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An examination of the economic significance of stock return predictability in UK stock returns Jonathan Fletchera,*, Joe Hillierb a

Department of Accounting and Finance, University of Strathclyde, Curran Building, 100 Cathedral Street, Glasgow G4 0LN, UK b Glasgow Caledonian University, Britannia Building, Cowcaddeus Rd., Glasgow G4 0BA, UK Received 26 February 2001; received in revised form 27 November 2001; accepted 17 April 2002

Abstract We explore the out-of-sample performance of domestic UK asset allocation strategies that use forecasts of expected returns from a linear predictive regression and those that are implied by asset pricing models such as the capital asset pricing model (CAPM) or arbitrage pricing theory (APT). Our findings suggest that using forecasts of expected returns from the predictive regression generate significant benefits in out-of-sample performance. We find the performance of the strategies using expected return forecasts implied by the CAPM or APT is lower than the predictive regression strategy. However, with binding investment constraints, the performance of the APT matches that of the predictive regression. D 2002 Elsevier Science Inc. All rights reserved. JEL classification: G12 Keywords: Predictability; Asset allocation; Asset pricing

1. Introduction Recent empirical research shows that financial asset returns in the United States (e.g., Fama, 1991) and other markets (e.g., Harvey, 1995; Solnik, 1993) are partly predictable over

* Corresponding author. Tel.: +44-141-548-3892; fax: +44-141-552-3547. E-mail address: [email protected] (J. Fletcher). 1059-0560/02/$ – see front matter D 2002 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 9 - 0 5 6 0 ( 0 2 ) 0 0 1 3 8 - 7

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time by common information variables such as the market dividend yield or interest rates. The standard approach to assess predictability is to use linear regression analysis. Excess asset returns over a given time horizon are regressed on a set of variables that are known to investors at the start of the time horizon. The significance of stock return predictability has been examined on a statistical basis (e.g., Ang & Bekaert, 2001; Bossaerts & Hillion, 1999, among others) and economic basis (e.g., Fletcher, 1997; Grauer, 2000; Handa & Tiwari, 2001; Harvey, 1994; Solnik, 1993).1 The existence of predictable stock returns is controversial in the academic literature. Predictable stock returns can be consistent with market efficiency if it can be explained by rational time variation in expected returns (e.g., Fama, 1991; Kirby, 1998). Ferson and Harvey (1991, 1993) and Ferson and Korajcyzk (1995) show that most of the time-series predictability in U.S. and international stock returns can be explained by multifactor asset pricing models. A recent study by Kirby (1998) shows that any candidate asset pricing model makes testable predictions about the values of the coefficients and R2 in the predictive regression. Kirby (1998) finds that the predictability in U.S. stock returns is greater than can be explained by a wide range of asset pricing models (see also Fletcher, 2001, for UK stock returns). We use the framework of Kirby (1998) to estimate out-of-sample forecasts of expected returns from the predictive regression that is consistent with a given asset pricing model. We use these estimates of expected returns as inputs into solving mean-variance optimal portfolios using UK industry portfolios. We estimate expected returns for a range of asset pricing models. The models include the capital asset pricing model (CAPM), arbitrage pricing theory (APT), and a three-factor model similar to Fama and French (1993). We evaluate the out-of-sample performance of the different asset allocation strategies using a range of performance measures. Our main focus is to compare the performance of the strategies that are based on asset pricing models to that of the strategy that uses expected return estimates from the predictive regressions. If the observed predictability in UK stock returns is consistent with any of the asset pricing models we evaluate, we expect that the performance of the two sets of strategies should be similar. Our study complements and extends the evidence of whether predictable stock returns is consistent with different asset pricing models provided by Fletcher (2001), Ferson and Harvey (1991, 1993), Ferson and Korajcyzk (1995), and Kirby (1998) among others. We provide out-of-sample evidence on this issue. Our study differs from Kirby (1998) in that we focus on exploring the impact of different out-of-sample forecasts of expected returns given from the predictive regression or implied by asset pricing models. This differs from Kirby (1998) who focuses on testing in sample predictability. Our study also complements that of studies that examine the economic significance of predictability. We provide evidence of the economic significance of predictability in UK stock returns over a longer time period than that of Fletcher (1997).

1 A number of studies also examine the implications of optimal portfolio choice for long term investors when stock returns are predictable over time (see Barberis, 2000; Campbell & Viceira, 1999, among others).

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We present three main findings. First, the strategy that uses forecasts of expected returns from the linear predictive regression provides significant benefits in out-of-sample performance for the domestic asset allocation strategy. Second, strategies that use forecasts of expected returns from the linear predictive regression that are consistent with the CAPM or APT generally produce lower performance compared to the strategy that uses the forecasts of expected returns from the predictive regression. However, we find that the performance of the strategy that uses forecasts that is consistent with the APT matches the performance of the strategy that uses the forecasts from the predictive regression whenever investors face binding investment constraints. Third, we find that the performance of the strategies that uses forecasts of expected returns that are consistent with the APT perform better than the strategy that uses forecasts that are consistent with the CAPM. Our results suggest that most of the predictability in UK stock returns can be captured by multifactor asset pricing models. The article is organized as follows: Section 2 describes the method used in the study. Section 3 discusses the data and the construction of the CAPM and APT models. Section 4 reports the empirical results and Section 5 concludes.

