Estimating Betas in Thinner Markets: The Case of the Athens Stock Exchange

Transcription

1 International Research Journal of Finance and Economics ISSN Issue 13 (2008) EuroJournals Publishing, Inc Estimating Betas in Thinner Markets: The Case of the Athens Stock Exchange George Diacogiannis Department of Financial Management and Banking University of Piraeus, Greece: , Visiting scholar School of Management, the University of Bath, UK Paraskevi Makri MSc in Financial Management and Banking Abstract This paper examines the intervalling-effect bias in ordinary least squares beta estimates and appraises the relative ability of models to estimate betas for securities traded on the Athens Stock Exchange. The results reveal the presence of the intervalling effect bias and the dependence of its intensity on the market value of the firms. Additionally, the results using the model of Hawawini (1993) show a good performance for estimating the betas for longer return intervals using shorter return intervals. Finally, betas are estimated employing the models proposed by Scholes and Williams (1977) and Cohen et al (1983a) and compared with those produced running the market model with the ordinary least squares method. The results reveal that there are no statistically significant differences between the mean beta estimated using the market model with the ordinary least squares method and the models of Scholes and Williams and Cohen et al. Keywords: Beta, market model, intervalling effect bias. Introduction The relevant measure of a securitys systematic risk is beta, the slope parameter of a simple linear regression equation relating the securitys returns 1 to the temporally corresponding market returns. Over the past years beta has received considerable attention by being widely applied in several topics including the measurement of the cost of equity for capital budgeting and firm valuation and the evaluation of security or portfolio performance. The method of the ordinary least squares (OLS) is typically employed to estimate security beta. However, a number of beta estimates can emerge for a security depending on various considerations such as the choice of the market index, the length of the return measurement interval, and the sample period. Studies by Hawawini (1980), Handa et al (1989), Corhay (1992), and Brailsford & Josev (1997) reported the sensitivity of the beta to the length of the return interval utilized to estimate it. This effect is known as the intervalling-effect bias in beta. More specifically, previous research has found that OLS beta estimates of securities with small (large) market values or portfolios comprising of 1 In this paper it is used returns to briefly say rates of return

2 International Research Journal of Finance and Economics - Issue 13 (2008) 109 securities having small (large) market values increase (decrease) as the return measurement interval is lengthened. Scholes and Williams (1977) and Dimson (1979) also reported the bias in the OLS beta estimation when the securities are subject to thin trading and the use of short return measurement intervals. Additionally they proposed procedures for correcting the bias in the OLS estimated betas 2. In essence, this paper is concerned with the effect of the length of the return measurement interval on the estimated betas for securities listed on the Main Market of the Athens Stock Exchange (ATSE). It was chosen to examine the case of the ATSE mainly for two reasons. Firstly, we would like to investigate the intervalling-effect bias in OLS estimated betas in the case of a small emerging market with different characteristics from those of the developed and mature markets of the United States and other major European countries which after the year 2000 passed from the status of high risk markets to the status of more developed ones 3. Secondly, during the four-year period examined in our study, the ATSE experienced a huge fall of share prices and thus a considerable number of infrequent trading securities has emerged. For these reasons we believe that the implications of our results may be particularly useful to academics and also to institutional investors. The objective of this article is threefold. First, it tests whether the beta estimates change as the return measurement interval lengthens and if the magnitude of this effect is related to the market value of the firms. Second, it examines the predictive ability of the model proposed by Hawawini (1980). Third, it estimates betas employing the models proposed by Scholes and Williams (1977) and Cohen et al (1983a) and compares these estimates with those produced running the market model with the OLS method. The importance of our work lies in the fact that, for the first time, a variety of issues related to the intervalling-effect bias in beta was thoroughly examined in the small emerging market of the ATSE and this examination was conducted during its evolution to maturity. The remaining contents of the paper are organized as follows. Section I presents a review of the previous studies related to the return interval effect on estimated betas. Section II describes the data employed in the empirical tests and the research methodology. Section III discusses the empirical results obtained. Finally, Section IV contains a summary of the paper. I. Review of the Previous Studies To begin the discussion it seems appropriate to introduce some early studies of the return interval effect on estimated betas. Fama (1970), Pogue and Solnic (1974), Altman, Jacquillat and Lavasseur (1974), Smith (1978), Hawawini (1980), and Ng (1981) reported that beta estimates via the market model vary systematically with the length of the return interval used to calculate security returns 4. Cohen et al. (1980) documented that security beta estimates calculated employing the OLS method are sensitive to changes in the return measurement interval. This sensitivity was attributed to friction in the trading process which can cause price-adjustment delays 5 that differ systematically across securities. They argued that delays in the adjustment of a security price to a change of information produce crosscorrelations among the security returns, which in turn results in autocorrelation in the returns of the market index and biased estimates of beta. In view of their argument infrequent traded securities have greater adjustment delays than frequently traded securities 6. Roll (1981) noted that the betas of shares that are traded infrequently are underestimated when short returns measurement interval are used. Several procedures have been developed for correcting the intervalling-effect bias in OLS beta. These include Scholes and Williams (1977), Cohen et al (1983a), and Dimson (1979) 7. Scholes and Other research has focused on the stability of beta (Blume (1971, 1975), Eubank and Zumwalt (1979), Emanuel (1980), Dimson and March (1983) and Diacogiannis (1989)), on the stochastic properties of beta (Bos and Newbold (1984) and Brooks et al. (1992)) and the method of estimation (Castagna et al. (1984), and Chan and Lakonishok (1992)). See the Morgan Stanley Capital International World Index. Hill and Schneeweis (1979) examined the intervalling effect in estimated betas using bond returns. The price adjustment delays was first discussed by Fisher (1966). Levhari and Levy (1977) illustrated theoretically and with the aid of empirical results that beta is dependent upon the return measurement interval when returns are measured in discrete time. Theobald (1980) offered a beta estimator which, under certain conditions, approaches that of Scholes and Williams (1977) as the differencing interval is lengthen.

