are industry stock returns predictable?

CFA Institute

Are Industry Stock Returns Predictable?Author(s): Kenneth R. Beller, John L. Kling and Michael J. LevinsonSource: Financial Analysts Journal, Vol. 54, No. 5 (Sep. - Oct., 1998), pp. 42-57Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4480108 .

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Are Industry Stock Returns Predictable?

Kenneth R. Beller, John L. Kling, and Michael J. Levinson

We investigated in-sample and out-of-sample predictability of equal- weighted and capitalization-weighted quarterly excess returns for 55 industries over the 1973-95 period. The in-sample analysis supported predictabilityfor about 80 percent of the cap-weighted industries and about 90 percent of the equal-weighted industries. The out-of-sample analysis provided strong evidence that theforecasting modelsfor industry returns combined with mean-variance optimization criteria are useful for portfolio selection.

y ost recent studies investigating return predictability have concluded that security returns are predictable from information investors can easily

obtain. But with few exceptions, the researchers studied broad classes of financial assets, such as aggregate stock and bond market returns. Industry return predictability has received little attention. This gap in the research is surprising in light of the presumed importance of industry analysis in the investment process.

As taught in the popular investment texts (for example, Sharpe, Alexander, and Bailey 1995 and Fabozzi 1995), the investment process consists of three steps: asset allocation, group rotation, and security analysis. In the asset allocation step, funds are allocated globally to broad asset classes on the basis of forecasts of the overall economic and market environment. Most studies of stock and bond predictability fall into this category. In the group rotation step, managers attempt to identify economic sectors and industries that stand to gain or lose relative to the overall market. Nevertheless, most of the research in finance that considers industry returns focuses on industry "factors" or risks in security returns. An important early study of industry factors is King's 1966 work. King found that security price changes can be broken down into market and industry components. King's study was followed in the late 1960s and early 1970s by several other studies that also demonstrated the importance of industry factors in security returns.

Reilly and Drzycimski (1974) provided a review of these studies and extended King's work by showing substantial divergence in relative performance among industries during any given time period and considerable variability in relative performance over time. They also found substantial variation in risk across industries (as measured by the betas of industry returns relative to the S&P 500 Index) but found that the risks were reasonably stable over time.

Recent studies have continued the focus on industry differences or industry factors to explain the variance of asset returns. Breeden, Gibbons, and Litzenberger (1989) found predictable differences in consumption betas among 12 industry groups (actually, economic sectors).1 Industries that pro- duce goods with high income elasticity of demand have high consumption betas. Kale, Hakansson, and Platt (1991) expanded the work of Rosenberg (1974), who treated company attributes as factor loadings in multifactor models of security returns, by comparing industry attributes (the company's proportion of sales from specific industries) to nonindustry attributes (earnings/price, dividend yield, size, etc.) in explaining the variance of asset returns. They found that industry attributes are substantially more important than nonindustry attributes. Boudoukh, Richardson, and Whitelaw (1994) investigated the cross-sectional relationship between industry-sorted stock returns and a measure of expected inflation. They found that noncyclical industries have a positive contemporaneous covariation with expected inflation whereas cyclical industries have a negative covariation. Although the Boudoukh, Richardson, and Whitelaw study was not predictive, it has predictive implications to the extent that expected inflation is predictable.

We have found no studies with the primary purpose of investigating industry return predict-

Kenneth R. Beller is an assistant professor at Washing- ton State University at Tri-Cities. John L. Kling holds the Gary P. Brinson Chair in Investment Management at Washington State University at Pullman. Michael J. Levinson is a senior consultant with BARRA, Inc.

42 ?Association for Investment Management and Research




ability, but a few studies do use industry groups as a form of classification. Two of the studies focused on industry rotation strategies, and the other three included industry returns in broader investigations of predictability.

First, Sorensen and Burke (1986) ranked 44 industries through relative strength analysis.2 Equal-weighted industry portfolios were formed each quarter based on the top-ranked industries. Sorensen and Burke reported significant abnormal returns relative to the S&P 500 Index for this strategy over the 1972-82 period. Their results provide some support for momentum-based industry predictability, but the time period is too short to make strong claims. Second, Grauer, Hakansson, and Shen (1990) applied multiperiod portfolio theory to the construction and rebalancing of portfolios com- posed of 12 industry indexes. The inputs to their optimizer were based on the "simple probability assessment" approach, which uses the past 32 quarters of returns to estimate means, variances, and covariances of returns. This approach is equivalent to assuming that the log wealth index for each industry follows a random walk with drift process, where the drift term and error covariance change slowly over time. Grauer, Hakansson, and Shen reported significant abnormal returns relative to their benchmarks for the 1966-86 time period.

The study that gave the most focus to industry return predictability is Fama and French (1988a). Their investigation of the mean-reverting behavior of long-horizon security returns added "to the mounting evidence that stock returns are predictable" (p. 247). Acknowledging that the mix of random walk and stationary components in stock prices can differ among stock groups, they estimated autoregressive models for returns of portfolios based on industry and size-decile classifications. For the overall time period (1926-1985), they found significant long-horizon autoregressive coefficients for most industry portfolios, but for the recent subpe- riod (1941-1985), few of the industry autoregressive coefficients were significantly different from zero.

