# Capital Asset Pricing Model (CAPM) in Emerging Markets

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This study establishes that the main prediction of the Sharpe, Lintner and Black model, a positive relationship between returns and beta is rejected for the Johannesburg Securities Market. The relationship between portfolio returns and beta is weak and inconsistent. However this paper finds a consistent and highly significant conditional relationship between beta and portfolio returns. There is an expected positive relationship between risk and return during periods of positive market excess returns. Likewise there is an expected negative relationship between risk and return in periods when market excess returns are negative. Furthermore, the model does not capture all aspects of reality including the residual variance of stocks; unsystematic risk has explanatory power on returns.

## Introduction

Investors have long been trying to devise a model that will enable them to make effective investment decisions. To make such effective decisions an investment manager needs to be able to explain the differences in return across assets; he/she needs to be able to explain the relationship between risk and expected returns. For this Sharpe (1964) and in subsequent years Lintner (1965) and Mossin (1966) developed the well-known Capital Asset Pricing Model (CAPM). The model states that there is positive risk-return trade-off; assets with higher systematic risk are associated with higher returns. The difference in asset returns is solely as a result of the difference in each assetâ€™s relationship to the systematic risk measured by asset beta. The CAPM asserts that expected return on assets above risk-free rate is linearly related to systematic risk.

In the last decade, emerging equity markets have drawn attention from investors looking to diversify their portfolios globally and academics looking to test models that have been tested in developed markets. The attention has come about because of the very large positive and sometimes very large negative returns realised in these markets. Such characteristics induce the testing of models used for the portfolio construction process.

The Johannesburg securities market poses as a peculiar market to investigate because it is the largest, best regulated and most advanced in Africa, with higher relative trade volumes and thus provides data that is not obsolete. Despite its emerging market status, there is significant institutional investor participation and ownership is well diversified in contrast to other African markets [1] . Literature on testing the validity of the CAPM in emerging markets is not yet as potent as testing in developed markets; moreover tests of the validity of the CAPM in the Johannesburg Securities Market is believed to be non-existent in the literature.

To test the validity, tests are conducted for the sample period of 2001-2009. The Fama and MacBeth (1973) three-step method is used to test the unconditional risk - return relationship. Portfolios are constructed and then portfolio returns from a time period are regressed against estimated betas from the previous period. T- tests are then conducted to test if the hypotheses of the model are rejected. Secondly the Pettengill et al (1995) method is used to test for a conditional relationship between risk and returns. This method is identical to the three-step but in the final step of the method the data set is divided into positive and negative market risk premium subsets.

The CAPM predicts that the market portfolio is efficient and the security market line (SML) which describes the relationship between risk and expected return accurately describes the risk-return trade-off. The second prediction implies that the model should predict an expected return which is equal to the actual realised return, so in equilibrium prices are accurately priced. This study has a central limitation, being that the hypothesized market portfolio is unobservable. This limitation is discussed in great depth in a discussion now called â€œRollâ€™s critiqueâ€ by Roll (1977). The discussion in short says the â€œmarket portfolioâ€ used in the model, for example an equity index contains all assets that can be held by investors. However the market portfolio would consists of many more assets like bonds, real estate, foreign assets and human capital, some of which may be traded thinly or untradeable such as human capital. The author asserts also that even small deviations from efficiency in the market portfolio can produce large deviations away from the risk-return relationship described by the Security Market Line. The general reaction to these criticisms has been to develop an alternative model for pricing assets. However a study by Jagannathan and Wang (1993) argue that the CAPM is a valid pricing model but it lacks support because of the inappropriateness of the underlying assumptions.

Infrequent trading is a feature prominent in a lot of emerging markets so it is likely that this market will be characterised by this. Infrequent trading may cause estimated beta from the OLS to exhibit a downward biased estimate for less frequently traded stocks as these stocks may not be perfectly synchronised with the market return. Dimson (1979), Scholes and Williams (1977) explore the effect of infrequent trading in Ordinary Least Squares regression estimates and provide several alternatives for correcting bias. One of the statistical effects according to Strebel (1977) is that the probability density function (p.d.f) is leptokurtic, or highly peaked (the distribution of returns has a large proportion values that lie in the extremes). There have been many other authors who have concluded that indeed, returns even in developed markets do not follow a normal distribution.

The sample data was tested for these properties and the risk premiums are leptokurtic (they exhibit excess kurtosis [2] , however to test formally, the Jarque - Bera [3] test was implemented. The tests showed returns for all stocks in the sample are not generated from a normal distribution. The JB statistic calculated for the sub period 2004-2009 which includes all stocks used in the sample is presented in the appendix.

