An Introduction to the Capital Asset Pricing Model

Capital Asset Pricing Model (CAPM) is based on the Markowitz' portfolio model, and its development is associated with the work of Sharpe, Lintner, and Mossin, is mostly used to in finance to measure the stock returns. It requires estimates of risk-free rate, expected rate of return on market portfolio and estimate of beta to measure stock return. The risk-free rate and expected return on the market portfolio can be established on the historical estimates. CAPM is given by: CAPM is subject to criticism by many researchers due to the empirical evidences they found do not precisely describe the average stock returns provided by CAPM. Prominent among those who questioned CAPM are Fama and French (1992, 2004), they argue that empirical evidences are not supportive to Sharpe-Lintner-Black (SLB) model. Other researchers like Banz (1981) found that size effect explain the cross-section of average returns provided by market, later Bhandari (1988) discovered risk and expected return to be associated with leverage. In addition, Stattman (1980) and Rosenberg, Reid in US securities and Chan, Hamao, and Lakonishok (1991) in Japanese securities detected average returns on U.S. stocks are positively related to the ratio of a firm's book value of common equity to its market value.

On the other hand, researchers like Chan and Lakonishok (1993) remained inconclusive regarding rejection relationship between returns and betas because they believe that results obtained are impacted by very noisy and constantly changing environment generating stock returns. The further emphasized that return may not only determined by the beta but other behavioral and institutional factors of equity markets might also drive returns. They concluded "it is of course possible that beta is very poor measure of risk, and much better risk measure exit but have not been covered". Black (1993) also supported beta as being a valuable investment tool and emphasized the existence of beta, contrary to those who believe beta as dead (Brailsford, 1997). Despite of being excessively criticized it is still most widely used model to estimate stock returns. Within the CAPM, systematic or non-diversifiable risk of security, known as beta (), is the only factor that differentiates between cross-sectional rates of return. The user of betas are very wide, most important of them are analysts, corporate managers, practitioners, and portfolio managers CAPM postulates that returns and betas are related. Beta is calculated as follows: The CAPM empirical counterpart is known as market model that is simply an expression of a statistical relationship about the relationship between realized security returns and realized security returns on a market index.

The beta of stock in CAPM can be estimated by using market model. The standard specification of the market model is given as follows: The beta of a security is obtained by regressing historical stock returns against historical returns from the proxy for market return using the market model via Ordinary Least Squares (OLS). OLS estimates requires an assumption of normally and independently distribution errors. There are several issues that relate to the research design issues of beta estimation, using market model, and is classified into two categories namely measurement assumptions and the assumptions which relate to the regression. This project will focus on investigation of some of the issues relating to both measurement assumptions and regression assumptions. These include: The effect of choosing alternative return measures: it compares and demonstrate the impact of using raw discrete and raw continuous return measures and raw continuous and excess continuous return measures, The effect of varying sample interval: different sampling intervals (daily, weekly, and monthly) are chosen to consider their effect on beta estimates, The effect of varying length of estimation period: affect of changing length of estimation period on beta estimates from 200 to 250 and also to 450 days; from 30 months to 60 months period are investigated respectively, The effect of outlier observations, Diagnostic analysis of market model regression residuals: standard residuals of weekly excess continuous returns are tested for the autocorrelation and basic descriptive statistics is also given, The issue of beta stability for individual securities and portfolios, and The issue day-of-the-week effect: it highlights the Friday effect on beta estimation. All of these issues listed above are discussed individually in the light of the test conducted on the 40 companies (for the time period of 1994 - 2005) and results observed, influential literature on the topic, and the empirical evidences found by the other researchers. The last section of this paper concludes the findings of the project.

Different return measures:

Raw discrete and raw continuous return:

