Ann Oper Res (2010) 176: 153–178 DOI 10.1007/s10479-008-0470-7

Economically relevant preferences for all observed epsilon Haim Levy · Moshe Leshno · Boaz Leibovitch

Published online: 20 November 2008 © Springer Science+Business Media, LLC 2008

Abstract The common investment decision rules, Markowitz’s Mean-Variance (MV) rule and the non-parametric Stochastic Dominance (SD) rules, suffer from one severe drawback: there are pairs of prospects where experimentally 100% of the subjects choose one prospect, yet these rules are unable to rank the two prospects—a paradoxical result. Thus, the set of all preferences corresponding to these decision rules is too large, because it contains theoretical preferences that are not encountered in practice. Based on 400 subjects’ choices we define the economically relevant set of preference and the corresponding new decision rules, which avoid the paradoxical results. The results are very robust and are almost unaffected by the magnitude of the outcomes and the structure of the prospects under consideration. 1 Introduction The most commonly employed investment decision rules are the mean-Variance (MV) rule and the non-parametric Stochastic Dominance (SD) rules. All these rules have one major deficiency: there are cases where these rules are unable to rank two prospects even though in any sample of investors that one takes a clear cut ranking is revealed. To illustrate, suppose that an investor considers two mutually exclusive prospects A and B which involved the same initial investment. Each prospect is worthwhile to take but the investor must choose one of them. Prospect A yields $900 with a probability of 1/2 and $100,000 with a probability of 1/2. Prospect B yields $1,000 with certainty. We suspect that all investors will choose A, yet A does not dominate B neither by the MV rule nor by SD rules.1 Thus, the MV and SD

1 The reason for the no dominance of A is that there is a mathematically legitimate utility function for which

B is preferred. For example, take the utility function u(x) = x for x ≤ 1,000 and u(x) = 1,000 for x ≥ 1,000. It is easy to verify that EB u > EA u, hence prospect A does not dominate prospect B by FSD. H. Levy () · B. Leibovitch The Hebrew University, Jerusalem, Israel e-mail: [email protected] M. Leshno Tel Aviv University, Tel Aviv, Israel

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rules are too loose, because they consider also preferences that are unrealistic which will be considered in the rest of the paper as “economically irrelevant”. This paper experimentally reveals the set of economically irrelevant preferences, hence solve paradoxes like the one mentioned above. One may doubt the above claim and argue that in the above choice indeed some investors may prefer B, hence the no dominance is justified. For those doubtful readers let us make the following changes: A yields $999 with a probability 1/100 and $1 million with probability of 99/100 and B yields $1,000 with certainty. We suspect that even those who choose B in the previous example would shift to A with the new figures. Yet, even with those changes there is no dominance by MV and by SD rules, a clear deficiency of these common investment decision criteria. Of course we present above an extreme deficiency of SD and MV rules, but in practice, as we shall see in this paper, such deficiencies exist with much less extreme choices. The reason for deficiencies of the SD and MV rules like the once given above, is that these investment decision rules correspond to a wide set of preferences which generally contain also economically irrelevant preferences. As we shall see below the decision rules corresponding to a given set of preferences with some bounds on preferences are different from the existing decision rules, but coincide with them when the bounds on preferences are relaxed. Therefore the decision rules employed in this paper refer theoretically to a subset of all investors. Yet, in practice these decision rules refer to all or “almost” all investors. To see this recall that the set of all rational preferences is the set U1 such that u ∈ U1 with u ≥ 0 and there is some range where u > 0. The set of all risk averse preferences is the set U2 where u ∈ U2 ⊂ U1 , with u ≤ 0 and for some range u < 0. First degree Stochastic Dominance (FSD) and second degree Stochastic Dominance (SSD) corresponds to U1 and U2 , respectively. If the distributions are normal and u ∈ U2 the MV rule coincides with SSD (see Hanoch and Levy 1969). Though FSD and SSD are generally accepted, sometimes, they imply no clear cut ranking of prospects, where all subjects clearly prefer one of the prospects, as illustrated above. Namely, in the above choices though it is reasonable to assume that experimentally all investors would choose option A, theoretically we cannot say that A is preferred over B for all u ∈ U1 as A does not dominate B by FSD (note that option A does not dominate option B also by Markowitz’s MV rule).2 Recently, Leshno and Levy (2002) (hereafter L&L) define “almost” all rational investors where these investors are characterized by the set of non-decreasing monotone functions u ∈ U1 where U1 ⊂ U1 , when some constraints are imposed on the variability of u . Intuitively, a sharp decline (see example in footnote 1) or increase (see footnote 2) in u are not allowed. By the same token, “almost” all risk averters are characterized by the set U2 where U2 ⊂ U2 , when some constraints are imposed on the variability of u and u . Once again, the constraints on preferences are imposed to avoid paradoxes like the one illustrated above. Thus, instead of FSD corresponding to U1 we have “almost” FSD denoted by FSD∗ corresponding to the bounded set U1 , and “almost” SSD denoted by SSD∗ corresponding 2 The above example demonstrates the deficiency of SD and MV rules due to a sharp decline in the range

