پیش بینی نشانه های بازده بر اساس ریسک شرطی: شواهدی از شاخص بازار سهام بریتانیا

Recent theoretical works have found a link between return sign forecastability and conditional volatility. This paper compares the predictive performance of the conditional country risk and the conditional residual risk in forecasting the direction of change in the return on the UK stock market index. The conditional country risk and the conditional residual risk are estimated using the bivariate BEKK-GARCH technique and the direction of change in the UK stock market index is modelled using the binary logit approach. Both the in-sample and the out-of-sample predictions suggest that, as a predictor, the conditional residual risk is superior to the conditional country risk. Our findings support the residual risk model while contradicting the traditional capital asset pricing model (CAPM). Moreover, our tactical asset allocation simulations show that when the conditional residual risk is used in conjunction with multiple-threshold trading strategies to guide the investment decisions, the actively managed portfolio achieves greater returns than the return on a buy and hold portfolio.

In their paper, Christoffersen and Diebold (2006) demonstrate a link between the forecastability of asset return sign and asset return volatility. Their findings suggest that information on conditional volatility can be used to produce direction-of-change forecasts of asset returns. Other authors, including Leung et al., 2000 and Nyberg, 2011 for example, provide empirical evidence demonstrating that return sign forecasts are possible and are potentially useful for asset managers who actively allocate their assets between risky and risk-free investments. Although research on asset return sign predictability is still in its infancy, several research papers in this area are worth mentioning. For example, Christoffersen and Diebold, 2006 and Christoffersen et al., 2007 demonstrate a direct link between asset return volatility and asset return sign forecastability. The latter paper also provides empirical evidence suggesting that conditional variance, skewness and kurtosis are all capable of predicting the direction of change in the asset return. Leung et al. (2000) evaluate the efficacy of several multivariate classification techniques relative to a group of level estimation approaches. The classification techniques, which are employed in their work to estimate the probabilities of the direction of change of the return, include linear discriminant analysis, logit, probit and probabilistic neural networks. The authors find that the classification models outperform the level estimation models in terms of predicting the direction of stock market movement and maximising returns from investment trading. Another author who demonstrates that return signs are predictable is Nyberg (2011) who adopts the binary dynamic probit model to investigate the sign predictability of the US stock market. In his paper, the author includes a binary recession indicator, estimated in the first stage using the binary probit model, which is subsequently included in a set of regressors employed in the second stage of the estimation. His findings suggest that the direction of change in the US stock market is predictable with the binary recession indicator being the most useful predictor. This paper takes the work of Christoffersen et al. (2007), which uses a univariate technique to model conditional volatility, one step further by modelling the conditional asset return volatility using the bivariate BEKK-GARCH technique to capture the impact of time-varying asset return correlation on conditional variance. Specifically, we investigate the connection between asset return sign forecastability and conditional asset return volatility using data on the UK stock market index. Our technique also improves upon the work of Nyberg (2011) which calculates the volatility of stock returns as a 1-month rolling window of sum of squared daily stock returns. The bivariate GARCH approach employed in this article has several attractive features that are worth mentioning. Firstly, it takes into account the impact of correlation between the UK stock market and the global stock market on the conditional return volatility thereby enabling us to capture the ‘spill-over’ effects between the two markets on the estimated conditional volatility. Secondly, the conditional return volatility and the conditional return correlation are jointly estimated using maximum likelihood estimation which permits asset return volatility to be a function of its own past values as well as its correlation with the other asset return volatility, thereby allowing correlation to change over time. The implication of time-varying correlation cannot be understated as a large body of research has shown that correlation is indeed unstable and changes over time (see Fabozzi and Francis, 1978, Bos and Newbold, 1984, Faff et al., 1992 and Brooks et al., 2002, for example). Thirdly, as the approach allows for time-varying correlation, it is possible for us to examine whether conditional country risk, which is a measure of time-varying co-movement between the return on a country’s stock market and the return on the world stock market, can be used to forecast the direction of change in the asset return. In order words, we examine whether the main result of the Capital Asset Pricing Model (CAPM), which suggests that a return on the asset should reflect only the undiversifiable risk component (i.e., the systematic risk), is valid when the direction of change in the asset return instead of the level is modelled. Our research method involves two stages. In the first stage, the bivariate BEKK-GARCH is fitted to the data to estimate the conditional variance–covariance matrices. The conditional country risk (also called the conditional beta or the time-varying beta) and the conditional volatility (also called the conditional residual risk or the conditional idiosyncratic risk) are computed using the information obtained from the conditional variance–covariance matrices. The two measures of risk are subsequently used as explanatory variables in the second-stage estimation. In the second stage, we employ the binary logit model to predict the direction of change in the UK stock market return. In our model, only two possible states for the change in the return are possible: ‘upwards’ and ‘downwards’. A set of diagnostic tests are performed to select a benchmark model whose predictive performance serves as a yardstick to which the predictive performance—both in-sample and out-of-sample—of the other models may be compared. In addition, using the out-of-sample return sign forecasts, we simulate investment returns generated by different trading strategies: the passive ‘buy and hold’ strategy and the active asset allocation strategies guided by the benchmark model, the conditional country risk model and the conditional residual risk model. This type of return simulation is potentially beneficial to asset managers whose job is to allocate their managed assets between risky and risk-free rate investments. Incorporated into our simulations are transaction costs and threshold trading rules employed by Leung et al. (2000). These rules are designed to better utilise the probability forecasts, generated by the binary logit estimations, by taking into account ‘degrees of confidence’ in the forecasting models’ predictive strength. Our results suggest that both the conditional country risk and the conditional volatility are capable of predicting directional changes in returns on the UK stock market. Specifically, the conditional residual risk possesses statistically significant predictive power, generating superior return sign forecasts both in-sample and out-of-sample. Diagnostic tests based on log-likelihood functions and outcome-based predictive performance evaluation both confirm the usefulness of the conditional residual risk in predicting the direction of change in the return on the UK stock market index. Our findings add to the growing literature on the relevance of residual risk to asset pricing. Furthermore, our investment return simulation results suggest that the return on a portfolio managed actively using the probability forecasts obtained from the conditional residual risk model exceeds the return on a passive ‘buy and hold’ portfolio provided that multiple threshold trading rules are employed. The article proceeds as follows. We briefly present the theoretical relationship between asset return volatility and return sign forecastability as well as relevant literature on usefulness of residual risk in Section 2. We then outline our methodology—the BEKK-GARCH technique and the binary logit model—in Section 3. While Section 4 explains the data, we present both the BEKK-GARCH and the binary logit estimation results in Section 5. The in-sample and out-of-sample results, as well as forecast evaluations, are discussed in Section 6. Section 7 deals with trading simulations. Finally, Section 8 concludes the paper.