The starting point in the estimation of security's beta is the choice which researcher or investor has to make is the selection between discrete or continuously compounded return measures. It is up to the choice of the researcher or investor whether to use raw discrete of raw continuous returns. Both of these return measures formulae are shown below: Discrete return is calculated as: On the other hand continuously compounded return is calculated as: Where Pit = Price of stock i at the end of the measurement interval t, and = Price of stock i at the at the interval t-1. It is widely accepted that in a stock market trading take place at discrete intervals, i.e. in week days, but returns are continuously generated through calendar time. However, some ideologues view returns are generated at discrete intervals because of stock market trading at discrete intervals. For both the discrete and continuously compounded returns the returns should be adjusted for capitalization changes and dividends (Correia, C. et. al., 2007). The table 1.1, below shows difference in the beta estimates due to different return measures used i.e. raw discrete and raw continuous returns. The daily estimates of beta showed positive movement in the beta estimates of about 85% of securities, when the estimates of the beta is changed from raw discrete return measure to raw continuous measure. However, for weekly data this increased to 87.5% of securities as compared to the daily return measure difference. On the other hand, the difference in these estimates due to the change in the return measures of the monthly data declined and 30 companies out of 40 companies' beta estimate differences showed positive or no differences, which is 80%. The average difference of these estimates of daily beta this figure is 0.0013 whereas for monthly beta is 0.0157, which is 12 times higher than differences of daily beta estimates and for weekly beta this figure is 5 times higher than daily beta estimates differences. The higher difference found between daily and monthly beta estimates could be due to the fact that beta estimates of daily data are mostly less than one. Also, more or less all beta estimates based on raw continuous returns have higher values as compared with their counterparts; this positive biasness can be linked to the low returns of continuous compounded returns.

Table 1.1

Raw return vs. excess return:

Another issue regarding the beta estimate is the choice between the raw return and excess return. In calculating excess return a benchmark return is deducted from the raw return, and then to estimate beta excess return is run against excess return on market. This benchmark asset is usually risk-free asset; in this case we have used annual UK Treasury bill rate as a proxy rate. This proxy rate is then converted into the rate in relation to the different interval lengths by utilizing the formula below: Table 1.2 shows that the difference between average daily raw continuous and daily excess continuous return is approximately negligible, in all the cases. However, this difference increases with the change in the time period from daily to weekly and monthly. The average difference between these beta estimates is 0.0009 and 0.0060 for weekly and monthly data. It could be implied from the figure from the average difference of beta estimate from daily returns that beta is unaffected by raw and excess continuous returns. The results obtained coincided with the study of BartholdyA and PeareA (2001) in which they used data from 1970 to 1996. The empirical evidences from their study also proved that the difference in beta estimates between using excess and raw returns is minimal, regardless of the index and data frequency used. Thus, it can be deduced from the results obtained that either of raw return or excess returns can be utilized for estimation of beta.

Table 1.2

The sampling interval:

The interval effect, could be defined as the changes in the beta estimates due to change in the return interval, is widely debated issue in the area of beta estimation. The beta estimates can vary depending on the sampling interval used and the selection of the sampling interval varies. The most commonly used intervals for the estimation of the betas used by the researchers and investors are daily, weekly, and monthly data returns. Also, in practice intraday, bi-weekly, and quarterly return intervals data are also utilized. Various researchers have found that return interval and the estimation period have substantial impact on the estimation of the beta. Pogue and Solnik (1974) reported that the betas estimated from daily returns are lower than betas estimated from monthly returns. They pointed out that this biasness in the estimation is due to the "lags in the adjustment of stock prices to changes in market levels" and measurement errors. Also, according to them if this price adjustment in the stock prices is slow it represents the large differences in betas estimated on the basis of monthly and daily returns, in an efficient market. They found that measurement errors diminish in as the return interval increases. Hawawini (1983) demonstrated that beta estimates depend on the length of the return interval. He analyzed that securities with large market value of shares outstanding (MVSO) relative to the market average have an increasing beta i.e. their betas are upward bias whereas those securities with low MVSO relative to market average has decreasing beta i.e. these are biased downwards.

He suggested that this shifts in estimated betas are due to the presence of non-contemporaneous cross-correlations between the daily returns of securities and those of general market. According to Brown, the volatility of return is the function of the return interval. Furthermore, he pointed out that volatility on an intraday basis is highest (Brown, 1990). Also, Brailsford (1995) conducted the research on the volatility of returns. To study the volatility in returns he used five minutes, hourly, daily and weekly return intervals. He noted that "as the length of the return interval increases, the ratio of the coefficient estimate on lagged conditional volatility to the coefficient estimate on past squared errors monotonically increases from 0.89 for the five minute return series to 2.1 for the daily return series and 8.11 for the weekly return series". His reports concluded that the variance is highest for intraday returns as compared with daily and the weekly returns. Table 1.3 shows that the beta estimated based on different return intervals for the same length of period from 1994 -2005. The least average differences in beta estimates is 0.1985 between daily and weekly beta estimates whereas this difference increases between the beta estimates of daily and monthly intervals, which clearly demonstrates increased variability. As shown in the column of range, which is difference between monthly and daily betas, that 62.5% of individual securities beta differences are greater than 0.5 whereas out of these 25 figures 12% have variability greater than 1. Also, the cross-sectional mean t-stat figures of daily, weekly, and monthly decreased from 17.4855to 9.7751to 7.0512 respectively. This decline in the average t-stat values clearly shows the reduction of precision in estimates of betas obtained as a result of the increase in the return interval. An explanation of the changes in the beta estimates caused by the changes in return interval is provided by Handa et al. (1989). They pointed out to an explanation relating to lack of statistical precision caused due to standard error of the beta estimates which increases as the length of return interval increases.

