تجزیه و تحلیل تجربی از تجارت بازده ریسک نامطلوب در فرکانس روزانه

This paper considers the downside-risk aversion of investors as an explanation for the risk-return trade-off. We test empirically this hypothesis using intraday data along with the recent measure of downside-risk called realized semivariance developed in Barndorff-Nielsen et al. (2010). The empirical analysis over the period 1996–2008 provides evidence of a significant relation between semivariance and excess returns at the daily frequency. To gain better understanding of the relation between returns and downside-risk, we investigate the statistical relation between a new measure of conditional asymmetry, namely the ratio of the downside realized semivariance over the variance, and obtain a revealing pattern using a rolling window framework able to link asymmetry in the distribution to future returns. In particular, the asymmetry measure becomes significant when the past realized variance is not significant any more thereby providing insights about a possible change in the behavior of investors.

The risk-return trade-off is one of the most pressing question in financial economics. Ghysels et al. (2005) even coin it as the “first fundamental law in finance”. This is also one of the main puzzles in finance. The puzzle arises from the fact that the ICAPM is not supported empirically. For instance, French et al. (1987) using the data from 1920 to 1984 do not provide evidence of a statistically significant relation between risk and returns. Recently, it has been suggested that the relation may only exist in the (very)-long term (Lundblad, 2007 and Bandi and Perron, 2008, etc.). Conversely, Bali and Peng (2006) have shown that it could exist in the (very)-short term, namely at the daily level. In this paper, we examine the predictive power of the semivariance estimator for future returns. We show that this estimator allows to establish a robust relation between (downside-)risk and return in the short run. We thus show that investors' downside-risk aversion may partly explain the empirical puzzle and confirm this finding using a new measure of asymmetry for the distribution of returns. The problem of the empirical non-recovering of a significant relation for the theoretical ICAPM may have different sources: (i) the inadequacy of the ICAPM itself, (ii) risk preferences of agents who may not only be risk-averse but also exhibit preferences towards higher moments and/or (iii) the inaccuracy of the volatility proxy and the methodology to forecast volatility considered in the literature. In this paper, as in most of the existing literature, we assume the validity of the ICAPM and suggest an empirical approach which deals with the two other issues, by considering both semivariance based on intraday data and a recent method for volatility forecasting. We then test the ICAPM using this nonparametric estimator1of conditional semivariance relying on intraday data. Thus, we both consider the downside-risk aversion of investors and the measure of downside-risk itself along with an adequate model of volatility forecasting. We finally combine this measure with the original measure of realized volatility to estimate the asymmetry of the returns' distribution and show that this measure is significant most of the time in our sample. Importantly, it is more significant when the realized volatility is not. The financial literature has stressed the importance of considering higher order moments in pricing securities (see Dittmar, 2002, Friend and Westerfield, 1980 and Lim, 1989 among others). The literature on the relation between the asymmetry in the distribution and expected returns is rooted in the seminal contribution of Arditti (1967) who showed using a Taylor series expansion how to derive the price risk of the third moment. The preference for a positive skewness in returns modifies standard portfolio choice theory when only first and second moments are under consideration. In a series of papers, alternative portfolio choice theories are developed which consider moments beyond variance.2This may also modify the equilibrium under consideration as in Kraus and Litzenberger (1976), Friend and Westerfield (1980) or Lim (1989). This leads investors to purchase less diversified portfolio than recommended by mean-variance based theory (see Mitton and Vorkink, 2007). Recently, Chabi-Yo et al. (2008) have derived a four-moment model to extend the two-fund separation theorem of Tobin. The recent contribution by Barberis and Huang (2008) illustrates the trade-off between risk and skewness for investors. In their model, agents have preferences following prospect theory along the lines of Kahneman and Tversky (1979, 1992). Assets with positive skewness are favored by investors who consider “stocks as lotteries” thereby holding nonoptimal portfolios in the Markowitz sense and favoring high-reward/low-probability assets. Brunnermeier et al. (2007) conclude that “The desire for skewness can also impact the market return. If bad aggregate states have low probabilities, as for disasters or peso problems, then it is possible for the desire for skewness to increase the equity premium as investors seek to avoid negative skewness.”(p. 164). Our estimation strategy, which relies on intraday (transaction data) and the estimator developed in Barndorff-Nielsen et al. (2010) (BNKS hereafter), has the strong advantage to use a very robust proxy for the latent volatility as shown in a series of papers since Andersen and Bollerslev (1998). Estimation of volatility using intraday data has only a short but very intense history. Zhou (1996) extended the idea by French et al. (1987) following Merton's (1980) argument to build the first estimator. Barndorff-Nielsen and Shephard (2002) gave theoretical foundations to this literature. The main idea is to use the theory of quadratic variation which states that the sum of squared intraday returns delivers an estimate of the current variance of a stochastic process which is more and more precise the finer is the partition used to compute the intraday returns (infill asymptotics). BNKS (2010) use the same idea but only consider signed (negative or positive) intraday returns to estimate the downside realized semivariance. In words, the concept of downside semivariance is related to the variability of a given process when only signed changes of this process are considered, i.e. when the process exhibits downside moves. These “realized” estimators of volatility are now widely used in various economic contexts (see Tauchen and Zhou, 2011; or Wright and Zhou, 2009 among many others) and for various financial markets (see for instance Liu and Wan, 2012 who investigate the empirical properties of price volatility in the Shanghai fuel oil futures). Another way to estimate volatility is naturally by using option prices. A number of recent studies have investigated the relation between returns and skewness using model-free implied skewness or asymmetry either at the cross-sectional level (Conrad et al., forthcoming; Cremers and Weinbaum, 2010, Rehman and Vilkov, 2012 and Xing et al., 2010) or, as we do, at the aggregate level (Atilgan et al., 2010 and Santa-Clara and Yan, 2010).3In these papers, the relative price of put options vis-à-vis call options is shown to provide some information about future returns.4The so-called “smirk” (difference between the out-of-the-money put and the at-the-money call implied volatility), a measure of ex ante asymmetry in the return distribution, is agood predictor of future returns thereby proving the relevance of risk-neutral measure for the risk-return debate. The important role of VIX in estimating a risk-return relationship has recently been highlighted in Kanas (2012). Our empirical results for the S&P 500 index futures over the period 1996–2008 indicate that downside-risk aversion is important in predicting future returns, at least as much as risk aversion. In particular, in the same vein as Patton and Sheppard (2011), we obtain comparable estimates for the realized variance and for the downside realized semivariance. This has implications for our understanding of the risk-return tradeoff as well as for the significance of the various components of realized variance – negative and positive realized semivariances being two of them.5To further test the downside-risk aversion, we use a new measure of conditional asymmetry derived from realized semivariance and variance and show that this measure help in recovering the risk-return tradeoff in recent years. The rest of the paper is organized as follows. In the next section, we present shortly the estimators that we use along with the intraday dataset. In Section 3, empirical results are presented for the different specifications using the full sample and then rolling windows. Section 4 proposes the new measure for asymmetry in the distribution of returns and provides an estimation of its significance in a ICAPM-like regression. Section 5 concludes.