Ferson and Harvey (1991) and Lo and MacKin- lay (1996) investigated sources of predictability in security returns. Ferson and Harvey provided evidence that predictability is caused mainly by time variation in the risk premiums in multifactor asset- pricing models. Lo and MacKinlay investigated lin- ear combinations of asset classes that would give "maximally predictable portfolios" in terms of maximizing R2. In both of these works, the researchers investigated industry groups together with size deciles and bond portfolios, but neither work focused on the predictability of industry returns.

The claims of predictability in Fama and

French (1988a), Ferson and Harvey (1991), and Lo and MacKinlay were based only on in-sample test statistics. These researchers did not perform out-of- sample testing.

The Study We investigated in-sample and out-of-sample predictability of equal-weighted and capitalization- weighted quarterly excess returns for 55 industries over the period from the first quarter of 1973 through the fourth quarter of 1995.3 We predicted returns with Bayesian multivariate regression models, which allowed us to incorporate predictor variables as well as prior information. In specify- ing the prior information, we assumed that expected returns are constant over time. This assumption reduces somewhat the data-snooping bias inherent in most ex post investigations of predictability. As with other studies, we tested in- sample predictability by investigating the regression coefficients of industry models estimated over the entire sample.

To investigate out-of-sample predictability, we used the Bayesian regression models to estimate the expected return vector and covariance matrix each quarter over the 1981-95 period. Each quarter, we formed six portfolios based on the 55 industries weighted equally or by capitalization: One portfolio was the benchmark, three portfolios were based on grouping industries only on their predicted returns, and two portfolios were optimized through the use of mean-variance analysis. We used several statistical methods to evaluate the ex post performance of the portfolios, and the reported results include familiar measures such as Sharpe ratios, differences in mean returns, and Jensen's alphas. We also present descriptive statistics for the portfolio weights and returns.

Industry Returns and Predictor Variables. We estimated Bayesian multivariate regression

models for the equal- and cap-weighted quarterly returns to each of the 55 industries of BARRA's U.S. Equity Model Version 2.4 We created the industry returns from the individual-security returns and obtained the primary industry classifications from the BARRA U.S. Equity database. This database is free of survivorship bias and consists of companies that are (or were) relatively large and have (or had) strong institutional interest. The industry returns were calculated for companies representing the BARRA HICAP Index, which consists of the largest 1,000 companies based on equity market value plus additional companies as needed to provide a minimal amount of industry representation. (Gen-

September/October 1998 43



Financial Analysts Journal

erally, the smaller companies are added only when fewer than 20 companies represent any given primary industry group.) Each month in the January 1973 through December 1995 period of this study (the BARRA universe starts in January 1973), we assigned each BARRA HICAP Index company to one BARRA industry group on the basis of the company's Primary Industry Group classification.

We first determined the quarterly industry returns by calculating quarterly holding-period returns for each company in the equity universe. Next, for each quarter and each industry, we equal- weighted the quarterly returns for all HICAP companies representing that industry as based on the primary industry classification for the month preceding the last month of the previous quarter. 6 (The only difference we followed for cap-weighted returns was to weight the individual returns by capitalizations for the end of the month preceding the month of the return.) For example, quarterly industry returns for the first quarter of the year were based on the industry and HICAP classifications of November the previous year. We calculated aggregate market returns in a similar fashion. This procedure resulted in a set of equal-weighted and cap-weighted quarterly returns for the 1973-95 period for each of the 55 BARRA industry classifications and the BARRA HICAP index. As a final step, we converted all return series to excess returns by subtracting from them the three-month U.S. T-bill return.7

The predictor variables are similar to those investigated by other researchers (for example, Fama and French 1989; Ferson and Harvey 1991, 1993; and Whitelaw 1994): term spread (TPREM), default spread (DPREM), commercial paper-T-bill spread (CPTBILL), aggregate dividend yield (DYLD), ex ante real rate of interest (RRATE), and expected inflation (EXPINFL).8 These variables capture important time-varying components in expected returns. To account for the possibility of January seasonality, we also included a dummy variable, FQTRt, that takes a value of 1 if period t is the first quarter of the year and a value of zero otherwise.

For the term spread, we used the difference in monthly yields between the 10-year and 1-year U.S. Treasury constant-maturity yields. Yields on Trea- sury securities at constant maturity are interpo- lated by the U.S. Treasury Department from the daily yield curve. The default spread was calculated monthly as the difference between Moody's Investors Service's Baa and Aaa seasoned bond yields. The commercial paper-T-bill spread is the

difference between the six-month commercial paper rate and the six-month T-bill rate. The expected inflation and real rate series were calculated in a manner similar to the method in Fama and Gibbons (1984). Subtracting the one-month inflation rate (taken from the U.S. Consumer Price Index) from the one-month T-bill return provided a monthly real return. A 12-month moving average of the 1-month real return was considered the ex ante real rate of interest. For the one-year expected inflation rate as of month t, we subtracted the ex ante real rate for month t - 1 from the one-year constant-maturity Treasury yield for month t. All of the monthly interest rate variables are monthly averages of daily rates.

