The Dimensions of Stock Returns: 2007
September 2007
Average stock returns are related to firm size and relative price. Firm size is measured by the market capitalization of equity (price times shares outstanding). When stocks are ranked by size, small stocks tend to have higher average returns than large stocks. This is known as the size effect.
Relative price is often measured by a firm's book-to-market ratio (BtM). When stocks are ranked by relative price, high BtM (or "value") stocks tend to have higher average returns than low BtM (or "growth") stocks. This is known as the relative-price or value effect.
Many studies document the existence of size and relative-price effects in equity markets in the US and many other countries. These findings have important implications for equity allocation. Investors may be able to increase the expected returns of their portfolios by holding small capitalization stocks and value stocks in greater than market-capitalization proportions. Such portfolios are said to be "tilted" toward small cap and value stocks.
Portfolio design can matter. A common way to build a tilted portfolio is to combine separate asset class funds. This building-block approach may generate trading costs that lower net returns.
To help reduce the costs of maintaining tilted portfolios, Dimensional introduced Core Equity funds. The Core Equity funds are integrated, marketwide portfolios designed to have lower trading costs than conventional tilted portfolios. Dimensional offers Core Equity portfolios for the US, international, and emerging markets.
The Size and Relative-Price Effects
Figure 1 displays the annualized averages and standard deviations of the monthly returns of five Fama/French indexes: market, value, growth, small, and large. The market portfolio holds all NYSE, Nasdaq, and Amex stocks except for ADRs, closed-end funds, and tracking stocks.1 The value portfolio is composed of NYSE, Nasdaq, and Amex stocks with positive BtMs in the top 30% of the NYSE BtM distribution. The growth portfolio is composed of stocks with positive BtMs in the bottom 30% NYSE BtM. The small portfolio is composed of NYSE, Nasdaq, and Amex stocks with market capitalizations in the bottom 30% of NYSE market cap. The large portfolio is composed of stocks with market caps in the top 30% of NYSE market cap. The sample period is July 1926 through December 2006.
| Figure 1 |
| Fama/French US Equity Indexes Annualized Averages of Monthly Rates of Return |
| July 1926-December 2006 |
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| Fama/French data provided by Fama/French. |
The relative-price and size effects are apparent in Figure 1. The difference between the average returns of value and growth stocks is 4.79% per year (Table 1). The difference between the average returns of small cap and large cap stocks is 4.69% per year. Both of the average return differences are reliably different from zero.
| Table 1 | |||||||||||||||
| Fama/French US Equity Portfolios Differences in Annualized Monthly Rates of Return |
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| July 1926-December 2006 | |||||||||||||||
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Portfolio Composition
A portfolio's composition determines its potential to capture the size and relative-price effects. The eligible market portfolio is the capitalization-weighted combination of all eligible NYSE, Amex, and Nasdaq stocks.2 Figure 2 shows the allocation of the market portfolio to cells in a two-dimensional style grid as of December 29, 2006. The grid's dimensions are size and book-to-market ratio.
| Figure 2 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups Dimensional US Eligible Market |
| December 29, 2006 |
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The size dimension has six rows. The largest 250 stocks (mega-cap) are in the first row. Other large stocks (251-600) are in the second row. Mid cap stocks (601-1000) are in the third row. The next three rows are groups of small cap stocks with stocks smaller than the 2,500th firm in the last row.
The relative-price dimension has six columns. The BtM quintile breakpoints for the 1,000 largest stocks determine the boundaries of the first five columns. Stocks with the lowest BtM (extreme growth) are in the first column, and stocks with the highest BtM (extreme value) are in the fifth column. The sixth column ("other") contains utilities as well as firms with missing or negative book equity.
The height of each bar in Figure 2 indicates the proportion of market value held in a grid cell. By definition, the market portfolio is neutral with respect to size and relative-price. Thus, the size and book-to-market allocations of the market portfolio, such as those in Figure 2, may serve as relative benchmarks for assessing a portfolio's size and value tilts and are not exact. Other portfolios can be compared to the market portfolio to identify their relative overweighting or underweighting of small cap and value stocks.
Figure 3 shows the composition of the S&P 500 Index. Dark cells indicate overweighting relative to the market. Light cells indicate underweighting. Relative to the market, the S&P 500 tilts strongly to very large cap stocks. Almost all S&P 500 stocks are in the two largest size categories, and none is in the two smallest groups.
