# 2: Facts: time-variation and business cycle correlation of expected returns.

We start with the facts. What is the pattern by which expected returns vary over time and across assets? What is the variation on the left-hand side of (1.3) that we want to explain by understanding the marginal value of wealth on the right-hand side of (1.3)?

First, a number of variables forecast aggregate stock, bond, and foreign exchange returns. Thus, expected returns vary over time. The central technique is simple forecasting regression: If we find |b| > 0 in [R.sub.t+1] = a + [bx.sub.t] + [[epsilon].sub.t+1], then we know that [E.sub.t]([R.sub.t+1]) varies over time. The forecasting variables [x.sub.t] typically have a suggestive business cycle correlation. Expected returns are high in "bad times," when we might well suppose people are less willing to hold risks.

For example, Table 2.1 reproduces a table from Cochrane [35] that reports regressions of returns on dividend price ratios. A one percentage point higher dividend yield leads to a five percentage point higher return. This is a surprisingly large number. If there were no price adjustment, a one percentage point higher dividend yield would only lead to a one percentage point higher return. The conventional "random walk" view implies a price adjustment that takes this return away. Apparently, prices adjust in the "wrong" direction, reinforcing the higher dividend yield.

The second set of regressions in Table 2.1 is just as surprising. A high dividend yield means a "low" price, and it should signal a decline in future dividends. Not only do we not see a decline, the point estimate (though insignificant) is that dividends rise.

Both of these numbers are subject to substantial uncertainty, of course. But given there is not a shred of evidence that high prices forecast higher subsequent dividends, Cochrane [35] shows how the return-forecasting coefficient must be at least two.

Second, expected returns vary across assets. Stocks earn more than bonds, of course. In addition, a large number of stock characteristics are now associated with average returns. The book/market ratio is the most famous example: stocks with low prices (market value) relative to book value seem to provide higher subsequent average returns. A long list of other variables including size (market value), sales growth, past returns, past volume, accounting ratios, short-sale restrictions, and corporate actions such as investment, equity issuance and repurchases are also associated with average returns going forward.

This variation in expected returns across stocks would not cause any trouble for traditional finance theory, if the characteristics associated with high average returns were also associated with large market betas. Alas, they often are not. Instead, the empirical finance literature has associated these patterns in expected returns with betas on new "factors."

(Cochrane [38] is an easily accessible review paper that synthesizes current research on both the time-series and the cross-sectional issues. Chapter 20 of Asset Pricing [40] is a somewhat expanded version, with more emphasis on the relationship between various time series representations. Campbell [22] also has a nice summary of the facts.)

2.1 History of return forecasts

Return forecasts have a long history. The classic view that "stocks follow a random walk," meaning that the expected return is constant over time, was first challenged in the late 1970s. Fama and Schwert [69] found that expected stock returns did not increase one-for-one with inflation. They interpreted this result to say that expected returns are higher in bad economic times, since people are less willing to hold risky assets, and lower in good times. Inflation is lower in bad times and higher in good times, so lower expected returns in times of high inflation are not a result of inflation, but a coincidence.

To us, the association with inflation that motivated Fama and Schwert is less interesting, but the core finding that expected returns vary over time, and are correlated with business cycles (high in bad times, low in good times), remains the central fact. Fama and Gibbons [68] added investment to the economic modeling, presaging the investment and equilibrium models we study later.

In the early 1980s, we learned that bond and foreign exchange expected excess returns vary over time--that the classic "expectations hypothesis" is false. Hansen and Hodrick [83] and Fama [56] documented the predictability of foreign-exchange returns by running regressions of returns on forward-spot spread or interest rate differentials across countries. If the foreign interest rate is higher than the domestic interest rate, it turns out that the foreign currency does not tend to depreciate and thus an adverse currency movement does not, on average, wipe out the apparently attractive return to investing abroad.