We used the procedure outlined in Fama and French (1988b, p. 6) to calculate the aggregate dividend yield from the Tbbotson Associates Large Company Stocks return index with and without dividends. The result was a monthly series of annual dividend yields where the numerator is the trailing 12-month dividend and the denominator is the current value of the large-stock portfolio-that is, Dt/Pt. We determined quarterly values for the predictor variables by using the monthly value for the second month of each quarter.9 The predictor variables are stated in percentage terms and cover the period 1960 through 1995.

The Forecasting Model. We forecasted industry returns with Bayesian multivariate regression models. In notational form, the model for a single industry, say Industry 1, is

rt = 01,1 + 01,2FQTRt + 01,3RRATEt-I

+ 01,4 EXPINFLti + 01,5 TPREMt-, + 01,6 CPTBILLt-

+ 01,7 DPREMt-, + 01,8 DYLDt-I + vl, t, (la)

where

r1,t = the excess return for In- dustry 1 at time t

01,j with j = 1, ... , 8 = the regression coefficients for Industry 1 and variable j

Vi,t = the residual error for Industry 1

Note that the predictor variables, RRATE, EXPINFL, TPREM, CPTBILL, DPREM, and DYLD, are lagged by one quarter, so the model can be used to forecast one quarter ahead. In matrix notation, the model for all 55 industries is





FQTR,

RRA TE

r1, t = EX IN rl_ 2 55)+ v 1, , t

EXPINFL 1 =TPREM, (l'120 55)+ ' (b

r55 t CPTBILL t1 V55, t

DPREM1

DYLD1

where Oi is the column vector of regression coefficients for industry i and the symbol ' (prime) stands for transpose. Equation lb is actually a system of 55 equations with one equation for each industry. In compact matrix notation, Equation lb, together with the distribution assumption for the residual error, can be written as

Y= Ft'O + v,', v, - N [O,], (ic)

where Yt = the vector of industry returns at time t Ft = the vector of predictor variables at time

t, which includes the lagged values in Equation la

e = the matrix of regression coefficients Vt = the vector of residual errors

Note that vt has a multivariate nornal distribution with a mean vector of zero and covariance matrix l.

One of the main advantages of the Bayesian approach to multivariate regression is that prior information can be imposed on the model. Prior information can lead to improved forecasts and is especially useful in situations where data are lim- ited. Another useful feature of the Bayesian approach, one that is necessary for portfolio for- mation, is that predicted returns and covariances are calculated each period as steps in the coefficient updating. With our particular analysis, once the prior information for the matrix of regression coefficients, 8, and residual covariance matrix, ;, were specified as of the end of 1972, the model could be updated automatically at each date in the 1973-95 period as new information on the returns and predictor variables became available. For example, when return forecasts were made as of fourth quarter 1972 with the regression model (Equation lb), prediction errors for each industry were generated when the actual returns for first quarter 1973 became available. These errors were translated into a revision of the predictive power of the various predictor variables used to make the forecasts. In Bayesian jargon, the "priors" became "posteriors." With the updated regression coeffi-

cient and covariance matrix (that is, the "beliefs"), the regression model was ready to make a forecast for the second quarter of 1973. This process was continued through the end of 1995. The actual revision of the beliefs (conditional probability statements) was performed by a set of recursive equations that are outlined in West and Harrison (1997, pp. 602-604).

Specification of priors is often a difficult process because precise information on coefficient values and covariance matrixes is usually not available. To specify the prior means for the regression coefficients, we used the prevailing view of the late 1960s and early 1970s that expected returns are constant and that the best forecast of a return is its historical mean.10 The belief that a historical mean is the best forecast for returns can be incorporated in the regression model for each industry by spec- ifying a nonzero prior mean for the intercept term and a prior mean of zero for all other information variables. For an intercept prior for each industry, we used 2.64 percent, which is simply the average quarterly return for the S&P 500 minus the T-bill return for the 1926-72 period. This prior is an informative prior that treats all industries as equal.

In addition to the priors for the regression coefficient means, we had to specify priors for the standard deviations. Here, we used the concept of "tightness priors" popularized by Litterman (1979, 1986) and Doan, Litterman, and Sims (1984).11 Basi- cally, the prior standard deviations for the eight regression coefficients in each of the 55 equations are

03 04 05 (6 07 08

In this specification, X is a tightness parameter and 03, .. , 08 are the standard deviations of the predictor variables RRATE, EXPINFL, TPREM, CPTBILL, DPREM, and DYLD. Because of differences in variability between the industry retums and the predictor variables, 03, ... ., 0 are needed to scale X. (We estimated 03, .. ., a8 over the 1960-72 period.) Val- ues of X reflect the degree of confidence a researcher has in the prior means of the regression coefficients. Small values reflect a high degree of confidence; large values reflect little or no confidence. Ordinary least squares (OLS) regression is equivalent to spec- ifying prior means of zero for all regression coefficients and tightness parameters approaching infinity. We used a value of X = 0.2 in this analysis because practice has shown that values of X of this magnitude work well in forecasting applications (see Doan 1992; Litterman 1979, 1986; Doan, Litter- man, and Sims; and Kling and Bessler 1985). With such a strong prior belief in constant expected





returns, the data had to be very informative for the model to indicate with high probability that returns are predictable in in-sample tests of predictability.

