The cross-section of returns in the U.K. stock market

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The cross-section of returns in the U.K. stock market

Abstract

This paper examines the ability of six different asset pricing models to explain the cross-section of returns in the U.K. stock market (2002M04-2014M06) on a monthly and quarterly basis. The paper of Kang et al. (2011) is followed and their two conditional models are tested. Overall all six models are poor with the Carhart (1997) four-factor model as the best model. The Carhart (1997) four-factor model is able to explain 14.6%-59.3% of the variance of the cross-section of returns in the U.K. stock market. U.K. specific asset pricing models are needed which need to be developed in the future.

JEL classification: G12

Keywords: asset pricing, CAPM, CCAPM, three-factor model, four-factor model, macroeconomic variables, cross-section of returns

Studentnr: s2015773 Author: T.I.M. Fierkens

Email: timfierkens@gmail.com Phone: +31 621849549 Study Program: MSc Finance

Supervisor: prof. dr. T.K. (Theo) Dijkstra Date: 14th of January 2015

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1. Introduction

This paper examines the ability of six different asset pricing models and the conditional Consumption Capital Asset Pricing Model (CCAPM) developed by Kang et al. (2011)1 in particular, to explain the cross-section of returns in the U.K. stock market (2002M04-2014M06).

Over the last decades, researchers have proposed many models in order to predict expected (excess) returns of stocks and portfolios, such as the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT) and multi-factor models. Probably the most famous model is the CAPM developed by Sharpe (1964) and Lintner (1965). The CAPM is a popular asset pricing model based on the relative risk of the respective asset with respect to the market risk.

However, in later years the CAPM has received fierce criticism as the model lacks the ability to describe the cross-section of returns (Fama and French, 1993). Therefore, conditional versions of the CAPM were developed which are more capable to explain the cross-section of returns. Therefore, several researchers have proposed other factors which should be able to predict expected returns of stocks, for example Jagannathan and Wang (1996). The most widely used factor models in asset pricing literature are the Fama and French (1993) three-factor model, who add a size and a value factor to the CAPM and the Carhart (1997) four-factor model which adds a momentum factor to the aforementioned three-factor model.

Furthermore, researchers have advocated the use of several other factors to predict returns and to explain the cross-section of returns. Frequently studied factors are dividend (yield) and consumption, as they are expected to have a strong predictive power (KEA). In the recent years, consumption as an added factor to the CAPM, creating the CCAPM, introduced by Breeden (1979), has gained more support as a successful asset pricing model. This is due to the fact that the results of these models are comparable (Lettau and Ludvigson 2001b) in explaining the cross-section of returns to the three-factor model of Fama and French (1993) in the U.S. stock market. The consumption-based models are also able to show that value stocks are riskier than growth stocks at certain times of the business cycle (KEA).

Recently KEA developed a conditional CCAPM which uses four macroeconomic variables to explain the cross-section of returns: the dividend yield, the term spread, the default spread and the short-term interest rate. These variables together form their conditioning variable ‘coin’ and thus their model ‘coin CCAPM’. The results look very promising for the U.S. stock market, as their coin CCAPM is able to explain 74% of the variance in the cross-section of stock returns. These results are comparable to the three-factor model of Fama and French (1993) in the U.S.. However, it is uncertain

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3 whether the model is also successful in other stock markets, such as Japan and the U.K.. I extend existing literature by testing the conditional CAPM and CCAPM of KEA in the U.K. stock market.

The U.K. stock market is chosen as Cheng (1998) found that the economies of the U.K. and the U.S. are closely related to each other. He also found a statistically significant relation between the stock markets of the two countries, where the U.S. economy has a high influence on the U.K. economy. Therefore, the conditional variable of KEA is also likely to have a countercyclical relationship with the U.K. business cycle. Hence, the two conditional models of KEA should be able to explain the cross-section of stock returns in the U.K. stock market.

This paper investigates whether the conditional CAPM and CCAPM by KEA are indeed able to explain the cross-section of stock returns in the U.K. stock market, as well as how the two conditional models perform compared to four other popular asset pricing models: the standard CAPM, the standard CCAPM, the Fama and French (1993) three-factor model and the Carhart (1997) four-factor model. The Fama and MacBeth (1973) method is used to estimate the cross-sectional regressions.

The results are that the proposed models by KEA appear to be weak performers in the U.K. stock market. The four-factor model from Carhart (1997) proves to be overall the best model to explain the cross-section of returns in the U.K. stock market during the period 2002M04-2014M06 and best when using the Fama and French 25 size and book-to-market portfolios. Furthermore, KEA found a negative relationship of the conditioning variable coin with the U.S. business cycle. In this paper, a negative relationship of coin with the U.K. business cycle has not been found.

This paper provides insights for portfolio managers, investors and academics as to what models can be best used in the U.K. stock market to explain the cross-section of stock returns. In particular, how the two conditional models from KEA perform in a different market than the U.S. stock market in explaining the cross-section of stock returns.

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2. Literature review

On the ideas of Markowitz (1959) the Capital Asset Pricing Model (CAPM), which was the first asset pricing model in modern finance, is created by Sharpe (1964) and Lintner (1965). The CAPM is one of the most famous models in the finance literature and there have been a lot of empirical testing and improvements in the last decades. The CAPM is based on three assumptions. Firstly, the relationship between risk and the expected return is linear. Secondly, the risk measure beta is a complete measure of risk. Thirdly, a higher risk should be associated with higher returns.

Later, the cross-section of returns became a more researched topic when Fama and French (1992) found that ‘’two easily measured variables, size and book to-market equity, seem to describe the cross-section of average stock returns’’ in the U.S. stock market. Furthermore, a year after their previous study Fama and French (1993) found that the CAPM lacks the ability to explain the cross-section of returns. When studying the cross-cross-section of returns it is investigated how average returns change over different stocks or portfolios, whereas a time-series study studies how average returns change over time (Cochrane, 2005). Fama and French (1993) came with their famous three-factor model, which was a more extended version of their 1992 research. Fama and French (1993) found common risk factors in the returns of stocks and bonds. These common risk factors were incorporated into their three-factor model and, moreover, this model appeared to explain the average returns of stocks and bonds more successfully than previous models.