3 110 International Research Journal of Finance and Economics - Issue 13 (2008) Williams (1977) have shown that beta estimates are biased downward for securities trading infrequently and beta estimates are biased upward for securities trading very frequently. To correct this bias they proposed a consistent estimator of beta given by the following equation: 1 o 1 ( ) i i i, (1) i (1 2 ) 1m where 1 = an estimate of the parameter derived from the simple regression between the observed i o security return and market index return with one lag, = an estimate of the parameter derived from i the simple regression between the observed security return and the corresponding market index return, 1 i = an estimate of the parameter derived from the simple regression between the observed security return and the market index return with one lead, and = the first-order serial correlation 1m coefficient of market returns. Moreover, using daily data from the New York and American Stock Exchange, Scholes and Williams (1977) formed for each year in their study portfolios of securities according to their total trading volume during the year. Their results revealed that the OLS beta estimates for securities trading infrequently are biased downward and for securities traded frequently are biased upwards. The theoretical work of Cohen et al (1983a) proposed a model that uses many leads and lags of the markets return as opposed to the Scholes and Williams model, which is based only upon one lead and one lag of the markets return 8. Their model provides the following consistent estimator of beta 9 : N N i 1 i n 1 i n n 1 i n N N n 1 m,m n n 1 m,m n where betas of the security i are obtained by separate regressions using the OLS method, and m,m n, m,m n = the n-order serial correlation of market returns with n and +n implying a lag and lead of n, respectively 10. Next, McInish and Wood (1986) examined the ability of the models of Scholes and Williams (1977) and Cohen et al (1983a) to reduce the bias in beta estimates. They concluded that each model provides a slight reduction of the amount of bias in beta estimates. Dimson (1979) argues that in a thin market beta can be estimated by aggregating the slope coefficients of the following multiple regression: L R a R u (3) it i k L i k mt k it where R it the return of security i over the return interval t, t=1,2,3,,t, R mt k = lagged, (2) contemporaneous and leading market index returns, k=-l,, L, and u it The beta estimator of Dimson can be expressed as: L i k L i k a random variable. (4) See Cohen et al (1983a), Equation 5, p Cohen et al (1983b) presented two approaches that eliminate the intervalling-effect bias in betas estimated using short return measurement intervals. Fowler and Rorke (1983) derived Equation (2) for N =2.

4 International Research Journal of Finance and Economics - Issue 13 (2008) 111 The values of L are chosen employing information about the degree of thinness of the security and/or the market index. Dimson proposed that the coefficient i k, k=-l,,l, all be simultaneously estimated using Equation (3) as opposed to the models of Scholes and Williams (1977) and Cohen et al (1983a) where the coefficients are estimated independently 11. Using monthly returns of shares traded on the London Stock Exchange he found in OLS beta estimates biases generated by infrequently trading. In another study, Hawawini (1983) presented some evidence indicating that security beta estimates change as the return measurement interval is lengthened and provided a simple model showing that security beta estimates depend upon the length of the return measurement interval. The model estimates beta as follows: (5) where i(t) = the securitys i estimated beta over return intervals of T-day length, i(1) = the securitys i estimated beta over return intervals of 1-day length, p im-1, p im, p im+1 = the intertemporal crosscorrelation coefficient of order 1 (lag), 0, and +1 (lead), respectively, between the security and the market returns measured over intervals of 1-day length, and p mm+1, p mm-1 = the autocorrelation coefficients of order +1 and -1, respectively, on the market daily returns 12. In point of fact, from Equation (5) it can be seen that the estimated beta over return measurement intervals of 1-day can be utilized to estimate beta for a T-period return measurement interval. Also it is evident from Equation (5) that the estimated security beta will be invariant to the length of the return measurement interval only if the intertemporal cross-correlations between the security and market returns are zero and if the market returns provide a zero autocorrelation. Lastly, his model predicted the size and the direction of change in beta when the return interval changes 13. Handa, Kothari & Wasley (1989) used security data from the NYSE and provided market-value ranked portfolio beta estimates via the market model with returns measured over one day, one week, one month, two months, one quarter, four months, six months, and one year. They have reported results revealing that portfolio betas change with the return measurement interval. In particular, they found that beta estimates of small market-value portfolios increase with the return measurement interval, while beta estimates of large market-value portfolios decrease with the return measurement interval. Simply stated, they concluded that such changes occurred because the covariances of security returns with the market index returns and the variance of the market index returns do not change proportionately as the return interval is changed 14. In a subsequent study, Corhay (1992) examined the intervalling-effect bias in beta using securities of the Brussels Stock Exchange and three adjacent periods each comprising of three years. For each period, he formed ten market value portfolios and considered various lengths of returns measurement intervals. He estimated portfolio beta employing the market model and he performed statistical tests for the equality between portfolio beta means. His results indicated the existence of an intervalling effect. To be specific, the effect is quite large for short return measurement intervals and it decreases when he used longer return measurement intervals The estimator of Dimson (1979) shown in Equation (4) is incorrect and cannot generally produce a consistent beta estimate (see Cohen et al (1980), footnote p.274, and Fowler and Rorke (1983)). The corrected form of the estimator of Dimson is the model proposed by Cohen et al (1983a). The model of Hawawini (1983) assumes zero leads and lags of higher orders than one. Also Hawawini (1980) provides a generalization of Equation (5) above. This can be seen by considering the first differential of i(t) with respect to T. Handa, Kothari & Wasley (1993) tested the impact of the return interval on the CAPM and rejected the model when monthly returns are used but when annual returns are employed they failed to reject the model.

