the dimensions of stock returns

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The Dimensions of Stock Returns: 2007 http://www.dfaus.com/library/articles/dimensions_stock_returns_2

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The Dimensions of Stock Returns: 2007

By Truman A. Clark

September 2007

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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 IndexesAnnualized Averages of Monthly Rates of ReturnJuly 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 PortfoliosDifferences in Annualized Monthly Rates of ReturnJuly 1926-December 2006

Value -Growth

Small -Large

Mean Return 4.79 4.68

Standard Error 1.62 2.00

t-Statistic 2.95 2.35

Probability 0.00 0.02

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 GroupsDimensional US Eligible MarketDecember 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 stronglyto 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 Capin Size and Book-to-Market GroupsS&P 500 IndexDecember 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 GroupsUS Large Cap Value PortfolioDecember 29, 2006

Figure 5

Allocation of Aggregate Market Capin Size and Book-to-Market Groups

US Small Cap Value PortfolioDecember 29, 2006




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Figure 6

Allocation of Aggregate Market Capin Size and Book-to-Market GroupsUS Micro Cap Value Portfolio

December 29, 2006

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 Capin Size and Book-to-Market GroupsUS Core Equity 1 PortfolioDecember 29, 2006




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Figure 8

Allocation of Aggregate Market Capin Size and Book-to-Market GroupsUS Core Equity 2 Portfolio

December 29, 2006

Figure 9

Allocation of Aggregate Market Capin Size and Book-to-Market GroupsUS Vector Equity PortfolioDecember 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 blendedportfolio 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 Capin Size and Book-to-Market GroupsBlended PortfolioDecember 29, 2006

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




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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 IndexesAnnualized Averages of Monthly Rates of ReturnJanuary 1975-December 2006

Indexes are net of withholding taxes on dividends. MSCI data copyright MSCI 2007, all rights reserved.

Table 2

MSCI International Equity IndexesDifferences in Annualized Monthly Rates of ReturnJanuary 1975-December 2006




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EAFE Value -EAFE Growth

Mean Return 3.78

Standard Error 1.20

t-Statistic 3.14

Probability 0.00

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 IndexesAnnualized Averages of Monthly Rates of ReturnJanuary 1989-December 2006

Fama/French data provided by Fama/French.

Table 3

Fama/French Emerging Markets Equity IndexesDifferences in Annualized Monthly Rates of ReturnJanuary 1989-December 2006

Value -Growth

Small -Large

Mean Return 6.19 3.09

Standard Error 2.45 2.16t-Statistic 2.52 1.43

Probability 0.01 0.15

Fama/French Emerging Markets Indexes compiled by Fama/French from IFC securities data. IFC Market isEmerging Markets Composite Investables (EMCI) Index (gross dividends), provided by InternationalFinance 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




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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




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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 annualizedstandard 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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materials presented are compiled from sources believed to be reliable and current, but accuracy

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