Abstract
 

This paper, we study the significance of the four-factor asset pricing model (market factor, size factor, book-to-market factor and momentum factor) in explaining the cross-sectional variation in average stock returns in the United Kingdom. Our findings show that the four-factor model does work well and significant to explain the cross-sectional variation in average stock returns for July 2000 to June 2007 period. Our empirical findings indicate that on average annual premium for the market factor, size factor, book-to-market factor and momentum factor is minus 3.64 percent, 5.20 percent, 3.12 percent and 26.52 percent. Our results suggest that the small firms produce higher returns than big firms. For the high book-to-market equity stocks more likely to perform higher returns than low book-to-market equity stocks. SMB and HML are proxy for sensitivity to risk factors that capture cross-sectional variation in average stock returns. We also find that there is statistically significant in momentum factor. Moreover, we observe that the four-factor model in the UK stock market works well in explaining the variation in the stock returns no matter the market is going up or down.
 

1.0 Introduction
 

In 1970’s, (Sharpe 1964), (Lintner 1965) and (Black 1972) introduce The Capital Asset Pricing Model (CAPM) which is the first asset pricing model in finance. This model has becomes popular in explaining the relationship between return and market beta on risky financial assets. According to CAPM, as the capital market in equilibrium, investors’ willingness to bear higher risk to obtain higher expected return. Investors face only one source of uncertainty which is the performance of the market as a whole. They bear only market risk (market beta) and no other risk factor. There is a linear, positively sloped relationship under risk and expected return on risky financial assets, noted by (Sharpe 1964). In other words, the expected return of a security or portfolio is greater, the greater its market risk.

1.1 Research Background

However, recent empirical tests in the area of asset pricing indicate that the cross-sectional of average stock returns does not have relation or lack of supportive evidence to the market beta of the CAPM. (Fama and French 1992) state that beta is nearly worthless as an explanation of a stock’s relative performance over time. They show that strategies based on investing in stocks with small market equity (ME) and high book-to-market equity (BE/ME) tend to produce better long-term performance than strategies based on market beta. If stocks are priced rationally in stock market, then different risk factors should able to explain systematic differences in average stock returns. (Fama and French 1993) developed a three-factor asset pricing model which includes two additional risk factors – size (SMB) and book-to-market ratio (HML) has found to be significantly correlated with returns. SMB is the three small stock portfolios average returns minus the three big stock portfolios average return. HML is the two high book-to-market stock portfolios average returns minus the two low book-to-market stock portfolios average returns. Empirical studies show that three factor model has been tested and confirm has high explanatory power than one factor CAPM. Therefore, the Fama and French three-factor model seems to become a popular asset pricing model compare to the CAPM in finance. (Fama and French 1995) attempt to test the three factor model toward fundamentals shock (earnings and sales). They find that the Fama and French three-factor model has high explanatory power in relation to the behavior of earnings. However, they show that their results are weak because there is lack of supportive evidence to confirm returns respond to the book-to-market factor in earnings. (Fama and French 1993) argue that the three-factor model can capture the cross-sectional variation of the stock returns except momentum effect. (Jegadeesh and Titman 1993) and (Carhart 1997) indicate that momentum factor also another risk factor which can capture the cross-sectional variation of the stock returns. They document that stocks with high returns over the past twelve months continue to perform well compare to stocks with low returns tend to perform poor over the same time. (Carhart 1997) constructs a four-factor model by combine the Fama and French three factor with momentum factor as additional risk factor. He finds that the four-factor model capable to explain persistence in equity mutual fund performance. The robustness of multifactor asset pricing model works well in explaining realized returns for the US portfolios. In recent years, many empirical studies have been tested outside the US portfolios. For instance, (Chan, Hamao and Lakonishok 1991) indicate that “fundamental” variables (size, book-to-market ratios, earnings yield and cash flow yield) capture the cross-sectional differences in returns on Japanese stocks. (Chan, Karceski and Lakonishok 1998) find that the performance of size factor and book-to-market factor work well in capturing the covariance in the UK stock returns. (Morelli 2007) also examines the Fama and French three-factor model for the period 1980 to 2000 and finds that market return and book-to-market premium in explaining the cross-sectional realized returns in up (Bull) and down (Bear) markets. However, he states that there is insignificant explanatory power for size factor in up and down markets. Other recent empirical evidence in the UK done by (Gregory and Michou 2009) point out that the SMB and HML factor slopes show considerable variability through time except momentum factor. Their findings are consistent with Fama and French three-factor model empirical results which present momentum factor cannot capture cross-sectional variation in average stock returns. In summary, there has been very limited research on the four-factor model relating to the UK stock market. Evidence of the four-factor model in the UK stock market has been inconsistent and mix findings.1.2 Research Aim #p#分頁標題#e#

In this study, we find that there is not sufficient amount of empirical studies regarding the four-factor model in the UK. Therefore, examining the effectiveness of this asset pricing model is assumed crucial. This study aim, we want to test the feasibility of the robustness four-factor model for the stocks traded on the London Stock Exchange (LSE) during July 2000 to June 2007. In other words, we want to investigate whether other risk factors (size effect, book-to-market effect and momentum effect) besides market beta are also explanatory variables of the variation in average returns in the UK stock market. Most of the empirical studies has focused on the US portfolios, if (Fama and French 1998) findings for the US portfolios capable to confirm outside the US market, then we assume our results can contribute to both individual and institutional investors who can form their optimal portfolios according to the significant measures in the UK. For example, (Fama and French 1996) state that size premium and value premium have a strong explanatory power to the cross-sectional in average stock returns. Therefore, we believe that with a better and accurate benchmark, individual or institutional investors able to construct their portfolios based on these risk factors and estimate their expected stock returns. Another motivation for this paper is that we investigate for the (Carhart 1997) four-factor model by adding a momentum factor into the Fama and French three-factor model. We want to examine whether momentum risk can takes account of the fact that past stock prices will have a significant impact on the returns made. There is limited documenting for examine the four-factor model in the UK stock market. Therefore, we suggest our analysis results can give valuable information to individual or institutional investors for investment purposes or future research in the UK stock market. 

1.3 Research Objectives

In this study, our paper has different from other research papers which investigate the four-factor model in the UK. First, our data collection contains highly capitalised company from Financial Times Stock Exchange (FTSE 100) listed on the London Stock Exchange. We use the recent sample periods (2000-2007) which can help us to test the four-factor model with most recent past returns data. Another point is most of the empirical evidences are tested in monthly interval. For our study, we collect the sample data in weekly interval. The reason we use shorter interval because we want to observe the volatility within the period chosen and it can give more of an overview of the direction the market is moving. We want to examine the four-factor model by weekly interval and observe whether this asset pricing model is adequate to explain the cross-section of expected returns. Second, we also test the asset pricing model in different market conditions (Bull and Bear market). According to (Pettengill, Sundaram and Mathur 1995), they state that there is significant impact to the relationship between beta and realized return during different market conditions. For instance, when the stock market is up, there should present positive excess market return. When the stock market is down, there should present negative excess market return. (Campbell, Lo and Mackinlay 1997) indicate that the performance of the asset pricing model can be tested with accuracy by collect data in different sample periods and different markets. Therefore, our research study is consistent with these two important points.1.4 Research Questions#p#分頁標題#e#

We specifically ask:

1) Can the Four-Factor model (Market factor, Size factor, Book-to-market factor and Momentum factor) show a significant value in explaining the average stock returns in the UK stock market? (Fama and French 1995) state that if stocks are priced rationally, the multi-factor model should able to give explanation the cross sectional variation in stock returns. For example, size and book-to-market become common risk factors which sensitive in returns. 

2) Does the Four-Factor model works well in different capital markets (UK) other than the U.S portfolios?

3) What are the outcomes from the Four-Factor model when we test in different market conditions (Bull and Bear market)?

In this paper, we would like to organize the structure of contents as follows: Chapter 2 presents a brief of literature review. Chapter 3 shows the data collection and details of methodology employed. Chapter 4 presents the data analysis and discusses research findings. Chapter 5 contains summary and conclusion about the implications of the research and recommendations for future research.

 

2.0 Literature Review
 

The Capital Asset Pricing Model (CAPM) introduced by (Sharpe 1964), (Lintner 1965) and (Black 1972) assume that there is a linear relationship between the excess return of the asset portfolio and the excess return of the market portfolio. They argue that the only risk factor which is market beta can use to capture the cross-sectional variation in average stock returns. A number of empirical studies have been tested and stated that higher risk should be associated with higher return. In other words, investors should happy to accept a lower expected return from defensive securities (low risk) and more willing to accept a higher expected return from aggressive securities (high risk). This model becomes popular and widely used in pricing the risky financial assets by investment managers since 1970’s. According to CAPM equilibrium, the expected rate of return on the asset portfolios can be written as following. 