However, the findings of Fama and French (1992, 1993) for the U.S. were not supported by researchers in the U.K. stock market. Chan and Chui (1996) did the study for the U.K. and found no significant effect of the size factor on average returns for the U.K. stock market.

In the same year Jegadeesh and Titman (1993) formed the basis of the momentum factor, also known as the winner minus loser (WML) factor. They found that the strategy of buying past winners and selling past losers yields significant positive returns. Years later, Carhart (1997) used this factor in addition to the three-factor model of Fama and French (1993), thereby creating the four-factor model.

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5 latter is more realistic in our world, whereas the static CAPM is very inaccurate according to Fama and French (1992). Both studies advocate using time-varying betas and risk premiums.

Another variation on the CAPM is the CCAPM, proposed by Breeden (1979), who added a consumption factor to the existing model. The CCAPM has the added benefit of being intuitively appealing (Cochrane, 2005). Nonetheless, consumption factor models have performed poorly as an asset pricing model (Cochrane, 1996). Mankiw and Shapiro (1986) found that the average return in their sample was more closely related to the market beta than to the beta measured with respect to consumption growth. A reason could be that consumers have a rather passive role on the stock market, as their wealth is usually stuck in either a savings account or a pension fund. Still, Mankiw and Shapiro (1986) maintain that the model could work effectively for the small, more actively involved, consumer group. Faff and Oliver (1998) also compared the performance of the traditional market betas with the consumption betas for the Australian market and found results similar to those of Mankiw and Shapiro (1986), although when they lag consumption the model performs significantly better.

Another problem of CCAPM is the lack of data: much of the consumer data is published monthly or quarterly, whereas data on e.g. stocks are available daily and often also over a longer time period. As a consequence, most CCAPM studies are quarterly studies with a rather short horizon. Despite these problems, Breeden et al. (1989) found that the performance of their CCAPM and the traditional CAPM are similar. Their version of the CCAPM is tested ‘’using betas based on both consumption and the portfolio having the maximum correlation with consumption’’. Furthermore, Breeden et al. (1989) found a significantly positive market price of risk with their version of the CCAPM, i.e. they found that the return increases when consumption risk increases. Moreover, Campbell and Cochrane (1999) developed another consumer based model able to explain some asset pricing issues. Their model uses a surplus consumption ratio and is able to explain the pro-cyclical variation of stock prices, the countercyclical variation of stock market volatility and the long horizon predictability of excess stock returns. Hereafter, more and more researchers considered the CCAPM as a possible model to explain the cross-section of returns instead of the CAPM.

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6 unconditional models and approximately as well as the Fama and French (1993) three-factor model in explaining cross-sectional returns, using their consumption-wealth ratio as conditioning variable. Furthermore, Lettau and Ludvigson (2001b) suggested that a multifactor version of the CCAPM should be able to explain a large portion of the cross-section of expected stock returns.

More recently, Yogo (2006) showed that the marginal utility of consumption is able to explain ‘’the trade-off between risk and return reflected in the size premium, value premium and the time-varying equity premium’’. Jagannathan and Wang (2007) found that investors are more likely to review their consumption and investment plans at the end of the calendar year and rather when the economy is in expansion than in contraction. Jagannathan et al. (2012) found that the performance of the CCAPM is indeed better at the end of the tax year in the U.S., the U.K. and Japan as suggested by Jagannathan and Wang (2007). Cochrane (2008) still believes that consumption-based models must be right ‘’if economics is to have any hope of describing stock markets’’. These findings are interesting, however this study will not investigate these effects for the U.K. stock market.

Ultimately, KEA extended the work of Lettau and Ludvigson (2001a,b) by building a conditional version of the CCAPM using the conditioning variable ‘coin’ from the co-integrating relation among four macroeconomic variables (dividend yield, term spread, default spread and the short-term interest rate). The conditioning variable of KEA possesses a strong power to predict future U.S. stock market returns and performs approximately as well as the Fama and French three-factor model (1993) in explaining the cross-section of average equity returns. It performs better than the conditioning variable introduced by Lettau and Ludvigson (2001b) for the U.S. stock market. The work of KEA has not been tested in other geographic areas yet, whereas the work of Lettau and Ludvigson (2001a,b) already has been. Gao et al. (2008) found that the model was able to explain the cross-section of returns in both the U.K. and Japan.

3. Methodology

In this study six asset pricing models are used. The central model in this paper is the model developed by KEA, is a conditional version of the CCAPM. The CCAPM is an asset pricing model derived from the CAPM, a popular asset pricing model in the finance literature, which has been developed by Sharpe (1964) and Lintner (1965). The CAPM is a linear pricing model described as:

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7 in the CAPM is the usual regression coefficient. The error terms for all models are possibly heteroskedastic and autocorrelated. We will calculate standard errors that take these issues in account, see Section 4. Following the notations by KEA the vector of factors in the stochastic discount factor is denoted as . For the respective models the stochastic discount factors are the factor(s) used in the Fama and Macbeth (1973) cross-sectional regressions. For the CAPM, there is only one factor:

where is the excess market portfolio return. Since it is very likely that excess market returns can be forecasted, as has been demonstrated by empirical studies, the CAPM does not hold whereas a conditional CAPM does (KEA). As explained by Campbell and Cochrane (1999), and are time-varying in conditional models.

The conditional CAPM is based on the standard CAPM, but it has different factors. The discount factor consists of the conditioning variable coin from KEA, the excess market return and the interaction of coin with the excess market return. The conditioning variable is explained in more detail in Section 4. The factors for the conditional CAPM are:

where is the conditioning variable at time t and is the excess market portfolio return at time t+1.