x ≥ 1,000. As U1 includes also convex functions, let us illustrate such possible deficiency also with a convex function with a sharp increase in u . Suppose that option A provides $14,999 with certainty and B provides $1,000 with probability 0.99 and $15,000 with a probability 0.01. We suspect that in any experiment with these two choices one would find that all subjects would prefer option A. Yet option A does not dominate B by FSD as the two cumulative distributions cross. The reason for such a no-dominance result is that there is a mathematically legitimate convex function u ∈ U1 with u = 0 for x ≤ 14,999 and u > 0 and very large for x > 14,999. Thus, the possible extreme changes in u assigns a very high utility weight to the last dollar in the $15,000 outcome, which in turn, creates such a deficiency with this convex preference.

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to the bounded set U2 . By the same argument we have “almost” MV denoted by MV∗ corresponding to U2 and normal distribution of returns.3 These new rules which substitute for FSD, SSD and MV, respectively, by construction (see below) avoid the scenario when two prospects are unranked, yet all subjects prefer one of them. What constraints on u and u should be imposed to eliminate the deficiency of the existing SD and MV rules?4 What preferences are economically irrelevant? To illustrate the path we take in answering these questions, let us consider the following example where option A yields either $1 or x = $1 million with an equal probabilities and option B yields $2 with certainty. We suppose that all investors prefer prospect A, and as there is no FSD, a deficiency of FSD criterion is revealed. Assume now that with option A rather than x = $1 million we have x = $10,000 with probability of 1/2. Is it still considered a deficiency of FSD as A and B cannot be ranked by it? What if it is only x = $1,000? To answer such a question we design three experiments. We define a clear cut deficiency of the existing investment rules by means of the smallest value x for which 100% of the subjects (composed of many groups) prefer one prospect over the other despite of the fact that by FSD (or SSD or MV) the two prospects cannot be ranked, hence are mistakenly included in the efficient set. In this study we experimentally find the set of preferences of U1 and U2 and the corresponding FSD∗ , SSD∗ (and MV∗ ) investment criteria. If one is willing to assume a specific α family of preferences e.g. u(x) = xα , then to solve the ranking problem illustrated above some constraints on α are imposed. This case is dealt with in Appendix A. However, in this paper we focus on the general sets of preferences U1 and U2 . The mathematical relationship between Ui and Ui (i = 1, 2) and SD and SD∗ (i.e. almost SD) have been developed by Leshno and Levy (2002), where, for any arbitrary constraints imposed on the variability u or u , L&L determine whether there is FSD∗ or SSD∗ . The paper of L&L is purely theoretical which relates to any arbitrary constraints on preference. However, L&L did not investigate what constraints are reasonable to impose from the investors’ point of view. We focus in this paper on SD and SD∗ rules and only some discussion is devoted to MV and MV∗ rules. However, it can be shown (for brevity sake the detailed analysis is not included in the paper) that MV∗ rule eliminates from the efficient set the lower part of Markowitz’s MV efficient frontier. It is interesting that this result is similar to the results of Baumol (1963) and Levy and Levy (2004), who used a completely different approaches. In particular, Levy and Levy (2004) find that a similar relegation of lower segment in the MV efficient frontier to the inefficient set is obtained by considering Prospect Theory preferences. When Normal distribution is assumed it can be shown that once one adds the riskless asset, employing the Sharpe (1966) ratio as a criterion for selecting among two choices, all investors who employ the MV∗ rule will select a portfolio located on the Capital Market Line (CML) and the Capital Asset Pricing Model (CAPM) is intact. Note that the CAPM is theoretically5 robust under several extensions in expected utility framework as well as in other frameworks. Berk (1997) has shown that normality is not necessary as the CAPM holds under the general assumption of Elliptic distributions. Levy and 3 The MV rule is optimal also under a set of Elliptic distributions and normality is not necessary (see Berk