Tests of the model will be carried out using robust standard errors to correct the effect of non-normality in returns. According to Huber (1981), robust standard errors produce t-statistics that are resistant to errors in the results, produced by deviations from assumptions; in this case, of normality. If the assumptions of normality are met but not entirely and undisputedly, the robust estimator will still have someÂ efficiency. It will also be asymptotically unbiased (bias tends to zero as sample size tends to infinity).

This study has a two-fold purpose: firstly, it seeks to establish whether the main prediction of the model which says that there is a positive relationship between returns and beta, is accepted. Secondly, it seeks to determine whether cross sectional regression tests provide support for a systematic but conditional relationship between beta and realised returns. This test was first carried out by Pettengill et al (1995). In addition, this study investigates whether the CAPM captures all aspects of reality including the residual variance of stocks.

Upon completion of this study, the modelâ€™s fundamental statement that higher systematic risk (beta) is associated with higher level of return will be supported or rejected for the case of the Johannesburg Securities Market. Moreover, the study aims to help investors in this market decide whether the CAPM is an appropriate model to use in portfolio management/construction or whether an alternative or an augmented version of the model should be adopted.

Section two covers the literature review of previous empirical analyses, section three and four the theoretical background and data description while the sections following these contain the methodologies and results.

## Literature Review

Black, Jensen, and Scholes (1972) study used stocks on the NYSE in the period 1931-1965 and using monthly data; they estimated beta coefficients for five-year periods and ranked these in decreasing order to form 10 portfolios. Their tests suggest the slope and intercept of the fitted line is significantly different from the hypothesized values. Although this may seem to disprove the CAPM, consider the problem of model specification and measurement error that may be caused by the use of a market proxy such as a stock index instead of an actual market. Such errors bias the fitted line slope towards zero and its intercept away from zero. Black et al (1972) conclude that the data are consistent with Blackâ€™s (1972) version of the model.

Fama and MacBeth (1973) included all stocks in the NYSE from 1962 to 1968 in their analysis. They used the three step approach, which divided the total sample period into 9 overlapping analysis hey used a method called three-step approach. Each analysis period, in turn, was divided into three sub-periods: a four-year portfolio formation period, a five year beta estimation period and a 5-year testing period. Individual securities were ranked based on the beta estimated in the first sub-period and then 20 portfolios were formed. Then the betas of the portfolios formed were re-estimated using the subsequent periodâ€™s data. Portfolio returns during the testing period were regressed against the betas calculated the estimation period. This study was the first to test the linearity hypothesis of the model and results did not reject the condition.

Banz (1981) tested the hypothesis that the CAPM captures all important determinants of risk. The study tested whether the size of firms explain the residual variance in average returns which is unexplained by beta. The study provided evidence that size of firms have significant explanatory power in particular, average returns on stocks of firms with low market value were higher than the average returns on stocks of firms with high market value. Many more variables have been used to try to explain the risk-return trade-off. Some of these are the earnings yield (Basu [1977]) and leverage (Rosenberg, Reid and Lanstein [1983]).

Pettengill et al (1995) attempted to overcome the problem of observed negative market and portfolio risk premiums in conducting CAPM tests. The CAPM suggests a positive risk-return trade-off and so market return must be higher than the risk free rate; however market returns that are lower than the risk free rate can be realised. Basically, if return on the market is less than the risk free rate then stocks with high-betas should produce lower return that low-beta stocks. Their study asserts that although this doesnâ€™t create any problems for in the estimation of beta, it weakens the ex-post relationship between betas and risk premiums. If neither group of returns is only a negligible fraction of total number of observations, the slope of regression line will imply that there is no meaningful relationship between beta and risk premium as predicted by the security market line. The method is an altered version of the Fama and MacBeth (1973) three-step method the difference being that the final step was altered taking into consideration positive The results of the conditional test showed a significant positive relationship between beta and risk premiums for periods that exhibited positive market excess returns and an inverse relationship for periods that exhibit negative market excess returns.

Sandoval, E.A., Saens R.N (2004) tested the conditional and unconditional model in emerging markets in Latin America using data from Brazilian, Chilean, Mexican and Argentine stock markets. They also tested the hypothesis that the CAPM captures all determinants of returns by constructing an augmented CAPM which included additional variables such as size, book to market ratio and the degree of market integration. They adopted a strategy to manage the effects of infrequent trading in these markets. They regressed individual security returns against lagging, matching and leading market returns calculated both from the Latin American Stock Market Index and S&P 500. The sample period chosen for testing spanned 1995-2002. The portfolio betas were obtained for each two year period and then these were included as explanatory variables in the following year. Finally, based on the Black (1972) model regressions were perform but with panel data. Their results of the conditional test showed a significant positive risk-return trade off during â€˜up-marketsâ€™ and a significant but negative beta-return relationship during â€˜down-marketsâ€™.

