The Dimensions of Stock Returns

Abbreviated from here.
By Truman A. Clark

An ongoing objective of financial research is to explain the behavior of stock returns. Factors are sought that explain both differences among the returns of individual stocks in any given time period and the variation of stock returns through time. If a factor does both, it is said to explain the common variation of returns. In addition, if a factor is related to non-diversifiable risk and possesses explanatory power independent of other factors, the factor is considered a "dimension" of stock returns.

Fama and French (1992) found that two factors related to company size and book-to-market ratio (BtM) together explain much of the common variation of stock returns and that these factors are related to risk. Small cap stocks have higher average returns than large cap stocks, and high BtM (or "value") stocks have higher average returns than low BtM (or "growth") stocks. Based on Fama and French's findings, size and BtM are dimensions of stock returns.

Fama and French also investigated a market factor. A market factor is needed to distinguish stocks from fixed income securities, and it is important in explaining the variation of stock returns through time. But, among stocks in a given time period, differences in their sensitivities to the market factor are unrelated to differences in their average returns, so the market factor is not a dimension of stock returns.

The Fama/French results have important implications for domestic equity portfolio design. Large capitalization growth stocks constitute large portions of traditional "market-like portfolios" based on indexes such as the S&P 500, the Russell 3000 and the Wilshire 5000. Domestic equity portfolios with greater commitments to small cap stocks and value stocks offer higher average returns than conventional market-like portfolios.

Size and BtM also are dimensions of international and emerging markets stock returns. This confirms Fama and French's interpretation of size and BtM effects as rewards for bearing risk that cannot be eliminated by diversification.

Risk and Return
Controlling for differences in BtM by comparing large cap value to small cap value and comparing large cap growth to small cap growth, small cap stocks had higher average returns than large cap stocks. Controlling for differences in size by comparing large cap growth to large cap value and comparing small cap growth to small cap value, value stocks had higher average returns than growth stocks. The higher average returns of small cap and value stocks represent rewards for bearing risk.

If standard deviation were a complete measure of risk, average returns would increase as standard deviations increase. Controlling for differences in BtM, a direct relation between average returns and standard deviations is found when large cap stocks are compared to small cap stocks. But, controlling for differences in size, a discrepancy appears. Small growth stocks had a lower average return and a higher standard deviation than small value stocks. Since greater standard deviations are not consistently associated with higher average returns, standard deviation is not a reliable measure of risk.

Size, Book-to-Market and Earnings
Seeking a risk-based explanation for the relations of size and BtM to average returns, Fama and French (1995) investigated the behavior of the earnings of stocks grouped by size and BtM. Measuring profitability by the ratio of annual earnings to book value of equity, Figure 2 illustrates the evolution of profitability over long periods before and after stocks are classified by size and BtM. BtM is associated with persistent differences in profitability. On average, low BTM stocks are more profitable than high BtM stocks of similar size for at least five years before and after portfolio formation. Low BtM indicates sustained high earnings that are characteristic of firms that are growing and financially robust. High BtM indicates protracted low earnings that are typical of firms experiencing financial distress.

Expected Returns and the Cost of Capital
Financial markets channel funds from suppliers of capital to users of capital. Expected returns are the rewards investors anticipate for supplying capital. Investors require a higher rate of return (or risk premium) for bearing greater risk. Risk is something that investors collectively shun and that cannot be eliminated by diversification.

The cost of capital is the price users of capital must pay to obtain financing. Competition forces users of capital to bid higher prices to obtain funding for more risky ventures.

In equilibrium, the expected rate of return and the cost of capital are determined jointly as the price at which the demand for and supply of capital are equal. In financial markets that function efficiently, investors only receive risk premiums for bearing risk. As risk increases, the expected rate of return and the cost of capital increase.

High BtM and small size often indicate companies that are experiencing some degree of financial distress. On average, they have higher costs of capital because they tend to be riskier than companies with low BtM and large market capitalization. The higher average returns of small stocks and value stocks reflect compensation for exposure to non-diversifiable risk factors.

The Three-Factor Model
The findings of Fama and French suggest that much of the variation in stock returns is explained by three systematic risk factors.