The CCAPM does not focus on the market returns but focuses on consumption growth rates. This linear model is determined by the correlation between the consumptions growth rates and the return of an asset (KEA). The CCAPM is a linear model and is estimated by:

where is the total net return of portfolio i at time t, is the risk-free rate, is the risk parameter for consumption risk, is the risk premium for consumption risk at time t and is the error term of the model. in the CCAPM is the usual regression coefficient. The standard CCAPM has only one factor: the log consumption growth rate (KEA) which is defined as:

where is the log consumption growth rate at time t+1.

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8 conditioning variable coin and the interaction of coin with the log consumption growth rate are new factors in the discount factor. The factors of the conditional CCAPM are:

where is the conditioning variable at time t and is the log consumption growth rate at time t+1. This is the key discount factor in this paper, since it is used to test the models ability to explain the cross-section of returns in the U.K. stock market.

A more contemporary model used frequently in asset pricing literature to explain the cross-section of stock returns is the three-factor model from Fama and French (1993). The three-factor model is an extended model from the CAPM and is denoted as:

where is the total net return of portfolio i at time t, is the risk-free rate, , and measure the exposure of portfolio i to the market, size and value factors at time t, is the risk premium for market risk at time t and is the error term of the model. The model adds two factors to the original CAPM: SMB and HML. SMB, small minus big, is the size factor and HML, high minus low, is the value factor. The factors of the three-factor model are therefore:

where is the excess portfolio market return at time t+1, is the size factor at time t and is the value factor at time t.

As an extension to the work of KEA this paper also tests the Carhart (1997) four-factor model, which adds a momentum factor (WML) to the Fama and French (1993) three-factor model of. The Carhart (1997) four-factor model is as follows:

where is the total net return of portfolio i at time t, is the risk-free rate, , , and measure the exposure of portfolio i to the market, size, value and momentum factors at time t, is the risk premium for market risk at time t, and are the Fama and French (1993) size and value factors, is the momentum factor at time t (Carhart, 1997) and is the error term of the model. Therefore, the factors of the four-factor model are:

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9 The method used to explain the section of returns is the Fama and MacBeth (1973) cross-sectional regression. Because of the use of the cross-cross-sectional regression method, the results are easy to compare to those of other asset pricing literature (since this method is frequently used in asset pricing models) (KEA). The Fama and MacBeth (1973) method is a two-stage regression model and is used to test all the six competing asset pricing models. In the first stage of the Fama and MacBeth (1973) method the multivariate time-series regressions for the portfolios are run to estimate the beta coefficients of the relative factors:

where is the return of portfolio i at time t+1 and are the previously described factors of the respective models at time t+1. The full sample betas are estimated as in Lettau and Ludvigson (2001a, b) and KEA instead of using rolling betas. This is acceptable since full sample betas have been used more frequently since the 2000s (KEA). The second stage of the method for every t, the excess returns of the 25 portfolios are regressed on a constant and the estimated betas:

2 where is the intercept and γ’ are regression coefficients to determine the risk premium of the respective factor . This discount factor is different for the six different asset pricing models. KEA compared their conditional model with two previous developed conditional CCAPM (Lamont, 1998 and Lettau and Ludvigson, 2001a). In this paper only the proposed conditional CAPM and CCAPM of KEA are tested, since KEA have proven that their conditional model outperforms the two other conditional models in explaining and predicting excess returns in the U.S. stock market.

4. Data and descriptive statistics

Most needed data is available on a monthly, quarterly and annual basis, such as consumption growth. In contrast to e.g. the returns of equities which are available daily. The required data to calculate coin is only available from 2002M04 onwards. Therefore, I use monthly data for my main results instead of the more frequently used quarterly data, in order to have substantial more data points for all the statistical analyses. As quarterly data is more frequently used, for robustness purposes the tests will also be conducted with quarterly data in Section 6.

Moreover, the Fama and French 25 size and book-to-market value-weighted portfolios for

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10 the U.K. market (Gregory et al., 2013) are used, which can be obtained from the Exeter University3. Value-weighted portfolios are chosen as they better reflect the markets than equally-weighted portfolios (Gregory et al., 2013). The Fama and French 25 size and book-to-market sorted portfolios are used as they are frequently used in empirical research and therefore the results are easier comparable to other empirical studies. Furthermore, these portfolios are designed to investigate two effects. Firstly, the size effect: on average firms with a small market capitalisation have higher returns than firms with a higher market capitalisation. Secondly, the value effect: on average firms with a high book-to-market ratio have higher returns than firms with low book-to-market ratios (Fama and French, 1993). The market return, the risk-free rate, risk premium, SMB, HML and WML for the U.K. market are also obtained from the Exeter University.

It is assumed that there exists a risk-free asset, denoted as rf. The market portfolio return, is the total return on the FT All Share Index and the risk-free rate is the monthly return on the three-month U.K. Treasury Bills (Gregory et al., 2013). The excess market portfolio returns ( ) are calculated by simply subtracting the risk-free rate from the market returns. The construction of the Fama and French (1993) factors (SMB and HML), the Carhart (1997) factor (WML) and portfolios are presented in Fama and French (1993), Carhart (1997) and adapted by Gregory et al. (2013) for the U.K. stock market.

For the log consumption growth rate KEA are followed, which implies that healthcare, education and housing are excluded as the expenditure of these sector are not entirely for personal consumption and are subject to large adjustment costs. The data on household consumption and its components are provided quarterly by the Office for National Statistics (ONS) in their so called ‘consumer trends’4.

The descriptive statistics of the monthly factors are presented in Table 1. There are some differences in the descriptive statistics in comparison to those of KEA. The mean of the conditioning variable coin is positive whereas at KEA the variable is negative. Furthermore, the excess returns are lower and the correlation of coin with the excess market return and SMB is negative whereas in the study of KEA this is positive. The Jarque-Bera statistics, which test for normality of the data, are high for the factors, but according to Brooks (2008) the violation of the normality assumption for OLS estimation is negligible. Lastly, there are no signs of multicollinearity, as there are no correlation coefficients which have a higher value than 0.7 (Brooks, 2008).

