Simple risk measure calculations for sums of positive random variables

- A multivariate generalized beta distribution is presented for positive losses.
- Marginals follow a second kind beta distribution and can be are heavy-tailed.
- Sums of dependent losses are easily derived in this model.
- Risk measures for the sum of marginals have simple expressions.
- Spreadsheet calculation is illustrated using operational risk data.

Closed-form expressions for basic risk measures, such as value-at-risk and tail value-at-risk, are given for a family of statistical distributions that are specially suitable for right-skewed positive random variables. This is useful for risk aggregation in many insurance and financial applications that model positive losses, where the Gaussian assumption is not valid. Our results provide a direct and flexible parametric approach to multivariate risk quantification, for sums of correlated positive loss distributions, that can be readily implemented in a spreadsheet.