Should you Tilt to Small Cap Equities?

Everybody has someone like Cousin Charlie in their life. He (or, more rarely, she) is the one who, in the middle of the party, likes nothing better than to boast of his investing prowess, while loudly sharing his market insights with everyone in earshot. Most recently, you've probably heard him talking about his "killing" in "small cap stocks." Unfortunately, claiming you're an index investor doesn't slow him down one bit. He simply points out, for example, that in 2003 the exchange traded fund that tracks the Russell 2000 (a small cap index) returned 46.2%, compared to only 28.8% by the Vanguard Viper ETF that tracks the broad Wilshire 5000 index. If he is in a particularly nasty mood, he might also point out that the Bridgeway Ultra Small Company Market Fund (which tracks so-called "microcap" stocks that in aggregate represent the smallest 10% of the total market capitalization of companies whose shares trade on the New York, American and NASDAQ Stock Exchanges) was up an astounding 79.4% in 2003. In short, "why aren't you putting your money in small caps?"

What can you say in response? Actually, quite a lot. As is usually the case, Charlie either isn't telling or does not know the full story.

Let's start with a deceptively simple question: what determines the current fair value of a stock? Broadly speaking, four variables are at work. The first is the dividend the stock currently pays. Let's assume this equals $5 per share. The second is the rate at which these dividends are expected to grow in the future. To arrive at current value, we have to discount this stream of expected future dividends at a rate which reflects their riskiness compared to other investments. The third variable is the first part of the discount rate -- the real rate of return on a risk-free government bond (for simplicity, we'll leave inflation out of this discussion for now). The fourth variable is the additional premium that you as an investor require above the return on government bonds to induce you to hold a risky equity investment. In theory, changes in these variables over time generate the annual returns you experience as an investor in a specific stock.

Over time, many researchers have tried to simplify this model still further, and identify a smaller number of "factors" that simultaneously affect the valuations of large numbers of stocks. A logical starting point for these efforts are economic factors that are widely available and theoretically easy to link to the basic stock valuation model. For example, in "Economic Forces and the Stock Market" by Chen, Roll, and Ross, the authors try to link future stock returns to industrial production, the spread between short and long term government bond rates (the "maturity premium"), and the spread between the rates on high and low credit quality corporate bonds (the "default premium"). These variables are thought to be proxies for expected changes in dividends and the expected dividend growth rate, as well as expected changes in the rate at which they are discounted (e.g., declining industrial production and an increasing maturity premium signal an oncoming recession and decline in cash flows, while rising maturity and default premiums signal increases in the required rate of return). Surprisingly, the results of this straightforward approach were mixed, with changes in economic variables unable to explain a substantial proportion of equity returns.

This led to an alternative approach to the problem, which used historical returns data to predict future stock returns. The earliest and most famous example of this methodology is the "Capital Asset Pricing Model" or CAPM. The theory behind this model is straightforward: since diversification eliminates the significance of company-specific risks (which offset each other in a large portfolio), the only risk factor that matters when forecasting future returns is the extent to which the return on a stock varies with the return on the overall market. This relationship is called the stock's "beta" (more specifically, beta is coefficient in a regression of the stock's returns on the market's returns). If a stock's beta is less than one, its return varies less than the return on the market, and if it is greater than one, it varies by more than the market. To determine the expected future return on a company's stock, you simply estimate the future return on the market (defined as the current risk free government bond rate plus the appropriate equity market risk premium), and then multiply this times the stock's beta.

Unfortunately, the future returns forecast by the CAPM model didn't always turn out to be accurate (and gave rise to many journal articles on "the death of beta"). In their search for explanations for these "anomalies" (which, in economist speak, is anything you repeatedly encounter in reality that doesn't match the predictions of your model), researchers identified a number of systematic (that is, predictable) forecasting errors that occurred when using the CAPM approach. One of the earliest of these was "The Relationship Between Return and Market Value of Common Stocks" by Rolf Banz. This 1981 study popularized the "size effect", or the tendency of stocks with smaller market capitalizations to earn different returns than their betas would predict. Because most of these studies found these differential returns to be higher than those on large stocks, the size effect is often referred to as the "size premium".

