optimizationofConditionalValue-at-Risk.pdf_VaR（Valueatrisk）资源-CSDN文库

CVAR

需积分: 30 147 浏览量 2020-10-22 13:34:02 上传评论收藏 236KB PDF 举报

资源推荐

资源详情

资源评论

Optimization of Conditional Value-at-Risk

R. Tyrrell Ro ckafellar

and Stanislav Uryasev

A new approach to optimizing or hedging a p ortfolio of nancial instruments to reduce risk is

presented and tested on applications. It fo cuses on minimizing Conditional Value-at-Risk (CVaR)

rather than minimizing Value-at-Risk (VaR), but portfolios with lowCVaR necessarily havelow

VaR as well. CVaR, also called Mean Excess Loss, Mean Shortfall, or Tail VaR, is anyway

considered to b e a more consistent measure of risk than VaR.

Central to the new approach is a technique for p ortfolio optimization which calculates VaR

and optimizes CVaR simultaneously. This technique is suitable for use byinvestment companies,

brokerage rms, mutual funds, and any business that evaluates risks. It can be combined with

analytical or scenario-based metho ds to optimize p ortfolios with large numb ers of instruments, in

which case the calculations often come down to linear programming or nonsmo oth programming.

The metho dology can be applied also to the optimization of p ercentiles in contexts outside of

nance.

Septemb er 5, 1999

Correspondence should b e addressed to: Stanislav Uryasev

UniversityofWashington, Dept. of Applied Mathematics, 408 L Guggenheim Hall, Box 352420, Seattle, WA

98195-2420, E-mail: rtr@math.washington.edu

University of Florida, Dept. of Industrial and Systems Engineering, PO Box 116595, 303 Weil Hall, Gainesville,

FL 32611-6595, E-mail: uryasev@ise.u.edu

, URL: http://www.ise.u.edu/uryasev

1 INTRODUCTION

This pap er introduces a new approach to optimizing a p ortfolio so as to reduce the risk of high

losses. Value-at-Risk (VaR) has a role in the approach, but the emphasis is on Conditional

Value-at-Risk (CVaR), whichisknown also as Mean Excess Loss, Mean Shortfall, or Tail VaR.

By denition with resp ect to a sp ecied probability level



, the



-VaR of a p ortfolio is the

lowest amount



such that, with probability



, the loss will not exceed



, whereas the



-CVaR

is the conditional exp ectation of losses ab ove that amount



. Three values of



are commonly

considered: 0.90, 0.95 and 0.99. The denitions ensure that the



-VaR is never more than the



-CVaR, so p ortfolios with lowCVaR must havelowVaR as well.

A description of various methodologies for the mo deling of VaR can be seen, along with re-

lated resources, at URL http://www.gloriamundi.org/. Mostly, approaches to calculating VaR

rely on linear approximation of the p ortfolio risks and assume a joint normal (or log-normal) dis-

tribution of the underlying market parameters, see, for instance, Due and Pan (1997), Jorion

(1996), Pritsker (1997), RiskMetrics (1996), Simons (1996), Beder (1995), Stambaugh (1996).

Also, historical or Monte Carlo simulation-based to ols are used when the p ortfolio contains non-

linear instruments such as options (Bucay and Rosen (1999), Jorion (1996), Mauser and Rosen

(1999), Pritsker (1997), RiskMetrics (1996), Beder (1995), Stambaugh (1996)). Discussions of

optimization problems involving VaR can be found in pap ers by Litterman (1997a,1997b), Kast

et al. (1998), Lucas and Klaassen (1998).

Although VaR is a very p opular measure of risk, it has undesirable mathematical charac-

teristics such as a lack of subadditivity and convexity, see Artzner et al. (1997,1999). VaR is

coherent only when it is based on the standard deviation of normal distributions (for a normal

distribution VaR is prop ortional to the standard deviation). For example, VaR asso ciated with a

combination of two portfolios can be deemed greater than the sum of the risks of the individual

p ortfolios. Furthermore, VaR is dicult to optimize when it is calculated from scenarios. Mauser

and Rosen (1999), McKay and Keefer (1996) showed that VaR can b e ill-behaved as a function

of p ortfolio p ositions and can exhibit multiple lo cal extrema, which can be a ma jor handicap

in trying to determine an optimal mix of p ositions or even the VaR of a particular mix. As an

alternative measure of risk, CVaR is known to have b etter prop erties than VaR, see Artzner et al.

(1997), Embrechts (1999). Recently, Pug (2000) proved that CVaR is a coherent risk measure

having the following properties: transition-equivariant, positively homogeneous, convex, mono-

tonic w.r.t. sto chastic dominance of order 1, and monotonic w.r.t. monotonic dominance of order

2. A simple description of the approach for minimization of CVaR and optimization problems

with CVaR constraints can be found in the review pap er by Uryasev (2000). Although CVaR

has not b ecome a standard in the nance industry, CVaR is gaining in the insurance industry,

see Embrechts et al. (1997). Bucay and Rosen (1999) used CVaR in credit risk evaluations. A

case study on application of the CVaR metho dology to the credit risk is describ ed by Andersson

and Uryasev (1999). Similar measures as CVaR have b een earlier introduced in the sto chastic

programming literature, although not in nancial mathematics context. The conditional exp ec-

tation constraints and integrated chance constraints describ ed by Prekopa (1995) may serve the

same purp ose as CVaR.