Fama [57] documented the predictability of short-term bond returns, and Fama and Bliss [59] the predictability of long-term bond returns, by running regressions of bond returns on forward-spot spreads or yield differentials. The latter findings in particular have been extended and stand up well over time. (Stambaugh [148] extended the results for short-term bonds and Cochrane and Piazzesi [42] did so for long-term bonds. Both papers ran bond returns from t to t + 1 on all forward rates available at time t, and substantially raised the forecast [R.sup.2]. The Cochrane and Piazzesi bond return forecasting variable also improves on the yield spread's ability to forecast stock returns.) Shiller, Campbell, and Schoenholtz [147] and Campbell and Shiller [26] rejected the expectations hypothesis by regressions of future yields on current yields; their regressions imply time-varying expected returns. Campbell [19] is an excellent summary of this line of research.

While the expectations hypothesis had been rejected previously, (1) these papers focused a lot of attention on the problem. In part, they did so by applying a simple and easily interpretable regression methodology rather than more indirect tests: just forecast tomorrow's excess returns from today's yields or other forecasting variables. They also regressed changes in prices (returns) or yields on today's yield or forward-rate spreads. A forecast that tomorrow's temperature equals today's temperature would give a nice 1.0 coefficient and a high [R.sup.2]. To see a good weather forecaster, you check whether he can predict the difference of tomorrow's temperature over today's. A report of today's temperature will not survive that test.

During this period, we also accumulated direct regression evidence that expected excess returns vary over time for the stock market as a whole. Poterba and Summers [135] and Fama and French [60] documented that past stock market returns forecast subsequent returns at long horizons. Shiller [146] Campbell and Shiller [25] and Fama and French [61] showed that dividend/price ratios forecast stock market returns. Fama and French really dramatized the importance of the D/P effect by emphasizing long horizons, at which the [R.sup.2] rise to 60%. This observation emphasized that stock return forecastability is an economically interesting phenomenon that cannot be dismissed as another little anomaly that might be buried in transactions costs. Long horizon forecastability is not really a distinct phenomenon; it arises mechanically as the result of a small short horizon variability and a slow-moving right-hand variable (D/P). It also does not generate much statistical news since standard errors grow with horizon just as fast as coefficients.

Fama and French [62] is an excellent summary and example of the large body of work that documents variation of expected returns over time. This paper shows how dividend-price ratios and term spreads (long bond yield less short bond yield) forecast stock and bond returns. The paper emphasizes the comforting link between stock and bond markets: the term spread forecasts stock returns much as it forecasts bond returns. Since stock dividends can be thought of as bond coupons plus risk, we should see any bond return premium reflected in stock returns. Most importantly, Fama and French show by a series of plots how the variables that forecast returns are associated with business cycles.

These papers run simple forecasting regressions of returns at time t + 1 on variables at time t. The forecasting variables are all based on market prices, though, which seems to take us away from our macroeconomic quest. However, as emphasized by Fama and French [62], the prices that forecast returns are correlated with business cycles. A number of authors, including Estrella and Hardouvelis [54] and more recently Ang, Piazzesi and Wei [4], documented that the price variables that forecast returns also forecast economic activity.

A related literature, including Campbell and Shiller [25] and Cochrane [32] (summarized compactly in Cochrane [39]) "New Facts in Finance," connects the time-series predictability of stock returns to stock price volatility. Iterating and linearizing the identity 1 = [R.sup.-1.sub.t+1][R.sub.t+1] we can obtain an identity that looks a lot like a present value model,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

where small letters are logs of capital letters, and k and [rho] = (P/D)/[1 + (P/D)] [approximately equal to] 0.96 are constants related to the point P/D about which we linearize. If price-dividend ratios vary at all, then, then either (1) price-dividend ratios forecast dividend growth (2) price-dividend ratios forecast returns or (3) prices must follow a "bubble" in which the price-dividend ratio is expected to rise without bound.