In their seminal papers ("The Cross Section of Expected Stock Returns", "Common Risk Factors in the Returns on Stocks and Bonds", and "Size and Book-to-Market Factors in Earnings and Returns"), Eugene Fama and Kenneth French pulled these various ideas about anomalies together into a comprehensive approach. They showed how future returns on a stock could be more accurately predicted using not one, but rather three factors. Like the CAPM, the first of these factors was the expected return on the market as a whole. In addition to this, they identified factors that were based on the difference between the return on small capitalization stocks less the return on large capitalization stocks (known as SML), and the difference between stocks with high book/market ratios and low book/market ratios (known as HML). In a subsequent paper ("On the Persistence of Mutual Fund Performance"), Mark Carhart added a fourth factor: the return on recent winning stocks less the return on recent losing stocks (WML).

The net result of these efforts is an asset pricing model which (using more popular terms) claims that a stock's future return is a function of the return on the overall equity market, as well as a stock's relative exposure to three additional factors: size, value and momentum. For example, this four factor model predicts that a small stock with a high book to market ratio which has recently enjoyed attractive returns will earn higher returns in the future than a large capitalization stock with a low book to market ratio whose recent returns have been poor.

As one might imagine, this model has provoked a great deal of debate. With respect to the size effect, three broad issues have been raised. First, does the size effect really exist, or does it simply reflect the misestimation of beta? Second, assuming it exists, what causes it? And finally, can an investor earn superior risk-adjusted returns from investing in small capitalization stocks rather than a broad market index?

Is the Size Effect Real?

However, a number of other papers have questioned the strength of this evidence. For example, in their paper "Estimates of Small Stock Betas are Much Too Low", Ibbotson, Kaplan and Peterson find that because they are traded less often, small stocks returns have a higher degree of autocorrelation than those of large stocks. Translated into English, this means that the relationship between a small stock's return in two successive periods is stronger than it is in the case of large stocks (which are effectively independent). Regressions that don't take this into account underestimate small stocks' betas. After this adjustment is made, the size effect shrinks, but doesn't completely disappear.

Along similar lines, in their paper "Equilibrium Cross-Section of Returns", Gomes, Kogan, and Zhang find that when beta is allowed to vary over time conditional on economic conditions, the size effect disappears, and beta alone is sufficient to predict future returns. The authors conclude that the strength of the size (and value) effects in models which don't allow beta to vary over time reflects these factors' correlation with the "true" conditional beta.

Other authors take a different approach, and look more closely at what actually causes the size effect to appear in the data. In their paper "On the Robustness of Size and Book to Market in Cross Sectional Regressions", Knez and Ready find that "the SMB disappears when the one percent most extreme observations are trimmed each month from the data set used by Fama and French." Similarly, in "Size and Book to Market Effects in Australian Share Markets", Halliwell, Heaney and Sawicki find that "small cap excess returns are the result of a small number of firms performing extremely well. In contrast, the typical (median) small firm has a lower rate of return than a typical large firm." A paper by Hsu and Chou ("Robust Measurement of Size and Book to Market Premia") uses U.S. data, and also finds that the size effect is due to a small number of firms performing extremely well. They conclude that in fact, "size has an asymmetric effect on returns. It is inverse for very small firms [i.e., small size indicating relatively higher returns], but positive for larger firms [i.e., beyond a certain point, returns actually increase with the size of the firm]." Finally, in "The Structural Characteristic Evidence on Risk Factors", Guidi and Davies note that their "evidence shows that the characteristics of newly listed and marginal firms are imbedded in and are important to the [SMB premium]. We find evidence that newly listed firms and marginal firms are the main characteristics that influence the S component in the SMB factor."