E(Ri) = Rf + βi (E(Rm)-Rf) (1)

E(Ri) is the expected return of security i, Rm is the rate of return on the market portfolio, Rf is the risk free rate, βi is the beta coefficient of the i security. 

However, recent empirical research has brought into question the robustness of the CAPM in explaining the cross-sectional variation in average stock returns. (Fama and French 1992) argue that the critical evidence with regard to the insufficiencies of CAPM, they tested CAPM on the basis of return of assets and observed a non-linear relationship between average return and beta coefficient. The expected rate of return from some investment portfolios which are based on firm characteristics (size, earnings/price, book-to-market ratio, past performance on sales growth, long term and short term stock returns) cannot be explained by the CAPM beta. For example, (Banz 1981) indicates that there is a strong average returns on small firms compare to big firms. (Rosenberg, Reid and Lanstein 1985) finds that average returns on firms with high book-to-market ratio (value stocks) outperform than those with low book-to-market ratio (growth stocks). Their findings are consistent with (Fama and French 1992) empirical works who argue that the portfolio with size (size effect) and book-to-market ratio (book-to-market effect) capture the cross sectional variation in average security returns. (Fama and French 1993) argue that the three-factor model does well in explain the cross section of returns on US stocks. Their model indicates that the excess return on an asset portfolio is explained by: i) market premium – the expected return on a market portfolio minus risk free rate; ii) size premium – the return on a portfolio of small stock minus the return on a portfolio of big stock; small minus big (SMB) iii) book-to-market premium – the average returns on two high book-to-market stock portfolios minus the average returns on two low book-to-market stock portfolios ; high minus low (HML).#p#分頁標題#e#

(Fama and French 1996) point out that ‘these patterns which are used in average stock returns are not explained by the CAPM, they are typically called anomalies.’ They find that the three-factor model is capable to capture the size and book-to-market effects, but not momentum effect which remains the greatest challenge to their model. Momentum effect is another variable which been tested by (Jegadeesh and Titman 1993) in empirical work and show a significant result in explanation the cross sectional variation in stock returns. They find that stocks with higher returns over the past 12 months (winners stocks) tend to gain higher returns in the future compare to lower returns over the past 12 months (losers stocks) only perform poorly returns in the future. A study on momentum effect is well-documented in the US and UK. In the US portfolios, (Jegadeesh and Titman 2001) state that both winners and losers are more likely to be small firms because small firms have more volatile and high extreme return. The average size of winner portfolio is larger than the average size of loser portfolio. Based on this portfolio characteristic, smaller firms portfolios tend to outperform larger firms portfolio. This is consistent with (Fama and French 1992) hypothesis which explain small firms need higher risk-adjusted return in order to bear higher risk. (Debondt and Thaler 1985) argue that the momentum effect is not related to size effect. However, their empirical results show that on average winner stocks are two times bigger than loser stocks. In the UK portfolios, (Tonks and Hon 2003) indicate that momentum effect is exists because there is positive and significant value on winner minus loser portfolios over the short to medium term horizons. In general, momentum still remains something of a puzzle. Even though a recent paper by (Liu 2006) provides evidence consistent with US momentum returns but no such evidence exists for the UK. If momentum effect is anomaly, we might expect this risk factor to explain the cross sectional of returns within any given time period. 

There have two financial theories which trying to explain those pricing anomalies: non-risk-based explanations versus risk-based explanations. One of the non-risk-based explanations is mispricing of assets. (Lakonishok, Vishny and Shleifer 1994) argue that the book-to-market effect caused by investors’ overreaction rather than compensation for risk bearing. They further explain that investors systematically overreact to recent companies’ announcement and naively extrapolate past earnings growth into the future when evaluating a firm’s prospects. Therefore, they suggest that book-to-market premium (high book-to-market stocks for high expected returns versus low book-to-market for lower expected returns) does not caused by risk. According to (La Portal and Shleifer 1997) who point out that market investors underestimate future earnings for high book-to-market stock and overestimate future earnings for low book-to-market stocks. In other words, investors more likely to buy “good” stocks with high earnings and good management compare to “bad” stocks with poor growth in the past and perform relatively slow in terms of earnings.On the other side, risk-based explanations argue that these anomalies arise because the CAPM does not capture all the systematic risks in the macroeconomic conditions. Thus, it may leave part of the systematic risks to firm characteristics. For instance, (Fama and French 1993) contend that the higher excess returns of size portfolios and book-to-market portfolios as a compensation for market risk. #p#分頁標題#e#

Small firms are more likely to have high book-to-market ratio and in fact the high book-to-market ratio firms tend to be small (they have low market equity). This is because they are possible to have problem in cash flow and high financial leverage. According to (Chan and Chen 1991), they argue that small firms tend to be more volatile to economy changes and exposed to adverse economic conditions. For example, small firms tend to perform poorly in stock market due to they have higher costs or higher financial leverage. These restrict them for accessibility to external financing especially during tight credit period. From this imperfect information on financing affect the company stock prices. When stock prices have fallen, this firms market equity tend to decrease and book-to-market ratio tend to increase. (Fama and French 1992) state that average return for high book-to-market ratio firms and small firms are high because these firms tend to bear financial distress risk like poor earnings income. They need higher return as a compensation for this risk. They also observe that high book-to-market factor have positive slopes because these firms are distress and weak. The firms are strong and high earnings have negative slopes on the book-to-market factor. Their hypothesis is consistent with (Chan and Chen 1991) who indicate that high book-to-market ratio firms are related to relative-distress effect. In other words, the firms have a large amount of leverage because the market judges that these firms are poor. Therefore, the stock prices discounted relative to book value. 

Fama and French three factor model have been critised by (Kothari, Shanken and Sloan 1995). They argue that the value premium does not affect by risk factor but comes from survivor bias. In other words, the book-to-market results are influenced by a combination of survivorship bias which affects the high and low book-to-market stocks performance. A study of research by (Black 1993) who challenges that Fama and French three factor model rely heavily on past returns data to estimate returns for those factors. For example, he suggests that in estimating expected return by past average return as estimate normally have a highly inaccurate estimate. The reason causes the estimate inaccurate because they think the factors as rationally priced and unwilling to hear about theory which suggesting that certain factors and securities are mispriced. Furthermore, (Chan, Jegadeesh and Lakonishok 1995) observed that selection bias cannot adequate to explain the difference in returns between ‘value’ stocks and ‘growth’ stocks. In recent studies, another challenged by (Daniel and Titman 1997) who argue that firms have a common characteristics such as similar size and book-to-market can help to explain cross-sectional variation in average stock returns rather caused by risk-based factors. For instance, firms that under similar size and book-to-market more likely to perform good or poorly together because size and book-to-market characteristics are similarly exposed to certain shocks. Recent empirical evidence by (Davis, Fama and French 2000) defence their statements towards (Daniel and Titman 1997) findings. They observed that a) the Fama and French multifactor model provides a better explanation for the relation between book-to-market and average returns compare to the characteristic-based model; b) the robustness of Fama and French multifactor model capable captures the average returns on US portfolios formed on size, book-to-market and other variables (earnings/price, cashflow/price) which cannot explain by the CAPM; c) the characteristics-based model of Daniel and Titman need to be tested with more extensive data rather than sample period. A recent research evidence by (Stambaugh and Pastor 2000) who make the argument on the robustness of characteristics-based model versus factor-based model pricing model. They argue that there is no difference between the Daniel and Titman characteristics-based model and the Fama and French risk-based model because both models lead to similar portfolio selections in investment. In addition, (Lakonishok, Vishny and Shleifer 1994) argue that the value premium might be genuine to explain the cross-sectional in average stock returns but irrational. They suggest that investor overreaction causes the value stocks underpriced and the growth stock overpriced. Therefore, investor overreaction makes the value premium exist in the stock market. In other words, investors overreact to the stocks which perform relatively well in the past and buy them. These stocks become overpriced. For the stocks that have performed relatively poorly in the past, investors tend to sell them. Therefore, these stocks become underpriced. (Mackinlay 1995) who debate that the explanation of Fama and French multifactor model towards value premium is caused by data snooping. They make further explanation that the best way to evaluate the data snooping hypothesis is to use different time periods and in different countries. They challenge most of the Fama and French empirical studies on multifactor model are tested in the US market. Their findings indicate that risk factors capture the cross sectional variation in average security returns in the US markets.The vast majority of empirical studies on three-factor model have been conducted using US data. They find that this model does work well in explaining the variation in average stock returns in the US portfolios. (Fama and French 1998) find that the annually book-to-market premium is 7.68 percent in international markets for the sample period 1975-1995. In the same year, (Arshanapali, Coggin and Doukas 1998) state that the three factor model is not only applicable in the US stock market, but also efficient in most of the international stock markets. Another study which conducted by (Maroney and Protopapadakis 2002) in international stock markets point out that the returns of the portfolios constructed according to size premium (SMB) and book-to-market premium (HML) have a significant correlation and high explanatory power (R2 ~ 1). (L'her, Masmoudi and Suret 2004) report that on average annual premium for the market factor 4.52 percent, size factor 5.08 percent , book-to-market factor 5.09 percent and momentum factor 16.07 percent on the Canadian stock market. Their findings are consistent with present results from (Liew and Vassalou 2000) who examination of 10 major stock markets on the four factor model in Canada from 1978 to 1996 periods and report positive and significant in returns. However, (Durand, Limkriangkrai and Smith 2006) find that the Fama-French model does not provide supportive evidence to explain Australian stock returns. They suggest that the exchange rate and the US market factors are the reason to fail the three-factor model to explain stock returns.#p#分頁標題#e#

