Tail moments and tail joint moments for multivariate generalized hyperbolic distribution

In this paper, we investigate two novel risk measures under the weighted risk aggregation model: Tail Moment (TM) and Tail Joint Moment (TJM). These measures encompass numerous classical risk measures and are capable of quantifying higher-moment risks such as tail skewness and tail kurtosis. Considering the asymmetric and heavy-tailed properties typical of financial and insurance data, we employ the Multivariate Generalized Hyperbolic (MGH) distribution to model risk variables. Within this framework, we derive analytical expressions for the TM and TJM. These results facilitate the precise assessment of portfolio tail risk as well as the tail dependence between risk assets. Furthermore, we present two applications to highlight the benefits and robustness of TM and TJM in risk management and portfolio selection. In the first example, we utilize tail conditional skewness (TCS) and tail conditional kurtosis (TCK) to evaluate the extreme loss risks of assets, which are not typically captured by conventional risk measure such as marginal expected shortfall (MES) and tail variance (TV). In the second example, we concentrate on the dependence of risks in a downside market. Specifically, we use Tail Correlation (TCOR) and Tail Co-skewness (TCOS) to analyze the risk relationships between stocks and the market index during downturns. These risk measures provide crucial insights for portfolio tail risk assessment and hedging against downside market risk.