More generally, other authors have raised caution flags about the strength of the statistical tests that have been used to identify the size effect. Two good examples of these papers include "Tests of Multifactor Pricing Models, Volatility Bounds, and Portfolio Performance" by Wayne Ferson, and "Cross Sectional Determinants of Expected Returns" by Brennan, Chordia and Subrahmanyam.

On the issue of whether or not the size effect actually exists, perhaps we should leave the last word to some of the people who made it famous. In their paper "Characteristics, Covariances, and Average Returns: 1929-1997", Davis, Fama and French concluded that "the size premium is…weaker and less reliable than the value premium." This conclusion was confirmed by Dimson and Marsh, who compared the size effect in Germany, Japan, the UK and US between 1955 and 1999 (in their paper "UK Financial Market Returns 1955-2000"). They noted that "while all four equity markets have exhibited a positive size effect, this has been surprisingly modest (1.2% in Germany, 2.7% in Japan, 2.0% in the UK, and 1.0% in the US), given the prominence the small firm premium continues to receive in the literature."

Let's now move on to what might cause the size effect, assuming it exists.

What Causes the Size Effect?

Explanations of why a company's market capitalization is related to its future returns fall into three broad categories: behavioral explanations, risk-based explanations, and characteristic based explanations.

The behavioral explanation for the size effect is potentially very important, because if true, it suggests the possibility of earning higher returns while taking on less risk than one would by investing in a broad equity market index. In this argument, small companies receive less attention from investors than large companies. This is important, because investors dislike uncertainty and ambiguity, and these feelings are strengthened (making investors relatively more pessimistic) when less information is available. This leads investors in small cap companies to systematically underestimate their future growth and overestimate their risk, while making the opposite mistakes with respect to large cap companies. In other words, small cap companies are systematically undervalued (giving rise to higher returns), while large cap companies are systematically overvalued (giving rise to lower returns). However, this theory also needs to explain why rational investors have not exploited the mistakes of the behaviorally challenged, and eliminated the size premium over time. In other words, a durable obstacles to arbitrage must exist. In this case, the thinly traded markets for small company shares could play this role, as they might expose arbitrageurs to substantial amounts of risk relative to their potential returns, and thus cause them to focus their energies on other markets (e.g., those for large stocks). An important implication of this behavioral explanation for the size effect would therefore seem to be that it should generate a relatively consistent positive premium -- that is, additional returns (alpha) for investors who tilt their portfolios towards small company stocks and away from the broad market index -- with relatively little additional risk (that is, a high information ratio). As we shall see, those predictions are not born out in recent data.

A much larger number of explanations view the size premium as the rational compensation an investor earns in an efficient market for bearing additional risk. The debate here is over the nature of this risk. One school of thought links the size effect to the statistical higher moments of stock returns. Let's look at some very interesting examples of this view. In his paper "The Pricing of Coskewness and Cokurtosis in International Size and Momentum Strategies", Daniel Hung looks at the returns on 44,290 stocks in 20 countries and finds that co-skewness (the tendency of the distribution of returns of two stocks to simultaneously be tilted in the same direction above or below their respective means) and co-kurtosis (the tendency of two stocks to simultaneously experience returns that are very far away from their long term averages) substantially explain the size effect. Chung, Johnson and Schill reach a very similar conclusion in their paper "Asset Pricing When Returns are Non-Normal." These conclusions clearly line up with the idea that risk is a multi-dimensional concept that goes well beyond the simple standard deviation of returns (that is, the dispersion of annual returns around their average) used in many asset allocation models. In particular, investors are (according to Kahneman and Tversky's Prospect Theory) approximately twice as concerned with avoiding losses as they are with maximizing gains. This means that downside risk measures are important to them. Statistically, skewness (the tendency of the median return to be either above or below the average return) and kurtosis (the tendency of returns to be significantly different from the average) provide such measures. Hence we should expect them to be related to a stock (or, indeed, any asset's) expected return. Another paper ("Expected Options Returns") by Coval and Shumway takes this logic a step further, and shows that (a) the standard deviation of returns is not stable over time, but varies between regimes of low and high volatility; (b) stocks differ in the extent to which their standard deviations vary between these two states; (c) investors receive compensation for bearing this risk; and (d) in firms with small market capitalizations this risk is higher than average.

