Estimation Of Beta And The Testing


02 Nov 2017

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The Estimation of Beta and the Testing of Non-linearity in Returns: The case of Mauritius

BSC (Hons) Economics and Finance 3

Mungroo Deevya



The study analyses the Capital Asset Pricing Model (CAPM) on the Mauritian Stock market. Monthly stock returns from 34 listed companies were taken into consideration in the timeframe of January 2002 to December 2007 that is a 6 year observation period. These 34 securities are grouped into 3 portfolios in an attempt to diversify risk and thus magnifying the precision of the beta estimates. Under the means of the two step procedure model by Fama & Macbeth (1973), the findings of the study are partly supporting the CAPM’s basic theory that high level of risk (beta) is liaised to high level of returns.

Moreover, the theory for a zero intercept and a slope equal to the excess returns on the market portfolio does not hold in our analysis but the non-linearity condition does uphold that is the beta and expected returns in the study portrays a linear relationship. Furthermore, a test to investigate if the model captures the residual variance of returns as well as all important determinants was conducted but residual variance did not seem to be affecting the data set.

Results may be in contradiction to CAPM but they do not provide support to an alternative model as well.

The Stock Exchange of Mauritius

A stock market acts as an indicator to the financial health of an economy. Its role in the economic development is highly valuable as it helps in developing the country through savings, efficient allocation of investment resources and attracting foreign investment. The stock exchange of Mauritius is fairly newly instated in comparison to many countries.

The Stock Exchange of Mauritius Ltd (SEM) has been operating for more than 20 years. Incorporated in March 1989 as a privately owned organization with a public mandate, the SEM became a member of the World Federation of Exchanges (WFE) in 2005 and is committed to becoming a World class stock exchange.

The SEM operates two markets, namely the official market (1989) with 5 listings since its inception to 40 listed companies as at December 2004 with a market capitalization of nearly USD 92 million and the DEM (2006) with 50 listed companies. The DEM is mostly designed for Small and Medium-sized Enterprises (SMEs) which demonstrate a good growth potential. Since its inception, the SEM developed the financial architecture to keep pace with the economy and to provide investors with the necessary platform to trade a variety of new financial products.

The listed companies are listed under 7 main categories: Banks and Insurance, Industry, Investment, Sugar, Commerce, Leisure & Hotels and Transport. Listing rules on the SEM are revised with the South African Development Community (SADC) countries and it is ranked as the second best African Stock Exchange due to its flexible regulatory environment.

The Stock exchange is supervised by the Financial Services Commission under the Stock Exchange Act 1988 that formed the Stock Exchange Commission which controls and oversees stock exchange operations.

After the lifting of exchange control in 1994, the stock market was opened to foreign investors and in 1997, daily trading was enacted. 11 stock broking companies are in operation with an order-driven trading and with the successful implementation of the Central Depository System (CDS) in 1997, prompt, efficient clearing and settlement of trades in the market were brought about. Furthermore, the Automated trading System (SEMATS) launched in 2001 put an end to the traditional patterns and brought in the third generation technology.

Chapter one


Considerable amount of research have been accomplished to understand the behavior of asset prices under uncertainty conditions. The necessity was felt to develop theories which would explain the risk associated with financial transactions and hence marked the breakthrough of the mean variance model pioneered by Sharpe (1964) and Treynor (1965) which became the cornerstone of asset pricing literature. The model was further developed by Lintner (1965) and Black (1972).

The model which revolutionized the financial economics world is referred to as the Capital Asset Pricing Model (CAPM) which is the subject of this study. The roots of this model go back to the 1960’s when significant studies were aimed at developing Markowitz’ portfolio-theory. In its almost 40 years of existence, the CAPM has been in the spotlight for research but has been subjected to criticism as well.

The Sharpe-Lintner and Black (SLB) model narrates a theory that stipulates the linear relationship of an asset’s expected return and systematic risk (beta). The investors are assumed to eliminate diversifiable risk to allow the calculation of non-diversifiable risk or systematic risk via the beta. The beta coefficient is said to measure the volatility of a stock or the portfolio to the market risk.

Behind the fundamentals in calculating expected return, CAPM puts emphasis on two important facts that an investor should know that is, the risk premium of the overall portfolio and the security’s beta versus the market.

The effectiveness of CAPM is not only restricted to the sphere of investments of securities but also to the estimation of the cost of capital of firms. Literally, the study of Graham and Harvey (2001) displayed that chief financial officers prefer CAPM as a primary tool to assess the cost of equity capital.

Despite its solid stand on the financial ground, the CAPM model has been targeted to opposite remarks since many researchers argue on the partial validity of the theory regarding the beta concluding that the theory is not effective as an investment method and thus disregard the importance of beta. Questions were also raised on its so called "unrealistic" assumptions.

1.2 Scope of the study

The study will present tests on the coherence of the CAPM framework on the Stock Exchange of Mauritius (SEM) by covering a period of 6 years that is from January 2002 to December 2007. Monthly data of 34 companies are analyzed selected over the wide spectrum of the economic sectors: Banks and Insurance, Industry, Investment, Sugar, Commerce, Leisure & Hotels and Transport. Since Mauritius is a growing developing market, it is of great significance to view how far the western portfolio theories explain return on financial assets.

1.3 Objective of the study

The aim of my study is to analyze thoroughly whether the CAPM holds in the capital market of Mauritius through the use of empirical validity. The areas of the CAPM theory that this study points out are:

To revisit the validity of CAPM by using portfolios.

If high returns in the market is related with high risk that is denoting a high beta.

Whether the intercept equals zero or the average risk free rate and the slope of the SML is in line with the average risk premium.

Find proof about the linearity relation between expected return and the stock beta (systematic risk).

Finding whether returns incorporate residual risk.

1.4 Outline of the study

The study is organized in 6 chapters. Chapter one is the introductory part of the study which is presented in a brief manner. Chapter Two illuminates in depth the various theories and assumptions attached to the model. Chapter Three gives an overview of the literature in a theoretical and empirical manner.

A rather detailed account of the methodology along with the data selection is given in Chapter Four whilst Chapter Five commemorates the analysis of the empirical tests. At last, Chapter six summarizes the conclusion of the overall study and suggests the path for further research.

The last section is reserved for references which are the sources used in the making of this study.

