Portfolio optimization with conditional value-at-risk objective and constraints

Krokhmal, Pavlo A.; Uryasev, tanislav; Palmquist, Jonas

doi:10.21314/jor.2002.057

Cited by 518 publications

(355 citation statements)

References 27 publications

Supporting

Mentioning

340

Contrasting

Unclassified

Order By: Relevance

“…5 See Jorion, 2001. 6 See Basak andShapiro, 2001;Krokhmal et al, 2002;Rockafellar and Uryasev, 2002;Alexander and Baptista, 2004; and Gaivoronski and Pflug, 2005. 7 Gaivoronski and Pflug (2005 Cai and Tan (2007).…”

mentioning

confidence: 99%

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

Tan

Weng

2011

Geneva Risk Insur Rev

View full text Add to dashboard Cite

The quest for optimal reinsurance design has remained an interesting problem among insurers, reinsurers, and academicians. An appropriate use of reinsurance could reduce the underwriting risk of an insurer and thereby enhance its value. This paper complements the existing research on optimal reinsurance by proposing another model for the determination of the optimal reinsurance design. The problem is formulated as a constrained optimization problem with the objective of minimizing the value-at-risk of the net risk of the insurer while subjecting to a profitability constraint. The proposed optimal reinsurance model, therefore, has the advantage of exploiting the classical tradeoff between risk and reward. Under the additional assumptions that the reinsurance premium is determined by the expectation premium principle and the ceded loss function is confined to a class of increasing and convex functions, explicit solutions are derived. Depending on the risk measure's level of confidence, the safety loading for the reinsurance premium, and the expected profit guaranteed for the insurer, we establish conditions for the existence of reinsurance. When it is optimal to cede the insurer's risk, the optimal reinsurance design could be in the form of pure stop-loss reinsurance, quota-share reinsurance, or a combination of stop-loss and quota-share reinsurance. The Geneva Risk and Insurance Review (2012) 37, 109-140. doi:10.1057/grir.2011; published online 23 August 2011Keywords: value-at-risk (VaR); optimal reinsurance; expectation premium principle; linear programming in infinite dimensional spaces IntroductionThe importance of sound risk management for financial institutions and insurance enterprises has been dramatically highlighted by the subprime crisis. Risk professionals are constantly seeking better risk measures to quantify risks associated with market, credit, operational, catastrophic, and many others. Risk measures such as value-at-risk (VaR) and conditional VaR or conditional tail expectation (CTE) have been proposed. Among these risk The Geneva Risk and Insurance Review, 2012, 37, (109-140) 9 and Zhou and Wu 10 demonstrated that employing VaR as a constraint could assist an insured in determining his or her optimal insurance policy. By minimizing VaR or CTE of the total risk exposure of an insurer, Cai and Tan 11 and Cai et al. 12 derived explicitly the optimal reinsurance treaties for the insurer. 13 While analytic solutions have been derived in the above reinsurance models, these results can be criticized on the ground that the optimality is based exclusively on minimizing an insurer's risk exposure. In practice, an insurer is concerned not only with its exposure to risk but also its profitability of insuring the underlying risks. To elaborate this point, let us first note that when an insurer uses reinsurance to cede (or transfer) part of its loss to a reinsurer, the insurer is liable to pay reinsurance 1 Artzner et al. (1999). 2 Basak and Shapiro (2001). 3 Yamai and Yoshiba (2005). 4 See Basle Commi...

show abstract

mentioning

confidence: 99%

Enhancing Insurer Value Using Reinsurance and Value-at-Risk Criterion

Tan

Weng

2011

Geneva Risk Insur Rev

View full text Add to dashboard Cite

show abstract

“…For example, consider an application in which a constraint must be satisfied within a specific confidence level α ∈ (0, 1]. Then the corresponding α-VaR value is the lowest value ζ such that with probability α, the loss does not exceed ζ [14]. In economic terms, VaR is simply the maximum amount at risk to be lost from an investment.…”

Section: Statistical Measures Of Losses For Optimization Problems Undmentioning

confidence: 99%

“…, S, [16]. Since L(x, y) is linear with respect to x,F α (x, ζ) is convex and piecewise linear [14]. If we assume that each scenario is equally likely, that is π s = 1 S , ∀ s = 1, 2, .…”

Section: Robust Minimum Cost Flow Problem Under Uncertainty With Cvarmentioning

confidence: 99%

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

2009

View full text Add to dashboard Cite

ABSTRACT. We propose LP-based solution methods for network flow problems subject to multiple uncertain arc failures, which allow finding robust optimal solutions in polynomial time under certain conditions. We justify this fact by proving that for the considered class of problems under uncertainty with linear loss functions, the number of entities in the corresponding LP formulations is polynomial with respect to the number of arcs in the network. The proposed formulation is efficient for sparse networks, as well as for timecritical networked systems, where quick and robust decisions play a crucial role.

show abstract

“…There is a very large literature on risk measures under uncertainty, see [GF04] for some recent work and a review. A central idea is that of value-at-risk (VaR), and the related concept, conditional value at risk (CVaR) (see [RU00], [KPU02], and references therein). Our definition of VaR follows that of [GI04].…”

Section: Var and Cvar Modelsmentioning

confidence: 99%