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http://business-school.exeter.ac.U.K./research/areas/centres/xfi/research/famafrench/files/

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11 Following KEA the conditioning variable, coin, which consists of four macroeconomic variables is defined as:

where is the dividend yield at time t, is the term spread at time t, is the default spread at time t and is the short-term interest rate at time t. The intuitive interpretation behind the conditioning variable ‘coin’ can be found in detail in KEA. The four variables are the most commonly used in asset pricing literature and are therefore used by KEA. As the results of KEA were promising for the U.S. stock market I have replicated their conditioning variable exactly for the U.K. stock market. The dividend yield ( ) is the dividend yield of the FT All Share Index, the term spread is the difference between the yields of a 10-year and a 1-month U.K. government gilt, the default spread ( is the difference between the yields of a year S&P BBB and a 10-year S&P AAA U.K. corporate bond and the U.K. short-term interest rate ( ) set by the Bank of England5.

Table 1: Descriptive statistics of the monthly factors for the U.K. market

The table shows the mean, standard deviation, skewness, kurtosis, Jarque-Bera and the first-order autocorrelation statistics of the conditioning variable coin, the log consumption growth, excess market return, the two Fama and French factors (SMB and HML) and the momentum factor (WML). Furthermore, the correlation coefficients are displayed. The sample period is 2002M04-2014M06. Factors coin Δc rm SMB HML WML Mean 0.047 0.002 0.004 0.003 0.001 0.008 Standard deviation 0.013 0.020 0.042 0.032 0.025 0.048 Skewness 0.000 -0.803 -0.703 0.330 0.188 -1.702 Kurtosis 1.925 2.334 3.958 6.823 4.351 10.440 Jarque-Bera 7.081 18.514 17.716 92.182 12.044 4.073 Autocorrelation 0.971 0.572 0.072 0.165 0.099 0.360

Correlation matrix coin Δc rm SMB HML WML

coin -0.010 -0.251 -0.169 -0.078 0.027 Δc 0.038 -0.098 0.078 -0.087 rm - rf 0.131 0.447 -0.101 SMB 0.215 -0.376 HML -0.403 WML

The descriptive statistics of the four variables compromising coin are presented in Table 2. The dividend yield is skewed, has a high kurtosis and therefore a high Jarque-Bera statistic indicating a non-normal distribution. The other three macroeconomic variables have a more normal distribution.

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12 Again, the Jarque-Bera statistics are relatively high but can be ignored when used in statistical testing according to Brooks (2008). Regarding the correlations of the four macroeconomic variables with coin it can be seen that only the dividend yield has a positive correlation with coin, thus showing different signs than in equation (13). There are signs of multicollinearity, but this is not odd as the four variables are used to construct the conditioning variable coin.

Table 2: Descriptive statistics of the four monthly variables which constitute coin

The table shows the mean, standard deviation, skewness, kurtosis and Jarque-Bera statistics. Furthermore, the correlation coefficients of the conditioning variables are displayed. DIV is the dividend yield, TERM is the term spread, DEF is the default spread, RF is the short-term interest rate and coin is the conditioning variable of KEA. The sample period is 2002M04-2014M06.

Variables DIV TERM DEF RF coin

Mean 0.034 0.012 0.014 0.027 0.047

Standard deviation 0.005 0.013 0.005 0.021 0.013

Skewness 2.152 0.190 0.944 -0.008 0.000

Kurtosis 8.567 1.786 3.985 1.203 1.925

Jarque-Bera 303.311 9.921 27.280 19.773 7.081

Correlation matrix DIV TERM DEF RF coin

DIV 0.235 0.747 0.167 0.200

TERM 0.474 0.550 -0.843

DEF 0.333 -0.135

RF -0.672

coin

In Figure 1 the excess returns are plotted over time, where the vertical grey bars are the times where the OECD recession indicator indicated a recession6_{. Figure 1 displays that the recession indicator}

shows four recession periods most notably the 2007 worldwide financial crisis and the more recent European sovereign debt crisis. What is surprising is that the excess returns during the recession period in 2004 are quite stable without high deviations whereas in the 2007 financial crisis the returns are highly fluctuating.

Following KEA the Johansen Cointegration test is conducted to test the co-integrating relation of the four macroeconomic variables compromising the conditioning variable coin. The results of the Johansen Cointegration test are presented in Table 3.

Both test statistics, Trace and max, indicate that there is one co-integrating equation at the 5% significance level. Thus, the four macroeconomic variables together have a co-integrated relation. This is in line with the findings of KEA, who also found that the four macroeconomic variables have a co-integrating relationship.

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13 Figure 1: Monthly excess returns and the OECD based recession indicator (RI) for the U.K. provided by the central bank of St Louis for the period 2002M04 until 2014M06 in the U.K..

-.15 -.10 -.05 .00 .05 .10 .15 02 03 04 05 06 07 08 09 10 11 12 13 14 RI Rm - Rf re tu rn time

Table 3: The Johansen Cointegration test of the four conditioning variables using the Trace and max statistic

A linear trend in the conditioning variables and an intercept are allowed and the number of lags is processed automatically using the Akaike Information Criterion (five in this case).

Null Hypothesis Trace max

Test statistic 95% Critical value Test statistic 95% Critical value

r = 0 75.691* 47.856 50.313* 27.584 r ≤ 1 r ≤ 2 r ≤ 3 25.379 10.765 2.750 29.797 15.495 3.841 14.614 8.015 2.750 21.132 14.265 3.841 *statistically significant at α=0.05, both statistical tests indicate one co-integrating equation.

As suggested by Lewellen et al. (2010) a good way to improve empirical testing is to expand the set of test portfolios, especially for improving robustness testing. This is particularly the case for my dataset, which has a relative short horizon compared to other asset pricing literature, different portfolios could provide different insights and more robust results. Therefore, the 25 value-weighted size and momentum portfolios, which are also used by KEA, are used as an extra set of test portfolios. Because KEA also used these portfolios, the results of the cross-sectional analyses are comparable. These portfolios are also available via the Exeter University for the U.K. stock market and the creation of these portfolios is described by Gregory et al. (2013).

