying volatility

There are many ways to do volatility weighting. Here are two that I have looked at:

1. Weighting a strategy by its own volatility: you look at some investment strategy and estimate its volatility in some form and then scale how much you invest in the entire strategy.
2. Weighting the underlying assets: Here you consider some investment strategy based on a set of assets. Now replace the assets with a set of assets that have been volatility weighted. I call this using normalised returns.
Both of the above forms of volatility weighting appear to be effective, the latter probably more so.