We then examined how constructing a diversified factor portfolio might impact those odds of underperformance. Today we’ll tackle factor diversification from another angle, by looking at the annualized returns and annual standard deviation of two simple portfolios.

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It turns out market gurus were right about one thing: The DJIA futures market crashed by more than 800 points in the first moments after it became clear that Trump was going to come out ahead.

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The research from a trio of studies on the U.S. stock market has found that this irrational preference is explained by individual investors who are searching for a cheap bet, as with lottery tickets. Therefore, such investors find lower-priced stocks attractive. The research also confirms the view that individual investors may see low-priced stocks as being closer to zero and farther from infinity. Thus, they are perceived to have more upside potential and less to lose.

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One of the benefits of adding the value factor (the tendency for relatively cheap assets to outperform relatively expensive ones) to asset pricing models was that its inclusion went a long way toward explaining the superior performance of “superstar” investors from the value school of Benjamin Graham and David Dodd.

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This is a big problem for the efficient markets hypothesis, as there’s no coherent risk-based explanation for momentum’s performance. Not only has there been a sizable momentum premium in stocks (larger even than the market beta premium), but its Sharpe ratio has been higher.

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There are two important takeaways from this data, which covers the period 1927 through 2015. See if you can identify them:

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Because the level of a company’s stock price is arbitrary—it can be manipulated, for example, by firms via adjustments in the number of shares outstanding—if markets were fully efficient, nominal stock prices should be randomly chosen by firms. However, until just recently, the average nominal price of U.S. stocks had remained around $30 over the last several decades.

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For instance, the research shows that individual investors tend to hold lower-priced stocks than institutions. And there’s also evidence demonstrating that the number of small shareholders in a stock increases following a split to a lower price level.

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But financial innovation didn’t end there. Today the literature contains more than 600 investment factors, a number so great that John Cochrane called it a “zoo of factors.” However, as my co-author Andrew Berkin and I explain in our recently released book, “Your Complete Guide to Factor-Based Investing,” only a small number of exhibits within this factor zoo are required to explain almost all the differences in returns between diversified portfolios.

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Their study seeks to investigate whether the market price of risk and the market price of uncertainty are significantly positive, and whether they may help explain the cross-sectional and time-series variation in stock returns. According to the authors’ model, the premium on equity is made up of two separate terms. The first term compensates for standard market risk. The second term represents an additional premium for variance risk.

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