Bart Heijnen

How to estimate the Value at Risk under incomplete information

A key problem in financial and actuarial research, and particularly in the field of risk management, is the choice of models so as to avoid systematic biases in the measurement of risk. An alternative consists of relaxing the assumption that the probability distribution is completely known, leading to interval estimates instead of point estimates. In the present contribution, we show how this is possible for the Value at Risk, by fixing only a small number of parameters of the underlying probability distribution. We start by deriving bounds on tail probabilities, and we show how a conversion leads to bounds for the Value at Risk. It will turn out that with a maximum of three given parameters, the best estimates are always realized in the case of a unimodal random variable for which two moments and the mode are given. It will also be shown that a lognormal model results in estimates for the Value at Risk that are much closer to the upper bound than to the lower bound.

Best upper and lower bounds on modified stop loss premiums in case of known range, mode, mean and variance of the original risk

Abstract If the stop-loss treaty is modified in such a way that the liability of the reinsurer is limited, and if the range, the mode, the mean and the variance of the original risk are known, the best possible upper and lower bounds for the reinsurance premiums are calculated, using only the given information.

General restrictions on tail probabilities

Abstract When limited information on the distribution of a positive random variable X (continuous or discrete) is known (e.g., mode, mean, variance), the tail probability P(X⩾t) cannot be chosen independently. In this paper supremum and infimum for P(X⩾t) will be calculated over the set of positive random variables with unique mode, mean and/or variance given.

A Recursive Scheme for Perpetuities with Random Positive Interest Rates. II: The Impenetrable Wall

In some former contributions, the authors investigated actuarial quantities with stochastic interest rates. In a first model, the randomness is modelled by means of an ordinary Wiener process, and as a consequence negative interest rates are possible. A second model provides a tool to avoid these negative interest rates, which can be necessary in particular situations. This paper wants to present an alternative solution to the problem of negative interest rates. This new model will be implemented to the case of an annuity certain and of a perpetuity.

Perturbation calculus in risk theory: Application to chains and trees of reinsurance

Abstract Several risk exchange models in reinsurance are presented, using ‘chain’ and ‘tree’ hierarchies. To get explicit solutions, a method is used which is a stochastic generalization of perturbation calculus in phycics.

Bounds for the mean system size in M/G/1/K-queues

Contrary to their infinite capacity counterparts, the moments of the distribution of the number in a M/G/1/K-system cannot be determined by means of the Pollaczek-Khinchine equation. If the finite capacity K is small the distribution under study can be obtained as the steady-state probability distribution related to the transition probability matrix. For larger capacities, we derive upper and lower bounds on the mean system size in an M/G/1/K-queue for which the first two moments of the number in the system of the infinite capacity queue are known. Numerical examples for the M/D/1/1-and M/D/1/3-queues are given.