Portfolio Optimization with Higher Moment Risk Measures

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1 Portfolio Optimization with Higher Moment Risk Measures Pavlo A. Krokhmal Jieqiu Chen Department of Mechanical and Industrial Engineering The University of Iowa, 2403 Seamans Center, Iowa City, IA Version: April 2006 Abstract The paper considers modeling of risk-averse preferences in stochastic programming problems using risk measures. We utilize the axiomatic foundation of coherent risk measures and deviation measures in order to develop simple representations that express risk measures via solutions of specially constructed stochastic programming problems. Using the developed representations, we introduce a new family of higher-moment coherent risk measures that measures, and, in particular, the second-moment coherent risk measure (SMCR). It is demonstrated that the SMCR measure is compatible with the second order stochastic dominance, and can be efficiently used in portfolio optimization, especially by investors with aggressive risk preferences. Keywords: Risk measures, portfolio optimization, stochastic programming, stochastic dominance 1 Introduction Research and practice of portfolio optimization is driven to a large extent by tailoring the measures of reward (satisfaction) and risk (unsatisfaction/regret) of the investment venture to the specific preferences of an investor. While there exists a common consensus that an investment s reward may be adequately associated with its expected return, the methods for proper modeling and measurement of an investment s risk are subject to much more pondering and debate. In fact, the risk-reward or mean-risk models constitute a major part of the areas of portfolio optimization and, more generally, decision making under uncertainty. The cornerstone of modern portfolio analysis was set up by Markowitz (1952, 1959), who advocated identification of the portfolio s risk with the volatility (variance) of its returns. More importantly, Markowitz s work led to formalization of the fundamental view that any decision under uncertainties may be evaluated in terms of its risk and reward. The seminal Markowitz s ideas are still widely used today in many areas of decision making, and the entire paradigm of bi-criteria risk-reward optimization received extensive development in both directions of increasing the computational efficiency and enhancing the models for risk measurement and estimation. For example, the linear Corresponding author. krokhmal@engineering.uiowa.edu 1

2 Mean-Absolute Deviation model (Konno and Yamazaki, 1991) targeted simplifications in dealing with risk-constrained problems, where risk is estimated similarly to the Markowitz Mean-Variance (MV) methodology. At the same time, it has been recognized that the symmetric attitude of the classical MV approach, where both the positive and negative deviations from the expected level are penalized equally, does not always yield an adequate estimation of risks induced by the uncertainties. Hence, significant effort has been devoted to the development of downside risk measures and models. Replacing the variance by the lower standard semi-deviation as a measure of investment risk so as to take into account only negative deviations from the expected level, has been proposed as early as by Markowitz (1959); see also more recent works by Ogryczak and Ruszczyński (1999, 2001, 2002). Among the popular downside risk models we mention the lower partial moment, or Expected Regret, in stochastic programming also known as integrated chance constraint (Bawa, 1975; Dembo and Rosen, 1999; Testuri and Uryasev, 2003; van der Vlerk, 2003). Widely known in finance and banking industry is the Value-at-Risk measure (JP Morgan, 1994; Jorion, 1997; Duffie and Pan, 1997; Larsen et al., 2002). Being simply a quantile of loss distribution, the Value-at-Risk (VaR) concept has its counterparts in stochastic optimization (probabilistic, or chance programming, see Prékopa, 1995), reliability theory, etc. Yet, minimization or control of risk using the VaR measure proved to be technically and methodologically difficult, mainly due to VaR s notorious non-convexity as a function of decision variables. A downside risk measure that circumvents the shortcomings of VaR while offering the same quantile approach to estimation of risk, is the Conditional Value-at-Risk measure (Rockafellar and Uryasev, 2000, 2002; Krokhmal et al., 2002a). Risk measures that are similar to CVaR and/or in some cases coincide with it, are Expected Shortfall and Tail VaR (Acerbi et al., 2001; Acerbi and Tasche, 2002), see also Conditional Drawdown-at-Risk (Chekhlov et al., 2005; Krokhmal et al., 2002b). A simple yet effective risk measure closely related to CVaR is the so-called Maximum Loss, or Worst-case risk (Young, 1998; Krokhmal et al., 2002b), also known as the robust optimization concept (see, e.g., Kouvelis and Yu, 1997), etc. In the last few years, the formal risk theory received a major impetus from the works of Artzner et al. (1997, 1999, 2001) and Delbaen (2000), who introduced an axiomatic approach to definition and construction of risk measures by developing the concept of coherent risk measures. Among the risk measures that satisfy the coherency properties, there are Conditional Value-at-Risk, Maximum Loss (Pflug, 2000; Acerbi and Tasche, 2002), coherent risk measures based on one-sided moments (Fischer, 2003), etc. Recently, Rockafellar et al. (2002, 2004) have extended the formal theory of risk measures to the case of deviation measures, and demonstrated a close relationship between coherent risk measures and deviation measures; spectral measures of risk have been proposed by Acerbi (2002). An approach to decision making under uncertainty, different from the risk-reward paradigm, is embodied by the von Neumann and Morgenstern (1944) utility theory, which utilizes a mathematically sound axiomatic approach to description of risk-averse and construction of corresponding decisions. Along with its numerous modifications and extensions, the vnm utility is widely adopted as a basic model of rational choice, especially in economics and social sciences (see, among others, Fishburn, 1970, 1988; Karni and Schmeidler, 1991, etc). Thus, substantial attention has been paid in the literature to development of risk-reward optimization models and risk measures that are consistent with the vnm utility theory. In particular, it has been shown that under certain conditions the Markovitz MV framework is consistent with the von Neumann & Morgenstern theory (Kroll et al., 1984). Ogryczak and Ruszczyński (1999, 2001, 2002) developed mean-semideviation models that are 2

