lower incomplete gamma function, is the Euler-Mascheroni constant. Generalized hyperbolic secant (GHS) distribution If the payoff of a portfolio follows GHS distribution with the p.d.f. and the c.d.f. then the left-tail TVaR is equal to , where is the Spence's function, is the imaginary unit. Johnson's SU-distribution If the payoff of a portfolio follows Johnson's SU-distribution with the c.d.f. then the left-tail TVaR is equal to , where is the c.d.f. of the standard normal distribution. Burr type XII distribution If the payoff of a portfolio follows the Burr type XII distribution with the p.d.f. and the c.d.f. , the left-tail TVaR is equal to , where is the hypergeometric function. Alternatively, . Dagum distribution If the payoff of a portfolio follows the Dagum distribution with the p.d.f. and the c.d.f. , the left-tail TVaR is equal to , where is the hypergeometric function. Lognormal distribution If the payoff of a portfolio follows lognormal distribution, i.e. the random variable follows normal distribution with the p.d.f. , then the left-tail TVaR is equal to , where is the standard normal c.d.f., so is the standard normal quantile. Log-logistic distribution If the payoff of a portfolio follows log-logistic distribution, i.e. the random variable follows logistic distribution with the p.d.f. , then the left-tail TVaR is equal to , where is the regularized incomplete beta function, . As the incomplete beta function is defined only for positive arguments, for a more generic case the left-tail TVaR can be expressed with the hypergeometric function: . If the loss of a portfolio follows log-logistic distribution with p.d.f. and c.d.f. , then the right-tail TVaR is equal to , where is the incomplete beta function. Log-Laplace distribution If the payoff of a portfolio follows log-Laplace distribution, i.e. the random variable follows Laplace distribution the p.d.f. , then the left-tail TVaR is equal to . Log-generalized hyperbolic secant (log-GHS) distribution If the payoff of a portfolio follows log-GHS distribution, i.e. the random variable follows GHS distribution with the p.d.f. , then the left-tail TVaR is equal to , where is the hypergeometric function. References Actuarial science Financial risk modeling Monte Carlo methods in finance