O ne of the fundamental tenets of modern finance has been that expected stock returns are determined by their corresponding level of systematic risk, or the beta factor. For more than 30 years, studies have found support for a positive and linear relation between return and risk known as the capital asset pricing model, or CAPM. Since the early 1980s, however, a growing number of studies have documented the presence of persistent patterns in stock returns that do not support CAPM.
The evidence suggests that betas of common stocks do not adequately explain cross- sectional differences in stock returns. Instead, other variables with no basis in current theoretical models seem to have a more significant predictive ability than beta. These
other variables include: company size, as measured by market capitalization of common stock; the ratio of book-to-market value: the accounting value of a company’s equity divided by its market capitalization; earnings yield: a company’s reported accounting net profits divided by price per share; a company’s prior return performance.
Many interpret the evidence as providing convincing support of market inefficiency: if stock returns can be predicted on the basis of historical factors such as market capitalization, book-to-market value and prior return performance, then it is difficult to characterize stock markets as informationally efficient. On the other hand, the rejection may be due to a test design based on an incorrect equilibrium model. The fact that so many of these regularities have persisted for more than 30 years suggests that our benchmark models may provide incomplete descriptions of equilibrium price formation.
An alternative explanation is that the failure of the evidence to provide unambiguous support for the CAPM is not necessarily proof of the model’s invalidity but may simply reflect our inability to measure beta risk accurately. For example, one can argue that stocks with higher ratios of book-to-market value have higher average returns than stocks with lower ratios of book-to-market because they are riskier in a beta sense. If we could measure beta risk with less error, then the reported positive relation between book-to-market and size and beta-risk-adjusted returns may disappear.
There are additional empirical puzzles. Not only does the evidence indicate that future stock returns are related to factors such as size, book-to-market value and prior return performance, but these relations are usually more significant (and often only significant) during January than during the other 11 months of the year. The presence of this January effect raises a further challenge to current financial theory. Why would a return-generating factor (risk or any thing else) manifest itself during one month?
The central prominence of beta in the asset pricing paradigm came into question with the first tests of other alternatives to the CAPM in the late 1970s. The earliest of these tests found that the price-to-earnings ratio (P/E) and the market capitalization of common equity (company size) provided considerably more explanatory power than beta. Other studies have extended the list of predictive factors to include, among others, the ratio of book-to-market value, price per share and prior return performance. Combined, these studies have produced convincing evidence of cross-sectional return predictability that vitiates the marginal explanatory power of beta found in the earlier studies. Notably absent in these studies, however, is any supporting theory to justify the choice of factors. Nevertheless, the findings collectively represent a challenge for alternative asset pricing models.
Much of the research on cross-sectional predictability of stock returns has focused on the relation between returns and the market value of common equity, commonly referred to as the size effect.