1997). For example, for Logistic distributions MV rule is optimal. 4 Leshno and Levy (2002) impose constraints on u for FSD∗ and on u for SSD∗ . While the constraint on u is given in L&L, we focus in this study only on the constraint on u (not given in L&L) which allows us to show that FSD∗ ⇒ SSD∗ . 5 The CAPM is empirically criticized by numerous studies. However, a recent study by Levy and Roll (2008)

revealed that when little errors in the estimation of the mean returns are accounted for, the CAPM can not be empirically rejected.

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Levy (2004) show that the CAPM is intact also under Cumulative Prospect Theory (CPT) as long as normality is assumed. It can be shown that the CAPM holds also when choices are selected by MV∗ provided that the riskless asset prevails. In a completely different framework Quiggin and Chambers (2004) show that with invariant risk attitudes the properties of the CAPM which can be expressed in terms of mean and standard deviation may be extended to the case of general invariant preferences. In their general preferences they derive a risk-return formula which leads to “asset pricing results entirely analogous to the standard CAPM model as derived from mean-standard deviation preferences” (see p. 109). Maccheroni et al. (2006) develop a decision model where there is a utility on outcomes and an ambiguity index on the set of probabilities. They show that Markowitz’s Mean-Variance preferences is a special case of their general preference model. The concept introduced in this paper is applicable to any extension and generalization of expected utility as long as partial ordering is involved. For example to avoid the famous Allais paradox (1953), Prospect Theory (PT), (see Kahneman and Tversky 1979), Cumulative Prospect Theory (CPT) (see Tversky and Kahneman 1992), Rank Dependent Expected utility (RDEU) (see Quiggin 1982, 1993) and other studies suggest employing decision weights (DW). With DW the choices are explained and the paradoxes are solved. Our MV∗ and SD∗ criteria are intact regardless of whether objective probabilities or DW are employed, as long as all decision makers employ the same set of DW, like those DW suggested by CPT. For example, with CPT Levy and Levy (2002) develop Prospect Stochastic Dominance (PSD) and Markowitz Stochastic Dominance (MSD). Yet, paradoxes in choices as presented in this paper exist also with PSD and MSD, hence one can develop PSD∗ and MSD∗ (similar to SD∗ ) which eliminates economically irrelevant preferences, hence solves the observed paradoxes which may exit also with PSD and MSD criteria. The contribution of this paper is as follows: a) We prove SSD∗ when only a constraint on u (and not on u as done by Leshno and Levy 2002) is imposed. b) We design an experimental study by which we are able to determine the set of economically irrelevant utility functions. c) We introduce the concept of “allowed area violation” (in contrast to actual area violation given in Leshno and Levy 2002) and analyze the magnitude of FSD, SSD and MV “allowed area violation”. d) We show that practitioners implicitly employ the SD∗ or MV∗ rules in their investment recommendation and in particular in selling the Life Cycle Mutual Funds. The structure of this paper is as follows: Sect. 2 provides the various decision rules with numerical examples revealing the deficiency of FSD, SSD and MV, and explain the source of this deficiency. We analyze MV, FSD and SSD as well as MV∗ , FSD∗ and SSD∗ investment criteria. Section 3 provides the experiment and the results. Section 4 discuss the importance of MV∗ and SD∗ rule in practice. Concluding remarks are given in Sect. 5. 2 Decision rules and paradoxes 2.1 The MV rule The most common rule employed in investment decision making under uncertainty is Markowitz’s (1952) mean-variance (MV) rule. Denote two investment options by F and G. F dominates G by the MV rule if and only if, EF (x) ≥ EG (x),

σF ≤ σG

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with at least one strict inequality. (If F and G are normally distributed and u ∈ U2 , SSD and MV coincide.) The MV rule is based on a relatively confining assumption (e.g., a quadratic preference or normal distribution and risk-aversion, see footnote 3). The MV rule suffers from the same drawbacks of FSD and SSD as discussed above. To see this suppose that x and y are normally distributed and, Ex = 1,

σx = 1,

Ey = 106 ,

σy = 2.

