An examination of the economic significance of stock return predictability in UK stock returns Jonathan Fletcher a, * , Joe Hillier b 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 1059-0560/02/$ – see front matter D 2002 Elsevier Science Inc. All rights reserved. PII:S1059-0560(02)00138-7 * Corresponding author. Tel.: +44-141-548-3892; fax: +44-141-552-3547. E-mail address: [email protected] (J. Fletcher). International Review of Economics and Finance 11 (2002) 373–392
Transcript
An examination of the economic significance of stock
return predictability in UK stock returns
Jonathan Fletchera,*, Joe Hillierb
aDepartment of Accounting and Finance, University of Strathclyde, Curran Building,
100 Cathedral Street, Glasgow G4 0LN, UKbGlasgow 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
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).
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392374
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 per-
formance 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 D0iuZZZZZZt�1 þ uit ð1Þ
where rit is the excess return of asset i in period t, Diu0 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:
EðriÞ ¼ D0iuZZZZt�1: 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
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392 375
conditional asset betas are constant2 as in Kirby (1998). The conditional CAPM or APT
implies the following relationship:
Eðrit j ZZZZZt�1Þ ¼XKk¼l
bikEðrkt j ZZZZZZt�1Þ ð3Þ
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 ¼XKk¼l
bbkbik ð4Þ
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:
EðriÞ ¼ D0irZZZZZt�1 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 � bimrmtÞrmt ð6aÞ
u2t ¼ ZZZZ 0t�1ðrit � D
0iuZZZZZZt�1Þ ð6bÞ
u3t ¼ ZZZZ 0t�1ðbimrmt � D
0irZZZZZZt�1Þ 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.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392376
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 mean-
variance 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.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392 377
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 birmt þ eit ð7Þwhere eit is a random error term with E(eit) = 0 and E(eitrmt) = 0. The bi coefficient is the betaof 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:
biðZZZZt�1Þ ¼ bi þXLl¼1
Dilzlt�1 ð8Þ
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 birmt þXLl¼1
Dilrmtzlt�1 þ eit ð9Þ
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.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392378
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=TÞXTt¼1
XNi¼1
ritðwit � wit�kÞ ð10Þ
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:
CWMt ¼ EXNi¼l
ðwit � wbitkÞðrit � EðrijZZZZt�1ÞÞjZZZZt�1
" #ð11Þ
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.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392 379
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 assump-
tions 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 CWMþ ;0t�1 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. Mineral extraction—group numbers between 123 and 165.
2. Building and chemicals—group numbers between 210 and 255.
3. Engineering—group numbers between 261 and 270.
4. Printing and textiles—group numbers between 282 and 297.
5. 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.
z
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392380
6. Distribution, leisure, and media—group numbers between 412 and 436.
7. Retailers and other services—group numbers between 440 and 490.
8. Utilities—group numbers between 620 and 680.
9. Financials—group numbers between 710 and 794.
10. 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.
10 The FTA index is a value-weighted index of the largest companies on the London Stock Exchange.
9 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.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392 381
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 Data-
stream), 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.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392382
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
Table 1
Summary statistics of industry portfolios
Panel A
Mean s Minimum Maximum
1 � 0.132 5.853 � 33.48 20.42
2 0.285 5.437 � 27.57 33.18
3 0.214 5.211 � 29.60 25.08
4 0.215 4.710 � 25.10 21.38
5 0.384 4.611 � 25.54 28.46
6 0.394 4.878 � 24.87 24.29
7 0.383 4.855 � 24.94 29.61
8 1.165 5.433 � 20.64 18.32
9 0.266 5.139 � 30.01 29.41
10 0.498 4.752 � 27.43 28.91
Panel B: Correlations
1 2 3 4 5 6 7 8 9
2 .728
3 .717 .955
4 .684 .943 .939
5 .714 .947 .919 .924
6 .721 .946 .937 .938 .926
7 .734 .954 .924 .923 .948 .943
8 .481 .554 .559 .565 .557 .573 .556
9 .743 .923 .893 .886 .912 .916 .937 .552
10 .709 .872 .844 .831 .855 .834 .868 .551 .869
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.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392 383
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.
Table 2
Tests of in-sample predictability
Industry CAPM Wald HJ APT Wald HJ
1 33.38 * 0.241 13.58 * 0.208
2 54.43 * 0.284 23.89 * 0.262
3 55.25 * 0.283 28.37 * 0.269
4 66.85 * 0.313 33.69 * 0.289
5 68.44 * 0.308 35.45 * 0.268
6 65.17 * 0.314 28.14 * 0.285
7 52.55 * 0.269 25.60 * 0.241
8 74.91 * 0.305 43.48 * 0.254
9 45.20 * 0.263 19.70 * 0.223
10 84.46 * 0.280 41.78 * 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%.
J. Fletcher, J. Hillier / International Review of Economics and Finance 11 (2002) 373–392384
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
Table 3
Performance of asset allocation strategies: no investment restrictions
