Conditional value-at-risk for general loss distributions

Conditional value-at-risk for general loss distributions R. Tyrrell Rockafellar a, Stanislav Uryasev b,* a Department of Mathematics, University of Washington, P.O. Box 354350, Seattle, WA 98195-4350, USA b Risk Management and Financial Engineering Lab, Department of Industrial and Systems Engineering, University of Florida, P.O. Box 116595, Gainesville, FL 32611-6595, USA Abstract Fundamental properties of conditional value-at-risk (CVaR), as a measure of risk with sig- nificant advantages over value-at-risk (VaR), are derived for loss distributions in finance that can involve discreetness. Such distributions are of particular importance in applications be- cause of the prevalence of models based on scenarios and finite sampling. CVaR is able to quantify dangers beyond VaR and moreover it is coherent. It provides optimization short-cuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach. The numerical efficiency and stability of such calcula- tions, shown in several case studies, are illustrated further with an example of index tracking. � 2002 Elsevier Science B.V. All rights reserved. JEL classification: C0; C1; G0; N2 Keywords: Value-at-risk; Conditional value-at-risk; Mean shortfall; Coherent risk measures; Risk sam- pling; Scenarios; Hedging; Index tracking; Portfolio optimization; Risk management 1. Introduction Measures of risk have a crucial role in optimization under uncertainty, especially in coping with the losses that might be incurred in finance or the insurance industry. Loss can be envisioned as a function z ¼ f ðx; yÞ of a decision vector x 2 X � Rn rep- resenting what we may generally call a portfolio, with X expressing decision con- straints, and a vector y 2 Y � Rm representing the future values of a number of Journal of Banking & Finance 26 (2002) 1443–1471 www.elsevier.com/locate/econbase *Corresponding author. E-mail address: uryasev@ufl.edu (S. Uryasev). URL: http://www.ise.ufl.edu/uryasev. 0378-4266/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved. PII: S0378-4266 (02 )00271-6 variables like interest rates or weather data. When y is taken to be random with known probability distribution, z comes out as a random variable having its distri- bution dependent on the choice of x. Any optimization problem involving z in terms of the choice of x should then take into account not just expectations, but also the ‘‘riskiness’’ of x. Value-at-risk, or VaR for short, is a popular measure of risk which has achieved the high status of being written into industry regulations (see, for instance, Jorion, 1996; Pritsker, 1997). It suffers, however, from being unstable and difficult to work with numerically when losses are not ‘‘normally’’ distributed – which in fact is often the case, because loss distributions tend to exhibit ‘‘fat tails’’ or empirical discrete- ness. Moreover, VaR fails to be coherent in the sense of Artzner et al. (1999). A very serious shortcoming of VaR, in addition, is that it provides no handle on the extent of the losses that might be suffered beyond the threshold amount indicated by this measure. It is incapable of distinguishing between situations where losses that are worse may be deemed only a little bit worse, and those where they could well be overwhelming. Indeed, it merely provides a lowest bound for losses in the tail of the loss distribution and has a bias toward optimism instead of the conservatism that ought to prevail in risk management. An alternative measure that does quantify the losses that might be encountered in the tail is conditional value-at-risk, or CVaR. As a tool in optimization modeling, CVaR has superior properties in many respects. It maintains consistency with VaR by yielding the same results in the limited settings where VaR computations are tractable, i.e., for normal distributions (or perhaps ‘‘elliptical’’ distributions as in Embrechts et al. (2001)); for portfolios blessed with