Pavlo A. Krokhmal

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.

Higher moment coherent risk measures

The paper considers modelling of risk-averse preferences in stochastic programming problems using risk measures. We utilize the axiomatic foundation of coherent risk measures and deviation measures in order to develop simple representations that express risk measures via specially constructed stochastic programming problems. Using the developed representations, we introduce a new family of higher-moment coherent risk measures (HMCR), which includes, as a special case, the Conditional Value-at-Risk measure. It is demonstrated that the HMCR measures are compatible with the second order stochastic dominance and utility theory, can be efficiently implemented in stochastic optimization models, and perform well in portfolio optimization case studies.

Risk Management for Hedge Fund Portfolios A Comparative Analysis of Linear Rebalancing Strategies

This article applies formal risk management methodologies to optimization of a portfolio of hedge funds (fund of funds). We compare recently developed risk management methodologies: conditional value-at-risk and conditional drawdown-at-risk with more established mean-absolute deviation, maximum loss, and market neutrality approaches. The common property of considered risk management techniques is that they admit the formulation of a portfolio optimization model as a linear programming (LP) problem. LP formulations allow for implementing efficient and robust portfolio allocation algorithms, which can successfully handle optimization problems with thousands of instruments and scenarios. The performance of various risk constraints is investigated and discussed in detail for in-sample and out-of-sample testing of the algorithm. The numerical experiments show that imposing risk constraints may improve the “real” performance of a portfolio rebalancing strategy in out-of-sample runs. It is beneficial to combine several types of risk constraints that control different sources of risk.

Random assignment problems

Analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problems solutions, feasible region, and optimal values, especially in large-scale cases. A class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems, and many important problems in operations research and computer science can be formulated as assignment problems. This paper presents an overview of the recent results and developments in the area of probabilistic assignment problems, including the linear and multidimensional assignment problems, quadratic assignment problem, etc.

Robust Decision Making: Addressing Uncertainties in Distributions

This paper develops a general approach to risk management in military applications involving uncertainties in information and distributions. The risk of loss, damage, or failure is measured by the Conditional Value-at-Risk (CVaR) measure. Loosely speaking, CVaR with the confidence level α estimates the risk of loss by averaging the possible losses over the (1 - α) · 100% worst cases (e.g., 10%). As a function of decision variables, CVaR is convex and therefore can be efficiently controlled/optimized using convex or (under quite general assumptions) linear programming. The general methodology was tested on two Weapon-Target Assignment (WTA) problems. It is assumed that the distributions of random variables in the WTA formulations are not known with certainty. The total cost of a mission (including weapon attrition) was minimized, while satisfying operational constraints and ensuring destruction of all targets with high probabilities. The risk of failure of the mission (e.g., targets are not destroyed) is controlled by CVaR constraints. The case studies conducted show that there are significant qualitative and quantitative differences in solutions of deterministic WTA and stochastic WTA problems.

Risk optimization with p-order conic constraints: A linear programming approach

The paper considers solving of linear programming problems with p-order conic constraints that are related to a certain class of stochastic optimization models with risk objective or constraints. The proposed approach is based on construction of polyhedral approximations for p-order cones, and then invoking a Benders decomposition scheme that allows for efficient solving of the approximating problems. The conducted case study of portfolio optimization with p-order conic constraints demonstrates that the developed computational techniques compare favorably against a number of benchmark methods, including second-order conic programming methods.

Detection of temporal changes in psychophysiological data using statistical process control methods

We consider the problem of detecting temporal changes in the functional state of human subjects due to varying levels of cognitive load using real-time psychophysiological data. The proposed approach relies on monitoring several channels of electroencephalogram (EEG) and electrooculogram (EOG) signals using the methods of statistical process control. It is demonstrated that control charting methods are capable of detecting changes in psychophysiological signals that are induced by varying cognitive load with high accuracy and low false alarm rates, and are capable of accommodating subject-specific differences while being robust with respect to differences between different trials performed by the same subject.

An algorithm for online detection of temporal changes in operator cognitive state using real-time psychophysiological data

Abstract We consider the problem of on-the-fly detection of temporal changes in the cognitive state of human subjects due to varying levels of difficulty of performed tasks using real-time EEG and EOG data. We construct the Cognitive State Indicator (CSI) as a function that projects the multidimensional EEG/EOG signals onto the interval [0,1] by maximizing the Kullback–Leibler distance between distributions of the signals, and whose values change continuously with variations in cognitive load. During offline testing (i.e., when evolution in time is disregarded) it was demonstrated that the CSI can serve as a statistically significant discriminator between states of different cognitive loads. In the online setting, a trend detection heuristic (TDH) has been proposed to detect real-time changes in the cognitive state by monitoring trends in the CSI. Our results support the application of the CSI and the TDH in future closed-loop control systems with human supervision.

Asymptotic behavior of the expected optimal value of the multidimensional assignment problem

The Multidimensional Assignment Problem (MAP) is a higher-dimensional version of the Linear Assignment Problem that arises in the areas of data association, target tracking, resource allocation, etc. This paper elucidates the question of asymptotical behavior of the expected optimal value of the large-scale MAP whose assignment costs are independent identically distributed random variables with a prescribed probability distribution. We demonstrate that for a broad class of continuous distributions the limiting value of the expected optimal cost of the MAP is determined by the location of the left endpoint of the support set of the distribution, and construct asymptotical bounds for the expected optimal cost.

A sample-path approach to optimal position liquidation

We consider the problem of optimal position liquidation where the expected cash flow stream due to transactions is maximized in the presence of temporary or permanent market impact. A stochastic programming approach is used to construct trading strategies that differentiate decisions with respect to the observed market conditions, and can accommodate various types of trading constraints. As a scenario model, we use a collection of sample paths representing possible future realizations of state variable processes (price, trading volume etc.), and employ a heuristical technique of sample-path grouping, which can be viewed as a generalization of the standard nonanticipativity constraints.