Conditional generalized quantiles as systemic risk measures: Properties, estimation, and application

The conditional $L^1$-quantile, or simply conditional quantile, is vital for measuring systemic risk, i.e., the risk that the distress experienced by one or more financial markets spreads to the others. One may formulate conditional quantile-based value-at-risk (CoVaR), but it depends only on the probability of loss occurrence. Alternatively, one may define conditional expectile-based value-at-risk (CoEVaR) or $L^2$-CoVaR, but it is too sensitive and thus unrobust to extreme losses. In this paper, we aim to construct a generalized measure of systemic risk, called $L^p$-CoVaR, based on conditional $L^p$-quantiles when the conditioning risks are measured using $L^p$-VaR, where $p\ge 1$. We find that the $L^p$-VaR and $L^p$-CoVaR are coherent for all linear portfolios of elliptically distributed losses and are asymptotically coherent at high confidence level for independently and identically distributed losses with heavy right-tail. In addition, we determine their estimators and the respective asymptotic properties. In particular, we perform the $L^p$-CoVaR estimation using multivariate copulas, enabling us to link marginal risk models and capture their complex dependence. Our Monte Carlo simulation study demonstrates that the $L^p$-VaR and $L^p$-CoVaR estimators with $1<p<2$, respectively, exhibit relatively better (conditional) coverage performance than the VaR and CoVaR as well as EVaR and CoEVaR estimators. Furthermore, our empirical study based on cryptocurrency return data with the best-fitting dependence model having heavy-tailed margins and tail dependence structures validates this result. It also confirms that the $L^p$-VaR and $L^p$-CoVaR with $1<p<2$ are, respectively, not as insensitive as VaR and CoVaR and not as sensitive as EVaR and CoEVaR to an extreme loss. Their less conservativity compared to the respective VaR and CoVaR at high confidence level is practically important for determining required capital reserves.