3 consistent with stochastic dominance concepts (Fishburn, 1964; Rothschild and Stiglitz, 1970; Levy, 1998, etc.) Optimization with stochastic dominance constraints was considered by Ruszczyński and Dentcheva (2003), etc. In this paper we aim to offer an additional insight into the properties of axiomatically defined risk measures by developing a number of representations that express risk measures via solutions of stochastic programming problems. Using the developed representations, we construct a new family of highermoment coherent risk measures that are compatible with the vnm utility theory via the second-order stochastic dominance. In particular, we focus on the second-moment coherent risk measure (SMCR) that quantifies risk using second moments of the tail of the loss distribution, and is of particular interest in the context of portfolio optimization. The conducted case study indicates that the SMCR measure has a high potential for practical application in portfolio selection problems, and may be particularly attractive to investors with aggressive risk preferences. 2 Modeling of risk measures as stochastic programs The discussion in the Introduction section has illustrated the variety of approaches to definition and estimation of risk. Arguably, the recent advances in risk theory are associated with the axiomatic approach to construction of risk measures pioneered by Artzner et al. (1997, 1999, 2001). The present endeavor essentially exploits this axiomatic approach in order to devise simple computational recipes for dealing with several types of risk measures by representing them in the form of stochastic programming problems. These representations can be used to create new risk measures to be tailored to specific risk preferences, as well as to incorporate these preferences in stochastic programming problems. In particular, we present a new family of Higher -Moment Coherent Risk measures (HMCR), and discuss in detail its member, the Second-Order Coherent Risk measure (SMCR). It will be shown that the SMCR measure is well-behaved in terms of theoretical properties, and demonstrates very promising performance in test applications. Within the axiomatic framework of risk analysis, risk measure R(X) of a random outcome X from some probability space (, F, µ) may be defined as a mapping R : X R, where X is a linear space of F -measurable functions X : R. In a more general setting one may assume X to be a separated locally convex space; for our purposes it suffices to consider X = L p (, F, P), 1 p, where the particular value of p shall be clear from the context. Traditionally to convex analysis, we call function f : X R proper if f (X) > for all X X and dom f, i.e., there exists X X such that f (X) < + (see, e.g., Rockafellar, 1970; Zălinescu, 2002). In the remainder of the paper, we confine ourselves to risk measures that are proper and not identically equal to +. Throughout the paper, it is assumed that X represents a loss function, i.e., small values of X are good, and large values are bad. 2.1 Coherent risk measures A coherent risk measure, according to Artzner et al. (1999) and Delbaen (2000), is defined as a mapping R : X R that further satisfies the next four properties (axioms): (A1) monotonicity: X 0 R(X) 0 for all X X, 3

4 (A2) sub-additivity: R(X + Y ) R(X) + R(Y ) for all X, Y X, (A3) positive homogeneity: R(λX) = λr(x) for all X X, λ > 0, (A4) translation invariance: R(X + a) = R(X) + a for all X X, a R. Since axioms (A2) and (A3) immediately yield convexity of coherent risk measures, little will be lost in the generality of the above definition if one replaces the sub-additivity requirement (A2) with the stronger requirement of convexity: (A2 ) convexity: R ( λx + (1 λ)y ) λr(x) + (1 λ)r(y ), X, Y X, 0 λ 1. From the axioms (A1) (A4) one can easily derive the following useful properties of coherent risk measures (see, for example, Delbaen, 2000): (C1) R(0) = 0 [A1+A2 or A3], (C2) X Y R(X) R(Y ) [A1+A2], (C3) R(a) = a, a R [A4+C1], (C4) X a R(X) a, a R [C2+C3], (C5) R ( X R(X) ) = 0 [A4+C3], (C6) R(X) is continuous in its effective domain, where the expressions in brackets indicate the axioms or properties that support the given statement. Also, throughout the paper the inequalities X a, X Y, etc., are assumed to hold almost surely. From the definition of coherent risk measures it is easy to see that, for example, EX and ess.sup X, where min{ η R X η }, if { η R X η }, ess.sup X =, otherwise, are coherent risk measures; more examples can be found in Rockafellar et al. (2002). Below we present simple computational formulas that aid in construction of coherent risk measures and their incorporation into stochastic programs. Namely, we execute the idea that one of the axioms (A3) or (A4) can be relaxed and then reinforced by solving an appropriately defined mathematical programming problem. In other words, one can construct a coherent risk measure via solution of a stochastic programming problem that involves a function φ : X R satisfying only three of the four axioms (A1) (A4). First we present a representation for coherent risk measures that is based on relaxation of the translation invariance axiom (A4). The next theorem shows that if one selects a function φ : X R satisfying axioms (A1) (A3) along with additional technical conditions, then there exists a simple stochastic optimization problem involving φ whose optimal value would satisfy (A1) (A4). 4

5 Theorem 1 Let function φ : X R satisfy axioms (A1) (A3), and be a lsc function such that φ(η) > η for all real η 0. Then the optimal value of the stochastic programming problem ρ(x) = inf η η + φ(x η) (1) is a proper coherent risk measure, and the infimum is attained for all X, so inf η in (1) may be replaced by min η R. Proof. Convexity, lower semicontinuity, and sublinearity of φ in X imply that function φ X (η) = η + φ(x η) is also convex, lsc, and proper in η R for each fixed X X. For the infimum of φ X (η) to be achievable at finite η, its recession function has to be positive: φ X 0 + (±1) > 0, which is equivalent to φ X 0 + (ξ) > 0, ξ 0, due to the positive homogeneity of φ X. By definition of the recession function (Rockafellar, 1970; Zălinescu, 2002) and positive homogeneity of φ, we have that the last condition holds if φ(ξ) > ξ for all ξ 0: φ X 0 + η + τξ + φ(x η τξ) η φ(x η) (ξ) = lim = ξ + φ( ξ). τ τ Hence, ρ(x) defined by (1) is a proper lsc function, and minimum in (1) is attained at finite η for all X X. Below we verify that ρ(x) satisfies axioms (A1) (A4). Indeed, properness of representationlinear. (A1) Let X 0. Then φ(x) 0 as φ satisfies (A1), which implies min η R η + φ(x η) 0 + φ(x 0) 0. (A2) For any Z X let η Z arg min η R {η + φ(z η)} R, then ρ(x) + ρ(y ) = η X + φ(x η X ) + η Y + φ(y η Y ) η X + η Y + φ(x + Y η X η Y ) η X+Y + φ(x + Y η X+Y ) = ρ(x + Y ). (A3) For any fixed λ > 0 we have ρ(λx) = min η R (A4) Similarly, for any fixed a R, ρ(x + a) = min η R { η + φ(λx η) } = λ min η R = a + min η R { } η + φ(x + a η) { η/λ + φ(x η/λ) } = λρ(x). (2) { (η a) + φ ( X (η a) )} = a + ρ(x). (3) Thus, ρ(x) defined by (1) is a proper coherent risk measure. Remark 1.1 It is all-important that the stochastic programming problem (1) is convex, due to the convexity of function φ. Also, note that one cannot substitute a coherent risk measure itself for function φ in (1), as it will violate the condition φ(η) > η of the Theorem. 5