It is easy to see that neither x nor y dominates the other by MV rule as well as by SD rules. For example, take the mathematically legitimate risk-averse utility function as follows:  u(t) =

t t0

t ≤ t0 t > t0

with t0 very small such that for t < t0 the cdf of y is above the cdf of x. Such a utility function reveals a preference for x. Similarly, one can easily find many risk averse functions showing a preference for y.6 Therefore, paradoxically both x and y are MV efficient. What is the remedy to the MV deficiency in this case, where probably 100% of investors in practice would choose y, yet the MV rule does not reveal this preference? We suggest to derive the adjusted MV rule, the MV∗ rule, which corresponds to a narrower set of preferences suitable for almost all investors, ruling out preferences like the one given above, and hence revealing of preference for y. 2.2 Stochastic dominance decision rules As we use in the experiment reported in this study SD and SD∗ rules, let us first define SD rules and demonstrate intuitively the concept of SD∗ with two examples, and then provide a formal definition of SD∗ rules. We show that in these two examples there is no stochastic dominance relationship, yet, “most” decision makers prefer one prospect over the other. In the examples below, we relate to First-degree Stochastic Dominance (FSD) and Seconddegree Stochastic Dominance (SSD), respectively. Therefore, let us first define these decision rules. Let x and y be two random variables, and F and G denote the cumulative distribution functions of x and y, respectively. 1. FSD definition: F dominates G by FSD (F 1 G) if F (x) ≤ G(x) for all x ∈ R. Thus, F 1 G iff EF u ≥ EG u for all u ∈ U1 , where U1 is the set of all non-decreasing differentiable real-valued functions. x 2. SSD definition: F dominates G by SSD (F 2 G) if −∞ [G(t) − F (t)]dt ≥ 0, for all x ∈ R. Thus, F 2 G) iff EF u ≥ EG u for all u ∈ U2 , where U2 is the set of all nondecreasing real-valued functions such that u < 0.  6 Denote the cumulative distribution of x by F , and of y by G. Then  ≡ E u − E u = ∞ [G(z) − F G −∞ F (z)]u (z)dz. Two cumulative normal distributions intersect at most once. Denote the intersection point by t0 . Then for t < t0 , G(z) > F (z) and as u ≥ 0, we have  > 0, i.e. x is preferred over y for this specific utility function (recall that u = 0 for t > t0 ).

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Fig. 1 Two hypothetical cumulative distribution functions

For proofs and discussion of SD rules, see Fishburn (1964), Hanoch and Levy (1969), Hadar and Russell (1969), and Rothschild and Stiglitz (1970).7 For a survey of SD rules and further analysis, see Levy (2006).8 2.3 Almost FSD (FSD∗ ) and almost SSD (SSD∗ ) We introduce the notion of FSD∗ and the intuitive difference between FSD and FSD∗ by means of the following example. Consider the following two distributions:  x=

1 106

probability 0.1 probability 0.9,

y = 2 with probability 1

The cumulative distribution Fx and Fy are given in Fig. 1. Because of the small “area violation”, denoted by “N ” in Fig. 1, F (x) does not dominate F (y) by FSD, SSD or MV rules. The mathematical reason for the no dominance is that there are some utility functions which assign a large utility weight to area “N ” and very small or zero weight to area K (see Fig. 1). As in practice almost all investors would choose prospect x, we would like to establish another decision rule which reveals a dominance of x over y despite the fact that FSD does not prevail. With this new rule we eliminate preferences assigning a very large utility weight to area N relative to the utility weight given to area K. By allowing this FSD violation, we obtain Almost FSD rule (FSD∗ ) corresponding to almost all investors. By a similar way we allow SSD area violation to derive SSD∗ rule. Let us now define formally the area violation illustrated in Fig. 1. The two cumulative distribution functions under consideration (F and G), are assumed to have finite support, say [a, b] (−∞ < a < b < ∞). Suppose that F is a candidate to dominate G. However, 7 SD criteria are not as well developed as the MV rule, in particular in their application to portfolio diversifi-

cation. However, recently, Post (2003) developed a linear programming procedure where he uses SD criteria to test whether one can rationalize the market portfolio efficiency by various set of preferences (see also Kuosmanen 2004). Also note that we define FSD and SSD such that binary relation is reflexive that is F dominates F by definition. In finance FSD and SSD are sometimes written without the reflexive requirement, namely a strong inequality is required for some value x. However, using either of the two definitions does not change the results of this paper. 8 In all the above investment criteria it is assumed that F and G are known. When F and G are empirical distributions, one has to test also for significance. See for example Barrett and Donald (2003).