To complete the prior specification for Equa- tion 1, we also had to specify a prior for the residual covariance matrix ? and a prior degrees of freedom, no, reflecting the confidence we placed in the prior for l. As with the regression coefficients, we used the informed prior for I that all industries are identical-that is, identical residual variances and pairwise correlations. The initial specification for fourth quarter 1972 had each diagonal element set to 172.65 and each off-diagonal element set to 86.325, with no equal to 32. The variance of 172.65 is the sample variance for quarterly S&P 500 excess returns over the 1926-72 period, and the covariance of 86.325 reflects a correlation of 0.5 between each pair of industries.12 The choice of no = 32 reflects the fact that the out-of-sample analysis starts in first quarter 1981; that is, the model is updated over the 32 quarters (1973-1980) prior to forming portfolios. Thus, we placed as much confidence in the initial prior for I as if it had been based on the first 32 observations.

Portfolio Selection. Given the prior distribution as of the end of 1972 and data for each quarter for the 1973-95 period, we updated the model parameters for each quarter. At the end of each quarter t, we determined the predictive distribution for the 55 industry returns for quarter t + 1. Specifically, the predictive distribution for the vector of industry returns, Yt+i, was the multivariate T distribution with nt degrees of freedom:

(Y,+jID, -

Tn, +t1, Qt+1,

where ft + i = the 55 x 1 vector of predicted returns

for period t + 1 Dt = information available at period t Qt+l = the 55 x 55 forecast error covariance

matrix for period t + 1 Note that bothft+j and Qt+l are conditional solely on information available only through time t.

As an example of one of the components of ft+i, consider the forecasting equation for Indus- try 1:

r1' t+1 = O1,1 + 01,2 FQTRt1+ + 61,3 RRATE,

+ 01,4 EXPINFL + 01,5 TPREMt

+ 01,6 CPTBILL, + 01,7 DPREM,

+ , D YLD, (2)

where r,+ is the predicted return for Industry 1

for period t + 1 and 61 j with j = 1, ... , 8 are the estimated regression coefficients for Industry 1, given information through time period t.

The covariance matrix Qt+i depends on the estimate of the residual error covariance matrix ? and the estimated covariance matrix for the regression coefficients. Thus, Qt?l incorporates the uncertainty in the regression residuals as well as regression parameter uncertainty.13

Prior to forming the first portfolio, we generated initial parameter estimates by updating the model for approximately one-third of the sample, 1973 through 1980. At the end of 1980, the first set of portfolios was based on the predictive distributions for the first quarter of 1981. We then updated the models recursively through third quarter 1995 and rebalanced portfolios at the end of each quarter on the basis of the predictive distributions for the subsequent quarter.

Six portfolios were formed: One portfolio was the benchmark, three portfolios (HIRET, LORET, and HI-LO) were chosen based only on predicted returns, and two portfolios (MAXU and TMV) were optimized through mean-variance analysis. The definitions are as follows: * BENCH. For the equal-weighted industry

benchmark, we equal-weighted the 55 industries; for the cap-weighted industry benchmark, we cap-weighted the 55 industries. Note that the cap-weighted benchmark is actually the cap-weighted market return.

* HIRET (LORET). This portfolio is the quintile of industries with the highest (lowest) predicted returns, where each industry is assigned an equal weight in the portfolio. HIRET (LORET) is applied to both equal-weighted and cap-weighted portfolios.

* HI-LO. This arbitrage portfolio is long the HIRET industries and short the LORET industries.

* MAXU. This portfolio maximizes utility in mean-variance space. For this portfolio, wt is the 55 x 1 vector of portfolio weights at the end of period t based on expectations for period t + 1 and 1 is the 55 x 1 unit vector.14 The risky portfolio that maximizes the negative exponential utility function with absolute risk aversion parameter a* = 2.0 is

Qt+1j w [Q1 (a* ft )( Q'J

)3

+~~~~~~~~~+

[( )Qt1 ft+1

where 1'wt = 1.15 * TMV. This portfolio minimizes variance in





mean-variance space. With the same definitions of Wt and 1 given for MAXU, the optimized risky portfolio wt that minimizes variance given a target expected return of Jp = w't ft+i = 2.64 percent is

1- t+l'JQt+lft+ij Qt+1]

+ [(1 Q2+lQ(l -ft+l))Qt+ift+1j

C(+'Q lXt+1Qt+1 ft+1) ('Qt+1 ft+1 S

where 1'wt = 1. Note that 2.64 percent is the mean- return prior used to initialize the models.

Empirical Results We present in this section the results of our investigations of in-sample and out-of-sample predictability of equal-weighted and cap-weighted quarterly excess returns for the 55 industries over the 1973-95 period.

In-Sample Tests. The in-sample analysis supported predictability for about 80 percent of the cap-weighted industries and about 90 percent of the equal-weighted industries. Coefficient estimates and test statistics are in Table 1 for the equal- weighted industry return model and Table 2 for the cap-weighted industry return model. Because the models are Bayesian, the coefficient estimates are actually the elements of the mean matrix of the posterior distribution for 0 as of fourth quarter 1995 and reflect the priors discussed previously. For positive (negative) coefficient estimates, boldface numbers indicate at least a 90 percent probability that the true coefficient values are greater (less) than zero and underlined boldface numbers indicate probabilities of at least 95 percent.