6 Corollary 1.1 The set arg min η R { η + φ(x η) } R of optimal solutions of (1) is closed. Example 1.1 (Conditional Value-at-Risk) A famous special case of (1) is the optimization formula for Conditional Value-at-Risk (Rockafellar and Uryasev, 2000, 2002): CVaR α (X) = min η R η + (1 α) 1 E(X η) +, 0 < α < 1, (4) where (X) ± = max{±x, 0}, and function φ(x) = (1 α) 1 E(X) + evidently satisfies the conditions of Theorem 1. The space X in this case can be selected as L 2 (, F, P). One of the many appealing features of (4) is that it has a simple intuitive interpretation: if X represents loss/unsatisfaction, then CVaR α (X), is, roughly speaking, the conditional expectation of losses that may occur in (1 α) 100% of the worst cases. In the case of a continuously distributed X, this rendition is exact: CVaR α (X) = E [ X X VaR α (X) ], where VaR α (X) is defined as VaR α (X) = inf { ζ P[X ζ ] > α }, i.e., the α-quantile of X. In the general case, the formal definition of CVaR α (X) becomes more intricate (Rockafellar and Uryasev, 2002), but the representation (4) still applies. Example 1.2 A generalization of (4) can be constructed as R α,β (X) = min η R η + α E(X η)+ β E(X η), (5) where, in accordance with the conditions of Theorem 1, one has to put α > 1 and 0 β < 1. Example 1.3 (Maximum Loss) If the requirement of finiteness of φ in (1) is relaxed, i.e., the image of φ is (, + ], then the optimal value of (1) still defines a coherent risk measure, but the infimum may not be achievable. An example is served by the so-called MaxLoss measure, MaxLoss(X) = ess.sup X = inf η η + φ (X η), where φ (X) = { 0, X 0,, X > 0. It is easy to see that φ is positive homogeneous convex, non-decreasing, lsc, and satisfies φ (η) > η for all η 0, but is not finite. Example 1.4 (Higher Moment Coherent Risk Measures) Let X = L p (, F, P), and for some 0 < α < 1 consider φ(x) = (1 α) 1 (X) + p, where X p = ( E X p) 1/p. Clearly, φ satisfies the conditions of Theorem 1, thus one can define a family of higher-moment coherent risk measures (HMCR) as HMCR p,α (X) = min η R η + (1 α) 1 (X η) + p, p 1, α (0, 1). (6) Of special interest is the case p = 2 that defines a second-moment coherent risk measure (SMCR): SMCR α (X) = min η R η + (1 α) 1 (X η) + 2, 0 < α < 1. (7) We will see below that SMCR α (X) is quite similar in properties to CVaR α (X), yet it measures the risk in terms of the second moments of losses. Implementation-wise, the SMCR measure can be incorporated in a mathematical programming problem via the second order cone constraints (see Section 3). The second order cone programming (SOCP) is a well-developed topic in the area of convex optimization, and a number of commercial off-the-shelf software packages are available for solving convex problems with second-order cone constraints. 6

7 Example 1.5 (Composition of risk measures) Formula (1) readily extends to the case of multiple functions φ i, i = 1,..., n, that are cumulatively used in measuring the risk of element X X and conform to the conditions of Theorem 1. Namely, one has that ρ n (X) = min η i R, i=1,...,n n ( ) η i + φ i (X η i ), (8) i=1 is a proper coherent risk measure. The value of η that delivers minimum in (1) does also possess some noteworthy properties as a function of X. In establishing of these properties the following notation is convenient. Assuming that the set arg min x R f (x) is closed for some function f : R R, we denote its left endpoint as Arg min x R f (x) = min { y y arg min x R f (x) }. Theorem 2 Let function φ : X R satisfy the conditions of Theorem 1. Then function η(x) = Arg min η R η + φ(x η) (9) exists and satisfies properties (A3) and (A4). If, additionally, φ(x) = 0 for every X 0, then η(x) satisfies (A1), along with inequality η(x) ρ(x), where ρ(x) is the optimal value of (1). Proof. Conditions on function φ ensure that the set of optimal solutions of problem (1) is closed and finite, whence follows the existence of η(x) in (9). Property (A3) is established by noting that for any λ > 0 equality (2) implies η(λx) = Arg min η R { } η + φ(λx η) { } = Arg min η R η/λ + φ(λx η/λ), from which follows that η(λx) = λη(x). Similarly, by virtue of (3), we have { } { η(x + a) = Arg min η + φ(x + a η) = Arg min (η a) + φ ( λx (η a) )}, η R η R which leads to the sought relation (A4): η(x + a) = η(x) + a. To validate the remaining statements of the Theorem, consider φ to be such that φ(x) = 0 for every X 0. Then, (C2) immediately yields φ(x) 0 for all X X, which proves By the definition of η(x), we have for all X 0 η(x) η(x) + φ ( X η(x) ) = ρ(x). η(x) + φ ( X η(x) ) 0 + φ(x 0) = 0, or η(x) φ ( X η(x) ). (10) Assume that η(x) > 0, which implies φ ( η(x) ) = 0. From (A2) it follows that φ ( X η(x) ) φ(x) + φ ( η(x) ) = 0, leading to φ ( X η(x) ) = 0, and, consequently, to η(x) 0 by (10). The contradiction furnishes the statement of the theorem. 7