While these studies certainly provide us with interesting insights about the nature and pricing of different statistical risks, they do not answer the question of what actually causes one stock's skewness or kurtosis to be different from another's (or small stocks' to be different from large stocks'). From this perspective, the statistical measures are still proxies for the real economic risks that are driving returns.

This brings us to another line of research that has attempted to identify the economic risks that drive the small stock effect. In their paper "Can Book to Market, Size, and Momentum Be Risk Factors that Predict Economic Growth?", Liew and Vassalou answer their question affirmatively, based on an examination of returns in ten developed country equity markets. In general, high returns on Fama and French's SMB and HML factors precede growth in an economy. However, they caution that the size effect only "produces statistically significant returns in Canada, France, Japan, and the United States, and marginally significant returns in the UK. It does not appear to be profitable in countries whose markets are typically smaller, less liquid, and dominated by a few large capitalization stocks."

Another related economic explanation for the size effect is that it reflects a higher degree of default or financial distress risk on the part of firms with small market capitalizations (a risk that would only increase when economic growth slows). A simple quantitative example illustrates this view. Consider two firms, one with a book value of $100, and the other with a book value of $1,000. Both firms pay dividends equal to 5% of their book value. However, the smaller firm is expected to grow at 6% in the future, while the larger firm is expected to grow at only 4% . Assume the real rate of interest equals 3.0%, and the equity market risk premium equals 4.0%. The market value of the company is equal to the discounted present value of the expected future dividend stream. Mathematically, the formula is Dividend/(Real Rate + Equity Risk Premium - Expected Growth Rate). The company with book value of $100 has a current market value of $500 [$5/(3% + 4% - 6%)]. The company with book value of $1,000 has a current market value of $1,667 [$50/(3% + 4% - 4%). Now consider what happens when the real interest rate increases to 4%. Assuming this increases the interest payments each company must make, while also reducing their sales (due to the economy slowing down), you could reasonably conclude that their risk of default or bankruptcy has also increased. However, as shown in the following table, the price of the smaller company's stock falls to $250, a loss of (50%), while the price of the larger company's stock falls to $1,250, a loss of only (25%). Why do we see this difference in returns between the small and the large company? This happens because the small company has a higher percentage of its starting value in the form of "growth options" that will only produce cash flows (e.g., dividends) in the future, while a higher proportion of the large company's value comes from cash flows that will be received in the near future. When the discount rate increases, growth options lose a greater percentage of their value than dividend cash flows.

In their paper "Bad Beta, Good Beta", Campbell and Vuolteenaho take an interesting approach to this difference in the reaction of small and large cap companies to changes in the discount rate. They break down the traditional beta into two component parts: the sensitivity of a company's returns to changes in the discount rate, and their sensitivity to changes in the expected dividend growth rate. They find that between 1963 and 2001, "small stocks have much higher discount rate betas than large stocks" and that this difference " is sufficient to explain most of the size premium."

Other papers have explored this issue from different perspectives. In "An Empirical Investigation of Risk and Return Under Capital Market Imperfections", Hahn and Lee find that "changes in the default and term spreads…contain most of the pricing implications of Fama and French's size (SMB) and book to market factors (HML)." In "Default Risk in Equity Returns", Vassalou and Xing compute default probabilities for individual firms and "find a strong size effect, but one that is only present in the 20% of the market with the highest default probability. These are typically the smallest of the small caps. There is no size effect in the remainder of the market." They conclude that the Fama French SMB and HML factors "contain some default related information, but this isn't the main reason they can explain the cross-section of equity returns…SMB and HML [also] appear to contain important priced information unrelated to default risk." Roberto Guitierrez, Jr. comes to the same conclusion, though by a wholly different route. In his paper "Book to Market Equity and Size in the Cross Section of Corporate Bond Returns," he notes that "since corporate bonds are priced in part according to default risk…size should be a determinant of the cross section of corporate bond returns if it captures distress risk." Indeed, Gutierrez does "find a strong size effect in bond returns; in fact, size is found to subsume book to market in bond returns." However, he also "finds that size is priced differently in the stock and bond markets. [This] finding that the reward for size is different across assets of the same firms suggests that it may be inappropriate to consider size as a sensitivity to a specific [default] risk factor."

