Internet Appendix for “On the performance of volatility-managed portfolios”

This internet appendix provides material that is supplemental to the paper “On the performance of volatilitymanaged portfolios. ” Section 1 presents additional empirical results. Section 2 provides analytical results on spanning regression alphas for volatility-managed portfolios. ∗Corresponding author. E-mail addresses: cederburg@email.arizona. edu (Scott Cederburg), odohertym@missouri.edu (Michael S. O’Doherty), wangf10@miamioh.edu (Feifei Wang), xuy219@lehigh.edu (Xuemin (Sterling) Yan). Preprint submitted to Journal of Financial Economics May 2, 2019 1. Supplementary empirical results Section 1.1 presents results on the performance of each of the 103 volatility-managed trading strategies considered in the paper. Section 1.2 details the bootstrap test used in the direct comparisons of volatilitymanaged and original portfolios in Section 3 of the paper. Section 1.3 considers reverse spanning regressions. Section 1.4 examines whether the poor performance for real-time combination strategies is attributable to 5 structural instability in the risk-return relation for the various factors and anomaly portfolios. 1.1. Results for individual trading strategies Table IA1 shows the mean, standard deviation, and Sharpe ratio for the original and volatility-managed versions of 103 trading strategies. We also present results from tests of whether the Sharpe ratio differences are statistically significant. The results in Table IA1 are summarized in Table 2 of the paper. 10 Table IA2 presents results from spanning regressions of volatility-managed portfolio returns on returns for the corresponding original portfolio. The table also details the portfolio properties of the ex post optimal combinations of volatility-managed and original portfolios. The spanning regression results are summarized in Table 4 of the paper. 1.2. Bootstrap analysis 15 Section 3 of the paper focuses on direct performance comparisons of volatility-managed and original strategies. In particular, we examine the Sharpe ratios of volatility-managed and original versions of the factors and anomaly portfolios. Overall, the volatility-managed portfolio outperforms the original portfolio in 53 of 103 comparisons as shown in Table 2. In this section, we describe a bootstrap analysis to examine whether observing 53 positive Sharpe ratio differences across the 103 strategies is statistically significant 20 relative to a null hypothesis. The null hypothesis has a data generating process in which volatility is persistent (as in the data) but there is no risk-return relation. Our bootstrap design generates 100,000 bootstrap samples under the null and tabulates the number of positive Sharpe ratio differences across the 103 strategies for each bootstrap sample. We then examine whether the number of positive performance differences that we observe in the data (53) is statistically significant relative to the bootstrap distribution 25 of the number of positive differences. To better understand the performance of volatility-managed portfolios under the null hypothesis, we introduce a decomposition of the difference between the average returns of the volatility-scaled and original portfolios. Given that the variance of the volatility-scaled portfolio is equal to that of the original portfolio by construction, a positive Sharpe ratio difference is equivalent to the condition that the volatility-scaled portfolio earns a higher average return compared with the original portfolio. The difference in average returns can be decomposed as f̄σ,t − f̄t = cov(wt, ft) + f̄t(w̄t − 1), (IA1)