8 Remark 2.1 If φ satisfies all conditions of Theorem 2, the optimal solution η(x) of the stochastic optimization problem (1) complies with all axioms for the coherent risk measures, except (A2), thereby failing to be convex. Example 2.1 (Value-at-Risk) A well-known example of two risk measures obtained by solving a stochastic programming problem of type (1) is again provided by formula (4) due to Rockafellar and Uryasev (2000, 2002), and its counterpart VaR α (X) = Arg min η R η + (1 α) 1 E(X η) +. The Value-at-Risk measure VaR α (X) (JP Morgan, 1994; Jorion, 1997; Duffie and Pan, 1997, etc), despite being adopted as the de facto standard for measurement of risk in finance and banking industries, is notorious for its poor behavior in risk estimation and control. In fact, the recent developments in axiomatic risk theory can be viewed as attempts of constructing risk management concepts that circumvent the failings of VaR. Example 2.2 (Second-Moment Coherent Risk Measure) For higher-moment coherent risk measures, the function φ in (9) is taken as φ(x) = (1 α) 1 (X) + p, and the corresponding optimal η p,α (X) satisfies the equation (1 α) 1/p = ( X η p,α (X) ) + p ( X η p,α (X) ) + p 1, p > 1. (11) Although the optimal η p,α (X) in (11) is determined implicitly, Theorem 2 ensures that it has properties similar to VaR α (monotonicity, positive homogeneity, etc). Moreover, using (11) with p = 2, the second-moment coherent risk measure (SMCR) can be presented in the form that involves only the first moment of losses in the tail of the distribution: SMCR α (X) = η 2,α (X) + (1 α) 2 ( X η 2,α (X) ) + 1 = η 2,α (X) + (1 α) 2 E ( X η 2,α (X) ) +. (12) Note that in (12), the second-moment information is concealed in the η 2,α (X). Further, by taking a CVaR measure with the confidence level α = 2α α 2, we can write SMCR α (X) = η SMCR α E( X η SMCR ) + η CVaR α E( X η CVaR ) + = CVaRα (X), (13) where η SMCR = η p,α (X) as in (11) with p = 2, and η CVaR = VaR α (X). In other words, with the above selection of α and α, expressions for SMCR α (X) and CVaR α (X) differ only by the choice of η that delivers minimum to the corresponding expressions (6) and (4). For Conditional Value-at- Risk, it is the α-quantile of the distribution of X, whereas the optimal η 2,α (X) for SMCR measure incorporates the information on the second moment of losses X. The developed results can be efficiently applied in the context of stochastic optimization, where the random outcome X is a function of the decision vector x R m, X = X (x, ω). Firstly, representation 8

9 (1) allows for efficient minimization of risk in stochastic programs. For a function φ that complies with the requirements of Theorem 1, denote (x, η) = η + φ ( X (x, ω) η ) and R(x) = ρ ( X (x, ω) ) = min η R (x, η). (14) Then, clearly, min ρ( X (x, ω) ) min x C (x,η) C R (x, η), (15) in the sense that both problems have the same optimal objective values and optimal vector x. Secondly, the representation (1) also admits implementation of risk constraints in stochastic programs. Namely, let C R m be a convex closed set, and g(x) a function that is convex on C. Then, the following two problems are equivalent, as demonstrated by Theorem 3 below: min { g(x) R(x) c}, (16a) x C min { g(x) (x, η) c}. (16b) (x,η) C R Theorem 3 Optimization problems (16a) and (16b) are equivalent in the sense that their objectives achieve the same minimum values. Further, if risk constraint in (16a) is active, (x, η ) achieves the minimum of (16b) if and only if x is an optimal solution of (16a) and η arg min η (x, η). The proof relies on the Kuhn-Tucker necessary and sufficient conditions for convex optimization problems (Pshenichnyi, 1971, Theorem 2.5) Theorem 4 (Kuhn-Tacker) Let ψ i (x), i = m,..., n, be functionals defined on a linear space E such that the ψ i are convex for i 0 and linear for i > 0. Then, in order for a point x E to deliver the minimum to the problem min x C ψ 0 (x) s. t. ψ i (x) 0, i = m,..., 1, ψ i (x) = 0, i = 1,..., n, where C is a given convex set in E, it is necessary that there exist constants λ i, i = m,..., n, such that n n λ i ψ i (x ) λ i ψ i (x) for all x C. i= m i= m Moreover, λ i 0 for each i 0, and λ i ψ i (x ) = 0 for each i 0. If λ 0 > 0, then these conditions are also sufficient. Proof of Theorem 3: Let (x, η ) be an optimal solution of (16b). Theorem 4 maintains that in this case there exists λ 0 such that g(x ) + λ (x, η ) g(x) + λ (x, η) for all (x, η) C R, λ ( (x, η ) c ) = 0. (17) For z C denote η(z) arg min η (z, η), then, obviously, (x, η(x )) (x, η ). On the other hand, for λ > 0 from (17) it follows that g(x ) + λ (x, η ) g(x ) + λ (x, η(x )), 9

10 whereby η arg min η (x, η). Then, for any x C, g(x ) + λ R(x ) = g(x ) + λ (x, η ) g(x) + λ (x, η(x)) = g(x) + λ R(x), λ ( (x, η ) c ) = λ ( R(x ) c ) = 0, proving that x solves (16a). In a similar fashion, if x is an optimal solution of (16a) and η arg min η (x, η), there exists λ 0 such that for any x from C g(x ) + λ R(x ) g(x) + λ R(x), λ ( R(x ) c ) = 0, Then for any pair (x, η) C R one has g(x ) + λ (x, η ) = g(x ) + λ R(x ) g(x) + λ R(x) = g(x) + λ ( x, η(x) ) g(x) + λ (x, η), and λ ( (x, η ) c ) = λ ( R(x ) c ) = 0. Hence, (x, η ) delivers the minimum to problem (16b). Now, observe that formula (1) in Theorem 1 is analogous to the operation of infimal convolution, well-known in convex analysis: ( f g)(x) = inf y f (x y) + g(y). Continuing the analogy between representation (1) and the operation of infimal convolution, consider the operation of right scalar multiplication (φη)(x) = η φ(η 1 X), η 0, where for η = 0 we set (φ0)(x) = (φ0 + )(X). If φ is proper and convex, then it is known that (φη)(x) is a convex proper function in η 0 for any X dom φ (see, for example, Rockafellar, 1970). Interestingly enough, this fact can be pressed into service to formally define coherent risk measure as the optimal value of stochastic programming problem ρ(x) = inf η 0 η φ( η 1 X ), (18) if function φ, along with some technical conditions similar to those of Theorem 1, satisfies axioms (A1), (A2 ), and (A4), but not (A3). Note that excluding the positive homogeneity (A3) from the list of properties of φ denies also its convexity, thus one must replace (A2) with (A2 ) to ensure that (18) is a convex programming problem. In the terminology of convex analysis the function ρ(x) defined by (18) is known as the positively homogeneous convex function generated by φ. Likewise, by direct verification of conditions (A1) (A4) it can be demonstrated that ρ(x) = sup η > 0 η φ(η 1 X), (19) is a proper coherent risk measure, provided that φ(x) satisfies (A1), (A2), and (A4). By (C1), axioms (A1) and (A2) imply that φ(0) = 0, which allows one to rewrite (19) as ρ(x) = sup η > 0 φ(ηx + 0) φ(0) η = φ0 + (X), (20) 10