Note in Tables 1 and 2 that the coefficient estimates for FQTR, TPREM, DPREM, and DYLD are generally positive for all of the industries whereas the estimates for RRATE, EXPINFL, and CPTBILL are generally negative. The information variables associated with the highest probability values are DPREM, RRATE, and EXPINFL; the estimates of largest magnitude are in the consumer cyclical, industrial, technology, and basic material sectors. Furthermore, noticeable differences exist among industries in the first-quarter dummy (FQTR), which may indicate an industry-specific January effect. Although most of the coefficients not in bold

have probability values of at least 70 percent, in conventional OLS t-tests, these coefficients would not be viewed as significant. Recall, however, that these estimates are Bayesian estimates, which reflect a strong prior view against predictability. In addition, the correlations between the predictor variables can lead to collinearity problems with coefficient estimates, which tends to understate t- ratios.

To gain some insight into the overall explana- tory power of the set of information variables, and as a test of in-sample predictability, we present the significance levels for the F-tests of the null hypothesis that the vector of coefficients for RRATE, EXPINFL, TPREM, CPTBILL, DPREM, and DYLD is equal to zero for each industry. In Bayesian ter- minology, this significance level is actually the probability density of the complement of the highest posterior density region that just includes the zero vector. For most of the industries, the probability values are small, which indicates low probability that the true values of the coefficient vectors are zero. For the equal-weighted return results, only 7 industries have significance levels greater than 0.05, whereas the cap-weighted returns provide 12. For most of the industries, then, the F-tests indicate a high degree of in-sample predictability. The Box-Pierce Q-statistic significance levels, also presented in Tables 1 and 2, test the null hypothesis that each industry model's prediction errors are noise. Note that these numbers are generally large, which indicates no serious problem with autocorrelation and suggests that these models are ade- quate representations of the data.

Out-of-Sample Tests. The out-of-sample analysis provided strong evidence that the forecasting models for industry returns combined with mean-variance optimization criteria are useful for portfolio selection. The out-of-sample portfolio performance statistics for the 1981-95 period are presented in Table 3 for equal-weighted industry returns and Table 4 for the capitalization-weighted returns.16 The first column of statistics in these tables presents the findings for the benchmark portfolio with each industry equal weighted (Table 3) or cap weighted (Table 4). The next columns contain the statistics for the six portfolios described previously.

The top panel of Table 3 provides evidence of out-of-sample predictability for portfolios formed only on the basis of expected returns (LORET, HIRET, and HI-LO). For the HI-LO portfolio, the mean excess return of 1.689 percent is significantly different from zero, with a Jensen's alpha of 1.735 percent and a beta not significantly different from zero. As evidenced in the LORET column, the pre-





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Table 3. Out-of-Sample Test Results by Portfolio: Equal-Weighted Industry Returns, 1981-95 Equal-

Weighted MAXU TMV (target ,p= Statistic BENCH HIRET LORET HI-LO (a* = 2.0) 2.64%)

Mean returns and ratios Mean return (%) 2.087 2.839 1.150 1.689 3.820 3.786

Significance 0.020 0.015 0.185 0.041 0.000 0.000 Portfolio mean - benchmark (%) na 0.752 -0.937 - 0.398 1.733 1.699

Significance na 0.079 0.045 na 0.000 0.000 Jensen's alpha (%) na 0.757 -0.978 1.735 3.055 2.996

Significance na 0.076 0.017 0.024 0.000 0.000 Beta na 0.998 1.020 -0.022 0.367 0.379

Significance na 0.955 0.798 0.818 0.000 0.000 Sharpe ratio 0.231 0.295 0.111 0.253 0.662 0.654

Significance na 0.181 0.074 0.453 0.001 0.003 Percent of returns > 0 0.633 0.633 0.550 0.633 0.717 0.700 Percent of returns > benchmark na 0.617 0.450 0.517 0.650 0.633 Minimum return (%) -26.986 -26.660 -26.782 -16.840 -10.863 -10.853 Maximum return (%) 22.111 22.700 32.481 17.253 14.063 14.803

Mean returns and Sharpe ratios with turnover costs: Purchases and sales = 0.5 percent; short position initiation = 1.0 percent Mean return (%) 2.087 2.455 0.818 0.807 3.548 3.458

Significance 0.020 0.030 0.263 0.200 0.000 0.000 Jensen's alpha (%) na 0.381 -1.308 0.865 2.772 2.656