It would be lovely if variation in price-dividend ratios corresponded to dividend forecasts. Investors, knowing future dividends will be higher than they are today, bid up stock prices relative to current dividends; then the high price-dividend ratio forecasts the rise in dividends. It turns out that price-dividend ratios do not forecast aggregate dividends at all. This is the "excess volatility" found by Shiller [144] and LeRoy and Porter [109]. However, prices can also be high if this is a time of temporarily low expected returns; then the same dividends are discounted at a lower rate, and a high price-dividend ratio forecasts low returns. It turns out that the return forecastability we see in regressions is just enough to completely account for the volatility of price dividend ratios through (2.1). (This is a main point of Cochrane [32].) Thus, return forecastability and "excess volatility" are exactly the same phenomenon. Since price-dividend ratios are stationary [47] and since the return forecastability does neatly account for price-dividend volatility, we do not need to invoke the last "rational bubble" term.

However, the fact that almost all stock price movements are due to changing expected returns rather than to changing expectations of future dividend growth means that we have to tie stock market movements to the macroeconomy entirely through harder-to-measure time-varying risk premia rather than easier-to-understand cashflows.

2.2 Lettau and Ludvigson

Martin Lettau and Sydney Ludvigson's [112] "Consumption, Aggregate Wealth and Expected Stock Returns" is an important recent extension of stock return forecastability. Lettau and Ludvigson find that the ratio of consumption to wealth forecasts stock returns. In essence, consumption along with labor income form a more useful "trend" than the level of dividends or earnings.

Cochrane [35] showed that consumption provides a natural "trend" for income, and so we see long-run mean reversion in income most easily by watching the consumption-income ratio. I also showed that dividends provide a natural "trend" for stock prices, so we see long-run mean-reversion in stock prices most easily by watching the dividend-price ratio. Lettau and Ludvigson nicely put the two pieces together, showing how consumption relative to income and wealth has a crossover prediction for long-run stock returns.

A number of other macroeconomic variables also forecast stock returns, including the investment-capital ratio in Cochrane [33], the dividend-earnings ratio in Lamont [107], investment plans in Lamont [108], the ratio of labor income to total income in Menzly, Santos and Veronesi [126], the ratio of housing to total consumption in Piazzesi, Schneider and Tuzel [133], and an "output gap" formed from the Federal Reserve capacity index in Cooper and Priestley [46]. It's comforting that these variables do not include the level of market prices, removing any suspicion that returns are forecastable simply because a "fad" in prices washes away.

Lettau and Ludvigson [114] show that the consumption-wealth ratio also forecasts dividend growth. This is initially surprising. So far, very little has forecast dividend growth; and if anything does forecast dividend growth, why is a high dividend forecast not reflected in and hence forecast by higher prices? Lettau and Ludvigson answer this puzzle by noting that the consumption-wealth ratio forecasts returns, even in the presence of D/P. In the context of (2.1), the consumption-wealth ratio sends dividend growth and returns in the same direction, so its effects on the price-dividend ratio offset. Thus, on second thought, the observation is natural. If anything forecasts dividend growth it must also forecast returns to account for the fact that price-dividend ratios do not forecast dividend growth. Conversely, if anything has additional explanatory power for returns, it must also forecast dividend growth. And it makes sense. In the bottom of a recession, both returns and dividend growth will be strong as we come out of the recession. So we end up with a new variable, and an opening for additional variables, that forecast both returns and cash flows, giving stronger links from macroeconomics to finance.

2.3 Fama and French and the cross-section of returns

Fama and French's [65] "Multifactor Explanations of Asset Pricing Anomalies" is an excellent crystallization of how average returns vary across stocks. Fama and French start by summarizing for us the "size" and "value" effects; the fact that small stocks and stocks with low market values relative to book values tend to have higher average returns than other stocks. (2) (See the average returns in their Table I, panel A, reproduced as Table 2.2 below.) Again, this pattern is not by itself a puzzle. High expected returns should be revealed by low market values (see (2.1)). The puzzle is that the value and small firms do not have higher market betas. As panel B of Fama and French's Table I shows, all of the market betas are about one. Market betas vary across portfolios a little more in single regressions without hml and smb as additional right-hand variables, but here the result is worse: the high average return "value" portfolios have lower market betas.

