MANAGEMENT SCIENCE informs ® Vol. 55, No. 1, January 2009, pp. 71–84 doi 10.1287/mnsc.1080.0929 issn 0025-1909 eissn 1526-5501 09 5501 0071 © 2009 INFORMS
Satisficing Measures for Analysis of Risky Positions
David B. Brown Fuqua School of Business, Duke University, Durham, North Carolina 27705, [email protected] Melvyn Sim NUS Business School, NUS Risk Management Institute, and Singapore MIT Alliance (SMA), National University of Singapore, Singapore 117592, [email protected]
n this work we introduce a class of measures for evaluating the quality of financial positions based on their Iability to achieve desired financial goals. In the spirit of Simon (Simon, H. A. 1959. Theories of decision- making in economics and behavioral science. Amer. Econom. Rev. 49(3) 253–283), we call these measures satisficing measures and show that they are dual to classes of risk measures. This approach has the advantage that aspi- ration levels, either competing benchmarks or fixed targets, are often much more natural to specify than risk tolerance parameters. In addition, we propose a class of satisficing measures that reward diversification. Finding optimal portfolios for such satisficing measures is computationally tractable. Moreover, this class of satisficing measures has an ambiguity interpretation in terms of robust guarantees on the expected performance because the underlying distribution deviates from the investor’s reference distribution. Finally, we show some promising results for our approach compared to traditional methods in a real-world portfolio problem against a competing benchmark. Key words: satisficing; aspiration levels; targets; risk measures; coherent risk measures; convex risk measures; portfolio optimization History: Accepted by David A. Hsieh, finance; received July 12, 2007. This paper was with the authors 1 1 months for 2 revisions. Published online in Articles in Advance October 7, 2008. 2
1. Introduction framework readily applies to other settings in which One of the key principles from Simon’s (1955) risk is an important issue. bounded rationality model is that, rather than In particular, our goal is to provide a framework for formulating and solving complicated optimization measuring the quality of risky positions with respect problems, real-world agents often can choose the first- to their ability to satisfy (i.e., achieve the aspiration available actions, which ensure that certain aspiration level). Aspiration level models based in the spirit of levels will be achieved. In other words, given the com- satisficing have been previously explored but the con- putational difficulties in the rational model paradigm, nection is usually drawn via probability measures, a more sensible (and descriptively accurate) approach i.e., the probability of achieving at least an aspira- may in fact be to view profit not as an objective to be tion level. Here we axiomatize a more general class of maximized, but rather as a constraint relative to some measures, called satisficing measures, which depend on given aspiration level. Simon (1959) coined the term the investor’s aspiration level. Probability measures satisficing1 to describe this approach. are one special case of these. The focus of this paper is to utilize the concept One important advantage of this approach is that of aspiration levels from satisficing as a means of aspiration levels are often very natural for investors quantifying the desirability of investment opportuni- to specify, whereas traditional models based on risk ties with uncertain payoffs. As such, we replace the measures or utility functions depend critically on term “agent” with “investor,” and we refer to the tolerance parameters, which are often difficult for investment opportunities as “positions.” We restrict investors to intuitively grasp and even harder to ourselves to the scalar case, i.e., the case when each appropriately assess (certainly, at least, relative to 2 position will realize a payoff representable by a sin- assessing aspiration levels). gle, real number. Although our primary focus in this paper is on a model useful for financial positions, the 2 We concede here that we avoid entirely the issue of ascertaining the “right” aspiration level and assume it is a given primitive; for some recent work that does address this issue, see Bearden and 1 A blend of the words “satisfy” and “suffice.” Connolly (2007).
71 Brown and Sim: Satisficing Measures for Analysis of Risky Positions 72 Management Science 55(1), pp. 71–84, © 2009 INFORMS
As we have stated, the idea of aspiration levels reference-dependent utility develops reference points is not new in the decision theory literature (which (targets in our language) endogenously (e.g., Shalev often uses the nomenclature “targets”). Some recent 2000 in a game theoretic setting) and allows the work focuses on the probability of achieving an aspi- reference point to be stochastic (e.g., Sudgen 2003). ration level as a way of comprehensively and rigor- Köszegi and Rabin (2006, 2007) present a framework ously discussing decision theory without using utility that allows the reference points to be both stochas- functions (see Castagnoli and LiCalzi 1996, Bordley tic and endogenously determined by preferences. One and LiCalzi 2000). Tsetlin and Winkler (2007) con- issue in this setting is that the statistical dependence sider the case of using aspiration levels along multiple between prospects and targets is ignored. De Giorgi dimensions within utility functions. From the descrip- and Post (2007) extend the framework to account for tive standpoint, a number of studies have concluded the (often very strong) dependence between prospects that aspiration levels play an important role in real- and targets, this has benefits computationally and can world decision-making behavior. Lanzillotti’s (1958) be critical from a modeling perspective if the target study interviews executives of 20 large companies is, for instance, a stochastic benchmark against which and concludes that these managers are primarily con- the decision maker is competing. cerned about target returns on investment. In another Our approach is a bit different than the reference- study, Payne et al. (1980, 1981) illustrate that man- dependent utility literature. For starters, our frame- agers tend to disregard investment possibilities that work does not rely in any way on the axioms of are likely to underperform against their target. Simon utility theory. Instead, we start with a very simple set (1959) also argued that most firms’ goals are not of axioms that depend solely and in simple fashion maximizing profit but attaining a target profit. In an upon performance relative to a target. Our approach empirical study by Mao (1970), managers were asked assumes nothing about the stochastic structure of the to define what they considered as risk. From their target (it can certainly be random), but we do take the responses, Mao (1970, p. 353) concluded that, “risk target as exogenously specified. As discussed above, is primarily considered to be the prospect of not however, we feel that targets