Common risk factors in the returns on stocks and bonds

TL;DR: In this article, the authors identify five common risk factors in the returns on stocks and bonds, including three stock-market factors: an overall market factor and factors related to firm size and book-to-market equity.

About: This article is published in Journal of Financial Economics.The article was published on 1993-02-01 and is currently open access. It has received 24874 citations till now. The article focuses on the topics: Low-volatility anomaly & Returns-based style analysis.

Summary

1. Introduction

• When stock-market factors are also included in the regressions, all of their stock portfolios load in about the same way on the two term-structure factors and on the market factor in returns.
• Seem to come largely from the term-structure factors.
• Stock returns have shared variation due to the three stock-market factors, and they are linked to bond returns through shared variation in the two term-structure factors.
• Except for low-grade corporate bonds, only the two term-structure factors seem to produce common variation in the returns on government and corporate bonds.
• The authors first introduce the inputs to the time-series regressions: the explanatory variables and the returns to be explained (sections Z and 3).

2. The inputs to the time-series regressions

• The explanatory variables fall into two sets, those likely to be important for capturing variation in bond returns and those likely to be important for stocks.
• Segmenting the explanatory variables in this way sets up interesting tests of whether factors important in stock returns help to explain bond returns and vice versa.

2.1 .I. Bond-mnrket factors

• Roll, and Ross (1986) use TERM and a variable like DEF to help explain the cross-section of average returns on NYSE stocks.
• They use the Fama and MacBeth (1973) cross-section regression approach: the cross-section of average stock returns is explained with the cross-section of slopes from timeseries regressions of returns on TERM, a default factor, and other factors.
• The default factor is the most powerful factor in average.
• The authors also find that the two variables dominate the common variation in government and corporate bond returns.

2.1.2. Stock-market fuctors

• Using value-weighted components is in the spirit of minimizing variance, since return vririances are negatively related to size (table 1. below).
• More important, using value-weighted components results in mimicking portfolios that capture the different return behaviors of small and big stocks.

Market

• The authors proxy for the market factor in stock returns is the excess market return, R.M-RF.
• R&l is the return on the value-weighted portfolio of the stocks in the six size-BE'ME portfolios, plus the negative-BE stocks excluded from the portfolios.

The returns to he espkuiwd

• -The set of dependent variables used in the time-series regressions includes the excess returns on two government and five corporate bond portfolios.
• The excess returns on these 25 portfolios for July 1963 to December 1991 are the dependent variables for stocks in the time-series regressions.
• ME, in December oft -I. A portfolio's earnings/price ratio (E P) for year I is the sum ofequity income for the firms in the portfolio for the fiscal year ending in calendar year t -1. divided by the sum of their market equity in December of r -1.
• Table 1 shows that, because the authors use NYSE breakpoints to form the 25 size-BE, ,CIE portfolios, the portfolios in the smallest size quintile have the most stocks (mostly small Amex and NASDAQ stocks).
• The portfolio of stocks in both the largest size and lowest BE/ME quintiles (big successful firms) alone accounts for more than 30% of the combined value of the 25 portfolios.

3. The playing field

• Table 2 summarizes the dependent and explanatory returns in the time-series regressions.
• The average excess returns on the portfolios that serve as dependent variables give perspective on the range of average returns that competing sets of risk factors must explain.
• The average returns on the explanatory portfolios are the average premiums per unit of risk (regression slope) for the candidate common risk factors in returns.

3.1. The dependent retwxs

• -In contrast to the stock portfolios, the average excess returns on the government and corporate bond portfolios in table 2 are puny.
• All the average excess bond returns are less than 0.15% per month, and only one of seven is more than 1.5 standard errors from 0.
• There is little evidence in table 2 that (a) average returns on government bonds increase with maturity, (b) long-term corporate bonds have higher average returns than government bonds, or (c) average returns on corporate bonds are higher for lower-rating groups.
• The flat cross-section of average bond returns does not mean that bonds are uninteresting dependent variables in the asset-pricing tests.
• Bonds are good candidates for rejecting asset-pricing equations that predict patterns in the cross-section of average returns based on different slopes on the common risk factors in returns.