The first set of columns in Figure 1 reports the average monthly returns for ten value-weighted portfolios of New York Stock Exchange (NYSE) and American Stock Exchange (Amex) stocks for the period April 1962 to December 1994, along with corresponding values for portfolio beta and average market capitalization of the stocks
|
Fig.1 Monthly |
percentage returns |
HH |
ti/is |
_ |
A KHH |
. |
|||||||||
|
Size (Market Capital) |
Earnings-to-price Cash flow-to-price ratio ratio** |
Price-to-book ratio |
Prior return |
||||||||||||
|
Porfolio* |
Size ($m) |
Return (%) |
Beta |
E/P |
Return (%) |
Beta CF/P |
Return (%) |
Beta |
P/B |
Return (%) |
Beta |
Prior Return return (%) (%) |
Beta |
||
|
1 |
10 |
1.56 (0.37) |
1.11 |
19.39 |
1.21 (0.27) |
1.01 |
52.08 |
1.47 (0.33) |
1.00 |
0.57 |
1.43 (0.28) |
1.04 |
53.1 |
1.18 (0.29) |
1.13 |
|
2 |
26 |
1.41 (0.34) |
1.14 |
12.88 |
1.25 (0.23) |
0.93 |
27.75 |
1.32 (0.29) |
0.90 |
0.84 |
1.42 (0.25) |
0.97 |
24.9 |
1.24 (0.26) |
1.05 |
|
3 |
48 |
1.25 (0.31) |
1.10 |
11.26 |
1.08 (0.23) |
0.88 |
23.02 |
1.17 (0.29) |
0.91 |
1.02 |
1.06 (0.23) |
0.92 |
16.7 |
1.09 (0.24) |
1.02 |
|
4 |
83 |
1.23 (0.31) |
1.15 |
10.09 |
1.02 (0.23) |
0.95 |
19.91 |
0.94 (0.31) |
0.99 |
1.18 |
1.05 (0.21) |
0.84 |
11.2 |
1.03 (0.23) |
0.02 |
|
5 |
104 |
1.22 (0.28) |
1.10 |
9.08 |
0.96 (0.23) |
0.94 |
17.37 |
1.14 (0.31) |
1.01 |
1.35 |
1.00 (0.22) |
0.90 |
6.5 |
0.88 (0.22) |
0.96 |
|
6 |
239 |
1.12 (0.26) |
1.04 |
8.14 |
0.77 (0.23) |
0.99 |
15.05 |
0.87 (0.30) |
0.99 |
1.56 |
0.79 (0.22) |
0.91 |
2.3 |
0.91 (0.22) |
0.93 |
|
7 |
402 |
1.09 (0.25) |
1.06 |
7.19 |
0.83 (0.22) |
0.96 |
12.96 |
1.12 (0.30) |
1.03 |
1.86 |
0.84 (0.23) |
0.98 |
-1.9 |
0.85 (0.23) |
0.95 |
|
8 |
715 |
. i (0.24) |
1.05. |
6.13 |
0.89 (0.24) |
1.04 |
10.85 |
1.05 (0.32) |
1.08 |
2.30 |
0.91 (0.24) |
1.03 |
-6.6 |
0.92 (0.24) |
0.96 |
|
9 |
1,341 |
1.03 (0.23) |
1.03 |
4.78 |
0.88 (0.25) |
1.06 |
8.40 |
0.89 (0.31) |
1.06 |
3.10 |
0.82 (0.25) |
1.11 |
-13.1 |
0.62 (0.27) |
1.05 |
|
10 |
5,820 |
0.83 (0.21) |
0.95 |
2.49 |
0.82 (0.26) |
1.08 |
4.77 |
0.80 (0.33) |
1.07 |
10.00 |
0.90 (0.25) |
10.00 |
-29.6 |
0.83 (0.28) |
1.15 |
Table caption* Portfolio 1 (portfoliolO) is the portfolio with the smallest (largest) market capitalization, highest (lowest) E/P and CF/P, lowest (highest) P/B, and largest (smallest) 6-month prior returns.
Table caption** All results for the cash flow-to-price portfolios are the period April 1972-December 1994.
in the portfolio. Note that the portfolio betas decline with increasing size though the differences are small.
The portfolio evidence from international equity markets is summarized in Figure 2 for the stock markets of Australia, New Zealand, Canada, Mexico, Japan, Korea, Singapore, Taiwan and eight European countries. The monthly size-premium is defined as the difference between the average monthly return on the portfolio of smallest stocks and the average monthly return on the portfolio of largest stocks.
In all countries, except Korea, the size premium is positive during the sample period. As expected, its magnitude varies significantly across markets. It is most pronounced in Australia and Mexico and least significant in Canada and the UK. As is the case for US data, differences in beta across size portfolios cannot explain differences in returns. (Note that the data in Figure 2 are likely to be sensitive to differences in sample dates and lengths).
There are, however, significant differences across the 15 markets in the spread between the size of the largest and smallest portfolios as indicated by the ratios of the average market capitalization of the largest portfolio to that of the smallest one, reported in Figure 2. There does not seem to be a relation between the magnitude of the size premium and the size ratio.
Earnings-related strategies have long been popular in the investment community. The most frequently used of these strategies, which calls for buying stocks that sell at low multiples of earnings, can be traced back at least to the pioneering work of Benjamin Graham and David Dodd.