are generally quite natu- meeting some target rate of return.” The notion of ral in many applications, so giving the decision maker success probability has also been applied widely in this freedom is often sensible practically. We will see the mathematical finance literature (see, for instance, by virtue of examples that utility theory models do Browne 1999, Müller and Baumann 2006, Föllmer and fall into our framework, but they are not the only ones Leukert 1999). that do so. Of course, one of the drawbacks of maximizing We summarize our primary contributions as the success probability alone is that it tacitly assumes follows: that the modeler is indifferent to the level of losses and gains. It does not address how catastrophic losses 1. We define axiomatically the concept of a satis- can be (or how exceptional gains can be) when ficing measure, which depends on a position’s per- “extreme” (i.e., low probability) events occur. A num- formance relative to a given aspiration level. These ber of studies have in fact suggested that subjects are satisficing measures can be thought of as a general- not completely insensitive to such magnitude vari- ization of the success probability approach. We also ations, particularly with respect to losses (see, for show that satisficing measures have a dual connection instance, Payne et al. 1980). More recently, Diecidue to risk measures; in particular, we prove a representa- and van de Ven (2008, p. 687) have argued that a model tion theorem, which shows that every satisficing mea- that solely maximizes the success probability is “too sure can be written in terms of a parametric family of crude to be normatively or descriptively relevant.” risk measures. It is certainly relevant to place our work in the 2. In general, a satisficing measure need not reward context of reference-dependent utility, a framework diversification. When we impose an additional prop- that arose from the development of prospect the- erty of quasiconcavity on satisficing measures, how- ory (Kahneman and Tversky 1979, Tversky and ever, diversification is rewarded. Moreover, the repre- Kahneman 1992). This theory was aimed at correcting sentation theorem in this case holds over a parametric perceived inadequacies of expected utility theory family of convex risk measures introduced by Föllmer from a descriptive perspective; for instance, tradi- and Schied (2002). Most quasiconcave satisficing mea- tional expected utility theory does not account for sures that we explore are not insensitive to the magni- framing effects and the fact that decision makers tude of losses below (or gains above) aspiration lev- may be significantly more sensitive (especially locally) els, which is in stark contrast to probability measures. to losses around a reference point than they are In addition, the quasiconcavity property is important to gains of comparable magnitude. These and other computationally because it readily admits the use of criticisms are often borne out in empirical data of efficient algorithms for optimization over satisficing real human decisions. Some of the recent work in measures. This class of satisficing measures has an Brown and Sim: Satisficing Measures for Analysis of Risky Positions Management Science 55(1), pp. 71–84, © 2009 INFORMS 73 alternate interpretation in terms of expected values for all ∈ . Similarly, X>Y denotes strict statewise over ambiguous probability distributions, which, inter- dominance (X > Y for all ∈ ). estingly, connects us back to Simon’s contention that We consider the situation in which the investor has probabilities are often not known exactly. an aspiration level , which she hopes to achieve via 3. Finally, when we impose an additional property these positions. We assume is a random variable of scale invariance on a satisficing measure, our rep- on as well, which, of course, includes the case resentation theorem then ties into to the well-known of being a constant. This encompasses the cases of class of coherent risk measures popularized by Artzner both individual investors, who have their own finan- et al. (1999). Furthermore, we show that a “separation cial goals (often constant targets) to meet, as well as theorem” holds for portfolio optimization problems the case of professional managers, who often compete over this class of satisficing measures. against alternative, risky positions (benchmarks). Before we proceed, we comment briefly on our Given an uncertain payoff X ∈ , we define the choice of nomenclature. The idea of defining a satis- target premium V to denote the excess payoff above ficing measure as something over which we can opti- the aspiration level, i.e., V = X − ∈ , where is mize may seem to be somewhat of an oxymoron to also a set of random variables on . Without loss of purists. Indeed, Simon’s original presentation of sat- generality and for notational convenience, we assume isficing was a computationally feasible alternative to = . In other words, we will assume each of the the potentially onerous approach of optimizing. On payoffs X ∈ already has the aspiration level embed- the other hand, we are not the first to use the aspira- ded within it, and therefore we will suppress the nota- tion level idea from satisficing within decision theo- tion in everything that follows. Thus, a position retic approaches based on optimization (e.g., Bordley achieves the aspiration level if and only if the realized and LiCalzi 2000, Charnes and Cooper 1963). More- target premium X satisfies X ≥ 0. over, the rapid proliferation of massive computational In the spirit of satisficing, one measure of satisfac- capabilities has radically changed the way decisions tion with respect to a target premium X ∈ is are made in many industries, particularly finance; Simon et al. (1986) were obviously quite aware of X X ≥ 0 (1) and interested in the impact of high-speed com- puting. Our choice of the word satisficing largely Obviously, such a measure is equivalent to the reflects a connection to the concept of aspiration lev- expected value of a 0 1-utility function u x , which els as a means of quantifying satisfaction with a risky is one if and only if x ≥ 0. position. We now define a more general notion of satisfic- The outline of this paper is as follows. In §2, ing measures, of which probability measures are one we motivate and define the axioms of satisficing example. We are attempting to get the best of both measures and state our most general representation worlds: on one hand, we are exploiting the fact that theorem relating these to risk measures; in §3, we con- aspiration levels are simple to understand and (often) sider imposing additional properties on the satisficing natural to specify; on the other hand, by imposing measures and then connect them to convex and coher- additional properties in the following section, we will ent risk measures. Finally, §4 considers portfolio opti- circumvent some of the main difficulties with proba- mization with satisficing measures, and §5 explores bility measures (e.g., its inability to reward diversifi- a simple debt management example. We refer the cation or distinguish magnitudes). reader to the e-companion3 for all proofs as well as for Definition 1. A function → 0 ¯ , where ¯ ∈ a computational example illustrating satisficing mea- 1 ,isasatisficing measure defined on the target sures in the context of portfolio optimization. premium if it satisfies the following axioms for all X Y ∈ : 1. Attainment content:IfX ≥ 0, then X =¯ . 