Mario Brandtner

Kohärente Risikomessung versus individuelle Akzeptanzmengen — Anmerkungen zum impliziten Risikoverständnis des “Conditional Value-at-Risk”

ZusammenfassungDas Konzept kohärenter Risikomessung basiert auf individuellen Akzeptanzmengen, die das individuelle Risikoverständnis des Entscheidungsträgers abbilden. In letzter Zeit hat das spezielle kohärente Risikomaß des Conditional Value-at-Risk zunehmende Beachtung gefunden, insbesondere als Surrogat für den “Industriestandard” des Value-at-risk in der Banken- und Versicherungsaufsicht. Dabei wird regelmäßig nicht geprüft, inwieweit sich das individuelle Risikoverständnis der Entscheidungsträger über den Conditional Value-at-risk abbilden lässt. Der Beitrag charakterisiert die Gestalt der vom Conditional Value-at-risk induzierten Akzeptanzmengen und zeigt, dass der Conditional Value-at-risk wichtige Formen individuellen Risikoverständnisses nicht abbilden kann.AbstractCoherent measurement of risk is based on individual acceptance sets which portray the decision makers individual risk perception. As a specific coherent risk measure, Conditional Value-at-Risk has gained significance in the recent past with a special focus on banking and insurance regulation, and as a surrogate for traditional Value-at-Risk. However, it can not be guaranteed that individual acceptance sets are modelled by Conditional Value-at-Risk. The contribution characterizes the shape of acceptance sets corresponding to Conditional Value-at-Risk, and the restrictions w.r.t. individual risk perception which are induced thereby.

On the (Mis)Use of Conditional Value-at-Risk and Spectral Risk Measures for Portfolio Selection - A Comparison with Mean-Variance Analysis

We study portfolio selection using Conditional Value-at-Risk and, as its natural extension, spectral risk measures instead of the variance. We do not focus only on the derivation of the efficient frontiers, but also consider the choice of optimal portfolios within an integrated framework. We find that spectral risk measures tend towards corner solutions. If a risk free asset exists, diversification is never optimal. Similarly, for risky assets we obtain only limited diversification. The reason is that spectral risk measures are based on a regulatory concept of diversification that differs fundamentally from the reward-risk tradeoff underlying the traditional mean-variance framework.

Entropic risk measures and their comparative statics in portfolio selection: Coherence vs. convexity

Abstract We conduct a decision-theoretic analysis of optimal portfolio choices and, in particular, their comparative statics under two types of entropic risk measures, the coherent entropic risk measure (CERM) and the convex entropic risk measure (ERM). Starting with the portfolio selection between a risky and a risk free asset (framework of Arrow (1965) and Pratt (1964)), we find a restrictive all-or-nothing investment decision under the CERM, while the ERM yields diversification. We then address a portfolio problem with two risky assets, and provide comparative statics with respect to the investor’s risk aversion (framework of Ross (1981)). Here, both the CERM and the ERM exhibit closely interrelated inconsistencies with respect to the interpretation of their risk parameters as a measure of risk aversion: for any two investors with different risk parameters, it may happen that the investor with the higher risk parameter invests more in the riskier one of the two assets. Finally, we analyze the portfolio problem “risky vs. risk free” in the presence of an independent background risk, and analyze the effect of changes in this background risk (framework of Gollier and Pratt (1996)). Again, we find questionable predictions: under the CERM, the optimal risky investment is always increasing instead of decreasing when a background risk is introduced, while under the ERM it remains unaffected.

Consistent modeling of risk averse behavior with spectral risk measures: Wächter/Mazzoni revisited

Wachter and Mazzoni (2013) (W/M) have proposed a procedure to consistently link traditional expected utility (EU)-theory with modern spectral risk measures (SRMs). They construct a corresponding W/M-risk measure which induces the same preference ordering as the decision maker’s initial utility function does. In this note, we revisit W/M’s procedure and show that it violates the axiomatic foundation of the underlying decision rules: Within the general part of the procedure that builds on an auxiliary equivalent probability measure, the emerging W/M-risk measure does not satisfy the axiomatic properties of SRMs. The specific part of the procedure that singles out a preferred equivalent probability measure links two decision rules of which neither does the initial one respect the axioms of EU-theory, nor is the emerging one in line with the SRM-axioms.