استراتژی های مومنتوم مبتنی بر پاداش-خطر شاخص انتخاب بورس

In this paper, we analyze momentum strategies that are based on reward–risk stock selection criteria in contrast to ordinary momentum strategies based on a cumulative return criterion. Reward–risk stock selection criteria include the standard Sharpe ratio with variance as a risk measure, and alternative reward–risk ratios with the expected shortfall as a risk measure. We investigate momentum strategies using 517 stocks in the S&P 500 universe in the period 1996–2003. Although the cumulative return criterion provides the highest average monthly momentum profits of 1.3% compared to the monthly profit of 0.86% for the best alternative criterion, the alternative ratios provide better risk-adjusted returns measured on an independent risk-adjusted performance measure. We also provide evidence on unique distributional properties of extreme momentum portfolios analyzed within the framework of general non-normal stable Paretian distributions. Specifically, for every stock selection criterion, loser portfolios have the lowest tail index and tail index of winner portfolios is lower than that of middle deciles. The lower tail index is associated with a lower mean strategy. The lowest tail index is obtained for the cumulative return strategy. Given our data-set, these findings indicate that the cumulative return strategy obtains higher profits with the acceptance of higher tail risk, while strategies based on reward–risk criteria obtain better risk-adjusted performance with the acceptance of the lower tail risk.

A number of studies document the profitability of momentum strategies across different markets and time periods (Jegadeesh and Titman, 1993, Jegadeesh and Titman, 2001, Rouwenhorst, 1998 and Griffin et al., 2003). The strategy of buying past winners and selling past losers over the time horizons between 6 and 12 months provides statistically significant and economically large payoffs with historically earned profits of about 1% per month. The empirical evidence on the momentum effect provides a serious challenge to asset pricing theory. There is so far no consistent risk-based explanation and, contrary to other financial market anomalies such as the size and value effect that gradually disappear after discovery, momentum effect persists. Stock selection criteria play a key role in momentum portfolio construction. While other studies apply simple cumulative return or total return criterion using monthly data, we apply reward–risk portfolio selection criteria to individual securities using daily data. A usual choice of reward–risk criterion is the ordinary Sharpe ratio corresponding to the static mean–variance framework. The mean–variance model is valid for investors if (1) the returns of individual assets are normally distributed or (2) for a quadratic utility function, indicating that investors always prefer the portfolio with the minimum standard deviation for a given expected return. Either one of these assumptions are questionable. Regarding the first assumption, there is overwhelming empirical evidence that invalidates the assumption of normally distributed asset returns since stock returns exhibit asymmetries and heavy tails. In addition, further distributional properties such as kurtosis and skewness are lost in the one-period mean–variance approach. Various measures of reward and risk can be used to compose alternative reward–risk ratios. We introduce alternative risk-adjusted criteria in the form of reward–risk ratios that use the expected shortfall as a measure of risk and expectation or expected shortfall as a measure of reward. The expected shortfall is an alternative to the value-at-risk (VaR) measure that overcomes the limitations of VaR with regards to the properties of coherent risk measures (Arztner et al., 1999). The motivation in using alternative risk-adjusted criteria is that they may provide strategies that obtain the same level of abnormal momentum returns but are less risky than those based on cumulative return criterion. In previous and contemporary studies of momentum strategies, possible effects of non-normality of individual stock returns, their risk characteristics, and the distributional properties of obtained momentum datasets have not received much attention. Abundant empirical evidence shows that individual stock returns exhibit non-normality, leptokurtic, and heteroscedastic properties which implies that such effects are clearly important and may have a considerable impact on reward and riskiness of investment strategies. The observation by Mandelbrot, 1963, Fama, 1963 and Fama, 1965 of excess kurtosis in empirical financial return processes led them to reject the normal distribution assumption and propose non-Gaussian stable processes as a statistical model for asset returns. Non-Gaussian stable distributions are commonly referred to as “stable Paretian” distribution due to the fact that the tails of the non-Gaussian stable density have Pareto power-type decay. When the return distribution is heavy tailed, extreme returns occur with a much larger probability than in the case of the normal distribution. In addition, quantile-based measures of risk, such as VaR, may also be significantly different if calculated for heavy-tailed distributions. As shown by Tokat et al. (2003), two distributional assumptions (normal and stable Paretian) may result in considerably different asset allocations depending on the objective function and the risk-aversion level of the decision maker. By using the risk measures that pay more attention to the tail of the distribution, preserving the heavy tails with the use of a stable model makes an important difference to the investor who can earn up to a multiple of the return on the unit of risk he bears by applying the stable model. Thus, consideration of a non-normal return distribution plays an important role in the evaluation of the risk-return profile of individual stocks and portfolios of stocks. Additionally, the distributional analysis of momentum portfolios obtained on some stock ranking criteria provides insight in what portfolio return distribution the strategy generates. The evidence on the distributional properties of momentum datasets in the contemporary literature is only fragmentary. Harvey and Siddique (2000) analyzed the relation between the skewness and the momentum effect on the momentum datasets formed on cumulative return criterion. They examine cumulative return strategy on NYSE/AMEX and Nasdaq stocks with five different rankings (i.e., 35 months, 23 months, 11 months, 5 months and 2 months) and six holding periods (i.e., 1 month, 3 months, 6 months, 12 months, 24 months and 36 months) over the period January 1926 to December 1997. Their results show that for all momentum strategy definitions, the skewness of the loser portfolio is higher than that of the winner portfolio. They conclude that there exists a systematic skewness effect across momentum portfolio deciles in that the higher mean strategy is associated with lower skewness. Although we consider a much shorter dataset than Harvey and Siddique, our conclusions are similar for an extended set of alternative reward–risk stock selection criteria. We extend the distributional analyses in that we estimate the parameter of the stable Paretian distribution and examine the non-normal properties of the momentum deciles. Stable Paretian distributions are a class of probability laws that have interesting theoretical and practical properties. They generalize the normal (Gaussian) distribution and allow heavy tails and skewness, which are frequently seen in financial data. Our evidence shows that the loser portfolios have the lowest tail index for every criterion, and that the tail index of the winner portfolio is higher than that of loser portfolio for every criterion. In addition, we also find a systematic skewness pattern across momentum portfolios for all criteria, with the sign and magnitude of the skewness differential between loser and winner portfolios dependent on the threshold parameter in reward–risk criteria. We interpret these findings as evidence that extreme momentum portfolio returns have non-normal distribution and contain an additional risk component due to heavy tails. The part of momentum abnormal returns may be compensation for the acceptance of the heavy-tailed distributions (with the tail index less than that of the normal distribution) and negative skewness differential between winner and loser portfolios. We examine our alternative strategies based on various reward–risk criteria on a sample of 517 S&P 500 firms over the January 1996 to December 2003 period. The largest monthly average returns are obtained for the cumulative return criterion. We also evaluate the performance of different criteria using a risk-adjusted independent performance measure which takes the form of reward–risk ratio applied to resulting momentum spreads. On this measure, the best risk-adjusted performance is obtained using the best alternative ratio followed by the cumulative return criterion and the Sharpe ratio. Following our analysis, we argue that risk-adjusted momentum strategy using alternative ratios provides better risk-adjusted returns than the cumulative return criterion although it may provide profits of lower magnitude than those obtained using the cumulative return criterion. Regarding the comparison of performance among various reward–risk criteria, we find that all alternative criteria obtain better risk-adjusted performance than the Sharpe ratio for our momentum strategy. A likely reason is that the alternative ratio criteria capture better the non-normality properties of individual stock returns than the traditional mean–variance measure of the Sharpe ratio. An important implication of these results concerns the concept of risk measure in that the variance as a dispersion measure is not appropriate where the returns are non-normal and that the expected tail loss measure focusing on tail risk is a better choice. The remainder of the paper is organized as follows. Section 2 provides a definition of risk-adjusted criteria as alternative reward–risk ratios. Section 3 describes the data and methodology. Section 4 conducts distributional analysis of the momentum portfolio daily returns obtained using applied criteria and evaluates the performance of resulting momentum strategies on an independent risk-adjusted performance measure. Section 5 concludes the paper.

