Common risk factors in the returns on stocks and bonds-Fama&

Common risk factors in the returns on

本文论证了股票和债券的回报率五种常见的危险因素。有三个 是股市的因素：整体市场因素，相关企业规模和账面市场因素。有两个债券市场的因素，市场成熟度和违约风险。股票收益有相同的变化，同时它们通过股票市场因素变化而变化。除了小型企业，债券市场收益共同变化。最重要的。五大因素似乎也解释了股票和债券的平均收益。 1. 简介

在美国普通股平均回报率的横截面显示相关的市场与夏普和林特纳的资本资产定价模型或布莱登的消费资本资产定价模型（CCAPM）等并无关系。【如： Reinganum(198 1)和布来登,吉本斯,Litzenberger(1989)】另一方面,资产定价理论没有特别变量来解释横截面的平均回报率。实证确定平均回报的列表变量包括规模(ME市值),财务杠杆，市盈率(E / P)和 账面值市值比【参见Banz(1981)。班达里(1988)。巴苏(1983)。罗森博格,里德Lansteinof Breeden】

法马-佛伦奇(1992年)在研究市场β市值、市盈率,财务杠杆,账面值对市值股票对股票平均回报率横截面的联合作用,。他们发现变量,β（回归股票市场回报率)无论单独使用或与其他的变量共同作用都不会影响股票平均水平的回报率。单独使用时，规模、E / P,财务杠杆和账面值市值比就有了话语权。而与其他组合使用时，市值大小(ME)和账面市值比(BE\\ME)似乎代替了财务杠杆和市盈率对股票平均水

平回报率的影响。最根本的结果是, 两个实证变量——市值大小和账面值对市值股票——解释了1963-1990期间纽约证交所、美国证交所和纳斯达克股市平均回报率的横截面。

本文扩展了法马-佛伦奇测试资产定价的三个方式。

（a）我们加深了对资产收益解释。法马-佛伦奇认为唯一是资产普通股。那么在集成市场,单一模型也应该解释债券的回报率。此次还包括了对美国政府和公司债券及股票的测试。

（b）同时我们通过扩大组变量来解释回报率。法马-佛伦奇理论中提及的市值和账面值对市值是针对股市而言的。所以我们增加了在债券市场易用的结构术语。我们的目标是检验对债券回收率重要的变量是否可以解释股票收益,反之亦然。依据的理论是,如果市场集中, 那么在债券和股票收益的过程中很可能有一些重叠之处。

(c) 也许最重要的是,测试资产定价模型的方法不同。法马-佛伦奇(1992)使用法马和麦克白提出的横截面回归法(1973):用股票收益横截面的回归变量假设来解释平均回报率。当解释像市值和账面值对市值股票对政府和公司债券没有明显意义时，它很难通过添加横截面回归债券以来解释变量。

本文运用了黑人詹森-斯科尔斯(1972)的时间序列回归方法。每月的股票和债券回报率回归于在收益中的股票市场投资组合,模拟投资组合规模，账面值对市值股票，期限结构风险因素时间序列回归载荷的不是市值或BE\\ME而是事实，有一个对债券或股票的风险因素敏感性明确的解释。

时间序列回归也方便学习两个重要的资产定价问题。

(a) 我们主要议题包括,资产定价合理时, 像账面值对市值股票、市值这样的相关平均收益变量必须可以应对法马-佛伦奇提及的常见敏感危险因素（共享,因此相似) 。在这个问题上的时间序列回归提供了直接的证据。尤其是，在与其他因素无关时，斜坡和R价值理论表明，无论模仿组合风险因素是否与市值大小或BE/ME相关，股票和债券收益有着相同的变量。

The r-statistics are in parentheses below the slopes; the R? is 0.38. This regression demonstrates that the market return is a hodgepodge of the common factors in returns. The strong slopes on TER,LI and DEF produced by RM-RF (the excess return on a proxy for the portfolio of stock-market wealth) are clear evidence that the two term-structure factors capture common variation in stock returns.

The sum of the intercept and the residuals in (l), call it RMO, is a zero-investment portfolio return that is uncorrelated with the four explanatory variables in (I). We can use RR/IO as an orthogonalized market factor that captures common variation in returns left by SR;IB, HML, TERM, and DEF. Since the stock-market returns, S&fB and HML, are largely uncorrelated with the bond-market returns, TERM and DEF (table 2). five-factor regres-sions that use R,ClO, SMB, HML, TERM. and DEF to explain bond and stock returns will provide a clean picture of the separate roles of bond- and stock-market factors in bond and stock

returns. The regressions are in table 8.

The story for the common variation in bond returns in table 8b is like that in table 7b. The bond-market factors, TER.Ll and DEF, have strong roles in bond returns. Some bond portfolios produce slopes on the stock-market factors that are more than two standard errors from 0. But this is mostly because TERM and DEF produce high R? values in the bond regressions, so trivial slopes can be reliably different from 0. As in table 7b. only the low-grade bond portfolio (LG) produces nontrivial slopes on the stock-market factors. Otherwise, the stock-market factors don?t add much to the shared variation in bond returns captured by TERM and DEF.

For the stock portfolios, the slopes on R,CIO in the five-factor regressions of table 8a are identical (by construction) to the large slopes on RM-RF in table 7a. The slopes on the size and book-to-market returns in table 8a shift somewhat (up for S;LIB, down for HXIL) relative to the slopes in table 7a. But the spreads Table 7a

Kcgressions of L?XCCSS stock returns on 25 stock portfolios formed on size and book-to-market equity (in percent) on the stock-market returns, Rhf -RF, Shff3, and IIML. and the bond-market returns, 7?EHM and DEF: July 1963 to December 1991, 342 months.”

R(r) -RF(r) = LI + b[Rhl(r) -RF(rj] + sSMB(r) + hIfML(r) + mTERM(r) + dDEF(t) + e(r)

Book-to-market equity (BE/ME) quintiles Size quintile Low 2 3 4 High Low 2 3 4 High h l(h) Small I.06 1.04 0.96 0.92 0.98 35.97 47.65 54.46 54.51 53.15

2 1.12 I .06 O.YX 0.94 I.10 47.1?) 54.95 49.0 I 54.19 5Y.W 3 1.13 I.01 0.Y7 0.95 I .0x 50.93 46.95 44.57 47.59 46.92 4 I .07 I .07 I.01 1.00 I.17 48.18 47.55 44.83 41.02 41.02 Iiig O.Y6 I .02 O.YX I so I.10 53.x7 51.01 41.35 4X.2Y 35.96 s r(s) Small 1.45 I .26 1.20 I.15 I.21 37.02 43.42 50.89 51.36 49.55 2 I .Ol O.YX 0.x?) 0.74 0.x?) 32.06 3x. IO 33.6X 32.12 35.7?)

3 0.76 0.66 0.00 0.4Y 0.6X 25.X2 22.Y7 20.x3 IX.54 22.32 4 0.3x 0.34 0.30 0.26 0.42 12.71 I I .36 Y.YY x.05 I I .07 Big -------1.44

17 -0.11 0.23 0. I7 0.06 -7.03 4.07 7.3 I 6.07 /I r(h)

Smdl -0.27 0.10 0.27 0.40 0.63 -5.Y5 2.90 9.82 15.47 22.27 2 -0.51 0.02 0.25 0.44 0.71 -14.01 0.69 8.1 I 16.50 24.61 3 -0.37 ~ 0.00 0.31 0.50 0.69 -IO.81 -0.1 I 9.28 16.18 19.34 4 -0.42 0.04 0.29 0.53 0.75 -12.09 I.10 8.37 14.20 16.88 Big -0.46 0.01 0.21 0.58 0.78 -16.85 0.38 5.70 18.16 16.59

E.F. Famo and K.R. French, Common risk factors in stock and bond relurns zswc,* ddddd I I I I I I I I J.F E.-B Table 7b

Regresstons of cwess stock returns on gobernment and corporate bonds tin percent) on the

stock-market returns. R.Lf-RF. S.\\fB. and H.UL. and the bond-market returns. T&R.\\! and DEf:

July 1963 to December 1991. 342 months.”