2. Method 2.1. Stock return predictability and asset pricing The standard approach to evaluate predictability in stock returns is to use the linear predictive regression model. Define Zt 1 as an (L*1) vector (which includes a constant) of information variables. The predictive regression is given by: rit ¼ D0iu Z t1 þ uit

ð1Þ

0 where rit is the excess return of asset i in period t, Diu is (1*L) vector of unrestricted coefficients, and uit is a random error term. From Eq. (1), the expected excess return of asset i (E(ri)) is given by:

Eðri Þ ¼ D0iu Z t1 :

ð2Þ

To calculate expected returns from Eq. (2), we require estimates of Diu. We follow Handa and Tiwari (2001) and Solnik (1993) and estimate expected returns at time t as follows. We estimate the predictive regression in Eq. (1) using data from a prior historical period. We then multiply the unrestricted coefficient vector by the current values of Zt 1 to get the expected excess returns of asset i. Kirby (1998) shows that for the predictability to be consistent with a given asset pricing model, the coefficients in Eq. (1) will have certain values. Kirby (1998) derives the testable restrictions within the stochastic discount factor and expected return/beta formulations of the asset pricing models. We use the expected return and beta framework and assume that

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conditional asset betas are constant2 as in Kirby (1998). The conditional CAPM or APT implies the following relationship: Eðrit j Z t1 Þ ¼

K X

bik Eðrkt j Z t1 Þ

ð3Þ

k¼l

where bik is the constant conditional beta of asset i with respect to factor k, E(rktjZt 1) is the conditional risk premium of factor k at time t, and K is the number of factors. Kirby (1998) shows that the constant conditional beta version of linear asset pricing models implies that the coefficient vector in the predictive regression of Eq. (1) should be: Diu ¼ Dir ¼

K X

bk bik

ð4Þ

k¼l

where bk is the (L*1) coefficient vector from the regression of factor k on a constant and the information variables and Dir is the (L*1) restricted coefficient vector from the predictive regression implied by the candidate asset pricing model. According to Eq. (4), the expected excess return on asset i is given by: Eðri Þ ¼ D0ir Z t1

ð5Þ

Kirby (1998) shows that the restrictions in Eq. (4) implied by the CAPM on the predictive regression can be estimated by the following system of equations: u1t ¼ ðrit bim rmt Þrmt

ð6aÞ

0 ðrit D0iu Z t1 Þ u2t ¼ Z t1

ð6bÞ

0 ðbim rmt D0ir Z t1 Þ u3t ¼ Z t1

ð6cÞ

where rmt is the excess return on the market index in period t. We estimate the system of equations in Eqs. (6a)–(6c) by generalized method of moments (GMM) (Hansen (1982)). Eq. (6a) identifies the constant conditional beta. The next L equation estimates the unrestricted coefficients from the predictive regression. The final L equation estimates the restricted coefficients implied by the CAPM. The system of equations in Eqs. (6a)–(6c) can be extended to incorporate multifactor models. 2

We use the expected return and beta framework because the constant price of risk stochastic discount factor formulation of the models tends to perform poorly in explaining stock return predictability (see Kirby, 1998). The assumption of constant betas is probably reasonable as Ferson and Harvey (1991, 1993) show that nearly all of the time-series predictability in asset returns captured by asset pricing models is due to changing risk premiums. We did experiment with time-varying betas with the CAPM where the betas were a linear function of the information variables. This tended to yield similar forecasts as the constant beta version of the CAPM.

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To calculate the expected excess returns from Eq. (5), we use the following approach. We estimate the system of equations in Eqs. (6a)–(6c) using data from the prior historical period. We then multiply the restricted coefficient vector by the current values of Zt 1 to get the expected excess return of asset i. We use the forecasts of expected returns from Eqs. (2) and (5) as inputs to solving a meanvariance portfolio problem in a domestic UK industry asset allocation setting. We construct strategies using the different models to forecast expected returns. The majority of prior studies have examined the economic significance of stock return predictability by using the forecasts from Eq. (2) as inputs to expected returns. If a candidate asset pricing model can explain stock return predictability, then Diu = Dir. This relation suggests that the expected excess returns of the strategies should be similar and we expect the performance of the strategies to be similar. We construct the asset allocation strategies as follows. At the start of each month, we estimate expected excess returns using data over the prior 60 months for the different models. We use the sample covariance matrix of the industry portfolio excess returns over the prior 60 months as the input for the covariance matrix, which is the same across all models.3 A mean-variance optimal portfolio is selected among the 10 UK industry portfolios and a risk-free asset for a given level of ‘‘mean-variance’’ risk tolerance4 (t) (see Best & Grauer, 1990). We solve the optimal portfolio where the investor faces no investment restrictions and where constraints are imposed. We assume the investment constraints are no short selling is allowed in the risky assets and an upper bound limit of 20% in each risky asset. Many institutional investors face no short selling constraints. The upper bound constraints ensure more diversification across the industry portfolios. We assume the investor is allowed unrestricted risk-free lending or borrowing. Using the optimal portfolio weights, we calculate the actual monthly excess returns. We repeat this process each month for the different models. This generates a time-series of monthly portfolio excess returns for each strategy. 2.2. Performance measures We evaluate the performance of the mean-variance strategies using a wide range of performance measures. This includes the Sharpe (1966) measure, the returns-based measures of Ferson and Schadt (1996) and Jensen (1968), and the weight-based measures of Ferson and Khang (2002) and Grinblatt and Titman (1993). We calculate the Sharpe (1966) measure as the mean excess return divided by the standard deviation of excess

3

It is well known that expected return inputs are more unstable than the covariance matrix estimates (Merton, 1980). The framework of Kirby (1998) is most usefully explored to get expected return estimates. Chan, Karceski, and Lakonishok (1999) examine the forecasting power of different models of the covariance matrix. Jagannathan and Ma (2001) show that when portfolio constraints are imposed, the performance of the sample covariance matrix is as good as other models of the covariance matrix for the global minimum variance portfolio. 4 We set t equal to 0.1, but using various levels of mean-variance risk tolerance has no impact on the analysis.