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there are some “area violation” which prevent such dominance. Define the following “area violation” regions that will be used later on in the experiments: S1 (F, G) = {t ∈ [a, b] : G(t) < F (t)},    t  t G(x)dx < F (x)dx , S2 (F, G) = t ∈ S1 (F, G) : a

(1) (2)

a

where F and G are two cumulative distribution functions of the returns on two investments under consideration, respectively. As we shall see below, (1) and (2) describe the “area violation” regions of FSD and SSD, respectively. For two cumulative distribution functions b F and G we denote the total area between the two distributions by F − G = a |F (t) − G(t)|dt . The integral over S1 of [|G(t) − F (t)|] and over its complement, S1 , gives the total area enclosed between the two cumulative distributions. Leshno and Levy (2002) use the concept of actual area violation, εi , and we use in this paper the concept of “allowed” area violation, ε  . To see the difference of these two concepts let us first see how L&L make the connection between actual area violation and the corresponding subset of preferences. We first show below the relationship between εi and Ui (i = 1, 2) for FSD∗ and SSD∗ respectively, where εi (i = 1, 2) are the actual relative area violation corresponding to FSD∗ and SSD∗ , respectively and Ui is the set of economically relevant preferences. Then, we provide a formal relationship between εi and εi (i = 1, 2) which is essential for the discussion of our experiments, where εi is the “actual” area violation and εi is the “allowed” area violation. 1. Almost FSD (FSD∗ ) For any two cdf F and G we have,  EF (u) =



b

u(x)dF (x) =

[u(x)F (x)]ba

a

b

−





b

u (x)F (x)dx = u(b) − a

u (x)F (x)dx.

a

Therefore 

b

≡ EF (u) − EG (u) = 

u (x)[G(x) − F (x)]dx

a

u (x)[G(x) − F (x)]dx +

= S1



u (x)[G(x) − F (x)]dx,

S1

where over S1 , F (x) > G(x) and S1 is the compliment of S1 . Thus, > 0 iff      u (x)[G(x) − F (x)]dx = u (x)[F (x) − G(x)]dx < u (x)[G(x) − F (x)]dx. − S1

S1

S1

This inequality holds if,   sup{u (x)} [F (x) − G(x)]dx < inf{u(x)} [G(x) − F (x)]dx S1

S1

or

 

S1 [G(x) − F (x)]dx

sup{u (x)} < inf{u(x)} 

S1 [F (x) − G(x)]dx

.

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Define

 ε1 = 

S1 [F (x) − G(x)]dx

S1 [F (x) − G(x)]dx

+



S1 [G(x) − F (x)]dx

.

To obtain that the condition for > 0 given by, 

1 −1 sup{u (x)} < inf{u (x)} ε1 



or u (x) < inf{u (x)}





 1 −1 . ε1

(3)

Where ε1 is the actual relative area violation that is determined solely by F and G. Thus, for a given actual area violation ε1 all utility functions obeying the above inequality belongs to U1 ⊂ U1 . 2. Almost SSD (SSD∗ ) Leshno and Levy (2002) derive SSD∗ by imposing constraints on u . In this paper we derive SSD∗ by imposing constraint on u . We can add also a constraint on u , hence with u and u one can obtained better theoretical results relative to the one reported by Leshno and Levy (2002). However, imposing only a constraint on u makes the proof very intuitive and transparent and also allows a simple comparison between FSD∗ and SSD∗ . For any two cdf F and G we have, 



b

EF (u) =

b

u(x)dF (x) = [u(x)F (x)]ba − a

u (x)F (x)dx = u(b) −

a



b

u (x)F (x)dx.

