How Math Is Sufficient To Explain Small Stock Outperformance

Takeaways from Diversification, Volatility, and Surprising Alpha by Fernholz et al. (Link)


  • It has been widely observed that capitalization-weighted indexes can be beaten by surprisingly simple, systematic investment strategies including equal and random-weighted portfolios.
  • This outperformance is generally attributed to beneficial factor exposures.
  • It turns out this outperformance needn’t invoke factors. It can be explained by stochastic math where correlation and variance play a larger and more predictable roles than returns.
  • Portfolio logreturns can be decomposed into an average growth and an excess growth component. They argue the excess growth component plays the major role in explaining the outperformance of naıve portfolios.

Some basics

Let’s establish some basic definitions.

Stock Returns

There are 3 types of returns commonly used to describe growth rates. But they are not equal.

Arithmetic returns > Geometric Returns > Log Returns

This is important because only logarithmic returns are an unbiased estimate of expected long term returns.  In other words, arithmetic and geometric returns will overestimate expected growths in wealth.

  • Logreturn of an asset= Arithmetic return – .5 * variance
  • .5 * variance is known as the volatility drag or variance drain. I’ve discussed this here in simple terms.

Portfolio Returns

The logreturn of a portfolio can be decomposed into 2 components:

Weighted avg of stock logreturns  + “excess growth rate” (aka EGR)

Understanding the EGR

EGR = (weighted average stock variance – portfolio variance) / 2

The relationship between stock variance and index variance

Now this part is not in the paper, but taking from my index options experience:

Portfolio variance = weighted average stock variance * average cross-correlation of the stocks

This is a common identity used to price index options. It makes intuitive sense.

  • If all components of the portfolio has a correlation of 1 the portfolio variance would be the same as the underlying stocks.
  • If you had a 2 stock portfolio and the correlation were -1 the portfolio variance would be zero. Iimagine a basket comprised of 50% SPY and 50% inverse SPY. It would never move in price (assuming no fees, frictions, etc) regardless of how high SPY variance was.
  • For an average correlation < 1,  the portfolio variance must be less than the average weighted stock variance.

The key insight: the lower the average correlation between the components the wider the spread between the portfolio and weighted average stock variances!

Back to the paper…

Observations about the EGR

Looking at the formula again:

EGR = (weighted average stock variance – portfolio variance) / 2

  • EGR boosts the portfolio returns beyond that of its components since portfolio variance < weighted average stock variance
  • EGR boosts portfolio returns with lower correlations
  • EGR boosts portfolio returns with high stock variance

Relationships Between Market Cap, Logreturns, and Variance

The authors then use a rank based computation to show:

  • Logreturns of individual stocks do not vary by market cap.
  • Variances of individual stocks do vary by market cap. Smaller stocks are more volatile.

This prompts the great reveal:

Small stocks don’t have higher returns but have higher variances which boost EGRs. The volatility and interaction of the stocks is boosting the portfolios that contain them without any need to rely on factors! The increased volatility of the individual stocks did not earn them a risk premium when considered in isolation, but at the portfolio level they contributed to excess growth.


  •  The authors contend that the excess growth component can be estimated relatively easily, since its value depends only on variances, or relative variances, which are not difficult to determine in practice. The average growth component, however, is more difficult to estimate. 
  • Small stocks are riskier and while this might mean higher single period arithmetic returns long term investors care about logreturns. In logreturn space, individual stocks don’t contribute excess returns. This is at odds with conventional wisdom.
  • Instead, the excess returns are coming at the portfolio level via the small stocks’ contribution to “excess growth rates” (EGRs).
  • They tested the expectations of this stochastic portfolio math on 5 commonly employed weighting strategies, some more diversified and some less diversified than the capitalization-weighted portfolio, confirmed these insights. In general, the more diversified portfolios outperform and the single less diversified portfolio underperforms, because the more diversified portfolios have a higher excess growth rate. This arises from the higher variances associated with the smaller stock exposure in these more diversified portfolios, and not because such stocks have inherently higher returns. This higher excess growth rate, in turn, increases the portfolios’ logarithmic return.

My Comments

  • The role of low correlation was not emphasized enough in the paper considering it drives the EGR by setting the gap between portfolio and average weighted stock variances.
  • You should read the paper if you’d like a refresher on computing arithmetic, geometric, and logreturns.
  • You should read the paper to see how they computed rankings since this work established that there was no relationship between logreturns and the market cap of a stock.

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