such simple distributions, working with CVaR, VaR, or minimum variance (Markowitz, 1952) are equivalent (cf. Rockafellar and Uryasev, 2000). Most importantly for applications, however, CVaR can be expressed by a remarkable minimization formula. This formula can readily be incorporated into problems of optimization with respect to x 2 X that are designed to minimize risk or shape it within bounds. Significant shortcuts are thereby achieved while preserving crucial problem features like convexity. Such computational advantages of CVaR over VaR are turning into a major stim- ulus for the development of CVaR methodology, in view of the fact that efficient al- gorithms for optimization of VaR in high-dimensional settings are still not available, despite the substantial efforts that have gone into research in that direction (Ander- sen and Sornette, 1999; Basak and Shapiro, 1998; Gaivoronski and Pflug, 2000; Gourieroux et al., 2000; Grootweld and Hallerbach, 2000; Kast et al., 1998; Puelz, 1999; Tasche, 1999). CVaR and its minimization formula were first developed in our paper (Rockafellar and Uryasev, 2000). There, we demonstrated numerical effectiveness through sev- eral case studies, including portfolio optimization and options hedging. In follow- up work in Krokhmal et al. (in press), investigations were carried out with the minimization of CVaR subject to a constraint on expected return, the maximization of return subject to a constraint on the CVaR, and the maximization of a utility func- tion that balances CVaR against return. Strategies for investigating the efficient fron- tier between CVaR and return were considered as well. In Andersson et al. (2000), the 1444 R.T. Rockafellar, S. Uryasev / Journal of Banking & Finance 26 (2002) 1443–1471 approach was applied to credit risk management of a portfolio of bonds. Extensions in Checklov et al. (in press) have centered on a closely related notion of conditional drawdown-at-risk (CDaR), in the optimization of portfolios with draw-down con- straints. In these works, with their focus on demonstrating the potential of the new ap- proach, discussion of CVaR in its full generality was postponed. Only continuous loss distributions were treated, and in fact, for the sake of an elementary initial jus- tification of the minimization formula so as to get started with using it, distributions were assumed to have smooth density. In the present paper we drop those limitations and complete the foundations for our methodology. This step is needed of course not just for theory, but because many problems of optimization under uncertainty in- volve discontinuous loss distributions in which the discrete probabilities come out of scenario models or the finite sampling of random variables. While some conse- quences of our minimization formula itself have since been explored by Pflug (2000) in territory outside of the assumptions we made in Rockafellar and Uryasev (2000), an understanding of what the quantity given by the formula then represents in the usual framework of risk measures in finance has been missing. For continuous loss distributions, the CVaR at a given confidence level is the ex- pected loss given that the loss is greater than the VaR at that level, or for that matter, the expected loss given that the loss is greater than or equal to the VaR. For distri- butions with possible discontinuities, however, it has a more subtle definition and can differ from either of those quantities, which for convenience in comparison can be designated by CVaRþ and CVaR�, respectively. CVaRþ has sometimes been called ‘‘mean shortfall’’ (cf. Mausser and Rosen (1999), although the seemingly identical term ‘‘expected shortfall’’ has been interpreted in other ways in Acerbi and Nordio (2001); Acerbi and Tasche (2001), with the latter paper taking it as a synonym for CVaR itself), while ‘‘tail VaR’’ is a term that has been suggested for CVaR� (cf. Artz- ner et al., 1999). Here, in order to consolidate