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14 Carhart (1997) when testing the size and momentum portfolios. Moreover, because Fama and French (2012) tested their three-factor model and the four-factor model of Carhart (1997) in Europe, Japan, North-America and Asia-Pacific, but not specifically for the U.K. stock market.

Finally, a cross-sectional regression analysis will be performed on the 25 size and book-to-market portfolios in the pre financial crisis period, that is from 2002M04 until 2008M03, since the crisis in the U.K. started in 2008Q2. The aforementioned cross-sectional regression analyses are also performed on quarterly data from 2002Q2 until 2014Q2 for the 25 size and book-to-market portfolios and 25 size and momentum portfolios respectively. The quarterly analysis on the pre-crisis period is from 2002Q2 until 2008Q1. The additional cross-sectional analyses for robustness can be found in Section 6.

In the next section results are presented according to previous asset pricing literature. Therefore the results of the models are displayed using the adjusted R2 as model assessment, to accommodate an easy comparison between the performance of the models with former academic empirical research and in particular with the U.S. results of KEA. The adjusted R2 has nice features which make it commonly used. When a regressor is added to a model the adjusted R2 only increases when the F-statistic of the model is greater than one. If the added regressor has a F-F-statistic of one the adjusted R2 remains the same and when it is smaller than one, the adjusted R2 decreases7. This is not the case when using the standard R2, since this measure always increases when a regressor is added to the model (Brooks, 2008). Therefore, the adjusted R2 measure is more insightful (Brooks, 2008).

All t-statistics in the results section are based on the Newey-West (1987) technique, which adjust the standard errors for heteroskedasticity and autocorrelation. Heteroskedasticity occurs when the error term is not homoscedastic, i.e. the error term does not have a constant variance (Brooks, 2008). When the covariance between error terms is not equal to zero over time or cross-sectionally, this is called autocorrelation (ibidem), thus error terms need to be uncorrelated to each other. If the standard errors are heteroskedastic and/or autocorrelated OLS estimation cannot be used (ibidem). The Newey-West (1987) heteroskedasticity and autocorrelation corrected standard errors take these two problems in account and can be easily measured using Eviews8.

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The algebraic intuition behind this is explained by Dave Giles and can be found here: http://davegiles.blogspot.nl/2013/07/the-adjusted-r-squared-again.html

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5. Results

In this section it is firstly investigated whether the conditioning variable coin has a countercyclical relationship with the U.K. business cycle. Secondly, the cross-sectional analyses for the six different asset pricing models on the Fama and French 25 size and book-to-market portfolios are presented and discussed.

5.1 The countercyclical relationship of coin with the U.K. business cycle

Before looking at the cross-sectional output the relation of the conditioning variable coin and the macroeconomy has to be examined. As KEA have found a significant relation between coin and the future stock market returns, it is interesting to investigate how the conditioning variable coin relates to other macroeconomic variables. Following KEA, five variables that have proved to reflect the real business environment are to investigate the aforementioned relationship. The five variables are: the recession indicator (Recession) which has been mentioned in the data section, the monthly growth rate of the U.K. GDP (GDPG), the monthly growth rate of consumption (CG), the monthly growth rate of labour income (LIG) and monthly inflation (INF)9. The used data is non-annualized data, except for the recession indicator which has only absolute values of zero and one. KEA found that the variable coin is countercyclical to the business cycle. They found a statistically significant negative relationship between coin and GDPG, CG, and LIG and a significant positive correlation with Recession. No significant relation between coin and inflation was found. In Table 4 the results of the regression on the relation between coin and the five monthly U.K. macroeconomic variables are displayed.

Table 4: Relationship of the conditioning variable coin and the five monthly macroeconomic variables

The time-series OLS estimates are presented in the first row and the Newey-West (1987) fixed t-statistics in the second row, the adjusted R2 is in parentheses. For the number of lags the Akaike Information Criterion (AIC) is used. In the final row the correlations are given. The sample period is from 2002M04 until 2014M06.

Recession GDPG CG LIG INF

Intercept 0.047 0.046 0.047 0.021 0.054

3.243** 4.593** 3.431** 4.915** 8.821**

Variable -0.001 0.823 -0.009 10.865 -3.556

-0.500 1.285 -0.107 8.378** -0.547 Adj. R2 (-0.006) (0.016) (-0.007) (0.702) (0.048)

Correlations Recession GDPG CG LIG INF

coin -0.028 0.152 -0.014 0.839** -0.233**

*statistically significant at α=0.05, **statistically significant at α=0.01

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16 Table 4 shows that the conditioning variable seems not to be countercyclical to the U.K. business cycle, as was found by KEA of coin with the U.S. business cycle. The variable coin has a significant relationship with only two variables: labour income growth (LIG) and inflation (INF). The correlation between labour income growth and coin is especially striking, since it is a statistically significant, positive relationship, whereas KEA found a negative relationship. Moreover, LIG as independent variable is able to explain over 70% of the variability of the conditioning variable and has a positive correlation value of 0.839 with coin.

Testing the conditioning variable as proposed by Santos and Veronesi (2006), who use a labour income to consumption ratio as a conditioning variable, could provide interesting results for the U.K. stock market. The statistically significant negative relationship between inflation and coin could be due to the relationship of inflation with interest rates, as coin consists of three related variables: the short-term interest rate, the default spread and the term spread. The results from Table 4 definitely lack support to the findings of KEA that coin has a countercyclical relation with the five macroeconomic variables.

5.2 Cross-sectional analysis of the six asset pricing models

In this paragraph the cross-section of returns is evaluated. As mentioned in the methodology section the Fama and MacBeth (1973) sectional regression method is used. The results of the cross-sectional regressions for each of the six models with respect to the Fama and French 25 size and book-to-market portfolios are presented in Table 5.