Chapter Two

2.1 The Literature Review

This chapter discusses the studies conducted on the Capital Asset Pricing Model (CAPM), both theoretically and empirically. Most of the early investigations on the model were done mainly on the USA market. However, presently this study has been dispersed around the globe where investors worldwide study the model. A summary of the "landmark" studies of CAPM in the USA market, the European market, the Asian Pacific market and the African market is reviewed.

2.2 The Theoretical Literature

A considerable amount of financial economics literature is devoted to the concept of asset pricing and their implications. The breakthrough of the Capital Asset Pricing Model (CAPM) was built on by Harry Markowitz’s (1952, 1959) mean-variance portfolio model which assumes that risk averse investors care only about the mean and variance of their one period investment return when choosing a portfolio. The investors keen on mean-variance efficiency focus on the two assumptions namely to minimize their portfolio return variance, S2 (Rpt), given expected return, E (Rpt), and maximize expected return given variance.

The model is aimed at constructing optimal portfolios which is based on the idea that there is a positive relationship between risk and return. Markowitz managed to prove that investors, through the creation of their portfolios, could obtain maximum level of return for a given level of risk, or a minimum level of risk for a given level of return.

Markowitz in his theory also portrayed that stocks are related to each other and that through diversification, the risk can be decreased. For instance, if the correlation coefficient of two stocks is calculated, the coefficient will give a value less than one and the overall risk will decrease given the two stocks are translated into a portfolio.

Despite being spectacular and useful in its field, the theory of Markowitz had some incommodes. The model takes into account a discrete concept of utility used in Economics which in practice is very difficult or even impossible to grasp. Moreover, the mathematics of the Mean-Variance is very sophisticated and complex which makes the application very difficult when the portfolio is consisted of a numerous number of shares. In addition, estimating the benefits of diversification would require the calculation of the covariance of returns between every pair of assets, which is quite cumbersome. The critics on the model continues on the fact that it is a static one, which makes results bias.

Fama and French quoted in their study in 2003 that "the way assets combine to produce efficient portfolios provides the template for the relation between expected return and risk in the CAPM. "

Applied to a portfolio, the linear equation of CAPM becomes a relation between expected return and risk that must hold in market equilibrium where asset prices are to clear the market of all securities.

Two key assumptions were added to the Markowitz’s model in the identification of an efficient portfolio for the market to clear by Sharpe and Lintner.

Given market clearing prices at t-1, investors agree on the joint distribution of asset returns from t-1 to t and it is the true distribution, that is, the distribution from which the returns we use to test the model are drawn.

There is borrowing and lending at a risk free rate, Rf, which is the same for all investor and does not depend on the amount borrowed or lent.

In 1964, Sharpe and in 1965, Lintner, developed from the work of Markowitz, the famous capital asset pricing model (CAPM). The basic model explains the differences in risk premium across assets that are the differences generated from the riskiness of assets. Meaning that the higher the level of risk in an asset, the higher the risk premium demanded by investors. Sharpe aims to use the theory of portfolio selection to build up a market equilibrium theory of asset prices under conditions of risk and highlights that his model sheds light on the relationship between the price of an asset and the different elements of its overall risk.

A very important sequel of this model is the separation theorem propounded by Tobin in 1958. It says that in the capital markets, risk is divided into diversifiable (unsystematic) risk and non-diversifiable (systematic) risk. For pricing purposes, only the systematic risk is significant as investors can easily get rid of the unsystematically risk through diversification. Sharpe and Lintner explain that the true measure of risk is the well-known coefficient beta.

Through their work, the CAPM became very famous in the modern portfolio theory. Depending on their appetite towards risk, investors would choose the stocks in their portfolio according to the value of beta. It is noted that stocks with a beta value less than 1 are considered passive stocks and those with a beta value greater than 1 were said to be aggressive and risky.

Yet, the first criticism in literature came from Fama and Macbeth. In 1992 they discovered a negative relation between risk and return. Important debates popped up on the validity of the beta and the true nature and measure of risk. In the lights of the doubts, Fama and Macbeth came up with the conclusion that a more realistic approach to the risk in the market is the multi-index models. They argued on the significant influence of the size of the firm and the book to market value on the performance on a stock.

However Sharpe defended his model by arguing that even the multi-index model does not eliminate beta but adds on some other variables and to think that stock’s return is not linked to market portfolio’s returns is misleading. Though CAPM does not fully reflect the reality of the market, it does provide an important guide to the investors.

After the publication of the Sharpe (1964), Lintner (1965) and Mossin (1966) articles, there was a wave of papers seeking to relax the strong assumptions that underpin the original CAPM. The most frequently cited modification is by Black (1972) who pioneered the zero-beta CAPM. The zero beta CAPM was laid on the grounds that the portfolio is uncorrelated with the market. He shows how the model changes when riskless borrowing is not available.

Black presents his equation in the traditional which is also called the expected EXCESS return form. On relaxing his previous assumption, he assumed a no constraint on short sales. His version of this transformed theory thus postulates that the ex-ante expected return on a security is liaised with the ex-ante expected return to the market shown by equation (2) in the methodology chapter.

2.3 Empirical Literature

The USA market

The early empirical studies of the CAPM were mostly done on the USA market. The first tests on individual stocks have been conducted by Lintner (1965) and Douglas (1968) in the excess return form. An intercept value much larger than the risk-free rate of return (rf) was found along with a statistically significant but with a lower value of the beta coefficient. It also shows that residual risk has effect on security returns. Despite the fact that their results show a contradiction to the CAPM, the studies of both Douglas and Lintner appear to suffer from various statistical weaknesses that might explain their anomalies results.

In their studies, Black, Jensen & Scholes (Black et al) (1992) used the equally-weighted portfolio of all stocks traded on the New York Stock Exchange (NYSE) as their proxy for the market portfolio and calculated the relationship between the average monthly return and the betas between 1926 and 1966. The cross sectional tests of significance resulted in misleading information caused by the measurement errors in the estimation of the security risk measures (beta).

The time series regressions of the portfolio excess returns on the market portfolio excess returns showed high beta securities with significant negative intercepts and low beta securities with significant positive intercepts that is; the intercept term is not different from zero and is time variant which proves to be a violation of the traditional form of the Capital Asset Pricing Model.

However, when the Two Factor Model was tested, the results gave support to the linearity hypothesis. Yet, on the whole evidence in support of the CAPM was found rather weak.