2.1 Prior Evidence of the Four-Factor Model in the UK

There are few empirical studies testing the role of Fama-French three-factor model and momentum effect to the UK stock market. Although the empirical results are mixed findings but most of the findings suggest that the robustness of multifactor models outperform the CAPM in explaining cross-sectional variation in expected returns. For example, (Chan and Chui 1996) find that the book-to-market factor is consistently significant in explaining the cross sectional variation average returns in the UK market over the period 1971-1990. They state that high book-to-market firms more likely to have financial distress and uncertain for future earnings. For low book-to-market firms are associated with sustained profitability and small firms tend to be less profitable than large firms. Therefore, the returns for high book-to-market stocks are higher as a compensation for holding the riskier stocks. However, they argue that size factor is statistically insignificant effect on average returns. Their results are inconsistent with (Levis 1985) findings that explore the size effect and reports size premium gains an average 6.5 percent per annum for smaller UK firms over the period January 1958 to December 1982. One of the reasons for size factor is diminishing in recent years because significant changes of the UK economy affect the small firms’ premium become vanish. For instance, (Levis and Liodakis 2001) point out that a dramatic reversal of small companies’ performance and report the average return differential has declined to 3.6 percent per annum for the period 1955 to 2000. He also provides supportive evidence that smaller UK firms appear to have lower betas than larger firms even after adjustment for thin trading. Recent studies in the UK have used to counter the findings of (Fama and French 1992). (Fletcher 1997) finds that the relationship between beta and returns is significant in the UK stock market over 1975- 1994 periods. However, his findings argue that this significant relationship does not symmetric between up and down markets. His results contradict with (Strong and Xu 1997) for the period 1973 to 1992 who report that there is insignificant relationship between beta and return in the UK stock returns. (Fama and French 1998) examine a value premium using the UK data. They find that the UK reports 4.62 percent per annum in value premium. The sample size is large because they use an average 185 firms from the UK market. The majority of the firms are from the MSCI database, which contains big market capitalisation firms. This is similar with our sample data which collect from FTSE 100 and those firms have large market capitalization. Other studies confirmed similar findings for value premium in the UK (Levis and Liodakis 2001). The momentum anomaly is not only restricted to the US. (Richards 1997) tests the momentum effect by using monthly returns from MSCI indices of sixteen markets for the period 1970 to 1995, he finds that momentum effect is exists and becomes stronger in six months horizon where winners stocks outperform losers stocks by an annualized 3.4 percent. (Liu, Strong and Xu 1999) examines the momentum effect in the UK stock market over the period 1977 to 1996. They find that controlling for systematic risk, size and book-to-market ratio did not eliminate momentum profits. However, (Clare, Morgan and Thomas 2002) find that there is weak evidence of momentum in the UK at the 12 month horizon, even though winners stocks tend to perform better than losers stocks but the momentum premium is insignificantly. A recent research paper by (Gregory and Michou 2009) find that their results on momentum factor unable to show anything significant to contribute in the UK stock market.#p#分頁標題#e#
2.2 Hypotheses

Before we start to analyse the multifactor asset pricing anomalies, we want to identify the hypothesis for each risk factor. This research paper is prepared to determine whether three risk factors (size, book-to-market and momentum) can be used to capture much of the cross-sectional variation in average stock returns.

Hypothesis 1

According to (Fama and French 1992, 1993, 1995), they indicate that small firms tend to have higher expected returns compare to large firms. This is because small firms are relatively financial distress risk. Small firms tend to receive higher returns as compensation because they need to bear higher risk. If our results are consistent with size effect hypothesis, size premium should generate positive return. 

Ho: The expected return is equal in small and big firm.

H1: The expected return is high in Small firm compare to large firm (small minus big).

Hypothesis 2

According to (Fama and French 1992, 1993, 1995), they state that high book-to-market firms provide higher expected returns compare to low book-to-market firms. They argue that high book-to-market firms are more risky due to financial distress. Therefore, higher returns on high book-to-market firms are as a compensation for this risk. If our results are consistent with book-to-market effect hypothesis, value premium should generate positive return. 

Ho: The expected return is equal in high and low book-to-market firms.

H2: The expected return is high in high book-to-market firms compare to low book-to-market firms (high book-to-market stock minus low book-to-market stock).

Hypothesis 3

(Jegadeesh and Titman 1993) argue that winners stocks in the past 12 months end to gain higher returns in future compare to lower returns in the past 12 months (losers stocks) only perform poorly returns in future. If our results are consistent with momentum effect hypothesis, momentum premium should generate positive return. 

Ho: The expected return is equal in winners and losers stocks.

H1: The expected return is high in winners stocks compare to losers stocks (winner minus loser).

 

3.0 Data and Methodology
 

3.1 Research Data

In this study, we use data available from Reuters 3000 Xtra for the period July 2000 to June 2007. The sample data contains 64 highly capitalized UK companies listed on the London Stock Exchange. We exclude all firms from the financial sector in the database because financial firms tend to have high leverage which usually indicates distress. We employed weekly return data on non-financial firms with appropriate adjustment for capital changes. A maximum 364 sampling data is available for each weekly return based on these periods. If some of the company’s do not have any trading record onto the stock market during our date of the study period or the date after, this company will be disregarded. This selection criterion is to eliminate extremely thin trading records because it can help to avoid very different return characteristics. We want to ensure the analysis result of sample data is accurate and comply with (Fama and French 1995) empirical studies. The risk-free rate used in this study is the yield on a 10-year AAA rated UK bonds as at 14/07/2009.#p#分頁標題#e#

In order to avoid look-ahead bias, as in (Fama and French 1995) show that all the accounting data at the fiscal year-end t-1 are matched to stock returns for the period between July of year t to June of year t+1. Firm size (ME) is defined as closing price times number of shares outstanding. It’s measured by market capitalization or market equity at the end of June in year t. The book to market equity (BE/ME) is calculated as the ratio between a firm’s book equity (BE) at the fiscal year end t-1 and it’s market equity (ME) at the end of year t-1. 

For each year from July of year t-1 to June of year t, stocks are grouped into two portfolios of size, Small (S) and Big (B) based on their June market equity (ME). The same stocks are allocated in an independent rank to three book-to-market equity Low (L), Middle (M) and High (H). According to (Fama and French 1995) breakpoints, three book-to-market equity groups are formed by the bottom 30 percent (Low), middle 40 percent and top 30 percent (High) of the ranked values of BE. We construct six equally-weighted portfolios (S/L, S/M, S/H, B/L, B/M, B/H) from the intersection of the two size portfolios and three book to market equity portfolios. For example, S/L portfolio consists of small market capitalization with low book to market stocks. While B/H contains stocks that are big market capitalization with high book to market ratio. We exclude negative BE firms while forming BE/ME portfolios because this kind of firms does not provide meaningful explanations. We only include firms that show positive BE in order to construct portfolios. Weekly equally-weighted returns on the portfolios and return of market are calculated by logarithmic return Rt = ln(Pt/Pt-1) from July of year t+1 to June of year t+1. 

SMB (Small minus Big) is the three small stock portfolios average returns minus the three big stock portfolios average return.