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returns. We use the Sharpe measure to rank the performance across strategies and relative to a domestic market index. The performance measures of Ferson and Khang (2002), Ferson and Schadt (1996), Grinblatt and Titman (1993), and Jensen (1968) estimate the abnormal returns of the strategies that can be used to assess the statistical and economic significance of the strategies. Positive abnormal returns are usually interpreted as superior performance and negative abnormal returns as inferior performance. Under the null hypothesis that the strategy exhibits no abnormal performance, then the abnormal returns should equal zero.5 The Ferson and Schadt (1996) and Jensen (1968) performance measures only require information on the portfolio returns of the mean-variance strategy. The Jensen measure is an unconditional performance measure that assumes that the portfolio beta is constant through time. The Ferson and Schadt measure is a conditional performance measure that allows the portfolio betas to vary through time as a function of lagged common information variables. Conditional performance measures assume that strategies based on publicly available common information should not generate superior performance. We estimate the Jensen (1968) measure by the following regression: rit ¼ ai þ bi rmt þ eit

ð7Þ

where eit is a random error term with E(eit) = 0 and E(eitrmt) = 0. The bi coefficient is the beta of portfolio i to the market index. The intercept ai is the Jensen performance measure. We refer to the performance from Eq. (7) as the Jensen measure. The Ferson and Schadt (1996) performance measure assumes that the portfolio beta is a linear function of the information variables used by investors. Within a CAPM framework, we can write this as: Zt1 Þ ¼ bi þ bi ðZ

L X

Dil zlt1

ð8Þ

l¼1

where zlt 1 are the de-meaned (deviations from mean) values of the l-th information variable at time t 1, the dil coefficients capture the response of the conditional beta to the L information variables, and bi is the average conditional beta. Ferson and Schadt show that assuming the linear conditional beta function implies we can estimate the conditional performance measure by the following regression: rit ¼ ai þ bi rmt þ

L X

Dil rmt zlt1 þ eit

ð9Þ

l¼1

The intercept ai is the Ferson and Schadt measure. The additional term’s rmt zlt 1 captures the covariance between the conditional beta and market risk premiums. We refer to the performance from Eq. (9) as the FS measure. 5

The interpretation of performance is controversial in the academic literature. See Grinblatt and Titman (1989) as an example.

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We estimate the Jensen and FS measures of the asset allocation strategies using the domestic market index as the benchmark portfolio. We can view the Jensen and FS measures as the abnormal returns of the strategies compared to an alternative passive strategy that invests in the risk-free asset and the domestic market index with the same risk characteristics as the strategy (see Elton & Gruber, 1995). The performance measures of Ferson and Khang (2002) and Grinblatt and Titman (1993) require information on portfolio weights. Weight-based measures have the advantage that they do not require a benchmark portfolio as returns-based measures do. The essence of weight-based measures is to estimate the covariance between changes in asset weights and future asset returns (abnormal returns). Informed investors who can correctly forecast future asset returns will adjust portfolio weights accordingly. This should result in a positive covariance between changes in asset weights, and future asset returns (abnormal returns). The Grinblatt and Titman measure is an unconditional performance measure that assumes that expected returns are constant through time for uninformed investors. Grinblatt and Titman show that the average (over time) covariance between changes in asset weights and future asset returns (portfolio change measure, PCM) can be estimated as: PCM ¼ ð1=T Þ

T X N X

rit ðwit witk Þ

ð10Þ

t¼1 i¼1

where wit is the investment weight of asset i at the beginning of period t, wit k is the investment weight of the i-th asset t k periods earlier, and T is the number of time-series observations. We set the lag equal to 1 month because the mean-variance strategies are estimated each month. Ferson and Khang (2002) extend the weight-based measure of Grinblatt and Titman (1993) to a conditional framework where expected returns can vary through time as a function of common information variables.6 The conditional weight measure (CWM) of Ferson and Khang measures the conditional covariance between changes in asset weights and subsequent asset abnormal returns, where abnormal returns are measured as the difference between actual excess returns and conditional expected excess returns. The CWM in period t is given by: "

N X ðwit wbitk Þðrit Eðri jZ Zt1 ÞÞjZ Zt1 CWMt ¼ E

# ð11Þ

i¼l

where wbitk is the benchmark weight of asset i at the start of period t, E(rijZt 1) is the conditional expected excess return of asset i, and Zt 1 is a vector of information variables that are assumed to capture publicly available information. The benchmark weight wbitk is 6

Ferson and Khang (2002) also propose an unconditional performance measure similar to Grinblatt and Titman (1993) with some minor modifications. We experiment with this but find similar inferences with the PCM measure and so we do not report the results here.

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the actual lagged weight of the i-th asset t k periods earlier that have been updated with a buy-and-hold strategy. In this context, k is the number of periods until the lagged weights become public information. As with the PCM we use a 1-month lag. Ferson and Khang (2002) use the average (over time) conditional covariance as their performance measure. To implement the CWM performance measure, we require assumptions about the functional form of the conditional expected excess returns and CWMt. We follow Ferson and Khang and assume that the conditional expected excess returns of assets can be proxied by a linear function of the lagged information variables and CWMt is also a linear function of the information variables given by: CWMt ¼ CWM þ ;0zt1

ð12Þ

where CWM is the average conditional covariance between changes in asset weights and the future asset abnormal returns, zt 1 is an (L*1) vector of de-meaned values of information variables that excludes the constant and ; is an (L*1) vector of slope coefficients. These slope coefficients measure the response of the CWM to the lagged information variables. In our context, the use of the CWM performance measure is particularly appropriate because the model of conditional expected excess returns is the same as that of the predictive regression. This provides useful insights into the source of any potential benefits of using stock return predictability in asset allocation strategies. Ferson and Khang (2002) show how to estimate the CWM measure by GMM. We estimate the two portfolio weight measures by GMM. The test statistics of all the performance measures are corrected for the effects of heteroskedasticity using White (1980).