Overall all models perform poorly when comparing the results with the results found by KEA for the U.S. stock market. Furthermore, all models have statistically significant constants at a 5% significance level, indicating that not all variability of the cross-section of returns are explained by the six models. In accordance to Fama and French (1993) the CAPM is not able to explain the cross-section of returns, with an adjusted R2 value of -4.2%. Furthermore, the conditional CAPM is not much better in explaining the cross-section of returns, demonstrated by an adjusted R2 value of only 9.0%. It is noteworthy that the consumption-based asset pricing models have substantially lower adjusted R2 values than the three-factor and four-factor models. This result is strikingly different from the findings of KEA, where the consumption-based models perform significantly better in explaining the cross-section of returns. In my results the consumption-based models are very poor (both models have a negative adjusted R2 value).

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17 stock market for the Fama and French 25 size and book-to-market sorted portfolios.

The three-factor and four-factor models are also the only models which contain factors with significant t-values at a 5% statistically significance level, thus factors which have a significant explaining power in explaining the cross-section of returns. The factors which have this impact for both models are the excess market returns on the FT All Share Index (rmt+1)and the Fama and French

(1993) size effect, the small minus big factor (SMB).

Therefore, I find no empirical support that coin as a conditioning variable is able to explain excess market returns and to explain the cross-section of stock returns in the U.K. stock market. Therefore it can be concluded that the three-factor and four-factor models perform significantly better in this study than the other four asset pricing models using the Fama and French 25 size and book-to-market portfolios.

Table 5: The Fama and MacBeth (1973) monthly cross-sectional regressions on the Fama and French 25 size and book-to-market portfolios

The monthly cross-sectional estimation results of the excess returns on the FT All Share index for the six different models are displayed. The coefficients are displayed in percentages. The t-values are based on the Newey-West (1987) technique. The sample period is from 2002M04 until 2014M06.

Model Factors R2 Adj. R2

CAPM Constant

Estimate 0.755 -0.102 0.001 -0.042

t-value 6.423** -0.137

coin CAPM Constant *

Estimate 0.372 -0.316 -0.884 -0.030 0.204 0.090 t-value 2.142* -1.611 -1.370 -0.883

CCAPM Constant

Estimate 0.708 -0.360 0.032 -0.010

t-value 10.898** -0.929

coin CCAPM Constant *

Estimate 0.559 -0.115 -0.290 -0.012 0.056 -0.079 t-value 3.223** -0.730 -0.766 -0.647 FF3 Constant SMB HML Estimate 0.560 -2.808 0.312 0.023 0.631 0.578 t-value 7.888** -4.080* 6.905** 0.309 Carhart 4F Constant SMB HML WML Estimate 0.588 -3.140 0.325 -0.011 -0.523 0.661 0.593 t-value 6.897** -4.362** 6.689** -0.172 -1.132

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18

6. Robustness

In this section the results presented in Section 5 are tested on robustness. Firstly the relationship of coin with the U.K. business cycle using quarterly data is examined and is presented in Section 6.1. Secondly, in Section 6.2 two other monthly cross-sectional analyses are presented: one using the Fama and French 25 size and momentum portfolios (2002M04-2014M06) and one using the Fama and French 25 size and book-to-market portfolios in the pre-financial-crisis period (2002M04-2008M03). Finally, in Section 6.3 three quarterly cross-sectional analyses are presented. These three analyses are the same as the former three monthly-based analyses, however this time quarterly data is used for the six used asset pricing models. Once more, the method used for the cross-sectional regressions is the Fama and MacBeth (1973) method.

6.1 The countercyclical relationship of coin with the U.K. business cycle using quarterly data

The results of Table 4 are now analysed with quarterly data are presented in Table 6. So, the five macroeconomic variables which are believed to represent the business cycle are tested for their relationship with the conditioning variable coin.

Table 6: Relationship of the conditioning variable coin and the five quarterly macroeconomic variables

The time-series OLS estimates are presented in the first row and the Newey-West (1987) t-statistics in the second row, the adjusted R2 is in parentheses. For the number of lags the Akaike Information Criterion (AIC) is used. In the final row the correlations of coin with the five macroeconomic variables are given. The sample period is from 2002Q2 until 2014Q02.

Recession GDPG CG LIG INF

Intercept 0.047 0.047 0.047 0.020 0.054

3.423** 3.127** 4.680** 4.553** 8.821**

Variable 0.000 -0.823 -0.003 3.738 -3.556

0.019 1.285 -0.191 7.120** -0.547

Adj. R2 (-0.021) (-0.008) (-0.021) (0.745) (0.048)

Correlations Recession GDPG CG LIG INF

coin -0.028 0.152 -0.014 0.839** -0.233

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19 6.2 Two other monthly cross-sectional regressions

To test the robustness of the cross-sectional regressions in the previous Section, firstly another test asset is used as advised by Lewellen et al. (2010). The other assets are the Fama and French 25 size and momentum portfolios adapted by Gregory et al. (2013) for the U.K. stock market. Secondly, the Fama and French 25 size and book-to-market portfolios are tested in the pre-financial crisis period (2002M04-2008M03) to investigate the potential impact of this crisis on the results.

The Fama and French 25 size and momentum portfolios are chosen as an additional test asset because KEA also used these portfolios to test the robustness of their results, therefore the results are easily comparable. The results of the cross-sectional analyses on these portfolios using the Fama and MacBeth (1973) method are presented in Table 7.

Similar to the results of Table 5, the results of Table 7 display that the six models have a poorer performance than the U.S. results of KEA when they used the Fama and French 25 size and momentum portfolios to explain the cross-section of returns. The CAPM has again no ability to explain the cross-section of returns. The main differences with the results of Table 5 are the performance improvements of both consumption-based models (CCAPM and the coin CCAPM) and the performance decline of the Fama and French (1993) three-factor model and the Carhart (1997) four-factor model.