ideas and reduce the potential for con- fusion, we speak of CVaRþ and CVaR� simply as ‘‘upper’’ and ‘‘lower’’ CVaR. Generally CVaR�6CVaR6CVaRþ, with equality holding when the loss distribu- tion function does not have a jump at the VaR threshold; but when a jump does occur, which for scenario models is always the situation, both inequalities can be strict. On the basis of the general definition of CVaR elucidated below, and with the help of arguments in Pflug (2000), CVaR is seen to be a coherent measure of risk in the sense of Artzner et al. (1999), whereas CVaRþ and CVaR� are not. (A direct alter- native proof of this fact has very recently been furnished by Acerbi and Nordio (2001).) The lack of coherence of CVaRþ and CVaR� in the presence of discreteness does not seem to be widely appreciated, although this shortcoming was already noted for CVaR� by the authors Artzner et al. (1999). They suggested, as a remedy, still another measure of risk which they called ‘‘worst conditional expectation’’ and proved to be coherent. That measure is impractical for applications, however, be- cause it can only be calculated in very narrow circumstances. In contrast, CVaR is not only coherent but eminently practical by virtue of our minimization formula for it. That formula opens the door to computational techniques for dealing with risk far more effectively than before. R.T. Rockafellar, S. Uryasev / Journal of Banking & Finance 26 (2002) 1443–1471 1445 Interestingly, CVaR can be viewed as a weighted average of VaR and CVaRþ (with the weights depending, like these values themselves, on the decision x). This seems surprising, in the face of neither VaR nor CVaRþ being coherent. The weights arise from the particular way that CVaR ‘‘splits the atom’’ of probability at the VaR value, when one exists. Besides laying out such implications of the general definition of CVaR and its as- sociated minimization formula, we put effort here into bringing out properties of CVaR that enhance the usefulness of this approach when dealing with fully discrete distributions. For such distributions, we furnish an elementary way of calculating CVaR directly. We show how a suitable specification of the confidence level, depend- ing on the finite, discrete distribution of y, can ensure that CVaR ¼ CVaRþ regard- less of the choice of x. For confidence levels close enough to 1, we prove that CVaR, CVaR� and VaR coincide with maximum loss, and again this can be ensured inde- pendently of x. We go over the optimization shortcuts offered by CVaR and extend them to mod- els where risk is shaped at several confidence levels. As part of this, CVaR is proved to be stable with respect to the choice of the confidence level, although other pro- posed measures of risk are not. Finally, we illustrate the main facts and ideas with a numerical example of port- folio replication with CVaR constraints. This example demonstrates how the in- corporation of such constraints in a financial model may improve both the in-sample and the out-of-sample risk characteristics. The calculations confirm that CVaR methodology offers a management tool for efficiently controlling risks in practice. Broadly speaking, problems of risk management with VaR and CVaR can be clas- sified as falling under the heading of stochastic optimization. Various other concepts of risk in optimization have earlier been studied in the stochastic programming lit- erature, but not in a context of finance (see Birge and Louveaux, 1997; Ermoliev and Wets, 1988; Kall and Wallace, 1994; Kan and Kibzun, 1996; Pflug, 1996; Prek- opa, 1995; Rubinstein and Shapiro, 1993). The reader interested in applications of stochastic optimization techniques in the finance area can find relevant papers in Zenios, 1993; Ziemba and Mulvey, 1998. For elucidation of the many statements in this paper that rely on background in convex optimization, we refer the reader to the book Rockafellar (1970) (or Rocka- fellar and Wets, 1997). Additional properties of CVaR, including a powerful result on estimation, are available in the new paper of Acerbi and Tasche (2001). 