Imre Csiszár

Measuring Distribution Model Risk

We propose to interpret distribution model risk as sensitivity of expected loss to changes in the risk factor distribution, and to measure the distribution model risk of a portfolio by the maximum expected loss over a set of plausible distributions defined in terms of some divergence from an estimated distribution. The divergence may be relative entropy or another f‐divergence or Bregman distance. We use the theory of minimizing convex integral functionals under moment constraints to give formulae for the calculation of distribution model risk and to explicitly determine the worst case distribution from the set of plausible distributions. We also evaluate related risk measures describing divergence preferences.

Stress Tests: From Arts to Science

Stress tests with handpicked scenarios might misrepresent risks either because the scenarios considered are too implausible or because some dangerous scenarios are not considered. Systematic search for the worst case within some set of plausible scenarios is introduced to overcome these two pitfalls. For arbitrary loss functions we determine explicitly the worst case scenario over Kullback-Leibler spheres of plausible scenarios. Practical implementations of this method do not require any numerical optimisation. The method is illustrated in a number of example applications: linear and quadratic portfolios, stressed credit default probabilities, stressed rating transition correlations.

On convergence rates of finite memory estimators of infinite memory processes

Stationary ergodic processes with finite alphabets are approximated by finite memory processes based on an n-length realization of the process. Under the assumptions of summable continuity rate and non-nullness, a rate of convergence in d̅-distance is obtained, with explicit constants. Asymptotically, as n → ∞, the result is near the optimum.