In this paper, we are interested in the empirical evaluation of the risk-return trade-off using alternative measures of risk, namely downside-risk. We show that realized semivariance provides similar results than realized variance in the day-ahead framework. Other proxies could be considered. For instance, in a recent paper, Anderson et al. (2009) suggest that uncertainty, measured as the degree of disagreement between professional forecasters may help to recover the relation between risk and return. Bollerslev et al. (2009b) show that the variance risk premium has the leading explanatory power for the risk-return trade-off. Atilgan et al. (2010) have shown that a risk neutral measure of skewness is significant most of the time and Chang et al. (2013) provide similar findings for the cross-section of stock returns. Santa-Clara and Yan (2010) used option implied moments and provide similar evidence. All these measures aim at validating the ICAPM without relying on additional explanatory variables beyond the empirical moments (see the discussion in Lettau and Ludvigson, 2009 about such additional variables). A possible extension of the present work might consider such variables to increase the explanatory power of our regressions which remains, as in previous similar analysis, very low. In addition, several macroeconomic variables have forecasting power for future volatility (see recent evidence in Paye, 2012 and Christiansen et al., 2012) which may help in recovering a risk-return trade-off empirically. Also, considering information embedded in options is now common practice (see Chang et al., 2013; or Conrad et al., forthcoming) and may be viewed as a complementary analysis to the present one that is based on moments from the real (empirical) distribution only. This is also one limit of our study. Another possible extension would be to consider several international markets as in Ghysels et al. (2011) or Bollerslev et al. (2011) to improve further our understanding of the relationship between downside-risk and returns.