Realized Semicovariances

Abstract

We propose a decomposition of the realized covariance matrix into components based on the signs of the underlying high‐frequency returns, and we derive the asymptotic properties of the resulting realized semicovariance measures as the sampling interval goes to zero. The first‐order asymptotic results highlight how the same‐sign and mixed‐sign components load differently on economic information related to stochastic correlation and jumps. The second‐order asymptotic results reveal the structure underlying the same‐sign semicovariances, as manifested in the form of co‐drifting and dynamic “leverage” effects. In line with this anatomy, we use data on a large cross‐section of individual stocks to empirically document distinct dynamic dependencies in the different realized semicovariance components. We show that the accuracy of portfolio return variance forecasts may be significantly improved by exploiting the information in realized semicovariances.

From zero to hero: Realized partial (co)variances

“…These measures naturally link to up-and down-side variances, and the long history of these types of measures in finance (e.g., Warren (1972, 1974), Fishburn (1977) and Ang, Chen and Xing (2006)). Bollerslev et al (2020a) extend the univariate semivariance measures to the multivariate context with the notion of realized semicovariances.…”

“…Meanwhile, the challenge posed by Frank in his No Hesitation blog relates explicitly to the forecast improvements afforded by the use of the semicovariance measures reported in Bollerslev et al (2020a). Hence, to directly address Frank's questions, we next extend the above univariate analyses to a multivariate context.…”

“…The complications associated with market microstructure "noise" is further compounded in the multivariate setting due to asynchronous prices, and the so-called Epps effect. To address this issue, for the stoicks analyzed here, which includes some smaller-cap and less liquid stocks, Bollerslev et al (2020a) advocated the use of a coarse 15-minute sampling frequency in the calculation of their realized semicovariance measures. While this sparse sampling ef-fectively alleviates the biases stemming from market microstructure "noise," it obviously also reduces the accuracy of the estimates compared to the estimates that would obtain with more finely sampled returns in the absence of any "noise."…”

“…Underscoring the practical usefulness of this decomposition from a volatility forecasting perspective, Patton and Sheppard (2015) find that embedding the realized semivariances within the HAR modeling framework results in significantly improved volatility forecasts for aggregate equity index returns, albeit only modest gains for individual equities. Bollerslev et al (2020a) extend this idea to a multivariate setting, and show how realized covariance matrices may be similarly decomposed into four realized semicovariance components based on the signs of the underlying high-frequency returns. They also go on to empirically demonstrate how this four-way decomposition may be used in the construction of improved covariance matrix and portfolio variance forecasts.…”

Tail Heterogeneity for Dynamic Covariance-Matrix-Valued Random Variables: the F-Riesz Distribution

“…Such data is ever more abundant in today's data rich environment, particularly in financial markets. For recent examples, see for instance Andersen et al (2003); Barndor↵-Nielsen and Shephard (2004); Chiriac and Voev (2011); Lunde et al (2016); Callot et al (2017); Bollerslev et al (2018Bollerslev et al ( , 2020 and the references cited therein. Most of the models currently available for these data are highly restrictive.…”