3.2. The explanatory returns

• In the time-series regression approach to asset-pricing tests, the average risk premiums for the common factors in returns are just the average values of the explanatory variables.
• The average value of RXJ-RF (the average premium per unit of market p) is 0.43% per month.
• The authors shall find, however, that the slopes on SAJB for the 25 stock portfolios cover a range in excess of 1.7, so the estimated spread in expected returns due to the size factor is large, about 0.46% per month.
• And H&IL, are risk factors in the sense that they capture common (shared and thus undiversifiable) variation in stock and bond returns.
• In section 5 the authors use the intercepts from the time-series regressions to test whether the average premiums for the common risk factors in returns explain the cross-section of average returns on bonds and stocks.

4. Common variation in returns

• In the time-series regressions, the slopes and R' values are direct evidence on whether different risk factors capture common variation in bond and stock returns.
• The authors first examine separately the explanatory power of bond-market and stock-market factors.
• The purpose is to test for overlap between the stochastic processes for stock and bond returns.
• The authors then examine the joint explanatory power of the bondand stock-market factors, to develop an overall story for the common variation in returns.

42. Stock-market f&ton

• Adding SMB and HML to the regressions has an interesting effect on the market ps for stocks.
• At the other extreme, the univariate /I for the portfolio of stocks in the biggest-size and highest-BE;.CfE quintiles is 0.89.
• In the three-factor regressions of table 6. the fls for these two portfolios are 1.04 and 1.06.
• Adding .S,LfB and H&IL to the regressions collapses the ps for stocks toward 1.0: low gs move up toward 1.0 and high ps move down.
• This behavior is due. of course, to correlation between the market and SMB or H&IL.

4.3. Stock-mnrkrt and bond-market factors

• 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 average-return tests center on the intercepts in the time-series regressions.
• The explanatory variables are excess returns (R.WRF and TERXJ) or returns on zero-investment portfolios (R,bf 0, S.CJB, HALI L. and DEF).
• Since the stock portfolios produce a wide range of average returns, the authors examine their intercepts first.
• The authors are especially interested in whether the mimicking returns S,CIB and HML.
• Absorb the size and book-to-market effects in average returns, illustrated in table 2.

3.1. The cross-section oj'acerage stock returns

• 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 ).
• 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.
• 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).

WE portfolios (S,'H and B H) and the average of the returns on the two low-BE .bfE portfolios (S L and B LI. TER.W is LX-RF.

• Where LX is the long-term government bond return.
• The table also shows the probability that a value drawn from an F-distribution is smaller than the empirical estimate.
• 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 TER.Ll and DEF returns (the average risk premiums for the term-structure factors) are puny, 0.06% and 0.02% per month.

5.2. The cross-section of average hod retwrrs

• 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.

j.3. Joiut tests on the regression intercepts

• The authors 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.
• 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 three stock-market factors, R&I-RF, SMB, and HML, produce the best-behaved intercepts.
• 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).
• TER.Cf and DEF, to regressions that also use R.11-RF, S.bfE, and H.ClL as explanatory variables increases F.

6.4. Portfolios fbrtwd on E, P

• 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.
• 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.

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.
• In contrast, the three-factor model that uses RM-RF, SMB.
• In spite of the portfolio's high average excess return (0.72% per month).
• 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.

7. Interpretation and applications

• This paper studies the common risk factors in stock and bond returns and tests whether these shared risks capture the cross-section of average returns.
• There are at least five common factors in returns.
• Except for low-grade corporate bonds, the stock-market factors have little role in returns on government and corporate bonds.
• The stock and bond markets are linked, however.

7.1. Interprrttrtion

• At a minimum, their results show that five factors do a good job explaining (a) common variation in bond and stock returns and (b) the cross-section of average returns.
• The authors think there is appeal in the simple way they define mimicking returns for the stock-market and bond-market factors.
• But the choice of factors, especially the size and book-to-market factors, is motivated by empirical experience.
• Without a theory that specifies the exact form of the state variables or common factors in returns, the choice of any particular version of the factors is somewhat arbitrary.
• Thus detailed stories for the slopes and average premiums associated with particular versions of the factors are suggestive.

7.2. Applicutions

• The authors conjecture that the persistent negative abnormal returns of acquiring firms are a book-to-market effect.
• The authors guess that acquiring firms tend to be successful firms that have high stock prices relative to book value and low loadings on HAIL.
• In their three-factor model, low loadings on HML would reduce the average stock returns of acquiring firms.
• And produce persistent negative abnormal returns in tests that adjust only for market and size factors.