Variance Risk: A Bird’s Eye View∗ Fabian Hollstein† and Chardin Wese Simen‡ January 11, 2018 Abstract Prior research documents a significant variance risk premium (VRP) for the S&P 500 index but only for few equities. Using high-frequency data, we show that these results are affected by measurement errors in the realized variance estimates. We decompose the index VRP into factors related to the VRP of equities and the correlation risk pre- mium. The former mostly drives the variations in the index VRP while the latter mainly captures the level of the index VRP. The two factors predict excess stock returns in the time-series and cross-section, but at different horizons. Together, they improve the return predictability. JEL classification: G11, G12 Keywords: Correlation Swaps, Realized Variance, Return Predictability, Variance Swaps ∗We are grateful to Arndt Claußen, Maik Dierkes, Binh Nguyen, Marcel Prokopczuk, Roméo Tédon- gap and seminar participants at Leibniz University Hannover for helpful comments and discussions. Part of this research project was completed when Chardin Wese Simen visited Leibniz University Hannover. †School of Economics and Management, Leibniz University Hannover, Koenigsworther Platz 1, 30167 Hannover, Germany. Contact: [email protected]. ‡ICMA Centre, Henley Business School, University of Reading, Reading, RG6 6BA, UK. Contact: [email protected]. 1 Introduction Carr & Wu(2009) and Driessen et al.(2009) document a significantly negative av- erage variance swap payoff (VSP) for the U.S. stock market index.1 However, they find significant VSPs only for a small number of individual equities that make up the stock index. This finding appears surprising at first glance since the stock market index is simply a portfolio of the constituent stocks and raises the question: where does the index VSP come from? This paper provides a bird’s eye view of the pricing of S&P 500 variance risk. We make four contributions to the literature. First, we investigate whether the findings of the aforementioned studies might be driven by statistical issues. In particular, we formulate and evaluate explanations based on (i) the power of the statistical test, (ii) measurement errors in the variance swap rate and (iii) measurement errors in the realized variance estimate. We find evidence in support of the third explanation. Indeed, when realized variance is computed using daily data, as is done in Carr & Wu(2009), Buraschi et al. (2014b) and Dew-Becker et al.(2017), the average 1-month VSP of individual equities is positive and statistically indistinguishable from zero (t-stat = 0.82). However, when realized variance is accurately measured using high-frequency data, the average VSP of individual equities becomes negative and statistically significant (t-stat = −1.99). Additionally, the proportion of stocks for which the null of an insignificant VSP can be rejected rises from around 20 % to nearly 40 % when we compute realized variance using high-frequency, as opposed to noisy daily, data. We thus provide evidence of an upward bias in the realized variance estimates based on daily data. This bias is more pronounced 1Throughout this paper, we refer to the variance swap payoff as the difference between the realized variance, computed ex-post, and the risk-neutral expectation of future variance. We also analyze the variance risk premium, defined as the spread between the physical and risk-neutral expectations of future variance. 1 for firms with low market capitalizations, high leverage and illiquid stocks. Second, we decompose the index VSP into (i) a factor that depends on the variance swap payoffs of the individual equities that make up the index and (ii) a factor that is a function of the correlation swap payoff. Our decomposition enables us to quantify the contribution of each of the two factors to the (i) level and (ii) variance of the index VSP. Empirically, the two factors have distinct dynamics, with the first factor being less persistent and more volatile than the second factor. The factors are only weakly correlated, suggesting that they capture fundamentally different information. The factor related to the correlation swap payoff accounts for most of the average level of the index VSP (70.4 %) while the factor linked to the VSP of individual stocks captures the lion’s share of the variations (64 %) in the index VSP. Third, we go beyond the index variance swap payoff and study the index variance risk premium (VRP). We decompose the index VRP, which we model as in Bollerslev et al.