SMB = 1/3 (S/L+S/M+S/H) – 1/3 (B/L+B/M+B/H) (1)

HML (High minus Low) is the two high book-to-market stock portfolios average returns minus the two low book-to-market stock portfolios average returns. HML = 1/2 (S/H+B/H) – 1/2 (S/L+B/L) (2)

According to (Carhart 1997), momentum factor can explain considerable variation in returns. The four-factor model extends the Fama-French three factor model by adding fourth factor, momentum factor. We follow (L'her, Masmoudi and Suret 2004) model to construct momentum factor. In the process of construction portfolios, we rank the stocks based on size for each month from July of year t-1 to June of year t. The size is based on the values of ME at the end of June in year t-1 and prior performance is based on the previous 11-month stock return lagged one month. (Jegadeesh and Titman 2001) point that the reason excludes the most recent month return because to attenuate the continuation effect which caused by bid-ask spread. Winner (WR) contains the top 30% of the total stocks with the highest average prior performance whereas Loser (LR) consists the bottom 30% of the total stocks with lowest average prior performance. Neutral (N) is the remaining 40% of the stocks. We construct six equally-weighted portfolios (S/LR, S/N, S/WR, B/LR, B/N and B/WR) at the intersection of size group and past return performance group.WML (Winner minus Loser) is the two winner stock portfolios average returns minus the two loser stock portfolios average returns on.#p#分頁標題#e#

WML = 1/2(S/WR+B/WR) – 1/2 (S/LR+B/WR) (3)

 

3.2 Research Methodology

The empirical model states that the relationship between the expected return of a certain portfolio in excess of the risk-free rate, (Rpt) - Rf, is explained by (i) the market premium, (ii) the size premium, SMB and (iii) the book-to-market premium, HML. 

Rpt-Rf = ai + bi(Rmkt-Rf) + siSMB + hiHML (4)

Rpt is the average return of six portfolios (S/L, S/M, S/H, B/L, B/M and B/H). Rf is the risk-free rate of AAA rated UK bonds. SMB (Small minus Big) is the three small stock portfolios average returns minus the three big stock portfolios average return. HML (High minus Low) is the two high book-to-market stock portfolios average returns minus the two low book-to-market stock portfolios average returns. The factor sensitivities bi,si,hi are the slope coefficients in the time-series regressions. 

The three-factor model works better in explaining for the anomalies, except for the short-term momentum strategy presented by (Jegadeesh and Titman 1993), (Carhart 1997) construct 4-factor model by adding a momentum anomaly with (Fama and French 1993) 3-factor model. Carhart’s fourth factor is based on long on the best-return stocks and short on the worst-return stocks over the previous year. In order to mimic such momentum factor, WML is defined as the return on a portfolio of winner-stocks minus the return on a portfolio of loser-stocks. The four-factor model is written as follow:

Rpt-Rf = ai + bi(Rmkt-Rf) + siSMB + hiHML + wiWML + εi

(Rp-Rf) is portfolio excess returns, εi is the disturbance term and uncorrelated with other variables such as factor sensitivities, bi, si, hi, wi are the beta coefficients in the time-series regressions. 

(Fama and French 1993) find that the market beta has poor explanatory power to explain the cross-section of average stock returns in the US portfolios. They find that the Fama and French three factor model does work well in captures all CAPM average return anomalies. They also state that small size firms with continually low earnings are tend to have higher book-to-market ratio and positive slopes on HML factor whereas, big size firms with continually high earnings are tend to have lower book-to-market ratio and negative slopes on HML factor. Based on their empirical results, they indicate that the HML captures the variation of the risk factor which is related to earnings performance. 

Interpretation on small stocks with high book to market ratio are the firms which perform poorly and have problem in financial distress. Therefore, the risk premium is high in order to compensate the high risk for bankruptcy. However, (Kothari, Shanken and Sloan 1995) argue that distress premium does not affect by risk factor but comes from survivor bias. (Mackinlay 1995) challenge that Fama and French study are involved data snooping. In this study, if our empirical results show that the robustness of four –factor model does work well in explaining the cross-sectional variation in average stock returns in the UK other than the US market then the issue of survivor bias seems irrelevant.#p#分頁標題#e#

Time-series is run to examine the four-factor asset pricing model and tested 6 size-B/M portfolios. (Fama and French 1993) suggest that time-series regressions method is functional for examine multifactor model. Time-series regressions can test on the issue of whether market factor, firm size, book to market equity effect and momentum effect must proxy for systematic risk factors in returns. In other words, if asset are priced rationally, those risk factors must proxy for sensitivity to common risk factors in returns. In this study, if the four-factor model is correct, we can expect the coefficients (bi, si, hi, wi) to be significantly different from zero. This is because the coefficients of the regression can measure the magnitude of compensation that should pay for such risk factors.

 

4.0 Empirical Results and Discussion
 

4.1 Summary Statistics

Table 1: Summary statistics for the Weekly Percent Four-Factor Model Explanatory Returns.

Rm is the return on FTSE 100 index. Rf is the risk-free rate based on yield of 10-year AAA rated UK bond at 14 July 2009. Stocks are categorised to two groups (S or B) based on the June market capitalisation. Stocks are allocated in an independent sort to three book-to-market ratios groups based on (Fama and French 1995) breakpoints for the bottom 30 percent, middle 40 percent and top 30 percent. Six portfolios (S/L, S/M, S/H, B/L, B/M, B/H) are constructed based on the intersections of the two size and three book-to-market ratios groups. Equal-weight weekly returns on the portfolios are calculated based on July of year t to June of year t+1. SMB is (S/L+S/M+S/H)/3 - (B/L+B/M+B/H)/3. HML is (S/H+B/H)/2-(S/L+S/H)/2. WML is (S/WR+B/WR)/2-(S/LR+B/LR)/2. The number of firms in our sample is 64 high capitalised companies. This sample size is tested throughout the time period (full sample, bear market and bull market). 

7/00-6/07: 364 weeks 7/00-6/02: 105 weeks 7/04-6/06: 103 weeks 

Full sample Bear market Bull market

Mean Std t(Mean) Mean Std t(Mean) Mean Std t(Mean)

Rm-Rf -0.07% 2.07% -0.64 -0.40% 2.91% -1.41 0.13% 1.35% 0.98

SMB 0.10% 1.18% 1.62 0.05% 1.45% 0.35 0.21% 2.21% 0.96

HML 0.06% 1.26% 0.91 0.08% 1.48% 0.55 0.15% 1.74% 0.87

WML 0.51% 1.89% 5.15 0.63% 2.63% 2.45 0.36% 2.13% 1.72

 

S/L-Rf 0.12% 2.59% 0.88 -0.30% 3.71% -0.82 0.17% 1.76% 0.98

S/M-Rf 0.08% 2.56% 0.60 -0.48% 3.64% -1.35 0.30% 1.76% 1.73

S/H-Rf 0.22% 2.03% 2.07 -0.08% 2.24% -0.36 0.28% 1.98% 1.44

 

B/L-Rf 0.08% 1.74% 0.88 -0.22% 2.37% -0.95 0.17% 1.16% 1.49

B/M-Rf -0.06% 2.40% -0.48 -0.49% 3.44% -1.46 0.10% 1.59% 0.64

B/H-Rf 0.11% 2.10% 1.00 -0.29% 2.74% -1.08 0.27% 1.75% 1.57

Before we take a step to do analysis on four-factor model, we would like to focus at the mean returns on the four risk factors (Rm-Rf, SMB, HML and WML). In table 1, the market premium, Rm-Rf for the full sample period (7-year) is -0.07 percent per week (t-statistic = -0.64). The market premium for the Bear market is -0.40 percent per week versus 0.13 percent per week for the Bull market. Thus, there is a surprisingly market premium in returns is not very strong. Our market premium results in Bear and Bull market are symmetrical with (Pettengill, Sundaram and Mathur 1995) who point out that the market excess return in negative (Bear) and positive (Bull). However, our results contradict with (Fletcher 1997) findings in the UK stock market over 1975-1994 periods. He shows the market premium in the Bear market is positive and higher than Bull market. He suggests that his findings are inconsistent with (Pettengill, Sundaram and Mathur 1995) and the relationship between beta and return ‘in puzzle.’ (Graph of market returns can be visualized in Appendix.)#p#分頁標題#e#

Furthermore, we can see that there is a positive return in size premium (SMB) over the sample period. The mean return for SMB is large in full sample, July 2000 to June 2007 (0.10 percent per week) and (0.21 percent per week) in Bull market, July 2004 to June 2006. The annualised SMB standard deviation (8.51 percent) is lower than the market factor standard deviation (14.93 percent). Thus, we suggest that SMB factor is riskless compare to market factor. Next, our results show that book-to-market premium (HML) in positive returns. The HML mean returns is (0.06 percent per week) for full sample, July 2000 to June 2007. In Bear market, the HML mean returns is (0.08 percent per week) which slightly higher than SMB average returns (0.05 percent per week). In Bull market, it gains 0.15 percent per week. Our SMB and HML mean returns results are approximately similar with (Liew and Vassalou 2000) findings in Bear and Bull market. They report annually returns for HML and SMB are 4.37 percent, 5.96 percent respectively in Bull market and 2.85 percent, minus 4.61 percent respectively in Bear market. (Graph of SMB returns and HML returns can be visualized in Appendix.)

