Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection

  title={Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection},
  author={Li Chen and Simai He and Shuzhong Zhang},
  journal={Oper. Res.},
In this paper we develop tight bounds on the expected values of several risk measures that are of interest to us. This work is motivated by the robust optimization models arising from portfolio selection problems. Indeed, the whole paper is centered around robust portfolio models and solutions. The basic setting is to find a portfolio that maximizes (respectively, minimizes) the expected utility (respectively, disutility) values in the midst of infinitely many possible ambiguous distributions… 

An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution

This paper derives a closed-form expression for the optimal portfolio strategy to the robust mean-multiple risk portfolio selection model under distribution and mean return ambiguity (RMP), and analytically and numerically shows that the resulting portfolio weight converges to the minimum variance portfolio when the level of ambiguity aversion is in a high value.

Robust ranking and portfolio optimization

Robust portfolio optimization with risk measures under distributional uncertainty

ROBUST PORTFOLIO OPTIMIZATION WITH RISK MEASURES UNDER DISTRIBUTIONAL UNCERTAINTY A. Burak Paç Ph.D. in Industrial Engineering Advisor: Mustafa Ç. Pınar July 2016 In this study, we consider the

Data-driven robust mean-CVaR portfolio selection under distribution ambiguity

An extension to the distributionally robust optimization model for a robust mean-CVaR portfolio selection model is developed that allows the model to capture a zero net adjustment via a linear constraint in the mean return, which can be cast as a tractable conic programme.

Robust portfolio selection with polyhedral ambiguous inputs

Ambiguity in the inputs of the models is typical especially in portfolio selection problem where the true distribution of random variables is usually unknown. Here we use robust optimization approach

Robust portfolio choice with CVaR and VaR under distribution and mean return ambiguity

We consider the problem of optimal portfolio choice using the Conditional Value-at-Risk (CVaR) and Value-at-Risk (VaR) measures for a market consisting of n risky assets and a riskless asset and

Computationally Tractable Counterparts of Distributionally Robust Constraints on Risk Measures

This paper shows that the derivation of a tractable robust counterpart can be split into two parts: one corresponding to the risk measure and the other to the uncertainty set, and provides the computational tractability status for each of the uncertaintySet-risk measure pairs that the authors could solve.

Robust portfolio rebalancing with cardinality and diversification constraints

A robust conditional value at risk (CVaR) optimal portfolio rebalancing model under various financial constraints to construct sparse and diversified rebalance portfolios is developed and a distributed-version of the Alternating Direction Method of Multipliers (ADMM) algorithm is developed.

Distributional Robust Portfolio Construction based on Investor Aversion

: In behavioral finance, aversion affects investors' judgment of future uncertainty when profit and loss occur. Considering investors' aversion to loss and risk, and the ambiguous uncertainty

Robust VaR and CVaR Optimization under Joint Ambiguity in Distributions, Means, and Covariances

Abstract We develop robust models for optimization of the VaR (value at risk) and CVaR (conditional value at risk) risk measures with a minimum expected return constraint under joint ambiguity in



Worst-Case Value-At-Risk and Robust Portfolio Optimization: A Conic Programming Approach

The problem of computing and optimizing the worst-case VaR and various other partial information on the distribution, including uncertainty in factor models, support constraints, and relative entropy information can be cast as semidefinite programs.


This paper derives exact and approximate optimal trading strategies for a robust (maximin) expected utility model, where the investor maximizes his worst‐case expected utility over a set of ambiguous distributions.

Robust Portfolio Selection Problems

This paper introduces "uncertainty structures" for the market parameters and shows that the robust portfolio selection problems corresponding to these uncertainty structures can be reformulated as second-order cone programs and, therefore, the computational effort required to solve them is comparable to that required for solving convex quadratic programs.

Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization

The proposed method results in the optimization of a modified VaR measure, Asymmetry-Robust VaR (ARVaR), that takes into consideration asymmetries in the distributions of returns and is coherent, which makes it desirable from a financial theory perspective.

Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems

This paper proposes a model that describes uncertainty in both the distribution form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance matrix) and demonstrates that for a wide range of cost functions the associated distributionally robust stochastic program can be solved efficiently.

Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management

The application of the worst-case CVaR to robust portfolio optimization is proposed, and the corresponding problems are cast as linear programs and second-order cone programs that can be solved efficiently.

On the Relation Between Option and Stock Prices: A Convex Optimization Approach

Convex and semidefinite optimization methods, duality, and complexity theory are introduced to shed new light on the relation of option and stock prices based just on the no-arbitrage assumption, and it is shown that it is NP-hard to find best possible bounds in multiple dimensions.


Robust Mean-Covariance Solutions for Stochastic Optimization

  • I. Popescu
  • Mathematics, Computer Science
    Oper. Res.
  • 2007
It is proved that for a general class of objective functions, the robust solutions amount to solving a certain deterministic parametric quadratic program, and a general projection property for multivariate distributions with given means and covariances is proved.

Robust Dynamic Programming

  • G. Iyengar
  • Mathematics, Economics
    Math. Oper. Res.
  • 2005
It is proved that when this set of measures has a certain "rectangularity" property, all of the main results for finite and infinite horizon DP extend to natural robust counterparts.