In this study, we apply the alternative reward–risk criteria to evaluate a risk-return profile of individual stocks and construct the momentum portfolio. These criteria are based on the coherent risk measure of the expected tail loss and are not restricted to the normal return distribution assumption. Key distinctive properties of alternative ratios are that they only assume finite mean of the individual stock return distribution and can model different levels of risk aversion via different parameters for significance level of the ETL measure that considers different parts of downside risk. Additionally, reward–risk ratio criteria values are computed using daily data which enable them to better capture the distributional properties of stock returns and their risk component at the tail part of distribution. Alternative ratios drive balanced risk-return performance according to captured risk-return profiles of observed stocks in a sample. For the examined 6/6 strategy, although the cumulative return criterion provides the highest realized annualized return of 15.36%, the alternative R-ratio provides a high annualized return of 10.32% and much better risk-adjusted performance than the cumulative return and traditional Sharpe ratio criterion. Distributional analysis of the momentum deciles within a framework of general stable distributions indicates that the stable Paretian distribution hypothesis provides a much better fit to momentum portfolio returns. Moreover, extreme winner and loser decile portfolios have unique characteristics with regards to stable parameter estimates of their returns. We observe a systematic pattern of index of stability for the winner and loser deciles that generally have lower tail index than that of middle deciles. It is not surprising that those assets should also exhibit the highest tail-volatility of the return distributions. This suggests that the winner and loser portfolio returns imply a substantial risk component due to heavy tails when compared to other deciles. The implication is that momentum strategies require acceptance of heavy-tail distributions (with a tail index below two). As a consequence, an investor who considers only the cumulative return criterion for momentum strategy needs to accept heavier tail distributions and greater heavy tail risk than an investor who follows strategies based on alternative reward–risk criteria. Alternative reward–risk strategies exhibit better risk-adjusted returns with lower tail risk. Furthermore, strategies using cumulative return, Sharpe ratio, and STARR ratio criteria require acceptance of negative skewness. The results for risk-adjusted performance of alternative strategies and cumulative return benchmark strategy using the STARR99% ratio for daily spreads confirm that the alternative R-ratio and STARR ratios capture the distributional behavior considerably better than the classical mean-variance model underlying the Sharpe ratio. The Sharpe ratio criterion underperforms based on the cumulative profit and independent risk-adjusted performance measure. The reason behind the better risk-adjusted performance of the alternative ratios lies in their compliance with the coherent risk measure’s ability to capture distributional features of data including the component of risk due to heavy tails, and the property of parameters in the R-ratio to adjust for upside reward and downside risk simultaneously.