Moreover, our statistical results show that the mean return on the WML premium is very large, (0.51 percent per week, t-statistic 5.15) over the 2000-2007 period. We can observe that the WML premium is the largest premium among other factors premium in the UK stock market. These results are similar with the empirical findings in the Canadian market by (L'her, Masmoudi and Suret 2004) over the period of July 1960- April 2001 (1.34 percent per month, t-statistic 6.66). Their results report momentum return is the highest compare to other three risk factors (market, size and book-to-market). We observe that the momentum return in Bear and Bull market is 0.63 percent per week, 0.36 percent per week respectively. It means that momentum return in Bear market performs better than Bull market. The results contradict with (Chordia and Shivakumar 2002) who indicate that momentum return is high during market upturns. One of the possible reasons is momentum effect works well for short-term period (3 to 12 months). According to (Carhart 1997) empirical results, he suggests that securities with high returns last year have higher return in one year period but not in years thereafter. He clarify that last year’s winners frequently become next year’s losers and vice versa. Our sampling data for Bear and Bull market is 2-year. We suggest that in Bull market, investors tend to buy the winner’s stocks and become loser’s stocks in years thereafter. In Bear market, investors tend to sell the loser’s stocks and become winner’s stocks in years thereafter. Therefore, we can see that higher return in Bear market compared to Bull market. Our results still provide supportive evidence towards Carhart’s empirical results. (Graph of WML returns can be visualized in Appendix.)

In general, the results we collected on the size, book-to-market and momentum premiums are approximately the same as the empirical studies in the UK equity market by (Gregory and Michou 2009) over the period 1975 to 2005. They reported SMB average return (0.01 percent per month), HML premium (0.44 percent per month) and WML premium (0.14 percent per month). However, the market excess returns differ significantly. For the rest of the three factors such as size, book-to-market and momentum provide positive premium and help to explain the cross-sectional variation in the average stock returns in the UK stock market. Furthermore, we compare our statistical results with Fama and French (FF) Benchmark Factors for the period 2000-2007. Our results show that market premium (minus 0.07 percent per week) versus FF market premium (0.03 percent per week), SMB premium (0.10 percent per week) versus FF SMB premium (0.15 percent per week) and HML premium (0.06 percent per week) versus FF HML premium (0.27 percent per week). We record a somewhat lower market premium, a higher SMB premium and a considerably lower HML premium in our study. The SMB premium and HML premium in positive returns have confirm that the premium of small stock and value (high book-to-market) stock returns are bigger than the premium of large stock and growth (low book-to-market) stock returns. This is reliable with the hypothesis of (Fama and French 1995). Therefore, we suggest that there is a strong SMB and HML premium in the UK stock market. (Fama and French 1998) find that value stocks have higher returns than growth stocks not only persist in the US but in different capital markets.Table 2(a): Excess Returns on the six portfolios sorted by size and book-to-market ratios from #p#分頁標題#e#

July 2000 to June 2007. (Full Sample) 

Based on the table 2 (a), we observe that the three small size portfolios (S/L, S/M, S/H) tend to outperform than the three big size portfolios (B/L, B/M, B/H). Refer to the statistical results in Table 1, the total excess returns on the three small stocks portfolios gain 21.84 percent per annum. For the three big stocks portfolios only generate total of return at 6.76 percent per annum. There is a clear inverse relationship between size and average return. We suggest that small stocks generate higher returns than big stocks. This is because small stocks are more risky assets and need higher returns as compensation to bear higher risk. Apart of size factor findings, we also notice that there is a direct relationship between book-to-market and average returns. In other words, the high book-to-market ratio stocks (value stocks) tend to perform relative well (higher returns) compare to low book-to-market ratio stocks (growth stocks). We look into the high book-to-market portfolios (S/H, B/H), the excess returns is 17.16 percent per annum. For the low book-to-market portfolios (S/L, B/L), we calculate the excess return is 10.40 percent per annum. The reason value stocks outperform than growth stocks because value stocks are more risky and need higher returns in order to bear higher risk. Our statistical results are consistent with (Fama and French 1996) empirical results. They state that small stocks and value stocks generate higher excess returns compare to big stocks and growth stocks. 

Table 2(b): Excess Returns on the six portfolios sorted by size and book-to-market ratios from July 2000 to June 2002. (Bear market) 

Table 2(b) shows that all of the six portfolios perform relatively poor excess returns in Bear market. In this period, we notice that even though the portfolios are in negative returns, but the total returns of three small stocks (-44.72 percent per annum) still perform better than the three big stocks (- 52 percent per annum). In other words, small stocks portfolios lose less money than big stocks portfolios. We also examine the high book-to-market portfolios and find that value stocks (- 19.24 percent per annum) still outperform than growth stocks (-27.04 percent per annum).

Table 2(c): Excess Returns on the six portfolios sorted by size and book-to-market ratios from July 2004 to June 2006. (Bull market) 

In Bull market, we examine that all risk factors premium are positive excess returns. We look into the three small stock portfolios which perform well and generate 39 percent per annum. For the three big stock portfolios, they only generate 28.08 percent per annum. There is a clear inverse relationship between size and average return. We can suggest that size factor does exist in the Bull market because small stocks tend to achieve higher returns than big stocks. Moreover, we also notice that there is a clear direct relationship between book-to-market and average returns. The table2 (c) shows that the value stocks (28.60 percent per annum) have higher excess return than growth stocks (17.68 percent per annum). We suggest that firms carry risk premium. In summary on the statistical results, we find that size factor and book-to-market factor does work well in full sample period, bear market and bull market. The results are consistent with Fama and French hypotheses.#p#分頁標題#e#

4.2 Data Analysis

4.2.1 Full Sample Regression

Table 3: Regression on the Six Portfolios Regressed for Market (Mkt), Size (SMB), Book-to-Market (HML) and Momentum (WML) Factors from July 2000 to June 2007. 

Rpt-Rf = ai + bi(Rmkt-Rf) + siSMB + hiHML + wiWML + εi

In line with the procedure used by (Fama and French 1993), the six portfolios are constructed by size and book-to-market ratios portfolios. Rpt is the equal-weighted weekly return on each of the six portfolios. Rf is the risk-free return. (Rmkt-Rf) is the equal-weighted market return. SMB is the three small stock portfolios average returns minus the three big-stock portfolios average returns. HML is the two high book-to-market portfolios average returns minus the two low book-to-market portfolios average returns. WML is the two winner-stock portfolios average returns (the highest performance stock return in the past 11 months lagged one month) minus the two loser-stock portfolios average returns (the lowest performance stock return lagged one month). t() indicates t-statistic. Adjusted R2 is the adjusted coefficient of determination. 

? Significant different from zero at the 5% level.