Significance na 0.336 0.090 na 0.001 0.004

Summary statistics on portfolio weights Average sum of positive weights 1.000 1.000 1.000 1.000 2.246 2.211 Maximum sum of positive weights 1.000 1.000 1.000 1.000 2.340 2.354 Minimum sum of positive weights 1.000 1.000 1.000 1.000 2.028 2.027 Average number of positive weights 55 11 11 11 33 33 Maximum number of positive weights 55 11 11 11 35 36 Minimum number of positive weights 55 11 11 11 30 30 Average positive weight 0.018 0.091 0.091 0.091 0.068 0.068 Median positive weight 0.018 0.091 0.091 0.091 0.053 0.052 Average maximum positive weight 0.018 0.091 0.091 0.091 0.248 0.258 Maximum positive weight 0.018 0.091 0.091 0.091 0.280 0.310 Average minimum positive weight 0.018 0.091 0.091 0.091 0.004 0.004 Minimum positive weight 0.018 0.091 0.091 0.091 0.000 0.000 Average sum of negative weights na na na -1.000 -1.246 -1.211 Maximum sum of negative weights na na na -1.000 -1.028 -1.027 Minimum sum of negative weights na na na -1.000 -1.340 -1.354 Average number of negative weights na na na 11 22 22 Maximum number of negative weights na na na 11 25 25 Minimum number of negative weights na na na 11 20 19 Average negative weight na na na -0.091 -0.057 -0.054 Median negative weight na na na -0.091 -0.048 - 0.045 Average maximum negative weight na na na - 0.091 -0.004 -0.003 Maximum negative weight na na na -0.091 0.000 0.000 Average minimum negative weight na na na -0.091 -0.166 -0.170 Minimum negative weight na na na -0.091 -0.189 -0.198

na = not applicable.

Note: Bayesian multivariate regression model.

dictor variables and statistical model reliably identify low-expected-return industries; the mean excess return for the bottom-quintile industries is only 1.15 percent, with a significant Jensen's alpha of -0.978 percent and beta of 1.02. The top-quintile industries (HIRET) have an average return 0.752

percent greater than the equal-weighted industry portfolio and a Jensen's alpha of 0.757 percent, but with significance levels less convincing than for the LORET portfolio. The Sharpe ratios are larger for the HIRET and HI-LO portfolios than for the benchmark portfolio but are not significantly different.





Table 4. Out-of-Sample Test Results by Portfolio: Capitalization-Weighted Industry Returns, 1981-95 Cap-Weighted MAXU TMV (target

Statistics BENCH HIRET LORET HI-LO (a*= 2.0) ip = 2.64%) Mean returns and ratios Mean return (%) 1.937 2.364 1.853 0.510 2.624 2.720

Significance 0.025 0.025 0.052 0.290 0.000 0.000 Portfolio mean - benchmark (%) na 0.427 -0.084 -1.426 0.687 0.783

Significance na 0.187 0.442 na 0.049 0.027 Jensen's alpha (%) na 0.190 -0.217 0.407 1.563 1.657

Significance na 0.354 0.318 0.321 0.000 0.000 Beta na 1.122 1.069 0.054 0.548 0.549


Significance na 0.452 0.204 na 0.009 0.014 Percent returns > 0 0.617 0.650 0.633 0.500 0.733 0.700 Percent returns > benchmark na 0.633 0.433 0.433 0.500 0.467 Minimum return (%) -23.866 -27.335 -25.436 -14.900 -17.058 -17.082 Maximum return (%) 19.255 21.706 28.551 15.740 18.609 17.760

Mean returns and ratios with turnover costs: Purchases and sales = 0.5 percent; short position initiation = 1.0 percent Mean return (%) 1.937 2.015 1.504 -0.363 2.380 2.368

Significance 0.025 0.047 0.095 na 0.001 0.002 Jensen's alpha (%) na -0.156 -0.566 -0.464 1.307 1.292

Significance na na 0.111 na 0.000 0.000 Sharpe ratio 0.250 0.218 0.159 -0.056 0.408 0.403

Significance na na 0.129 na 0.015 0.033

Summary statistics on portfolio weights Average sum of positive weights 1.000 1.000 1.000 1.000 2.085 2.071 Maximum sum of positive weights 1.000 1.000 1.000 1.000 2.226 2.293 Minimum sum of positive weights 1.000 1.000 1.000 1.000 1.890 1.909 Average number of positive weights 55 11 11 11 35 35 Maximum number of positive weights 55 11 11 11 38 39 Minimum number of positive weights 55 11 11 11 33 31 Average positive weight 0.019 0.091 0.091 0.091 0.060 0.059 Median positive weight 0.012 0.091 0.091 0.091 0.048 0.048 Average maximum positive weight 0.077 0.091 0.091 0.091 0.238 0.245 Maximum positive weight 0.097 0.091 0.091 0.091 0.259 0.328 Average minimum positive weight 0.001 0.091 0.091 0.091 0.003 0.003 Minimum positive weight 0.000 0.091 0.091 0.091 0.000 0.000 Average sum of negative weights na na na -1.000 -1.085 -1.071 Maximum sum of negative weights na na na -1.000 -0.890 -0.909 Minimum sum of negative weights na na na -1.000 -1.226 -1.293 Average number of negative weights na na na 11 20 20 Maximum number of negative

weights na na na 11 22 24 Minimum number of negative weights na na na 11 17 16 Average negative weight na na na -0.091 -0.054 -0.054 Median negative weight na na na -0.091 -0.049 -0.049 Average maximum negative weight na na na -0.091 -0.004 -0.004 Maximum negative weight na na na -0.091 0.000 0.000 Average minimum negative weight na na na -0.091 -0.126 -0.125 Minimum negative weight na na na -0.091 -0.154 -0.152 na = not applicable.