5 112 International Research Journal of Finance and Economics - Issue 13 (2008) The empirical work of Frankfurter et al. (1994) obtained data from the Center for Research in Security Prices NYSE-AMEX daily file and estimated security betas using daily, weekly, monthly quarterly, semiannually and annually returns. Using the whole data set of securities they presented evidence showing that the mean beta and the standard deviation of betas increase as the return measurement interval increases. The discussion proceeds with the work of Beer (1997) who investigated the ability of three beta estimation models, the market model using the OLS method, the model proposed by Scholes and Williams (1977) and the model of Dimson (1979). She employed securities from the Brussels Stock Exchange (BSE) and estimated security betas with the OLS method and the whole sample period. After grouping the securities in 10 portfolios according to their market capitalization she computed the beta for each portfolio as the mean of the betas of its securities. Her results revealed statistically significant portfolio betas which decrease substantially as we move from large capitalization portfolios to those of small capitalization. This non-trading effect bias is similar to previous studies using data from other stock exchanges 15. Using the method of Scholes and Williams (1977) she found that the values of the lagged betas do not increase with the infrequency of trade. These findings contradict the results of Scholes and Williams. Employing the method of Dimson (1979) and the Bayesian correction introduced by Vasicek (1973) she found less biased betas only in the case when one lead and one lag are used. Her results suggested that the market model with the OLS method may be the best technique to produce beta estimates of securities traded on the BSE 16. The impact of the return measurement interval on estimated betas was examined by Brailsford & Josev (1997) using daily, weekly and monthly data of securities listed on the Australian Stock Exchange for a 4-year period from January 1988 to December After forming a low-cap portfolio and a high-cap portfolio they presented evidence showing that the mean beta of the former rises while the mean beta of the latter falls as the return measurement interval is lengthened. The ability of the Hawawini model to predict beta was also tested and they concluded that his model performs well in approximating OLS betas especially in predicting the OLS betas for the high-cap portfolio 17. Finally, the study conducted by Daves, Ehrhardt & Kunkel (2000) investigated the return interval and estimation period the financial managers should employ for the beta estimation. They used security returns from CRSP NYSE/AMEX databases and they found that daily returns produce a smaller standard error of the estimated beta than the weekly, bi-weekly or monthly returns. In view of these finding they concluded that the financial managers should use daily returns since they provide greater precision in the estimation of beta. They also reported the mean standard errors of estimated betas for eight estimation periods ranging from one-year to eight years. Their results showed that the utilization of an estimation period of three years captures most of the maximum reduction in the standard error of the beta estimate as compared to other periods. II. Sample and Research Methodology The present study employs a time-series sample of security returns. We confined our attention to continuously listed firms on the main market of the ATSE during a sample period beginning on January 2001 and ending on December Given this objective 187 securities were initially selected with continuous data during the 4-year period. The longer the period we select for estimation the more information we have and consequently the more precise the estimate of beta. However, by going back further in time, the likelihood that structural characteristics of the firm have changed (e.g. firms during time recapitalize, spin- off or acquire divisions e.g.) increases and so the beta we obtain may have little relevance with the systematic A similar result was found earlier by Hawawini and Michel (1974) employing securities from the Brussels Stock Excahnge Fung et al. (1985) used Paris Bourse data and argued that the model of Scholes and Williams (1977) under-adjusts for the intervalling effect bias when beta are estimated from daily returns. Fowler at al. (1980) also investigated thin trading and beta estimation problems using data from the Toronto Stock Exchange. This sample criterion probably introduces a survival bias. We believe that this survival bias is unlikely to produce a systematic bias on the results.

