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181
Optimization of Conditional ValueatRisk
 Journal of Risk
, 2000
"... 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 ValueatRisk (CVaR) rather than minimizing ValueatRisk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR ..."
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Cited by 396 (26 self)
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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 ValueatRisk (CVaR) rather than minimizing ValueatRisk (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 scenariobased 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.
Conditional valueatrisk for general loss distributions
 Journal of Banking and Finance
, 2002
"... Abstract. Fundamental properties of conditional valueatrisk, as a measure of risk with significant advantages over valueatrisk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence o ..."
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Cited by 356 (28 self)
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Abstract. Fundamental properties of conditional valueatrisk, as a measure of risk with significant advantages over valueatrisk, are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications because of the prevalence of models based on scenarios and finite sampling. Conditional valueatrisk is able to quantify dangers beyond valueatrisk, and moreover it is coherent. It provides optimization shortcuts which, through linear programming techniques, make practical many largescale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calculations, shown in several case studies, are illustrated further with an example of index tracking. Key Words: Valueatrisk, conditional valueatrisk, mean shortfall, coherent risk measures, risk sampling, scenarios, hedging, index tracking, portfolio optimization, risk management
From Stochastic Dominance to MeanRisk Models: Semideviations as Risk Measures
, 1997
"... Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. The former is based on an axiomatic model of riskaverse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid f ..."
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Cited by 104 (20 self)
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Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. The former is based on an axiomatic model of riskaverse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible tradeoff analysis, but cannot model all riskaverse preferences. In particular, if variance is used as a measure of risk, the resulting mean–variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean–risk model consistent with the second degree stochastic dominance, provided that the tradeoff coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome–Risk diagram,
Heuristics for cardinality constrained portfolio optimisation
, 2000
"... In this paper we consider the problem of finding the efficient frontier associated with the standard meanvariance portfolio optimisation model. We extend the standard model to include cardinality constraints that limit a portfolio to have a specified number of assets, and to impose limits on the pr ..."
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Cited by 91 (4 self)
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In this paper we consider the problem of finding the efficient frontier associated with the standard meanvariance portfolio optimisation model. We extend the standard model to include cardinality constraints that limit a portfolio to have a specified number of assets, and to impose limits on the proportion of the portfolio held in a given asset (if any of the asset is held). We illustrate the differences that arise in the shape of this efficient frontier when such constraints are present. We present three heuristic algorithms based upon genetic algorithms, tabu search and simulated annealing for finding the cardinality constrained efficient frontier. Computational results are presented for five data sets involving up to 225 assets.
Portfolio Optimization with Conditional ValueatRisk objective and Constraints
, 1999
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Optimization with stochastic dominance constraints
 SIAM Journal on Optimization
"... We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for the ..."
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Cited by 52 (6 self)
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We consider the problem of constructing a portfolio of finitely many assets whose returns are described by a discrete joint distribution. We propose a new portfolio optimization model involving stochastic dominance constraints on the portfolio return. We develop optimality and duality theory for these models. We construct equivalent optimization models with utility functions. Numerical illustration is provided.
On consistency of stochastic dominance and mean–semideviation models
 MATHEMATICAL PROGRAMMING
, 1997
"... We analyse relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. The concept of αconsistency of these approaches is defined as the consistency within a bounded range of mean–risk tradeoffs. We show that mean ..."
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Cited by 43 (11 self)
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We analyse relations between two methods frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean–risk approaches. The concept of αconsistency of these approaches is defined as the consistency within a bounded range of mean–risk tradeoffs. We show that mean–risk models using central semideviations as risk measures are αconsistent with stochastic dominance relations of the corresponding degree if the tradeoff coefficient for the semideviation is bounded by one.
Markowitz revisited: meanvariance models in financial portfolio analysis
 SIAM Rev
, 2001
"... Abstract. Meanvariance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of singleperiod variants, including semivariance models. Particular emphasis is laid on avo ..."
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Cited by 35 (1 self)
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Abstract. Meanvariance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of singleperiod variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.
Multiple Criteria Linear Programming Model for Portfolio Selection
 Annals of Operations Research
, 2000
"... The portfolio selection problem is usually considered as a bicriteria optimization problem where a reasonable tradeoff between expected rate of return and risk is sought. In the classical Markowitz model the risk is measured with variance, thus generating a quadratic programming model. The Markowit ..."
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Cited by 32 (11 self)
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The portfolio selection problem is usually considered as a bicriteria optimization problem where a reasonable tradeoff between expected rate of return and risk is sought. In the classical Markowitz model the risk is measured with variance, thus generating a quadratic programming model. The Markowitz model is frequently criticized as not consistent with axiomatic models of preferences for choice under risk. Models consistent with the preference axioms are based on the relation of stochastic dominance or on expected utility theory. The former is quite easy to implement for pairwise comparisons of given portfolios whereas it does not offer any computational tool to analyze the portfolio selection problem. The latter, when used for the portfolio selection problem, is restrictive in modeling preferences of investors. In this paper, a multiple criteria linear programming model of the portfolio selection problem is developed. The model is based on the preference axioms for choice under risk. Nevertheless, it allows one to employ the standard multiple criteria procedures to analyze the portfolio selection problem. It is shown that the classical meanrisk approaches resulting in linear programming models correspond to specific solution techniques applied to our multiple criteria model.
Stochastic Constraint Programming: A ScenarioBased Approach
 SUBMISSION TO CONSTRAINTS
"... To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. ..."
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Cited by 30 (4 self)
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To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (nonstochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management.