In the second set of columns in Figure 1 we find that the portfolio returns confirm the E/P effect documented in previous studies. There is, however, less evidence of an
Fig.2 The size effect: international e
Size premium
Australia
Belgium
Canada
Finland
France
Germany
Ireland
m
Korea Mexico New Zealand Singapore Spain Switzerland Taiwan UK US
I
|
Monthly size |
Test |
Number |
Largest* size/ |
|
premium (%) |
period |
of portfolios |
smallest size |
|
1.21 |
1958-81 |
10 |
NA |
|
0.51 |
1969-83 |
5 |
188 |
|
0.44 |
1973-80 |
5 |
67 |
|
0.76 |
1970-81 |
10 |
133 |
|
0.90 |
1977-88 |
5 |
NA |
|
0.49 |
1954-90 |
9 |
NA |
|
0.47 |
1977-86 |
5 |
NA |
|
1.20 |
1965-87 |
10 |
NA |
|
-0.40 |
1984-88 |
10 |
62 |
|
| 4.16 |
1982-87 |
6 |
37 |
|
0.51 |
1977-84 |
5 |
60 |
|
0.41 |
1975-85 |
3 |
23 |
|
0.56 |
1963-82 |
10 |
228 |
|
0.52 |
1973-88 |
6 |
99 |
|
0.57 |
1979-86 |
5 |
17 |
|
0.61 |
1973-92 |
10 |
182 |
|
0.61 |
1951-94 |
10 |
490 |
-0.5
0
0.5
1.0
4.0
* ratio based on average,market values (median in Singapore) over sample period, except for United Kingdom where it is calculated in 1975 and Finland in 1970. NA = not available.
E/P effect in markets outside the US. This is partly due to a lack of computerized accounting databases available for academic research. The evidence is also more varied than that for the size effect. The evidence from six markets outside the US indicates that in the UK, Japan, Singapore and Taiwan there is a significant E/P effect similar to that found in the US market. There is no evidence, however, of a significant E/P effect in New Zealand and Korea. Given the small size and relatively short sample period for the cases of Taiwan, New Zealand and Korea, it is difficult to draw definitive conclusions from the evidence regarding these three markets.
One alternative to the E/P ratio is the ratio of cash flow to price (CF/P), where cash flow is defined as reported accounting earnings plus depreciation. Its appeal lies in the fact that accounting earnings may be a misleading and biased estimate of the economic earnings with which shareholders are concerned. There is evidence of a CF/P effect in the US and Japan. The US evidence is summarized in the third set of columns in Figure 1, which report average returns and other portfolio characteristics for 10 decile portfolios based on annual rankings of NYSE and Amex securities on the ratio of cash flow per share to price per share for the period 1972-94.
The ratio of price-per-share to book-value-per-share (P/B) has received considerable attention recently for its significant predictive power. As is the case for the other
variables we have discussed, there is no theoretical model that says why P/B should be able to explain the cross-sectional behavior of stock returns. However, investment analysts have long argued that the magnitude of the deviation of current (market) price from book price per share is an important indicator of expected returns.
A succession of studies have documented a significant inverse relation between P/B and stock returns. To provide some perspective on the magnitude of the P/B effect, the fourth set of columns in Figure 1 reports average monthly returns and other portfolio characteristics for 10 decile portfolios drawn from the same data we used to examine the size, E/P and CF/P effects in the US market. The average monthly returns in Figure 1 indicate a significant negative relation between P/B and returns.
There is some evidence of a P/B effect outside the US. A P/B effect has been documented for stocks trading on the Tokyo Stock Exchange, the London Stock Exchange and also on stock exchanges in France, Germany and Switzerland. The reported magnitude of the P/B effect in these markets is smaller than that observed in the US and only marginally significant.
Finally, there is evidence that the prior performance of stock returns can explain the cross-sectional behavior of common stock returns. The literature documents two seemingly unrelated phenomena. The first is the existence of return reversals (past ‘losers’ become ‘winners’ and vice versa) over both long-term horizons (three to five years) as well as very short-term periods (a month and shorter). The second is the presence of an opposite effect over horizons of intermediate lengths: when prior returns are measured over periods of six to 12 months, ‘losers’ and ‘winners’ retain their characteristic over subsequent periods. There is, in this case, return momentum rather than reversal.
The last set of columns in Figure 1 provides evidence of return momentum in our sample (we do not examine reversal strategies). We measure prior returns over the six months prior to the portfolio formation month, and then hold the portfolio over the next 12 months. Consistent with previous research, portfolios with the highest prior returns (the winners) earn, on average, higher subsequent returns. Also, portfolios with the lowest prior returns (the losers) earn, on average, the lowest subsequent returns.