However, Stambaugh (1982) used the Lagrange Multiplier (LM) test to estimate the market model which is a slightly different methodology from Black and found evidence which supports the version of CAPM propounded by Black. But the results did not conform to the validity of the Sharpe-Lintner CAPM. Gibbons (1982) reached the conclusion that both the standard and zero betas CAPM are invalid while using the same method as Stambaugh but not through various ratio tests.

In regards to the tests of CAPM conducted on portfolios, one classic example is from Fama & Macbeth (1973) who followed the footsteps of Black et al (1992) in their studies that is evaluated stocks traded on the NYSE with a similar period (1926-1968). Combining the time series and cross sectional steps, they investigate whether the risk premium of the factors in the regression are non-zero. There seems to have evidence of a positive tradeoff between risk and return which gives support to the implications of CAPM. But Schwert (1983) argues that Fama and Macbeth (1973) arguments on a positive linear relationship between risk and return is quite weak as the tradeoff is not significant across sub periods. Doubtlessly, Fama and Macbeth (1973) showed a statistically insignificant coefficient of beta and a continuous small value for many sub-periods. Traces of residual risk which has no influence on security returns were found which contradicts directly the model proposed by Lintner. Nonetheless, the intercept surpasses the risk free rate by a great deal and the results show that CAPM might not hold.

In contrast to the proposed failure of the CAPM, the study Fama and Macbeth (1973) is among one of the building blocks of the study. It has been revised, replicated and cited many times.

Furthermore, if some research works followed the same hypotheses of Fama and Macbeth (1973), others used their cross sectional methodology to give support to the CAPM framework. Considerately, Blume and Friend (1973) tested the two factor model again on securities listed on the NYSE by employing the famous cross sectional methodology of Fama and Macbeth (1973). The authors deployed a more complicated return generating mechanism to explain the model but the CAPM did not explain for differentials returns on all financial assets. Estimates of beta seemed to work better under diversified portfolio than for individual securities. However, grouping of securities shrunk the range of beta and reduced statistical power as expected return differed from the average risk-free rate. Also, the linear model was deemed better for the empirical relationship between ex-post return and risk.

Among those who were inspired by the methodology of Fama and Macbeth (1973), Lakonishok & Shapiro (1986) examined the monthly returns of all stocks traded on the NYSE in the period 1962 to 1980. The resulting findings were that return on individual security is not specifically related to its beta, but is significantly related to the market capitalization value.

Lakonishok & Shapiro (1986) using the cross sectional methodology, tested the stocks of small firms and in order to explain average portfolio returns, they included firm size. They comprehended that the SLB model which assumes the complete diversification for all investors did not hold since investors could only diversify to a limited extent due to transaction cost and other barriers to trade. The concluding comment was that only size could significantly explain the cross sectional variations in returns in contrast to the traditional beta and the alternative (residual standard deviation) risk measure. This result highlighted the fact that cross section of expected returns is significantly explained by the existence and relative importance of various factors other than the market beta (β). Somehow these factors gave birth to the so called "Anomalies Literature" of the Sharpe-Lintner Model in the late 1970’s from which emerged less favorable evidence for the CAPM.

Anomalies in asset pricing are empirical results that seem to be inconsistent with maintained theories of asset-pricing behavior (Nair et al, 2009). When the realized average returns of an asset differs significantly from the returns predicted by an asset pricing model, the presence of an anomaly is detected.

Reinganum (1981) and Ball (1978) find that anomalies are caused due to model misspecification of static CAPM (Fama, 1970) rather than market inefficiency. The important anomalous factors identified in literature are the size of the firm, the value (ratio of book to market equity (BE/ME)), leverage and earnings to price ratio.

In the same line of study, Basu (1977) found that stocks bearing low Price-Earnings (P/E) ratios yield relatively high risk adjusted rates of returns and a positive relation between expected return and Earnings to Price (E/P). However in his study conducted in 1983, Basu refuted on the linkage between stock returns and the E/P and size. Similarly, Banz (1981) shows that size of a stock do explain expected returns, given market betas. That is, small stocks bear very high expected returns and large stocks have too low expected returns. Bhandari (1988), on the other hand investigated successfully the positive relationship between leverage (Debt to Equity ratio) and expected returns.

In addition, Fama and French (1992) propounded that other than beta, extraction of information in prices about risk and expected returns could be done by the variables like size (market value of equity), E/P, D/E (leverage) and BE/ME. They conducted the test on NYSE, AMEX and NASDAQ (1963-1990) and used the same method as Fama and Macbeth (1973). Size and return seem to have a strong negative relation while return and beta displayed a strong positive relation which coincides with the SLB model prediction of a direct relationship between beta and average return. But this evidence was shortly distorted due to high correlation between size and betas of portfolios.

A major turning point in empirical tests of Sharpe-Lintner-Black (SLB) model was the criticism of Roll (1977). Roll demonstrated that the market is not a single equity market but a market index of all wealth which includes bonds, property, foreign assets, human capital and anything else, tangible or intangible that adds to the wealth of mankind (SLB model) (J. Petros, 2009), thus defiling the portfolio used by Black, Jensen & Scholes. He concluded that by using other than the true market portfolio, tests were done for the efficiency of the proxy portfolio rather than actually testing for the CAPM. However, Stambaugh (1982) reasoned that Roll’s criticism is too strong and that the SLB model is not sensitive to market proxy since when one expands the composition of their market portfolio proxy, the results are not materially affected.

The European market

Among the recent studies on the developed countries, Nikolaos (2009) performed the two steps regression procedure on the UK stock market with a sample of 39 stocks during 18 years (1980-1998). Arguments in favor of the CAPM were found as the beta proved out to be a significant coefficient of measuring returns. Nevertheless, the second step of regression, a slope of the Security Market Line (SML) was found to be different from the slope of the SML portrayed by the traditional CAPM. In 1990, Green also found out on the unreliability of the SLB model in UK. But, while investigating the German stock market data, Saeur and Murphy (1992) demonstrated CAPM as a good model in explaining stock.

