On lower partial moments for the investment portfolio with variance-gamma distributed returns (Q2113612)

The author considers a portfolio of investments \(x_1,\ldots, x_n\) into \(n\) financial assets with value \(X=\sum_{j=1}^n x_jR_j\), where the returns \(R_j\) are variance-gamma distributed, i.e.,\ \(R_j=r_j+\theta_jX_j+\sigma_j\sqrt{X_j}N_j\) with \(r_j,\theta_j\in\mathbb R\), \(\sigma_j\geq0\), and gamma distributed random variables \(X_j\) independent of the (possibly correlated) standard normal distributed random variables \(N_j\). The author derives analytic expressions of the lower partial moments of \(X\) in terms of generalized hypergeometric functions, and as a consequence provides formulas for the value at risk, expected shortfall, and the target semideviation of \(X\).