The difference in return between the interpreted as risk premiums. Under the hypothesis that the variables we have dis-cussed are proxies for separate risk ‘factors’, then the premiums should be uncorrelated across variables. Figure 3 reports the pairwise correlations between the monthly premiums. Inconsistent with the hypothesis, all of the correlations are large (in absolute value) and are significantly different from zero.
extreme portfolios in Figure 1 can be loosely
Fig.3 Correlations
Table captionApril 1962 - December 1994
|
Earnings/ Price |
Cash flow/ Price |
Price/ Book |
Prior Return |
|
|
Size |
0.265 |
0.444 |
0.472 |
-0.017 |
|
Earnings/Price |
0.727 |
0.590 |
-0.230 |
|
|
Cash Flow/Price |
0.760 |
-0.212 |
||
|
Price/Book |
-0.172 |
Beta, size and price/book: three risk measures or one?
Interestingly, the prior return premiums are negatively correlated with the premiums associated with the other variables, suggesting that prior return captures a characteristic of stock returns that is quite different from the other variables. Otherwise, the significant correlations indicate a high degree of commonality among the effects. The significant correlation of the premiums partly reflects the fact that these effects are most pronounced in January. Specifically, the average premiums during January tend to be positive and are usually significantly larger than the average premiums measured during the rest of the year. Internationally, most of the evidence on January seasonality has been related to the size premiums. A significant January seasonal has been reported in Belgium, Finland, Taiwan and Japan. Countries in which the January size premium is insignificant include France, Germany and the UK. Studies in Japan report a significant January seasonal for E/P, P/B, and CF/P in Japan but no January seasonal for size.
The evidence above suggests a great deal of commonality among the various effects. The consensus from research is that the relation between market capitalization and average returns is robust. Variables such as E/P, P/B, and prior return seem to provide additional explanatory power for cross-sectional differences in average returns beyond the influence of size, although the
evidence on E/P has recently been argued to be weak. Correspondingly, we compare the interaction between size, the P/B effect and the 6-month prior-return performance, using our sample of NYSE and Amex stocks to compute size-adjusted returns for portfolios created on the basis of both P/B and prior returns.
The 25 portfolios in Figure 4 are constructed as follows: first, for each stock in our sample we compute a size-adjusted return in which the influence of size is removed. Then we divide the sample into five groups of stocks based on P/B, and divide these groups further into five additional subgroups based on prior return. The stocks in each of the 25 groups are value-weighted to form a portfolio that is held for the following 12 months. Figure 4 reports average returns for the 25 portfolios separately for January and for the rest of the year.
First consider the P/B effect, which can be detected by reading down any column (within which
iff*
Table captionNYSE and AMEX stocks ranked first by price-to-book ratio (P/B) and then by prior return
|
Prior Return |
|||||
|
A. January |
Lowest |
2 |
3 |
4 |
Highest |
|
Low P/B |
3.71 |
3.13 |
2.13 |
2.15 |
1.34 |
|
(1.32) |
(0.69) |
0.76 |
0.53 |
0.55 |
|
|
2 |
0.96 |
1.28 |
1.41 |
1.10 |
0.38 |
|
(0.57) |
(0.50) |
(-0.53) |
(0.54) |
(0.52) |
|
|
3 |
0.35 |
0.66 |
0.05 |
-0.16 |
-1.07 |
|
(0.63) |
(0.71) |
(0.36) |
(0.44) |
(0.52) |
|
|
4 |
0.92 |
-0.36 |
-0.03 |
-0.92 |
-0.83 |
|
(0.53) |
(0.56) |
(0.40) |
(0.47) |
(0.74) |
|
|
High P/B |
-0.08 |
-0.06 |
-1.03 |
-1.02 |
-0.84 |
|
(0.57) B. February-December |
(0.44) |
(0.45) |
(0.55) |
(0.55) |
|
|
Low P/B |
-0.16 |
0.10 |
0.10 |
0.33 |
0.36 |
|
(0.20) |
(0.15) |
(0.15) |
(0.13) |
(0.16) |
|
|
2 |
-0.21 |
-0.06 |
-0.11 |
0.04 |
0.39 |
|
(0.15) |
(0.15) |
(0.13) |
(0.12) |
(0.15) |
|
|
3 |
-0.16 |
-0.16 |
-0.09 |
0.10 |
0.27 |
|
(0.15) |
(0.13) |
(0.11) |
(0.11) |
(0.14) |
|
|
4 |
-0.36 |
-0.28 |
-0.08 |
0.03 |
0.42 |
|
(0.14) |
(-0.12) |
(-0.10) |
(0.12) |
(0.15) |
|
|
High P/B |
-0.43 |
-0.09 |
0.21 |
0.40 |
0.10 |
|
(0.15) |
(0.12) |
(0.11) |
(0.13) |
(0.17) |
Table captionThe size-adjusted monthly return for a security is defined as the return for that security minus the monthly portfolio return for the size docile in which the security is a member. P/B and prior return portfolios in the table are value-weighted combinations of these monthly size-adjusted returns. All portfolios are formed on March 31 of each year using year- end accounting values and March 31 market prices. Stocks with negative P/B values are excluded from the sample.