2. Defining Satisficing Measures 2. Nonattainment apathy:IfX<0, then X = 0. Let be a measure space and let be a 3. Monotonicity:IfX ≥ Y , then X ≥ Y . set of random variables on , i.e., a set of func- 4. Gain continuity: lima↓0 X + a = X . tions X →. Each X ∈ represents the payoff (or These axioms are rather straightforward to moti- return) of a different, risky position. Throughout the vate in light of satisficing. Attainment content reflects paper, we will use the notation X ≥ Y for X Y ∈ the fact that satisficing is primarily concerned with to represent statewise dominance, i.e., X ≥ Y achieving the aspiration level; if a position always achieves this, then we are always satisfied with the outcome, and therefore fully “content.” On the flip 3 An electronic companion to this paper is available as part of the online version that can be found at http://mansci.journal.informs. side, a position that never reaches the aspiration level org/. does not satisfy us in the slightest (nonattainment Brown and Sim: Satisficing Measures for Analysis of Risky Positions 74 Management Science 55(1), pp. 71–84, © 2009 INFORMS apathy). Monotonicity is quite clear as well: if one that the induced relation is convex if and only if position never underperforms another, then we must the function is quasiconcave, i.e., for all X Y ∈ , be at least as satisfied with it. Finally, gain continu- ∈ 0 1, X + 1 − Y ≥ min X Y . ity is simply right continuity of ; if we augment For perhaps the simplest example of probabil- our position with an infinitesimally small but positive ity’s failure to satisfy quasiconcavity (and hence amount, in the limit we cannot improve our satisfac- failure to induce convex preferences), consider a tion.4 In other words, we are indifferent to small gains target premium X distributed symmetrically about zero, so X ≥ 0 = 1 . Now consider an alterna- in the payoff, but not necessarily small losses (in par- 2 ticular, if we are right at the aspiration level). tive position Y =−X − , where >0. If is small Remark 1. An immediate criticism of this frame- enough and the distribution of X is continuous, then Y ≥ 0 will be close to 1 as well. The posi- work is that, if the target premia are all above (below) 2 tion Z = 1 X + Y , however, is −/2 with probability zero, then any satisficing measure cannot distinguish 2 among any of the positions. On one hand, if aspira- one, so Z ≥ 0 = 0. Therefore, this satisficing mea- tion levels are purely preference-dependent and in no sure says we are worse off by diversifying between way influenced by the available opportunities, then these positions as opposed to investing into either it is true that the satisficing measures will not be individually. very useful in such circumstances. On the other hand, One could object that this is a rather silly exam- it is not unreasonable to assume that aspiration lev- ple; indeed, if aspiration levels are really all we care els are often affected by the opportunities available about, then the probability measure is “doing its job” to the investor. In such cases, one could argue that just fine. Even in the limit as → 0−, however, the the investor is too pessimistic (optimistic) and should probability measure prefers positions X or Y , both of be reassessing her aspiration level. In fact, Simon which can have arbitrarily large losses, to the sure (1955) provides an example of selling a house and the position Z, even if it is only a fraction of a cent below agent’s aspiration level is determined, in part, via an our target. This does not seem reasonable. “exploration” phase in which she learns about the cli- Moving beyond this illustrative example to a gen- mate of her housing market. In short, the satisficing eral discussion, there are a number of arguments that measures we define are most useful when the aspi- strongly support quasiconcavity of a satisficing mea- ration levels are set such that positions can fall both sure. The first is the widely accepted tenet that diver- above and below this level. sification of positions is almost always a good thing. Clearly, X ≥ 0 is one example of a satisficing Indeed, we can go back to Markowitz (1952, p. 77), measure; X > 0, however, violates gain continuity, who boldly declares, “Diversification is both observed and is therefore not a satisficing measure. and sensible; a rule of behavior which does not imply We also point out that our definition allows for sat- the superiority of diversification must be rejected both as a hypothesis and as a maxim.” Since this work, isficing measures to map to either 0 1 or all of +. We will give examples of both cases in §2.2. it has become almost a fundamental law of invest- ing that approaches to portfolio choice that do not 2.1. Quasiconcave and Coherent lead to diversified positions are generally undesirable Satisficing Measures because of their tendency to expose investors to large The probability measure X ≥ 0 is the most obvi- positions. ous example of a satisficing measure. On one hand, A second and related reason quasiconcavity is it embodies the concept of satisficing in that it reflects a desirable property is for robustness of positions. very directly the performance of a position relative to In particular, given that probability assessments of the aspiration level. On the other hand, unfortunately, the real world are inherently inexact, it behooves the it suffers a rather critical flaw from an economic per- investor to split positions up and thereby reduce sen- spective in that it does not reflect convex preferences sitivity to errors in modeling the real world. In con- (i.e., does not value diversification). trast, satisficing measures that are not quasiconcave Clearly, any satisficing measure induces a prefer- (like probability measure) can lead us to large posi- ence relation with, for any X Y ∈ , X Y if and tions in which expected performance is highly depen- only if X ≥ Y . Recall that is a convex pref- dent on the specifics of our model for that position. erence if, for all X Y Z ∈ , the implication Y X, In some sense, this also links us to Simon’s bounded Z X ⇒ Y + 1− Z X ∀ ∈ 0 1 holds. It is clear rationality, which posits that models of the real world are largely approximations subject to errors. Finally, we would ultimately like to exploit the 4 Note that left continuity is critically different and need not hold. In simplicity of satisficing measures and use them as particular, clearly probability measure X ≥ 0 is a satisficing mea- sure. Now consider the case when X is a constant; right-continuity, a method for selection of positions, and their fail- but not left-continuity, holds at X = 0. ure to satisfy quasiconcavity is tantamount to having Brown and Sim: Satisficing Measures for Analysis of Risky Positions Management Science 55(1), pp. 71–84, © 2009 INFORMS 75 to select over potentially nonconvex spaces of posi- such that X ∈ is either a constant or a random vari- tions. From a computational standpoint, this is in gen- able with infinite support. This includes a set of inde- eral very difficult when the number of opportunities pendently distributed random variables with infinite (or available asset classes in a portfolio setting) is support such as those with normal distributions. The large. Although tractability of an approach cannot by following Sharpe ratio, itself form a foundation for an axiomatic theory, it is a fair to say that an approach to risk management that S X = sup a −Ɛ X + X ≤ 0 is not amenable to rapid computation will be of lim- a∈0 1 1 − a ited use in practice. (or zero if no such a exists) is a coherent satisfic- In short, the desire to ensure that satisficing mea- ing measure on . Moreover, S X ≤ X ≥ 0. Note sures recognize diversification appropriately, both for that the function g a = a/ 1 − a is an increasing economic and computational reasons, motivates the function of a ∈ 0 1 , hence, whenever X > 0 and following. Ɛ X ≥ 0, g S X = Ɛ X / X , which is the stan- Definition 2. A function → 0 ¯ , where ¯ ∈ dard definition of the Sharpe ratio (without a risk- 1 ,isaquasiconcave satisficing measure QSM free asset). Note that the Sharpe ratio is generally not defined on the target premium if, in addition to Defi- a coherent satisficing measure on random variables nition 1, it satisfies the following for all X Y ∈ : with finite support. For instance, consider the random√ 1. Quasiconcavity:If ∈ 0 1, then X + variable X ∼ U0 1, which has the Sharpe ratio 3 1 − Y ≥ min X Y . and hence, S X = 3/4. However, since X ≥ 0, the sat- If, in addition, satisfies isficing measure violates the axioms of attainment 2. Scale invariance:Ifk>0, then kX = X ;we S content and monotonicity (since S X < S 0 = 1). say is a coherent satisficing measure CSM . This satisficing measure is explored in more detail by Note that, in the simple example we gave, a QSM Pinar and Tütüncü (2005). would have to satisfy min X Y ≤ 0 i.e., either Example 3 (Optimized Sharpe Ratio). Let X = 0or Y = 0, which is not the case for proba- n+1 bility measure. = X ∃ 0 n ∈ X=0 +1X1 +···+nXn The scale invariance axiom arises in situations in where X1 Xn are random variables defined on which, like with probability measures, we are indiffer- with positive definite covariance, and without loss ent to symmetric scalings of the payoffs with respect of generality, zero means. The following optimized to our aspiration level. Note that quasiconcavity alone Sharpe ratio, implies, for any ∈ 0 1, X = X + 1 − 0 ≥ min X 0 = X , since, by definition, 0 =¯ OS X = sup a min −Ɛ X − V for a satisficing measure. For a CSM, this bound is a∈0 1 V ∈ tight. In other words, for a CSM, any decrease in a potential losses below the aspiration level is offset + X− V V ≥ 0 ≤ 0 by symmetrically increased gains above the aspiration 1 − a level. We can therefore view CSMs as diversification- (or zero if no such a exists) on is a coherent satis- rewarding measures that retain the “target-oriented” ficing measure on . Moreover, nature of probability measures. Or, put another way, CSMs are quasiconcave “relaxations” of probability S X ≤ OS X ≤ X ≥ 0 measures. Note that we can express !a X = minV ∈ −Ɛ X − V + a/ 1 − a X −V V ≥ 0 as an optimization prob- 2.2. Examples lem over an n-dimensional convex cone. Moreover, We now provide some examples of satisficing mea- its objective approaches to infinity whenever Ɛ V or sures. We intentionally provide a number of examples V approaches infinity. Because the covariance of here to illustrate how the concepts of QSMs and CSMs X X is positive definite, the optimal solution cut across a variety of approaches to decision mak- 1 n must be bounded and achieved. ing under uncertainty, including risk measures from In the same example in which X ∼ U0 1, we note finance and ideas from utility theory. Some of these that for all a ∈ 0 1 , examples require proofs, which we have relegated to the e-companion. !a X ≤ min −Ɛ X − V Example 1 (Probability Measure). As we have V ∈ mentioned, X = X ≥ 0 is a satisficing measure. a Note that it satisfies all the axioms of coherence except + X− V V ≥ 0 X= V = 0 quasiconcavity. 1 − a
Example 2 (Sharpe Ratio). Let be a linear space Hence, OS X = 1. Thus, OS provides a better bound of a finite set of random variables including constants to probability measures than S . Brown and Sim: Satisficing Measures for Analysis of Risky Positions 76 Management Science 55(1), pp. 71–84, © 2009 INFORMS
Example 4 (Maximized Risk Aversion). Consider This quantity can be interpreted as the smallest the function Ca X = sup m ∈ Ɛ ua X − m ≥ 0, amount of capital t necessary to add to X to ensure where uaa>0 is a family of concave, nonde- that the augmented portfolio X + t breaks even with creasing utility functions nonincreasing in a>0. The probability at least 1 − %. Thus, all other things being scalar a is a risk aversion parameter, with larger a equal, a portfolio with a lower value-at-risk at a pre- reflecting greater risk aversion. Ca X is a certainty specified % is preferred. In addition, value-at-risk is equivalent and may be interpreted as the maxi- decreasing with %, i.e., lower % corresponds to more mum buying price of position X. Then the function conservatism.
X = sup a > 0Ca X ≥ 0 (or zero if no such When the distribution has discontinuities, there are a exists) is a QSM. As an example of this, when various definitions of VaR that take into account ua x = 1/a 1 − exp −ax , we have potential limiting conditions at these discontinuities. 1 VaR has received considerable attention among both X = sup a>0 − ln Ɛ exp −aX ≥ 0 practitioners and academics alike; see, for instance, a Duffie and Pan (1997) for one treatment of value-at- (or zero if no such a exists). It maximizes the risk risk. aversion so that its certainty equivalence of X ∈ In fact, probability measures as defined in (1) are under an exponential utility achieves the break-even simply dual forms of VaR. In particular, it is not hard point (this QSM is explored in Chua et al. 2007; inter- to see that estingly, it is also the reciprocal of the “economic index of riskiness” recently developed by Aumann X ≥ 0 = sup 1 − % VaR % X ≤ 0 (3) and Serrano 2008). Example 5 (Family of Concave Utilities). Let Indeed, observe that VaR1 X =− and for %<1, f → be a continuous concave nondecreas- the infimum in problem (2) is achievable since X + t ≥ 0 is a right continuous, nondecreasing function ing function with f 0 = 0 and limx→ f x = 1. Then X = lim sup Ɛ f a X + is a coherent with respect to t. Hence, ↓0 a∈+ satisficing measure. Moreover, 0 ≤ X ≤ X ≥ 0. sup 1 − % VaR X ≤ 0 The choice of function, f x = min x 1 has recently % been explored by Chen and Sim (2008). = sup 1 − % inf t t + X ≥ 0 ≥ 1 − % ≤ 0 Most of the examples we have provided in this sec- = sup 1 − % ∃t ≤ 0 t + X ≥ 0 ≥ 1 − % tion are of the form of choosing the