Fama and French then explain the variation in mean returns across the 25 portfolios by variation in regression slope coefficients on two new "factors," the hml portfolio of value minus growth firms and the smb portfolio of small minus large firms. Looking across the rest of their Table I, you see regression coefficients b,s,h rising in panel B where expected returns rise in panel A. Replacing the CAPM with this "three-factor model" is the central point of Fama and French's paper. (Keep in mind that the point of the factor model is to explain the variation in average returns across the 25 portfolios. The fact that the factors "explain" a large part of the return variance--the high [R.sup.2] in the time-series regressions of Table I--is not the central success of an asset pricing model.)

This argument is not as circular as it sounds. Fama and French say that value stocks earn more than growth stocks not because they are value stocks (a characteristic) but because they all move with a common risk factor. This comovement is not automatic. For example, if we split stocks into 26 portfolios based on the first letter of the ticker symbol and subtracted the market return, we would not expect to see a 95% [R.sup.2] in a regression of the A portfolio on an A-L minus M-Z "factor," because we would expect no common movement between the A, B, C, etc. portfolios.

Stocks with high average returns should move together. Otherwise, one could build a diversified portfolio of high expected return (value) stocks, short a portfolio of low expected return (growth) stocks and make huge profits with no risk. This strategy remains risky and does not attract massive capital, which would wipe out the anomaly, precisely because there is a common component to value stocks, captured by the Fama-French hml factor.

Fama and French go further, showing that the size and book to market factors explain average returns formed by other characteristics. Sales growth is an impressive example, since it is a completely non-financial variable. Stocks with high sales growth have lower subsequent returns ("high prices") than stocks with low sales growth. They do not have higher market betas, but they do have higher betas on the Fama-French factors. In this sense the Fama-French three-factor model "explains" this additional pattern in expected returns. In this kind of application, the Fama-French three-factor model has become the standard model replacing the CAPM for risk adjusting returns.

The Fama-French paper has also, for better or worse, defined the methodology for evaluating asset pricing models for the last 10 years. A generation of papers studies the Fama-French 25 size and book to market portfolios to see whether alternative factor models can explain these average returns. Empirical papers now routinely form portfolios by sorting on other characteristics, and then run time-series regressions to see which factors explain the spread in average returns.

Most importantly, where in the 1980s papers would focus entirely on the p value of some overall statistic, Fama and French rightly got people to focus on the spread in average returns, the spread in betas, and the economic size of the pricing errors. Remarkably, this, the most successful model since the CAPM, is decisively rejected by formal tests. Fama and French taught us to pay attention to more important things than test statistics.

(The rejection of the three-factor model in the 25 portfolios is caused primarily by small growth portfolios, and Fama and French's Table I shows the pattern. Small growth stocks earn about the same average returns as large growth portfolios--see Table I "means", left column--but they have much larger slopes s. A larger slope that does not correspond to a larger average return generates a pricing error a. In addition, the [R.sup.2] are so large in these regressions, and the residuals correspondingly small, that economically small pricing errors are statistically significant. [alpha]'[[SIGMA].sup.-1][alpha] is large if a is small, but [SIGMA] is even smaller. A fourth "small growth-large value" factor eliminates this pricing error as well, but I don't think Fama and French take the anomaly that seriously.)

The Fama-French model seems to take us away from economic explanation of risk premia. After all, hml and smb are just other portfolios of stocks. However, it provides a very useful summary device. To the extent that the Fama-French three-factor model is successful in describing average returns, macro-modelers need only worry about why the value (hml) and small-large (smb) portfolio have expected returns. Given these factors, the expected returns of the 25 portfolios (and any other portfolios that are explained by the three-factor model) follow automatically. Macro-factor papers tend to evaluate models on the Fama-French 25 portfolios anyway, but they don't really have to, at least unless they want to take the small-growth puzzle seriously.