Medvedev, A. (2004) augmented the CAPM by including ambiguity as a second variable in the two-factor CAPM model to determine the effects of ambiguity on equilibrium asset prices. The volatility is assumed to lie within known boundaries. The coefficient of the ambiguity variable was derived by taking the average of the standard deviations of the residual returns of the industry portfolios created by Fama and French (1992). Monthly returns of these portfolios and market excess returns for the period 1973 through 2003 were used for the analysis. Results showed that the ambiguity factor had more explanatory power on returns than beta.

More studies on the pricing of asset returns are based on the argument that volatility of returns is not constant over time. Hence, the estimates of returns over a sample period provide unconditional estimates because variance is treated as constant over time. Schwert and Seguin (1990) provide evidence from their study that there is heteroscedasticity in residuals of returns.

Academics widely use the Generalised Autoregressive Conditional Heteroscedasticity (GARCH) model developed by Robert Engle to estimate the conditional (time varying) mean and variance of assets in their analyses.

## Theoretical Background

The Markowitz (1959) portfolio model was developed on the premise of a perfect capital market. The CAPM was later developed and based on the same following assumptions. Each investorâ€™s wealth endowment is small compared to the total endowment of all investors. Investors have no market power such that they behave as though their trades have no effect on prices. Investors can only purchase assets that are publicly traded financial instruments and investors are limited to fixed, risk-free borrowing or lending. The probability distribution function of the one-period returns on all assets is assumed to be normally distributed. Investors incur no taxes, transaction costs nor information costs. All investors are rational mean-variance optimizers who use the Markowitz portfolio selection model and select among portfolios to maximize utility. Finally, investors share the homogenous expectations.

The resulting equilibrium conditions are as follows. All investors will hold the market portfolio which is the optimal risky portfolio. This market portfolio contains all securities with the proportion of each security given as a percentage of the total market value. Risk premium on the market depends on the average risk aversion of all participants while risk premium on an individual asset is a function of its covariance with the market.

In equilibrium prices are adjusted such that excess demand is zero and market equilibrium is reached where tangency portfolio, optimal risky portfolio becomes the market portfolio. The assumption of investors having homogenous beliefs means that all investors have the same linear efficient set â€“ the capital market line.

## Deriving the CAPM

A portfolio consisting of a% in risky asset i and (1-a) % in the market M will have the following return and risk:

; expected return of the portfolio is a weighted average of expected return of risky asset i and the market

## ;

; Portfolio variance is calculated as function of the weights squared of the risky asset and the market multiplied by their respective standard deviations plus the covariance between asset i and the market

The next step is to evaluate the rate of change in expected return and standard deviation with respect to the percentage change of the portfolio a, invested in i:

In equilibrium, market portfolio contains all assets, and it will have a weight wi of asset i. Therefore, in equilibrium, the fraction a must be excess demand and excess demand in equilibrium is zero. So, evaluating rates of change at a=0:

and

Slope of the efficient set is given by

But at M;

Equation the slope of the efficient with the slope of the capital market line [] and writing the covariance term as:

Rewriting as ; gives the CAPM or the Security Market Line

The required return on any asset i, is equal to the risk-free rate of return plus a risk premium. The risk premium is the price of risk multiplied by the quantity of risk. The price of risk is the excess return on market portfolio and the quantity of risk is the beta of the asset:

## Model Implications

The relationship between these variables poses certain implications for testing the relationship between beta and returns. The assumption of positive risk return trade-off implies that market return must be greater than risk free rate otherwise investors will hold the risk free investment. Since the difference in market return and risk free rate must be positive then the expected return to a risky portfolio is a positive function of beta.

However market return in fact, may be below risk free rate. Pettengill et al (1995) argue that investors must, in conjunction with an expectation of market return being higher than risk free return, believe a nonzero probability that the realised market return will be less than the risk-free return. Therefore if investors are certain that the market return will always yield a higher return than the risk free, then investors will never hold the risk free asset. This suggests that the relationship between beta and realised return is different from the relationship between beta and expected return. Pettengill et al (1995) states: â€œthe model does not provide a direct indication of the relationship between portfolio beta and portfolio returns when realised market returns is less than risk - free returnâ€. Further analysis of the relationship between portfolios returns and portfolio risk gives insight into the relationship.

Portfolios with higher betas have higher risk and therefore higher returns. In order for a high beta portfolio to have higher risk, there must be some a level of realised return such that the probability of obtaining that return is greater for the low beta portfolio than for the high beta portfolio. Otherwise, no rational investor will hold a low bet a portfolio. Therefore the model also requires the expectation that, there is some probability that the realised returns of high beta portfolios will be less than that of low beta portfolios. One could draw inference from this that returns for high beta portfolios are less than returns for low beta portfolios, when the realised market return is less than risk free return.