Note: Bayesian multivariate regression model.

The Sharpe ratio for the LORET portfolio is about half that of the benchmark and significantly different at the 0.074 level. Finally, the percentages of returns greater (less) than zero and greater (less) than the benchmark in the top panel of Table 3 are consistent with the findings on average returns:

The HIRET portfolio has returns higher than the benchmark almost 62 percent of the time, whereas the LORET portfolio beats the benchmark only 45 percent of the time.

The results for quintile portfolios formed on the basis of returns alone from the cap-weighted





industry returns, shown in Table 4, are qualita- tively similar to those for the equal-weighted industry returns, but the significance levels are not convincing. For example, the HI-LO portfolio had an average return of only 0.51 percent, with a significance level of 0.29. This return is less than one- third the average return for the same strategy applied to the equal-weighted industry returns. Similar comparisons are found for the high- expected-return and low-expected-return portfolios, although the percentages of returns greater than the benchmark are almost identical to the comparable returns in Table 3. A probable reason for the statistical differences between portfolios based on equal-weighted returns and portfolios formed on cap-weighted returns is that the cap- weighted industries are dominated by the largest companies in the industries. The larger companies' operations are more diversified across industries than are the operations of smaller companies. Therefore, to the extent that genuine differences among industries exist in return predictability relative to our information set, this predictability is more likely to be captured in equal-weighted than cap-weighted industry portfolios.

The most impressive statistics showing evidence of out-of-sample predictability are those reported in Tables 3 and 4 for the optimized portfolios, MAXU and TVM. Interestingly, the results are almost identical for both methods of optimization. The mean retums and Sharpe ratios are significantly greater than those of the benchmarks for both the equal- and cap-weighted groups. The Jensen's alphas for the portfolios of equal-weighted industry returns (top panel, Table 3) are about 3.0 percent, with betas about 0.37. The mean retums and Jensen's alphas are smaller for the cap-weighted scheme reported in Table 4 than for the equal- weighted scheme but are still highly significant. Note that in Tables 3 and 4, the minimum and maximum returns for the optimized portfolios are much smaller in magnitude than for the benchmark portfolios and for the HIRET and LORET portfolios.

The mean return of the equal-weighted benchmark portfolio is not much greater than that of the cap-weighted benchmark portfolio (2.087 versus 1.937). The mean returns, Sharpe ratios, and alphas for the optimized portfolios when industries were equal weighted are about 50 percent greater than those of the optimized portfolios when industries were cap weighted. Again, the probable reason for this disparity is the impact of large firms in the cap- weighted scheme. Because large firms are more diversified than small firms, some of the diversification advantages of applying mean-variance optimization to equal-weighted industries are lost

when those methods are applied to cap-weighted industries.

Evidence needed to evaluate the practicality of implementing portfolio strategies based on these findings is in the second and third panels of Tables 3 and 4. In the second panels are the mean returns, Jensen's alphas, and Sharpe ratios after deducting turnover costs. The third panels display descriptive statistics for the average, maximum, and minimum sums of positive and negative weights in addition to the averages, medians, max- imums, and minimums of the actual positive and negative weights. Turnover costs were imposed when the ex ante weights changed from the ex post weights. Specifically, at time t, we formed a vector of weights, Wt t based on the predictive distribution for time t + 1. The actual vector of portfolio weights at time t, wt-1, t, depended on the ex ante weight vector, and security price changes. We imposed a trading cost of 0.5 percent when weight changes resulted in ordinary purchases or sales and a cost of 1.0 percent when weight changes resulted in a short sale. For example, suppose the ex post weight for industryj at time t was wj,ti,,t = 0.2 and the ex ante weight at time t was Wj t t+1 = -0.1. The trading cost for industry j at the end of quarter t was then (0.5 x 10.21) + (1.0 x 1-0.11) = 0.2 percent. We summed the trading costs across industries at each date t and subtracted the costs from the return earned by the portfolio over the per; ad t - 1 to t.

Obviously, large and volatile w,ight changes resulted in large trading costs that substantially reduced portfolio returns. Trading costs eliminated most of the gains from the strategies based only on expected returns (please refer to the HIRET, LORET, and HI-LO columns of Tables 3 and 4). On the other hand, the mean returns, Jensen's alphas, and Sharpe ratios for the optimized portfolios remained relatively large and highly significant. The average returns for the optimized portfolios with turnover costs are only about 0.3 percent lower than without turnover costs, which indicates that changes in portfolio weights were neither volatile nor constant over time.

Given that short sales were allowed, the portfolio weights for the optimized portfolios shown in the third panels of Tables 3 and 4 do not seem excessive. The average sum of positive weights is about 2.0 and the average sum of negative weights is about -1.0. Even the extreme weights for any individual industry are, on average, between - 0.17 and 0.25, with the median negative and positive weights between -0.05 and 0.06. In principle, then, the optimized portfolios formed using inputs from the Bayesian regression models could be imple- mented in dynamic investment strategies.