Fama and French speculate suggestively on the macroeconomic foundations of the value premium (p. 77):
```   One possible explanation is linked to human capital,
and important asset for most investors. Consider an
investor with specialized human capital tied to a growth
firm (or industry or technology). A negative shock to
the firm's prospects probably does not reduce the value
of the investor's human capital; it may just mean that
employment in the firm will expand less rapidly. In
contrast, a negative shock to a distressed firm more
likely implies a negative shock to the value of specialized
human capital since employment in the firm is more
likely to contract. Thus, workers with specialized human
capital in distressed firms have an incentive to avoid
holding their firms' stocks. If variation in distress is
correlated across firms, workers in distressed firms have
an incentive to avoid the stocks of all distressed firms.
The result can be a state-variable risk premium in the
expected returns of distressed stocks.
```

Much of the work described below tries to formalize this kind of intuition and measure the required correlations in the data.

The Fama-French paper closes with a puzzle. Though the three-factor model captures the expected returns from many portfolio sorts, it fails miserably on momentum. If you form portfolios of stocks that have gone up in the last year, this portfolio continues to do well in the next year and vice-versa (Jegadeesh and Titman [98]; see Fama and French's Table VI). Again, this result by itself would not be a puzzle, if the "winner" portfolio had higher market, smb, or hml betas than the loser portfolios. Alas (Fama and French, Table VII), the winner portfolio actually has lower slopes than the loser portfolio; winners act, sensibly enough, like high-price growth stocks that should have low mean returns in the three-factor model. The three-factor model is worse than useless at capturing the expected returns of this "momentum" strategy, just as the CAPM is worse than useless at explaining the average returns of book-to-market portfolios.

Now, the returns of these 10 momentum-sorted portfolios can be explained by an additional "momentum factor" umd of winner stocks less loser stocks. You cannot form a diversified portfolio of momentum stocks and earn high returns with no risk; a common component to returns shows up once again. Yet Fama and French did not take the step of adding this fourth factor, and thus claiming a model that would explain all the known anomalies of its day.

This reluctance is understandable. First, Fama and French worry (p. 81) whether the momentum effect is real. They note that the effect is much weaker before 1963, and call for more out-of-sample verification. They may also have worried that the effect would not survive transactions costs. Exploiting the momentum anomaly requires high-frequency trading, and shorting small losing stocks can be difficult. Second, having just swallowed hml and smb, one might naturally be reluctant to add a new factor for every new anomaly, and to encourage others to do so. Third, and perhaps most importantly, Fama and French had at least a good story for the macroeconomic underpinnings of size and value effects, as expressed in the above quotation. They had no idea of a macroeconomic underpinning for a momentum premium, and in fact in their view (p. 81) there isn't even a coherent behavioral story for such a premium. They know that having some story is the only "fishing license" that keeps one from rediscovering the Roll theorem. Still, they acknowledge (p. 82) that if the effect survives scrutiny, another "factor" may soon be with us.

In the time since Fama and French wrote, many papers have examined the momentum effect in great detail. I do not survey that literature here, since it takes us away from our focus of macroeconomic understanding of premia rather than exploration of the premia themselves. However, momentum remains an anomaly. "New Facts in Finance" [38] presents a simple calculation to show that momentum is, like long-horizon regression, a way to enhance the economic size of a well-known statistical anomaly, as a tiny positive autocorrelation of returns can generate the observed momentum profits.