## Testable Implications

Consider the ex-post Security Market Line (SML) first estimated by Black (1972) is the relationship:

Rip = y0t +y1t*ip,+ Ñ”t (1)

where,

Rip,t is portfolio risk premium

y0t is the intercept of the SML

y1t is ip,t-1 coefficient, the market price of risk, risk premium for taking on a unit of market risk (the slope of the SML)

Ñ”t is the error term

The equation above has four testable implications: (i) the SML intercept is zero; a zero beta portfolio yields a return of zero. (ii) there is a positive risk-return trade-off. (iii) the relationship between risk and return is linear. (iv) the model captures all important determinants of returns.

More formally,

y0 is not statistically different from zero (its absolute t value is not greater than 2)

y1>0, there is a positive risk-return trade-off (y1 is the market portfolio risk premium)

y2=0, the SML has no non-linear properties

y3=0, residual risk does not affect return (CAPM captures all important factors that explain the difference in asset returns)

## The Conditional Relationship

The SML equation estimates the beta risk for each portfolio using realised returns for the portfolio and the market which provides the proxy for beta. Since the modelâ€™s underlying assumption is that betas estimated from past a time period as proxy for betas in the test period, the model can be used to test for a positive risk-return relationship. As argued by Pettengill et al (1995), this method may test the suitability of beta as a measure of risk but it does not test the validity of Blackâ€™s (1972) SML equation.

The model developed by Black requires a direct and unconditional relationship between beta and expected returns, and it requires the expectation that the relationship between realised returns and betas will be varied. Equation 1 provides the condition under which realised return to high beta portfolios are expected to be lower than that of lower beta portfolios. The relationship between the return to high and low beta portfolios is conditional on the relationship between realised market returns and risk free rate. If Rm<Rf, then p * (Rmt â€“ Rft) <0; the predicted portfolio return includes a negative risk premium that is proportionate to beta. Therefore if the realized market return is less than the risk free rate then the relationship between beta and predicted return is inverse. Higher beta portfolios will result in return lower than low beta portfolios.

Extant literature prescribes a systematic and positive trade-off between beta and expected return, but the above argument reveals a relationship that is conditional; a positive relationship when market excess return is positive and a negative relationship during periods of negative market excess return. A negative market excess return will be insignificant in the beta- return relationship if it wasnâ€™t a common feature of financial markets. In later sections, this occurrence in the Johannesburg Market is examined in some more detail. A large number of negative market excess return observations means that studies that havenâ€™t adapted the model to capture this segmentation in the relationship are biased against finding a systematic relationship.

## Data

The sample period extends from January 2001 to November 2009. This study uses weekly data as it enables the regression to obtain better estimates of the value of the beta coefficient. High frequency data such as daily data that span a short and stable time span can result in the use of noisy data which yield inefficient estimates. On the other hand returns calculated using a longer period such as monthly or annual might result in changes of beta over the examined period introducing biases of estimates.

Weekly adjusted closing prices of all stocks listed on the JSE/FTSE stock exchange are collected and weekly returns calculated. The yields of 1year South African government bond for the period mentioned were collected and 3-month yield be estimated to act as proxy for the risk free rate. Weekly JSE/FTSE equity price index acts as market portfolio and is used to calculate market returns. The data is obtained from DataStream.

## Methodologies

## 5.1 Fama and MacBeth 3-step traditional approach

Since data for 3-month Treasury bill yields were unavailable, the 1-year bond was divided by 4 to obtain the weekly 3-month â€œTreasury billâ€ yields. From this point on the methodology followed the Fama and MacBethâ€™s three step traditional approach. The analysis period 2001-2009 was divided into four 6-year sub periods with one overlapping year in each sub period. Overlapping is expected to alleviate possible volatility of beta coefficients in each sub period. The sub-periods were then further divided into three 2-year periods which are the portfolio formation period, beta estimation period and testing period.

Table 1: Division of Analysis Period, and Number of stocks included in analysis

## Sub-Periods

## Â

1

2

3

4

## 2001-2006

## 2002-2007

## 2003-2008

## 2004-2009

## Portfolio Formation Periods

2001-2002

2002-2003

2003-2004

2004-2005

## Portfolio Beta Estimation Periods

2003-2004

2004-2005

2005-2006

2006-2007

## Testing Periods

2005-2006

2006-2007

2007-2008

2008-2009

Number of JSE/FTSE Stocks Available at the beginning of Portfolio Formation Period

125

127

131

133

The raw data of adjusted closing prices of stocks and JSE/FTSE market index were manipulated to calculate returns of each stock and the market. The returns were calculated as the difference in the log of prices at time t and the log of lagged prices (logged prices at time t-1):

rit = Ln(price it) - Ln(price it-1); (2)

rit is the return of stock i at time t

Ln(price it) is log of price at time t

Ln(price it-1)is the log of price in the previous week

In the portfolio formation period of each sub period, beta coefficients, of each individual stock was estimated by regressing weekly risk premium which is the difference in returns and the risk free rate of return (rit-rft) against weekly risk premiums of the JSE/FTSE index (rmt- rft). The estimation model was:

Rit = y_ot + Rmt + Ñ”t; (3)

Rit is weekly risk premium on stock i

y_ot is the intercept

is estimated beta

Rmt is the return on the market index

Ð„t is the error term (noise)

The (since true beta, is unknown) were then ranked in decreasing order and using this ranking, 12 portfolios with a minimum of 10 stocks each were formed. A minimum of 10 stocks was taken, as the diversification of firm specific risk is assumed more important in obtaining reliable results than the number of portfolios. A summary is presented in Table 2. Blume (1970) showed that for a portfolio defined by weights, if errors in are significantly less than perfectly positively correlated, the estimated betas of portfolios can estimate more precisely. This is because the disturbance term in portfolios being much less than for individual securities. Creating portfolios has the advantage of adequately spreading assetsâ€™; if measurement error is too great, then portfolios with different may be estimated as having the same beta because of a large measurement error.

The second 2-year slot, the beta estimation period is used to re-compute of the stocks allocated in portfolios. Beta of portfolios was then calculated as the average of of stocks in the respective portfolios. A different period was used to estimate because high observed tends to be above the corresponding likewise low observed tend to be below. The result therefore of creating portfolios using ranked betas is that large will be an overestimate while low will be an underestimate.

The third 2-year period of the sub period was used as the testing period. To test, portfolio risk premiums are calculated. Firstly the average of weekly risk premiums of the stocks within respective portfolios is calculated. Then portfolio risk premium is obtained by the simple average of the average weekly risk premiums of the stocks within the portfolio. So, for each 6-year sub period, there were 12 portfolio risk premiums (obtained from the third 2-year slot) and 12 portfolio betas (obtained from the second 2-year slot).

The ex-post Security Market line (SML) is estimated by regressing portfolio risk premiums against portfolio betas. The estimated model is:

Rip,t = y0t +y1t*ip,t-1+ Ñ”t; (4)

Rip,t is portfolio risk premium

y0t is the intercept of the SML

y1t is ip,t-1 coefficient, the market price of risk, risk premium for taking on a unit of market risk (the slope of the SML)

Ñ”t is the error term

To test for nonlinearity between portfolio risk premiums and betas, betas are squared and included as an additional explanatory variable in the estimated model for the SML.

Rip,t = y0t +y1t*ip,t-1 + y2t ip,t-1 + Ñ”t; (5)

y2t is the ip,t-1 coefficient

According to the model, asset returns vary only because of the differences in . So a way to test the validity of the CAPM is to determine if the CAPM captures all important determinants of returns including the residual variance of stocks. The residual variance of portfolios is added as an explanatory variable to the estimated model 4 creating an augmented model. This allows for testing the explanatory power of non-systematic risk:

Rip,t = y0t +y1t*ip,t-1 + y2t ip,t-1 + y3t RVp + Ñ”pt; (6)

RVp = (Ñ”pt); residual variance of portfolio returns

To calculate the residual variance of portfolios, residual variance of each stock allocated in portfolios is generated and an equally weighted average of the residual variances is computed with the weights squared. Finally, this is repeated for all sub-periods.

The estimated coefficients allow for testing the hypotheses mentioned in the previous section:

y0 is not statistically different from zero (its absolute t value is not greater than 2)

y1>0, there is a positive risk-return trade-off (y1 is the market portfolio risk premium)

y2=0, the SML has no non-linear properties

y3=0, residual risk does not affect return (CAPM captures all important factors that explain the difference in asset returns)

## 5.2 Testing the systematic conditional relationship between Beta and Returns

The Pettengil et al approach was used to test here. The approach is identical to that of Fama and MacBeth but the only modification is an alteration in the final step of the Fama and MacBeth method. As explained earlier, if the market return is greater that the risk free rate, portfolio returns and betas should be positively related, however if market return is less than risk free rate, portfolio returns and betas should be inversely related. In the testing periods, market risk premium was negative in 73 out of 103 weeks for testing period 2005-2006; 80 out of 103 weeks for period 2006-2007; 84 out of 103 weeks for period 2007-2008 and 72 out of 96 weeks for period 2008-2009. The data sample was divided into positive and negative market excess return weeks. These two sub samples are referred to as the â€˜upmarketâ€™ and â€˜downmarketâ€™. For each sub sample, portfolio risk premiums are calculated using data in the testing period of each sub-period. The average of weekly risk premiums of the stocks within respective portfolios is calculated. Portfolio risk premium is obtained by the simple average of the average weekly risk premiums of the stocks within the portfolio. So, for each the 6-year sub period within a sub-sample (positive and negative market excess return samples), there are 12 portfolio risk premiums. As mentioned, the estimated portfolio betas computed earlier are used in the regression equation.