R(t) -RF(r) = u + h[R.U(o -RF(r)] + rS.CfB(rl i hH.lfL(O + mTER.U(rt + dDEF(fI + e(t) Bond portfoIl”

I-Xi 6-1OG Aaa .Aa A Baa LG h -0.02 -0.04 -0.01 0.00 0.00 0.02 0.18 C(h) 2.8-I -3 I4 2.96 0.06 I .05 I .99 7.39 0.00 -0.02

-0.0? -0.01 0.00 0.05 0.08 0.30 -I.12 2.28 2.42 0.40 3.20 2.34 0.00 -0.02 -0.02 -0.00 0.00 0.04 0.12

-

0.44 -1.29 2.46 -0.40 0.90 2.39 3.13 0.47 0.75 1.03 0.99 1.00 0.99 0.64

30.0 I 36 8-t 93.30 117.30 124.19 50.50 14.25 0.27 0.32 0.97 0.97 I .02 1.05 0.80 9.57 x.77 49.X 65.04 71.51 30.33 9.92 0.80 0.87 0.97 0.98 0.98 0.9 I 0.58 0.56 0.73 0.40 0.30 0.29 0.70 1.63

“R.M is the value-weighted monthly percent return on all stocks in the 25 size-BE ME portfolios. RF the negative-BE stocks excluded from the portfolios. RF is the one-month Treasury bill rate. observed at the beginning of month. RMB (small minus big) is the difference each month between the simple average of the returns on the three small-stock portfolios (S L. S .Lf. and SH) and the sample average of the returns on the three big-stock portfolios (FL. B .\\I. and B H). H.ML (high minus low)is the difference each month between the simple average of the returns on the two high-BE .ME portfohos (S H and 6 H) and the average of the returns on the two lo&-BE ME portfolios (5 L and B I!.). TERM is LTG-RF, where LTG is the long-term government bond return. DEF is CB-LTG. where CB is the return on a proxy for the market portfolio of corporate bonds.

The seven bond portfolios used as dependent variables in the excess-return regressions are 1-to 5-year and 6- to 10-Year governments ( I-5G and 6-10GJ)and corporate bonds rated Aaa, Aa. A, Baa.

and below Baa (LGJ by Moody?s. The 25 size-BE\\ME stock portfolios are formed as follows. Each year from 1963 to 1991 NYSE quintile breakpoints for size (ME, stock price times shares outstanding). measured at the end of June, are used to allocate NYSE. Amex. and NASDAQ stocks to tice size quintiles. Similarly. NYSE quintile breakpoints for BE \\ME are used to allocate NYSE. Amex. and NASDAQ stocks to five book-to-market quintiles. In BE .\\ME. BE is book common equity for the fiscal year ending in calendar year t -1. and ,ME is for the end of December of r -I. The 25 size-BE \\ME portfolios are the intersections of the five size and the fire BE \\ME groups. Value-weighted monthly percent returns on the portfolios are calculated from July of year f to June ofr+ I. R? and the residual standard, error, s(r). are adjusted for degrees of freedom. in the .SMB and HML slopes across the stock portfolios in table 8a are like those in table 7a, and .SMB and HML again capture strong shared variation in stock returns.

What changes dramatically in the five-factor regressions of table 8, relative to table 7. are the slopes on the term-structure factors for stocks. The slopes on TERM are more than 14 standard errors from 0; the DEF

slopes are more than seven standard errors from 0. The slopes on TERM and DEF for stocks are like those for bonds. Thus unlike table 7, the five-factor regressions in table 8 say that the term-structure factors capture strong common variation in stock and bond returns.

How do the tracks of the term-structure variables get buried in the five-factor regressions for stocks in table 7a? Table 8a says that stocks load strongly on RMO, TER.CI. and DEF, but there is little cross-sectional variation in the slopes on these factors. All the stock portfolios produce slopes on TER,V and DEF close to 0.81 and 0.79, the slopes produced by the excess market return in (I).

And the stock portfolios all produce slopes close to 1.0 on R.MO in table 8a, and thus on RM-RF in table 7a. Tables 7a and 8a then say that because there is little cross-sectional variation in the slopes on RJI-RF, RM-RF, TERM, and DEF, the excess market return in table 7a absorbs the common variation in stock returns associated with RMO. TERM and DEF. In short, the common variation in stock returns related to the term-structure factors is buried in the excess market return in table 7a. Is there any reason to prefer the five-factor regressions in table 8 over those in table 7? Only to show that, in addition to the three stock-market factors. there are two bond-market factors in stock returns. Otherwise, the two sets of regressions produce the same R? values and thus the same

estimates of the total common variation in returns. And the two sets of regressions produce the same intercepts for testing the implications of five-factor models for the cross-section of average stock returns. 5. The cross-section of average returns

The regression slopes and R? values in tables 3 to 8 establish that the Stock-market returns. SMB, HML, and RM-RF? (or R.MO), and the bond-market returns, TERM and DEF. proxy for risk factors. They capture common variation in bond and stock returns. Stock returns have shared variation related to three stock-market factors, and they are linked to bond returns through shared variation in two term-structure factors. We next test how well the average premiums for the five proxy risk factors explain the cross-section of average returns on bonds and stocks. The average-return tests center on the intercepts in the time-series regressions. The dependent variables in the regressions are excess returns. The explanatory variables are excess returns (R.WRF and TERXJ) or returns on zero-investment portfolios (RMO, SMB, HM L. and DEF). Suppose the explanatory returns have minimal variance due to firm-specific factors, so they are good mimicking returns for the underlying state variables or common risk factors of concern to investors. Then the multifactor asset-pricing models of Merton (1973) and Ross. Table 8a

Regressions ofexcess stock returns on 25 stock portfolios formed on size and book-to-market equity (in percent) on the stock-market returns, KMo, SMB, and ffML, and the bond-market relurns, TEKM and DEF: July 1963 IO December 1991, 342 months.?

K(r) -RF(f) = u + bRMO(r) + sSMB(r) + hIIML(r) + ruTERM(r) + dDEF(~) + r(f)

Book-to-market equity (BE/ME) quintiles Size quinlile Low 2 3 4 High Low 2 3 4 High b t(b)

Sn1all I .06 I.04 0.96 0.92 0.98 35.97 47.65 54.48 54.51 53.15 2 I.12 I .06 0.98 0.94 1.10 47.19 54.95 49.01 54.19 59.00 3 1.13 I.01 0.97 0.95 I.08 50.93 46.95 44.57 47.59 46.92 4 1.07 I .07 1.01 I.00 1.17 48.18 47.55 44.83 41.02 41.02 Big 0.96 I.02 0.98 1.00 1.10 53.87 51.01 41.35 48.29 35.96 s r(s)

SlIldl I.92 1.72 I .62 1.56 1.64 51.96 62.88 73.21 73.72 71.32 2 1.50 I.45 1.33 1.16 1.3x 50.66 59.80 53.02 53.20 58.79 3 I.26 I.11 I .03 0.9 1 I.16 45.37 40.94 37.83 36.47 40.24 4 0.85 0.8 I 0.75 0.70 0.94 30.49 28.84 26.42 23.02 26.22 Big 0.26 0.34 0.20 0.28 0.43 II.56 13.69 6.85 10.62 II.17 II t(h)