One can also begin to imagine macroeconomic stories for momentum. Good cash-flow news could bring growth options into the money, and this event could increase the systematic risk (betas) of the winner stocks. Of course, then a good measure of "systematic risk" and good measurements of conditional betas should explain the momentum effect. Momentum is correlated with value, so it's tempting to extend a macroeconomic interpretation of the value effect to the momentum effect. Alas, the sign is wrong. Last year's winners act like growth stocks, but they get high, not low, average returns. Hence, the component of a momentum factor orthogonal to value must have a very high risk premium, and its variation is orthogonal to whatever macroeconomic effects underlie value.

In any case, the current crop of papers that try to measure macroeconomic risks follow Fama and French by trying to explain the value and size premium, or the Fama-French 25 portfolios, and so far largely exclude the momentum effect. The momentum factor is much more commonly used in performance evaluation applications, following Carhart [28]. In order to evaluate whether, say, fund managers have stock-picking skill, it does not matter whether the factor portfolios correspond to real risks or not, and whether the average returns of the factor portfolios continue out of sample. One only wants to know whether a manager did better in a sample period than a mechanical strategy.

I suspect that if the momentum effect survives its continued scrutiny, macro-finance will add momentum to the list of facts to be explained. A large number of additional expected-return anomalies have also popped up, which will also make it to the macro-finance list of facts if they survive long enough. We are thus likely to face many new "factors." After all, each new expected-return sort must either fall in to one of the following categories. (1) A new expected-return sort might be explained by betas on existing factors, so once you understand the existing factors you understand the new anomaly, and it adds nothing. This is how, for example sales growth behaves for the Fama-French model. (2) The new expected-return sort might correspond to a new dimension of comovement in stock returns, and thus be "explained" (maybe "summarized" is a better word) by a new factor. (3) If a new expected-return sort does not fall into (1) and (2), it corresponds to an arbitrage opportunity, which is most unlikely to be real, and if real to survive longer than a chicken in a crocodile pond. Thus, any expected return variation that is both real and novel must correspond to a new "factor."

2.4 Liew and Vassalou

A large body of empirical research asks whether the size and book to market factors do in fact represent macroeconomic phenomena via rather a-structural methods. Liew and Vassalou [117] is a good example of such work. It is natural to suppose that value stocks--stocks with low prices relative to book value, thus stocks that have suffered a sequence of terrible shocks--should be more sensitive to recessions and "distress" than other stocks, and that the value premium should naturally emerge as a result. Initially, however, efforts to link value stocks and value premia to economic or financial trouble did not bring much success. Fama and French [66, 67], were able to link value effects to individual cash flows and "distress," but getting a premium requires a link to aggregate bad times, a link that was notoriously absent, as emphasized by Lakonishok, Shleifer and Vishny [106]. However, in the 1990s and early 2000s, value stocks have moved much more closely with the aggregate economy, so more recent estimates do show a significant and heartening link between value returns and macroeconomic conditions. In this context, Liew and Vassalou [117] show that Fama and French's size and book to market factors forecast output growth, and thus are "business cycle" variables.

(1) Evidence against the expectations hypothesis of bond yields goes back at least to Macaulay [121]. Shiller, Campbell and Schoenholtz generously say that the expectations hypothesis has been "rejected many times in careful econometric studies," citing Hansen and Sargent [86], Roll [136], Sargent [140, 141], and Shiller [143]. Fama says that "The existing literature generally finds that forward rates ... are poor forecasts of future spot rates," and cites Hamburger and Platt [80], Fama [55], and Shiller, Campbell and Shoenholtz.