The Pettengill et al (1995) method includes a condition in the regression such that the equation is as thus:

Rip,t = y0t +y1t*ð›¿*ip,t-1 + y2t*(1- ð›¿)*ip,t-1 + Ñ”pt; (7)

Where ð›¿ =1, if (rmt- rft) > 0 (when market excess returns are positive), and ð›¿ =0, if (rmt- rft) < 0 (when market excess returns are negative). The above relationship will be examined for for each testing period by estimating either y1t or y2t, depending on the sign for market excess returns. [4]

Since y1t will be estimated in periods that exhibit positive market excess returns, the coefficient is expected to be positive. Therefore the following hypotheses will be tested:

Ho: y1t = 0,

Ha: y1t > 0

Since y2t will be estimated in periods that exhibit negative market excess returns, the coefficient is expected to be negative. Therefore the following hypotheses will be tested:

Ho: y2t = 0,

Ha: y2t < 0

Pettengill et al (1995), state that a systematic conditional relationship between beta and realised returns is supported if, in both cases, the null hypothesis is rejected in favour of the alternate.

## Empirical results

## 6.1 Fama and MacBeth 3-step traditional approach

Results are presented for each of the 6-year sub periods; 2001-2006, 2002-2007, 2003-2008 and 2004-2009. Results are presented for three different versions of the risk-return regression equation. For each sub period, the table shows: yj, the regression coefficient estimates; se(yj), the standard error of regression coefficient estimates; s(Rp-Rf), the standard deviation of portfolio risk premiums. The t-statistics for testing the hypothesis that yj=0 are calculated and presented and were calculated as:

t =

Tests are conducted at 5% significance level

Following the explanation of the effects on thin trading on in performing regressions, the assumption of the distribution of one-period returns on all assets and portfolios being normally distributed was suppressed and regressions were conducted with robust standard errors. Evidence from Fama (1965a) and Blume (1970) is that the probability distribution function of returns of stocks is different from that of the normal distribution. The distribution of returns is characterised by â€œthick tailsâ€ (excess kurtosis) and they suggest that it conforms better to nonnormal stable distributions than to the normal. Fama and Babiak (1968) explains that the consequences of this when interpreting large t-statistics under such an assumption is that the significance levels may be an overestimate. This has obvious consequences for determining whether the hypotheses except y1>0, (there is a positive risk-return trade-off) are rejected or not. Upward biases of the probability levels lead to biases towards rejecting the hypotheses of the model. Using robust standard errors helps to improve test results and eliminates the possibility of putting the blame of hypothesis rejection on the assumption that returns follow a normal distribution.

The initial remarks on securities are that all portfolios constructed in all sub-periods have negative risk premiums; this market is characterized by negative portfolio returns on average. This is a result of persistent negative market portfolio risk premium during the sample period. In all sub-periods except 2001-2006, portfolios with higher betas generally had higher returns; in particular the portfolio with the highest beta had the highest return.

If the CAPM is valid the hypotheses that y0=0 and y1>0 cannot be rejected. The results reject the hypothesis that y0=0 (not statistically different from zero). Although y0 in all sub-periods is close to zero, the absolute values of t(y0) in all sub-periods are greater than 2. This provides some evidence against the model, a zero-beta portfolio in this market does not yield zero returns. In fact y0 is negative in all sub-periods.

Table 2: Estimates of intercepts and slope coefficients

Rip,t = y0t +y1t*ip,t-1+ Ñ”t

## Sub-period

## 2001-2006

## 2002-2007

## 2003-2008

## 2004-2009

## Testing period

## 2005-2006

## 2006-2007

## 2007-2008

## 2008-2009

## y0

## -0.0143

## -0.0172

## -0.0239

## -0.0241

## y1

## 0.0019

## 0.0003

## -0.0039

## -0.0001

## robust se (y0)

0.0007

0.0011

0.0011

0.0009

## robust se (y1)

0.0009

0.0023

0.0015

0.0018

## s(Rp-Rf)

0.0015

0.0016

0.0016

0.0011

## t(y0)

## -20.58

## -15.04

## -22.59

## -25.8

## t(y1)

## 2.11

0.15

## -2.6

-0.05

Source: STATA 10 regression output

Considering the hypothesis that y2=0 which says the relationship between and return is linear, the results of the t-statistic presented do not reject this hypothesis. The only sub-period that refutes this hypothesis is 2002-2007 with a coefficient of .013 which is quite small.