Sm:~ll -0.94 -0.56 -0.34 -0.18 0.01 -22.65 -18.19 -13.67 -7.49 0.57

2 1.22 -0.65 -0.37 0.15 0.0 I -36.52 -23.89 -13.09 -6.22 0.51 3 -1.0x -0.64 -0.30 -0.10 0.00 -34.6X -21.18 -9.x2 -3.61 0.16 4 -1.09 -0.64 -0.35 -0.10 0.00 -34.85 -20.12 -10.93 -2.83 0.10 Big -1.07 -0.63 -0.4 1 -0.05 0.09 -42.62 -22.46 -12.30 -1.75 2.06 Sn1all 0.75 0.73 0.73 0.71 0.73 15.66 20.60 25.32 25.67 24.24

2 0.85 0.82 0.86 0.89 0.84 22.08 25.96 26.40 31.68 27.57

3 0.88 0.84 0.84 0.86 0.88 24.21 23.85 23.73 26.34 23.52

4 0.85 0.87 0.90 0.98 0.94 23.24 23.77 24.35 24.76 20.11 Ijig 0.x0 0.79 0.79 0.77 0.73 27.M) 24.17 20.42 22.83 14.66 cl f@)Small 0.67 0.63 0.66 0.78 0.79 7.25 9.20 11.90 14.81 13.73 2 0.76 0.72 0.81 0.19 0.79 10.23 11.94 12.96 16.36 13.57 3 0.80 0.78 0.83 0.84 0.69 II.53 Il.64 12.25 13.53 9.63 4 0.74 0.74 0.84 0.91 0.80 10.56 10.48 11.88 12.01 X.98 nig 0.81 0.66 0.75 0.72 0.68 14.56 10.62 IO.15 II.04 7.15 HZ s(e)

Small 0.94 0.96 0.97 0.97 0.96 1.93 1.43 I.16 1.11 1.20

2 0.95 0.96 0.95 0.95 0.96 1.55 1.27 I.31 1.13 1.23

3 0.95 0.94 0.93 0.93 0.93 1.45 1.41 1.43 1.31 1.50

4 0.94 0.93 0.91 0.90 0.89 1.46 1.47 1.48 1.59 1.88 Big 0.94 0.92 0.87 0.90 0.83 1.17 1.31 1.55 1.36 2.00 \ Table 8b

Regressions of excess returns on government and corporate bonds (in percent1 on the stock-market returns. RMO, SMB, HM L. and DEF and the bond-market returns. TERM and DEF July 1963 to December 1991. 342 months.? Bond portfolio I-SG 66IOG .Aaa .-\\a .A Baa LG

h -0.03 -0.04 -0.02 0.00 0.00 0.0? 0.18 ribI -1.34 -3.14 2.96 0.06 I .05 I .99 7.39 5 -0.00 -0.03 -0.03 -0.01 0.00 0.06 0.16 r(s) -0.68 -2.30 -3.17 -2.55 0.80 1.09 5.09 /I 0.02 -0.00 -0.01 -0.00 0.00 0.03 0.00 f(lll 1.76 -0.00 -1.36 -0.17 0.52 1.72 0.12 V, 0.45 0.71 1.02 0.99 1.00 1.01 0.79

t(trll 32.09 39.55 102.65 130.93 139.1 I 57.34 19.56

Cl 0.15 0.29 0.95 0.97 I .02 1.07 0.94

t(ll) 9.16 8.25 50.04 67.05 74.00 31.77 I?.09

R2 0.130 0.87 0.97 0.9s 0.98 0.9 I 0.58 S(Z) 0.56 0.73 0.40 0.30 0.29 0.70 1.63

“RMO, the orthogonalized market return, is the sum of intercept and residuals from the repression of RM-RF on SMB HML TERM and DEF. RM is the value-weighted monthly percent return on all stocks in the 25 size-BE .RE portfolios. plus the negative-BE stocks excluded from the portfolios. RF is the one-month Treasury bill rate, observed at the beginning of the month. SMB(small minus big). the return on the mimicking portfolio for the common size factor in stock returns, is the difference each month between the simple aberage of the returns on the three small-stock portioitos (S L. S M and S H) and the simple average of the returns on the three big-stock portfoltos (BL. BM. and B H). HML(high minus low). the return on the mimicking portfolio for the common book-to-market equity factor m returns, is the difference each month between the simple average of the returns on the too high-BE .ME portfolios 1.5 H and B i-l) and the average of the returns on the two low-BE .ME portfoltos (S f. and B L). TERM is LTG-RF, where LX is the long-term government bond return. DEF IS CB-LTG, where CB is the return on a proxy for the market portfolto of corporate bonds.

The seven bond portfolios used as dependent variables m the excess-return regressions are 1-to5-year and 6-to10-year governments (1-5G and &6-10G) and bonds rated .Aaa. Aa. A. Baa, and belou Baa (LG, by Moody?s The 25 size-BE\\ME stock portfolios are formed as follows. Each year (from 1963 to 1991) NYSE quintile breakpoints for size (LIE. stock price times shares outstanding). measured at the end of June. are used to allocate SYSE. Amer. and NASD.A.Q stocks to fire size quintiles. NYSE quintile breakpoints for BE\\ME are also used to allocate NYSE. Amex and NASDAQ stocks to five-book-to-market equity quintiles. In BE\\ME.,BE is book common equity for the fiscal year ending in calendar year I -I. and .\\fE ts for the end of December of t -1. The 25 size-BE .ME portfolios are the intersections of the five size and the five BE .ME groups. Value-weighted monthly percent returns on the portfolios are calculated from July of year t to June of t+1.

R2 and the resdiual standard error. ste), are adjusted for degrees of freedom.

(1976) imply a simple test of whether the premiums associated with any set of explanatory returns suffice to describe the cross-section of average returns: the intercepts in the time-series regressions of excess returns on the mimicking portfolio returns should be indistinguishable from 0. Since the stock portfolios produce a wide range of average returns, we examine their intercepts first. We are especially interested in whether the

mimicking returns SmB and HML. absorb the size and book-to-market effects in average returns, illustrated in table 2. We then examine the intercepts for bonds. Here the issue is whether different factor models predict patterns in average returns that are rejected by the flat average bond returns in table 2.

5.1. The cross-section average stock returns

RM-RF -When the excess market return is the only explanatory variable in the time-series regressions, the intercepts for stocks (table 9a) show the size effect of Banz (1981). Except in the lowest-BE/ME quintile. the intercepts for the smallest-size portfolios exceed those for the biggest by 0.22% to 0.37% per month. The intercepts are also related to book-to-market equity. In every size quintile, the intercepts increase with BE/ME; the intercepts for the highest-BE/ME quintile exceed those for the lowest by 0.25% to 0.76% per month. These results parallel the evidence in Fama and French (1992a) that, used alone, market βs leave the cross-sectional variation in average stock returns that is related to size and book-to-market equity.

In fact, as in Fama and French (1992a), the simple relation between average return and β for the 25 stock portfolios used here is flat. A regression of average return on βyields a slope of -0.22 with a standard error of 0.31. The Sharpe (1964)-Lintner (1965) model (/I suffices to describe the cross-section of average returns and the simple relation

between β and average return is positive) fares no better here than in our earlier paper.

SMB and HML -The two-factor time-series regressions of excess stock returns on SMB and HML produce similar intercepts for the 25 stock portfolios (table 9a). The two-factor regression intercepts are, however, large (around 0.5% per month) and close to or more than two standard errors from 0. Intercepts that are similar in size support the conclusion from the cross-section regressions in Fama and French (1992a) that size and book-to-market factors explain the strong differences in average returns across stocks. But the large intercepts also say that S.LfB and HML. do not explain the average premium of stock returns over one-month bill returns.