(2009), into an individual VRP factor and a correlation risk premium (CRP) factor. The CRP factor accounts for most of the level of the index VRP, whereas the individual VRP factor is the primary force behind fluctuations in the index VRP. We explore the implications of the uncovered two-factor structure for the predictive power of the index VRP. In particular, we seek to clarify the contribution of the Individual VRP and the CRP factors to the predictability of S&P 500 excess returns over various horizons. Our analyses reveal that both factors are relevant for the predictability of S&P 500 excess returns, but at different horizons. The individual VRP factor is a highly significant predictor of excess returns over short horizons of up to 6 months whereas the predictive power of the CRP factor manifests itself also at longer horizons. Fourth, we explore the cross-section of excess stock returns. Stocks that have a high 2 exposure to the index VRP earn a significantly higher future risk-adjusted return than those that have a low exposure to the index VRP. This positive risk-adjusted return is highly significant at horizons of up to 12 months. Analyzing the two factors that make up the index VRP, we find that they both predict the cross-section of excess stock returns. However, they operate at different horizons. Dependent sort analyses reveal that the Individual VRP factor is a highly significant predictor for horizons of up to 12 months, whereas the CRP factor’s predictive ability is also discernible at horizons such as 24 months. These results suggest that analyzing the index VRP alone may not reveal the full scale of predictability in the cross-section of excess stock returns. We perform several robustness checks. To begin with, we consider alternative sam- pling frequencies for the computation of the realized variance estimates and reach similar conclusions. Additionally, we consider alternative models to forecast realized variance and obtain similar findings. Analyzing the term-structure of the VRP, we show that the CRP factor accounts for most of the level of the index VRP of various maturities. However, the contribution of the CRP factor to the variations of the index VRP rises with the maturity of the variance swap. Our study adds to the literature on measurement errors in the variance and correlation swap payoff estimates.2 Du & Kapadia(2013) establish that the fixed leg of variance swaps computed as in Carr & Wu(2009) is biased in the presence of jumps. 3 Faria & Kosowski(2016) compare the synthetic correlation swap rates to those available in the over-the-counter (OTC) market and show that the two are equal on average. We differ from these studies in that we explore the role of measurement errors in the floating 2Bekaert & Hoerova(2014) analyze the impact of measurement errors in the index variance risk premium. We complement this work by also analyzing realized payoffs. Furthermore, we study correlation swaps and variance swaps related to the index as well as individual stocks. 3See also the works by Andersen et al.(2015), Martin(2017) and the references therein. 3 (rather than fixed) leg of variance and correlation swaps. We leverage developments in the literature on high-frequency financial econometrics (Barndorff-Nielsen & Shephard, 2002; Andersen et al., 2003) to obtain more accurate estimates of the realized variance. Cleaner realized variance estimates lead to more significant VSPs for individual equities and affect the profitability of popular trading strategies.4 For instance, the average payoff to a strategy that sells 1-month index variance swaps each month rises by 80 %, from 0.85 % to 1.53 % when realized variance is accurately measured. Relatedly, the average payoff of a strategy that sells the 1-month correlation swap each month increases by 28 %, from 10.74 % to 13.75 %, when realized variance is accurately measured. These findings suggest that there is a large bias in the estimates reported in the literature. Our paper belongs to the growing literature that dissects the index variance risk pre- mium. Todorov(2010) and Bollerslev & Todorov(2011) decompose the index VRP into components associated with (i) smooth and (ii) discontinuous movements. Bollerslev et al.(2015) show that the compensation for the discontinuous movements plays an im- portant role in the predictive power of the index VRP for aggregate excess stock returns. Kilic & Shaliastovich(2015) and Feunou et al.(2017) decompose the index VRP into up- side and downside parts and study their implications for the predictability of aggregate excess returns. We decompose the index variance risk premium into factors linked to the variance risk premium of stocks and the correlation risk premium. Our findings are relevant for the literature on theoretical models of variance risk. For example, Driessen et al.(2013) propose a model with correlation risk only. The model counterfactually predicts that the variations in the index VRP are solely driven by the CRP factor. In 4The finding of significant average VSPs in stocks is relevant for studies that use implied variance to directly forecast the realized variance of stocks: it might be possible to improve the forecast accuracy by accounting for the variance risk premium in the spirit of Prokopczuk & Wese Simen(2014) and Kourtis et al.(2016).