3. Data and models We use monthly return data from the London Business School Share Price Database (LBS) between February 1955 and December 1995. We use 10 UK industry portfolios as the investment universe for the domestic asset allocation strategies. We form the industry portfolios using the industry classifications7 as recorded in the 1996 LBS handbook. The following classifications are used: 1. 2. 3. 4. 5.

Mineral extraction—group numbers between 123 and 165. Building and chemicals—group numbers between 210 and 255. Engineering—group numbers between 261 and 270. Printing and textiles—group numbers between 282 and 297. Consumer goods—group numbers between 320 and 380.

7 We also experiment with using 10 size portfolios as the investments universe. Each year, all stocks on the LBS database are ranked on the basis of their beginning of year market value. All securities with a nonzero market value are grouped into 10 portfolios and equal weighted monthly returns are estimated during the year. This process is repeated each year. We find using the size portfolios has no impact on the performance inferences reported in the paper.

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6. 7. 8. 9. 10.

381

Distribution, leisure, and media—group numbers between 412 and 436. Retailers and other services—group numbers between 440 and 490. Utilities—group numbers between 620 and 680. Financials—group numbers between 710 and 794. Investment trusts—group numbers between 801 and 980.

For each classification, we calculate equal weighted monthly excess returns for all securities with a return observation in a given month. The number of securities within each group ranges between 190 and 908. The monthly return on a 90-day UK Treasury bill is used as the risk-free asset. We choose this as the risk-free asset because the 30-day UK Treasury bill is not available for the whole sample period. We use four categories of models of expected returns in the analysis. This includes the following. 3.1. Conditional model (Cond) This is based on the statistical model of returns in Eq. (1). We use instruments that prior studies have found to be important in predicting asset returns (see Fletcher, 1997; Solnik, 1993, for evidence of UK stock return predictability among others). We include8 the lagged 1 month excess return on the Financial Times All Share (FTA) index, lagged 1 month risk-free return, lagged dividend yield on the FTA index (obtained from LBS), a January dummy that equals 1 if the month is January and 0 if otherwise.9 3.2. Single factor (CAPM) This is based on the restricted coefficients in Eq. (5) from the predictive regression implied by the CAPM. We use the excess returns on the FTA index10 as the single-factor portfolio. We also use the FTA index as the benchmark portfolio in estimating the Jensen and FS performance of the strategies. 3.3. Multifactor models This is based on the restricted coefficients in Eq. (5) implied by multifactor models. Our primary model uses the APT based on the asymptotic principal components technique of Connor and Korajcyzk (1986). Connor and Korajcyzk show that the K factor portfolios can 8

We also use these instruments in the conditional performance measures. The dividend yield and risk-free return instruments have an extremely high autocorrelation. This can create problems of finding spurious statistical relation between stock returns and these instruments (see Ferson, Sarkissian, & Simin, 2000). Ferson et al. (2000) show that the bias can be reduced by a simple form of stochastic detrending where we subtract from the actual value of the instruments the previous 12-month average value of the instruments. We experiment with this but found that this has no impact on our performance results. 10 The FTA index is a value-weighted index of the largest companies on the London Stock Exchange. 9

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be estimated from the first k eigenvectors of the cross-products matrix of excess returns. We use the first five eigenvectors of the estimated cross-products matrix of excess returns of all securities with return histories on LBS between 1955 and 1995 using the approach of Heston, Rouwenhorst, and Wessels (1995). We use five factors because Connor and Korajcyzk (1993) find that between three to six factors are important in U.S. stock returns. We refer to this model as the APT (Stat) model. We also construct two other multifactor models that are available between February 1976 and December 1995. The first is based on the APT and uses economic variables as the risk factors. We construct the APT model following the approach of Connor and Korajcyzk (1991). Connor and Korajcyzk develop an approach to form mimicking portfolios of prespecified economic factors (Chen, Roll, & Ross, 1986) using the principal components analysis of Connor and Korajcyzk (1986). We use the following five factors: (i) excess return on the FTA index, (ii) term structure—difference in monthly returns of 15-year UK government bonds (obtained from Datastream) minus the risk-free return. (iii) monthly percentage change in UK industrial production (obtained from Datastream), seasonally adjusted, (iv) monthly percentage change in UK inflation (obtained from LBS). (v) difference between risk-free return and inflation. Since factors (i) and (ii) are already portfolio returns, we do not require mimicking portfolios (see Shanken, 1992). We use the Connor and Korajcyzk (1991) technique to form mimicking portfolios for factors (iii) to (v). The first step in the Connor and Korajcyzk (1991) approach is to regress the de-meaned (actual factor realization minus the average value) factors on the five de-meaned eigenvectors obtained from the cross-products matrix. We then multiply the coefficients from the regression by the original eigenvectors to get the estimated factor portfolios for factors (iii) to (v). We refer to this model as the APT (Risk) model. The second multifactor model we use is similar to Fama and French (1993). This model includes additional factors beyond the stock market index. We use the excess returns on the FTA index as the stock market index. We measure the size factor as the difference in returns between the average monthly returns of the smallest three size portfolios described earlier and the largest three size portfolios. We measure the value/growth effect is captured by the difference in returns between the Morgan Stanley Capital International UK value index and UK growth index (obtained from Datastream). 3.4. Historical mean (Mean) This is based on the sample mean excess return of the asset of the prior 60 months. This provides a useful comparison for the other models of expected returns. Table 1 presents the summary statistics for the 10 industry portfolios. Panel A of the table includes the mean, standard deviation, and minimum and maximum monthly excess returns.