6 International Research Journal of Finance and Economics - Issue 13 (2008) 113 risk of the firm as it exists today. Facing such a dilemma, we believe that the chosen 4-year period is appropriate. Before proceeding further, it would be useful to pause briefly and discuss the technique of security return calculation. For each security in the sample daily, weekly and monthly prices were extracted and the return on a security i at the end of period t is calculated by: R ln(p d ) ln(p ) it it t it 1, where ln = the natural logarithm operator, P it = the last traded price for security i in period t (price were adjusted for capital changes), P it 1 = the last traded price for security i in period t-1 adjusted to the same base, and d it = the dividend for security i declared xd during the period t. The present study uses the Athens Stock Exchange Composite Index (ATSECI) which is a market value weighted index composed of 60 securities adjusted for capitalization changes and for dividend payments. The firms in the sample are sorted on the basis of market capitalisation on December 29 of 2000 and then are used to form two portfolios one comprising of 30 securities having the largest capitalization and the other containing 30 securities with lowest capitalization. We call these portfolios high-cap portfolio and low-cap portfolio, respectively. Incidentally, the average market capitalization of firms in the high-cap portfolio was 209,229,947 euros and in the low- cap portfolio was 23,239,443 euros 19. For each portfolio in our study and each return measurement interval the mean percentage zero returns and the corresponding standard deviations appear in Table 1. Virtually, t-tests reveal quite clearly that the mean percentage zero returns is significantly different between the high-cap and lowcap portfolios for all the returns measurement intervals at 1% level of significance. Namely, these results suggest that the low-cap portfolio constitutes a good proxy for a portfolio comprising of infrequent trading securities. Table 1: T-tests between the mean percentages of zero security returns for two portfolios and three return measurement intervals a b c d Daily return interval Biweekly return interval Monthly return interval High- cap portfolio a Mean percentage of zero returns 7.7% b 2.1% c 1.0% d Standard deviation 3.3% 1.6% 1.4% Low- cap portfolio a Mean percentage of zero returns 18.2% 4.2% 2.4% Standard deviation 7.3% 2.9% 2.3% T-test value For each security in the high-cap portfolio and low-cap portfolio its zero returns are calculated as a percentage of its total daily, biweekly and monthly return observations. Then for each portfolio and return measurement interval mean percentage zero returns and standard deviations are calculated. There are 996 observations. There are 105 observations There are 48 observations. Briefly, security beta is estimated via the following standard market model with the OLS method 20 : R R e, mt it i i it (6) The statistic of the t-test regarding the equality between these two averages of market capitalizations was Thus, the difference in average market capitalization between these two portfolios is significant at 1% level of significance. Daily estimates are based on 996 observations, biweekly estimates are based on 105 observations and monthly estimates are based on 48 observations.

7 114 International Research Journal of Finance and Economics - Issue 13 (2008) where R it the return on security i in period t, R mt = the return on the market index in period t,, the parameters of the model, and e the disturbance term. As usual, the disturbance terms i i it are assumed to be normally distributed, to have zero means, constant variances, Cov(R, e ) 0, mt it Cov(e, e ) 0, t = 1,2,3,T 21. it it 1 Basically, beta estimates are obtained using Equation (6) and the OLS method with alternatively daily, biweekly and monthly returns, the Scholes and Williams (1977) methodology (see Equation (1)) with daily returns, and the Cohen et al (1983a) methodology (see Equation (2)) with daily returns. The beta of each portfolio in our study is computed as the mean of its security betas. III. Empirical Results and Implications In this section special attention is devoted to the examination of the intervalling-effect bias in OLS betas estimates, the investigation of the usefulness of Hawawini (1983) model to estimate betas and the comparison of the abilities of the models of Scholes and Williams (1977), Cohen et al (1983a) and market model for beta estimation. IIIa. Beta estimation using the OLS technique and different return intervals For each security in the high-cap and low-cap portfolios betas are estimated using the entire sample period with daily, biweekly and monthly return measurement intervals and the technique of the OLS. Table 2 presents various descriptive statistics of the portfolio betas for each return measurement interval. Focusing on the high-cap portfolio, the mean beta estimates increase as the return measurement interval is lengthened but such changes are not statistically significant at the 1% level of significance (see Table 3). 21 In view of the market model, the coefficients i and are independent of the length of the return measurement interval. i

