Risk Management Recursive Copula Aggregation for Market and Credit Portfolios Risk aggregation is a central problem in financial risk management and regulatory capital assessment. While copula-based approaches are widely used due to their tractability and interpretability, standard implementations rely on flat dependence structures that may fail to capture hierarchical interactions across risk components. This paper introduces a hierarchical copula framework for risk aggregation on trees, in which dependence is specified locally at each aggregation node and propagated through recursive rank-based reordering. The approach preserves marginal distributions while enabling flexible and economically interpretable modeling of dependence across multiple aggregation levels. Copula parameters are calibrated from empirical data using Kendalls tau, and the methodology is illustrated using market and credit risk proxies. Robustness is assessed across alternative credit segments, including investment-grade and high-yield indices. Numerical results show that dependence strength increases under stress and varies systematically across credit quality, with important implications for risk measurement. The proposed framework combines the transparency of copula-based aggregation with the structural richness of hierarchical models, providing a practical and scalable tool for enterprise risk management. This contributes to bridging bottom-up and top-down approaches to risk aggregation. Keywords: hierarchical copulae; dependence modelling; Kendalls tau calibration; Gaussian and Clayton copulae; tail dependence; risk aggregation. Key Messages: - Recursive hierarchical copulas aggregate market and credit portfolios. - Empirical dependence calibration is combined with sample reordering. - The framework computes VaR and Expected Shortfall transparently. - Tail dependence significantly affects hierarchical risk aggregation. (author: Eric Dal Moro)

Risk Management Conditional value-at-risk under reward-penalty mechanism with applications to robust portfolio management In this paper, we present robust portfolio selection models by incorporating a reward and penalty mechanism into portfolio management. We assume that the joint distribution of the losses of the underlying risky assets in a portfolio is uncertain but lies within a multivariate distribution set. Our goal is to identify optimal portfolio allocations by minimizing the worst-case conditional value-at-risk (CVaR) of portfolio loss under the reward and penalty mechanism and distribution uncertainty. Our models can also be used to investigate the problem of how to balance portfolio losses with their associated downside risk in portfolio management. We first derive an explicit closed-form expression for the worst-case CVaR under the reward-penalty mechanism, which generalizes several existing models and results regarding the worst-case CVaR, such as those studied in Jagannathan (1977), Chen et al. (2011), and Cai et al. (2024). We then apply this expression to obtain optimal portfolio allocations that minimize the worst-case CVaR under both a classical mean-covariance-based multivariate distribution set and a generalized mean-covariance-based multivariate distribution set introduced in Kang et al. (2019). Additionally, we utilize real market data to illustrate the application of the proposed models and the corresponding optimal portfolio allocations in portfolio management. Our empirical experiments show that portfolios based on the proposed models have the potential to outperform those based on several existing related models. Furthermore, the results demonstrate that incorporating downside risk into portfolio loss helps better manage risk and can achieve higher investment returns than considering either downside risk or portfolio loss alone. Moreover, our experiments reveal the trade-off between improving expected portfolio return and controlling the worst-case portfolio CVaR. (author: Zhiqiao Song)

Risk Management Model Risk Analysis for Entropic Hedging Strategies Entropic risk optimisation is a general framework for pricing and hedging financial derivatives in incomplete markets that can be used to decompose P&L into market and model risk contributions. When the prices are quadratic Gaussian, the coupled equations for price and hedge ratios are solved in closed form. This enables comprehensive analysis of trading P&L, with a decomposition of the model value-at-risk into convexity, dimension and funding contributions that are attributed in the explanation of realised P&L. The equations of the quadratic Gaussian model are directly applicable when the underlying prices follow Gaussian processes, such as fractional Ornstein-Uhlenbeck processes. The model also provides simple parametric expressions for hedge ratios that can be used for regressions in deep hedging. (author: Paul McCloud)