Portfolio a b s h w R2

S/L 0.01 0.675 0.353 -0.235 0.054 0.787

(1.82) *(24.84) *(13.39) *(-8.81) (1.90) 

S/M 0.01 0.703 0.272 0.084 -0.094 0.694

(1.80) *(21.71) *(8.65) *(2.65) *(-2.78) 

S/H 0.01 0.742 0.304 0.301 0.214 0.784

(1.88) *(27.10) *(11.45) *(11.21) *(7.49) 

B/L 0.001 0.831 -0.402 -0.304 0.284 0.624

*(1.99) *(23.03) *(-11.48) *(-8.57) *(7.54) 

B/M 0.001 0.837 -0.09 0.007 -0.139 0.766

(1.65) *(29.40) *(-3.24) (0.26) *(-4.48) 

B/H 0.001 0.78 -0.202 0.357 0.089 0.649

*(1.99) *(22.38) *(-5.97) *(10.42) *(2.43)

 

4.2.1.1 Full Sample Analysis

We report the coefficient for the UK stock market of six portfolios on the (Fama and French 1993) three risk factors: market, size, book-to-market and the (Carhart 1997) momentum factor in Table 3. The intercept, a coefficient is statistically insignificant at the 5 percent level for other portfolios except B/L and B/H portfolios. We examine that the b coefficient is highly significant at the 5 percent level for the six portfolios. The t-values ranging from the lowest of 21.712 to the highest of 29.404 and the betas range from 0.675 to 0.837. According to (Fama and French 1992) , they argue that ‘beta is dead’ because market beta is not the main explanation of differences in rates of return between stocks. They find that when size and book-to-market factor are tested under the Fama and French three factor model. The results show that beta becomes statistically insignificant. However, (Carhart 1997) extends the Fama and French three factor model by added additional risk (momentum) which becomes the four-factor model. His findings show that beta becomes statistically significant. Therefore, our beta findings are consistent with their empirical results. (Boasson, Boasson and Cheng 2006) find that the market beta is found statistically significant for all funds in the US portfolios by the four-factor model. The overall average beta is close to 0.80. Our betas results show that the six portfolios can be expected to fluctuate less than the market as a whole. For instance, small stock with low book-to-market portfolio (S/L) has an average beta of 0.80, it will react 20 percent less strongly than the average ups and downs of the market. According to recent empirical studies by (Gregory and Michou 2009) in the UK stock market, they find that beta value is 0.90. Certainly, our results provide supportive evidence that the market factor (Rmkt-Rf) does play an important role in the four factor model. Other interesting findings, we observe that the average beta for small stock portfolios (0.71) is smaller than average beta for big stock portfolios (0.82). According to (Fama and French 1996), ‘β alone is unrelated to size.’ In other words, it does not mean that small firms are riskier and definitely there have higher beta. Even though our market betas show positive and statistically significant but we suggest that market beta is unrelated to firm size when explain average returns.#p#分頁標題#e#

For SMB factor, we observe that s coefficient is found positive and statistically significant for the three small stock portfolios (S/L, S/M and S/H). However, the three big stock portfolios (B/L, B/M and B/H) are shown negative and statistically significant different from zero at the 5 percent level. We notice that small stock portfolios have higher positive slope compare to big stock portfolios which have negative slope. So, we suggest that small stocks tend to have higher returns than big stocks. In other words, small stocks are more risky than big stock. Therefore, they need to higher compensation for higher risk. Another interesting part, we find that the coefficient decrease monotonically from positive values to negative values in the small stock portfolios and the big stock portfolios. The pattern on SMB in this study is similar to that documented by (Fama and French 1993).

For HML factor, the h coefficient is positive and statistically significant in high book-to-market stock portfolios (S/H and B/H). However, when we observe the low book-to-market stock portfolios, the h coefficient is shown negative and significant at the 5 percent level. We compare our h coefficient results with (Fama and French 1996), their findings show that high book-to-market stock portfolios have higher positive slope compare to low book-to-market stock portfolios which have negative slopes. Therefore, we believe that our regression results are consistent with their findings. This results support high book-to-market stock portfolios tend to outperform low book-to-market stock portfolios. For example, investors hold high book-to-market stocks can receive higher excess return compare to hold low book-to-market stocks. This is because high book-to-market stocks more likely to have high financial leverage and poor earnings in future. Therefore, those stocks need higher expected returns as compensation to bear higher risk. Apart from that, we also notice the coefficient of HML is increasing from low book-to-market stock portfolios to high book-to-market stock portfolios. Based on the regression analysis, we suggest that the SMB factor and HML factor do provide convincing evidence to capture the cross-sectional variation in average returns that fail captured by market factor. According to (Fama and French 1992), they indicate that size and the book-to-market ratio are the two missing factors use to explain the differences in stock return.

For the momentum factor, we examine that the w coefficient is positive appear in small stock portfolios (S/L, S/H) and big stock portfolios (B/L, B/H). These portfolios are statistically significant at 5 percent level except S/L portfolio. As most of them are statistically significant, the WML factor appears to have statistically ability to explain the cross-sectional variation in stock returns. For these two portfolios (S/M and B/M) coefficient are negative and statistically significant. One of the explanations, we suggest that most of the investors prefer to long winner stocks and short losers stocks in small stock portfolios or big stock portfolios rather than medium stock portfolios. In overall, our study shows that momentum factor acts as compensation for bearing risks. The reason is winners stocks are more risky than losers stocks. Therefore, investors tend to long winners stocks because they can gain higher returns for bearing higher risk. Our results are consistent with (Tonks and Hon 2003) which show that winners stocks tend to achieve superior returns than losers stocks because winners stocks have higher risk than losers stocks.In addition, we calculate that there is 72 percent in average R2 for the six portfolios. The adjusted R2 range from 0.624 to 0.787. We find that the small stock portfolios (S/L, S/M, SH) tend to have higher R2 than big stock portfolios (B/L, B/M, B/H). It seems that small stock portfolios have a larger explanatory power than the big stocks. The evidence suggests that the return variation is better explained by the four factors in small firms. Again, our empirical results are consistent with (Fama and French 1995) results. They show that small stock portfolios (R2=0.99) have higher explanatory power than big stock portfolios (R2=0.96). Moreover, we also observe that the high book-to-market stock portfolios have higher R2 compare to low book-to-market stock portfolios. #p#分頁標題#e#

4.2.1.2 Graph of the Full Sample Analysis 

Figure 1: Market Factor

In Figure 1, it shows that there is positive relationship between small stock portfolios with market excess. It means that the small stock portfolios tend to perform well (poor) when the market outperform (underperform). According to CAPM, small stocks more risky than big stocks. Therefore, small stocks tend to have high beta and receive higher return as compensation. However, our empirical results show that the market beta does not have any relationship with size factor. Our summary statistics in Table 1, it shows higher beta in big stock portfolios rather than small stock portfolios. The scatter points focus mostly in minus 0.05 percent and 0.05 percent range. The pattern of the graph shows in upward trend. 

Figure 2: Size Factor

In Figure 2, we observe that the relationship between big stock portfolios and size factor is negative relationship. In other words, holding the big stock portfolios tend to receive lower returns compare to the small stock portfolios in size factor. One of the possible reasons is small firms are risky than big firms due to poor earnings in future and cash flow problems. Therefore, investors need higher returns as compensation in order to bear higher risk. Another possible reason is due to the economic conditions. For example, during the market in upsizing, big stock portfolios have stable earnings and investors accept lower returns. During the market in downsizing, big stock portfolios have less growth and lower risks. Thus, the returns pay to investors become small. Our relationship is consistent with Fama and French (1992) findings. There is negative slope to big stock portfolios. The scatter points highly appear in the range of minus 0.02 percent and 0.02 percent. 

Figure 3: Book-to-market Factor

 

In Figure 3, there is positive relationship between high book-to-market (value) stocks with book-to-market factor. We notice that higher the book-to-market, they can receive higher returns. The reason is value stocks have higher risk than growth stocks due to financial distress in the company. Investors only want to hold this risky assets if higher returns as compensation to them. Our relationship is consistent with Fama and French (1992) findings. There is positive slope to high book-to-market stock portfolios. The scatter points mostly concentrate within the range of minus 0.04 percent and 0.04 percent.

Figure 4: Momentum Factor

In Figure 4, we observe that the relationship between low book-to-market (growth) stock portfolios with momentum factor is negative relationship. It means that low book-to-market stock portfolios tend to perform poorer than high book-to-market stock portfolios. (Jegadeesh and Titman 2001) state that both winners and losers are more likely to be small firms because small firms have more volatile and high extreme return. Smaller firms tend to have high book-to-market and large firms more likely to have low book-to-market. Therefore, we suggest that one of the possible reasons is growth stocks are less risky than value stocks because they do not have financial distress problems. Growth stocks have stable earnings and less volatile. Investors who hold low book-to-market tend to receive lower returns due to lower risk. Our regression results in Table 3 also show that the w coefficient is positive and statistically significant at the 5 percent level in value stock portfolios. The scatter points highly focus in the range of minus 0.05 percent and 0.05 percent.#p#分頁標題#e#

4.2.2 Bull Market Regression

Table 4: Regression on the Six Portfolios Regressed for Market (Mkt), Size (SMB), Book-to-Market (HML) and Momentum (WML) Factors from July 2004 to June 2006. 