the influence of prior return is held constant). The relation between returns and P/B in January (Panel A) is significant in every column. In February-December, though, the relation between P/B and returns is flat. Thus, after controlling for size and momentum effects, the P/B effect is evident in the data primarily during the month of January.
The story is quite different for the momentum effect, which can be detected by reading across any row (within which the influence of P/B is held constant). After controlling for both size and P/B, the relation between prior and subsequent returns during February - December has the same significant positive relation as noted previously. In January, though, the ‘momentum’ effect has the appearance of a reversal effect in that subsequent returns increase as prior returns decrease. That is, stocks that have recently declined (and therefore more likely candidates for trading at the end of the year based on taxes or window dressing) have the largest returns in January, particularly if they are low price/book (or low price) stocks. Thus, in January, it appears that end-of-year trading patterns tend to offset the momentum effect. However, unlike the other effects, the momentum effect persists throughout the rest of the year.
Many have argued that a multidimensional model of risk and return is necessary to explain the cross section of stock returns. That is, beta by itself is insufficient to characterize the risks of common stocks. Prominent in this category are Eugene Fama and Kenneth French, who suggest a three-factor equity-pricing model to replace the CAPM. Their three-factor model adds two empirically-determined explanatory factors: size (market capitalization) and financial distress (B/M). Others propose an additional factor, prior return performance.
Our findings suggest that such conclusions may be premature. Aside from the absence of a theory that says why such variables have a place in the risk-return paradigm, the evidence strongly indicates that the statistical relation between returns and variables like size and B/M derives primarily from the month of January. It is difficult to tell an asset-pricing story where risk manifests itself only during one month. An exception is momentum, where the influence on returns is spread more evenly throughout the year. This latter finding is difficult to reconcile with current asset pricing models and/or an informationally efficient market.
These points notwithstanding, one of the most significant contributions of this entire line of research is that is has sharpened our focus on potential alternative sources of risk. On the other hand, there are good reasons that make it difficult to argue that the evidence constitutes proof that the CAPM is ‘wrong.’ For example, no one has yet conclusively shown that variables like size and P/B are not simply proxies for measurement error in betas. Are we certain that variation in ratios of P/B is not picking up variation in leverage that is not reflected in betas that are typically estimated with 60 months of prior - and arguably stale - prices? The book is not closed; more research is necessary to resolve these issues.
There is also the question of believability: is the evidence as robust as the sheer quantity of results would lead us to believe? First, there is the issue of data snooping - many of the papers we have cited were predicated on previous research that documented the same findings with the same data. Degrees of freedom are lost at each turn and several authors have warned about adjusting tests of significance for these.
Also, the existence of these patterns in our experiments does not necessarily imply that they exist in the returns of implementable portfolios - returns net of transactions costs - for example market illiquidity and transactions costs may render a small stock strategy infeasible.
Finally, the persistence of these effects for nearly 100 years does not guarantee their persistence in the future. How many years of data are necessary to construct powerful tests? Research over the next 100 years will, we hope, settle many of these issues.
The article ‘An APT alternative to assessing risk’ highlighted one alternative to the Capital Asset Pricing Model. But as Gabriel Hawawini and Donald Keim explain in this article there are many cross-sectional patterns in stock returns - variables such as company size, the ratio of book-to-market value and a company’s prior return performance - that do not support the CAPM. The authors argue that the multidimensional model lacks a convincing theoretical underpinning and that the evidence which indicates a statistical relationship between certain variables and stock returns derives primarily - there is one important exception - from the month of January. It is difficult on this basis to argue that the CAPM is ‘wrong’ but at least the recent line of research has sharpened our focus on potential alternative sources of risk.