| Figure 3 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups S&P 500 Index |
| December 29, 2006 |
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Dimensional asset class funds concentrate their holdings in specific regions of the style grid. The DFA US Large Cap Value Portfolio (Figure 4), the DFA US Small Cap Value Portfolio (Figure 5), and the DFA US Micro Cap Portfolio (Figure 6) specialize in the regions that their titles imply. Much of each fund's weight is in its target zone.3
| Figure 4 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups US Large Cap Value Portfolio |
| December 29, 2006 |
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| Figure 5 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups US Small Cap Value Portfolio |
| December 29, 2006 |
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| Figure 6 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups US Micro Cap Value Portfolio |
| December 29, 2006 |
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In September 2005, Dimensional introduced two domestic Core Equity funds. Unlike the S&P 500 and the three asset class funds, each Core Equity fund seeks to make eligible for inclusion all stocks traded in US markets. Relative to the market, the Dimensional US Core Equity 1 Portfolio (Figure 7) holds smaller proportions of mega-cap stocks and large cap growth stocks and greater proportions of all other stocks. The Dimensional US Core Equity 2 Portfolio (Figure 8) has greater small cap and relative-price tilts than Core Equity 1. Dimensional introduced the US Vector Equity Portfolio (Figure 9) in December 2005. Unlike Core 1 and Core 2, Vector Equity does not target all eligible stocks. Instead, Vector Equity targets only a wide range of smaller-cap and value stocks.
| Figure 7 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups US Core Equity 1 Portfolio |
| December 29, 2006 |
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| Figure 8 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups US Core Equity 2 Portfolio |
| December 29, 2006 |
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| Figure 9 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups US Vector Equity Portfolio |
| December 29, 2006 |
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Conventional Tilted Portfolios
The traditional way to build tilted portfolios is by combining an S&P 500 or other large cap index fund with one or more asset class funds. Figure 10 shows the style composition of a blended portfolio holding 40% in the S&P 500 Index, 30% in the US Large Cap Value fund, 20% in the US Small Cap Value fund, and 10% in the US Micro Cap fund. This portfolio provides value and small cap tilts, but it is perhaps operationally inefficient.
| Figure 10 |
| Allocation of Aggregate Market Cap in Size and Book-to-Market Groups Blended Portfolio |
| December 29, 2006 |
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Each asset class fund has defined investment guidelines. As stock prices change, an asset class fund will sell stocks that no longer meet its guidelines. (For example, the Micro Cap fund will sell a stock that has appreciated sufficiently to exceed its maximum capitalization limit.) An asset class fund also will buy other stocks as they become eligible. For funds specializing in relatively illiquid small cap stocks, this adjustment trading can be quite costly.
Core Equity Funds: Improving Tilted Portfolios
The Core Equity funds are designed to reduce the costs of maintaining tilted portfolios. Unlike asset class funds, Core Equity funds target all eligible stocks, and trading is aimed at preserving portfolio balance.
Each Core Equity fund establishes its tilts through a unique process applied to the style grid.4 The process determines a target percentage of shares outstanding to be held for all stocks within each grid cell. When a stock's price changes, no trading is required if either the stock remains in the same cell or it drifts into another cell having the same target percentage of shares. Trading is required only when a stock moves into a cell with a different target percentage of shares. The trading that occurs adjusts the percentage of shares held to the new target. The trading does not result in total liquidation of a position or the establishment of a new position starting from zero holdings.5
A Core Equity fund's cell targets for percentages of shares are chosen with the intent to keep trading costs low. The targets channel turnover toward larger-cap stocks (where trading costs are relatively low) and away from smaller-cap stocks (where trading costs are relatively high). By working to reduce trading costs, Core Equity funds seek to produce higher net returns than conventional tilted portfolios of comparable risk.
International Equities: Size and Relative-Price Effects
Size and relative-price effects also are found in international equity markets.
The MSCI EAFE Index is a capitalization-weighted aggregate of large cap stocks from developed countries. MSCI divides EAFE into value and growth indexes. Figure 11 shows the annualized averages and standard deviations of the monthly rates of return of the EAFE market, value, and growth indexes for the January 1975-December 2006 period.6 EAFE Value has a higher average return than EAFE Growth. The difference in average returns (3.78% per year) is reliably different from zero (Table 2).
| Figure 11 |
| MSCI EAFE International Equity Indexes Annualized Averages of Monthly Rates of Return |
| January 1975-December 2006 |
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| Indexes are net of withholding taxes on dividends. MSCI data copyright MSCI 2007, all rights reserved. |
| Table 2 | ||||||||||
| MSCI International Equity Indexes Differences in Annualized Monthly Rates of Return |
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| January 1975-December 2006 | ||||||||||
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| Indexes are net of withholding taxes on dividends. MSCI data copyright MSCI 2007, all rights reserved. |
The Fama/French emerging markets portfolio is a capitalization-weighted collection of stocks from emerging markets countries. Fama and French divide their market portfolio into value and growth components and into small and big components. Figure 12 displays the annualized averages and standard deviations of the monthly rates of return of the Fama/French emerging markets portfolios for the 1989-2006 period. The average return of the value portfolio exceeds the average return of the growth portfolio. The difference between the average returns of emerging markets value and growth stocks (6.19% per year) is reliably different from zero (Table 3).
| Figure 12 |
| Fama/French Emerging Markets Equity Indexes Annualized Averages of Monthly Rates of Return |
| January 1989-December 2006 |
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| Fama/French data provided by Fama/French. |
| Table 3 | |||||||||||||||
| Fama/French Emerging Markets Equity Indexes Differences in Annualized Monthly Rates of Return |
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| January 1989-December 2006 | |||||||||||||||
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| Fama/French Emerging Markets Indexes compiled by Fama/French from IFC securities data. IFC Market is Emerging Markets Composite Investables (EMCI) Index (gross dividends), provided by International Finance Corporation. |
At present, reliable evidence of size effects in developed and emerging markets is difficult to find using multi-country indexes. MSCI introduced small cap total return indexes in 2001. These indexes do not currently provide enough observations to conduct powerful statistical tests of average return differences. In emerging markets, the Fama/French small cap portfolio has a higher average return than the large cap portfolio (Figure 12), but the difference in average returns (3.09% per year) is only marginally different from zero (Table 3).