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383

Table 1 Summary statistics of industry portfolios Panel A 1 2 3 4 5 6 7 8 9 10

Mean

s

Minimum

Maximum

0.132 0.285 0.214 0.215 0.384 0.394 0.383 1.165 0.266 0.498

5.853 5.437 5.211 4.710 4.611 4.878 4.855 5.433 5.139 4.752

33.48 27.57 29.60 25.10 25.54 24.87 24.94 20.64 30.01 27.43

20.42 33.18 25.08 21.38 28.46 24.29 29.61 18.32 29.41 28.91

1

2

3

4

5

6

7

8

9

.728 .717 .684 .714 .721 .734 .481 .743 .709

.955 .943 .947 .946 .954 .554 .923 .872

.939 .919 .937 .924 .559 .893 .844

.924 .938 .923 .565 .886 .831

.926 .948 .557 .912 .855

.943 .573 .916 .834

.556 .937 .868

.552 .551

.869

Panel B: Correlations 2 3 4 5 6 7 8 9 10

The table includes summary statistics of 10 industry portfolios between February 1955 and December 1995. Panel A includes the mean, standard deviation (s), and minimum and maximum monthly excess returns (%). Panel B reports the correlations between the 10 portfolios.

Panel B reports the correlations between the 10 portfolios over the period February 1955 and December 1995. The mean monthly excess returns in panel A of Table 1 show a fairly wide cross-sectional spread in values. The utilities group has the highest mean excess return and the mineral extraction group has the poorest excess returns. The mineral extraction group also has the highest standard deviation across the 10 portfolios. The correlations in panel B range between .481 and .955. Most of the industry portfolios are highly positively correlated with one another and in excess of .7. The main exception is the utilities group, which tends to have a lower correlation with all the other groups.

4. Empirical results We begin our analysis by exploring the ability of the CAPM and APT (Stat) models to explain the in-sample predictability of UK stock returns over the whole sample period. We

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Table 2 Tests of in-sample predictability Industry

CAPM Wald

HJ

APT Wald

HJ

1 2 3 4 5 6 7 8 9 10

33.38 * 54.43 * 55.25 * 66.85 * 68.44 * 65.17 * 52.55 * 74.91 * 45.20 * 84.46 *

0.241 0.284 0.283 0.313 0.308 0.314 0.269 0.305 0.263 0.280

13.58 * 23.89 * 28.37 * 33.69 * 35.45 * 28.14 * 25.60 * 43.48 * 19.70 * 41.78 *

0.208 0.262 0.269 0.289 0.268 0.285 0.241 0.254 0.223 0.288

The table reports tests based on Kirby (1998) that examine whether the observed predictability in UK industry portfolio excess returns are consistent with either the CAPM or APT under the constant price of risk assumption. The tests are implemented on the February 1955 to December 1995 period. The Wald test examines the hypothesis that the difference between the unrestricted coefficients in Eq. (1) and the restricted coefficients implied by the CAPM or APT are jointly equal to zero. The HJ is the diagnostic test based on Hansen and Jagannathan (1997). * Significant at 5%.

use the stochastic discount formulations of the models under the assumption of a constant price of risk as in Kirby (1998). Fletcher (2001) reports results for the constant conditional beta versions of the models. Table 2 reports the Wald test and the Hansen and Jagannathan (HJ, 1997) distance measure for each of the 10 industry portfolios (see Kirby, 1998, for details as to how these tests are implemented). The Wald test examines if the difference between the unrestricted coefficients in Eq. (1) and the restricted coefficients implied by the asset pricing model are jointly equal to zero. The HJ distance measure evaluates the ability of the asset pricing model to explain the predictable variation in stock returns. A lower value of the distance measure11 implies that the model is more able to explain return predictability. Table 2 shows that neither the CAPM nor APT (Stat) models are able to explain the observed predictability in the industry portfolio excess returns. The Wald test rejects the null hypothesis that the unrestricted coefficients and restricted coefficients implied by either the CAPM or APT (Stat) models are jointly equal to each other for every portfolio. This rejection is similar to Kirby (1998) for U.S. stock returns. The HJ distance measures show that the APT (Stat) model has a lower HJ distance measure for 9 out of the 10 industry portfolios compared to the CAPM. This finding suggests that the APT is more able to explain the time-series predictability in returns compared to the CAPM, which supports Fletcher (2001). We now examine the out-of-sample performance of the asset allocation strategies. Our first set of out-of-sample tests assumes that the investor faces no investment restrictions. Table 3 reports the out-of-sample performance results between February 1960 and December 1995 of

11 As pointed out by Kirby (1998), we can only use the HJ distance measure for the stochastic discount factor formulations of the models.

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Table 3 Performance of asset allocation strategies: no investment restrictions Panel A