More recent studies which questioned the investigation on CAPM have been conducted on emerging markets. Michailidis et al (2003) lent support from the Fama and Macbeth (1973) method to test the CAPM on the Greek’s emerging market where high beta did not result into high levels of risk but the linear structure of the CAPM equation and excess returns was explained for the period of January 1998 to December 2002. The results went hand in hand with the fact that Greece was experiencing intense return volatility at that time. Darasteanu (2001) studied the CAPM on the Bucharest Stock Exchange to check whether beta is a good measure of risk and in contrast to the study of Michailidis et al (2003), there seems to be a positive relation between beta and return. His analysis gave recommendation for the multi-index model of Fama & French (1992) for better results as well as the taking in consideration of the other factors that affect risk.

The African market

A series of empirical studies conducted in some emerging countries concluded on a wave of partial validation of the CAPM. Amongst was the work of Wakyiku in 2010. The paper analyzed the validity of the CAPM on the Ugandan Stock market for the period 1 March 2007 to 10 November 2009 (33 months) and tested the Black, Jensen and Scholes (1972) CAPM version which predicts a non zero-beta and the relation of higher returns to higher risk. The results obtained portrayed the portfolio beta coefficient to be statistically significant providing evidence that the traditional form of CAPM holds on the USE, despite the fact that the beta coefficient does not explain the relationship between risk and return and systematic risk on the USE. A serious issue was detected from the time series analysis: the zero beta rates for most of the stocks on the USE are not different from zero. However, this does not imply that CAPM does not hold as systematic risk is low and the portfolios do not offer higher risk-adjusted returns. This fact is consistent with the notion that most emerging markets are characterized by low risk.

Further research on the USE have been carried out by Lutwama (2006), Atuhairwe and Tarinyeba (2005) and Katto and Tarinyeba (2004) who are among those who carried Non- econometric discussion on the CAPM validity. Only Mayanja and Legesi (2007) attempted a computation of stock betas called the "covariance method" but came out with a sample period biasness.

Continuing the conquest of African countries, Petros examined the CAPM on the Zimbabwe Stock Exchange in 2009. Monthly stock returns of 28 firms were utilized ranging from January 2003 to December 2008 (6 years) but was not adjusted for dividends. The monthly Zimbabwean Treasury bill issued was used as a proxy for the risk-free rate. The approached methods as described by Black, Jensen and Scholes (1972) time series test and Fama and Macbeth (1973) cross sectional test were given a try only to come up with the confirmation that the study does not fully hold up with the CAPM. Two investigations were channeled; one is the use of diversification through portfolio formation, while the other does not put it forward. In any case the results matched each other that is the lack of evidence for a higher beta yielding higher return and the portrait of a negative and downward sloping security market security line. Also a difference between average risk free rate, risk premium and their estimated values were found. Nonetheless, a linear relationship between beta and return was established.

The Asian Pacific market

Yalcin and Ersahin (2010) on the other hand, debated on the conditional CAPM to bring forward evidence from the emerging market such the Istanbul Stock Exchange (ISE) in Turkey. A time frame of February 1997 to April 2008 was used on which the Lewellen and Nagel (LN) (2006) methodology was used for the empirical tests. A market model regression on quarterly returns was performed as well as a time series of estimated alphas to investigate whether the average conditional alpha is zero. The Data consisted of daily returns, market capitalization, and book value of equity and lagged returns relying majorly on portfolio formation procedures of anomalies literature. Results showed that the conditional version of CAPM fairs no better than the static counterpart in pricing assets that is it allows beta to vary overtime committing large pricing errors.

LN propounded that as long as the conditional CAPM holds, a stock’s unconditional alpha depends on covariance between its beta and market risk premium and between its beta and the market volatility but despite significant inter temporal variations in betas, the pricing errors due to time variation seem too small to explain the aggregate pricing error.

CAPM providing accurate results when applied to the Karachi stock market and assisting the investors in pricing the securities was researched by Muhammad Hannif and Uzair Bhatti in 2010. Conducted in the Pakistani institutional framework, the study covers a six years period ranging from 2003 to 2008. The methodology encloses the calculation of Beta through the variance/covariance approach in order to predict the required return to compare with actual return; the pricing of the security and risk calculation required by investors in portfolio composition.

Evidence shows that CAPM does not give accurate results when applied on KSE. Only a small fraction of the whole sample suggested differentiated the expected returns to the actual returns and the beta explained and measured the correct risk of the security.

Thus the findings suggest that the study is not fully applicable on the KSE and required returns calculated through CAPM equation cannot be used to make investment decisions by investors.

The authors proposed a further investigation on the CAPM testing with dividend (capital gains) and the use of more sophisticated tools like the GARCH model or the APT which is known as a multi factor model to understand the KSE pricing phenomenon.

Nair, Sarkar, Ramanathan and Subramanyam (Nair et al, 2009), on their part, analyzed the relevance of factors other than beta that affect asset returns in the Indian Stock Markets. The Fama and Macbeth (FM) cross sectional regression are conducted to test the significance of anomalous factors in the Indian Stock Markets. Other diagnostic tests such as examining the violation of the assumptions of Ordinary Least Squares (OLS) method of estimation was run where a problem of heteroskedasticity was found. The paper sheds light on the estimation of the relationship between asset return and select anomalous factors, using Least Squares Dummy Variables (LSDV) technique. Weekly price data of 82 non-financial firms on the BSE 100 index were analyzed from January 1993 to August 2004 revealing size and value as important anomalous factors. Beta of the asset was found to be not significant to explain the cross section of asset returns through the informal tests and the FM model as well as the existence of factors other than beta that explains asset returns. However, the pooled regression results show that asset returns are significantly explained by beta and size and earnings to price ratio.

Additionally, the LSDV analysis reveals insignificant beta, a negative size effect as well as a positive value effect. Also, while studying the significance of additional factors like size, value, leverage and earnings to price ratio, the present study improves upon the widely adopted FM methodology by using panel data approach. The results reveal size and value as significant factors anomalous to CAPM.

Further research on the Asian pacific stock markets, was viewed by Mingchen and Zhentaozhu (2011) on the Shanghai Stock market. According to the BJS and FM tests on stocks from 2002 to 2010 the traditional CAPM did not hold. Unsettling factors such as a higher risk premium than expected and an average rate of return not linearly related to systemic risk were found. The investors disregarded the effect of non-systematic risk and due to their lack of expertise in investment, high speculation rose in investment strategies.

