Capital allocation and tail central moments for the multivariate normal mean-variance mixture distribution

Battle Count: 133

The paper is highly relevant to risk management aspects of quantitative trading, particularly for portfolio-level capital allocation and tail risk assessment. It provides analytical tools for understanding how individual positions contribute to portfolio tail risk beyond what CTE captures. However, it does not directly address trading strategy development, signal generation, or execution. The insights about tail behavior differences between stocks (e.g., XOM showing negative TCM contribution indicating diversification benefit) are valuable for portfolio construction and risk budgeting in trading desks.

The analytical formulas require careful implementation of recursive computations involving hazard functions, survival functions, and modified Bessel functions (for GH/GIG distributions). The multivariate case requires computing coefficients a0,i, a1,i, a2,i and evaluating tail moments at multiple confidence levels. The numerical stability of recursive formulas for higher-order TCMs may be challenging. Fitting the multivariate GH distribution via EM algorithm adds complexity. However, the R package ghyp provides some infrastructure.

The paper provides complete analytical derivations with all formulas explicitly stated. Numerical illustration uses publicly available stock data (BA, AXP, XOM, CVX from Yahoo Finance) and R packages (quantmod, ghyp). The EM algorithm calibration is referenced to McNeil et al. (2015). All parameters of the fitted GH model are provided. However, no code repository is explicitly linked.