11 where the last inequality in (20) comes from the definition of the recession function (Rockafellar, 1970; Zălinescu, 2002). Note that quantities defined by (18) and (19) coincide when function φ is positive homogeneous, i.e., it is a coherent risk measure itself: inf η η > 0 φ(η 1 X) = sup η > 0 η φ(η 1 X) = φ(x). (21) Yet, the practical usefulness of representations (18) or (19) for coherent risk measures seems rather questionable, as (18) (19) would generally lead to non-convex programming problems, should X parametrically depend on a decision vector x R n. On the contrary, representation (1) yields a nice convex optimization problem, provided that X is convex in x. 2.2 Deviation measures Since being introduced in Artzner et al. (1999), the axiomatic approach to construction of risk measures has been repeatedly employed by many authors for development of other types of risk measures, tailored to specific preferences and applications (see Rockafellar et al., 2002, 2004; Acerbi, 2002; Ruszczyński and Shapiro, 2004). In this subsection we consider deviation measures as introduced by Rockafellar et al. (2002). Namely, a deviation measure is a mapping D : X [0, + ] that satisfies (D1) D(X) > 0 for any non-constant X X, whereas D(X) = 0 for constant X, (D2) D(X + Y ) D(X) + D(Y ) for all X, Y X, 0 λ 1, (D3) D(λX) = λd(x) for all X X, λ > 0, (D4) D(X + a) = D(X) for all X X, a R. Again, from axioms (D1) and (D2) follows convexity of D(X). In Rockafellar et al. (2002) it was shown that deviation measures that further satisfy (D5) D(X) ess.sup X EX for all X X, are characterized by the one-to-one correspondence D(X) = R(X EX) (22) with expectation-bounded coherent risk measures, i.e., risk measures that satisfy (A1) (A4) and an additional requirement (A5) R(X) > EX for all non-constant X X, whereas R(X) = EX for all constant X. Using this result, it is easy to provide an analog of formula (1) for deviation measures. Theorem 5 Let function φ : X R satisfy axioms (A1) (A3), and be a lsc function such that φ(x) > EX for all X 0. Then the optimal value of the stochastic programming problem D(X) = EX + inf η { η + φ(x η) } is a deviation measure, and the infimum is attained for all X, so inf η in (23) may be replaced by min η R. (23) 11

12 Proof. Since formula (23) differs from (1) by the constant summand ( EX), we only have to verify that R(X) = inf η { η + φ(x η) } satisfies (A5). As φ(x) > EX for all X 0, we have that φ(x η X ) > E(X η X ) for all non-constant X X, where η X arg min η {η + φ(x η)}. From the last inequality it follows that η X + φ(x η X ) > EX, or R(X) > EX for all non-constant X X. Thus, D(X) > 0 for all non-constant X. For a R, inf η { η + φ(a η) } = a, whence D(a) = 0. Given the close relation between deviation measures and coherent risk measures, it is straightforward to apply the above results to deviation measures. 2.3 Connection with utility theory and second-order stochastic dominance As it has been mentioned in the Introduction, substantial attention has been devoted in the literature to the development of risk models and measures compatible with the utility theory of von Neumann and Morgenstern (1944), which represents one of the cornerstones of the decision-making science. The vnm theory argues that when the preference relation of the decision-maker satisfies certain axioms (completeness, transitivity, continuity, and independence), there exists a function u : R R, such that an outcome X is preferred to outcome Y ( X Y ) if and only if E[u(X)] E[u(Y )]. If function u is non-decreasing and concave, the corresponding preference is said to be risk averse. Rothschild and Stiglitz (1970) have bridged the vnm utility theory with the concept of second-order stochastic dominance, by showing that X dominating Y by the second-order stochastic dominance, X SSD Y, is equivalent to the relation E[u(X)] E[u(Y )] holding true for all concave non-decreasing functions u, where the inequality is strict for at least one such u. Recall that a random outcome X dominates outcome Y by the second order stochastic dominance if z P[X t] dt z P[Y t] dt for all z R. Since coherent risk measures are generally inconsistent with the second-order stochastic dominance (see an explicit example in De Giorgi, 2005), it is of interest to introduce risk measures that comply with this property. To this end, we replace the monotonicity axiom (A1) in the definition of coherent risk measures by the requirement of SSD isotonicity (Pflug, 2000; De Giorgi, 2005): ( X) SSD ( Y ) R(X) R(Y ). Namely, we consider risk measures R : X R that satisfy the following set of axioms: (A1 ) SSD isotonicity: ( X) SSD ( Y ) R(X) R(Y ) for X, Y X, (A2 ) convexity: R ( λx + (1 λ)y ) λr(x) + (1 λ)r(y ), X, Y X, 0 λ 1, (A3) positive homogeneity: R(λX) = λr(x), X X, λ > 0, (A4) translation invariance: R(X + a) = R(X) + a, X X, a R. Again, it is possible to develop an analog of formula (1), which would allow for construction of risk measures with the above properties using functions that comply with (A1 ), (A2 ), and (A3). 12

