An examination of conditional effect on cross-sectional returns: Singapore evidence

Abstract

This article empirically examines the usefulness of beta, firm size, book-to-market equity ratio (B/M) and earnings-to-price ratio (E/P), as risk proxies in explaining the cross-sectional returns in the Singapore stock market under both unconditional and conditional frameworks based on up and down markets. Consistent with previous studies, though beta plays no role under unconditional framework, there is evidence of a significantly positive (negative) risk premium on beta during periods of up (down) markets, supporting for the continuous use of beta as a risk measure. Interestingly, our results show that firm size is the only significant variable in explaining average returns under the unconditional framework but its impact becomes much less under the up and down market conditions. However, significant conditional effect of E/P is found. Although, B/M alone is not significantly conditionally related to returns, in various combinations with beta, it becomes significant and the joint role of beta and B/M, due to an interaction effect between them, has an ‘amplified’ gain in the explanatory power. Our study suggests that beta does not suffice to explain the cross-sectional variations of returns, but it is possible that the joint effect of beta and B/M may be a surrogate as an underlying and more fundamental factor that is missing in the conditional SLB model. We also find evidence that investors in the Singapore stock market react virtually the same to these firm-specific factors and to beta during up and down markets. Our results are robust for both beta-size and size-beta sorting procedures and for both value- and equally weighted market proxies.

Notes

¹ This article does not include every anomaly found in the literature but the most important and well documented ones.

² As noted in Pettengill et al. (1995), one more condition is necessary for a positive trade-off between beta and return. It is that the realized market excess return should be significantly positive on average.

³ Three statistical measures, unsystematic risk, total risk and kurtosis, are significantly, with expected signs, related to realized returns in up market only and they are found to be asymmetric during up and down markets.

⁴ The remaining two statistical measures, beta-squared and skewness, are insignificantly related to realized returns in both up and down markets and also asymmetrical across up and down markets.

⁵ According to major similar studies, the empirical results are not materially affected by the choice of risk-free proxy.

⁶ The purposes of using portfolios instead of individual stocks are: (1) to reduce the errors-in-variables problem [see Blume (1970) for details] and (2) to mitigate the impact of large informational surprises [see Elton (1999) for details], in order to increase the stability of beta estimates.

⁷ Indeed, upon testing portfolios formed by univariate sort and bivariate independent-group sort, the results using independent-group sort are of no material difference to those using the two within-group sorts, while those using univariate and bivariate sorts are quite similar and significantly different in some cases only.

⁸ Fortunately, the estimates are still valid and multicollinearity causes no special problem for statistical inferences associated with the overall regression model.

⁹ In deciding on the number of portfolios formed, we have to insure that there are sufficient stocks in each portfolio during the entire test period. The number of stocks in the tested sample is between 45 and 107 only. Therefore, this portfolio formation can enable us to avoid too few stocks in some portfolios.

¹⁰ Klein and Bawa (1977) suggest that portfolios rebalanced monthly on the basis of firm size reduce estimation risk.

¹¹ One may suggest that running monthly cross-sectional regression based on individual stocks can solve the problem of too small degrees of freedom. However, using data in individual stocks instead of portfolios may encounter another statistical problem. Market beta is first computed on the basis of portfolio and then subsequently assigned to individual stocks within each portfolio. There is concern that this process may dilute the statistical power of beta. At the same time, assigning the precisely measured firm size and accounting data to individual stocks may enhance their role in capturing the cross-sectional variation in returns. Therefore, this study keeps the data in portfolios for analysis.

¹² To save space, these Durbin−Watson results are not reported here but available from the authors upon request.

¹³ For the sake of brevity, we only report the results from all univariate regression models and the bivariate regression models where variable BETA is included.

¹⁴ The other three results are omitted for the sake of brevity but available from the authors upon request.