However, studies of stock returns in many individual countries and regions report evidence of positive size effects. Rizova summarizes the findings of a number of these investigations.7
- In studies of developed markets, the estimated size effect is positive in nineteen instances and negative in three. Four of the nineteen positive estimates have t-statistics exceeding 2. None of the three negative size effect estimates has a t-statistic of this magnitude.
- In studies of emerging markets, the estimated size effect is positive in fourteen cases and negative in seven. Four of the fourteen positive size effect estimates have t-statistics greater than 2. One of the seven negative size effect estimates has a t-statistic of this magnitude.
Given the preponderance of this evidence, it is reasonable to assume that small cap stocks tend to have higher average returns than large cap stocks in developed and emerging markets.
Implications for International Equity Allocation
Investors may be able to increase their portfolios' average returns by tilting their developed and emerging markets holdings toward small cap and high BtM stocks. Tilted portfolios can be constructed with separate asset class funds, but, as with domestic equities, this may not be the most efficient approach. Because of stamp duties, custodial charges, and other fees, trading costs in international markets are generally higher than in US markets.
To attempt to reduce the costs of maintaining tilted international portfolios, Dimensional introduced International and Emerging Markets Core Equity funds. The Dimensional International Core Equity Portfolio began operations in September 2005. International Core Equity is broadly diversified through its holdings of stocks from twenty-two countries: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Hong Kong, Ireland, Italy, Japan, the Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden, Switzerland, and the United Kingdom.
Dimensional launched the Emerging Markets Core Equity Portfolio in April 2005. Emerging Markets Core Equity invests in stocks from sixteen countries: Brazil, Chile, the Czech Republic, Hungary, India, Indonesia, Israel, Malaysia, Mexico, the Philippines, Poland, South Africa, South Korea, Taiwan, Thailand, and Turkey. China joined the list of eligible countries in 2007. Dimensional also offers an Emerging Markets Social Core Equity Portfolio.
Concluding Comments
Size and relative-price effects are found in the US, international, and emerging markets. Small cap stocks tend to have higher average returns than large cap stocks. Value stocks tend to have higher average returns than growth stocks. Investors may be able to increase their portfolios' average returns by tilting their holdings toward small cap and value stocks, which historically have produced greater expected returns. Traditional tilted portfolios combine an S&P 500 or other large cap index fund with an assortment of asset class funds. Such tilted portfolios may be costly to maintain. The component asset class funds trade frequently to adhere to their investment guidelines. These transactions often are concentrated in small cap stocks that are relatively expensive to trade. In addition, investors must trade their asset class funds from time to time to maintain portfolio balance. Such adjustment trading may reduce the net returns of traditional tilted portfolios.
As alternatives to conventional tilted portfolios, Dimensional offers Core Equity portfolios: US Core Equity 1, US Core Equity 2, US Vector Equity, International Core Equity, Emerging Markets Core Equity, and Emerging Markets Social Core Equity. Each Core Equity fund is tilted toward small cap and value stocks within its eligible universe. These funds are designed to reduce trading costs by directing turnover toward larger-cap stocks (which are cheaper to trade) and away from smaller-cap stocks (which are costly to trade). With reduced trading costs, the Core Equity funds strive to provide higher net returns than standard tilted portfolios of comparable risk.
Dimensional continues to expand its offerings of core equity portfolios. Core equity funds designed for investors in Australia, Canada, and the UK are operational. Other core equity strategies are under development.
1 The annualized average return equals 12 times the average monthly return. The annualized standard deviation of returns equals the square root of 12 times the standard deviation of monthly returns.
2 Ineligible stocks include REITs, closed-end funds, limited partnerships, bankrupt firms, ADRs, and illiquid stocks that are very costly to trade.
3 Note that the ranges of the vertical axes in Figures 4-5 are greater than the ranges of Figures 2-3 and 6-10. Figures 4 and 5 require greater ranges because the portfolios they depict are concentrated highly in relatively few grid cells.
4 Dimensional Fund Advisors, "Dimensional Fund Advisors' Core Equity Technology" (white paper, Dimensional Fund Advisors, August 2006).
5 A Core Equity fund will liquidate its holdings of a stock that loses its market eligibility. A Core Equity fund will establish a new position in a stock that gains market eligibility.
6 Different time periods are used to compute the returns statistics for domestic (Figure 1), international (Figure 11), and emerging markets (Figure 12) equities. The sample periods are determined by data availability.
7 Savina Rizova, "International Evidence of the Size Effect" (white paper, Dimensional Fund Advisors, August 2006).
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