13 Theorem 6 Let function φ : X R satisfy axioms (A1 ), (A2 ), and (A3), and be a lsc function such that φ(η) > η for all real η 0. Then the optimal value of the stochastic programming problem ρ(x) = min η R exists and is a proper function that satisfies (A1 ), (A2 ), (A3), and (A4). η + φ(x η) (24) Proof. The proof of existence and all properties except (A1 ) is identical to that of Theorem 1. Property (A1 ) follows elementarily: if ( X) SSD ( Y ), then ( X + c) SSD ( Y + c), and consequently, φ(x c) φ(y c) for c R, whence ρ(x) = η X + φ(x η X ) η Y + φ(x η Y ) η Y + φ(y η Y ) = ρ(y ), where, as usual, η Z arg min η {η + φ(z η)} R, for any Z X. Obviously, by solving a risk-minimization problem min ρ( X (x, ω) ) x C where ρ is a risk measure that is both coherent and SSD-compatible in the sense of (A1 ), one obtains a solution that is SSD-efficient, i.e., is acceptable to any risk-averse rational utility maximizer, and also bears the lowest risk in terms of coherence preference metrics. Below we illustrate that functions φ satisfying the conditions of Theorem 6 can be easily constructed in the scope of the presented approach. Example 6.1 Let φ(x) = E[u(X)], where u : R R is a convex, positively homogeneous, nondecreasing function such that u(η) > η for all η 0. Obviously, function φ(x) defined in this way satisfies the conditions of Theorem 6. Since u( η) is concave and non-decreasing, one has that E[u(X)] E[u(Y )], and, consequently, φ(x) φ(y ), whenever ( X) SSD ( Y ). It is easy to see that, for example, function φ of the form φ(x) = αe(x) + βe(x), α (1, + ), β [0, 1), satisfies the conditions of Theorem 6. Thus, in accordance to Theorems 1 and 6, the coherent risk measure R α,β (5) is also consistent with the second-order stochastic dominance. A special case of (5) is the Conditional Value-at-Risk, which is known to be compatible with the second-order stochastic dominance (Pflug, 2000). Example 6.2 (Second-Moment Coherent Risk Measure) The SMCR and, in general, the family of higher-moment risk measures consitute another example of risk measures that are both coherent and compatible with second-order stochastic dominance. Indeed, function u(x) = ( (η) +) p is convex and non-decreasing, whence ( E[u(X)] ) 1/p ( E[u(Y )] ) 1/p for any ( X) SSD ( Y ). Thus, the HMCR family, defined by (24) with φ(x) = (1 α) 1 (X) + p, HMCR p,α = min η R η + (1 α) 1 (X η) + p, p 1, is both coherent and SSD-compatible, by virtue of Theorems 1 and 6. Implementation of such a risk measure in stochastic programming problems enables one to introduce risk preferences that are consistent with both concepts of coherence and second-order stochastic dominance. 13

14 Thus, the developed SMCR measure (7), and, generally, the family of higher-moment risk measures (6) possess all the nice properties that are sought after in the realm of risk management and decision making under uncertainty: compliance with the coherence principles, amenability for an efficient implementation in stochastic programming problems (via second-order cone programming), and compatibility with second-order stochastic dominance and utility theory. The question that remains to be answered is whether these superior properties translate into equally superior performance in practical risk management applications. The next section reports a pilot study intended to investigate the performance of the SMCR measure in real-life risk management applications. It shows that SMCR is a promising tool for tailoring risk preferences to specific needs of decision-makers, and can be compared favorably with some of the most widely used risk management frameworks. 3 Case study: Portfolio Optimization with SMCR measure In this section we illustrate the practical merits of the developed SMCR risk measure on the example of portfolio optimization, a typical testing ground for many risk management and stochastic programming techniques. To this end, we compare portfolio optimization models that use the SMCR risk measure against portfolio allocation models based on two well-established, and theoretically as well as practically proven methodologies, namely, the Conditional Value-at-Risk measure and Markowitz Mean-Variance framework. This choice of benchmark models is further supported by the fact that SMCR is similar in construction and properties to CVaR, but, while CVaR measures risk in terms of the first moment of losses residing in the tail of the distribution, the SMCR measure quantifies risk using the second moments, in this way relating to the MV paradigm. To illuminate the effects of taking into account the second-moment information in estimation of risks using the SMCR measure, as compared to CVaR measure that utilizes the first-moment information, in this case study we consider SMCR measure with parameter α = 0.90, and CVaR measure with confidence level α = 0.99, so that the relation α = 2α α 2 holds (see the discussion in Example 2.2). The portfolio optimization models used in this case study have the form min x R ( r, x ) (25a) s. t. e, x = 1, (25b) Er, x r 0, x 0, (25c) (25d) where x = (x 1,..., x n ) T is the vector of portfolio weights, r = (r 1,..., r n ) T is the random vector of assets returns, and e = (1,..., 1) T. The risk measure R in (25a) is taken to be either SMCR (7), CVaR (4), or variance of the negative of the portfolio return, r, x = X. The corresponding solutions of (25) are therefore denoted as SMCR, CVaR, and MV optimal portfolios, respectively. In the above portfolio optimization problem, (25b) represents the budget constraint, which, together with the no-short-selling constraint (25d) ensures that all the available funds are invested, and (25c) imposes the minimal required level r 0 for the expected return of the portfolio. Traditionally to stochastic programming, we modeled the distribution of random return r i of asset i using a set of J discrete equiprobable scenarios {r i1,..., r i J }. Then, the optimization problem (25) reduces to a linear pro- 14

15 gramming problem if CVaR is selected as the risk measure R in (25). Within the Mean-Variance framework, (25) becomes a quadratic optimization problem with the objective R ( r, x ) = n i,k=1 σ ik x i x k, where σ ik = 1 J 1 J (r i j r i )(r kj r k ), r i = 1 J j=1 J r i j. (26) j=1 In the case of R(X) = SMCR α (X), problem (25) reduces to a linear programming problem with one second-order cone constraint: min η t 1 α J n s. t. x i = 1, i=1 1 J J j=1 i=1 w j n r i j x i r 0, (27a) (27b) (27c) n r i j x i η, j = 1,..., J, (27d) i=1 t J w 2 j, j=1 (27e) x i 0, i = 1,..., n, (27f) w j 0, j = 1,..., J. (27g) The resulting mathematical programming problems have been implemented in C++, and we used CPLEX 10.0 for solving the LP and QP problems, and MOSEK 4.0 for solving the SOCP problem (27). It is necessary to emphasize that we have deliberately chosen not to include any additional trading or institutional constraints (transaction costs, liquidity constraints, etc.) in the portfolio allocation problem (25) so as to make the effect of risk measure selection in (25) more marked and visible. The data set used in this case study contained daily closing prices, from October 23, 2003 to Jan 16, 2006, of 100 stocks selected at random from the S&P 500 index (as of January 2006). For scenario generation we used 10-day historical returns over three hundred overlapping periods (J = 300). 15