Mean

s

Sharpe

Minimum

Maximum

Mean Cond CAPM APT (Stat) Market

1.655 8.370 0.277 4.159 0.369

10.91 18.38 4.74 12.06 5.84

0.152 0.455 0.058 0.345 0.063

65.61 63.68 51.55 59.53 31.29

43.91 76.43 27.04 67.95 42.14

Panel B

Jensen

FS

PCM

CWM

Mean Cond CAPM APT (Stat) c12 c22

1.512 (2.90) * 8.204 (9.26) * 0.334 (1.53) 4.147 (6.89) * 92.98 * 92.62 *

1.601 (3.24) * 7.851 (9.19) * 0.443 (2.55) * 3.992 (7.33) * 90.86 * 87.97 *

0.398 (4.86) * 1.883 (2.39) * 0.184 (1.05) 1.966 (3.47) * 31.74 * 9.65 *

0.475 (4.06) * 0.065 ( 0.09) 0.056 ( 0.36) 0.629 (1.15) 21.66 * 9.58 *

The out-of-sample performance of monthly mean-variance strategies is evaluated between February 1960 and December 1995. Expected returns are estimated from four models. This includes the historical mean (Mean), linear predictive regression (Cond), and the forecasts implied from the predictive regression by the CAPM and APT (Stat) models. Panel A reports summary statistics of performance for the models and the FTA market index. This includes the mean, standard deviation (s), Sharpe (1966) measure, and minimum and maximum monthly excess returns. Panel B reports the tests of abnormal returns of the strategies. This includes the Jensen (1968) measure, the conditional measure (FS) of Ferson and Schadt (1996), the portfolio change measure (PCM) of Grinblatt and Titman (1993), and the conditional weight measure (CWM) of Ferson and Khang (2002). The portfolio weight measures are estimated between March 1960 and December 1995. The t statistics are in parentheses. The c2 statistics examine the hypotheses that the estimated performance across the four strategies are jointly equal to zero (c12) or jointly equal to zero (c22). All of the test statistics are corrected for the effects of heteroskedasticity using White (1980). The analysis assumes no investment restrictions and all performance numbers are monthly percentage. * Significant at 5%.

the four asset allocation strategies. Panel A includes summary statistics of performance, that comprises the mean and standard deviation of excess returns, the Sharpe (1966) measure, and the minimum and maximum excess returns. The corresponding figures for the FTA market index are included. Panel B reports the four performance measures and corresponding t statistics.12 To test whether the estimated performance measures are statistically different across the asset allocation strategies, we estimate two joint (Wald) tests. The c12 statistic tests the null hypothesis that the estimated performance measures across the four strategies are jointly equal to zero. The c22 statistic tests the null hypothesis that the estimated performance measures across the four strategies are jointly equal to each other. Table 3 shows that the strategy that uses the predictive regression to forecast expected returns produces dramatic out-of-sample performance. The Cond model has the highest

12

The portfolio weight measures are estimated between March 1960 and December 1995.

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Sharpe performance in panel A across the four models, that is more than three times greater than the CAPM or Mean models and market index. High mean excess returns and standard deviation characterize the Cond model. The Sharpe performance of the APT (Stat) model is lower than the Cond model but outperforms the Mean and CAPM models and market index. High mean excess returns and standard deviation characterize the APT (Stat) model. The poorest Sharpe performance is for the CAPM model. This underperforms the Mean model and market index. The estimated performance measures in panel B of Table 3 are significantly positive for the Mean, Cond, and APT (Stat) models using the Jensen, FS, and PCM performance measures. The estimated performance measures are economically large. The estimated performance is highest for the Cond model using the Jensen and FS performance measures. The APT (Stat) model has the highest performance across the four models using the PCM and CWM performance measures. The small CWM performance for the Cond model shows that the superior performance of the Cond model relative to the other performance measures arises from the use of predictable stock returns. When we control for predictability, as we do in the CWM performance measure, the abnormal performance disappears. The CAPM model has the poorest performance across the four models. The c12 and c22 statistics reject the hypotheses that the estimated performance measures of the four strategies are jointly equal to zero or each other, for all the performance measures. These tests suggest that there are significant differences across the estimated performance measures of the asset allocation strategies. The significant positive performance of the Cond model supports the findings of Fletcher (1997), Grauer (2000), Harvey (1994), and Solnik (1993). The Cond model outperforms the Mean model whenever we measure performance by the Sharpe, Jensen, FS, and PCM performance measures. Using the Cond model provides significant benefits in out-of-sample performance. Table 3 also shows that the APT (Stat) model has significant positive performance for many of the performance measures. However, the performance of the APT (Stat) model is lower than the Cond model for the Sharpe, Jensen, and FS measures. This finding suggests that the APT (Stat) model does not fully capture the predictability in UK stock returns. The performance of the APT (Stat) model is better than the CAPM. This supports Ferson and Harvey (1991, 1993) and Ferson and Korajcyzk (1995) where multifactor models do a better job in capturing time-series predictability in stock returns. The results in Table 3 assume that the investor faces no investment restrictions. Do similar results occur when the investor faces binding investment restrictions? We estimate the asset allocation strategies where we impose the investment restrictions of no short selling in the risky assets and an upper bound restriction of 20% in each of the risky assets. Table 4 reports the outof-sample performance results. The table contains the same performance data as Table 3. Table 4 highlights that when we impose binding investment constraints, the Cond model still provides significant benefits in out-of-sample performance. The Cond model has a higher Sharpe performance measure than the other three models and the FTA market index. The Cond model provides significant positive performance for all measures except the CWM performance measure. The Cond model also outperforms the Mean model for all measures except the CWM performance measure.

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Table 4 Performance of asset allocation strategies: investment restrictions Panel A

Mean

s

Sharpe

Minimum

Maximum

Mean Cond CAPM APT (Stat)