Well if it is not solely default risk that is generating the small company effect, what else could it be? In their paper "Asset Pricing and the Bid-Ask Spread", Amihud and Mendelson suggest that firm size proxies for more illiquid markets and higher transaction costs. Under these conditions, an investor would have to earn a higher gross return on a small cap stock to end up with the same net return (after transaction costs) on a large cap stock. In a related paper ("Is There a Neglected Firm Effect?"), Beard and Sias reach a very similar conclusion. They find that because information about small capitalization companies is less available and/or more expensive to obtain, small cap stocks are harder to value, implying a relatively higher risk that an investor will find himself on the losing end of a trade with someone who is better informed.

To guard against this risk, investors provide less liquidity (e.g., the maximum number of shares an investor will buy or sell at a given price), leading to higher transaction and market impact costs for small cap stocks. This view gains further support in the paper "Is Information Risk A Determinant of Asset Returns?" by Easley, Hvidkjaer, and O'Hara, who find that it is. Also, in a recent paper ("Hedging Against Liquidity Risk and Short Sale Constraints"), Avramov, Chao, and Chordia find that the combination of beta on the market portfolio and a liquidity risk factor does a better job of predicting future returns than the SMB + HML + WML model. Chen and Jindra provide further evidence on this point in their paper "A Valuation Study of Stock Market Seasonality and Firm Size." They find that "in a typical month, small cap stocks show the widest valuation dispersion, implying that they are the hardest to value." Moreover, "the valuation dispersions for all stocks [whatever their capitalization] widens at year end -- overvalued stocks become more so, and undervalued stocks become more so." The authors attribute this to two causes: end of the year "window dressing" by institutional investors to enhance their reported returns, and tax loss selling by individual investors. Both of these lead to the "January effect", or the tendency of stock prices to show a sharp rise during the first month of the year. The authors find that the January effect is particularly strong for small cap stocks.

The underlying causes of these valuation fluctuations are analyzed in "Prospect Theory and Institutional Investors" by O'Connell and Teo. In this paper, the authors tested the application of Prospect Theory to the group of investors who collectively account for the majority of trading volume in most financial markets. They found "no evidence whatsoever of disposition effects [the tendency to sell winners too soon, and hold losers for too long]; rather the dominant characteristic [of the investors they studied] was aggressive risk reduction in the wake of losses." The also found that this phenomena was related to time (or, more accurately, the nearness to year-end and the final performance numbers that would determine bonuses). "Fund managers were conditionally more risk-tolerant in the first half of the year. Gains during this period lead to incremental risk taking, but there was no evidence of this during the second half of the year. Correspondingly, losses in the first half of the year produced very little risk reduction: it was only in the second half of the year that managers systematically cut risk following losses." Finally, they note that experience (learning) also plays an important role: "older, wiser funds did not take on more risk in the wake of gains, but cut risk more aggressively in the wake of losses."

The authors conclude that the modified version of prospect theory first proposed by Barberis, Huang, and Santos (in "Prospect Theory and Asset Prices") best explains the behavior they observed. According to this theory, rather than being a constant, an investor's degree of risk aversion changes in response to the evolution of gains and losses relative to some starting anchor value (reference point). As gains grow larger, an investor becomes less risk averse (i.e., he or she reduces his or her equity risk premium), which lowers their required rate of return and drives asset prices still higher. However, as losses grow, so too does risk aversion and the required rate of return, which further accelerates the decline in asset prices. In short, the model proposed by Barberis, et al, whose presence was tentatively confirmed by O'Connell and Teo, implies much more volatile asset prices (and returns) than would be the case if all investors were rational and only changed their valuation of an asset in response to new information about its future cash flows or a change in interest rates.