8 International Research Journal of Finance and Economics - Issue 13 (2008) 115 Table 2: Summary statistics of beta estimates and R 2 for two portfolios and three return measurement intervals a b c d e Daily return interval Biweekly return interval Monthly return interval High- cap portfolio Mean beta Standard deviation of beta estimates Mean standard error of beta estimates Maximum beta Minimum beta Range ,787 Coefficient of skewness of beta estimates a T-Test 2,286 0,966 1,301 Coefficient of kurtosis of beta estimates b T-Test 1,096-0,338 0,082 Mean R 2 c,d Low- cap portfolio Mean beta Standard deviation of beta estimates Mean standard error of beta estimates Maximum beta Minimum beta Range Coefficient of skewness of beta estimates T-Test -0,020-0,054-0,082 Coefficient of kurtosis of beta estimates T-Test -3,319-3,309-3,072 Mean R 2 e,g The coefficient of skewness is calculated by dividing the third central moment of security betas by the standard deviation to the power of 3. The variance of the coefficient of skewness is approximately 6/30. Hence, we calculate the t-value by dividing the coefficient of skewness by its standard deviation. The coefficient of kurtosis is calculated by dividing the forth central moment of security betas by the standard deviation to the power of 4. The variance of the coefficient of kurtosis is approximately 24/30. Hence, we calculate the t-value by dividing the coefficient of kurtosis by its standard deviation. For the high- cap portfolio the difference between the mean R 2 using daily returns and the mean R 2 using biweekly returns is not significant at 1% significance level (t = 1.92). For the high-cap portfolio the difference between the mean R 2 using daily (biweekly) returns and the mean R 2 using monthly (monthly) returns is significant at 1% significance level (t = 2.81 (t = 4.39)). For the low- cap portfolio all differences between the mean values of R 2 are significant at 1% significance level (for the comparison of the mean R 2 using daily returns and the mean R 2 using weekly returns, t = 4.19, between daily returns and monthly returns, t = 6.47, and between biweekly returns and monthly returns, t = 2.89). Looking closely at Table 2 we see that the mean R 2 values for the high- cap portfolio increase as the return measurement interval lengthens and these values are higher than the corresponding values of the low-cap portfolio. This result is consistent with the findings of Dimson (1979) and Brailsford and Josev (1997). What is more important, for the high-cap portfolio the mean R 2 obtained using daily or biweekly returns is significantly different from that calculated from monthly return at 1% level of significance, a result that also supports the presence of the intervalling effect 22. Inspection of Table 2 shows that the mean beta estimate for the low-cap portfolio rises as the return interval lengthens. This result is consistent with previous evidence from other markets. The mean beta estimate of this portfolio using daily returns is and its beta based on monthly returns is an increase of 23%. From Table 3 it can be seen that the difference between the mean beta estimates is significant only for the return measurement interval daily-to-monthly. This finding indicates the presence of the intervalling-effect in estimated betas when we use daily returns instead of monthly returns. For the low-cap portfolio, it is also observed that the ATSECI explains, on average, only 18.6% of the variation in returns when daily returns are used and rises to 36.3% when monthly 22 The mean R 2 of the high-cap portfolio is ranged from 42.8% to 52.6%. It is noted that the majority of the securities that form the high- cap portfolio are member of the ATSECI, so the correlation between the returns of the securities that compose the high- cap portfolio and those of the ATSECI is strong.

9 116 International Research Journal of Finance and Economics - Issue 13 (2008) returns are employed. For the low-cap portfolio every difference between two mean values of the R 2 is statistically significant at 1% level of significant, a result that provides the basis for inferring the existence of a strong intervalling effect 23. Table 3: Testing the equality of beta mean estimates between series for two portfolios a b Biweekly return interval Monthly return interval High-cap portfolio Daily Difference between the means T-test a a Biweekly Difference between the means T-test a Low-cap portfolio Daily Difference between the means T-test a b Biweekly Difference between the means T-test a The null hypothesis of equality of beta mean estimates cannot be rejected at 1% significance level. It is assumed both equal and unequal sample variances. The null hypothesis of equality of beta mean estimates is rejected at 1% significance level. It is assumed both equal and unequal sample variances. Note that the beta estimates for the high-cap portfolio and the low-cap portfolio are moving in the same direction. This result appears to contract the evidence provided by Brailsford and Josev (1977) showing that betas estimates of high capitalized firms decrease as the return measurement interval is lengthened while betas estimates of low capitalized firms increase as the return measurement interval is lengthened. We now turn to discuss the ranges of betas. Based on the results in Table 2 we clearly observe an increase in the range between the minimum and maximum values of beta as the return measurement interval is lengthened. In effect, the greater range is obtained in the case of the low cap- portfolio when monthly returns are used. This finding is consistent with the finding of Brailsford and Josev (1977). It might be useful at this stage, to note that the mean standard error of the beta estimates increase as the return measurement interval is lengthened for both portfolios (see Table 2). This result can be explained by noting that the number of observations used in the OLS regressions decreases when the length of the return measurement interval increases given the fixed sample period of four years. It is helpful to note that, similar results have been found by Handa, Kothari & Wasley (1989) for security data from the NYSE and by Brailsford and Josev (1977) for securities listed on the Australian Stock Exchange. To this end, another observation from the results obtained using the OLS method is that the magnitude of the intervalling effect is inversely related to the market value of the firms. Specifically, the OLS betas obtained under daily or even biweekly returns are underestimated for both portfolios but the rate of change in beta is greater for the low- cap portfolio. Thus, the results support the dependence and the intensity of the intervalling effect on the market value of the firms. A similar conclusion reached by Corhay (1992) for securities listed on the Brussels Stock Exchange. Additionally, in Table 2 are displayed some results related to the distributions of betas. For the high-cup portfolio except for daily beta estimates, all other beta distributions are not skewed. For the same portfolio the coefficients of kurtosis are not statistically significant at 1% level of significance. However, for the low-cup portfolio, all beta distributions are not skewed but all the coefficients of kurtosis are statistically significant at 1% level of significance. We finally present some concluding remarks. Table 2 reveals that the mean beta estimates and the mean R 2 for both portfolios increase as the return measurement interval increases indicating the 23 We employed the technique proposed by Newey and West (1987) in order to obtain a covariance estimator that is consistent in the presence of both heteroskedasticity or autocorrelation of unknown form on the residuals. Also, Durbin- Watson statistics from the OLS regressions are around 2 indicating no serial correlation on the residuals.

