Theory and - perhaps more importantly - financial common sense suggest that there should be a trade-off between a stock’s riskiness and its expected returns. On the one hand, standard asset pricing models suggest that systematic risk should be positively rewarded, i.e. stocks with higher betas should earn a higher expected return (see Ross’s Arbitrage Pricing Theory, 1976). Subsequently, research has underlined the explanatory power of stock-specific or so-called idiosyncratic risk for expected returns (Merton, 1987). Taken together, these results suggest that total volatility, which is the model-free sum of systematic volatility explained by a factor model, and idiosyncratic volatility, should also be positively rewarded (Martellini, 2008).

In contrast to this consensus regarding the existence of an unambiguously positive risk-return relationship from a theoretical perspective, a number of older as well as more recent papers have reported a number of puzzling, or at least, contrasted findings from an empirical perspective. First, the “low beta anomaly” stipulates that the relationship between systematic risk as measured by a stock beta and return is much flatter than predicted by the CAPM (see early papers by Black, 1972, Black, Jensen, and Scholes, 1972) and sometimes even inverted (paper by Haugen and Heins, 1975). More recently, Ang, Hodrick, Xing, and Zhang (2006, 2009) have drawn new attention to these results with a focus on the specific risk component, finding that high idiosyncratic volatility stocks have had "abysmally low returns" in longer U.S. samples and in international markets. This result is now widely known as the “idiosyncratic volatility puzzle”. Yet other papers have documented a rather flat or even negative relationship between total (as opposed to specific) volatility and expected return, an anomaly that some call the “total volatility puzzle” (Haugen and Baker, 2008, Blitz and Van Vliet, 2007, Baker, Bradley, and Wurgler, 2011).

Several attempts have been made to explain these puzzling empirical results. A number of recent papers have questioned the robustness of Ang et al.’s (2006, 2009) results. Among other concerns, the findings are not robust to changes in data frequency, portfolio formation period, to the screening out of illiquid stocks (Bali and Cakici, 2008), or to adjusting for short-term return reversals (Huang et al., 2010). Several other authors have changed the short-term measure of volatility in Ang et al. (2006) with conditional measures estimated from returns over longer calibration periods and found a positive relationship (Fu, 2009, Brockmann and Schutte, 2007).

In order to understand the risk-return relationship further, recent research at EDHEC-Risk Institute adopts a long-horizon perspective. Rather than looking at return realisations over a monthly horizon to proxy for expected returns, we look at longer investment horizons beyond one year, which are arguably closer to horizons relevant for a typical institutional equity investor. Taking a long-horizon perspective seems to be the natural approach since the theoretical predictions of standard asset pricing models relate to the relationship between a stock’s risk estimate and expected return on that stock across many varying market conditions. This can only be assessed by looking at a stock performance over long horizons. A similar approach is taken by Bandi and Perron (2008) in recent work on the long-term risk-return relationship, with a focus on the time-series perspective. In a related effort, from a cross-sectional perspective, Bandi et al. (2010) find empirical support for an approximate long-run version of the CAPM, where betas and returns are both measured over long horizons.

Below we report results obtained using a broad cross-section of US stocks over the period from July 1963 to December 2009. First, we replicate the short horizon findings in the early literature on the idiosyncratic volatility puzzle. To study the effect of idiosyncratic volatility on long horizon expected returns, we use a simple trading strategy similar to that used by Jegadeesh and Titman (1993). This trading strategy sorts stocks into portfolios every month by their volatility and holds the portfolios over longer horizons of up to three years to allow expected return differences to materialise¹.

Figure 1: Risk-return profile for portfolios using long and short horizon The following graphs show the arithmetic average return and average risk of decile portfolios built on idiosyncratic volatility (relative to Fama-French Factor exposures). The values plotted are the average values over all cross sections and are annualised. The bounds of the one-standard-deviation error are shown along the return axes, along with the average values. The top graph replicates the short horizon results of Ang et al. (2006) while the bottom graph uses a longer horizon of 24 months. The period of analysis runs from July 1963 to December 2009.