0.331 0.973 0.352 0.937

3.22 4.38 2.81 4.29

0.103 0.222 0.125 0.218

37.50 51.86 33.64 52.42

15.75 22.04 13.40 21.64

Panel B

Jensen

FS

PCM

CWM

Mean Cond CAPM APT (Stat) c12 c22

0.211 (1.52) 0.834 (4.16) * 0.281 (2.17) * 0.816 (3.99) * 32.96 * 32.38 *

0.223 (1.87) 0.771 (4.59) * 0.286 (2.80) * 0.744 (4.23) * 35.17 * 35.15 *

0.068 (1.99) * 0.520 (4.14) * 0.132 (1.21) 0.588 (4.25) * 18.56 * 16.81 *

0.024 (1.05) 0.033 (0.31) 0.010 ( 0.11) 0.091 (0.77) 2.61 2.14

The out-of-sample performance of monthly mean-variance strategies is evaluated between February 1960 and December 1995. Expected returns are estimated from four models. This includes the historical mean (Mean), linear predictive regression (Cond), and the forecasts implied from the predictive regression by the CAPM and APT (Stat) models. Panel A reports summary statistics of performance for the models. This includes the mean, standard deviation (s), Sharpe (1966) measure, and minimum and maximum monthly excess returns. Panel B reports the tests of abnormal returns of the strategies. This includes the Jensen (1968) measure, the conditional measure (FS) of Ferson and Schadt (1996), the portfolio change measure (PCM) of Grinblatt and Titman (1993), and the unconditional (UWM) and conditional (CWM) weight measures of Ferson and Khang (2002). The portfolio weight measures are estimated between March 1960 and December 1995. The t statistics are in parentheses The c2 statistics examine the hypotheses that the estimated performance across the four strategies are jointly equal to zero (c12) or jointly equal to zero (c22). All of the test statistics are corrected for the effects of heteroskedasticity using White (1980). The analysis assumes short selling and 20% upper bound constraints and all performance numbers are monthly percentage. * Significant at 5%.

Table 4 also shows that the APT (Stat) model is numerically close to the Cond model for all performance measures. The APT (Stat) model has a higher Sharpe performance measure than the Mean or CAPM models and the market index. The Jensen, FS, and PCM performance measures are all significantly positive. In contrast to the APT (Stat) model, the performance of the CAPM model is poorer. The c2 tests show that there are significant differences in the performance measures of the four strategies except CWM. Table 4 shows that the APT (Stat) model can capture all the benefits of incorporating predictable stock returns when investors face binding investment constraints. We next examine the impact of using different multifactor models in our analysis. In addition to the APT (Stat) model, we use the APT (Risk) and Fama and French (1993) (FF) models. We evaluate the out-of-sample performance of the asset allocation strategies between February 1981 and December 1995. Tables 5 and 6 report the out-of-sample performance results.13 Table 5 refers to the case where are no investment restrictions and Table 6 where there are investment restrictions. 13

We estimate the portfolio weight measures between March 1981 and December 1995.

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Table 5 Performance of asset allocation strategies: no investment restrictions (subperiod analysis) Panel A

Mean

s

Sharpe

Minimum

Maximum

Mean Cond CAPM APT (Stat) APT (Risk) FF Market

2.222 9.104 0.227 4.475 4.034 3.282 0.615

12.88 19.13 3.76 12.54 12.08 11.64 5.18

0.172 0.476 0.060 0.357 0.334 0.281 0.119

65.61 63.68 35.65 59.53 58.22 59.47 31.29

43.91 69.36 14.73 65.55 58.93 66.49 12.35

Panel B

Jensen

FS

PCM

CWM

Mean Cond CAPM APT (Stat) APT (Risk) FF c1 2 c2 2

1.859 (1.89) 8.678 (6.02) * 0.392 ( 1.26) 4.164 (4.04) * 3.699 (3.73) * 2.852 (3.26) * 51.77 * 47.40 *

2.835 (3.29) * 9.314 (7.20) * 0.284 ( 1.57) 4.573 (6.13) * 4.204 (5.71) * 3.278 (4.71) * 70.23 * 67.19 *

0.370 (2.75) * 1.895 (1.61) 0.173 ( 0.64) 1.594 (1.51) 1.425 (1.38) 1.383 (1.68) 12.82 * 6.72

0.493 (1.92) 0.371 ( 0.33) 0.221 ( 0.92) 0.023 ( 0.03) 0.004 ( 0.01) 0.314 (0.43) 7.89 6.88

The out-of-sample performance of monthly mean-variance strategies is evaluated between February 1981 and December 1995. Expected returns are estimated from five models. This includes the historical mean (Mean), linear predictive regression (Cond), and the forecasts implied from the predictive regression by the CAPM, APT (Stat), and APT (Risk) models. Panel A reports summary statistics of performance for the models and the FTA market index. This includes the mean, standard deviation (s), Sharpe (1966) measure, and minimum and maximum monthly excess returns. Panel B reports the tests of abnormal returns of the strategies. This includes the Jensen (1968) measure, the conditional measure (FS) of Ferson and Schadt (1996), the portfolio change measure (PCM) of Grinblatt and Titman (1993), and the unconditional (UWM) and conditional (CWM) weight measures of Ferson and Khang (2002). The portfolio weight measures are estimated between March 1981 and December 1995. The t statistics are in parentheses. The c2 statistics examine the hypotheses that the estimated performance across the six strategies are jointly equal to zero (c12) or jointly equal to zero (c22). All of the test statistics are corrected for the effects of heteroskedasticity using White (1980). The analysis assumes no investment restrictions and all performance numbers are monthly percentage. * Significant at 5%.

The results in Table 5 show a similar picture to Table 3. The Cond model provides significant benefits in out-of-sample performance. The Cond model has the highest Sharpe performance across the six strategies and has significant positive performance using the Jensen and FS performance measures. The Cond model has the highest estimated performance across the six strategies for all performance measures except CWM. The performance measures of the Cond model are higher over the subperiod compared to the whole sample period except for the CWM. This finding suggests that the benefits of the Cond model are robust to the sample period used when there are no constraints. The APT (Stat), APT (Risk), and FF models also provide significant benefits in out-ofsample performance. All three models have higher Sharpe measures than the Mean and CAPM models. In addition, all three models have significant Jensen and FS performance measures. The APT (Stat) model provides the most positive abnormal returns across the three

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Table 6 Performance of asset allocation strategies: investment restrictions (subperiod analysis) Panel A