Other studies related to CAPM

2.4.1 The Intertemporal CAPM

Further research have resulted into a more sophisticated version of the CAPM which concludes that there are some factors beside the market that systematically affects the returns on securities. Sharpe (1964) and Lintner (1965) propounded a single-period CAPM which claims the non diversifiable risk measured by the covariance of an asset return is proportional to the expected return of an asset above the risk free rate while Merton (1973) introduced the Intertemporal CAPM (ICAPM) with multiple periods (continuous time model). He states that an asset’s expected excess return is showed by a ‘multi-beta’ version of CAPM. The number of betas is equal to one plus the number of state variables. The multi factor ICAPM provides the researchers with another root of changing risk premium.

ICAPM was developed on the assumption that the demands for assets by the investors are influenced by uncertain changes such as wealth and the forecast of future changes in income. In this framework investors maximize utility (wealth) over the total investment horizon and investors prefer to hedge against adverse changes in future investment set thus pricing assets by their covariance, other than just their systematic risk (covariance with the market return).

Not much tests on the ICAPM were conducted despite the CAPM model failed to explain the cross section of average returns. Among the few empirical tests on the original ICAPM are Campbell (1993, 1996), Chen (2003), Breeden et al (2004) and Campbell & Vuolteenaho (2004) who found the common assumption that the coefficient of relative risk aversion associated with the original utility function is constant through time. In addition, Brennan et al. (2004) documented that the investment opportunity set in the ICAPM was completely described by the real interest rate and the Sharpe ratio in the US.

Nevertheless, Cochrane (2001) demonstrated a general ICAPM framework which produces a time varying relative risk aversion coefficient. Also, some of the papers cited ignore time variation in the volatility of stock returns which affect adversely investment opportunities.

However Merton’s ICAPM is criticized for assuming price to be exogenous which is not necessarily consistent with present value computation. The problem is that the asset price or future cash flow must adjust to show a change in expected return but the exogenous price process explains that asset price is not dependant on factors that impact time varying discount rates. Thus asset price cannot adjust at the same time as changes in discount rates. Therefore the corresponding change in future expected cash flows becomes the only way to implement the change in expected return giving rise to disequilibrium between expected return method and present value computation.

2.4.2 The Conditional CAPM

The criticism against Merton’s ICAPM becomes relevant on the empirical literature on tests of the Conditional CAPM (CCAPM) which is a dynamic implementation of the Sharpe-Lintner-Mossin CAPM where their classic formula is assumed to hold period by period. Jagannathan & Wang (1996) reviewed the performance of the CAPM using US data and found the explanatory factor of CAPM (beta is stationary) to be poor and thus approves the CCAPM which shows the effects of time varying beta and where the cross section of stock returns are taken into consideration. However according to Yalcin & Ersahin (2010), when the CCAPM was tested on the Istanbul Stock market, the variation of beta over time cased pricing errors and thus could not explain risk adjusted returns. The tests of CCAPM incorporate the variances and co variances and changes in the future period and have multiple beta coefficients. To distinguish itself from the static CAPM the CCAPM includes the human capital variable and thus is able to explain the expected cross section of returns. Such an inclusion causes the R2 value to rise abruptly.

Among the empirical studies conducted, a series of positive results were found. Durack et al (2009) tested the Conditional CAPM on the Australian market using monthly returns of the period January 1980 to December 2001 and found prove that the CCAPM outperforms the standard CAPM in asset pricing. Filho et al (2009) found the validity of the CCAPM on the Brazilian, Argentinean and German markets.

In the same line of study in Malaysia, Ismail & Shakrani (2003) found a relationship between beta and returns in cross sectional regression analysis while Kumar et al (2006) test a model where returns are multi variate and the market risk premium, the market volatility and systematic risk are responsive to information that affects the market’s conditional covariance of expected returns which supports the CCAPM

2.4.3 The International CAPM

Investment in the performance of local currency in respect to foreign currency bears a risk which asset pricing models should consider. In order to reap maximum returns on their portfolio, investors diversify and hold both domestic and foreign assets. De Santis & Gerard (1998) appraised a conditional version of an International Asset Pricing Model which supports the existence of a foreign exchange risk premium. This model allows time to vary in order to compensate for the exchange rate risk. The Fama & French (1998) CAPM three factor model is applied in the international environment which provides an argument that returns in a cross section of national value portfolios cannot be explained by the standard CAPM whereas the multi factor model can easily acquire the value premium in international returns.

Having been useful as an explanatory ability the international asset pricing model has received considerable support. Dahlquist & Sallstrom (2002) explains how an asset pricing model inclusive of foreign exchange risk is able to explain over 60% of the variation in average returns in their examination of whether International CAPM outperforms an international version of the three factor model. Alternatively, using assets from the US, the UK and Japan, Zhang (2006) explains that Exchange risk exposure contributes largely to international asset returns and that Conditional International CAPM performs at its best with exchange risk. However the model does not provide a good forecasting tool for investor’s guidance in portfolio selection as depicted by Wu (2008).

Balancing the portfolio to flexible exchange rates argues that uncovered interest rate does not hold in the study of Engel & Rodrigues (1989) that is expected return differential between comparable assets across countries is non-zero. The study does however favor the International CAPM model over the Static CAPM model.

2.4.4 The Consumption-CAPM

Investigation on the Consumption-based CAPM theory was originally conducted by Lucas (1978) in an attempt to explain return on risk-free asset and risky assets by using an intertemporal substitution channel of consumption. Breeden (1979) derived the Consumption-oriented CAPM on the assumption that it follows a discrete-time framework for a subset of assets which has jointly lognormally distributed returns with aggregate consumption. All assets are assumed to have returns and individuals’ optimal consumption paths follow a diffusion process. Considerable empirical evidences prove that consumption risk does matter in explaining asset returns (Lettau & Ludvigson (2001), Parker & Julliard (2005)).

Breeden et al (1989) on the other hand explain that the performance of the traditional CAPM and the Consumption-CAPM are similar which proves the general opinion of dissatisfactory empirical estimation of the model. The estimation is made after adjusting for measurement problems related to consumption data and the betas used are based on consumption and the portfolio having the maximum correlation with consumption.

On further investigation, Soderlind (2003) tried to see if the consumption-based CAPM explains the cross section on Sharpe ratio which linearly increases the asset’s correlation with aggregate consumption growth. The slope of this relation should be positive to approve the C-CAPM. The Sharpe Ratio is designed by Sharpe (1966) to measure the expected return per unit of risk for a zero investment strategy which experiences a high value if consumption is more volatile. However the study concluded that Sharpe ratios incorporate a large deal of variation which provided little support to the model and much to the classical CAPM.