In the opinion of other authors, the fact that so much of the small cap return premium tends to be earned in January raises doubts about the extent is really represents compensation for risk. In their paper "The Cross Section of Common Stock Returns: A Review of the Evidence and Some New Findings," Hawawini and Keim ask "if the [small cap] premium is compensation for risk, is there reason to believe the market is systematically more risky in January than during the rest of the year? Second, if the size and book/market premia are compensation for additional risks that are priced in the context of an international asset pricing model under conditions of integrated international capital markets, then the premia should be correlated (that is, move together) across markets, in much the same way that the market risk premium is significantly correlated across markets. Inconsistent with this hypothesis, we find that the premia correlations are insignificant across the 17 international markets in our sample. If these premia are uncorrelated across international markets, is it reasonable to characterize them as compensation for risk?" On the other hand, in their paper "UK Financial Market Returns 1955-2000", Dimson and Marsh "find no evidence of a year-end size effect in the UK, regardless of whether we look at the calendar or the tax year-end."

A final stream of research into the causes of the size effect starts with the critical observation that it applies only to relative market capitalization, and not to other measures of a firm's size, such as sales revenue, book assets, or employees. Indeed, like other researchers, Jonathan Berk (in "An Empirical Re-Examination of the Relation Between Firm Size and Return") finds that "when the market value of the firm is controlled for, [there is] a positive relation between the non-market value size measures and average returns." He notes that "if two firms have the same expected cashflow [e.g., dividend], the one with the higher discount rate will have the lower market value. Consequently, according to this view expected returns will always be negatively correlated with firm market value." This leads Berk to conclude that "rather than evidence of a 'size effect', the relation might be due solely to the endogenous inverse relation between the market value and discount rate of firms."
Two subsequent papers elaborated on this critical insight. In "Optimal Investment, Growth Options, and Security Returns", Berk, Green and Naik hold a firm's expected dividends constant, and show how changes in its discount rate over time (due to both changes in interest rates and its decisions to make investments with varying degrees of risk) cause predictable changes in its returns, market capitalization and book-to-market ratio. In "Corporate Investment and Asset Price Dynamics", Carlson, Fisher and Giammarino hold the discount rate constant, and show how unpredictable shocks to demand for the firm's products lead to changes in its expected growth rate, and to the same changes in returns, market capitalization and book-to-market found by Berk, Green and Naik. Both of these papers present simplified models; in the real world, firms often face simultaneous changes in interest and growth rates. Indeed, the two are not independent; a rise in the interest rate tends to slow economic growth.

A simplified example may help to integrate the points made by these two papers. Consider a small firm that starts Year 1 with an equity investment of $100 (the firm uses no debt). Let's say this firm earns a profit of 5%, and pays it all out as a dividend (assume there are no taxes in this marvelous country). Investors expect these dividends to grow in the future by 4% per year. Finally, assume that the real rate of interest is 3%, the equity market risk premium is 4%, and investors charge the firm an extra .5% because it is judged riskier than the market as a whole (perhaps because it is small). As you can see in the next table, this results in a year-end market value of $143, and an ending book value/market value ratio of .70. Investors' return on the initial $100 investment is 48% (a $5 dividend and a price change of $43). The next year, three things change. As the result of its initial success in the market for its product, the company's dividends are now expected to grow by 6% per year. As a result of these changes, the firm's market value rises to $333, its book/market ratio falls to .30, and investors earn a total return (dividend plus price change) of 137%.

In the third year, investors see the company as less risky, and reduce the equity risk premium they require from 4.5% to just 4%. This causes market value to rise to $500, book/market to fall to .20, and investors' total return to fall to a still not-too-shabby 51%. By the fourth year, competition has intensified, which has forced a cut in the company's expected growth to 4% per year. This causes a fall in market value to $167, a rise in book/market to .60, and a fall in investors' total return to (66%).