Table 3 : Summary of results for testing non-linearity

Rip,t = y0t +y1t*ip,t-1 + y2t ip,t-1 + Ñ”t

## Sub-period

## 2001-2006

## 2002-2007

## 2003-2008

## 2004-2009

## Testing period

## 2005-2006

## 2006-2007

## 2007-2008

## 2008-2009

## y0

-.0138

-.014

-.0207

-.0257

## y1

0.00

-.0141

-.0143

.0052

## y2

## .0013

## .0126

## .0074

## -.004

## robust se (y0)

.0015

.0008

.0026

.0018

## robust se(y1)

.0047

.0037

.0073

.0059

## robust se(y2)

.0027

.0039

.0047

.0044

## s(Rp-Rf)

0.0015

0.0016

0.0016

0.0011

## t(y0)

-9.16

-18.22

-8.07

-14.66

## t(y1)

0.00

-3.77

-1.96

0.89

## t(y2)

0.49

## 3.25

1.58

-0.91

Source: STATA 10 regression output

Results presented reject the hypothesis y3=0, which says that all measures of risk are captured by. For the sub-periods 2003-2008 and 2004-2009, y3 is significantly different from zero. Even for the two sub-periods that have an absolute t-statistic not greater than the critical, y3 (coefficient of portfolio residual variance) is 2.39 and -4.71 which is not close to zero. An explanation of this insignificance is the large standard errors in these periods 2.73 and 2.56 respectively which cause an underestimate of the t-statistic. Therefore it can be concluded that the model does not capture all aspects of reality, and in this case firm specific risk has explanatory powers on returns.

Table 4: Summary of results for testing non-systematic risk

Rip,t = y0t +y1t*ip,t-1 + y2t ip,t-1 + y3t RVp + Ñ”pt

## Sub-period

## 2001-2006

## 2002-2007

## 2003-2008

## 2004-2009

## Testing period

## 2005-2006

## 2006-2007

## 2007-2008

## 2008-2009

## y0

-.0157

-.0105

-.0075

-.0218

## y1

.0039

-.0241

-.0075

.0086

## y2

-.0008

.0204

.0013

-.0034

## y3

## 2.3901

## -4.7154

## 11.4061

## -20.7527

## robust s.e (y0)

.0026

.0023

.0021

.0022

## robust s.e(y1)

.0066

.0072

.0056

.0064

## robust s.e(y2)

.0037

.0056

.0039

.00412

## robust s.e(y3)

2.7364

2.5698

3.6929

7.9468

## s(Rp-Rf)

0.0015

0.0016

0.0016

0.0011

## t(y0)

-6.12

-4.61

-12.05

-9.76

## t(y1)

0.60

-3.35

-1.34

1.34

## t(y2)

-0.22

3.65

0.33

-0.83

## t(y3)

0.87

-1.83

## 3.09

## -2.61

Source: STATA 10 regression output

The main hypothesis of the CAPM is that the coefficient of ip,t-1, y1 is > 0. Thus, more weight is put on this test for coming to conclusions. If this is rejected then there is considerable evidence that the CAPM is invalid. In a market characterized with risk averters, participants taking on more risk should be compensated with higher returns. The test for a positive linear relationship between risk and return proves to reject the hypothesis in that y1>0 in all sub-periods except in sub-period 2001-2006. Existing literature, discussed in section two provides evidence against the systematic relationship between risk and returns; moreover in all periods, the slope of the SML (y1) is not equal to the market risk premium as suggested by the CAPM.

## Sub-periods

## 2001-2006

## 2002-2007

## 2003-2008

## 2004-2009

## y1

## 0.0019

## 0.0003

## -0.0039

## 0.00

## Average (Rm-Rf)

## -0.0132

## -0.0171

## -0.0254

## -0.0228

Table 5: Comparing y1 and market risk premium

Source: STATA 10

## 6.2 Systematic conditional relationship between Beta and Returns

Section 1 details the argument to support a hypothesis of the beta â€“ return relationship being conditional. The hypothesis of a positive relationship during periods of positive market excess returns and a negative relationship during periods where the market excess returns are negative are tested.

Results provide significant support for a systematic but conditional relationship between beta and realised returns. y1 is estimate for weeks that exhibit positive market return. During these periods, high beta portfolios should perform better than low beta portfolios. Indeed this is the case as y1 is .019759 .0235881 .0413994 and .052677 for the testing periods respectively which are all significantly differently from zero at the 0.05 level (t= 6.37, 2.72, 10.84 and 10.79 respectively).