This paper offers evidence of the usefulness of return volatility in forecasting the direction of change in the asset return. Both in-sample and out-of-sample forecast evaluation indicates that conditional volatility possess statistically significant predictive power and can be employed to forecast the direction of change in asset returns. Our findings are also in line with previously published research that provides support for the role residual risk plays in asset pricing, such as Glosten et al., 1993, Gonzalez-Rivera, 1996, Goyal and Santa-Clara, 2003, Guo and Savickas, 2003, Ghysels et al., 2004, Malkiel and Xu, 2004 and Fu, 2009. They also lend support to previous studies carried out by Bekaert and Harvey, 1995, Carrieri et al., 2007 and Guesmi and Nguyen, 2011 that individual stock markets and the global stock market are segmented as the impact of conditional residual risk on asset returns dominates the impact of conditional country risk. Our methodology involves a two-stage process starting with the estimation of the conditional volatility that takes into account co-movement between returns on the country stock market and returns on the global stock market. This approach yields improvements over other univariate and multivariate approaches that assume constant correlation since market interdependence tends to rise during periods of high uncertainty. Using data on the UK stock market index and the world market index between 1996 and 2011, the conditional volatility is modelled using the BEKK-GARCH approach. This technique ensures that the variance–covariance matrices are positive definite, thus guaranteeing convergence when the model is estimated by maximum likelihood. Using the information obtained from the estimated conditional variance–covariance matrices, we compute the conditional country risk and the conditional residual risk, which subsequently enter the binary logit estimation in the second stage as explanatory variables. In the second stage, we employ the binary logit model to investigate the ability of the conditional country risk and the conditional residual risk to predict the direction of change in the UK stock market index. The binary logit estimation results indicate that the parameters of the output gap, the dividend yield and the conditional residual risk are all statistically significant. The outcome-based in-sample forecast evaluation suggests that, while the conditional country risk possesses little predictive strength, the conditional residual risk appears to be a competent predictor. It correctly forecasts the sign of the return on the UK stock market 125 out of 185 occasions compared to 120 and 123 occasions for the benchmark model and the conditional country risk model. The results from the exclusion tests, which seek to examine the marginal predictive strength of the variable that is excluded from the model, show that the conditional country risk plays very little role in forecasting the direction of change in the return on the UK stock market whereas the conditional residual risk is shown to be an important explanatory variable and should not be excluded from the forecasting model. The out-of-sample forecast evaluation also suggests that the conditional country risk model dominates the other models when used to forecast the sign of return on the UK stock market out-of-sample. Finally, we conduct trading simulations which use the benchmark model, the conditional country risk model and the conditional residual risk model to produce trading recommendations. Our simulations are enhanced by the use of multiple thresholds and incorporate transaction costs that impose a penalty on the portfolio return when assets are shifted between the risky and the risk-free investments. Our simulation results indicate that an actively managed portfolio which follows a trading strategy guided by the conditional residual risk model outperforms portfolios guided by the other trading strategies. Moreover, the portfolio managed using the conditional residual risk model generates the highest Sharpe ratio, suggesting that the trading strategy succeeds in enhancing the portfolio return while reducing the overall level or risk.