For several years I managed quantitative equity portfolios at Freeman Investment Management. The firm had been a pioneer in creating low volatility strategies, both long-only and long-short.
In addition to managing low volatility portfolios, at Freeman we had been advocating using volatility indices instead of style indices, both as performance benchmarks and as explanatory variables in style analysis. We had created and maintained our own analogs to the Russell style indices in which we divided the Russell 1000 and 2000 universes into Low Volatility and High Volatility halves, reconstituting on the same schedule as Russell. The table below offers a performance summary.
| Return measure | Freeman 1000 Low Volatility | Freeman 1000 High Volatility | Freeman 2000 Low Volatility | Freeman 2000 High Volatility |
|---|---|---|---|---|
| Arithmetic mean | 11.93% | 11.96% | 14.62% | 12.31% |
| Geometric mean | 11.64% | 10.45% | 14.32% | 9.55% |
| Std. Deviation | 13.13% | 19.68% | 15.04% | 24.77% |
| Annualized monthly return statistics, 1979 – January 2010. Excludes IPOs. Source: Freeman Investment Management | ||||
As unimpressive as these results are for High Volatility stocks, keep in mind these are well diversified cohorts containing several hundred stocks. The results are even worse for the High Volatility stocks if you slice the universes into volatility deciles.
Given these results, why would a rational investor hold High Volatility stocks?
We used to ask clients, consultants, and prospects that question all the time. It was rhetorical, but I believe I now know the answer.
In September 2011 I wrote a post about modeling expected returns. In that post I wrote about the method described by Jacquier, Kane, and Marcus for obtaining an unbiased estimate of return over a forecast horizon of length H, given observations from a sample period of length T.
It surprises many to learn that the unbiased estimation formula is a weighted average of the Arithmetic mean and the Geometric mean of the sample return data, and that the weights depend on the ratio of the length of the future horizon, H, to the length of the sample period, T:
Expected log return over H periods =
(1- 
where 
So what does this have to do with rational investors holding volatile assets?
This formula implies that, given a long enough forecast horizon H, all assets with positive volatility have an unbiased expected return that is negative. Not just High Volatility assets, ALL assets. The Arithmetic Mean is always greater than the Geometric Mean, so if you keep extending the forecast horizon H, eventually the unbiased forecast return becomes negative.
When you rebalance a diversified portfolio, you are engaging in volatility capture. So long as you rebalance regularly, as a rational investor you should hold volatile assets in your portfolio; how much depends on your rebalancing and holding period horizons.