Conclusions We investigated the predictability of industry stock returns within the context of a Bayesian multivariate regression model with conditioning information. Our results provide evidence that industry returns are predictable-not merely in a statistical sense but also from the economically relevant standpoint of portfolio selection.

Our information set consisted of familiar predictor variables, and we investigated in-sample as well as out-of-sample predictability. The regression- based tests provide strong evidence of in-sample predictability for about 90 percent of the equal- weighted industries and about 80 percent of the capitalization-weighted industries. Industry return

predictability is also evident in the out-of-sample portfolio strategies. In the absence of trading costs, portfolios formed by simply choosing the quintile of industries with the highest predicted returns outper- formed the benchmarks. Trading costs, however, eliminated the trading profits. Optimized portfolios formed by using the predicted returns and predicted covariance matrixes of returns significantly outper- formed the benchmarks even in the face of trading costs.

We received helpful comments from Rick Sias, Harry Turtle, and Eric Weigel, andfrom Russ Fuller and par- ticipants of the RJF Behavioral Finance Workshop.

Notes 1. The consumption beta is the covariance of an asset's return

with the growth rate in aggregate per capita consumption divided by the variance of growth in aggregate per capita consumption.

2. Relative strength is defined as the current industry index value divided by the prior six months' average for the index.

3. In-sample predictability is usually tested by taking the entire sample of available data and estimating regression models of returns against a set of information variables. Predictability is then associated with a rejection of the null hypothesis that the regression coefficients are zero. This method of testing for predictability has it roots in the early economics literature on Granger causality testing (see Granger and Newbold, 1986, pp. 259-62, for a review). Out- of-sample predictability tests, on the other hand, use sequential estimates of regression models to forecast future returns based on information that would have been available at the time. The out-of-sample forecasts are then tested for accuracy relative to some criterion that, ideally, com- putes losses from decisions based on the forecasts. In-sample versus out-of-sample predictability has been discussed in the causality literature by Granger (1980), in the statistics literature by Geisser (1993), and in the investment literature by Fuller and Kling (1990,1994).

4. The BARRA U.S. Equity Model is one of the most popular multifactor models in applied use. This model, introduced in 1975 and revised in 1982, uses 13 company-specific attributes (size, earnings/price, etc.) and 5 industry classifications. See Rosenberg; Rudd and Rosenberg (1979); Rosenberg, Reid, and Lanstein (1985); and Rudd and Clas- ing (1988) for background and a description of the BARRA model.

5. BARRA analysts use various sources of information in con- junction with a proprietary econometric model to periodi- cally reevaluate the industry group classifications of all actively traded public companies within its database. Con- sequently, these historical industry classifications are subject to change from time to time. We have incorporated a contemporaneously correct history of these changes into our current analysis.

6. The restriction to previous-period classifications was intended to guard against look-ahead bias.

7. The quarterly T-bill return was calculated by compounding the one-month return for one-month T-bills over the three months of each quarter. T-bill returns were taken from the Ibbotson Associates Stocks, Bonds, Bills, and Inflation data-

base. 8. We used RRATE and EXPINFL instead of a single nominal

interest rate because Boudoukh, Richardson, and Whitelaw hypothesized the existence of cross-sectional differences among industries in the contemporaneous correlation between industry return and expected inflation.

9. We used the second instead of the last month of the quarter to take into account the data-reporting lags inherent in real- time forecasting and investing.

10. In his second review of the efficiency of capital markets, Fama (1991, p. 1578) stated, "In the pre-1970 literature, the common equilibrium-pricing model in tests of stock market efficiency is the hypothesis that expected returns are constant through time. Market efficiency then implies that returns are unpredictable from past returns or other past variables, and the best forecast of a return is its historical mean." Based on evidence from the early literature, Fama concluded, ". . . the hypothesis of market efficiency and constant expected returns is typically accepted as a good working model."

11. These priors were originally applied to Bayesian vector autoregressive models and are often referred to as "Minne- sota priors" because of their use in economic forecasting at the Federal Reserve Bank of Minneapolis.

12. This is a rather vague prior setting for X, but without access to industry return data preceding 1973, we had few reason- able choices.

13. The exact updating formulas for ft+i, X, Qt+i' and the regression coefficients are given in West and Harrison (pp. 602-604).

14. The mean-variance formulas can be found in Ingersoll (1987, Chapter 4).

15. We derived the solution to the negative exponential utility problem under the assumption that returns are multivariate normally distributed, and therefore, the solution is an approximation when the predictive distribution is multivariate T. Ferson and Siegel (1997) provide a solution for unconditional minimum-variance portfolios in the pres- ence of conditioning information. The solution is quantita- tively similar to the exponential utility solution that we used. When we applied the Ferson-Siegel solution, we got results almost identical to those we report here.

16. The significance level attached to the Sharpe ratio actually compares the squared Sharpe ratio of a particular portfolio with the squared Sharpe ratio of the benchmark portfolio. We tested the difference with generalized method of





moments estimators; the details are available from the authors. The significance levels for the betas tested the null hypothesis that beta equals 1 for the high (HIRET) and low (LORET) predicted return portfolios and zero for the other

portfolios. The Newey-West (1987) correction for autocorrelation and heteroscedasticity was applied to all hypothesis tests in this section.

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