2. General concept of conditional value-at-risk In everything that follows, we suppose the random vector y is governed by a prob- ability measure P on Y (a Borel measure) that is independent of x. (The indepen- dence could be relaxed for some purposes, but it is essential for key results about convexity that underlie the use of linear programming reductions in computation.) 1446 R.T. Rockafellar, S. Uryasev / Journal of Banking & Finance 26 (2002) 1443–1471 For each x, we denote by Wðx; �Þ on R the resulting distribution function for the loss z ¼ f ðx; yÞ, i.e., Wðx; fÞ ¼ P y jf ðx; yÞf 6 fg; ð1Þ making the technical assumptions that f ðx; yÞ is continuous in x and measurable in y, and that Efjf ðx; yÞjg <1 for each x 2 X . We denote by Wðx; f�Þ the left limit of Wðx; �Þ at f; thus Wðx; f�Þ ¼ Pfy jf ðx; yÞ < fg: ð2Þ When the difference Wðx; fÞ �Wðx; f�Þ ¼ Pfy jf ðx; yÞ ¼ fg ð3Þ is positive, so that Wðx; �Þ has a jump at f, a probability ‘‘atom’’ is said to be present at f. We consider a confidence level a 2 ð0; 1Þ, which in applications would be some- thing like a ¼ 0:95 or 0.99. At this confidence level, there is a corresponding VAR, defined in the following way. Definition 1 (VaR). The a-VaR of the loss associated with a decision x is the value faðxÞ ¼ minff jWðx; fÞP ag: ð4Þ The minimum in (4) is attained because Wðx; fÞ is nondecreasing and right-contin- uous in f. When Wðx; �Þ is continuous and strictly increasing, faðxÞ is simply the un- ique f satisfying Wðx; fÞ ¼ a. Otherwise, this equation can have no solution or a whole range of solutions. The case of no solution corresponds to a vertical gap in the graph of Wðx; �Þ as in Fig. 1, with a lying in an interval of confidence levels that all yield the same VaR. The lower and upper endpoints of that interval are a�ðxÞ ¼ Wðx; faðxÞ�Þ; aþðxÞ ¼ Wðx; faðxÞÞ: ð5Þ Fig. 1. Equation Wðx; fÞ ¼ a has no solution in f. R.T. Rockafellar, S. Uryasev / Journal of Banking & Finance 26 (2002) 1443–1471 1447 The case of a whole range of solutions corresponds instead to a constant segment of the graph, as shown in Fig. 2. The solutions form an interval having faðxÞ as its lower endpoint. The upper endpoint of the interval is the value fþa ðxÞ introduced next. Definition 2 (VaRþ). The a-VaRþ (‘‘upper’’ a-VaR) of the loss associated with a decision x is the value fþa ðxÞ ¼ infff jWðx; fÞ > ag: ð6Þ Obviously faðxÞ6 fþa ðxÞ always, and these values are the same except when Wðx; fÞ is constant at level a over a certain f-interval. That interval is either ½faðxÞ; fþa ðxÞÞ or ½faðxÞ; fþa ðxÞ�, depending on whether or not Wðx; �Þ has a jump at fþa ðxÞ. Both Figs. 1 and 2 illustrate phenomena that raise challenges in the treatment of general loss distributions. This is especially true for discrete distributions associated with finite sampling or scenario modeling, since then Wðx; �Þ is a step function (con- stant between jumps), and there is no getting around these circumstances. Observe, for instance, that the situation in Fig. 2 entails a discontinuity in the be- havior of VaR: A jump is sure to occur if a slightly higher confidence level is de- manded. This degree of instability is distressing for a measure of risk on which enormous sums of money might be riding. Furthermore, although x is fixed in this picture, examples easily show that the misbehavior in the dependence of VaR on a can effect its dependence on x as well. That makes it hard to cope successfully with VaR-centered problems of optimization in x. These troubles, and many others, motivate the search for a better measure of risk than VaR for practical applications. Such a measure is CVaR. Definition 3 (CVaR). The a-CVaR of the loss associated with a decision x is the value /aðxÞ ¼ mean of the a-tail distribution of z ¼ f ðx; yÞ; ð7Þ Fig. 2. Equation Wðx; fÞ ¼ a has many solutions in f. 1448 R.T. Rockafellar, S. Uryasev / Journal of Banking & Finance 26 (2002) 1443–1471 where the distribution in question is the one with distribution function Waðx; �Þ de- fined by Waðx; fÞ ¼ 0 for f < faðxÞ;½Wðx; fÞ � a�=½1� a� for fP faðxÞ: � ð8Þ Note that Waðx; �Þ truly is another distribution function, like Wðx; �Þ: it is nonde- creasing and right-continuous, with Waðx; fÞ ! 