· The market factor measured by the returns of stocks minus the returns of Treasury bills.
· The size factor measured by the returns of small stocks minus the returns of big stocks.
· The value factor measured by the returns of high-BtM stocks minus the returns of low-BtM stocks.

Portfolio Engineering
Many investors commit high proportions of their domestic equity holdings to portfolios resembling the S&P 500, Russell 3000 or other market-like proxies. Large cap growth stocks are the dominant holdings of the S&P 500 and the Russell 3000. As a result, market-like proxies are poor portfolio structures for investors seeking exposure to the size and/or value factors. Investors can get such exposure by increasing their relative holdings of small cap and/or value stocks.

The potential increases in expected returns due to these tilts can be estimated with the three-factor model. For purposes of illustration, it is assumed that the expected risk premiums are six percent per year for the market factor and three percent per year for both the size and value factors.

Words of Caution
Structured portfolios offer the prospect of higher long-term returns than market-like portfolios, but the expected risk premiums are not sure things. Factor premiums vary widely and randomly. For the 1927-2001 period, the standard deviations of the annual premiums were approximately 21% per year for the market factor, 14% for the size factor and 14% for the value factor. Owing to their high variability, it may take decades before rewards for bearing increased size and value risk are realized.

Cumulative premiums for each factor are computed by adding successive monthly premiums for the period January 1927 through December 2001. Although the cumulative premiums tend to rise over long periods of time, each moves erratically with lengthy episodes of downward drift. The market premium declined from December 1967 to July 1982—a period of more than 14 years. The size premium declined from December 1983 to December 2001—a period of 18 years (and still counting). The value premium declined from December 1987 to December 2000—a period of 13 years.

Structured portfolios are not appropriate for all investors. Structured portfolios have higher expected returns because they are riskier than market-like portfolios. Over periods of less than 20 years, structured portfolios often will have lower returns than market-like portfolios. It is only over periods of 20 years or more that it becomes more probable that structured portfolios will outperform market-like portfolios. Investors with short horizons or aversion to risk should stick with market-like portfolios. Structured portfolios only make sense for investors with long time horizons and sufficient tolerance for increased risk.

International and Emerging Markets Equities
Size and value effects are not confined to US equity markets. The MSCI EAFE Index represents a portfolio of international stocks from developed countries similar to the S&P 500. EAFE is composed predominantly of large cap growth stocks. During 1975-2001, international small cap stocks had a higher average return than EAFE indicating a size effect, and international large cap value stocks had a higher average return than EAFE indicating a value effect.Based on the limited amount of data available, size and value effects also appear in emerging markets. The IFC Investables Total Return Index represents a portfolio of tradable stocks in emerging markets countries that non-resident investors are permitted to own. During 1989-2001, emerging markets small cap stocks and value stocks had higher average returns than the IFC index.

The international findings are consistent with Fama and French's interpretation of the size and value effects as rewards for bearing non-diversifiable risk. If size and value effects were related to risk factors unique to the US, forming globally diversified portfolios could eliminate them. Instead, the existence of similar size and value effects in both domestic and international stock returns demonstrates that these effects are global phenomena reflecting exposures to ubiquitous sources of risk.

Implications for Global Equity Allocation
EAFE is the international equivalent of the S&P 500. EAFE returns, expressed in US dollars, are determined jointly by stock returns computed in local currencies and foreign-exchange gains or losses against the dollar. Because the two indexes contain stocks with similar size and value characteristics, it is reasonable to assume that the costs of capital of EAFE and the S&P 500 are approximately equal. If it is also assumed that currencies have zero expected returns, EAFE should have about the same expected gross rate of return as the S&P 500.

Concluding Comments
The identification of size and value factors by Fama and French has important implications for equity portfolio design. Relative to traditional market-like portfolios, portfolios with greater exposures to the size and value factors offer higher expected long-term rates of return.
Structured portfolios can be designed that provide targeted sensitivities to the size and value factors. International and emerging markets equity returns also exhibit size and value effects.
Structured portfolios only make sense for investors with long time horizons and sufficient tolerance for increased risk. For the right investors, structured portfolios are promising alternatives to old-fashioned market-like portfolios.