largest value of a parameter subject to some sort of risk constraint. = max 1 − % X ≥ 0 ≥ 1 − % In the case of Examples 2 and 3, for instance, we = X ≥ 0 are choosing the largest value of a scalar a such that −Ɛ X + g a X ≤ 0, which is a mean-variance = X risk constraint. In Example 4, we are choosing the largest value of a risk aversion parameter subject to Note that (3) is a dual of (2) in which , the aspiration its certainty equivalent under a concave utility func- level, is specified and %, the tolerance level, is chosen tion being nonnegative at that level. Although it is not to be as small as possible. VaR, on the flip side, has the immediately obvious, we can also write Example 5 in tolerance level fixed with the target (or augmenting a similar form. In fact, it turns out that we can write capital) chosen to be as small as possible. Thus, value- any satisficing measure in the form of the largest value at-risk measures and probability measures are dual of a parameter subject to a risk constraint, as we will forms. now show. One may ask whether this relationship extends to general satisficing measures; in other words, whether 3. Dual Representation of Satisficing any satisficing measure can be represented in dual form over a family of risk measures. The answer, as via Risk Measures we will show, is yes. First, we define formally the If we were to choose positions based on higher or concept of a risk measure. maximal values of the satisficing measure X ≥ 0, Definition 3. A function ! →is a risk measure intuition suggests that we are somehow embedding if it satisfies the following for all X Y ∈ : risk aversion into our selection process; we are hop- 1. Monotonicity:IfX ≥ Y , then ! X ≤ ! Y . ing to find portfolios that have a greater likelihood of 2. Translation invariance:Ifc ∈, then ! X + c = achieving the aspiration level. ! X − c. Indeed, probability measures do in fact connect to The interpretation of risk measures is, like value- the classical definition of risk, known as value-at-risk. at-risk, the smallest amount of capital necessary to The usual definition of value-at-risk is augment a position in order to make it “acceptable”
VaR % X inf t ∈ t + X ≥ 0 ≥ 1 − % (2) according to some standard. As such, the properties Brown and Sim: Satisficing Measures for Analysis of Risky Positions Management Science 55(1), pp. 71–84, © 2009 INFORMS 77 above are clear; if one position never performs worse all X Y ∈ : than another, then it cannot be any riskier. In addition, 1. Convexity:If ∈ 0 1, then ! X + 1 − Y ≤ if we augment our position by a guaranteed amount c, ! X + 1 − ! Y . then our capital requirement is reduced correspond- If, in addition, we have ingly by c as well. See, for instance, Föllmer and 2. Positive homogeneity:If ≥ 0, then ! X = Schied (2004) for more on risk measures.5 ! X ; we say that ! is a coherent risk measure. We are now ready for our first result (please see the It is known that every coherent risk measure may e-companion for all proofs of main results). be written in the form
Theorem 1. A function → 0 ¯ , where ¯ ∈ ! X = sup Ɛ −X (6) 1 , is a satisficing measure if and only if there exists ∈ a family of risk measures !kk∈ 0 ¯ , nondecreasing for a family of generating measures (see, for in k, and !0 =− such that instance, Huber 1981 for a proof of this in a different context). Föllmer and Schied (2002) show more gener- X = sup k ∈ 0 ¯ !k X ≤ 0 (4) ally that any convex risk measure may be written in the form Moreover, given , the corresponding risk measure is
! X = sup Ɛ −X − % (7) !k X = inf a X + a ≥ k (5) where % is a convex function. It is then not hard to Theorem 1 states that every satisficing measure can show that a convex risk measure is coherent if and be written as the largest value of a parameter sub- only if the corresponding % in its representation is an ject to a risk constraint at that parameter value over indicator function on a subset of measures . a parametric family of risk measures. Moreover, the We remark that we will assume, without loss of proof of the result is constructive, so given a satisfic- generality, that a finite convex risk measure is nor- ing measure, we can always specify exactly the struc- malized such that ! 0 = 0; this implies, for instance, ture of the corresponding, parametric family of risk that % ≥ 0 for all . It is without loss of gen- measures. erality because, when ! is convex, ! X − ! 0 is also The interpretation is that, rather than specifying convex. risk aversion parameters (e.g., choosing an % for VaR), As the names of QSMs and CSMs betray, they are the satisficing measure approach is always equivalent intimately related to these classes of risk measures. In to fixing an aspiration level, then choosing the largest fact, one can see that the satisficing measure X = value of the aversion parameter such that the risk sup k ∈ 0 ¯ !k X ≤ 0, !0 =− and the !k are a of the target premium evaluated at that parameter is family of convex risk measures nondecreasing in k,is acceptable. In other words, satisficing measures quan- indeed quasiconcave; moreover, is a CSM when the tify how risk averse a position allows us to be while !k are also coherent. still maintaining the goal of achieving our aspiration Guided by Theorem 1, we may wonder whether level. every QSM and CSM has such a representation; in fact, this is the case. 3.1. Representation of QSMs and CSMs Of course, without additional, structural properties Theorem 2. A satisficing measure is quasiconcave if and only if the family ! k∈ 0 ¯ in Theorem 1 is a on the satisficing measure, there are no other prop- k erties that the family of risk measures must satisfy. family of convex risk measures. Similarly, it is coherent if We now examine structural properties of the family and only if the family is a family of coherent risk measures. of risk measures when we are dealing with QSMs and Remark 2. We remark that using Theorem 1 in the CSMs, respectively. opposite direction, it is possible to obtain potentially To relate these satisficing measures to risk mea- new risk measures starting from reasonable satisfic- sures, we now formally define a class of risk measures ing measures.6 Consider, for instance, the satisficing introduced by Artzner et al. (1999) and then general- measure in Example 5, which provides a recipe for ized by Föllmer and Schied (2002). In particular, we constructing coherent satisficing measures based on have the following. appropriate choices of utility functions. From Theo- Definition 4. A function ! →is a convex risk rem 2 and using Equation (5), the risk measure measure if, in addition to Definition 3, it satisfies, for
!k X = inf t lim sup Ɛ f a X + + t ≥ k (8) ↓0 a∈ 5 These authors use the same definition but refer to risk measures + as “monetary measures of risk,” which is perhaps more descriptive; for convenience we’ll simply use the term “risk measures.” 6 We thank an anonymous referee for suggesting this idea. Brown and Sim: Satisficing Measures for Analysis of Risky Positions 78 Management Science 55(1), pp. 71–84, © 2009 INFORMS