(2) These expected-return findings go back a long way, of course, including Ball [6], Basu [10], Banz [9], DeBondt and Thaler [49], and Fama and French [63, 64].
```Table 1. OLS regressions of percent excess returns (value weighted
NYSE--treasury bill rate) and real dividend growth on the percent VW
dividend/price ratio

[R.sub.t [right arrow] + k]
= a + b ([D.sub.t]/[P.sub.t]

Horizon k
(years)      b     [sigma] (b)   [R.sub.2]

1            5.3      (2.0)        0.15
2           10        (3.1)        0.23
3           15        (4.0)        0.37
5           33        (5.8)        0.60

[D.sub.t [right arrow] + k]/[D.sub.t]
= a + b ([D.sub.t]/[P.sub.t]

Horizon k
(years)      b    [sigma] (b)   [R.sub.2]

1           2.0      (1.1)        0.06
2           2.5      (2.1)        0.06
3           2.4      (2.1)        0.06
5           4.7      (2.4)        0.12

[R.sub.t [right arrow] + k] indicates the k-year return. Standard
errors in parentheses use GMM to correct for heteroskedasticity and
serial correlation. Sample 1947-1996.

Table 2.1 Table 1 from Cochrane [40]

Table I

Summary Statistics and Three-Factor Regressions for Simple
Monthly Percent Excess Returns on 25 Portfolios Formed on Size
and BE/ME: 7/63-12/93, 366 Months

[R.sub.f] is the one-month Treasury bill rate observed at the beginning
of the month (from CRSP). The explanatory returns [R.sub.M], SMB, and
HML are formed as follows. At the end of June of each year t (1963-
1993), NYSE, AMEX, and Nasdaq stocks are allocated to two groups (small
or big, S or B) based on whether their June market equity (ME, stock
price times shares outstanding) is below or above the median ME for
NYSE stocks. NYSE, AMEX, and Nasdaq stocks are allocated in an
independent sort to three book-to-market equity (BE/ME) groups (low,
medium, or high; L, M, or H) based on the breakpoints for the bottom
30 percent, middle 40 percent, and top 30 percent of the values of
BE/ME for NYSE stocks. Six size-BE/ME portfolios (S/L, S/M, S/H, B/L,
B/M, B/H) are defined as the intersections of the two ME and the three
BE/ME groups. Value-weight monthly returns on the portfolios are
calculated from July to the following June. SMB is the difference, each
month, between the average of the returns on the three small-stock
portfolios (S/L, S/M, and S/H) and the average of the returns on the
three big-stock portfolios (B/L, B/M, and B/H). HML is the difference
between the average of the returns on the two high-BE/ME portfolios
(S/H and B/H) and the average of the returns on the two low-BE/ME
portfolios (S/L and B/L). The 25 size-BE/ME portfolios are formed like
the six size-BE/ME portfolios used to construct SMB and HML, except
that quintile breakpoints for ME and BE/ME for NYSE stocks are used to
allocate NYSE, AMEX, and Nasdaq stocks to the portfolios.

BE is the COMPUSTAT book value of stockholders' equity, plus balance
sheet deferred taxes and investment tax credit (if available), minus
the book value of preferred stock. Depending on availability, we use
redemption, liquidation, or par value (in that order) to estimate the
book value of preferred stock. The BE/ME ratio used to form portfolios
in June of year t is then book common equity for the fiscal year ending
in calendar year t -1, divided by market equity at the end of December
of t - 1. We do not use negative BE firms, which are rare prior to
1980, when calculating the breakpoints for BE/ME or when forming the
size-BE/ME portfolios. Also, only firms with ordinary common equity (as
classified by CRSP) are included in the tests. This means that ADR's,
REIT's, and units of beneficial interest are excluded.