Rpt-Rf = ai + bi(Rmkt-Rf) + siSMB + hiHML + wiWML + εi

In line with the procedure used by (Fama and French 1993), the six portfolios are constructed by size and book-to-market ratios portfolios. Rpt is the equal-weighted weekly return on each of the six portfolios. Rf is the risk-free return. (Rmkt-Rf) is is the equal-weighted market return. SMB (Small minus Big) is the three small stock portfolios average returns minus the three big stock portfolios average return. HML (High minus Low) is the two high book-to-market stock portfolios average returns minus the two low book-to-market stock portfolios average returns. WML is the two winner-stock portfolios average returns (the highest performance stock return in the past 11 months lagged one month) minus the two loser-stock portfolios average returns (the lowest performance stock return lagged one month). t() indicates t-statistic. Adjusted R2 is the adjusted coefficient of determination. 

? Significant different from zero at the 5% level.

Portfolio a b s h w R2

S/L 0 0.295 0.799 0.006 -0.051 0.656

(0.2) *(4.69) *(11.71) (0.08) (-0.61) 

S/M 0.001 0.346 0.665 0.121 0.074 0.686

(0.62) *(5.76) *(10.18) (1.66) (0.93) 

S/H 0 0.212 0.518 0.355 0.09 0.658

(0.29) *(3.38) *(7.60) *(4.65) (1.08) 

B/L 0.001 0.608 0.204 -0.155 0.215 0.455

(0.67) *(7.68) *(2.37) (-1.61) *(2.05) 

B/M 0 0.228 0.387 0.234 -0.13 0.239

(0.31) *(2.44) *(3.81) *(2.06) (-1.05) 

B/H 0.001 0.41 0.204 0.562 0.068 0.783

(0.48) *(8.21) *(3.76) *(9.26) (1.03) 

After we examine the four factor model in full sample period (July 2000 to June 2007), we continue to study this asset pricing model in different market conditions (Bull and Bear market). The reason is check whether the robustness of four factor model has any explanatory power of those risk factors against returns in different market conditions. According to (Pettengill, Sundaram and Mathur 1995), they measure market beta against returns in two different sampling periods (up and down markets). Their findings show that market beta is high against low returns during down market because market returns fall below the risk-free rate. In up market, the market beta is high against high returns because market returns rise above the risk-free rate. Therefore, we can notice that risk-free rate has significant relationship between portfolios returns and market beta in different market conditions. However, (Fama and French 1996) argue that there is weak relationship between beta and return in the US market. Other empirical study done by (Chan, Chen and Hsieh 1985) emphasize that the risk differences between small and large stocks arise from the differences in their market conditions to changes in the underlying risk factors.#p#分頁標題#e#

4.2.2.1 Bull Market Analysis

The sampling period is split into Bull (up) and Bear (down) market periods. There are 103 weeks in up market period and 105 weeks in down market period. In Table 4 shows the result for the time-series regressions under Bull market conditions. The intercept, a coefficient is statistically insignificant at the 5 percent level for six portfolios. We observe for the market factor, b coefficient is positive and statistically significant in all portfolios. According to (Carhart 1997), he states that the four-factor model shows beta is statistically significant. Our results also show that there is positive relationship between market betas against returns in Bull market. This is consistent with (Pettengill, Sundaram and Mathur 1995) findings. In other words, when the market return is positive, we should see high beta portfolios are more sensitive to Bull market performance, they should receive high returns than any low beta portfolios. The average of market betas in Bull market is only 0.35. It means that the portfolios are less volatile towards the market in general. According to (Levis 1985) who investigate firm’s size affect the stock returns, he finds that smaller stocks have lower beta than larger stocks. Certainly, we provide a supportive evidence that small stock portfolios (S/L, S/B, S/H) have on average lower beta occur in up (0.284) compared to big stock portfolios (B/L, B/M, B/H) have on average higher beta occur in up (0.415).

The SMB factor coefficient is positive and statistically significant at the 5 percent level for the six portfolios. We observe that t statistics highly significant value for the small stock portfolios (S/L, SM, S/H) compare to big stock portfolios (B/L, B/M and B/H). We compare our results are consistent with (Lam, Li and So 2009) regression results in Bull market period. We suggest that small stock portfolios have more explanation power than big stock portfolios. It means that small stocks tend to perform better than big stocks in terms of returns during Bull market. Investors are more likely to buy small stocks even though they are more risky than big stocks. This is because investors willing to accept higher reward to small stocks as compensation for higher risk. Our explanation is similar with (Liew and Vassalou 2000) who justify that high economic growth can affect the size effect. They show that invertors preferably to hold small stock portfolios when the market is rising. Our results in Bull market are similar with full sample regression results. We can see that the pattern of s coefficient decrease monotonically from small to big stock portfolios. In full sample period, Table 3, it shows that small stock portfolios have higher positive slope compare to big stock portfolios which have negative slope. So, the SMB factor has strong effect during the full sample period. 

Next, we identify for the HML factor. We find that the h coefficient is positive and statistically significant at 5 percent level only for high book-to-market stock portfolios (S/H, B/H). This finding is consistent with (Fama and French 1993) and (Chan and Chui 1996) empirical evidence. They find that the risk captured by book-to-market stock defines for distress factor. They interpret the return to book-to-market as compensation for financial distress. Financial distress means to high book-to-market firms that they more like to have problem on cash flow and poor expected earnings in future. These negative issues would cause the stock price of the firms become low. Therefore, most of the investors have to bear high risk when they hold high book-to-market stock portfolios. In order to attract the investors preferable on high book-to-market stock portfolios, they need to perform relatively well and generate higher expected returns as compensation compare to low book-to-market stock portfolios. We also notice that the h coefficient on low book-to-market stock portfolios (S/L and B/L) is statistically insignificant. This result confirms that HML factor in the four-factor model is significant to explain the cross-sectional variation in stock returns during Bull market. Other possible explanation is the macroeconomy conditions. (Liew and Vassalou 2000) state that high book-to-market stock portfolios perform better than low book-to-market stock portfolios because of the high economic growth in Bull market. In this case, most of the investors more likely to hold high book-to-market stock portfolios because they have high possibility to generate higher returns. Therefore, our results are consistent with his findings which show that there is a positive relationship between return and high book-to-market.Based on the summary statistics in Table 1, we calculate that the mean return for the momentum factor is positive returns. However, we observe that the w coefficient is statistically insignificant at 5 percent level in most of the portfolios. It shows the momentum factor does not explain the average stock returns during Bull market. According to (Liu 2006), he finds that the momentum factor fail to explain in the UK stock market. One of possible reasons is momentum driven by behaviour finance. During the Bull market, investors tend to overreact the information. In this case, the winners stocks receive the positive information should realise positive returns during that period. Similarly, the losers stocks receive the negative information should realise negative returns.#p#分頁標題#e#

We calculate that there is 58 percent in average R2 for the six portfolios. We find that the small stock portfolios (S/L, S/M, SH) tend to have higher R2 than big stock portfolios (B/L, B/M, B/H). It seems that small stock portfolios have a larger explanatory power than the big stocks. The evidence suggests that the return variation is better explained by the four factors in small firms. We also compare the R2 for book-to-market stock portfolios. We observe that on average R2 for high book-to-market stock portfolios is 72 percent .The average R2 for low book-to-market stock portfolios is 55 percent. We suggest that value stock portfolios have a larger explanatory power than the growth stock portfolios. This regression results are consistent with (Shum and Tang 2005) who examine the Taiwan stock market in Bull market. They find that small firms and high book-to-market firms have higher explanatory power in R2.

4.2.3 Bear Market Regression

Table 5: Regression on the Six Portfolios Regressed for Market (Mkt), Size (SMB), Book-to-Market (HML) and Momentum (WML) Factors from July 2000 to June 2002. 

Rpt-Rf = ai + bi(Rmkt-Rf) + siSMB + hiHML + wiWML + εi

In line with the procedure used by (Fama and French 1993), the six portfolios are constructed by size and book-to-market ratios portfolios. Rpt is the equal-weighted weekly return on each of the six portfolios. Rf is the risk-free return. (Rmkt-Rf) is is the equal-weighted market return. SMB (Small minus Big) is the three small stock portfolios average returns minus the three big stock portfolios average return. HML (High minus Low) is the two high book-to-market stock portfolios average returns minus the two low book-to-market stock portfolios average returns. WML is the two winner-stock portfolios average returns (the highest performance stock return in the past 11 months lagged one month) minus the two loser-stock portfolios average returns (the lowest performance stock return lagged one month). t() indicates t-statistic. Adjusted R2 is the adjusted coefficient of determination. 

? Significant different from zero at the 5% level.