Table 1.3

Beta estimation and length of estimation period:

Draper and Paudyal (1995) in their research paper pointed out the importance influence of the sample size on the estimation of the beta. They suggested controlling the substantial variations in the beta estimates by increasing the number of observations in the sample. Also, they mentioned that 100 numbers of observations are much more affected by large and sudden changes as compared to the large sample size and become more reliable and stable as the numbers of observations approaches to 400. Harrington (1983) research on the methods of forecasting beta remained inconclusive about the best method but it revealed that betas of the security will be better forecasted when it is based on longer time horizon. She mentioned that longer time duration is predicts betas of security much efficiently because in the long run short-term error terms will cancel each other out. Gonedes (1973) showed that the betas estimated on seven years period are superior to five years and three years. Bartholdy and Peare (2001) recommended the 5 years period reruns to estimate the beta. Moreover, Brailsford, Faff and Oliver (1997) suggested that it is generally accepted that approximately 50 data points are essential to obtain reliable OLS estimates and mentioned that however when dealing with monthly sampling interval, a five years of data is often considered as a guideline because beta estimates are relatively stable over this period. Alexander and Chervany (1980) suggested the estimation interval for the beta estimation to should be between 4 - 6 years. He used mean absolute deviation as a measure of beta stability and found that securities with extreme betas tend to be more volatile than the securities with less extreme betas, which are found to be stable.

The effect of change on the beta estimation is measured by changing the sampling interval of the daily and monthly data. These sampling intervals are constructed in such a way that the selection is non-overlapping. To measure the effect on the daily data the sample size consisting of non-overlapping data of 200, 250 and 450 days and with regard to monthly return intervals 30 and 60 months return interval is selected. The corresponding betas of securities and t-stat are shown in table 1.4 below. Some hypothesis testing is also conducted at both 5% and 1% significance level. The hypothesis is tested. Test statistic under H0 is estimated beta over standard error of beta which is also the t-statistic reported. For both the daily and the monthly data, if t-statistics value obtained for beta estimates fall between -1.96 and 1.96 it is concluded that relevant beta estimate is not significantly different from zero at 5% level of significance and if the value of t-stat is -2.58 and 2.58 it is concluded that relevant beta estimate is not significantly different from zero at 1% level of significance. From the table 1.4, it can be deduced that 47.5% of betas calculated using 200 days period are insignificant at both 5% and 1% significance level, however 35% of these are insignificant at 5% level of significance. On the other hand, when the betas are calculated using 450 days, 25% of beta remained insignificant at both 5% and 1% level of significance. Also, 55% of the betas estimated based on 30 months interval are insignificant whereas for 60 months period 30% of betas estimates are insignificant at both 5% and 1 % level of significance. The cross-sectional mean t-stat increased from 3.7744 to 6.8655 and from 2.599 to 4.677 when the estimation period is changed from 200 to 450 days and from 30 months to 60 months respectively. Thus, the increase in the average t-stats, in both observed in both cases, points out that beta estimates reliability in enhanced by increasing the length of estimation period.

Table 1.4

Distribution of stock returns:

The important assumption of OLS regression for estimating beta is the residuals have uniform variance and are uncorrelated with each other. The regression distribution might be homoscedastic or hetreroscedastic. The former states that regression distribution variances are uniform whereas latter states that regression distribution variances are non-uniform. In this section of report beta estimated using market-model regression are examined for normality by analyzing descriptive statistics of monthly and daily returns. Kurtosis is usually used to provide an indication of possible heteroscedasticity (Brailsford et al., 1997). Both skewness and kurtosis are used to analyze the normal distribution of residuals. Table 1.5 reports the results of descriptive statistics of daily and monthly excess continuous returns. The daily mean excess continuous return showed a minimum of -0.0002 and a maximum of 0.0010 whereas the monthly mean excess continuous return showed a minimum value of -0.0048 and maximum of 0.0199.

