Abstract: Evaluating risk measures, premiums, and capital allocation based on dependent multi-losses is a notoriously difficult task. In this paper, we demonstrate how this can be successfully accomplished when losses follow the multivariate Pareto distribution of the second kind, which is an attractive model for multi-losses whose dependence and tail heaviness are influenced by a heavy-tailed background risk. A particular attention is given to the distortion and weighted risk measures and allocations, as well as their special cases such as the conditional layer expectation, tail value at risk, and the truncated tail value at risk. We derive formulas that are either of closed form or follow well-defined recursive procedures. In either case, their computational use is straightforward.

Abstract: The financial equilibrium of pension funds relies on the appropriate computation of retirement benefits, taking account of future payments and discount rates. Short-term errors in the commitment for retirement benefits, ill-suited investment in the stock market, or improper mixture with pay-as-you-go payments have long-term consequences and may lead the pension fund to insolvency. The differential equation governing the current assets shows the respective weights associated with the error on the interest rate, the error on the extra bonus, and the error made in forecasting mortality. These weights are estimated through simulations. A short follow-up is sufficient to estimate the three errors. A threshold for the extra interest rate to be earned on the financial market is given to counter-balance the extra bonus when mortality is forecast correctly.