Minimizing CVaR of a p ortfolio is closely related to minimizing VaR, as already observed from

the denition of these measures. The basic contribution of this pap er is a practical technique of

optimizing CVaR and calculating VaR at the same time. It aords a convenientwayofevaluating



linear and nonlinear derivatives (options, futures);



market, credit, and op erational risks;



circumstances in any corp oration that is exp osed to nancial risks.

It can be used for such purp oses by investment companies, brokerage rms, mutual funds, and

elsewhere.

In the optimization of p ortfolios, the new approach leads to solving a sto chastic optimization

problem. Manynumerical algorithms are available for that, see for instance, Birge and Louveaux

(1997), Ermoliev and Wets (1988), Kall and Wallace (1995), Kan and Kibzun (1996), Pug (1996),

Prekopa (1995). These algorithms are able to make use of sp ecial mathematical features in the

p ortfolio and can readily be combined with analytical or simulation-based metho ds. In cases

where the uncertainty is modeled by scenarios and a nite family of scenarios is selected as an

approximation, the problem to b e solved can even reduce to linear programming. On applications

of the sto chastic programming in nance area, see, for instance, Zenios (1996), Ziemba and Mulvey

(1998).

2 DESCRIPTION OF THE APPROACH

Let

(

;

) be the loss associated with the decision vector

,tobechosen from a certain subset

, and the random vector

. (We use b oldface type for vectors to distinguish them

from scalars.) The vector

can be interpreted as representing a p ortfolio, with

as the set of

available portfolios (sub ject to various constraints), but other interpretations could be made as

well. The vector

stands for the uncertainties, e.g. in market parameters, that can aect the

loss. Of course the loss might b e negative and thus, in eect, constitute a gain.

For each

, the loss

(

;

) is a random variable having a distribution in

induced by that

. The underlying probability distribution of

will b e assumed for convenience to have

density, whichwe denote by

(

). However, as it will b e shown later, an analytical expression

(

)

for the implementation of the approach is not needed. It is enough to have an algorithm (co de)

which generates random samples from

(

). Atwo step pro cedure can b e used to derive analytical

expression for

(

) or construct a Monte Carlo simulation code for drawing samples from

(

)

(see, for instance, RiskMetrics (1996)): (1) mo deling of risk factors in

,(with

), (2)

based on the characteristics of instrument

;:::;n

, the distribution

(

) can be derived or

co de transforming random samples of risk factors to the random samples from density

(

) can

constructed.

The probabilityof

(

;

) not exceeding a threshold



is given then by

(

;

) =

(

;

)





(

)

(1)

As a function of



for xed

,(

;

) is the cumulative distribution function for the loss asso ciated

with

. It completely determines the b ehavior of this random variable and is fundamental in

dening VaR and CVaR. In general, (

;

) is nondecreasing with resp ect to



and continuous

from the right, but not necessarily from the left b ecause of the p ossibility of jumps. We assume

however in what follows that the probability distributions are such that no jumps o ccur, or in

other words, that (

;

) is everywhere continuous with resp ect to



. This assumption, like

the previous one about density in

, is made for simplicity. Without it there are mathematical

complications, even in the denition of CVaR, which would need more explanation. We prefer

to leave such technical issues for a subsequent paper. In some common situations, the required

continuity follows from prop erties of loss

(

;

) and the density

(

); see Uryasev (1995).

The



-VaR and



-CVaR values for the loss random variable asso ciated with

and any sp ecied

probability level



in (0

;

1) will b e denoted by





(

) and





(

). In our setting they are given by





(

) = min



: (

;

)





(2)

and





(

) = (1



)

(

;

)







(

)

(

;

)

(

)

(3)

剩余25页未读，继续阅读

评论收藏

内容反馈

Quant0xff

粉丝: 1w+
资源: 459

optimization of Conditional Value-at-Risk.pdf

最新资源

optimization of Conditional Value-at-Risk.pdf

optimization of conditional value-at-risk.pdf

optimization of conditional value at risk.pdf

credit risk optimization with conditional VaR criterion.pdf

Value at Risk

StatFacts_Conditional_Value_at_Risk.pdf

论文研究-评估Value-at-Risk模型——条件矩检验方法.pdf

Worst-Case_Conditional_Value-at-Risk_with_Applicat.pdf

论文研究-基于 Bayesian-SV-SGT 模型的原油价格‘Value at Risk’估计.pdf

Andersson2001_Article_CreditRiskOptimizationWithCond.pdf

Data Driven Mean-CVaR Portfolio Optimization Model.pdf

Python库 | qiskit-finance-0.2.1.tar.gz

CVaR_Robust_Mean-CVaR_Portfolio_Optimization.pdf

Robust Optimization - Princeton University Press

Python for analyze big financial data.pdf

Optimization with uncertain data.pdf

cvar代码matlab-A-novel-DRO-model-for-self-scheduling-problem:本研究使用带有条件风险值

Robust VaR and CVaR optimization under joint ambiguity .pdf

portfolio optimization based on risk measures and ensemble EMD.pdf

python for finance

matlab本研究采用带有条件风险值(CVaR)的分布鲁棒优化(DRO)算法求解自调度问题

Cvar Optimization Algorithms and Application.pdf

基于WOA优化的CVAR模型的售电公司的购电侧模型和售电侧模型matlab实现-源码

Introducing Monte Carlo Methods with R

CVaRoptimization_CVAR_portfolioCVAR_源码.zip

EMP.jl：Julia中的扩展数学编程

portfolio-optimization

MATLAB金融计算_matlab源码.rar

最新资源