Furthermore, the daily FTSE-All shares return index mean return remained at 0.0002 and for the monthly average return of FTSE-All shares return index is 0.0031. However, the average of these forty securities is 0.0003 and 0.0061 for daily mean continuous return and monthly mean continuous return respectively. The standard deviation of the daily returns found to be ranging from 0.0076 to of 0.0284 and monthly returns ranges from 0.0430 to 0.1489 where as the standard deviation of FTSE all share daily return is 0.0096 and for monthly return is 0.0388. It is also worth noting in the daily data that only 5 securities showed standard deviation less than FTSE standard deviation, whereas none of the security under monthly data showed standard deviation less than the FTSE All share return. Also, for the daily returns 97.5% of the companies have kurtosis value greater than 3, with the average kurtosis value of 19.7233. This result implies that daily returns distributions are significantly deviated from normal distribution. For monthly data only 6 out of 40 securities' kurtosis is found to be greater than 3, with the average value of 2.0512. Furthermore, the daily data on average is positively skewed, 25 out of 40 securities returns are negatively skewed whereas on average the monthly data is negatively skewed and 31 out of 40 securities' monthly returns are negatively skewed.

Table 1.5

Testing residuals for autocorrelation

There are several methods which can be used to test correlation of disturbances, mainly used are Durbin Watson d statistics test an Box-Pierce-Ljung test. Whether there exists the correlation among the residuals hypothesis are tested, against. The null hypothesis implies that and is therefore iid (0, AÆ’2) (Gujarati, 2003) under the alternative hypothesis the disturbance terms are correlated. The Durbin-Watson d statistic, mostly used to test the serial correlation, is conducted on weekly excess continuous returns in order to assess the autocorrelation. If DW statistic is less than 1.758, that is DL value of 200 observations taken from the Durbin-Watson d statistic table, with one explanatory variable, reject in favour of and conclude on this basis that there exists a positive autocorrelation among the residuals. The results of autocorrelation tests of DW stat and Corr(ut, ut-1) is shown in table 1.6. It is clear from the table that only 22.5% of the companies have shown Durbin Watson d statistics below 1.758 and rest of 31 companies DW stat demonstrate positive autocorrelation. However, 13 out of 40 companies have negative correlation co-efficient as shown by red fills in correlation co-efficient column. It is clear from the table below, 25% of companies' residuals exhibited positive autocorrelation. From the analysis conducted on the weekly data it can be concluded that observations are likely to be interdependent because the data demonstrated inertia.

Table 1.6

Effect of outlier on beta estimates:

Outliers are the values in the observations that are "genuinely extreme" by the virtue of their absolute size and are stimulated by the errors in the data. The beta estimated using the standard OLS procedure might suffer non-normality as a result of these outliers' presence in data. These outliers may substantially distort reliable beta estimates and thus requires exclusion from the data set, in order to find a reliable beta forecast. The magnitude of distortion in the beta estimates are to some extent depends on the magnitude of that outlier and the overall sample size (Brailsford et al., 1997). According to Martin and Simin (1999) mentioned that the beta estimates are affected by the outliers and pointed out that in the presence "influential outlier" nor OLS beta neither robust beta estimates are effective. However, they suggested the use of robust estimation method over both OLS estimation. To detect the impact of the outliers on beta estimates 5 companies are chosen randomly for both daily and monthly returns data. Then these companies' outliers are observed by means of plotting the standardized residuals of each of these companies individually. Only two extreme values are selected and removed from these outliers, which are 2.5 or 3 standard deviation away from their mean. For example, using the daily data of British Airways, two outliers were identified, i.e. 9/11/2001 and 11/15/2001, having standardized residuals of -7.6677 and 7.1590 respectively and then both of these are removed from the data to estimate beta. Table 1.7 shows the securities and their respective betas including outliers (full data) and excluding the outliers (reduced data), from daily data and monthly data. The percentage changes in the beta estimates due to the removal of the observations ranges from -0.17 to 3.32 and -12.03 to 3.79 for daily and monthly data respectively. The reduction in the data of outliers resulted in increase in the beta estimates, as represented by the positive differences, of 1 security and 3 securities for daily data and monthly data respectively. The tendency of beta estimates to decrease for monthly data and to increase for daily might be due to the reason that most of the estimate betas of monthly data are greater than 1 whereas for daily full data most of the beta estimates are less than one. This might have exerted more influence on the betas of the monthly data as compared with the daily data. Also, daily data numbers of observations are 22 times higher than the numbers of observations of monthly data and removal of 1 or 2 observations have less impact on the daily data.