Zakamulin (2011) portrayed that two different investment sets with the same number of risky assets (but with different values for expected returns and standard variations) do produce different minimum-variance frontiers of risky assets that generate the same investment opportunity set given the risky assets in the two sets have the same Sharpe ratios and correlation matrix. The use of Sharpe ratios re configures the CAPM relationship and adjusts the Jensen alpha to properly measure abnormal portfolio performance. Moreover, Empirically,

Brennan, Wang, and Xia (2004) show that including a measure of innovations to the maximum Sharpe ratio improves the performance of their pricing model significantly.

The conclusion of all these issues is that while the academic debate continues, the CAPM may still be useful for those interested in the long run.

Chapter Three

Components of CAPM

3.1 Assumptions of the CAPM theory:

Given the paradox between the complexities of the real world, in order to construct good models, those complexities having little effect of the model should be assumed away. A theory is usually validated when it is based on empirical accuracy of its predictions rather than on the realism of its assumptions.

The major assumptions of the CAPM are listed as follows:

Investors aim at the maximization of utility from holding wealth.

Investors’ selection criteria of investment opportunities are based on expected return and risk.

All investors have a risk adverse attitude and behave rationally.

Investors choose investment opportunities set based on expected return and risk.

Expected returns follow a normal distribution.

The lending and borrowing process is unlimited at a common interest rate.

No transactions costs are entailed in the trading of securities.

Taxes on dividends and capital gains are at similar rates.

3.2 The Capital Market Line

In order to represent the set of portfolios that investors would choose in equilibrium through the stated assumptions above, an opportunity set of all risky portfolios is drawn where with the inclusion of a risk free rate asset, the combination of the risk free asset with any risky portfolios is made possible.

Expected return Iii

Ii Capital market line

Rm Opportunity set



σm Standard deviation

Figure 1

In an equilibrium state, investors prefer a composition of the risk free asset and one risky portfolio that provides the maximum expected return for any given level of variance that is hold efficient portfolios. Such a condition is labeled M in the above diagram, where capital market line (the vertical line starting at the risk free rate of return) meets the opportunity set of risky assets.

One of the unparalleled characteristic of the CML is that investors would not want to move beyond point M. That is, investors cannot improve upon the alternatives by this set of portfolios otherwise the market would not be in equilibrium and arbitrage would occur.

3.3 The Security Market Line

Sharpe and Lintner developed a framework to describe the relationship between expected returns and the risk associated with securities with the following equation which is in its ex-ante form:

E (ri) = rf + β [E (rm) – rf] (3.1)

In simpler terms, the above equation (3.1) shows that the expected return on an asset which is equal to the risk free rate of return plus a risk premium. The risk premium is the price of risk (slope of the line) multiplied by the quantity of risk which is the systematic risk (β).

Equation (3.1) propounds that in the equilibrium state, an asset with zero systematic risk (β=0) will have expected return just equal to that on the riskless asset rf , and expected return on all risky securities (β >0) will be higher by the risk premium which is directly proportional to their risk as measured by β.

Such a relation is graphed through the Security Market Line (SML) in the below diagram (figure 2) with expected returns on the vertical axis and beta on the horizontal axis. The SML shows a positive linear relationship between beta and expected return and the intercept is equal to the risk free rate.

Expected Return

Security market line

Rm M


Figure 2 Beta

It important to note that efficient portfolios are usually plot on the CML and Figure 3 provides two diagrams which portrays the relationship between the CML and the SML.

Point A represents an efficient portfolio A which lies on the CML and point B is an inefficient portfolio outside the CML. However, both portfolios have similar expected return and beta value.

Note that the CML concentrates on portfolio standard deviation rather than beta.

E(R) E(R)

A A, B

E (RA) =E (RB) B

Rf Rf

Figure 3 σA σB βA = βB

An efficient portfolio is said to have lesser standard deviation of returns than the inefficient portfolio given equal expected returns. The excess standard deviation associated with the inefficient portfolio is called diversifiable risk or unsystematic risk. Investors are not compensated for this kind of risk because in a state of equilibrium, investors only hold efficient portfolios. Hence it can be perceived that the CAPM is effective in the pricing of all assets whether they demonstrate efficiency or not but the CML only prices efficient portfolios.

3.4 The transition from the ex-ante to the ex-post model

As incorporated in the overall study, equation (2)… is denoted as an ex-ante or a forward-looking model which uses entirely historical data for the testing purposes. However, a contradiction is formed upon this belief of historical data as there is no proof that the rates of return expected in the future will automatically be equal to realized rates of return over the past periods.

Moreover, it should be acclaimed that historical beta may or may not mirror expected future risk. Hence the need to traverse from the ex-ante principles to the ex-post is felt in order to better test for CAPM. The ex-post model specifies some return generating process by assuming that the rate of return on an asset follows a "fair game".

The "fair game" signifies that, on average, across a large number of samples the expected return on an asset equals its actual return and is explained as follows:

ri = E (ri) + βiδm + εi (3.2)

Where, δm = rm – E (rm)

E (δm) = 0

εi = a random error term

Cov (εi , δm ) = 0

Cov (εi , εi t-1 ) = 0

βi = systematic risk

Note that since CAPM assumes that asset returns are jointly normal, βi in the fair game model behaves exactly as in the CAPM model and the market model must hold.

Equation (3.2) assumes that if expected return is taken on both sides, the average realized return is equal to the expected return:

E (ri) = E (ri)

Substituting E (ri) from the CAPM into (3.2) yields:

ri = { rf + [ E (rm) ] βi + βi [ rm – E (rm) ] + εi

ri = rf + (rm – rf) βi + εi

Substracting the risk free rate from both sides:

ri – rf = (rm – rf) βi + εi (3.3)

Equation (3.3) represents the ex-post model of the CAPM. One important difference between the ex-ante theoretical model and the ex-post model is that the latter can present a negative slope while the former cannot because the theoretical CAPM must have a higher expected return on the market than the risk free rate of return.


The Data and Methodology chapter

This chapter lays down the rationale behind the time series analysis and the cross section analysis estimation techniques to actuate the relationship between risk and return in the financial market of Mauritius.

