Stanislav Uryasev

OPTIMIZATION OF CONDITIONAL VALUE AT RISK

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.

Portfolio Optimization with Conditional Value-at-Risk Objective and Constraints

Recently, a new approach for optimization of Conditional Value-at-Risk (CVaR) was suggested and tested with several applications. For continuous distributions, CVaR is defined as the expected loss exceeding Value-at Risk (VaR). However, generally, CVaR is the weighted average of VaR and losses exceeding VaR. Central to the approach is an optimization technique for calculating VaR and optimizing CVaR simultaneously. This paper extends this approach to the optimization problems with CVaR constraints. In particular, the approach can be used for maximizing expected returns under CVaR constraints. Multiple CVaR constraints with various confidence levels can be used to shape the profit/loss distribution. A case study for the portfolio of S&P 100 stocks is performed to demonstrate how the new optimization techniques can be implemented.

Generalized Deviations in Risk Analysis

General deviation measures are introduced and studied systematically for their potential applications to risk management in areas like portfolio optimization and engineering. Such measures include standard deviation as a special case but need not be symmetric with respect to ups and downs. Their properties are explored with a mind to generating a large assortment of examples and assessing which may exhibit superior behavior. Connections are shown with coherent risk measures in the sense of Artzner, Delbaen, Eber and Heath, when those are applied to the difference between a random variable and its expectation, instead of to the random variable itself. However, the correspondence is only one-to-one when both classes are restricted by properties called lower range dominance, on the one hand, and strict expectation boundedness on the other. Dual characterizations in terms of sets called risk envelopes are fully provided.

Conditional value-at-risk: optimization algorithms and applications

This article has outlined a new approach for the simultaneous calculation of value-at-risk (VaR) and optimization of conditional VaR (CVaR) for a broad class of problems. We have shown that CVaR can be efficiently minimized using LP techniques. Our numerical experiments show that CVaR optimal portfolios are near optimal in VaR terms, i.e., VaR cannot be reduced further more than a few percent. Also, CVaR constraints can be handled efficiently using equivalent linear constraints, which dramatically improves the efficiency of the optimization techniques.

Credit risk optimization with Conditional Value-at-Risk criterion

Abstract.This paper examines a new approach for credit risk optimization. The model is based on the Conditional Value-at-Risk (CVaR) risk measure, the expected loss exceeding Value-at-Risk. CVaR is also known as Mean Excess, Mean Shortfall, or Tail VaR. This model can simultaneously adjust all positions in a portfolio of financial instruments in order to minimize CVaR subject to trading and return constraints. The credit risk distribution is generated by Monte Carlo simulations and the optimization problem is solved effectively by linear programming. The algorithm is very efficient; it can handle hundreds of instruments and thousands of scenarios in reasonable computer time. The approach is demonstrated with a portfolio of emerging market bonds.

Drawdown Measure in Portfolio Optimization

A new one-parameter family of risk measures called Conditional Drawdown (CDD) has been proposed. These measures of risk are functionals of the portfolio drawdown (underwater) curve considered in active portfolio management. For some value of the tolerance parameter α, in the case of a single sample path, drawdown functional is defined as the mean of the worst (1 - α) * 100% drawdowns. The CDD measure generalizes the notion of the drawdown functional to a multi-scenario case and can be considered as a generalization of deviation measure to a dynamic case. The CDD measure includes the Maximal Drawdown and Average Drawdown as its limiting cases. Mathematical properties of the CDD measure have been studied and efficient optimization techniques for CDD computation and solving asset-allocation problems with a CDD measure have been developed. The CDD family of risk functionals is similar to Conditional Value-at-Risk (CVaR), which is also called Mean Shortfall, Mean Excess Loss, or Tail Value-at-Risk. Some recommendations on how to select the optimal risk functionals for getting practically stable portfolios have been provided. A real-life asset-allocation problem has been solved using the proposed measures. For this particular example, the optimal portfolios for cases of Maximal Drawdown, Average Drawdown, and several intermediate cases between these two have been found.