16 Expected return CVaR SMCR MV Standard deviation Figure 1: Expected return vs standard deviation for the SMCR-, CVaR-, and MV-optimal portfolios. The SMCR-optimal portfolio was calculated with R( ) = SMCR 0.90 ( ) in (25a), and the CVaR-optimal portfolio was calculated with R( ) = CVaR 0.90 ( ) in (25a). 3.1 In-Sample Optimization: Efficient Frontier The first set of experiments conducted in this case study involves construction of efficient frontiers for the portfolios defined by (25), and is otherwise known as in-sample optimization. Figures 1, 2, and 3 display the so-called risk-reward curves of the SMCR-, CVaR-, and MV-optimal portfolios, which allow for comparing the risk borne by each of the three portfolios in the same scale. The procedure for construction of these risk-reward curves is as follows. For a given level of expected return r 0, let x smcr (r 0), x cvar (r 0), and x mv (r 0) denote the corresponding SMCR-, CVaR-, and MV-optimal portfolios. Then, for each choice of risk measure R in (25a), the corresponding risk-reward curves are traced by pairs ( R [ x smcr (r 0) ], r 0 ), ( R [ x cvar (r 0 ) ], r 0 ), and ( R [ x mv (r 0 ) ], r 0 ), respectively; observe that one of the three risk-reward curves would necessarily represent an efficient frontier of one of the portfolios. For example, the dashed curve in Figure 1 represents the efficient frontier of the MV-optimal portfolio plotted in the (standard deviation, expected return) scale. The other two curves in Fig. 1 are traced by pairs ( σ [ x smcr (r 0) ], r 0 ) and ( σ [ x cvar (r 0 ) ], r 0 ), where σ [ ] is the standard deviation operator. In other words, the three curves in Fig. 1 report the standard deviations of the SMCR-, MV-, and CVaRoptimal portfolios that have equal expected return. Of course, it is not surprising that the SMCR- and CVaR-optimal portfolios exhibit higher standard deviation than the MV-optimal portfolio. Similarly, Figs. 2 and 3 demonstrate the risk borne by the SMCR-, CVaR-, and MV-optimal portfolios with equal expected return when the risk is measured using the SMCR measure with α = 0.90, and CVaR with 0.99 confidence level, respectively. Again, in Fig. 2 the SMCR curve represents the efficient frontier of the SMCR-optimal portfolio, and the CVaR curve is the efficient frontier of the CVaR-optimal portfolio in Fig. 3. From Figures 1 3 we observe that the risk-reward curves generated by the SMCR- and CVaR-optimal portfolios are very similar in all three cases. In fact, in Fig. 2 these two virtually coincide; however, the graphs in Figures 1 and 3 indicate that the SMCR- and CVaR-optimal portfolios are nevertheless different. Although these results are data-dependent, they generally confirm the conjecture that for certain values of parameters, namely, α = 2α α 2, the SMCR α and CVaR α measures may lead 16

17 Expected return CVaR SMCR MV SMCR Figure 2: Expected return vs SMCR (α = 0.90) for the SMCR-, CVaR-, and MV-optimal portfolios. The SMCR-optimal portfolio was calculated with R( ) = SMCR 0.90 ( ) in (25a), and the CVaR-optimal portfolio was calculated with R( ) = CVaR 0.90 ( ) in (25a) Expected return CVaR SMCR MV CVaR Figure 3: Expected return vs 99%-CVaR for the SMCR-, CVaR-, and MV-optimal portfolios. The SMCRoptimal portfolio was calculated with R( ) = SMCR 0.90 ( ) in (25a), and the CVaR-optimal portfolio was calculated with R( ) = CVaR 0.90 ( ) in (25a). 17

18 to close estimates of risk. Recall that for such values of parameters, the expressions for SMCR α and CVaR α take almost identical forms, differing only in the values of η that deliver the minima in the corresponding representations (7) and (4): in the case of SMCR, the optimal η contains the second-moment information via (11), whereas in the case of CVaR the corresponding η represents a quantile. In particular, the presented results suggest that in the in-sample setting the effects of second-moment information in risk estimates produced by the SMCR risk measure, as compared to those due to CVaR measure, are almost negligible. This, however, will change in the out-of-sample setting, as we will see below. 3.2 Out-of-sample tests The out-of-sample testing allows one to shed light on the potential real-life performance of a particular decision-making strategy under uncertainty. As the name suggests, the out-of- sample testing involves construction of a solution based on a given sample of the underlying stochastic data of the problem, and then testing this solution on a different, out-of-sample, data. In the context of the present work, we employ out-of-sample optimization to compare simulated historic performances of the three self-financing portfolio rebalancing strategies based on (25) with R being taken as SMCR 0.90 ( ), CVaR 0.99 ( ), or variance σ 2 ( ). It may be argued that in practice of portfolio selection, instead of solving (25), it is of more interest to construct investment portfolios that maximize the expected return subject to constraint(s) on risk, e.g., { max E r, x ( ) } R r, x c0, e, x = 1. (28) x 0 Indeed, many investment institutions are required to keep their investment portfolios in compliance with numerous constraints, including constraints on risk. However, the primary goal of this case study is to elucidate the effectiveness of employing the SMCR risk measure in portfolio optimization by comparing it against other well-established risk management methodlogies, such as the CVaR and MV frameworks. And since these three risk measures yield risk estimates on different scales, it is not evident what risk tolerance levels c 0 should be selected in (28) to make the resulting portfolios comparable. Thus, to have a fair apple-to-apple comparison, we construct self-financing portfolio rebalancing strategies by solving the risk-minimization problem (25), so that the resulting portfolios will all have the same level r 0 of expected return, and the success of a particular portfolio rebalancing strategy will depend on the actual amount of risk borne by the portfolio due to utilization of the corresponding risk measure. The out-of-sample experiments have been set up as follows. The initial SMCR-, CVaR-, and MVoptimal portfolios were constructed on Jan 03, 2005 by solving the corresponding variant of problem (25), where the scenario set consisted of 300 overlapping bi-weekly returns covering the period from Oct 23, 2003 to Jan 03, The duration of rebalancing period for all strategies was set at two weeks (10 business days). Thus, the next rebalancing date was Jan 17, at which the 10-day out-ofsample portfolio returns were observed for each of the three portfolios by plugging the vector ˆr of Jan 03 Jan 17 returns into ˆr, x, with x being the corresponding optimal portfolio configuration obtained on January 03. Then, all the three portfolios were rebalanced by solving (25) with an updated 18