RM-RF, SMB, and HML -Adding the excess market return to the time- series regressions pushes the strong positive intercepts for stocks observed in the two-factor (SMB and HML) regressions to values close to 0. Only three of the 25 intercepts in the three-factor regressions differ from 0 by more than 0.2 percent per month: 16 are within 0.1% of 0. Intercepts close to 0 say that the regressions that use R.M-RF. SMB. and HML to absorb common time-series variation in returns do a good job explaining the cross-section of average stock returns.‘This implication is only an approximation in the Ross (19761 model. Ser. for example. Shanken (1982).

Table 9a

intercepts from excess stuck return regressions for 25 stuck portfolios formed on size and book-to-market equity: July 1963 IO December IYYI, 342 months.”

Book-to-market equity (HE/ME) quintiles t, I(4 Size

quinlile Low 2 3 4 High Low 2 3 4 High

(i) K(r) -W?(r) = (I + OI~?ERM(I) + dDEF(f) + t$r) Small 0.31 0.62 0.71 0.80 0.92 0.75 1.73 2.20 2.61 2.87

2 0.35 0.63 0.77 0.75 0.93 0.93 I.91 2.60 2.85 3.03

3 0.34 0.58 0.60 0.73 0.89 I.00 I .9Y 2.28 3.01 3.1 I

4 0.4 1 0.27 0.4?) 0.69 0.96 1.34 1.01 1.96 2.X8 3.35 Uig 0.34 0.30 0.25 0.50 0.53 1.35 1.27 1.17 2.36 2.14 (ii) R(r) - RF(r) = u + ~[RM(I) -RF(r)] + ($0

sm;III -0.22 0.15 0.30 0.42 0.54 -O.YO 0.73 1.54 2.19 2.53 2 -0. IX 0.17 0.36 0.3?) 0.53 -1.00 I .05 2.35 2.7Y 3.01 3 -0. I6 0.1s 0.23 0.39 0.50 -1.12 1.25 1.n2 3.20 3.19

4 -0.05 -0.14 0.12 0.35 0.57 -0.50 -1.50 1.20 2.91 3.71 Big -0.04 -0.07 -0.07 0.20 0.21 -0.4Y -0.95 -0.70 1.89 1.41 (iii) X(r) -RF(I) = u + sSMtqr) + hlfML&) + t(r) SIll;Lll 0.24 0.46 0.49 0.53 0.55 0.97 I.92 2.24 2.52 2.49 2 0.52 0.58 0.64 0.58 0.64 2.00 2.40 2.76 2.61 2.56 3 0.52 0.61 0.52 0.60 0.66 2.00 2.58 2.25 2.66 2.61 4 0.69 0.39 0.50 0.62 0.79 2.78 1.55 2.07 2.51 2.85 Big 0.76 0.52 0.43 0.51 0.44 3.41 2.23 1.84 2.20 1.70

(iv) K(r) -W(r) = 0 + h[RM(r) -RF(r)] + sSM&I) + h/IML(f) + V(I) Smnll -0.34 -0.12 -0.05 0.01 0.00 -3.16 -1.47 -0.73 0.22 0.14 2 -0.11 -0.01 0.08 0.03 0.02 -1.24 -0.20 1.04 0.51 0.34 3 -0.1 I 0.04 -0.04 0.05 0.05 -1.42 0.47 -0.47 0.7 I 0.56 4 0.09 -0.22 -0.08 0.03 0.13 I.07 -2.65 -0.99 0.33 I .24 Big 0.21 -0.05 -0.13 -0.05 -0.16 3.27 -0.67 -1.46 0.69 -I.41

(v) H(l) -RF(f) = u + h[RM(r) -RF(r)] + sSMB(r) + hHML(r) + ntTERM(r) + dDEF(r) + &)

Sm;ill -0.35 -0. I3 -0.05 0.0 I 0.00 -3.24 -1.58 -0.79 0.20 0.09 2 -0.1 I -0.02 0.0x 0.04 0.02 ~ 1.29 -0.24 1.10 0.67 0.29 3 -0.12 0.04 -0.03 0.06 0.05 -1.45 0.48 -0.42 0.79 0.56 4 0.0X -0.22 -0.08 0.04 0. I3 1.04 -2.67 -0.94 0.47 1.23 Big 0.21 -0.05 -0.13 -0.06 -0.17 3.29 -0.72 -1.46 0.73 -I.51 ‘See footnote under table 9c.

Table 9b

portfolios. Jul> 1963 to December 1931. 343 months.? Bond portfolIo

1-K 6-IOG ?AU .A.l A B&I LG

II) R(r) -Rf(tl = tt 4 mTER.LIItI +

luil R(rl -RF(t) = LI + sS.tlB(t) + hH.LfLlt) + e(t) 0. I1 0. I6 0.07 0.07 0.0? 0.1 I 0.08 1.70 I.47 0.52 0.58 0.55 0.82 0.58

(1~) R(r) -RF(t) = LI + h[RM(t) -RF?ltl] + s.S.\\IBltl + ItH.VLltl + CICI 0.06 0.07 -0.07 -0.07 -0.08 -0.05 -0.1 I 0.8Y 0.62 -0.62 -0.64 -0.69 -0.41 -1.00

(\\I R~ri -RF(t) = (1 + h[R.\\l(t) -RFltl] t sS.\\/51t1 + hff.\\IL(t~ + ntTER.Lf(t) + riDEflt) + p(t) 0.09 0.1 I -o.Oil -0.00 -0.00 0.02 -0.07 2.8-l 1.77 -0.17 -0.25 -0.57 0.52 _ 0.77 ‘See footnote under table Yc.

There is an interesting story for the smaller intercepts obtained when the excess market return is added to the two-factor (SMB and HML) regressions. In the three-factor regressions. the stock portfolios produce slopes on R.&RF closed2 to I. The average market risk premium (0.33% per month) than absorbs the similar strong positive intercepts observed in the regressions of stock returns on .SMB and HML. In short, the size and book-to-market factors can Explain the differences in average returns across stocks. but the market factor is needed to explain why stock returns are on average above the one-month bill rate.

TERM and DEF -Table 9a shows that adding the term-structure factors, TER.11 and DEF. to the tim2-series regressions for stocks has almost no effect on the intercepts produced by the three stock-market factors. Likewise, in spite of the strong slopes on TERM and DEF when they are used alone to explain stock. Table 9c

F-statistics testing the intercepts in the excess-return regressions against 0 and matching probability levels of bootstrap and F-distributions.” Regression (from tables 9a and 9b) (i) (ii) (iii) (iv) (v)

F-statistic 2.09 !.91 1.75 1.56 1.66

Probability level

Bootstrap 0.998 0.996 0.985 0.95 1 0.971 F-distribution 0.999 0.996 0.990 0.96 I 0.975

“R.M is the value-weighted monthly percent return on all stocks in the 25 size-BE\\ME portfolios. plus the negative-BE stocks excluded from the 25 portfolios. RF 15 the one-month Treasury bill rate. observed at the beginning of the month. SMB(small minus big), the return on the mimicking portfolio for the common size factor in stock returns. is the difference each month between the simple average of the returns on the three small-stock portfolios (5 L, S?XI. and S H) and the simple average of the returns on the three big-stock portfolios IB L. B?.LI, and B H). H,ML (high minus low). the return on the mimicking portfolio for the common book-to-market equity factor in returns. is the difference each month between the simple average of the returns on the two high-BE .WE portfolios (S:H and B:H) and the average of the returns on the two low-BE\\ME portfolios (S L and B LI) TERM is LTG-RF. where LTG is the long-term government bond return. DEF is CB-LTG. where CB is the return on a proxy for the market portfoho of corporate bonds.

The seven bond portfolios used as dependent variables m the excess-return regressions are 1-to5-year and 6-to10-year governments (1-5G and &6-10G) and bonds rated .Aaa. Aa. A. Baa, and belou Baa (LG, by Moody?s The 25 size-BE\\ME stock portfolios are formed as follows.

