Optimization of conditional value-at-risk

Abstract

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 C V aR necessarily have l o w 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 i n vestment companies, brokerage rms, mutual funds, and any business that evaluates risks. It can becombined 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.

Stochastic optimal generation bid to electricity markets with emissions risk constraints

“…Classical risk management methods in portfolio optimization theory, include Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) [31,33]. Following [31], the VaR of a portfolio with respect to a specified probability β is the lowest amount α such that, with probability β, the loss will not exceed α. The CVaR is defined as the conditional expectation of losses above amount α.…”

“…The CVaR is defined as the conditional expectation of losses above amount α. [31] show that, under certain assumptions, the minimization of the CVaR of a given portfolio can be formulated as a continuous optimization problem where the value of VaR is computed endogenously during the optimization process. By analogy to the use of CVaR in the portfolio optimization, the Conditional Emission-at-Risk (CEaR) is proposed in this work as a tool for measuring and controlling the risk of violating the NERP emission limits.…”

“…That fact introduces two relevant differences between the use of CVaR models in portfolio optimization and that of CEaR in our formulation. The first one is that the value of VaR is no longer obtained implicitly as in the models in [31], but must be given explicitly as an exogenous parameter. Actually, as we will see, the equivalent to VaR in our models is the emission limits imposed by the NERP, SO 2 and N O x .…”

Polynomial-time identification of robust network flows under uncertain arc failures

“…Clearly, CVaR is a more conservative measure of risk than VaR. Rockafellar and Uryasev [16] proved several important results regarding optimization of CVaR, which make this risk measure rather attractive from the optimization viewpoint. In particular, CVaR has been shown to possess the properties that VaR lacks; in particular, it is coherent (which includes convexity among other properties).…”

Utility Maximization Under Solvency Constraints and Unhedgeable Risks

“…We use here well-known results from Rockafellar & Uryasev (2000) on optimisation under a Tail-VaR (or CVaR) constraint. First, we define the solvency function…”