Bayesian Updating and the Small Cap Premium

Bayes’s Theorem was developed by a French mathematician, and is named after Thomas Bayes, an English minister in the 18th century. Bayes earned his name on the theorem with a paper titled An Essay toward Solving a Problem is the Doctrine of Chances.

The theorem is simple and intuitive: the current odds of something happening are the past odds, adjusted for new information.

Small stocks have historically outperformed large stocks. However, for the past five years, investors holding a higher weight in Canadian and US small cap stocks have trailed the market by 0.49% and 0.60% respectively. Five years is a long time to trail the market; is it time to throw in the towel on small stocks?

	5-year Small Cap Premium as at 4/30/2018
Canada¹	-0.49%
US²	-0.60%

While five years may seem like a long time to watch an investment trail the market, the data behind the small cap premium is very difficult to disagree with. Unfortunately, humans are affected by the availability bias; we are happy to draw conclusions quickly based on recent information.

An expectation about the likelihood of an outcome (the posterior assessment) should include prior information combined with any new evidence since the prior information was collected. This way of thinking is most helpful when we have a strong prior belief. Understanding the prior information is an important part of developing a posterior assessment about the expected returns of small cap stocks.

In Canada, small stocks outperformed large stocks by an average of 0.21% from 1988 through 2017; small caps in Canada have notably exhibited a relatively small, but still existent, premium. Over the 241 overlapping rolling 10-year periods spanning the same years, Canadian small cap stocks outperformed Canadian large cap stocks 63% of the time³. In the US, small stocks outperformed large stocks by an average of 2.29% from 1928 through 2017. Over the 968 overlapping rolling 10-year periods spanning the same years, US small cap stocks outperformed US large cap stocks 72% of the time⁴.

Not only have small caps outperformed in Canada and the US, but International markets have shown an even stronger and more persistent small cap premium. International markets have been mostly left out of this post because their 5-year performance has beaten the market by 3.30%⁵ per year on average – nobody needs to be convinced to continue holding outperforming assets!

The prior evidence that small stocks have higher expected returns is very strong; not only have the long-term premiums been positive, but they have been pervasive across various markets and time periods.

If we hold a strong prior belief about something, then new evidence must be compelling for us to change that belief. The stronger new evidence is, the more our beliefs might shift. On the other hand, the stronger our prior belief is, the more compelling any new evidence must be to have a meaningful impact on our expectations about the future. With an extremely strong prior belief, new evidence must be truly overwhelming for us to change our posterior assessment.

When we think about something like the small cap premium, we do have an extremely strong prior belief. Small caps’ outperformance relative to large caps has been pervasive across regions, markets, and time periods. Beyond exhibiting outperformance, small caps have clear and consistent explanatory power in a model.

Based on the strength of our prior assessment that the small caps are likely to outperform over the long-term, it would take overwhelming evidence to make us believe otherwise.

Is five years of market-trailing performance overwhelming enough to change our posterior assessment? Probably not.

As useful as Bayesian thinking can be in making sensible decisions, it should not be mistaken as a way to avoid unwanted outcomes. In his book The Black Swan, Nassim Taleb offers the example of a well-fed turkey:

Consider a turkey that is fed every day. Every single feeding will firm up the bird’s belief that it is the general rule of life to be fed every day by friendly members of the human race “looking out for its best interests,” as a politician would say. On the afternoon of the Wednesday before Thanksgiving, something unexpected will happen to the turkey. It will incur a revision of belief.

Of course, the risk of underperforming is one of the reasons that a small cap investor expects higher returns – they are being compensated for taking additional risk. As we have seen, small stocks do underperform over some time periods. One of the most interesting aspects of targeting small and value stocks in portfolios is that their performance is likely to be different from the market at different times. This characteristic of factor diversification was explained by Antti Ilmanen and Jared Kizer in a 2012 paper titled The Death of Diversification Has Been Greatly Exaggerated:

The argument that we make for factor diversification partly rests on the expectation that the positive factor premia … will continue to persist. But the correlation (or relative lack thereof) of these premia with each other are at least as important.

Small stocks may underperform sometimes. As much as that may be a risk, it is also a by-product of diversification. Diversification is hardly a reason to abandon an investment strategy.

Will North American small stocks outperform again? We cannot know with certainty. All we can do is weigh new information against prior information to update our posterior assessment. In the worst case of long-term small cap underperformance, there is still a diversification benefit.

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