Each year (from 1963 to 1991) NYSE quintile breakpoints for size (LIE. stock price times shares outstanding). measured at the end of June. are used to allocate SYSE. Amer. and NASD.A.Q stocks to fire size quintiles. NYSE quintile breakpoints for BE\\ME are also used to allocate NYSE. Amex and NASDAQ stocks to five-book-to-market equity quintiles. In BE\\ME.,BE is book common equity for the fiscal year ending in calendar year I -I. and .\\fE ts for the end of December of t -1. The 25 size-BE .ME portfolios are the intersections of the five size and the five BE .ME groups. Value-weighted monthly percent returns on the portfolios are calculated from July of year t to June of t+1.

Regressions (I~+v) in table 9c correspond to the regressions tn tables 9a and 9b. The F-statistic is

F = (.-l?_r-??t)(.v -K -f + I),(L*(.V -K)*W,,,).

Where N= 342 observations. L = 31 regressions, K is 1plus the number of explanatory variables in the regression. A is the (column) vector of the 32 regression intercepts. Z (L x L) is the unbiased covariance matrix of the residuals from the 32 regressions. and o,.r is the diagonal element of (.Y?X)-r corresponding to the intercept. Gibbons, Ross. and Shanken (19891 show that this statistic has an F-distribution with L and N -K -f. i I degrees of freedom under the assumption that the returns and explanatory variables are normal and the true intercepts are 0.

In the bootstrap simulations, the slopes (with intercepts set to 0).explanatory variables. and residuals from the regressions for July 1963 to December 1991 in tables 3 to 7 are used to generate 342 monthly excess returns for the 25 stock and seven bond portfolios for each regression model. These model returns and the exclamatory returns. RM-RF. SMB. HML, TEM. and DEF, for July 1963 to December 1991, are the population for the simulations. Each simulation takes a random sample. with replacement. of 342 paired observations (the same set of observations for each of the five regression models) on the model returns and the explanatory variables. and estimates the regressions. For each model the table shows the proportion of 10.000 simulations rn which the F-Xdttstic is smaller than the empirical estimate. The table also shows the probability that a value drawn from an F-distribution is smaller than the empirical estimate. returns (table 3). the two variables produce intercepts close to the average excess returns for the 25 stock portfolios in table 2.

The reason for these results is straightforward. The average TEM and DEF returns (the average risk premiums for the term-structure factors) are puny, 0.06% and 0.02% per month. The high volatility of TER.Ll and DEF (table 2) allows them to capture substantial common variation in bond and stock returns in the two-factor regressions of table 3 and the

five-factor regressions of table 8. But the low average TER,LI and DEF returns imply that the two term-structure factors can?t explain much of the cross-sectional variation in average stock returns. 5.2. The cross-section of average bond returns

Tables 3,7b and 8b say that the common variation in bond returns is dominated by the bond-market factors, TERM and DEF. Oniy the low-grade bond portfolio (LG) has nontrivial slopes on the stock-market factors when TERM and DEF are in the bond regressions. Like the average values of TERM and DEF. the average excess returns on the bond portfolios are close to 0 (table 2), so it is not surprising that the intercepts in the time-series regressions for bonds (table 9b) are close to 0.

Do low average TERM and DEF premiums imply that the term-structure factors are irrelevant in a well-specified asset-pricing model? Hardly. TERM and DEF are the dominant variables in the common variation in bond returns. Moreover, Fama and-French (1989) and Chen (1991) find that the expected values of variables like TER;LI and DEF vary through time and are related to business conditions. The expected value of TER.CJ, the term premium for discount-rate risks, is positive around business cycle troughs and negative near peaks. The expected value of the default premium in DEF is high when economic conditions are weak and default risks are high, and it is low when business conditions are strong.

Thus, the common sensitivity of stocks and bonds to TERM and DEF implies interesting intertemporal variation in expected stock and bond returns.

5.3. Joint tests on the regression intercepts

We use the F-statistic of Gibbons, Ross. and Shanken (1989) to formally test the hypothesis that a set of explanatory variables produces regression intercepts for the 32 bond and stock portfolios that are all equal to 0. The F-statistics, and bootstrap probability levels, for the five sets of intercepts produced by the explanatory variables in tables 3 to 8 are in table 9c. The F-tests support the analysis of the intercepts above. The tests reject the hypothesis that the term-structure returns, TERM and DEF. suffice to explain the average returns on bonds and stocks at the 0.99 level. This confirms the conclusion, obvious from the regression intercepts in table 9a, that the low average TERM and DEF returns cannot explain the cross-section of average stock returns. The F-test rejects the hypothesis that RAWRF suffices to explain average returns at the 0.99 level. This confirms that the excess market return cannot explain the size and book-to-market effects in average stock returns. The large positive intercepts for stocks observed when SMB and HML are the only explanatory variables produce an F-statistic that rejects the zero-intercepts hypothesis at the 0.98 level.

In terms of the F-test.,the three stock-market factors, R&I-RF, SMB, and

HML, produce the best-behaved intercepts. Nevertheless, the joint test that all intercepts for the seven bond and 25 stock portfolios are 0 rejects at about the 0.95 level. The rejection comes largely from the lowest-BE?JIE quintile of stocks. Among stocks with the lowest ratios of book-to-market equity (growth stocks), the smallest stocks have returns that are too low ( - 0.34% per month, t = -3.16) relative to the predictions of the three-factor model, and the biggest stocks have returns that are too high (0.21% per month, t = 3.27). Put a bit differently, the rejection of a three-factor model in table 9c is due to the absence of a size effect in the lowest-BE,?ME quintile. The five portfolios in the lowest-BE/ME quintile produce slopes on the size factor SMB that are strongly negatively related to size (table 6). But unlike the other BE/ME quintiles, average returns in the lowest-BE/ME quintile show no relation to size (table 2).

Despite its marginal rejection in the F-tests, our view is that the three-factor model does a good job on the cross-section of average stock returns. The rejection of the model simply says that because R,WRF, S,LIB, and HML absorb most of the variation in the returns on the 25 stock portfolios (the typical R? values in table 6 are above 0.93). even small abnormal average returns suffice to show that the three-factor model is just a model, that is, it is false. To answer the important question of whether the model can be useful in applications, the interesting result

is that only one of the 25 three-factor regression intercepts for stocks (for the portfolio in both the smallest-size and the lowest-BE/ME quintiles) is much different from 0 in practical terms.

Indeed, our view is that the three-factor regressions that use RM-RF, SMB, and HML to explain average returns do surprisingly well, given the simple way the mimicking returns SMB and HML for the size and book-to-market factors are constructed. The regressions produce intercepts for stocks that are close to 0, even though SMB and HML surely contain some firm-specific noise as proxies for the risk factors in returns related to size and book-to-market equity.

Adding the term-structure returns, TERM and DEF, to regressions that also use RM-RF, SME, and HML as explanatory variables increases F. The larger F comes from bonds. The five-factor regression intercepts and R? values for stocks are close to those produced by the three stock-market factors. But for bonds, adding TERM and DEF results in much lower residual standard errors. and the increased precision pushes the five-factor intercepts for the two government bond portfolios beyond two standard errors from 0. The two intercepts are. However, rather small, 0.09O 0 and 0. I 1% per month. The three stock-market factors produce a lower F, but we think the five-factor Regressions provide the best model for returns and average returns on bonds and stocks. TERM and DEF dominate the variation in bond returns. And the variation in the expected values of

TERM and DEF with business conditions is an interesting part of the variation through time in the expected returns on stocks and bonds that is missed by the F-test, which is concerned only with long-term average returns. 6. Diagnostics

In this section we check the robustness of our inference that five common risk factors explain the cross-section of expected stock and bond returns. We first use the residuals from the five-factor time-series regressions to check that the regressions capture the variation through time in the cross-section of expected returns. We then examine whether our live risk factors capture the January seasonal in stock and bond returns. Next come split-sample regressions that use one set of stocks in the explanatory returns and another. disjoint, set in the dependent returns. These tests address the concern that the evidence of size and book-to-market factors in the regressions above is spurious, arising only because use size and book-to-market portfolios for both our dependent and explanatory returns. The last and most interesting tests examine whether the stock-market factors that capture the average returns on size-BE\\ME portfolios work as well on portfolios formed on other variables known to be informative about average returns. in particular, earnings, price and dividend, ?price ratios.