1 1 as f !1. The a-tail distribution referred to in (7) is thus well defined through (8). The subtlety of Definition 3 resides in the case where the loss with distribution function Wðx; �Þ has a probability atom at faðxÞ, as illustrated in Fig. 1. In that case the interval ½faðxÞ;1Þ has probability greater than 1� a, inasmuch as Wðx; faðxÞ�Þ < a6Wðx; faðxÞÞ when Wðx; faðxÞ�Þ < Wðx; faðxÞÞ; ð9Þ and the issue comes up of what really should be meant by the a-tail distribution, since that term presumably ought to refer to the ‘‘upper 1� a part’’ of the full distribution. This is resolved by specifying the a-tail distribution through the distribution function in (8), which is obtained by rescaling the portion of the graph of the original dis- tribution between the horizontal lines at levels 1� a and 1 so that it spans instead between 0 and 1. For the case shown in Fig. 1, the result is depicted in Fig. 3. The consequences of this maneuver will be examined in relation to the following variants in which the whole interval ½faðxÞ;1Þ or its interior ðfaðxÞ;1Þ are the focus. Definition 4 (CVaRþ and CVaR�). The a-CVaRþ (‘‘upper’’ a-CVaR) of the loss associated with a decision x is the value /þa ðxÞ ¼ E f ðx; yÞ jf ðx; yÞf > faðxÞg; ð10Þ whereas the a-CVaR� (‘‘lower’’ a-CVaR) of the loss is the value /�a ðxÞ ¼ E f ðx; yÞ jf ðx; yÞf P faðxÞg: ð11Þ Fig. 3. Distribution function Waðx; fÞ is obtained by rescaling the function Wðx; fÞ in the interval ½a; 1�. R.T. Rockafellar, S. Uryasev / Journal of Banking & Finance 26 (2002) 1443–1471 1449 The conditional expectation in (11) is well defined because Pff ðx; yÞ jf ðx; yÞP faðxÞgP 1� a > 0, but the one in (10) only makes sense as long as Pff ðx; yÞ j f ðx; yÞ > faðxÞg > 0, i.e., Wðx; faðxÞÞ < 1, which is not assured merely through our assumption that a 2 ð0; 1Þ, since there might be a probability atom at faðxÞ large enough to cover the interval 1� a�ðxÞ. As indicated in the introduction, (10) is sometimes called ‘‘mean shortfall’’. The closely related expression Eff ðx; yÞ � faðxÞ jf ðx; yÞ > faðxÞg ¼ /þa ðxÞ � faðxÞ ð12Þ goes however by the name of ‘‘mean excess loss’’; cf. Bassi et al. (1998); Embrechts et al. (1997). In ordinary language, a shortfall might be thought the same as an excess loss, so ‘‘mean shortfall’’ for (10) potentially poses a conflict. The conditional ex- pectation in (11) has been dubbed in (Artzner et al., 1999) the ‘‘tail VaR’’ at level a, but as revealed in the proof of the next proposition, it is really the mean of the tail distribution for the confidence level a�ðxÞ in (5) rather than the one appropriate to a itself. The ‘‘upper’’ and ‘‘lower’’ terminology in Definition 4 avoids such difficulties while emphasizing the basic relationships among these values that are described next. Proposition 5 (Basic CVaR relations). If there is no probability atom at faðxÞ, one simply has /�a ðxÞ ¼ /aðxÞ ¼ /þa ðxÞ: ð13Þ If a probability atom does exist at faðxÞ, one has /�a ðxÞ < /aðxÞ ¼ /þa ðxÞ when a ¼ Wðx; faðxÞÞ; ð14Þ or on the other hand, /�a ðxÞ ¼ /aðxÞ when Wðx; faðxÞÞ ¼ 1 ð15Þ (with /þa ðxÞ then being ill defined). But in all the remaining cases, characterized by Wðx; faðxÞ�Þ < a < Wðx; faðxÞÞ < 1; ð16Þ one has the strict inequality /�a ðxÞ < /aðxÞ < /þa ðxÞ: ð17Þ Proof. In comparison with the definition of /aðxÞ in (7), the /þa ðxÞ value in (10) is the mean of the loss distribution associated with Wþa ðx; fÞ ¼ 0 for f < faðxÞ; ½Wðx; fÞ � aþðxÞ�=½1� aþðxÞ� for f P faðxÞ; � ð18Þ whereas the /�a ðxÞ value in (11) is the mean of the loss distribution associated with W�a ðx; fÞ ¼ 0 for