Table 1.7

Beta Stability:

The CAPM is a single-period model whereas the beta estimates obtained using OLS is applied in a multi-period setting. An assumption is to be made regarding the beta estimates that these are constant through time, due to the shift from the single-to-multi-period environments. On the contrary, beta estimates are found to be volatile and thus breaching the assumption made. Several studies have shown that betas are not constant through time and points out that the market model is misspecified (Brailsford et. al., 1997). Fabozzi and Francis (1977) showed the relationship between systematic risk stability and bull and bear market conditions. The two issues that relates to beta stability are inter-period stability and intra-period stability. To enquire about the inter-period stability the analysis is usually done to test whether the beta is stable between the estimation period and the application period, which are non-overlapping.

This issue is dealt by allowing for the possibility of mean shifts in beta. Also, generally the reason of inter-period differences in beta estimates varies from firm-specific factors through to market wide factors but specifically these differences are due to mean reversion and structural breaks. However, to analyze the intra-period stability it is tested for the stability in the estimation period by incorporating the concept of time-varying beta (Brailsford et. al., 1997). As mentioned earlier that inter-period stability of beta estimates are influenced by the mean reversion issue, and structural breaks issues. "A structural break is a point in the sample at which there is clear delineation of groups of the data" example of structural breaks is changes in the market conditions. Due to these structural breaks beta estimates of one sample become incomparable with other estimates of beta obtained from another set of data. Therefore, it is important to find structural breaks, and divide initial sample into sub periods using structural breaks as the delineation date, which can be found by the use of Chow test (Brailsford et. al, 1997). Also, Empirical evidences postulate that both individual stocks and portfolio betas are time varying. The three models that are mostly used to observe these time variations are Random walk method, Random coefficient approach and Autoregressive process beta (Brailsford et. al., 1997). Different models have been proposed by different researchers.

Hildreth and Houck (1968) recommended the use of random-coefficient model. However, Faff et al. (2000), Sunder (1980), Simonds et al. (1986) stresses the use of random walk for observing time variation in beta estimates and suggested that it provides best characterization of time-varying beta. Brooks, Faff, and lee (1992), empirically found beta is time varying and also favoured Hildreth-Houck model over Rosenberg model for determination of appropriate form of time variation in beta. Blume (1975) showed that the regression tendency of beta estimates is to regress towards the grand mean of all betas over time. He mentioned in his study that betas estimated for the same portfolios of securities inclined closer to market beta of one or become less extreme than prior estimates of betas. Furthermore, he explained that the companies which are of very low or very high risk characteristics become less risky over the time because the companies' existing projects risk decline over time and new projects considered by the company are less risky as compared with existing projects. Brailsford et al. (1997) withstanding to the reasoning offered by Blume, further added that market portfolio beta has value of one, as the new stocks lists on the stock exchanges their betas are offset by movements in betas of existing stocks towards unity, thus maintain the market portfolio beta of one. Alexander and Chervany (1980) results reasserted Blume findings of tendency of beta to regress towards one. Furthermore, they suggested as number of securities in the portfolio increases the magnitude of intertemporal changes (time stability) in portfolio beta coefficients decreases or become substantially stable, regardless of how the portfolios are formed. He concluded that the portfolios of ten or more securities are found to stabalise these intertemporal changes (measured in mean absolute deviation).

