Fair estimation of capital risk allocation

  • 1 Department of Mathematical Stochastics, University of Freiburg, Eckerstr.1, 79104, Freiburg, Germany
  • 2 Department of Applied Mathematics, Illinois Institute of Technology, 10 W 32nd Str, Building REC, Room 220, Chicago, USA
  • 3 Institute of Mathematics, Jagiellonian University, Lojasiewicza 6, 30-348, Cracow, Poland
Tomasz R. Bielecki
  • Department of Applied Mathematics, Illinois Institute of Technology, 10 W 32nd Str, Building REC, Room 220, Chicago, IL 60616, USA
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, Igor CialencoORCID iD: https://orcid.org/0000-0002-5185-4765, Marcin PiteraORCID iD: https://orcid.org/0000-0003-2469-8864 and Thorsten SchmidtORCID iD: https://orcid.org/0000-0001-9254-4010

Abstract

In this paper, we develop a novel methodology for estimation of risk capital allocation. The methodology is rooted in the theory of risk measures. We work within a general, but tractable class of law-invariant coherent risk measures, with a particular focus on expected shortfall. We introduce the concept of fair capital allocations and provide explicit formulae for fair capital allocations in case when the constituents of the risky portfolio are jointly normally distributed. The main focus of the paper is on the problem of approximating fair portfolio allocations in the case of not fully known law of the portfolio constituents. We define and study the concepts of fair allocation estimators and asymptotically fair allocation estimators. A substantial part of our study is devoted to the problem of estimating fair risk allocations for expected shortfall. We study this problem under normality as well as in a nonparametric setup. We derive several estimators, and prove their fairness and/or asymptotic fairness. Last, but not least, we propose two backtesting methodologies that are oriented at assessing the performance of the allocation estimation procedure. The paper closes with a substantial numerical study of the subject and an application to market data.

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