a

Therefore 

b

u (x)[G(x) − F (x)]dx

≡ EF (u) − EG (u) = 

a



u (x)[G(x) − F (x)]dx +

= S2

u (x)[G(x) − F (x)]dx,

S2

where S2 is defined by equation (2) in the text (the region which if eliminated we would have SSD of F over G), and S2 is the compliment of S2 . Obviously S2 is a subset of S1 (see (1) and (2)). Note that as u ∈ U2 the integral over S2 is non-negative (see (2) and Fig. 3). However, if u ∈ U2 than this claim is incorrect and the integral over S2 may be negative. Thus, for u ∈ U2 > 0 iff, 

u (x)[G(x) − F (x)]dx =

− S2



u (x)[F (x) − G(x)]dx <

S2



u (x)[G(x) − F (x)]dx S2

This inequality holds if and only if, sup{u (x)}



 [F (x) − G(x)]dx < inf{u(x)}

S2

[G(x) − F (x)]dx. S2

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As u ∈ U2 (i.e. u < 0) we have sup{u (x)} = u (a) and inf{u (x)} = u (b) hence, the above condition is reduced to  S [G(x) − F (x)]dx   . u (a) < u (b)  2 S2 [F (x) − G(x)]dx Define the relative actual SSD area violation ε2 as follows,  S2 [F (x) − G(x)]dx  ε2 =  . S2 [F (x) − G(x)]dx + S2 [G(x) − F (x)]dx To obtain the condition for > 0 if for a given area violation ε2   1 u (a) < u (b) −1 . ε2

(4)

Thus, all preferences which fulfill the above inequality belong to U2 ⊂ U2 . Having the relationship between FSD∗ and U1 and SSD∗ and U2 , let us define ε1 and ε2 given above and ε1 and ε2 which are derived from our experiments. • ε1 = is the actual FSD relative area violation, i.e. it is the integral over the range S1 (F, G) divided by the total absolute area enclosed between F and G. It does not reflect preferences, and is purely determined by the characteristics of F and G. • ε1 = the allowed FSD relative area violation which reflects the investors’ preferences. Thus, while ε1 is objective, ε1 is subjective and may change from one experiment to another. As we shall see below if ε1 < ε1 , we have FSD∗ for the set U1 corresponding to ε1 . ε2 and ε2 corresponding to SSD∗ and risk aversion are defined in a similar way. To provide an intuitive explanation to the relationship between ε1 and ε1 , suppose that F is below G in most of the range of outcomes but it is above G in some range S1 , with a calculated actual area violation of ε1 = 1%. If all subjects allow a larger area violation, say, ε1 = 5%, and still all prefer F over G, such a preference a fortiori holds for a lower actual area violation ε1 = 1%. Thus, if ε1 < ε1 we have a dominance by FSD∗ . Therefore the FSD∗ and SSD∗ are defined in terms of ε1 and ε2 as follows: Definition (ASD): Let x and y be two random variables, and F and G denote the cumulative distribution functions of x and y, respectively. For 0 < ε1 , ε2 < 0.5 we define: FSD*: Suppose that all investors allow ε1 or more area violation. Then, F dominates G by ε1 -Almost FSD if and only if,  [F (t) − G(t)]dt ≤ ε1 F − G . (5) S1

Namely, dominance exists iff the area violation ε1 ≤ ε1 . SSD*: Suppose that all risk-adverse investors allows ε2 or more area violation. Then, F dominates G by ε2 -Almost SSD if and only if,  [F (t) − G(t)]dt ≤ ε2 F − G . (6) S2

Namely, dominance exists iff the area violation ε2 ≤ ε2 .

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Note that ε1 and ε2 are the “allowed” area violation by all rational investors and by all risk-averse investors, respectively, such that SD∗ exists. Also, for a given ε1 = ε2 = ε  , if (5) holds (6) holds, implying that FSD∗ ⇒ SSD∗ . The required conditions ε1 < 0.5 and ε2 < 0.5 imply that EF (x) ≥ EG (x) is a necessary condition for FSD∗ and SSD∗ .9 To sum up we have two approaches: By the first approach we check the actual area violation ε1 which defines the set of preferences U1 for which FSD∗ exists. By the second approach we find for a given group of subjects what is the allowed area violation by each investor and whether for all subjects belonging to this group the allowed area violation is greater than the actual area violation. If this is the case we can conclude that for this specific group there is FSD∗ . Obviously, in such a case all the subjects in this group have preferences which belong to U1 . As we shall see below we design experiments such that the subjects themselves determine the distributions under consideration, hence determine by their preference the actual area violation ε1 . But as their preference reflects the allowed area violation ε1 , we have in this specific experiments that ε1 = ε1 . The same argument holds in the case of SSD∗ hence, in our specific experiments we have also that ε2 = ε2 . However, recall that in general εi may differ from εi , where εi is objective and depends solely on F and G, while εi is subjective and depends on preferences.