Portfolio a b s h W R2

S/L 0.01 0.67 0.275 -0.217 -0.036 0.83

(0.30) *(10.22) *(6.26) *(-4.52) (-0.56)

S/M 0 0.712 0.198 0.191 -0.308 0.849

(0.06) *(11.52) *(4.78) *(4.22) *(-5.07) 

S/H 0.01 0.985 0.3 0.278 0.228 0.694

(0.48) *(11.21) *(5.09) *(4.33) *(2.64) 

B/L 0 0.82 -0.336 -0.213 0.125 0.714

(0.30) *(9.66) *(-5.90) *(-3.42) (1.49) 

B/M 0 0.774 -0.099 0.032 -0.226 0.877

(0.32) *(13.87) *(-2.65) (0.80) *(-4.12) 

B/H 0 0.84 -0.148 0.328 -0.109 0.695

(0.19) *(9.57) *(-2.52) *(5.12) *(2.00)

代寫英國dissertation

4.2.3.1 Bear Market Analysis

In Bear market, our regression results show that a coefficient is statistically insignificant in all portfolios. We observe that the market beta factor, b coefficient is positive and highly significant in the six portfolios. The average betas in Bear market are 0.80. It indicates that the market beta is significant and close to one in Bear market. For example, portfolios tend to have more volatile changes toward the market in general. We notice that the market betas in six portfolios are high but the market returns is negative (minus 0.40 percent). It means that there is negative relationship between market beta and market return in Bear market. This finding is consistent with (Pettengill, Sundaram and Mathur 1995) findings in Bear market. They suggest that higher beta stock portfolios tend to have less realised return when the stock market is perform badly. In other words, market returns in negative during Bear market, high beta portfolios tend to have more sensitivity towards the market and they more likely to perform poorly compare to low beta portfolios. #p#分頁標題#e#

Next, the s coefficient for the SMB factor is statistically significant at the 5 percent level in all six portfolios. We observe that there is positive slope to the three small stock portfolios (S/L, S/M, S/H) and negative slope to the three big stock portfolios (B/L, B/M, B/H). The pattern on SMB is similar to the full sample regression results. So, we suggest that small stocks gain higher reward as compensation for higher risk than big stocks. We also find that the coefficient decrease monotonically from positive values to negative values in the small stock portfolios and the big stock portfolios. The coefficients also display decreasing trends with increasing size within each book-to-market equity portfolios. Moreover, we compare the pattern of s coefficient with Bull market results. We observe that there is high statistically t-value in small stock portfolios and almost double than the small stock portfolios in Bear market. (Levis and Liodakis 2001) indicates that small stock portfolios tend to underperform in Bear market and outperform in Bull market. One of the possible reasons is small stocks more sensitive towards the economic environments. According to (Jensen, Johnson and Mercer 1998), they state that when economic in recession, the interest rate changes and monetary policy would affect the small stock performance.

For the HML factor, we notice that the h coefficient has positive and highly significant value in high book-to-market stock portfolios (S/H, B/H). Compare to low book-to-market portfolios (S/L, B/L), the h coefficient is negative but still statistically significant at 5 percent level. Again, we find that HML factor significant to explain long high book-to-market stock portfolios capable to generate higher returns than low book-to-market stock portfolios. Even though high book-to-market stock portfolios are more risky due to financial distress problem but they tend to generate higher returns to investor as compensation.

For the momentum factor, the regression results show that the w coefficient is positive and statistically significant at 5 percent level in value portfolios (S/H and B/H). We find that the w coefficient is insignificant in growth portfolios (S/L and B/L). We suggest that most of the winners stocks are value stocks than growth stocks in Bear market. It shows the momentum factor does explain the cross-sectional variation in stock returns in Bear market.

We calculate that there is 78 percent in average R2 for the six portfolios. When we compare Bear market regression results with Bull market results. There is improvement and stronger R2 as explanatory power in the four factor model. We suggest that the one of the reason is momentum factor has highly significant value in Bear market. Our high R2 is almost similar with (Lam, Li and So 2009) time series regression results. They show that 53 percent in average R2 in Bull market and 73 percent in Bear market. Besides average R2, their results also show momentum factor is insignificant during Bull market from July 1981 to June 2001. We also observe different size and book-to-market stock portfolios may have different explanatory power in R2. However, the results seem to suggest small (78 percent) and big stock (76 percent) portfolios have similar explanatory power in Bear market. For the high book-to-market stock (70 percent) portfolios tend to have less explanatory power in R2 than low book-to-market stock (77 percent) portfolios. #p#分頁標題#e#

4.2.4 Behaviour Finance Arguments:

However, (Debondt and Thaler 1985) argue that size effect is affected by irrational in stock market. For example, investors tend to overreact the new information such as financial analysts reports and economics forecast. For example, investors feel optimism on the large stock portfolios because high earnings with persistently. They preferable to hold this kind of stocks and the share price tend to overprice. If they have any bad news on large stock, they have large impact on share price loses. In other way round, investors feel pessimism on the small stock portfolios because poor earnings and cash flow problem. They don’t feel comfortable to hold this kind of stocks and the share price tend to underprice. If they have any bad news on small stocks, the impact on these stocks is smaller than large stocks. In next case, if small stocks receive any good news, the positive surprise makes the large impact on their share price. Therefore, we can notice that the small stock portfolios tend to outperform than large stock portfolios in Bear or Bull market. 

Another empirical result done by (Lakonishok, Vishny and Shleifer 1994) who argue that there is irrational on distress premium. This is because as investors overreact to the growth stocks (low book-to-market ratio) which perform relatively well in the past returns and cause the stock price overpriced. However, investors underreact to the value stocks (high book-to-market ratio) which perform relatively poor in the past returns and cause the stock price underpriced. According to (La Portal and Shleifer 1997) who point out that market investors underestimate future earnings for high book-to-market stock and overestimate future earnings for low book-to-market stocks. In other words, investors more likely to buy “good” stocks with high earnings and good management compare to “bad” stocks with poor growth in the past and perform relatively slow in terms of earnings.Moreover, other behaviour findings show that momentum factor is also caused by overreaction. (Debondt and Thaler 1985) find that “winner-loser” effect is consistent with overreaction. The portfolios of losers outperformed than the portfolios of winners by an average of 31.9% in 5 year periods. Investors tend to overreact to the “winner” stocks in the short term periods. When “winner” stocks price exceed from their underlying fundamental values, it starts to adjust the pricing until fair value. However, (Jegadeesh and Titman 1993) challenge that momentum factor has promise. Winners stocks have much higher average returns than losers stocks during six months to one year period. 
 

5.0 Summary and Conclusion
 

In our study, we examine the four-factor model in the UK stock market from July 2000 to June 2007. The four-factor model includes market factor (excess market return), size factor (SMB), book-to-market factor (HML) and momentum factor (WML). Based on our research results, we find that the four-factor model does well in explaining the average returns in the UK stock market. #p#分頁標題#e#

According to (Fama and French 1992), they state that the expected returns may not depend wholly on the beta values. In other words, they point out that this signal the “death of beta” in Fama and French three factor model. In our study, we observe that our market beta shows highly significant at 5 percent level in the four-factor model. This finding is consistent with (Carhart 1997) who shows that beta becomes statistically significant after added momentum factor in the multifactor model. Apart from the market beta is found highly significant, we also find that there is significant in size and book-to-market factor. We find that small stock portfolios seem to have performed better than big stock portfolios and high book-to-market stock portfolios (value) seem to have performed relatively well than low book-to-market stock portfolios (growth). Based on our findings, small firms and value firms tend to perform better in terms of realized returns because these firms carry a risk premium. In other words, small firms and value firms are more likely to have financial distress problem and poor earnings in future. They need to obtain higher returns as compensation for bearing higher risk. Our results are consistent with the (Fama and French 1996) findings. They state that small stocks and value stocks generate higher excess returns compare to big stocks and growth stocks. If the stock price is priced rationally, then other risk factors related to size and book-to-market capable to explain the average returns. Our results show that size premium and book-to-market premium gain average returns of 5.20 percent per annum and 3.12 percent per annum, respectively. However, market premium does not generate a positive return in our full sample period. The annual returns for the market portfolio are minus 3.64 percent per annum. The figures clearly show that size portfolio and book-to-market portfolio perform better than market portfolio. According to (Fama and French 1992), they state that the relationship between market beta and stock returns is ‘flat’ and stock returns are more likely to depend on size and book-to-market factor. In our view, the evidence that market beta does not suffice to explain expected return in the UK stock market.