The chapter is structured as follows:

Section 4.1 gives a brief explanation on the data used in this study and some unique characteristics of the data is pointed out in a trend analysis. Section 4.2 depicts the method in which returns have been calculated and Section 4.3 gives us an overview of the methodology in depth. Sub-sections 4.3.1 and 4.3.2 respectively explains the various facets of the tests conducted that is the portfolio grouping technique and the tests design. At last the Cross section tests design is summarized in Section 4.4.

4.1 Data and its selection criterion

The study uses share price of companies and market index data obtained from the Stock Exchange of Mauritius for a period of 6 Years. The recent years were not taken into consideration so as not to be hassled with the huge volatility caused from the lurking financial crisis at the start of 2008. 34 Monthly closing prices of stocks are examined for the period of January 2002 until December 2007. Monthly data have been chosen for analysis due to the fear of obtaining very noisy data and hence inefficient estimates caused by the use of high frequency observations such as daily or weekly prices.

The markets in Mauritius are characterized by thin trading and thus it is very important to take this matter into consideration when estimating systematic risk. A thin market is known for having high price volatility. One of the problems of this market is the lack of price data for the months with no trading in the stock. The share returns on the months with no trading are simply ignored. In addition, no trading means no change in stock prices.

When securities are thinly traded, applying the market model (ordinary least squares) to time series return causes biased estimates of the single factor, market model according to Bartholdy & Riding’s (1994) study. The occurrence of this biasness is due to returns being measured with an error that is correlated with the market index. Beta estimates are thus biased downwards.

Several correction methods are available for this issue among which two are widely used. These bias-correcting procedures were developed by Dimson (1979) and William (1977). However, Fowler et al (1989) found negative results from these techniques as he argues that they involve significant computational effort relative to the simple model.

The index employed as a proxy for the market is the SEMDEX. This choice has been made due to the fact that the market must be in an equilibrium state to validate the CAPM and the market proxy should be a value-weighted index which comprises the entire constituent of the market. The SEMDEX is a value-weighted index which is comprised of all the stocks listed on the SEM.

Trend Analysis

A trend analysis of the data obtained is carried out to show some unique characteristics of share prices when they were manipulated. The 34 companies selected follow a moreover similar pattern during the 6 years of observation period selected. It can be noted that at the start of 2002, only these 34 companies were listed on the official market of the Stock Exchange of Mauritius (SEM) and are some of the leading companies in their sectors nowadays in Mauritius. The listed prices of the companies showed an increasing pattern that is throughout the timeframe, they kept on rising. There may have been moments of momentary decrease but overall the situation has been thus.

Out of the 34 securities examined, 5 companies showed distinct patterns in the share prices which will be presented in five case studies. The progression of the 5 stocks to be examined are MCB (Mauritius Commercial Bank Ltd), MEI (Mauritius Eagle Insurance Co Ltd), ROGE (Rogers and co Ltd), HARF (Harel Freres ) and AIRM (Air Mauritius Ltd).

Case Study 1:

The Mauritius Commercial Bank Limited was among the first group of companies to be listed on the Stock Exchange of Mauritius.

The above trend shows the volatility in the share prices. At the end of March 2003, it can be noted that prices reached a peak of Rs119 but as at April 2003, a sudden fall is spotted to Rs 28.1. However hereafter, the observation depicts a rather upwards trend with no wild fluctuation. As at end of December 2007, a high price of Rs 162 is recorded.

The drastic decrease accounted in April 2003 can be explained by the exceptional loss of Rs 881.6 million incurred by the bank in February due to the discovery of a massive fraud. An extreme confusion period followed shortly due to unfound allegations against the charges of money laundering and failure to report suspicious transaction by the General Manager and his assistant. This hassle lead to the downgrading of the bank’s financial strength rating by Moody’s in the wake of June 2003.

Nonetheless, a notable recovery of the situation could be felt as share prices started to rise gradually again along with a steady growth in profits, dividends and share value. Moody’s recognized the capacity of the bank to absorb the loss comfortably and no further rating downgrade was recorded.

Case Study 2

We can notice a rather stable upwards trend without any huge volatility during the period Jan 2002 (Rs 44) to Oct 2005 (Rs 195). A small decrease in trend took place afterwards with a slight increase in November 2006 with the stated price of Rs190. The peak of the overall trend is at Rs 250 reached in March 2007 but the situation was reversed soon after with gradual decrease until a huge fall, recorded in November 2007 where the prices reached Rs 55.

The huge decline in the whole year of 2007 can be explained by the negative impact of the cyclone Gamede which was followed by the tidal wave in the same year. The accident claims experienced a big rise which had no good impact on the claims ratio. However this was partly compensated with the blooming motor business and the situation has been stabilized.

Case Study 3

Rogers and company Ltd showed no major fluctuations in prices. In fact it can be seen that the trend has been rather in a rising manner, from Rs 70 in the wake of 2003 to Rs 412 at end of 2007. The peak reached at end of 2007 is quite striking with a quote of Rs 415 in October. Among the selected stocks, only ROGE depicted such an impressive turn out of performance.

This performance is backed up by the fact that the company was in a restructuring and reorganizing period which revealed to be very successful. The group is now incorporated into three brands namely, Rogers, Cim and Cerena which specializes in Property, Investments and ICT. The effect was immediately seen through a rise of 27% in the group profits and improvements in the earnings per share.

Case Study 4

A rather smooth trend is depicted in the above diagram with very mild fluctuation until October 2006, where share prices reached its peak of Rs 258. However the following month experienced a drastic fall where the share price reached Rs 30. The graph kept its downward trend, with the lowest recorded price of Rs18.5 in December 2007.

The considerable increase in share prices in 2006 could be explained by the favorable economic conditions prevailing in the country. The high growth rate influenced the EPZ sector in an upbeat manner, however the sudden change of events was the direct consequence of the lurking financial crisis in the US since the dollar alongside the major currency were threatened to further depreciation. The sugarcane was fiercely affected by this event.

Case Study 5

The transport sector is represented by the Air Mauritius Ltd. The company has always been experiencing from lows, if not from a reduction in trading activities to effects on the international level. This sector has always been very vulnerable and the share prices also portray its situation.

During the whole timeframe, Air Mauritius seemed to keep a constantly low share price in the range Rs11.8 to Rs22.5. unlike the other companies who started off with low prices, but achieved considerate improvement with the passing time. The trend also shows major volatility in prices and the most remarkable was the sudden drop of prices to Rs 0 in October 2005.