19 scenario set. Namely, we included in the scenario set the 10 vectors of overlapping biweekly returns that realized during the ten business days from Jan 03 to Jan 17, and discarded the oldest 10 return vectors from October The process was repeated on January 31, 2005, and so on. In such a way, the out-of-sample experiment consisted of 25 biweekly rebalancing periods. We ran the outof-sample tests for different values of minimal required expected return r 0, and typical results are presented in Figures 4 and 5. Figure 4 reports the portfolio values (in percent of the initial investment) for the three portfolio rebalancing strategies based on (25) with SMCR 0.90 ( ), CVaR 0.90 ( ), and variance σ 2 ( ) as R( ), and the expected return level r 0 being set at 1%. One can observe that in this case the trajectories of SMCR and CVaR portfolios virtually coincide as they both outperform the MV portfolio; this situation is typical for smaller values of r 0. An insight as to why the SMCR and CVaR portfolios have very similar out-of-sample performances at lower r 0 can be gleaned from the in-sample experiments: indeed, as evidenced by Figures 1 3, at lower values of r 0 the variations of problem (25) with R( ) = SMCR 0.90 ( ) and R( ) = CVaR 0.99 ( ) tend to produce very similar portfolios. As r 0 increases and the rebalancing strategies become more aggressive, the differences between SMCR and CVaR portfolios and the corresponding risk-reward curves become more pronounced, which translates into differing out-of-sample performance. An illustration of the typical behavior of more aggressive rebalancing strategies is presented in Figure 5, where r 0 is set at 2.5%. As a general trend, an aggressive SMCR portfolio would outperform similarly aggressive CVaR portfolio, and they both would outdo the MV portfolio. For instance, at r 0 = the SMCR portfolio outperforms the CVaR portfolio by approximately as much as CVaR outperforms the MV portfolio (see Fig. 5). Similar situations are observed for values of r 0 ranging from 2.0% to 3.0% (on the dataset used in this case study, infeasibilities in (25) began to occur for values of r ). Notwithstanding the data-specific nature of the obtained results, the conducted out-of-sample simulations suggest that the SMCR risk measure may be preferred by investors with rather aggressive risk preferences Out-of-sample results: Deviation measures In addition to out-of-sample comparison of SMCR and CVaR as coherent risk measures, we also conducted similar studies for their deviation counterparts. Recall that under certain conditions, a deviation measure can be formed from a coherent risk measure by simply subtracting the expectation of the argument: D(X) = R(X) EX. Figure 6 displays the simulated historical trajectories of the DSMCR and DCVaR portfolios obtained from (25) by plugging the expressions SMCR 0.90 ( ) E( ) and CVaR 0.90 ( ) E( ), respectively, for R( ) in (25a). The required level of expected return in of portfolios in Fig. 6 is fixed at r 0 = Again, we observe a picture that is consistent with our previous findings: for higher values of r 0, the portfolio based on second-moment deviation measure ( DSMCR ) tends to outperform the corresponding portfolio based on the Deviation CVaR ( DCVaR ) measure, which is again a typical situation for higher values of expected return r 0. 19

20 130% 125% 120% Portfolio value 115% 110% 105% 100% CVaR SMCR MV 95% 90% 29-Dec Feb-05 8-Apr May Jul-05 5-Sep Oct Dec-05 2-Feb-06 Time Figure 4: Out-of-sample performance of self-financing portfolio rebalancing strategies based on SMCR, CVaR, and MV measures of risk (r 0 = 0.01) 170% 160% 150% Portfolio value 140% 130% 120% CVaR SMCR MV 110% 100% 90% 29-Dec Feb-05 8-Apr May Jul-05 5-Sep Oct Dec-05 2-Feb-06 Time Figure 5: Out-of-sample performance of self-financing portfolio rebalancing strategies based on SMCR, CVaR, and MV measures of risk (r 0 = 0.025) 20

21 180% 170% Expected return 160% 150% 140% 130% 120% DSMCR DCVaR 110% 100% 29-Dec Feb-05 8-Apr May Jul-05 Time 5-Sep Oct Dec-05 2-Feb-06 Figure 6: Out-of-sample performance of self-financing portfolio rebalancing strategies based on deviation measures (r 0 = 0.025) Portfolio optimization with SMCR constraint Finally, we present the results of the out-of-sample simulations for self-financing portfolio rebalancing strategies with SMCR constraints, which are based on solutions of (28), where R( ) = SMCR 0.90 ( ). Due to Theorem 3, problem (28) may be formulated, similarly to (27), as a linear programming problem with a single second-order cone constraint: min s. t. 1 J J j=1 i=1 n r i j x i j n x i = 1, i=1 (29a) (29b) η t c 0, (29c) 1 α J n w j r i j x i η, j = 1,..., J, (29d) i=1 t J w 2 j, j=1 (29e) x i 0, i = 1,..., n, (29f) w j 0, j = 1,..., J. (29g) The out-of-sample tests were conducted in the same fashion as described above, with problem (29) being solved to obtain the optimal portfolio configurations. Multiple runs of the out-of-sample simulations were performed for different values of the risk tolerance c 0 in (29c). The resulting simulated historical trajectories are presented in Figure 7. It is worth noting that at higher levels of risk tolerance c 0 the portfolio model with SMCR constraint allows for achieving considerably higher out-of-sample returns comparing to models where the SMCR objective is minimized, although at the expense of 21