10 International Research Journal of Finance and Economics - Issue 13 (2008) 117 presence of an intervalling effect. A possible explanation of why beta shifts as the return measurement interval lengthens is provided by the argument that the full impact of information is not immediately reflected into prices because of price adjustment delays, but the impact of this phenomenon is reduced with longer return measurement intervals since prices incorporate much of the relevant information (see also Scholes and Williams (1977) and Cohen et al. (1983a)). Consequently, it seems reasonable to conclude that when monthly returns for OLS beta estimation are used less bias is introduced. IIIb. Testing the Hawawini (1983) model Having examined the intervalling-effect bias in OLS estimated betas, we are now ready to apply the model of Hawawini (1983) to estimate security betas for biweekly and monthly return intervals using OLS betas estimated from daily returns. Indeed Table 4 shows the means of estimated betas for the high-cap and low-cap portfolios. The results indicate that the Hawawinis model estimates well the portfolio beta for longer return intervals for the high-cap portfolio. The difference of the mean betas obtained under OLS and estimated mean betas are not statistically significant at 1% level of significance. In addition, for the low-cap portfolio, the difference between OLS beta estimates and estimated betas over a return interval of two-weeks with Hawawinis model is statistically insignificant at 1% level of significance. Table 4: Predicted Hawawini betas for biweekly and monthly return intervals using OLS betas estimated from daily returns Biweekly return interval Monthly return interval High- cap portfolio Hawawini mean beta OLS mean beta T-statistic a b Low- cap portfolio Hawawini mean beta OLS mean beta T-statistic c d a,b,c The null hypothesis of equality of means can not be rejected at 1% significance level. d The null hypothesis of equality of means can not be rejected at 5% significance level, but it can be rejected at 1% significance level. Table 5: Estimating Hawawini betas for monthly return interval using OLS betas estimated from biweekly returns Monthly return interval High- cap portfolio Hawawini mean beta OLS mean beta T-statistic a Low- cap portfolio Hawawini mean beta OLS mean beta T-statistic b The null hypothesis of equality of means can not be rejected at 1% significance level. a,b The model of Hawawini is also used to estimate security betas for a return interval of a month employing betas estimated with the OLS method and biweekly returns. As it can be seen by inspecting the results of Table 4 the performance of Hawawinis model is good for both portfolios when biweekly returns and the OLS method are used. Going still further, Hawawini (1983) argues that his model can also predict the direction of change in beta estimates resulting from the change in the return interval. It is appropriate here to

11 118 International Research Journal of Finance and Economics - Issue 13 (2008) present the first differential of i(t) 24 with respect to T as ( ) (1)[q - q ] i i im m = dt 2 [T + (T - 1)q ] m (Hawawini (1983) p. 76) 25. Quite logically, then, if the difference between the securitys q im ratio and the markets q m ratio is positive (negative), beta will rise (fall) as longer return measurement intervals are used. According to Hawawini securities with large (small) market value will have low (high) q im ratios in comparison to the corresponding ratios of the market. At this point, taking into account biweekly returns we observe the following results. For the high- cap portfolio the difference q im - q m is negative. As a consequence of this fact, the model estimates a decreasing beta as we use longer return intervals. On the other hand, for the low- cap portfolio the difference q im - q m is positive, and thus the model estimates an increasing beta as we employ longer return intervals. When daily returns are used the following observations are made. For the low- cap portfolio the difference q im - q m is positive and thus the model estimates an increasing beta for each one of the longer return intervals. Also, for the high- cap portfolio the difference q im - q m is positive and thus the model estimates an increasing beta for each one of the longer return intervals. These findings are consistent with the results of Hawawini (1983). Before continuing with the last part of our investigation we are now in a position to conclude that the Hawawinis model estimates well the beta value for longer return intervals for both portfolios. However, when it is used to predict the direction of change in beta provides a better performance in the case of the low- cap portfolio as compared with that of the high-cap portfolio. IIIc. Beta estimation using daily return interval and the models proposed by Scholes and Williams (1977) and Cohen et al (1983a) In this section our aim is to investigate whether betas estimated with the market model via the OLS method differs significantly from those estimated via the models proposed by Scholes and Williams (1977) and Cohen et al (1983a). We applied these corrective procedures only to daily returns data for both portfolios since the thin trading problem is particularly serious with daily data and is reduced as longer return intervals are employed. For the high-cap (low-cap) portfolio the difference between the mean beta estimated using the OLS method for each security and the mean beta obtained using the model proposed by Scholes and Williams (1977) for each security is statistically insignificant (see Table 6). In addition to the model of Scholes and Williams, the model of Cohen et al (1983a) is applied with two (three, four) leads, two (three, lags) lags and one contemporaneous markets index return. The results included in Table 6 show that the difference between the mean beta estimated using the OLS method for each security and the mean beta provided employing the model of Cohen et al (1983a) is statistically insignificant. From our results we see that the methods of Scholes and Williams (1977) and Cohen et al (1983a) do not improve the biases of the OLS method when two (three, four) leads, two (three, four) lags and one contemporaneous markets index return are used. Thus, in these cases the OLS method may be the method to employ for beta estimation when considering infrequently traded securities in the Athens See Equation (5) in Section I. where q im p im +1 + pim -1 p im and q 2p. m mm 1