The market return [R.sub.M] is the value-weight return on all stocks in
the size-BE/ME portfolios, plus the negative BE stocks excluded from
the portfolios.

Book-to-Market Equity (BE/ME) Quintiles

Size     Low      2       3       4     High

Panel A: Summary Statistics

Means

Small    0.31    0.70    0.82    0.95    1.08
2        0.48    0.71    0.91    0.93    1.09
3        0.44    0.68    0.75    0.86    1.05
4        0.51    0.39    0.64    0.80    1.04
Big      0.37    0.39    0.36    0.58    0.71

Panel B: Regressions: [R.sub.i] - [R.sub.f]        = [a.sub.i] + [b.sub.i]([R.sub.M] - [R.sub.f]) +
[s.sub.l]SMB + [h.sub.i]HML + [e.sub.l],

a

Small   -0.45   -0.16   -0.05    0.04    0.02
2       -0.07   -0.04    0.09    0.07    0.03
3       -0.08    0.04    0.00    0.06    0.07
4        0.14   -0.19   -0.06    0.02    0.06
Big      0.20   -0.04   -0.10   -0.08   -0.14

b

Small    1.03    1.01    0.94    0.89    0.94
2        1.10    1.04    0.99    0.97    1.08
3        1.10    1.02    0.98    0.97    1.07
4        1.07    1.07    1.05    1.03    1.18
Big      0.96    1.02    0.98    0.99    1.07

s

Small    1.47    1.27    1.18    1.17    1.23
2        1.01    0.97    0.88    0.73    0.90
3        0.75    0.63    0.59    0.47    0.64
4        0.36    0.30    0.29    0.22    0.41
Big     -0.16   -0.13    0.25   -0.16   -0.03

h

Small   -0.27    0.10    0.25    0.37    0.63
2       -0.49    0.00    0.26    0.46    0.69
3       -0.39    0.03    0.32    0.49    0.68
4       -0.44    0.03    0.31    0.54    0.72
Big     -0.47    0.00    0.20    0.56    0.82

[R.sup.2]

Small    0.93    0.95    0.96    0.96    0.96
2        0.95    0.96    0.95    0.95    0.96
3        0.95    0.94    0.93    0.93    0.92
4        0.94    0.92    0.91    0.88    0.89
Big      0.94    0.92    0.87    0.89    0.81

Book-to-Market Equity (BE/ME) Quintiles

Size     Low       2       3       4     High

Panel A: Summary Statistics

Standard Deviations

Small     7.67    6.74    6.14    5.85    6.14
2         7.13    6.25    5.71    5.23    5.94
3         6.52    5.53    5.11    4.79    5.48
4         5.86    5.28    4.97    4.81    5.67
Big       4.84    4.61    4.28    4.18    4.89

Panel B: Regressions: [R.sub.i] - [R.sub.f]                 = [a.sub.i] + [b.sub.i]([R.sub.M] - [R.sub.f]) +
[s.sub.l]SMB + [h.sub.i]HML + [e.sub.l],

t(a)

Small    -4.19   -2.04   -0.82    0.69    0.29
2        -0.80   -0.59    1.33    1.13    0.51
3        -1.07    0.47   -0.06    0.88    0.89
4         1.74   -2.43   -0.73    0.27    0.59
Big       3.14   -0.52   -1.23   -1.07   -1.17

t(b)

Small    39.10   50.89   59.93   58.47   57.71
2        52.94   61.14   58.17   62.97   65.58
3        57.08   55.49   53.11   55.96   52.37
4        54.77   54.48   51.79   45.76   46.27
Big      60.25   57.77   47.03   53.25   37.18

t(s)

Small    39.01   44.48   52.26   53.82   52.65
2        34.10   39.94   36.19   32.92   38.17
3        27.09   24.13   22.37   18.97   22.01
4        12.87   10.64   10.17    6.82   11.26
Big      -6.97   -5.12   -8.45   -6.21   -0.77

t(h)

Small    -6.28    3.03    9.74   15.16   23.62
2       -14.66    0.34    9.21   18.14   25.59
3       -12.56    0.89   10.73   17.45   20.43
4       -13.98    0.97    9.45   14.70   17.34
Big     -18.23    0.18    6.04   18.71   17.57

s(e)

Small     1.97    1.49    1.18    1.13    1.22
2         1.55    1.27    1.28    1.16    1.23
3         1.44    1.37    1.38    1.30    1.52
4         1.46    1.47    1.51    1.69    1.91
Big       1.19    1.32    1.55    1.39    2.15

Table 2.2 (continued) Fama and French [65], Table 1, continued.
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