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MANAGEMENT SCIENCE Informs ® Vol 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.
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