From the short horizon results, we can see that the high volatility portfolio has negative returns over the following month. As the graph shows, the finding of a negative relation between risk and return can be largely attributed to the portfolios of high volatility stocks. The finding of a negative relation does not hold when comparing returns across the first six portfolios. This is similar to the findings of Ang et al. It is also well known (Huang et. al. 2010) that the short horizon underperformance of high volatility stocks can be attributed to the short-term return reversal effect. The cap-weighting scheme employed to weight stocks within the portfolio further increases the exposure to short-term return reversals as it overweights the past winners. Using equally-weighted portfolios (which are not sensitive to past returns) and holding the portfolios for longer horizons avoids short-term reversal effects and provides a clear positive risk-return relationship across all the portfolios.

In order to show the horizon effect more clearly we show the geometric mean returns of the high volatility and low volatility portfolios over different horizons in Table 1. The low volatility portfolio yields higher return over a one-month horizon (i.e. the risk-return relationship appears to be negative). But holding the stocks for longer horizons shows the opposite relationship (high volatility portfolios have higher returns and lower volatility portfolios have lower returns).

Table 1: Geometric mean returns for multi-portfolio analysis The following table shows the results using geometric averaging of portfolio returns over the whole analysis period. The annualised returns of the high and low portfolios are provided for both risk measures. The period of analysis runs from July 1963 to December 2009.

Taking the risk-return relationship beyond the mean-variance setting, theoretical models have also shown that investors are willing to accept lower expected returns and higher volatility compared to the mean-variance benchmark in exchange for higher skewness and lower kurtosis of returns. High skewness and low kurtosis have been shown to be associated with lower expected returns in theory (see e.g. Barberis and Huang, 2004, Mitton and Vorkink, 2007, among many others). The intuition behind this result is that investors like to hold portfolios with positive skew and low kurtosis. In terms of idiosyncratic and total risk measures, Boyer, Mitton and Vorkink (2010) and Conrad, Dittmar and Ghysels (2008) also provide converging empirical evidence that individual stocks’ skewness and kurtosis are indeed positively related to future returns.

To assess the impact of higher-order risk measures along with volatility we performed a multivariate regression analysis. This assessment is potentially important to take into account any link between volatility, skewness and kurtosis. Also, using such analysis we can test the joint impact of all of these risk measures. We run a monthly regression by using volatility, skewness and kurtosis as the independent variables. Table 3 shows the average slope coefficient estimate and R-squared, over all the cross sections, as well as the autocorrelation-adjusted t-statistics.

Table 2: Multi-variate regression results using three risk measures Each month we run stock-level regressions (similar to a Fama-MacBeth stock regression) using the stock returns as the dependent variable and the idiosyncratic/total risks (volatility, skewness, and kurtosis) as the independent variables. The stock returns used are the average monthly returns for next twenty-four months and the risk measures use the historical daily data of the previous twelve months. This table shows the average values of the coefficients of regression and R-squared values over all of the cross sections. Newey-West is used to correct the t-stats. The period of analysis runs from July 1963 to December 2009.

The regression results confirm the strong positive risk/return relationship for volatility and skewness . The effect of kurtosis is insignificant when used along with volatility and skewness². On the whole, these multivariate results suggest that the greatest effects of risk on expected stock returns stem from volatility and skewness.

Although the results of past research into the cross-sectional relationship between idiosyncratic/total risk and expected stock returns are puzzling, the results are not universal and the puzzle exists only as a short-term effect that depends on how we go about measuring volatility and its effects. Various authors have shown that the risk-return relation is positive, for example using value-at-risk instead of volatility (Bali et al., 2004) and after adjusting for reversal effects (Huang et al., 2010). Overall, the case for a negative relationship is not only contrary to common sense and theory but also weak empirically given the opposing evidence in empirical papers. Our findings suggest that the ambiguous nature of research into the risk/return trade-off may be accounted for in part by the horizon used in that research. Our results provide evidence that, although there may well be short-term anomalies of higher risk not leading to higher expected returns, the trade-off between risk and return plays out over longer horizons much as is posited by financial theory.

Footnotes

1. Multiple portfolios are held at the same time and only those that arrive at their horizon are rebalanced in a given month.
2. For skewness, we expect a negative relationship since negatively skewed stocks are riskier.

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