The regular CCAPM is now able to explain 8.2% and the coin CCAPM of KEA is able to explain 17.6% of the variance of the cross-section of U.K. stock returns. However, these percentages are nowhere close to the findings of KEA who used a longer time period and U.S. data, since KEA found an adjusted R2 value of 79% for the coin CCAPM when they tested the 25 size and momentum portfolios in the U.S. stock market.

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20 Table 7: The Fama and MacBeth (1973) monthly cross-sectional regressions on the Fama and French 25 size and momentum portfolios

CAPM Constant

Estimate 0.906 -0.509 0.017 -0.026

t-value 6.671** -0.595

Estimate 0.524 -0.386 -0.693 -0.040 0.156 0.036 t-value 2.205* -1.987 -0.968 -0.986

CCAPM Constant

Estimate 0.793 -0.586 0.121 0.082

t-value 11.507** -2.063

Estimate 0.767 -0.214 -0.290 -0.022 0.279 0.176 t-value 2.373* -0.957 -0.815 -1.467 FF3 Constant SMB HML Estimate 0.969 -0.017 0.202 -0.339 0.187 0.071 t-value 4.509** -0.025 1.425 -1.654 Carhart 4F Constant SMB HML WML Estimate 0.509 0.332 0.241 0.517 1.282 0.455 0.345 t-value 3.164** 0.458 2.182* 2.450* 3.988**

As the time horizon of the research is not as long as some other empirical studies, the worldwide financial crisis may have a large impact on the results. The 2007 financial crisis for example has no influence on the results of KEA, since the dataset of their study ends in 2005. As the financial crisis started 2008Q2 in the U.K. the sample period is now from 2002M04 to 2008M03.

The results of the cross-sectional analyses in the pre-financial crisis using the Fama and French 25 size and book-to-market portfolios adapted by Gregory et al. (2013) for the U.K. stock market are presented in Table 8. The results have to be interpreted carefully as the time horizon and therefore the amount of data points is limited. Again none of the six models shows to be a proper model to be used in the U.K. stock market to explain the cross-section of returns. Once more, all constant coefficients are statistically significant.

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21 results of Table 5. Again, the CAPM and the coin CAPM perform better than their consumption counterparts and the three factors in the coin CAPM are statistically significant at the 5% significance level. The coin CAPM is able to explain 21.5% of the variance of the cross-section of stock returns in the U.K. stock market in the pre-financial crisis period, making it the best performer of the four. For the two consumption-based models, again, no statistically significant factors are present.

Table 8: The Fama and MacBeth (1973) monthly cross-sectional regressions on the Fama and French 25 size and book-to-market portfolios in the pre-financial crisis period

CAPM Constant

Estimate 0.615 -1.251 0.164 0.127

t-value 10.783** -0.276

Estimate 0.573 -0.112 -1.394 -0.072 0.313 0.215 t-value 2.230* -2.673* -2.898** -2.729*

CCAPM Constant

Estimate 0.500 -0.009 0.000 -0.043

t-value 8.521** -0.020

Estimate 0.583 -0.096 0.614 0.031 0.236 0.127 t-value 3.121** -1.683 1.293 1.221 FF3 Constant SMB HML Estimate 0.372 -1.508 0.221 0.310 0.378 0.289 t-value 5.213** -4.891** 1.994 3.549** Carhart 4F Constant SMB HML WML Estimate 0.592 -1.453 0.210 0.357 1.035 0.470 0.364 t-value 6.410** -5.146** 2.190* 3.985** 3.072**

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22 power in contrast to the full investigation period, which includes the financial crisis. The shorter time horizon could be the cause, and/or it seems that the Fama and French (1993) three-factor model and the Carhart (1997) four-factor model perform better during highly fluctuating excess returns.

6.3 Quarterly cross-sectional regressions

As a final investigation, the three cross-sectional analyses are also conducted with quarterly data over the same respective time periods. Most consumption-based asset pricing literature uses quarterly data, therefore it might be that the consumption-based models perform better on a quarterly basis. Since the analyses are done on only 49 and 24 data points for the whole and pre-financial crisis period respectively, cautious interpretation is required.

Table 9 presents the results of the quarterly cross-sectional regressions on the Fama and French 25 size and book-to-market portfolios, adapted by Gregory et al. (2013) for the U.K. stock market (2002Q2-2014Q2). Once more, the six models are poor and have statistically significant constants. Table 9: The Fama and MacBeth (1973) quarterly cross-sectional regressions on the Fama and French 25 size and book-to-market portfolios

The quarterly cross-sectional estimation results of the excess returns on the FT All Share index for the six different models are displayed. The coefficients are displayed in percentages. The t-values are based on the Newey-West (1987) technique. The sample period is from 2002Q2 until 2014Q2.

CAPM Constant

Estimate 1.852 1.044 0.077 0.037

t-value 7.329** 1.709

Estimate 3.058 0.772 3.276 0.152 0.254 0.147

t-value 3.865** 1.593 2.434* 2.541* CCAPM Constant

Estimate 2.188 1.170 0.224 0.190

t-value 21.202** 5.949*

Estimate 3.127 0.357 1.861 0.092 0.333 0.238 t-value 4.095** 1.218 2.996** 3.317** FF3 Constant SMB HML Estimate 2.000 -1.383 0.312 -0.154 0.223 0.112 t-value 7.638** -1.482 3.383** -0.548 Carhart 4F Constant SMB HML WML Estimate 2.164 -1.136 0.573 -0.057 0.013 0.288 0.146 t-value 8.331** -1.296 2.872** -0.204 1.281

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23 The main differences of Table 9 and Table 5, are that the two consumption-based models are the two best models and contain factors which are statistically significant at the 5% significance level, whereas the Fama and French (1993) three-factor model and the Carhart (1997) four-factor model lose substantial explaining power. Now, the coin CCAPM is the best model since it is able to explain 23.8% of the cross-section of returns. Even the coin CAPM now performs better than the Fama and French (1993) three-factor model and the Carhart (1997) four-factor model, although all models are have poor adjusted R2 values.