Relaxation algorithms to find Nash equilibria with economic applications

Recent theoretical studies have shown that a relaxation algorithm can be used to find noncooperative equilibria of synchronous infinite games with nonlinear payoff functions and coupled constraints. In this study, we introduce an improvement to the algorithm, such as the steepest-descent step-size control, for which the convergence of the algorithm is proved. The algorithm is then tested on several economic applications. In particular, a River Basin Pollution problem is considered where coupled environmental constraints are crucial for the relevant model definition. Numerical runs demonstrate fast convergence of the algorithm for a wide range of parameters.

Stochastic optimization: algorithms and applications

Preface. Output analysis for approximated stochastic programs J. Dupacova. Combinatorial Randomized Rounding: Boosting Randomized Rounding with Combinatorial Arguments P. Efraimidis, P.G. Spirakis. Statutory Regulation of Casualty Insurance Companies: An Example from Norway with Stochastic Programming Analysis A. Gaivoronski, et al. Option pricing in a world with arbitrage X. Guo, L. Shepp. Monte Carlo Methods for Discrete Stochastic Optimization T. Homem-de-Mello. Discrete Approximation in Quantile Problem of Portfolio Selection A. Kibzun, R. Lepp. Optimizing electricity distribution using two-stage integer recourse models W.K. Klein Haneveld, M.H. van der Vlerk. A Finite-Dimensional Approach to Infinite-Dimensional Constraints in Stochastic Programming Duality L. Korf. Non-Linear Risk of Linear Instruments A. Kreinin. Multialgorithms for Parallel Computing: A New Paradigm for Optimization J. Nazareth. Convergence Rate of Incremental Subgradient Algorithms A. Nedic, D. Bertsekas. Transient Stochastic Models for Search Patterns E. Pasiliao. Value-at-Risk Based Portfolio Optimization A. Puelz. Combinatorial Optimization, Cross-Entropy, Ants and Rare Events R.Y. Rubinstein. Consistency of Statistical Estimators: the Epigraphical View G. Salinetti. Hierarchical Sparsity in Multistage Convex Stochastic Programs M. Steinbach. Conditional Value-at-Risk: Optimization Approach S. Uryasev, R.T. Rockafellar.

Algorithms for Optimization of Value-at-Risk

This paper suggests two new heuristic algorithms for optimization of Value-at-Risk (VaR). By definition, VaR is an estimate of the maximum portfolio loss during a standardized period with some confidence level. The optimization algo- rithms are based on the minimization of the closely related risk measure Conditional Value-at-Risk (CVaR). For continuous distributions, CVaR is the expected loss exceeding VaR, and is also known as Mean Excess Loss or Expected Shortfall. For discrete distributions, CVaR is the weighted average of VaR and losses exceeding VaR. CVaR is an upper bound for VaR, therefore, minimization of CVaR also reduces VaR. The algorithms are tested by minimizing the credit risk of a portfolio of emerging market bonds. Numerical experiments showed that the algorithms are efficient and can handle a large number of instruments and scenarios. However, calculations identified a deficiency of VaR risk measure, compared to CVaR. Minimization of VaR leads to an undesirable stretch of the tail of the distribution exceeding VaR. For portfolios with skewed distributions, such as credit risk, minimization of VaR may result in a significant increase of high losses exceeding VaR. For the credit risk problem studied in this paper, VaR minimization leads to about 16% increase of the average loss for the worst 1% scenarios (compared to the worst 1% scenarios in CVaR minimum solution). 1% includes 200 of 20000 scenarios, which were used for estimating credit risk in this case study.

Optimal Risk Path Algorithms

Analytical and discrete optimization approaches for routing an aircraft in a threat environment have been developed. Using these approaches, an aircraft’s optimal risk trajectory with a constraint on the path length can be efficiently calculated. The analytical approach based on calculus of variations reduces the original risk optimization problem to the system of nonlinear differential equations. In the case of a single radarinstallation, the solution of such a system is expressed by the elliptic sine. The discrete optimization approach reformulates the problem as the Weight Constrained Shortest Path Problem (WCSPP) for a grid undirected graph. The WCSPP is efficiently solved by the Modified Label Setting Algorithm (MLSA). Both approaches have been tested with several numerical examples. Discrete nonsmooth solutions with high precision coincide with exact continuous solutions. For the same graph, time in which the discrete optimization algorithm computes the optimal trajectory is independent of the number of radars. The discrete approach is also efficient for solving the problem using different risk functions.