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Distribution-Invariant Risk Measures, Entropy, and Large Deviations

Part of: Manifolds Sampling theory, sample surveys Mathematical economics Sufficiency and information

Published online by Cambridge University Press:  14 July 2016

Stefan Weber
Stefan Weber*
Affiliation:
Cornell University
*
∗ Postal address: School of Operations Research and Industrial Engineering, Cornell University, 279 Rhodes Hall, Ithaca, NY 14853, USA. Email address: [email protected]
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Abstract

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The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.

Keywords

Risk measureaverage value at riskshortfall riskMonte Carlolarge deviations principleSanov's theoremrelative entropy

MSC classification

Primary: 91B30: Risk theory, insurance 49Q20: Variational problems in a geometric measure-theoretic setting 62B10: Information-theoretic topics 62D05: Sampling theory, sample surveys
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 16 - 40
Copyright
Copyright © Applied Probability Trust 2007 

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