is a family of coherent risk measures on k ∈ 0 1. 3. Let += +1 + +2 /2. To make this expression more concrete, we consider 4. If S + is infeasible, update +2 = +. Otherwise, the interesting case when f x = exp −x , i.e., expo- update +1 = + and find Z ∈ S + . nential utility. To avoid excessive analysis, we con- 5. Go to Step 2. sider k>0 and assume that the limit in Equation (8) is achievable and that the supremum is attained at If the satisficing measure is given in its dual some a>0. We have form (4), we can evaluate the feasibility of S k by solving the following convex optimization problem: !k X = inf t ∃a>01−Ɛ exp −a X +t ≥k r k = min ! X X ∈ (10) = inf t ∃a>0ln Ɛ exp −a X +t −ln 1−k ≤0 k = inf t ∃a>0ln Ɛ exp −aX −ln 1−k ≤at Indeed, checking the feasibility of S k is the same if checking r k ≤ 0, which is finding the random vari- 1 1 = inf ln Ɛ exp −aX − ln 1−k able with the smallest risk under the convex risk mea- a>0 a a sure !k. which is easily verified as a coherent risk measure. In sum, we can find a *-optimal position X∗ to (9) in log b − a /* solutions of (10); provided we can In fact, the generating family for this coherent risk 2 measure is = ln d /d d ≤ ln k . This solve (10), then, we can also solve (9) to high precision is the coherent risk measure that protects equally without too much additional difficulty. against all measures close in relative entropy to the reference distribution . 3.3. An Ambiguity Robust Representation It would be interesting to see if a more general of QSMs Theorem 2 immediately implies an interesting inter- connection between utility theory and the theory of pretation for any quasiconcave satisficing measure. convex risk measures (often viewed as very differ- ent approaches) could be established; it seems that Corollary 1. A satisficing measure is quasiconcave the satisficing approach offers some clues into these if and only if there exists a family of convex functions connections. %kk≥0, nonincreasing on k ≥ 0, such that for all X ∈ and all , Ɛ X ≥−% X . 3.2. Optimization over QSMs This is a direct implication of Theorem 2 and the In contrast with maximizing the success probability representation theorem (7) for convex risk measures. over a convex set of random variables, which is com- This interpretation states that X can be inter- putationally intractable, we show that maximizing a preted as a “robustness level” that guarantees per- QSM can be reduced to solving a sequence of convex formance in expectation for distributions , which optimization problems. We consider the problem differ from our assessed distribution . We can visu- alize % as a family of functions measuring a scaled ∗ max X X ∈ (9) k notion of distance from to ;as X grows, we where is a convex set of random variables. From scale this distance by less, and therefore become closer a computational perspective, finding a feasible solu- to achieving our target in expectation even for other tion in a convex set is relatively easy compared to a distributions . Note that when is a CSM, the % will be indicator nonconvex one. In particular, given a QSM, , the set k functions on sets of probability measures. In partic- S k = X X ≥ k X ∈ is a convex set of ran- ular, in this case, there exists a nested family of sets dom variables. Indeed, if X ∈ S k and Y ∈ S k ,itis of probability measures k , i.e., k ⊆ k easy to see that X + 1 − Y ∈ S k for all ∈ 0 1. k≥0 1 2 for k ≥ k ≥ 0, such that, for all X ∈ , Ɛ X ≥ 0 Assuming that there exists a computationally efficient 2 1 ∀ ∈ X . In this case, we hit the target as long method for finding a feasible random variable in the as the “true” probability measure falls within the set set S k , and assuming that ∗ ∈ a b, a>0, and b< ¯, X , where X can be interpreted as measuring we propose the following binary search to obtain a the “size” of this set. solution, Z ∈ satisfying ∗ − * ≤ Z ≤ ∗. In short, this interpretation states that quasicon- Algorithm 1 (Binary Search). cave satisficers are expected value maximizers in an Input: A routine that returns a feasible random ambiguous world in which probabilities are not known variable in the set S k or reports infeasible; real, exactly and they value positions by how well they per- nonnegative numbers a, b, and *. form under other probability models, i.e., how insen- Output: A random variable Z. sitive a position is to changes in the underlying model.
1. Set +1 = a and +2 = b. This perspective connects directly back to Simon’s 2. If +2 − +1 <*, stop. Output: Z. stance that real-world agents do not know probability Brown and Sim: Satisficing Measures for Analysis of Risky Positions Management Science 55(1), pp. 71–84, © 2009 INFORMS 79 distributions exactly and therefore are making deci- We can illustrate this fact with some small exam- sions in inherently ambiguous environments. ples. Indeed, consider the random variable X ∼ Put another way, Corollary 1 implies that quasicon- U−1 2. Clearly, X ≥ 0 = 2 and a quick calcula- 3 cave satisficers are ambiguity averse in the sense that tion shows X = 1 . Now consider the random CVaR 3 they want to maximize how well they are protected in variable X = 1 2Y + Z , where Y ∼ U0 2 and Z ∼ 3 expectation against worst-case distributions that dif- U−2 0 and Y and Z are independent; the only fer from their underlying reference distribution. difference between X and X is that the left tail below zero has been spread out from −1 0 in X to 3.4. CVaR Measure −2 0 in X. Now, X ≥ 0 = 2 = X ≥ 0. On the In this section, we consider a CSM with some 3 √ √ other hand, we have X = 2 2 − 1 /3 < 1 = very interesting properties. This satisficing measure CVaR 3 X , so CVaR measure does in fact recognize the is a dual form of the following coherent risk mea- CVaR differences in the left-tail losses, whereas probability sure, popularized by Rockafellar and Uryasev (2000), measure does not. among others. A similar example can be used to illustrate that Definition 5. For any % ∈ 0 1, the coherent risk CVaR measure is sensitive to right-tail changes as well. measure Consider the same X; now let X = 1 Y +2Z , where 3 1 + Y ∼ U−1 0, Z ∼ U0 4, and Y and Z are inde- CVaR% X = inf - + Ɛ −- − X -∈ % pendent. X is the same as X with the right-tail (gains) is known as the conditional value-at-risk at level %. expanded. As before, X ≥ 0√= X ≥ 0 = 2/3, but now we find X = 4 − 2 /6 > 1 = X . It is well known that when X has a continu- CVaR 3 CVaR In fact, it turns out that CVaR measure is a special ous distribution, CVaR% X =−ƐX X ≤−VaR % X . Roughly speaking, we can interpret CVaR as the QSM when it comes to bounding probability measure; expected value over the lower %-tail of the distribu- we now define the following. tion. As it is a coherent risk measure, it induces a Definition 7. A satisficing measure on is law- CSM, which we now define. invariant if X = Y whenever X and Y have the Definition 6. The coherent satisficing measure same distribution under . → 0 1 This definition extends the notion of law-invariant risk measures (e.g., Föllmer and Schied 2004). Law- CVaR X = sup 1 − % CVaR% X ≤ 0 (11) invariance simply means that the satisficing measure depends only on the