Schneller (1983) states that return on portfolio will be impacted if naive beta is included in the portfolio construction causing the deviation in the portfolio of beta, known as "beta error risk". This  beta is the result of measurement error, while estimating beta, and can be diversified away by enlarging the portfolio size. It has been found that large portfolios, of more than 25securities, betas are stationary, less stationary for smaller portfolios and variable for individual securities. Also, as the period lengthens from 26 weeks the betas showed tendency to regress towards their means. This tendency appears stronger for high risk portfolios than for low risk portfolio. Porter and Ezzell (1975), found that inter-temporal stability of betas are sensitive to the method utilized in selecting portfolios. Furthermore, Porter and Ezzell (1975), Blume and levy agreed with regard to beta portfolio that betas of randomly selected portfolios are relatively unstable and unrelated to the number of securities in the portfolio. Inter-period stability is tested for both individual securities and portfolios. To test the inter-period stability using the weekly excess continuous returns, 11 year period from 1994 - 2005 is divided into three non-overlapping sub-periods of 191, 191 and 192 observations. As shown in the table below that percentage change in the beta estimates ranges between -1735.70% to 1159.42% and -1907.07% to 223.68 for period 1 and 2 and for period 2 and 3 respectively. The average estimates of beta increased from period 1 to 2 and to 3 with betas 0.5883, 0.6379 and 0.7310 respectively. However, 6, 3, and 2 betas estimates as shown in table by red fills are insignificant at 5% level of significance. It is also found that the significant variation in betas from period to period is associated with the insignificant betas, as shown clearly by color fills. However, exclusion of these insignificant betas do not impact the stability of the average beta of the respective period.

Table 1.8

Also to test inter-period stability of betas using weekly excess continuous returns, 2 equally weighted portfolios consisting of 20 securities each and 4 equally weighted portfolios consisting of 10 securities each are constructed. The portfolio is constructed by grouping 20 or 10 securities based on the portfolio design in a random manner. As, shown in the table 1.10 that the extreme percentage changes relates to the smaller size of portfolios of ten securities compared to the large size portfolio of twenty securities. The percentage change in the beta estimates of period 1 for 20 securities portfolio ranges from -0.73% to 14.08 and for second portfolio of 20 securities this change ranged from 4.86% to 21.49% in the period 1-2 and period 2-3. For portfolios comprising of the 10 securities the variability is higher than the portfolios constructed by including 20 random securities. The percentage change in the period 1 and 2 is found to vary between -23.40 to 41.47, however in period 2 and 3 it remained at -0.30 to 35.62.

Table1.9

Blume (1975) pointed in his article that for equally weighted portfolios, the larger the number of securities in portfolio more reliable will be the beta forecast. He concluded that "for an equally weighted portfolio of 100 securities, the standard deviation of the error in the portfolio beta would be 1/3 of the standard error of the estimated betas for individual securities". According to Harrington (1983), the beta forecasts based on the portfolios is superior to the forecasts of the single securities. She mentioned that this is due to the fact that error terms cancel each other out in the portfolio, leading to better forecasts. Furthermore, she demonstrated by using Mean square Error (MSE) test that the forecast improved as the size of the portfolio is enhanced. The MSE of portfolios decreased from 0.2013 to 0.06 as the portfolio size is increased from 5 securities to 15 securities. She suggested that this reduction in MSE is associated with the reduction in the random error.

Day-of-the-week effect - Friday effect:

The day-of-the-week effect is important from an investor's perspective because it could support in reaping substantial benefits by devolving trading strategy of buying stock on abnormally low returns day and selling of stock on abnormally high stock returns day. According to Drogalas et al. (2007), day-of-the-week effect means the average stock returns of Monday are negative, while the average stock returns of Friday are positive. The anomalousness of stock market due to day-of-the-week effect is therefore required investigation. To test the day-of-the-week effect a dummy variable is included and regression is run against the following equation The effect of the introduction of the dummy in the equation above will help in controlling the explanatory power of the-day-of-the-week i.e. Friday. Thus, the result in the coefficient estimate of the beta can be considered to be "fundamental" beta. All the securities daily excess continuous compounded returns are regressed against both the market excess returns and the dummy variable of Friday to determine whether there is an impact of Friday and also to find out significance of dummy in explaining the security returns. Table 1.11 shows that only two securities namely Redrow and Shaftesbury have significant Friday dummy betas at 5% significance level, marked in green fills, all other securities betas are insignificant at 5% level of significance. Furthermore, 50% of these Friday dummy have positive betas and 50% of these have negative betas.

The difference found by including Friday as dummy variable and excluding it from the data is found to be 0.00011 on average which is very minute. Moreover, the Friday dummy average is 0.0001 which shows slight positive biasness in the beta estimates. This slight positive effect of the beta could be the result of behavior and of trading patterns that are observed in the stock exchanges. Usually bad news are announced on Friday which are incorporated into the prices on the following Monday, thus creating large gaps between the closing prices of stocks on Friday and opening prices of stocks on Monday. According to the study conducted on emerging stock market, it is found that some of emerging stock markets displayed day-of-the-week effect like Philippines, Pakistan and Taiwan whereas it is absent in majority of the emerging stock markets. Also, it is observed by several studies that the United States and Canada finds that daily stock market returns tend to be lower on Mondays and higher on Fridays (Basher and Sadorsky, 2006). However, as mentioned earlier the Friday effect on beta estimates is found to be unsubstantial by regressing betas using the continuous daily excess return of companies provided, listed on London Stock Exchange.