There is evidence that stock and bond returns can be predicted using (a) Dividend yields (D\\P), (b) spreads of low-grade over high-grade bond yields (default spreads, DFS), (c) spreads of long-term over short-term bond yields (term spreads. TS),and (d) short-term interest rates. [See Fama (1991) and the references therein. ] If our five risk factors capture the cross-section of expected returns. the predictability of stock and bond returns should be embodied in the explanatory returns (the month-by-month risk premiums) in the five-factor regressions. The regression residuals should be unpredictable. To test this hypothesis. we estimate the 32 time-series regressions,

e,(t + 1) = k,, + k,D(r) P(r) -I- k2DFS(r) + k,TS(t) + k,RF(r)+ r/Jr + 1). (2) The r,(t + I)in (2) are the time series of residuals for our 25 stock and seven bond portfolios from the five-factor regressions of table 7. The dividend yield, D(t). P(t), is dividends on the value-weighted portfolio of NYSE stocks for the year ending in month t divided by the value of the portfolio at the end oft. The default spread, DFS(t), is the difference at the end of month r between the yield on a market portfolio of corporate bonds and the long-term government bond yield (from Ibbotson Associates). The term spread. TS(r), is the difference between the long-term government bond yield at the end of month c and the one-month bill rate, RF(r).

The estimates of (2) produce no evidence that the residuals from the five-factor time-series regressions are predictable. In the 31 regressions, 15 produce negative values of R? (adjusted for degrees of freedom). Only four of the 32 R? values exceed 0.01; the largest is 0.03. Out of 128(32 x 4) slopes in the residual regressions. Ten are more than two standard errors from 0; they are split evenly between positive and negative values. And they are scattered randomly across the 32 regressions and the four explanatory variables. The fact that variables known to predict stock and bond returns do not predict the residuals from our live-factor regressions supports our inference that the five risk factors capture the cross-section of expected stock and bond returns.

The residual tests are also interesting information on a key regression specification. Since we estimate regression slopes on returns for the entire 1963-1991 period, we implicitly assume that the sensitivities of the dependent returns to the risk factors are constant. If the true slopes vary through time, the regression residuals may be spuriously predictable. The absence of predictability suggests that the assumption of constant slopes is reasonable, at least for the portfolios used here.

Since the work Roll (1983) and Keim (1983). documenting that stock returns, especially returns on small stocks, tend to be higher in January, it is standard in tests of asset-pricing models to look for unexplained January effects. We are leery of judging models on their ability to explain

January seasonally. If the seasonal are, in whole or in part, sampling error, the tests can contain a data- snooping bias toward rejection [Lo and MacKinlay (1990)]. Nevertheless, we test for January seasonal in the residuals from our five-factor regressions. Despite our fears. We find that, except for the smallest stocks. Residual January seasonal are weak at best. The strong January seasonal in the returns on stocks and bonds are largely absorbed by strong seasonals in our risk factors.

Table 10 shows regressions of returns on a dummy variable that is 1 in January and 0 in other months. The regression intercepts are average returns for non-January months, and the slopes on the dummy measure differences between average January returns and average returns in other months. Table 10

Tests for January seasonals in the dependent returns, explanatory returns, and residuals from the five-factor regressions: July 1963 IO December 1991, 342 months.? K(r) = u + bJAN(r) + r u b w l(b) R2 u h I(4 ((4 R2

ICaclor Five-factor exptanalory returns HM-RF 0.31 I .49 I .22 I .67 0.00 KM0 0.40 1.19 2.03 I .70 0.00

SMB 0.05 2.74 0.30 4.96 0.06 t/ML 0.2 I 2.29 I .53 4.70 0.06 0.10 _

TERM 0.41 0.56 -0.69 -0.00 DEF -0.07 I.10 -0.8 I 3.56 0.03

Stock portfolio Excess stock returns Five-factor regression residuals Smattrst-size quintitc

t~t~/Art: Low -0. I3 6.31 -0.30 4.23 0.05 -0.12 I.51 -1.17 4.09 0.04 Mr/Are ? 0.24 5.62 0.03 4.27 0.05 -0.05 0.56 -0.57 2.01 0.00 NT/MI:3 0.31 5.91 0.90 4.93 0.06 -0.06 0.69 -0.81 3.06 0.02 UE/M E 4 0.37 6.29 1.14 5.55 0.08 -0.06 0.76 -1.02 3.57 0.03 BE/ME High 0.40 7.39 1.20 6.3 I 0.10 -0.09 1.13 -I.41 4.94 0.06 Size quintite 2

BE/ME Low 0.20 2.92 0.48 2.04 0.00 0.02 -0.23 0.21 -0.74 -0.00 BE/ME 2 0.37 4.17 I.04 3.34 0.03 0.00 -0.04 0.04 -0.15 -0.00 HE/ME 3 0.53 3.95 I .63 3.48 0.03 0.04 -0.55 0.62 -2.16 0.01 BE/M I: 4 0.4x 4.32 I.65 4.22 0.05 0.02 -0.22 0.2x -0.97 ~ 0.00 HE/ME High 0.55 5.76 1.66 4.99 0.07 -0.01 0.12 -0.14 0.49 -0.00 Size quinlite 3

BE/ME Low 0.24 2.35 0.62 I .78 0.00 0.04 -0.49 0.50 ~ 1.74 0.00 BE/ME 2 0.42 2.87 1.31 2.57 0.02 0.03 -0.41 0.42 -1.48 0.00 HE/At E 3 0.43 3.06 t.47 2.99 0.02 0.07 -0.x0 0.x3 -2.90 0.02

tlE/ME4 0.52 3.51 I.92 3.6X 0.04 0.04 -0.46 0.52 -1.80 0.00 nEjnlEHigh 0.60 4.53 t.9t 4.12 0.04 0.03 -0.34 0.33 -1.15 0.00 Size quintile 4

BE/ME Low 0.39 1.12 1.16 0.95 -0.00 0.04 -0.46 0.46 -1.60 0.00 BEfME 2 0.21 1.77 0.68 1.65 0.00 0.06 -0.73 0.73 -2.54 0.02 LX/ME 3 0.40 2.0X I .40 2.1 I 0.0 I 0.08 -0.93 0.93 -3.27 0.03 BE/ME 4 0.52 3.12 1.88 3.24 0.03 0.03 -0.37 0.34 -I.17 0.00 nE/ME High 0.68 4.45 2.15 4.00 0.04 0.00 -0.03 0.03 -0.09 -0.00 Biggest-size quintile

LIE/ME Low 0.37 0.34 1.34 0.35 -0.00 0.38 -0.48 I .67 0.00 NEjhfE 2 0.27 1.11 I .02 1.19 0.00 -0.00 0.00 -0.02 -0.00 BE/ME 3 0.23 1.11 0.92 1.28 0.00 -0.17 0.16 -0.57 -0.00 SE/ME 4 0.37 2.31 1.54 2.85 0.02 0.08 -0.09 0.31 -0.00 BE/ME High 0.32 3.38 1.17 3.59 0.03 0.25 -0.18 0.63 -0.00 Bond portfolio Excess bond returns Five-factor regression residuals I SC -0.04 0.12 -0.40 -0.00 6-I o<; -0.1 I 0.23 -0.79 --O.(K) Aa;l -0.17 0.62 -2.17 0.0 I A;1 -0.1 I 0.53 -I.85 0.00 A 0.12 -0.60 2.0x 0.0 I Baa 0.14 -0.29 I.01 O.lNJ LG 0.19 -0.17 0.58 -0.00

‘JAN(r) is a dummy variable that is I if month I is January and 0 otherwise. RMO is the sum of the intercept and residuals from the regression of RM-RF SMB HML TERM, and DEP. RM is the value-weighted monthly stock-market return. RF is the one-month Treasury bill rate, observed at the beginning of the month. SMU and f/ML are the returns on the mimicking portfolios for the size and book-to-market equity factors in stock returns.