underlying distribution of the (or zero if no such % exists) is called the conditional random variable.7 value-at-risk measure (or CVaR measure) on the target It is well known that CVaR is the smallest law- premium X ∈ . invariant convex risk measure which dominates VaR Given the interpretation of CVaR, we can, roughly speaking, interpret X as one minus the smallest for all X ∈ over the space = L . By this CVaR we mean not only do we have CVaR X ≥ VaR X left quantile such that the expected value of X over % % this quantile is nonnegative. If the distribution is con- for all X ∈ , but if ! is any other law-invariant convex risk measure, the implication ! X ≥ VaR X tinuous, this interpretation is exact. % ∀ X ∈ ⇒ ! X ≥ CVaR X ∀ X ∈ holds. We now We consider the interesting case when X < 0>0. % Noting that v + 1/% Ɛ −X − v + >0 for all v ≥ 0, show that this idea extends to satisficing measures, we can express CVaR measures in the form of an i.e., that CVaR measure is the largest lower bound to expected utility of a concave function as follows: probability measure over all law-invariant QSMs. Theorem 3. Consider the case when = L ; CVaR X for any X ∈ , we have CVaR X ≤ X ≥ 0. More- = sup 1 − % CVaR% X ≤ 0 over, for any law-invariant quasiconcave satisficing mea- 1 sure → 0 1 the following implication holds = sup 1 − % ∃v<0v+ Ɛ −X − v + ≤ 0 % X ≤ X ≥ 0 ∀ X ∈ = sup 1 − % ∃v<0 1 − % ≤ 1 − Ɛ −X/ −v + 1 + ⇒ X ≤ CVaR X ∀ X ∈ (12) = sup 1 − Ɛ −aX + 1 + a≥0 7 It is worth noting that any law-invariant risk measure preserves = sup Ɛ min aX 1 first-order stochastic dominance (Föllmer and Schied 2004), which a≥0 immediately implies that any law-invariant satisficing measure does as well. Every example of a satisficing measure in this paper As such, it is clear that CVaR measures do not fail to is law-invariant. In fact, it may be sensible to impose this property ignore the magnitude of losses, as opposed to proba- as one of the satisficing axioms up front, but we decided against bility measures. this for the purposes of exposition. Brown and Sim: Satisficing Measures for Analysis of Risky Positions 80 Management Science 55(1), pp. 71–84, © 2009 INFORMS
As suggested by our small example above, the gap X ∈ and any % ≥ 1, let X% denote the modification in this bound can be quite loose; for instance, for of X by scaling its left tail by the factor %. Specifically, a random variable symmetrically distributed around we have zero, we have X ≥ 0 = 1 , whereas X = 0. 2 CVaR X if X ≥ 0 Still, Theorem 3 states that, among law-invariant X% = QSMs, we cannot hope to do any better. %X otherwise. Despite the gap in this bound, if is a family of % normal random variables, then the set of X ∈ that Now let X + be a right-tail modification defined % maximize the probability measure is the same as the analogously; we have, for any % ≥ 1, X = −1 set of solutions that maximizes the CVaR measure. max X 0 + % min X 0 = % % max X 0 + %−1 Indeed, if X is normally distributed and Ɛ X > 0, min X 0 ≤ X + . Therefore, under a QSM, a left- tail scaling by a factor of % ≥ 1 is less satisficing to a 0 1−1 % right-tail scaling by a factor of 1/%. Given the option, CVaR% X =−Ɛ X + X % an investor who maximizes a QSM would find cutting 2 % their upside by a factor of % no worse than increasing their downside by a factor of %. In this sense, then, where 0 · is the density of a standard normal, we can view QSMs as more sensitive to losses than and 1 · is the corresponding cumulative distribution gains and can consider this somehow related to more function. Now, noting that 2 % is a decreasing func- classical definitions of loss aversion. tion of %, we see that any X ∈ that maximizes CVaR measure also maximizes the Sharpe ratio, Ɛ X / X , 4. Portfolio Optimization Using CSMs as well as the probability measure X ≥ 0 = We consider an investor who selects a portfolio return 1 Ɛ X / X . that maximizes her probability of achieving at least
3.5. Loss Aversion Characteristics of QSMs the risk-free rate, rf , which is the return she would In this section, we briefly discuss some concepts of have had if she puts all her wealth in the risk-free “loss aversion” and their relationship to QSMs. The asset. Her portfolio comprises of a nonnegative frac- classical definition of loss aversion is greater penal- tion, %, of her wealth in a selected risky asset with ties for losses than rewards for gains of the same size. random return X ∈ and the remaining portion in This is loss aversion in the sense of early prospect the risk-free asset. Hence, her portfolio has random theory (Kahneman and Tversky 1979); in expected return %X + 1 − % rf . The fraction % is chosen such utility approaches, this is equivalent to concavity of that the expected return of her effective portfolio the utility function. We do not discuss loss aversion achieves the level , with >rf . Clearly, if ≤ rf , she in the sense of a “kink” in the slope of the utility would invest only in the risk-free asset. Therefore, it function around the target. Although there are actu- suffices to focus on the set of random variables such ally many definitions of loss aversion in the deci- that Ɛ X > rf for all X ∈ . The investor solves the sion analysis literature (for a detailed discussion of following problem: the many definitions, see Köbberling and Wakker max %X + 1 − % rf ≥ rf 2005, §6) and we do not provide any formal results, subject to Ɛ %X + 1 − % r = the discussion seems relevant given the large litera- f (13) ture on reference-dependent utility and its focus on X ∈ %≥ 0 loss aversion characteristics. If the returns are normally distributed, then prob- It does seem difficult to make such a compari- lem (13) is equivalent to the following: son, because this classical definition of loss aversion is equivalent to a property of the underlying utility Ɛ %X − %r max 1 f function and is therefore not directly applicable to %X approaches that are not utility based. Therefore, our subject to Ɛ %X + 1 − % r = discussion can only be meaningful if we are willing f to work with other notions of loss aversion. X ∈ %≥ 0 As an example, and to some extent generalizing the where 1 · is the distribution function of a standard example discussed with CVaR measure in §3.4, we normal. Noting that 1 · is an increasing function, can show that QSMs at least weakly prefer reduced this is equivalent to gains to increased losses. First, consider a risky position X ∈ and now con- Ɛ %X − %r max f sider scaling the position by some factor 3 ≥ 1. For a %X QSM , we will have 3X ≤ X . To see this, we (14) subject to Ɛ %X + 1 − % r = note that X = 1/3 3X + 3 − 1 /3 0, hence, X ≥ f min 3X 0 = 3X . Now, for any position X ∈ %≥ 0 Brown and Sim: Satisficing Measures for Analysis of Risky Positions Management Science 55(1), pp. 71–84, © 2009 INFORMS 81
