Conditional Value-at-risk Minimization Research Articles

Conditional value-at-risk in portfolio optimization: Coherent but fragile

We evaluate conditional value-at-risk (CVaR) as a risk measure in data-driven portfolio optimization. We show that portfolios obtained by solving mean-CVaR and global minimum CVaR problems are unreliable due to estimation errors of CVaR and/or the mean, which are magnified by optimization. This problem is exacerbated when the tail of the return distribution is made heavier. We conclude that CVaR, a coherent risk measure, is fragile in portfolio optimization due to estimation errors.

Cost and risk management for a local distribution company in purchasing electricity

AbstractIn the competitive electricity markets, the allocation of electricity among multi‐markets for purchasers is worth concerning. Based on the portfolio optimization theory in the field of financial risk, a novel optimal electricity‐procurement model for the local distribution company (LDC) is presented, which can synthetically consider the risk and expected purchase cost. For the purpose of risk assessment, the conditional value at risk (CVaR) is implemented. This model is applied to determine the electricity allocation and efficient frontiers for the LDC among three markets. Then it is compared with the mean‐variance model. Simulation results demonstrate that the proposed model can guarantee the LDC to bear the minimum CVaR risk within an acceptable expected procurement cost. It provides an effective way for the LDC to manage the cost and risk in electricity‐purchase decision. Copyright © 2009 John Wiley & Sons, Ltd.

On the optimal product mix in life insurance companies using conditional value at risk

This paper proposes a Conditional Value-at-Risk Minimization (CVaRM) approach to optimize an insurer’s product mix. By incorporating the natural hedging strategy of Cox and Lin (2007) and the two-factor stochastic mortality model of Cairns et al. (2006b), we calculate an optimize product mix for insurance companies to hedge against the systematic mortality risk under parameter uncertainty. To reflect the importance of required profit, we further integrate the premium loading of systematic risk. We compare the hedging results to those using the duration match method of Wang et al. (forthcoming), and show that the proposed CVaRM approach has a narrower quantile of loss distribution after hedging—thereby effectively reducing systematic mortality risk for life insurance companies.

A robust approach based on conditional value-at-risk measure to statistical learning problems

In statistical learning problems, measurement errors in the observed data degrade the reliability of estimation. There exist several approaches to handle those uncertainties in observations. In this paper, we propose to use the conditional value-at-risk (CVaR) measure in order to depress influence of measurement errors, and investigate the relation between the resulting CVaR minimization problems and some existing approaches in the same framework. For the CVaR minimization problems which include the computation of integration, we apply Monte Carlo sampling method and obtain their approximate solutions. The approximation error bound and convergence property of the solution are proved by Vapnik and Chervonenkis theory. Numerical experiments show that the CVaR minimization problem can achieve fairly good estimation results, compared with several support vector machines, in the presence of measurement errors.

Portfolio optimization by minimizing conditional value-at-risk via nondifferentiable optimization

Conditional Value-at-Risk (CVaR) is a portfolio evaluation function having appealing features such as sub-additivity and convexity. Although the CVaR function is nondifferentiable, scenario-based CVaR minimization problems can be reformulated as linear programs (LPs) that afford solutions via widely-used commercial softwares. However, finding solutions through LP formulations for problems having many financial instruments and a large number of price scenarios can be time-consuming as the dimension of the problem greatly increases. In this paper, we propose a two-phase approach that is suitable for solving CVaR minimization problems having a large number of price scenarios. In the first phase, conventional differentiable optimization techniques are used while circumventing nondifferentiable points, and in the second phase, we employ a theoretically convergent, variable target value nondifferentiable optimization technique. The resultant two-phase procedure guarantees infinite convergence to optimality. As an optional third phase, we additionally perform a switchover to a simplex solver starting with a crash basis obtained from the second phase when finite convergence to an exact optimum is desired. This three phase procedure substantially reduces the effort required in comparison with the direct use of a commercial stand-alone simplex solver (CPLEX 9.0). Moreover, the two-phase method provides highly-accurate near-optimal solutions with a significantly improved performance over the interior point barrier implementation of CPLEX 9.0 as well, especially when the number of scenarios is large. We also provide some benchmarking results on using an alternative popular proximal bundle nondifferentiable optimization technique.

Minimizing CVaR and VaR for a portfolio of derivatives

Value at risk (VaR) and conditional value at risk (CVaR) are frequently used as risk measures in risk management. Compared to VaR, CVaR is attractive since it is a coherent risk measure. We analyze the problem of computing the optimal VaR and CVaR portfolios. We illustrate that VaR and CVaR minimization problems for derivatives portfolios are typically ill-posed. We propose to include cost as an additional preference criterion for the CVaR optimization problem. We demonstrate that, with the addition of a proportional cost, it is possible to compute an optimal CVaR derivative investment portfolio with significantly fewer instruments and comparable CVaR and VaR. A computational method based on a smoothing technique is proposed to solve a simulation based CVaR optimization problem efficiently. Comparison is made with the linear programming approach for solving the simulation based CVaR optimization problem.

On Relation Between Expected Regret and Conditional Value at Risk

The paper compares portfolio optimization approaches with expected regret and Conditional Value-at-Risk (CVaR) performance functions. The expected regret is defined as an average portfolio underperformance comparing to a fixed target or some benchmark portfolio. For continuous distributions, CVaR is defined as the expected loss exceeding α-Value-at Risk (VaR), i.e., the mean of the worst (1-α) 100% losses in a specified time period. However, generally, CVaR is the weighted average of VaR and losses exceeding VaR. Optimization of CVaR can be performed using linear programming. We formally prove that a portfolio with a continuous loss distribution, which minimizes CVaR, can be obtained by doing a line search with respect to the threshold in the expected regret. An optimal portfolio in CVaR sense is also optimal in the expected regret sense for some threshold in the regret function. The inverse statement is also valid, i.e., if a portfolio minimizes the expected regret, this portfolio can be found by doing a line search with respect to the CVaR confidence level. A portfolio, optimal in expected regret sense, is also optimal in CVaR sense for some confidence level. The relation of the expected regret and CVaR minimization approaches is explained with a numerical example.