Mean

s

Sharpe

Minimum

Maximum

Mean Cond CAPM APT (Stat) APT (Risk) FF

0.236 0.629 0.020 0.646 0.539 0.571

3.93 5.53 3.41 5.43 5.42 5.73

0.060 0.114 0.006 0.119 0.099 0.099

37.50 51.87 33.65 52.42 51.46 51.69

15.75 22.04 12.08 21.64 21.64 19.30

Panel B

Jensen

FS

PCM

CWM

Mean Cond CAPM APT (Stat) APT (Risk) FF c12 c22

0.085 ( 0.29) 0.202 (0.48) 0.236 ( 0.91) 0.269 (0.59) 0.154 (0.35) 0.136 (0.31) 6.97 4.34

0.315 (1.54) 0.523 (1.98) * 0.157 ( 1.02) 0.642 (2.75) * 0.584 (2.45) * 0.506 (2.05) * 24.19 * 22.06 *

0.105 (1.35) 0.322 (1.46) 0.047 ( 0.22) 0.421 (1.79) 0.320 (1.59) 0.314 (2.01) * 6.85 6.76

0.070 (1.29) 0.089 ( 0.47) 0.125 ( 0.69) 0.004 (0.02) 0.003 ( 0.20) 0.002 (0.01) 5.59 4.14

The out-of-sample performance of monthly mean-variance strategies is evaluated between February 1981 and December 1995. Expected returns are estimated from five models. This includes the historical mean (Mean), linear predictive regression (Cond), and the forecasts implied from the predictive regression by the CAPM, APT (Stat), and APT (Risk) models. Panel A reports summary statistics of performance for the models. This includes the mean, standard deviation (s), Sharpe (1966) measure, and minimum and maximum monthly excess returns. Panel B reports the tests of abnormal returns of the strategies. This includes the Jensen (1968) measure, the conditional measure (FS) of Ferson and Schadt (1996), the portfolio change measure (PCM) of Grinblatt and Titman (1993), and the unconditional (UWM) and conditional (CWM) weight measures of Ferson and Khang (2002). The portfolio weight measures are estimated between March 1981 and December 1995. The t statistics are in parentheses. The c2 statistics examine the hypotheses that the estimated performance across the six strategies are jointly equal to zero (c12) or jointly equal to zero (c22). All of the test statistics are corrected for the effects of heteroskedasticity using White (1980). The analysis assumes short selling and 20% upper bound constraints all performance numbers are monthly percentage. * Significant at 5%.

multifactor models. However, the performance of the multifactor models is lower than the Cond model. In contrast to the multifactor models, the CAPM performs poorly. The CAPM model has a negative Sharpe measure and negative abnormal returns irrespective of how we measure performance. These findings suggest that multifactor models do not fully explain the predictability in UK stock returns and multifactor models do a better job than the CAPM in explaining the predictability in returns. This supports our earlier findings for the whole sample period. Table 6 shows that when we impose investment constraints, the Cond model provides less benefit in out-of-sample performance. The Sharpe performance of the Cond model is higher than the Mean, CAPM, APT (Risk), and FF models but similar to the market index. The Cond model only provides significant positive performance using the FS performance measure. The abnormal returns for the Cond model are lower than the whole sample period in Table 4. The reduced significance in performance might be linked to Handa and Tiwari (2001). Handa and

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Tiwari find the performance of the Cond model in U.S. stock returns varies according to the sample period used. Over the 1985–1998 period, the performance of the Cond model is poorer than the Mean model. Our results differ from this since we find the performance of the Cond model is still better than the Mean model over our subperiod. Table 6 also shows that the APT (Stat) model matches the performance of the Cond model. The APT (Stat) model has a similar Sharpe performance to the Cond model. The APT (Stat) model also provides higher estimated performance than the Cond model for all four performance measures. The APT (Risk) and FF models yield lower performance compared to the APT (Stat) model. The CAPM model continues to perform poorly. There is less evidence of significant differences in performance across the six strategies as reflected in the c2 tests. The findings in Tables 5 and 6 support those in Tables 3 and 4. Multifactor models do not fully capture the predictability in UK stock returns as reflected by the differences in performance in Table 5 compared to the Cond model. However, with investment constraints, APT models can capture all the benefits of predictable stock returns. Our additional finding in Tables 5 and 6 is that the APT (Stat) model performs as well as other multifactor models even using a model similar to Fama and French (1993). This supports Ferson and Korajcyzk (1995) who find that APT-based models using either principal components or economic risk factors have similar power to explain the time-series predictability in returns.

5. Conclusions We examine the out-of-sample performance of domestic asset allocation strategies using forecasts of expected returns based on the predictability of returns. We use forecasts based on the unrestricted version of the linear predictive regression as well as the restricted version of the predictive regression that is consistent with a candidate asset pricing model (Kirby (1998)). We find that the Cond model provides significant benefits in out-of-sample performance, even when the investors face investment restrictions. The Cond model has significant positive performance for a range of performance measures. We can attribute this positive performance to the impact of predictable stock returns since when we use the CWM performance measure, this abnormal performance disappears. Our evidence supports Harvey (1994) and Solnik (1993) among others that there is a strong economic basis of stock return predictability. We also find that the performance of the APT (Stat) model provides significant benefits in out-of-sample performance. When there are no investment constraints, the performance of the APT (Stat) model is poorer than the Cond model. This poorer performance suggests that multifactor models are unable to fully explain the predictability in UK stock returns. However, with binding investment constraints, the performance of the APT (Stat) model matches that of the Cond model. This result suggests that the APT (Stat) model can capture the benefits of predictable stock returns when investors face realistic investment constraints. Our subsidiary findings show that the APT model based on statistical factors performs as well as models using economic risk factor or empirical factors as in Fama and French (1993).

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In addition, multifactor models outperform the CAPM and are more able to explain any timeseries predictability in stock returns.

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