Furthermore, Table 9 shows that the log consumption growth rate, , in both consumption-based models as well as the interaction factor in the coin CCAPM are statistically significant. Similar to previous results, the factor SMB is statistically significant in the Fama and French (1993) three-factor model and the Carhart (1997) four-factor model.

In Table 10 the results of the cross-sectional regressions on the Fama and French 25 size and momentum portfolios, adapted by Gregory et al. (2013) for the U.K. stock market, are presented. The results in Table 10 are compared to the results presented in Table 7, where the monthly cross-sectional regressions using the 25 size and momentum portfolios are presented.

Similar to previous findings, Table 10 shows that the Carhart (1997) four-factor model is the best performing model, with an adjusted R2 value of only 30.7%. Moreover, the momentum factor is now statistically significant. However, the excess return factor is also statistically significant and the factor SMB has lost its significant explaining power. Furthermore, the Fama and French (1993) three-factor shows to be a very weak model with respect to explaining the excess returns of the Fama and French 25 size and momentum portfolios.

Overall the consumption-based model performed better and the CAPM based models performed worse with respect to monthly data. Again the conditional CCAPM is the second best performer with a 6.5% performance improvement and is able to explain 24.1% of the variance of the cross-section of U.K. stock market returns. All six models show a substantial lower adjusted R2 value than the results of KEA for the U.S. stock market. It appears that the consumption-based models perform better when quarterly cross-sectional data is used on the Fama and French 25 size and momentum portfolios.

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24 Table 10: The Fama and MacBeth (1973) quarterly cross-sectional regressions on the Fama and French 25 size and momentum portfolios

CAPM Constant

Estimate 2.376 0.445 0.012 -0.031

t-value 4.436** 0.413

Estimate 1.766 -0.223 0.325 0.029 0.053 -0.083 t-value 2.338* -1.300 0.335 0.543

CCAPM Constant

Estimate 2.507 1.315 0.133 0.095

t-value 15.843** 1.622

Estimate 1.989 -0.150 1.434 0.107 0.336 0.241 t-value 2.858** -0.709 1.910 3.651** FF3 Constant SMB HML Estimate 2.891 -0.489 0.704 -1.337 0.155 0.034 t-value 3.623** -0.508 1.623 -1.283 Carhart 4F Constant SMB HML WML Estimate 2.594 1.906 0.730 -0.295 1.442 0.422 0.307 t-value 4.229** 2.124* 1.354 -0.348 2.354*

In the final cross-sectional analyses of this paper the Fama and French 25 size and book-to-market portfolios, adapted by Gregory et al. (2013) for the U.K. stock market, are tested with quarterly data on the pre-financial crisis period (2002Q2-2008Q1). The results are shown in Table 11 and are compared to the results presented in Table 8, where monthly data was used. As the analysis only comprises 24 data points the results are highly questionable.

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25 Table 11: The Fama and MacBeth (1973) cross-sectional regressions on the Fama and French 25 size and book-to-market portfolios in the pre-financial crisis period

CAPM Constant

Estimate 1.955 -1.640 0.054 0.013

t-value 6.391** -1.648

Estimate 1.721 -0.067 -2.320 -0.121 0.190 0.074 t-value 2.297* -1.826 -2.143* -2.026

CCAPM Constant

Estimate 1.741 0.959 0.036 -0.006

t-value 7.240** 1.136

Estimate 1.452 -0.017 0.295 0.009 0.201 0.087 t-value 2.938** -0.497 0.249 0.135 FF3 Constant SMB HML Estimate 1.646 -2.630 0.402 0.401 0.197 0.082 t-value 4.457** -2.457* 1.470 1.497 Carhart 4F Constant SMB HML WML Estimate 3.030 -2.423 0.339 0.513 2.390 0.336 0.204 t-value 4.696** -2.441* 1.343 1.990 2.756*

In conclusion, it can be noted that almost all constant coefficients are statistically significant at the 5% significance level for all the cross-sectional analyses, which implies that all the models do not contain all factors able to sufficiently explain the variance of the cross-section of returns in the U.K. stock market. Furthermore, it seems that the consumption-based models score better when using quarterly data, especially the coin CCAPM.

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26

7. Conclusions and recommendations

The study in this paper examined whether six different asset pricing models are able to explain the cross-section of returns in the U.K. stock market in the period 2002M04-2014M06. These six models were: the CAPM, the CCAPM, the Fama and French (1993) three-factor model, the Carhart (1997) four-factor model, the conditional CAPM of KEA and the conditional CCAPM of KEA in particular. As a start, it was investigated whether the conditioning variable ‘coin’ has a countercyclical relationship with the U.K. business cycle. Two sets of portfolios were tested: the 25 size and book-to-market portfolios and the 25 size and momentum portfolios. These sets are created by Fama and French and adapted for the U.K. stock market by Gregory et al. (2013). The applied cross-sectional method used was the one developed by Fama and MacBeth (1973).

As a main conclusion, all six asset pricing models were not able to sufficiently explain the cross-section of U.K. stock returns. Similar investigations with the same models for the U.S. stock market showed much better results. Even with the best model, the four-factor model of Carhart (1997), the adjusted R2 value varied between 34.5% and 59.3% for the investigated cases on a monthly basis. The five other models resulted in clearly lower and also highly varying adjusted R2 values. The results remained poor when focusing on a shorter time period without the financial crisis (2002M04-2008M03), and overall became even worse when investigating on a quarterly basis.

Furthermore, no significant countercyclical relationship of the conditioning variable coin with the U.K. business cycle has been found, in contrast to the results found by KEA for the U.S. business cycle. The conditioning variable ‘coin’ shows a statistically significant relationship with only two of the five macroeconomic variables: labour income (positive) and inflation (negative). These results are surprising and need further study.

This seems to indicate that every stock market is different in nature and therefore might need a set of model factors deviating from the U.S. situation for which the six models in this paper have been originally developed. A recommendation for future research is to develop an asset pricing model suitable for the U.K. stock market.

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27

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