Table 6: Conditional Test for â€˜Upmarketâ€™

Rip,t = y0t +y1t*ð›¿*ip,t-1 + y2t*(1- ð›¿)*ip,t-1 + Ñ”pt

## Sub-period

## 2001-2006

## 2002-2007

## 2003-2008

## 2004-2009

## Testing period

## 2005-2006

## 2006-2007

## 2007-2008

## 2008-2009

## y0

-.012978

-.0135427

-.0207831

-.0265162

## y1

## .019759

## .0235881

## .0413994

## .052677

## robust s.e (y0)

.0022144

.0041519

.0025786

.0029179

## robust s.e(y1)

.0030999

.0086844

.003819

.0048816

## t(y0)

-5.86

-3.26

-8.06

-9.09

## t(y1)

## 6.37

## 2.72

## 10.84

10.79 Source: STATA 10 regression output

y2 is estimated in each testing period for the weeks where market excess return is negative. The negative relationship anticipated should produce negative values for y2. The values of y2 in these periods are all negative: -0.0058, -.0063, -.0142, -.0177 and statistically different from zero at 0.05 level (t=-3.19, -6.63, -5.85, -9.91).

Table 7: Conditional Test for â€˜Downmarketâ€™

Rip,t = y0t +y1t*ð›¿*ip,t-1 + y2t*(1- ð›¿)*ip,t-1 + Ñ”pt

## Sub-period

## 2001-2006

## 2002-2007

## 2003-2008

## 2004-2009

## Testing period

## 2005-2006

## 2006-2007

## 2007-2008

## 2008-2009

## y0

-.0149

-.0182

-.0246

-.0233

## y2

## -.0058

## -.0063

## -.0142

## -.0177

## robust s.e (y0)

.0013

.0007

.0015

.0013

## robust s.e(y2)

.0018

.0010

.0024

.00179

## t(y0)

-11.50

-23.59

-16.61

-18.14

## t(y2)

## -3.19

## -6.63

## -5.85

## -9.91

Source: STATA 10 regression output

There is a highly significant relationship between beta and returns in each sub period. There is an expected positive relationship between risk and return during periods of positive market excess returns. Likewise there is an expected negative relationship between risk and return in periods when market excess returns are negative. Results from Lakonishok and Shapiro (1984) show a similar relationship; they find y1 as 0.0333 and y2 as -0.0333 for up and down markets respectively. The results are similar to Pettengill et al (1995) who also find this relationship with y1 as 0.0336 and y2 as -0.0337 and being both significantly different from zero. They also find a highly significant relationship between beta and returns in each sub period. These results suggest, quite strongly those high beta portfolios incur lower realised returns during down markets than low beta portfolios. In comparison to the results obtained from the Fama and MacBeth traditional approach, this method produces consistent relationships between beta and returns across sub periods as it has the ability to adjust for the conditional relationship between returns and beta.

## Conclusions

## 7.1 Summary

In sum, results of the Pettengill et al (1995) method support the conditional relationship hypothesis of the relationship between beta and realised returns. There is a highly significant relationship between beta and returns in each sub period. There is an expected positive relationship between risk and return during periods of positive market excess returns. Likewise there is an expected negative relationship between risk and return in periods when market excess returns are negative. These results suggest, quite strongly those high beta portfolios incur lower realised returns during down markets than low beta portfolios. In comparison to the results obtained from the Fama and MacBeth traditional approach, this method produces consistent relationships between beta and returns across sub periods as it has the ability to adjust for the conditional relationship between returns and beta.

The main hypothesis of the unconditional CAPM: the coefficient of market risk premium y1 is greater than zero is rejected for this sample. The test for a positive linear relationship between risk and return proves to reject the hypothesis for all sub-periods except in sub-period 2001-2006. The results reject the hypothesis that the intercept is zero. Although y0 in all sub-periods is close to zero, the absolute values of t(y0) in all sub-periods are greater than 2. This provides some evidence against the model, a zero-beta portfolio in this market does not yield zero returns. In fact y0 is negative in all sub-periods.

Considering the hypothesis that y2=0 which says the relationship between and return is linear, the results of the t-statistic presented do not reject this hypothesis. The only sub-period that refutes this hypothesis is 2002-2007 with a coefficient of .013 which is quite small.

Results presented reject the hypothesis y3=0, which says that all measures of risk are captured by. For the sub-periods 2003-2008 and 2004-2009, y3 is significantly different from zero. Even for the two sub-periods that have an absolute t-statistic not greater than the critical, y3 (coefficient of portfolio residual variance) is 2.39 and -4.71 which is not close to zero. An explanation of this insignificance is the large standard errors in these periods 2.73 and 2.56 respectively which cause an underestimate of the t-statistic. Therefore it can be concluded that the model does not capture all aspects of reality, and in this case firm specific risk has explanatory powers on returns.

## 7.2 Scope for further studies

## Acknowledgements

Firstly, I would like to thank the Almighty God for giving me the ability to complete this study successfully. Specifically, I thank Dr. Vinay Prasandjeet Nundlall for his oversight and supervision which have undoubtedly contributed to helping me write a stimulating paper. I thank Dr. Violetta Dalla for her help with particular challenges encountered whilst using statistical programs. I wish to use this opportunity to appreciate all those who have supported my endeavours through the different stages of this work particularly, fellow student Abdulkadir Nasir.

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