This eventual occurrence was due to the sudden unprecedented rise in oil prices. A rise of more than 40% was recorded which kept on escalating.

4.2 Quantification of returns

The analysis studies monthly returns. The market return and the return on a stock can both be computed as follows:

rt = Pit – Pt-1 X 100 (4.1)



rt: return in period t

Pt: price level of stock or index in period t

Another way of computing the return on stock or market is to use the logarithmic formula:

rt = ln (Pt / Pt-1) (4.2)

The rates of return as measured by equation (4.1) are discrete or occur over a particular period of time. Usually stock prices do not observe such kind of trend and to neutralize this problem, it is assumed that shareholder’s wealth grows continuously over the holding period. Equation (4.2) takes into account such a continuous compounded rate of return. In fact, recent studies have shown that that stock prices follow rather a log-normal distribution and returns a normal distribution.

Other than continuous compounding of return rates, there are other 2 reasons why logarithm is taken into consideration (Govindasamy 1984):

The variability of simple prices for a given stock is an increasing function of the stock’s price level. This is neutralizing the natural logarithm.

The change in logarithm prices is very close to the percentage price changes when it is less than ± 15% and it is often convenient to look at data in terms of percentage price changes.

Taking into account these advantages, the logarithmic method of computing returns is adopted in the study and as per the CAPM requirement; all the securities’ returns are adjusted for dividends.

4.3 Methodology

The methodology employed is the two-parameter model or two step procedure propounded by Fama & Macbeth (1973). The equilibrium relation of CAPM is stated in terms of expected returns. The two-parameter model is considered on two categories of implication on the conditions on expected returns. Firstly, in the two parameter world, investors hold efficient portfolios and secondly, the capital market is perfect in the sense that there isn’t the presence of neither transaction costs nor information costs.

The expected return is derived as follows which form an equation that describes the expected returns and risk relationship in its ex-ante form originally stated by Black (1972):

E (Ri) = E (R0) + {E (Rm) – E (R0)} βi (4.3)

In words, E (R0) is the expected return on security i that is riskless in the portfolio m, plus

{E (Rm) – E (R0)} βi which is a risk premium that is βi times the difference between E (Rm) and

E (R0).

This model is known as the zero beta CAPM since E (R0) is uncorrelated with the market. It is important to note that the expected return on this model should be less than the expected return on the market portfolio in order to not violate the risk aversion assumption. Also, investors prefer the minimum variance portfolio.

Equation (4.3) has three testable implications which reflect efficient portfolios:

(C1): The relationship between the expected return on a security and its risk in any efficient portfolio m is linear.

(C2): βi is a complete measure of the risk of security i in the efficient portfolio m

(C3): in a market of risk-averse investors, higher risk should be associated with higher expected return; E (Rm) – E (R0) > 0.

However our model uses data on period-by-period security and portfolio returns that allows us to use observed average returns to test the expected return conditions C1-C3. In other words, to test the ex-post model of the CAPM with historical data, the stochastic process is adopted which is given by:

Rit = γ0t + γ1t βi + γ2t βi2 + γ3t si + ηit. (4.4)

The transition from the ex-ante to the ex-post model is given in Chapter Three under section 3.4.

The subscript t refers to period t so that Rit is the one period percentage return on portfolio i at time t. The above equation allows γ0t and γ1t to vary stochastically from period to period. The hypothesis of condition C3 is that the expected value of risk premium γ1t which is the slope

{E (Rmt) – E (R0t)} in (3) is positive that is, E (γ1t) = E (Rmt) – E (R0t) > 0.

The variable βi2 is inserted in (4.4) to test for linearity. The hypothesis of condition C1 is E (γ2t) =0, although γ2t is allowed to vary stochastically from period to period.

Also, si in (4.4) is some measure of the unsystematic risk of portfolio i. The hypothesis of condition C2 is E (γ3t) =0, but γ3t can vary stochastically through time.

The disturbance term ηit is assumed to have zero mean and to be independent of all other variables in the above Equation (4.4).

Portfolio grouping

The tests are conducted on portfolios rather than individual securities. Estimating equation (3) with individual securities might give bias results since an estimated single stock beta is subject to measurement error. The individual stock betas are thus grouped into portfolios to reduce the measurement error. Organizing and grouping securities into portfolios is considered a strategy of partially diversifying away a portion of risk (firm-specific part of returns) whereby increasing the chances of a better estimation of beta since the spread in betas across portfolios is maximized. Thus the effect of beta upon return can clearly be determined. Diversification holds true because prices of different stocks do not move exactly together that is stock prices changes are less than perfectly correlated.

Fama & Macbeth (1973) explained the problem of measurement error as being an unavoidable "errors in the variables" problem. He explains that the efficiency condition or expected return-risk equation (2) is in terms of true values of the relative risk measure βi, but in empirical tests estimates, βi, must be used.

Blume (1970) shows that for any portfolio p, defined by weights xip of each individual security in the portfolio.


βp = ∑ xip βi (4.5)


Given the 34 securities, 3 portfolios are created. The formation of portfolios is based on ranked values of βi of the individual securities in the portfolio formation period, January 2002 to December 2003, which is according to the high-low beta criteria. Portfolio 1 contains a set of 12 securities with highest betas; Portfolio 2 has 11 securities as well with the middle ranked and finally the lowest 11 beta securities are placed in the last portfolio (portfolio 3).

4.3.2. The tests design

Considering the short observation period of 6 years, the investigation is divided into only three consecutive non-overlapping main periods. These periods are denoted as the portfolio formation period (Jan 2002-Dec 2003) which according to Dimson(1979) is called the pre ranking of beta. He also states that this procedure needs a minimum of 24 months to a maximum of 57 months. The estimation period (Jan 2004-Dec 2005) and the testing period (Jan 2006-Dec 2007), are both known as the post ranking of beta.

Portfolio Formation period

This is the first step of the procedure where time series tests are conducted to estimate a beta for each stock which will be used to form the 3 portfolios. The estimation of betas is carried out through the Ordinary Least Squares (OLS) regression on monthly return of individual stock during the time span of Jan 2002 to Dec 2003 against the market index’ return. The formula used goes as follows:

rit = ai + βi


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