12 International Research Journal of Finance and Economics - Issue 13 (2008) 119 Stock Exchange 26 (a similar conclusion reached also by Beer (1997) for securities traded on the Brussels Stock Exchange). Table 6: Comparing OLS beta estimates and betas estimated with the models of Scholes and Williams (1977) and Cohen et al (1983a), daily returns OLS Scholes and Williams (1 lead & 1 lag) Cohen et al (2 leads & 2 lags) Cohen et al (3 leads & 3 lags) High- cap portfolio Mean beta Standard deviation of beta T-statistic a c c c Low- cap portfolio Mean beta Standard deviation of beta Cohen et al (4 leads & 4 lags) T-statistic b d d d a,b The difference between the mean beta obtained under OLS employing daily returns and the mean beta obtained with the and Williams model with one lead, one lag and one contemporaneous markets index return is insignificant for both portfolios at 1% significance level. c,d The difference between the mean beta obtained under OLS employing daily returns and the mean beta obtained with the Cohen et al model with two leads (three leads, four leads), two lags (three lags, four lags), and one contemporaneous markets index return is insignificant for both portfolios at 1% significance level. According to Scholes and Williams (1977) the beta coefficients and the t-tests of the lagged (lead) betas decrease (increase) as we move from low-volume portfolios to high volume portfolios. To examine whether this result is valid in our data we divided the 30 securities of the high-cap (low-cap) portfolio into six distinct groups each comprising of five securities. The shares of each portfolio were ranked by their market capitalisation and assigned to distinct groups. The first group contains the largest market value securities and the last group are contains the smallest market value securities. From the results presented in Table 7, we see that the above inference made by Scholes and Williams (1977) is not validated with data from the ATSE. This result is in line with that of Beer (1997) employing a number of securities listed on Brussels Stock Exchange. Table 7: Lists of betas (, -1 and +1) that compose Scholes and Williams beta and their corresponding t- tests. Ranking of securities as of the Mean T- -1 Mean T- +1 Mean T- Mean Mean Mean capitalization at 29 December 2000 Statistic Statistic Statistic High- cap portfolio 1 st group a nd group rd group th group th group th group Low- cap portfolio 7 th group th group th group th group th group th group a The 1 st group includes the first five securities with the highest market capitalizations, the 12 th group contains securities with the lowest market capitalization. 26 This conclusion relies only on the number of leads and lags used in the present work.

13 120 International Research Journal of Finance and Economics - Issue 13 (2008) IV. Summary and Conclusions The present work investigates the intervalling-effect bias in OLS estimated betas for securities listed on the main market of the Athens Stock Exchange. This impact was tested by using two extreme portfolios one of high capitalized securities and one of low capitalized securities. The results support the presence of the intervalling-effect bias and the dependence of its intensity on the market value of the firms. Specifically, for the high-cap portfolio the change in mean beta as we use longer return intervals is not statistically significant. On the other hand, for the low- cap portfolio, the difference between the mean beta value using daily returns and monthly returns is statistically significant. The results using the Hawawinis model indicate a good performance for estimating the beta value for longer return intervals for high-cap portfolios. However, for the low- cap portfolio we observed a relatively poor performance when we employed daily returns to estimate monthly beta. Thus, the ability of the model using the low-cap portfolio is good only for near-term estimations. The Hawawinis model was also used to predict the direction of change in beta for longer return measurement intervals. The results are consistent with Hawawinis argument only in the case where biweekly returns are used. Lastly, using the Hawawini model and daily returns to predict the direction of change in beta as the return interval lengthens, we present findings that are consistent with Hawawinis argument only for the low- cap portfolio. Finally, the paper examines whether beta estimated via the market model with the OLS method (using daily returns) differs significantly from that estimated via the models proposed by Scholes and Williams (1977) and Cohen et. al (1983a). The results reveal that there are no statistically significant differences between the mean beta estimated using the OLS method and the mean beta obtained employing the model of Scholes and Williams (1977). Additionally, the market model with the OLS method and the model of Cohen et. al (1983a) do not provide statistically significant beta differences when two (three, four) leads, two (three, four) lags and one contemporaneous markets index return are used.

