Tail risk in financial markets reflects the probability of extreme financial losses and is typically measured by the tail index of the financial return distribution. A commonly employed method to assess the dynamics of the tail index is the dynamic Peaks over Threshold (PoT) approach, which assumes that exceedances over a high threshold follow a Generalized Pareto (GP) distribution with time-varying parameters. However, this method is sensitive to the choice of the threshold, and deviations from the GP distribution can introduce bias in tail index estimates. To address this issue, we extend the dynamic PoT model to incorporate an extended GP (EGP) distribution, which accommodates departures from the standard GP distribution. Through simulation studies and an empirical analysis using the S&P500 returns, we find that the EGP model provides more reliable assessments of tail risk dynamics, leading to improvements in the stability and accuracy of tail risk estimates in the dynamic setting.

We propose an l1-penalized estimator for high-dimensional models of Expected Shortfall (ES). The estimator is obtained as the solution to a least-squares problem for an auxiliary dependent variable, which is defined as a transformation of the dependent variable and a pre-estimated tail quantile. Leveraging a sparsity condition, we derive a nonasymptotic bound on the prediction and estimator errors of the ES estimator, accounting for the estimation error in the dependent variable, and provide conditions under which the estimator is consistent. Our estimator is applicable to heavy-tailed time-series data and we find that the amount of parameters in the model may grow with the sample size at a rate that depends on the dependence and heavy-tailedness in the data. In an empirical application, we consider the systemic risk measure CoES and consider a set of regressors that consists of nonlinear transformations of a set of state variables. We find that the nonlinear model outperforms an unpenalized and untransformed benchmark considerably.

Minimum-variance portfolios often incur high rebalancing costs and estimation risk, particularly in large asset markets. We propose a novel approach that targets the precision matrix by jointly estimating hedging portfolios under two complementary regularizations. The first controls the complexity of hedging portfolios, while the second shrinks hedging positions without imposing sparsity. Unlike sparse methods, our framework accommodates pervasive factors and multicollinearities, and it produces stable, diversified portfolios without extreme weights with low turnover. In empirical tests across both small and large cross-sections of assets, these portfolios achieve low out-of-sample volatility, and higher net Sharpe ratios than competing methods.