Optimization of conditional value-at-risk

TLDR: In this paper, a new approach to optimize or hedging a portfolio of financial instruments to reduce risk is presented and tested on applications, which focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value at Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well.

Abstract: A new approach to optimizing or hedging a portfolio of nancial instruments to reduce risk is presented and tested on applications. It focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value-at-Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is anyway considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage rms, mutual funds, and any business that evaluates risks. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can be applied also to the optimization of percentiles in contexts outside of nance.

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Integrated Risk-/Return-Management Approach for the Bank Portfolio
Ursula A. Theiler
Abstract
In an intensifying international competition banks are forced to place increased emphasis on enter-
prise wide risk-/return management. Financial risks have to be limited and managed from a bank
wide portfolio perspective. Risk management requirements have to be met from an internal as well
as from a regulatory point of view. Banks need to maximize their expected returns under these
constraints. This leads to a generalized portfolio optimization problem under different capital re-
strictions.
We pursue a two-step Risk-/Return Management Approach (“RRM-Approach”) [3]. At first we
formulate an optimization model that maximizes the expected returns of the bank portfolio to the
planning horizon under internal and regulatory loss risk limitations. The restriction on the internal
economic capital is based on the risk measure of Conditional Value at Risk (CVaR), that has been
proved to be appropriate for measuring bank wide loss risk [1]. The regulatory capital restrictions
represent the actual Basle Rules of risk limitation. The optimization model of step 1 of the RRM-
Approach is solved by an application of the CVaR-optimization approach by Rockafellar/Uryasev
[1].
In the second step, we derive a consistent risk-/return key ratio system from the optimum portfolio
of step 1. We estimate the risk and return contributions of each single asset in the portfolio and
achieve additive, linear representations of the expected returns and the regulatory and internal risk
contributions. The latter we obtain by an application of Euler’s Formula on CVaR [2]. We sum up
the risk and return contributions of the single assets on the business line level. In this way we de-
duce consistent return targets and capital limits of the economic and the internal capital for each
business line. These quantities represent basic planning information ensuring maximum return tar-
gets and an efficient capital allocation of the economic and the regulatory capital. The impact of the
RRM-Approach is shown by a brief application example.
References
[1] Rockafellar, R. T. and Uryasev, S.: Optimization of Conditional Value-At-Risk. The Journal
of Risk, Vol. 2, No. 3, 2000, 21-41.
[2] Tasche, D.: Risk Contributions and Performance Measurement. Working Paper, Technische
Universität München, June 1999.
[3] Theiler, U.: Integrated Risk-/Return-Management Approach for the Bank Portfolio (in Ger-
man). Ph.D. Thesis, Wiesbaden, 2002.

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Q1. What are the contributions in "Integrated risk-/return-management approach for the bank portfolio" ?

The authors estimate the risk and return contributions of each single asset in the portfolio and achieve additive, linear representations of the expected returns and the regulatory and internal risk contributions.