14 International Research Journal of Finance and Economics - Issue 13 (2008) 121 References [1] Altman E., Jacquillat B. and Lavasseur M. (1974), Comparative analysis of risk measures: France and the United States, Journal of Finance, 29, pp [2] Beer F. M. (1997) Estimation of risk on the Brussels Stock Exchange: Methodological Issues and Empirical Results, Global Finance Journal, 8, pp [3] Blume M., (1971), On the assessment of risk, Journal of Finance, 26, pp [4] Blume M. (1975), Betas and the regression tendencies, Journal of Finance, 30, pp [5] Bos T. and Newbold P. (1984) An empirical investigation of the possibility of stochastic systematic risk in the market model, Journal of Business, 57, pp [6] Brailsford J.T. and Josev T. (1997) The impact of the return interval on the estimation of systematic risk, Pacific-Basin Finance Journal, 5, pp [7] Brooks R., Faff R.and Lee J. (1992),The form of time variation of systematic risk: Some Australian evidence, Applied Financial Economics, 2, pp [8] Castagna A., Greenwood L., and Matolcsy Z. (1984), An evaluation of alternative methods for estimating systematic risk, Australian Journal of Management, 9, pp [9] Chan I. and Lakonishok J. (1992) Robust measurement of beta risk, Journal of Financial and Quantitative Analysis, 27, pp [10] Cohen K. J., Hawawini G. A.,, Maier S. F., Schwartz R. A., and Whitcomb D. K. (1980) Implications of Microstructure Theory for Empirical Research on Stock Price Behaviour, Journal of Finance, 35, pp [11] Cohen K. J., Hawawini G. A., Maier S. F., Schwartz R. A., and Whitcomb D. K. (1983a), Friction in the trading process and the estimation of systematic risk, Journal of Financial Economics, 12, pp [12] Cohen K. J., Hawawini G. A.,, Maier S. F., Schwartz R. A. and Whitcomb D. K. (1983b) Estimating and Adjusting for the Intervalling- Effect Bias in Beta, Management Science, 29, pp [13] Corhay A. (1992), The intervalling effect bias in beta: A note, Journal of Banking and Finance, 16, pp [14] Daves R. P., Ehrhardt C. M., and Kunkel A. Robert (2000) Estimating systematic risk: The choice of return interval and estimation period, Journal of Financial and Strategic Decisions, 13, pp [15] Diacogiannis G. P. (1989) Stationary of Beta Forecast: Some Evidence for the London Stock Exchange, Spoudai, pp [16] Dimson E. (1979) Risk measurement when shares are subject to infrequent trading, Journal of Financial Economics, 7, pp [17] Dimson E. and Marsh P. (1983) The stability of UK risk measures and the problem of thin trading, Journal of Finance, June, pp [18] Emanuel D.M. (1980), The market model in New Zealand, Journal of Business Finance and Accounting, Winder, pp [19] Eubank A. A. and. Zumwalt J.K, (1970), An analysis of the forecast error impact of alternative beta adjustments techniques and risk classes, Journal of Finance, June, pp [20] Fama E. F. (1970), The behavior of stock market prices, Journal of Business, 38, pp [21] Fisher L. (1966), Some new stock market indices, Journal of Business, 39, pp [22] Fowler D. J and Rorke H. C. (1983) Risk measurement when shares are subject to infrequent trading: Comment, Journal of Financial Economics, 12, pp [23] Fowler D. J., Vijay M.J., and Rorke H. C. (1980) Thin trading and beta estimation problems on the Toronto Stock Exchange, Journal of Business Administration, 1,pp [24] Frankfurter G, Leung W. and Brockman P. (1994) Compounding period length and the Market Model, Journal of Economics and Business, Volume 46, pp [25] Fung, W.K.H, Schwartz R.A. and Whitcomb, D.K. (1985), Adjusting for the intervalling effect bias in beta: A test using Paris Bourse data, Journal of Banking and Finance, 9, pp

15 122 International Research Journal of Finance and Economics - Issue 13 (2008) [26] Handa P., Kothari S. P. and Wasley C. (1989) The relation between the Return Interval and Betas: Implications for the size effect, Journal of Financial Economics, 23, pp [27] Handa P., Kothari S. P. and Wasley C. (1993) Sensitivity of Multivariate Tests of the Capital Asset- Pricing Model to the Return- Measurement Interval, Journal of Finance, 48, pp [28] Hawawini G. A. (1980), Intertemporal cross dependence in securities daily returns and the short-run intervalling effect on systematic risk, Journal of Financial and Quantitative Analysis,15, pp [29] Hawawini G. A (1983) Why beta shifts as the return interval changes Financial Analysts Journal, 39, pp [30] Hawawini G. A and Michel A. (1974) An assessment of risk in thinner markets: The Belgian case, Journal of Economics and Business, pp [31] Hill J. and Schneeweis T. (1979) The effect of interval selection on the parameters of the market model as applied to bond returns, Financial Review, Fall, pp [32] Klemkosky R, C. and Martin J. D. (1975), The adjustment of beta forecast, Journal of Finance, September, pp [33] Levhari D. and Levy H. (1977), The capital asset pricing model and the investment horizon, Review of Economics and Statistics, 49, pp [34] Newey W. and West K. (1987) A simple positive semi-define, heteroskedasticity and autocorellation consistent covariance matrix, Econometrica, 55, [35] Ng K. W. (1981) Theoretical and empirical evaluation of alternative procedures for estimating betas on nonsychronous data, M.S. thesis, University of Manchester Institute of Science and Technology, Manchester. [36] McInish T. H. and Wood R.A. (1986), Adjusting for beta bias: An assessment of alternative techniques; A note, Journal of Finance, 1, pp [37] Pogue G. and Solnic B. (1974), The market model applied to European common stock: Some empirical results, Journal of Financial and Quantitative Analysis,9, pp [38] Roll R. (1981), A possible explanation of the small firm effect, Journal of Finance, 36, pp [39] Scholes, M. and Williams, J. (1977) Estimating betas from non-synchronous data, Journal of Financial Economics, 5, pp [40] Smith K. V. (1978), The effect of intervalling on estimating parameters of the capital asset pricing model, Journal of Financial and Quantitative Analysis,13, pp [41] Theobald M. (1980), The analytic relationship between intervalling and non-trading effects in continuous time, Journal of Financial and Quantitative Analysis,18, pp