Note that Ɛ %X −%rf = −rf , which is a constant; we for an application of satisficing measures to portfolio can therefore reformulate problem (14) as the mean- optimization against a competing benchmark. variance Markowitz portfolio model as follows: Consider a firm attempting to manage a future lia- bility L (which may be random) by investing a cur- 2 min %X + 1 − % rf rent amount of capital C. For ease of notation, assume C = $1. The firm has at their disposal the option to subject to Ɛ %X + 1 − % rf = invest in a risk-free asset with guaranteed return rf as X ∈ well as n risky assets with return vector r˜ (not neces- sarily independent of L). We denote by y the fraction Hence, under normal distributions, Markowitz port- of the capital invested in the risk-free asset and by x folio model is one that maximizes the probability i the fraction of capital in the ith risky asset. of achieving the risk-free return rate, subject to the Here, the aspiration level for investment is very desired level of expected return. natural: it is the firm’s liability amount L, which Generalizing this idea, we consider an investor who they want to ensure they cover. On the other hand, maximizes a CSM, · as follows: although the firm may be risk averse, appropriate tol- erance levels could well be unclear. In this situation, max %X + 1 − % rf − rf a satisficing measure on the net surplus X = 1 + yrf + subject to Ɛ %X + 1 − % rf = (15) r˜x − L is quite sensible. X ∈ %≥ 0 Consider the use of an underlying exponential util- ity function. The satisficing measure for a fixed posi-
We now formalize the discussion above. tion X has the form X = sup a > 0Ca 1 + yrf + r˜x − L ≥ 0, where C X =− 1/a ln Ɛ exp −aX is Theorem 4. Suppose E X > r for all X ∈ . Let · a f the certainty equivalent of X under the exponential be a CSM and suppose the objective of problem (15) is utility function. strictly positive for a given >r . Then the corresponding f To illustrate this with some near-analytical expres- optimal asset position X∗ is also optimal to the following sions, we consider the simple case when L is a problem constant and the covering positions are bonds with independent default probabilities p and returns + ,in max X − rf i i (16) which + 1 − p − p ≥ r . We have subject to X ∈ i i i f C X = C 1 + yr + r˜x − L = 1 + yr − L + C r˜x Theorem 4 implies that, when optimizing over a a a f f a CSM with the risk-free aspiration level, we may find 1 n = 1 + yr − L − ln Ɛ exp −ar˜ x a single “tangent” portfolio and then mix it with f a i i i=1 the risk-free asset according to our desired expected 1 n return. = 1 + yr − L − ln p exp ax f a i i We remark that if the risky assets are normally dis- i=1 tributed, the problem + 1 − pi exp −a+ixi max X ≥ r f In this case, then, our satisficing measure reduces to subject to X ∈ X = sup a>0 1 + yr − L is equivalent to f Ɛ X − r 1 n max f ≥ ln p exp ax a i i X i=1 subject to X ∈ + 1 − pi exp −a+ixi which is the classical problem of choosing an asset that maximizes the Sharpe ratio. If we do not allow short positions, our satisficing problem reduces to the convex optimization problem: 5. Application Example: Debt n Management inf b fi b xi ≤ 1 + rf y − L i=1 In this section, we demonstrate the use and applica- bility of satisficing measures. We encourage the inter- n y + x = 1 y≥ 0 x ≥ 0 b>0 ested reader to also consult §EC.2 of the e-companion i i i=1 Brown and Sim: Satisficing Measures for Analysis of Risky Positions 82 Management Science 55(1), pp. 71–84, © 2009 INFORMS
xi/b −+ixi/b where fi b xi = b ln pie + 1 − pi e , which is Figure 1 Effect of Liability Size on Satisficing for the Debt 8 Management Example jointly convex in b xi . Therefore, we can find the optimal satisficing position very efficiently. 70 For a simple example, consider independent and n = 2 identically distributed (i.i.d.) bonds with default prob- n = 5 60 n = 10 abilities p and returns +; by symmetry and the qua- n = 20 siconcavity of the satisficing measure, it is satisficing 50 optimal for the risky bond investments to be split equally. Hence, x = 1 − y /n and the satisficing mea- 40 i ) X sure reduces to ( ρ 30
X = sup a>0 1 + yrf − L 20 n ≥ ln p exp a 1 − y /n a 10 + 1 − p exp −a+ 1 − y /n 0 1.060 1.065 1.070 1.075 1.080 1.085 1.090 Liability value (L) From our robustness interpretation of satisficing measures, this measure tells us something very con- crete. In particular, the risk measure optimal satisficing portfolio and find the optimal sat- isficing level. We can see the effect of larger L on the
!a X =−Ca X = 1/a ln Ɛ exp −aX optimal satisficing level for various n. Clearly, as L gets larger, it becomes more likely that the firm will is convex and generated by the penalty function fall short on its payments. Note that there is a distinct value of L<+ at which drops to zero; this occurs %a = 1/a ln d /d d when L is large enough such that the firm cannot ∈ even achieve L in expectation under the given distri- i.e., a−1 times the relative entropy. For this example bution. This cutoff value for L is given by L = 1 + thresh with n i.i.d. bonds, consider that the true distribution n i=1 xiƐri = 1 − p 1 + + , which, for this example, differs from our assessed distribution according to we obtain Lthresh = 1089. We also see that satisficing default = p for some ≥ 1. In other words, we are goes up with n; this reflects the diversifying effect of off by a factor of in assessing the default probability. increasing the number of available bonds, which thins Then %1 = np ln + n 1 − p ln 1 − p / 1 − p , the left tail of the distribution. and the expected value Ɛ of our net surplus X under Figure 2 shows how the lower bounds on the this actual distribution is bounded below: expected surplus as the modeling error parameter
! X X ≤ 0 Figure 2 Impact of Probability Error on ExpectedSurplus for the Debt ⇔ sup −Ɛ X − % X ≤ 0 Management Example
npln + 1−p ln 1−p / 1−p )) 0 ⇔ Ɛ X ≥− X (
X –0.1 i.e., the firm cannot miss its payments by too much in expectation for distributions with small relative –0.2 entropy compared to the satisficing measure X .
This satisficing measure, then, offers a worst-case sen- –0.3 sitivity guarantee with respect to model mis-spec- ification in the relative entropy sense. –0.4 We illustrate a numerical example in Figures 1 and 2. The parameters are + = 10%, p = 001, and –0.5 ρ = 0.5 rf = 5%. In Figure 1, we take L as a constant value, ρ = 1 which we vary. For each value of L, we compute the –0.6 ρ = 2 ρ = 10
8 ∗ –0.7 The optimal satisficing value is the reciprocal of the optimal Lower bound on actual expected surplus ( 100 101 value of this problem, i.e., ∗ = 1/b∗. Modeling error () Brown and Sim: Satisficing Measures for Analysis of Risky Positions Management Science 55(1), pp. 71–84, © 2009 INFORMS 83 varies (n = 2 here). Increasing the satisficing level this paper. The research of the second author was supported decreases our sensitivity with respect to such errors. by SMA, NUS Risk Management Institute, and NUS Aca- We can compare this to the much more standard demic Research Grant R-314-000-068-122. risk management practice of using a utility function and maximizing the certainty equivalent. In such an approach, one would first need to specify a risk toler- References ance level. There is a significant body of work in the Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. 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