7?k?RM is LTG-RF, where LTG is the long-term government bond return. DEF is Cn-LTG, where CB is the return on a proxy for the market portrolio of corporate bonds.

The seven bond portfolios are I-to 5-year and 6- to IO-year governments (I-5G and 6-10G) and bonds rated Aaa, Aa, A. Baa, and below Baa (LG) by Moody?s, The 25 size--BE/ME portfolios are formed as the intersections of independent sorts of stocks into size and book-to-market equity quinliles in June of each year from 1963-1991. The variables are described in more detail in table 8.

The table confirms that there are January season& in excess stock returns, And the seasonals are related to size. The slopes on the January dummy are all more than 1.91% per month and more than two standard errors from 0 for the portfolios in the two smallest size quintiles. Controlling for BE \\ME. the extra January return declines monotonically with increasing size. More interesting, the January seasonal in stock returns is also related

to book-to-market equity. In every size quintile. the slopes on the January dummy tend to increase with BE\\ME. The extra January return for the two highest-BE .\\JE portfolios in a size quintile is always at least 2.38% per month and 2.85 standard errors from 0.

January season& are not limited to stock returns. The slopes on the January dummy for corporate bonds increase monotonically from the Aaa to the LG portfolio. The extra January returns are 0.86?5b. 1.13%. and 1.56% per month for the A, Baa, and LG portfolios, and these extra average returns are at least 1.94 standard errors from 0.

If our five-factor time-series regressions are to explain the January seasonals in stock and bond returns. there must be January seasonals in the risk factors.

Table 10 shows that, except for TERJI. the risk factors have extra January returns in excess of I % per month and at least I .67 standard errors from 0. The season& in the size and book-to-market factors are especially strong. The average SMB and HML returns in January are 2.73% and 2.29% per month greater than in other months, and the extra January returns are 3.96 and 4.70 standard errors from 0. Indeed, like the excess returns on the 25 stock portfolios and the five corporate bond portfolios that are the dependent variables in the five-factor regressions. the extra January returns on the risk factors are generally much larger and more reliably different from 0 than the average returns for non-January months.

Finally, table 10 shows that the January seasonals in our risk factors largely absorb the seasonals in stock and bond returns. fn the regressions of the five-factor residuals on the January dummy. only the stock portfolios in the smallest-size quintile produce systematically posttive slopes: even these slopes are only one-quarter to one-tenth the positive January seasonals in the raw excess returns on the portfolios. If anything. the five-factor residuals for the remaining size quintiles show negative January seasonals, but the slopes on the January dummy for these stock portfolios. and for the bond portfolios, are small and mostly within two standard errors of 0. In short, whether spurious or real, the January seasonals in the returns on stocks and corporate bonds seem to be largely explained by the correspondin, u seasonals in the risk factors of our five-factor model. 6.3 Split-sample test

In the time-series regressions for stocks the dependent returns and the two explanatory returns SMB and HML are portfolios formed on size and book-to-market equity. Many readers worry that the apparent explanatory power of SMB and HML is spurious, induced by the regression setup. We think this is unlikely. given that the dependent returns are based on much finer size and BE\\ME sorts (25 portfolios) than the SMB and HML returns. It also seems unlikely that we have stumbled on two mimicking returns for size and BE/ME factors that (a) measure strong common

variation in the returns on 25 portfolios when really there is none, and (b) produce exactly the patterns in the regression slopes on SMB and HML needed to explain the size and book-to-market effects in the average returns on the 25 portfolios. Still, an independent test is of interest. We split the stocks in each of the 25 size-BE\\ME portfolios into two equal groups. One group is used to form the 25 dependent value-weighted portfolio returns for the time-series regressions. The other is used to form half-sample versions of the explanatory returns. RM-RF. SMB. and HML. The roles of the two groups are then reversed. and another set of regressions is run. In this way we have two sets of regressions. In each set, the explanatory and dependent returns are from disjoint groups of stocks. Without showing all the details, we can report that the results for the two sets of regressions of excess returns for 25 size-BE\\ME portfolios on disjoint versions of RM-RF. SMB, and HML are similar to the full-sample results in tables 6 and 9. The slopes on RM-RF, SMB and HML in the split-sample regressions are close to those in table 6, and the intercepts. like those for the full-sample three-factor regressions in table 9. are close to 0. In short, the split-sample regressions confirm that there are common risk factors in returns related to size and book-to-market equity. They also confirm that market, size, and book-to-market factors seem to capture the cross-section of average stock returns.

If anything, the split-sample regressions show less power to reject the

hypothesis that RM-RF, SMB, and H&IL capture the cross-section of average stock returns than the full-sample regressions. Since the 25 dependent portfolio returns in the split-sample regressions use half the available stocks. the portfolios are less diversified than those in table 6. Although the three-factor split-sample regressions produce high values of R? (mostly greater than 0.88), they are a bit lower than those in table 6 (mostly greater than 0.9). As a result. the F-tests of the zero-intercepts hypothesis are weaker for the split-sample regressions than for the full-sample regressions. 6.4. Portfolios formed on E, P

The most interesting check on our inferences about the role of size and book-to-market risk factors in returns is to examine whether these variables explain the returns on portfolios formed on other variables known to be informative about average returns. Table 11 shows summary statistics, as well as one-factor (RM-RF) and three-factor (R.m-RF. SMB, and HAIL) regressions for portfolios formed on earning\\price (E\\ P) and dividend/price (D\\P) ratios.

The average returns on the E/P portfolios have the U-shape documented in Gaffe, Keim, and Westerfield (1989) and Fama and French (1992a). The portfolio of firms with negative earnings and the portfolio of firms in the highest-E/P quintile have the highest average returns. For the positive-E/P portfolios, average return increases from the lowest-to the

highest-E/P quintile. This pattern is an interesting challenge for our risk factors. Table I I

Summary statistics for value-weighted monthly excess returns (in percent) on portfolios formed on

dividend price (D P 1and earnings, price (E, P), and regressions of excess portfolio returns on (i) the

excess market return (R.WRF) and (ii) the excess market return (R.Lf-RF) and the mimicking

returns for the size ISCUB) and book-to-market equity (H.VL) factors: July 1963 to December 1991. 342 months.”