Table 1.10: Friday Dummy variable

Summary and Conclusion:

The important findings regarding issues concerned of the project can be summarized as follows: Raw discrete average betas are slightly lower than raw excess continuous return. In fact both daily and weekly data has shown equal numbers of securities' betas where securities' betas of raw discrete is less than raw continuous securities' betas. However, it doubled as the betas from monthly data is considered for both raw discrete and raw continuous. It is also worth noting that average betas increase by 1.5 times as interval changes from daily to weekly to monthly for raw continuous average betas and 1.4 times for raw discrete average beta. The positive biasness can be linked to the low returns of continuous compounded returns. For the daily raw continuous and daily excess continuous return, it is found that average beta differences increases at decreasing rate as the interval changes from daily to weekly to monthly. Also, monthly betas of securities are found to be highest among monthly and weekly betas, mostly monthly securities' betas are equal to 1 or greater than.

It can be deduced from the results obtained that either of raw return or excess returns can be utilized for estimation of beta. It is found that differences in betas increase as the sampling interval changes from daily to weekly to monthly intervals. Beside it, betas of all securities are found to be significant at both 5% and 1% level of significance for daily data estimates of beta however for weekly and monthly data number of securities found to be significant at both 5% and 1% level of significance decreases and also are found to be same in numbers. Reason of this could be impact of standard error of beta increases as the length of the return interval increases. Hypotheses are tested for. More betas are found to be significant at both 5% and 1% level of significance as the estimation period increases from 200 to 250 to 450 days. Same is the case with the 30 months and 60 months estimation period, with more betas are found to be significant for 60 months data as compared with 30 months data, at both 5% and 1% level of significance. It clearly shows that beta estimates are more reliable when estimated over relative large estimation period.

The findings regarding distributional assumptions and autocorrelation of residuals are as under: Average beta estimates of daily data are lower than monthly data Average standard deviation in beta estimates is higher in monthly betas as compared with the daily betas. Daily data beta estimates are found to be positively skewed whereas monthly beta estimates are negatively skewed, both same in magnitudes but opposite. Overall, daily data has showed non-normality, as measured by average excess kurtosis. It might also impact reliability of beta estimates obtained by OLS, which assume normally distributed residuals. Durbin Watson d statistic test is conducted on weekly excess continuous returns to test autocorrelation. It is observed that 32.5% of securities have positive correlation coefficients and 25% of the securities showed positive autocorrelation. Thus, betas showed inertia on week to week basis and violated OLS regression assumption.

It is found that both daily and monthly data are affected by the outlier. However, monthly betas of securities as compared to daily betas are largely affected by the removal of outliers. This could be due to the fact that numbers of observations in daily data are more than monthly data and removal of same number of observations might not justifiable. Betas of portfolio showed more stability as compared with the individual securities betas, which demonstrated considerable variations in betas from one period to another period. Also, it is found that larger the portfolio more stable will be the beta estimates. Betas also depicted the tendency to regress towards one. Results regarding Friday effect on beta estimates showed no substantial differences between securities betas estimated without Friday dummy and including Friday dummy. All in all, the paper has clearly demonstrated estimates of beta are prone to different methods used for estimation as exhibited by the variations in these estimates. The limitation of the project is time constraint in general. Other limitations are beta stability is not tested for portfolios comprising more than 20 securities, which would have further revealed the stability of betas estimated, if larger portfolios are considered. Also, for seasonality only betas are tested for Friday effect. For regression assumptions about residuals only auto-correlation is examined.

If more time is provided I would have looked at the following issues regarding the beta estimation: Effect of using different market proxies e.g. value weighted, price weighted and equally weighted market indices on estimates of beta, Beta stability for portfolio comprising of more than 20 securities, Errors arising in the estimates of beta arising from the thinly traded shares in stock market, Estimation of fundamental beta by dealing with the specification error, Seasonality issues in context of January or July and days other than Friday, and Also, I would have tried to look at the interdependencies of several issues related to beta estimation.

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