Thomas Breuer

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, Maximum Loss, and Value at Risk

Some ten years ago, Value at Risk (VaR) set out to conquer the risk management community. Originally intended as a reporting tool for senior management, it soon entered other core areas of banking, such as capital allocation, portfolio optimisation and risk limitation. With its increasing importance, regulators also acknowledged VaR as a risk measure, when they allowed the calculation of capital requirements to be based on VaR. In this case, however, they required that a rigorous and comprehensive stress testing program be in place in order to complement the statistical model (Basle Committee on Banking Supervision, 1996). In this context, two questions arise: (1) Why is there a need to complement VaR models? and (2) What can be regarded as a rigorous and comprehensive stress testing program?

Subjective decoherence in quantum measurements

General results about restrictions on measurements from inside are applied to quantum mechanics. They imply subjective decoherence: For an apparatus it is not possible to determine whether the joint system consisting of itself and the observed system is in a statistical state with or without interference terms; it is possible that the apparatus systematically mistakes the real pure state of the joint system for the decohered state. We discuss the relevance of subjective decoherence for quantum measurements and for the problem of Wigners friend.

Using Quasi-Monte Carlo Scenarios in Risk Management*

We report on the use of quasi-random numbers in searching for worstcase scenarios of security portfolios. A systematic search for the worst-case scenario requires to find the global minimum of the portfolio-value function within a search domain of all plausible scenarios, which usually is an ellipsoid in the high dimensional space of risk factors.

Evolution on Trees: On the Design of an Evolution Strategy for Scenario-Based Multi-Period Portfolio Optimization under Transaction Costs

Abstract Scenario-based optimization is a problem class often occurring in finance, planning and control. While the standard approach is usually based on linear stochastic programming, this paper develops an Evolution Strategy (ES) that can be used to treat nonlinear planning problems arising from Value at Risk (VaR)-constraints and not necessarily proportional transaction costs. Due to the VaR-constraints the optimization problem is generally of non-convex type and its decision version is already NP-complete. The developed ES is the first algorithm in the field of evolutionary and swarm intelligence that tackles this kind of optimization problem. The algorithm design is based on the covariance matrix self-adaptation ES (CMSA-ES). The optimization is performed on scenario trees where in each node specific constraints (balance equations) must be fulfilled. In order to evaluate the performance of the ES proposed, instances of increasing problem hardness are considered. The application to the general case with nonlinear node constraints shows not only the potential of the ES designed, but also its limitations. The latter are basically determined by the high dimensionalities of the search spaces defined by the scenario trees.

An Information Geometry Problem in Mathematical Finance

Familiar approaches to risk and preferences involve minimizing the expectation \(E_{{\mathrm{I}\!\mathrm{P}}}(X)\) of a payoff function X over a family \(\varGamma \) of plausible risk factor distributions \({\mathrm{I}\!\mathrm{P}}\). We consider \(\varGamma \) determined by a bound on a convex integral functional of the density of \({\mathrm{I}\!\mathrm{P}}\), thus \(\varGamma \) may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution \({{\mathrm{I}\!\mathrm{P}}_0}\). Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing \(E_{\mathrm{I}\!\mathrm{P}}(X)\) subject to \({\mathrm{I}\!\mathrm{P}}\in \varGamma \)), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When \(\varGamma \) is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.

Information geometry in mathematical finance: Model risk, worst and almost worst scenarios

The mathematical problem addressed is minimising the expectation of a random variable over a set of feasible distributions P ϵ Γ, given as a level set of a convex integral functional. As special cases, Γ may be an f-divergence or f-divergence ball or a Bregman ball around a default distribution. Our approach is motivated by geometric intuition and relies upon the theory of minimising convex integral functionals subject to moment constraints. One main result is that all “almost minimisers” P ϵ Γ belong to a small Bregman ball around a specified distribution or defective distribution P, equal to the strict minimiser if that exists but well defined also otherwise.

Robustness in quantum measurements

Conditions are formulated under which a representation of an intrinsic C*‐ algebra of (often quasilocal) observables of an infinite system is appropriate to describe measurement‐type processes: such a representation should allow for the description of robust experiments, it should be separable, and the pointer observable should be in its weak closure. If the pointer values are discrete the existence of such a measurement representation can be proven. If the pointer can take continuously many values, then the existence can only be proven under the additional assumptions of having an asymptotically Abelian system or dealing with type I representations. In the constructed measurement representations the pointer observable turns out to be classical. The structure of the representation suggests that spontaneous symmetry breaking might be a physical explanation of the emergence of the classical pointer.

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.

Portfolio selection with transaction costs under expected shortfall constraints

An investor subject to proportional transaction costs allocates funds to multiple stocks and a bank account, to maximise the expected growth rate of the portfolio value under Expected Shortfall (ES) constraints. In a numerical example with ten time steps and one stock important innovations are caused by the introduction of the Expected Shortfall constraint: First, expected returns are reduced by less than one-tenth when the ES constraint is introduced. In comparison, economic capital as measured by ES, is reduced to amounts between one-half and three-quarters, when the ES constraint is introduced. Second, the dependence of expected return and ES on the initial portfolio, in particular when transaction costs are high, is largely removed by the introduction of the ES constraint.