Ii) R(t) -RF(r) = a + b[RM(t) -RF(t)] + e(t)

(ii) R(r) -RF(t) = tz + b[Rhf(r) -RF(r)] + sS.CfB(r) + hH.VfQt) + e(t) Portfolios formed on E P Portfolios formed on D P _

Purrfolio Mean Std. r(mn) Mean Std. t(mn) GO 0.72 7.77 1.72 0.48 7.36 1.20 Low 0.27 5.13 0.96 0.39 5.48 I .30 7

; 0.170.16 4.764.68 1.53I .82 0.440.47 4.534.65 1.681.87 4 0.55 4.4s 2.27 0.57 4.33 1.42 Htgh 0.86 -I.%+ 3.30 0.56 3.86 2.67 Portfolios formed on E?P Regression (i) Regression (ii) Portfolio ?1 h R? ll b 5 h R2

EPGO 0.13 1.37 0.64 -0.30 I.21 I.13 0.16 0.82 (0.50) (?4.70) ( - 1.68) (27.82) (17.42) (6.10) Low -0.10 1.10 0.9 I 0.04 0.99 -0.01 -0.50 0.96 ( - 2.35, (57.42) (0.70) (66.75) ( -0.55) t - 19.73) 2 0.03 I.01 0.91 0.03 1.01 0.02 -0.00 0.94 (0.46) (70.24) (OAO, (61.17) (1.01) ( - 0.081 3 0.04 0.99 0.91 -0.00 1 .oo 001 009 0.92 (0.50) 161.62) ( - 0.11) (55.16) (0.40) (3.86, 1 0.15 0.93 0.58 -0.02 0.98 0.05 0.33 0.9 I t1.761 (49.78) ( -0.28) (53.57) ( 1.95) (lO.UI High 0.46 0.91 0.78 0.08 I.03 0.24 0.67 0.91 (3.69) (34.733 11.01) (51.56) (8.34) (19.62)

Table I1 (continued)

Portfolios formed on D P

Regression (i) Regression (ii)

Portfolio U b R? 0 b s h R?

D*P=O -0.15 1.45 0.80 -0.23 I.20 0.99 -0.21 0.94 I - 0.86) (37.18) ( - 2.30) (49.45) (35.09) ( -5.17) Low -0.1 I I.15 0.9 I 0.11 I.03 0.09 -0.48 0.95 ( - 1.29) (59.15) (1.64) (65.09) (3.92) ( - 17.92)

2 -0.01 1.04 0.96 0.06 1.01 -0.01 -0.14 0.96 ( - 0.19) (85.34) (1.17) (77.07) ( - 0.66) ( - 6.49)

3 0.04 0.99 0.93 -0.03 I .02 0.02 0.14 0.94 (0.64) (69.14) ( - 0.44 (64.43) (0.72) (5.09) 4 0.17 0.9 I 0.9 1 0.04 0.98 -0.06 0.30 0.91 (2.45) (58.42) (0.59) (66.51) ( - 2.80) (12.00) High 0.24 0.72 0.73 -0.01 0.85 -0.05 0.54 0.8-t

(2.22) (30.16) (0.16) (40.08) ( - 1.77) (15.04)

“Portfolios are formed in June of year r, 1963-1991. The dividend yield (D P) for year f is the

dividends paid from July of I -1to June oft [measured using the procedure described in Fama and

French (1988)]. divided by market equity in June of r - 1. The earnings price ratio (E/P) for year t is

the equity income for the fiscal year ending in calendar year t - 1. divided by market equity in

December of t -I. Equity income is income before extraordinary items. plus income-statement

deferred taxes, minus preferred dividends. The quintile breakpoints for D.?P or E?P are determined

using only NYSE firms with positive dividends or earnings. Regression t-statistics are in parentheses.

See table 7 for definitions of R,Lf-RF. S.Wf3. and H&IL.

Table It confirms the evidence in Basu (1983) that the one-factor Sharpe-Lintner model leaves the relation between average return and E,?P largely unexplained. For the positive-E/P portfolios. the intercepts in the one-

factor regressions increase monotonically, from -0.20% per month (t = -2.35) for the lowest-E/P quintile to 0.46% (t = 3.69) for the highest. The

failure of the one-factor model has a simple explanation. The market j?s for the

positive-E/P portfolios are all close to 1.0, so the one-factor model cannot explain the positive relation between E/P and average return.

In contrast, the three-factor model that uses RM-RF, SMB. and HML to explain returns leaves no residual E/P effect in average returns. The three-factor

intercepts for the five positive-E P portfolios are within 0.1 of0 ( C?S from -0.12

to 1.01). Interestingly, the three-factor regressions say that the increasing pattern

in the average returns on the positive-E/P portfolios is due to their loadings on

the book-to-market factor HML. The lowest positive-E P quintile has an HML

slope, -0.50, like those produced by portfolios in the lowest-BEIME quintile in

the three-factor regressions in table 6. The highest-E/P quintile has an

HML

slope, 0.67, like those for portfolios in the highest-BE .UE quintile in table 6.

Table 1 confirms that there is also a positive relation between E?P and BE/ME

for our 25 portfolios formed on size and BE/ME.

Fama and French (1991bl find that lovv BE .tfE is characteristic of growth

stocks. that is. stocks vvith persistently high earnings on book equity that result

in high stock prices relative to book equity. High BE .\\fE. on the other hand, is

associated with distress. that is. persistently low earnings on book equity that

result in low stock prices. The loadings on H.1fL in the three-factor regressions

of table II then say that low-E P stocks have the low average returns typical of

(low-BE.?.CfE) growth stocks, while high-E P stocks have the high average

returns associated with distress (high-BE?.!IE).

The negative-E. P portfolio produces the only hint of evidence against the three-factor model. In spite of the portfolio?s high average excess return (0.72%

per month). the three-factor model says that its average return is O.??!/o per

month too low, given its strong loadings on .S.!fB (1.13. like the smallest-size

portfolios in table 6) and H.LfL (0.16. like the higher-BE .!fE portfolios in table

6). In other words, according to the three-factor model. the average return on

this portfolio should be higher because its return behaves like those of small.

relatively depressed. stocks. The three-factor intercept for the negative-E P

portfolio is, however. only I.65 standard errors from 0.

In short. E/P portfolios produce a strong spread in average returns, which seems to be absorbed by the three common risk factors in stock returns. The E,P

portfolios are thus interesting corroboration of our inferences that (a) there are

common risk factors in stock returns related to size and book-to-market equity.

and (bj R!tf-RF. S.LfB. and H.CfL. the mimicking returns for market. size, and

BE .tfE risk factors. capture the cross-section of average stock returns.

Table I 1 shows that. as in Keim ( 3983). average returns on portfolios formed

on D P are also U-shaped; they drop from the zero-dividend portfolio to the

lowest positive-D P portfolio. and then increase across the positive-D, P port-

folios. The U-shaped pattern. and the overall spread in average returns, are,

hovvever. much weaker for the D P portfolios than for the E/P portfolios.

Table I1 also confirms Keim?s (1983) finding that the one-factor Sharpe- Lintner model leaves a pattern in average returns that looks like a tax penalty

on dividends. The one-factor intercepts increase monotonically from the lowest-

to the highest-D P portfolios. This suggests that pre-tax returns on higher-D,P

stocks must be higher to equalize after-tax risk-adjusted returns.

But the apparent tax effect in average returns does not survive in the

three-factor regressions that use R.Lf-RF, S.LfB, and H.LfL to explain rc,urns.

The three-factor intercepts for the five positive-D P portfolios are close to 0 and

show no relation to D P. The three-factor regressions say that the increasing

pattern in the average returns on the positive-D P portfolios is due to the increasing pattern in their loadings on the book-to-market factor H.bfL. The

lovvest-(positive:)-D P quintile has a strong negative H.CfL slope, -0.18, and the

highest-D. P portfolio has a strong positive slope. 0.54. Again. the three-factor

model says that low-D P stocks have the low average returns typical of growth

stocks, whereas high-D P stocks have the high average returns associated with

relative distress. Table 1 confirms that there is also a positive relation between

D P and BE?ME for our 25 portfolios formed on size and BE,?JIE.

The zero-dividend portfolio produces the strongest evidence against the three-factor model. The three-factor model says that the high average excess

return on this portfolio (0.48% per month) is 0.13% too low (r = -2.30). given

its strong loading (0.99) on SSIB, the mimicking return for the size factor. In

other words, because the return on the zero-dividend portfolio varies like the

return on a portfolio of small stocks. the three-factor model says that the high

本文来源：https://www.bwwdw.com/article/w0np.html