3 The experiment and the results We conducted three experiments, each with different choices aiming to measure the allowed area violation εi (i = 1, 2) under various scenarios. The characteristics of the subjects and the experiments are as follows: • Experiment 1: n = 196 subjects. The choices involve only positive cash flows. There are six groups of students, five groups are composed of third year undergraduate students and one group of third year medical students. In two groups, the subjects where told that among the subjects who make consistent choices (for details see below) in all tasks, a lottery will be conducted and one of them will get a $100 prize. In the group of the medical students a 5% addition to the grade is rewarded to all those who make consistent choices. There where only slight difference between the results of those six groups, hence, for brevity sake we report the results of all subjects together. • Experiment 2: Here we have n = 88 MBA students. What distinguishes this experiment from the previous one is that the outcomes are relatively large, thousand of dollars rather than hundreds of dollars. This allows us to study the needed restriction on preferences “in the small” and “in the large”. • Experiment 3: We have n = 132 subjects of five groups of subjects, three of them included third year undergraduate business school students; one group included Medical Laboratory employees. The fifth group, includes financial analysts and mutual fund investment managers with many years of experience on the job. The choices involve positive as well as negative outcomes. We analyze the effect of the negative outcome on the perceived risk, hence on the allowed area violation. 9 To have dominance, we must have that ε ≤ ε  (i = 1, 2). Hence, also ε < 1 and ε < 1 . i 1 2 i 2 2 To see that this implies that EF (x) ≥ EG (x) is a necessary condition for dominance, note that b   EF (x) − EG (x) = a [G(x) − F (x)]dx = S [G(x) − F (x)]dx + S [G(x) − F (x)]dx. We have ε1 = 1 1  S1 [F (x)−G(x)]dx   < 12 . Cross multiplied to see that EF (x) ≥ EG (x). [G(x)−F (x)]dx S [F (x)−G(x)]dx+ 1

S1

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Reporting in details each group separately would require a large space and it is involved with many repetition. Therefore, we report in detail all the choices in Experiment 1 and we report only a summary results of Experiment 2 and 3, respectively. In the experiments reported below we measure experimentally the corresponding relative area violation ε1 and ε2 by each subject, and the smallest ε  across all subjects is the allowed area violation by 100% of the subjects. Note that if the smallest allowed area violation is, say, ε1 = 2%, then those who allow ε1 > 2%, a fortiori will allow ε1 = 2%. 3.1 Experiment 1 The subjects face two tasks, one designed to test the relationship of U1 and U1 and one is designed to test the relationship between U2 and U2 . 3.1.1 Task I Table 1 provides the choices in Task I. The subjects had to choose between prospects A and B five times. In the sixth choice, which is the crucial one for our study, they have to write what is the minimum value of $z such that they will choose prospect B. Note that names (option A or B) and the place (right and left columns) of the almost dominating option changes across the various choices to make sure that the subjects will not be biased in their selection by selecting one option mechanically. The choices in decisions 1 and 2 are conducted simply to check whether the subjects violate FSD. In the first choice A dominates B by FSD and in the second choice B dominates A by FSD. In the next three choices there is no FSD dominance. Table 1 Task I: In each decision you have to choose between Prospect A and B. Circle your choice. (Experiment 1) Decision

1

2

3

4

5

Prospect A

Prospect B

Probability

Outcome

Probability

Outcome

0.5

$100

0.5

$50

0.5

$200

0.5

$150

0.5

$50

0.5

$100

0.5

$200

0.5

$200

0.5

$50

0.5

$100

0.5

$250

0.5

$200

0.5

$100

0.5

$50

0.5

$200

0.5

$350

0.5

$50

0.5

$100

0.5

$400

0.5

$200

What is the minimal value z such that will make you prefer B over A? Please write z = $_ Decision

6

Prospect A

Prospect B

Probability

Outcome

Probability

Outcome

0.5

$100

0.5

$50

0.5

$200

0.5

$z

164

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Fig. 2 Cumulative distributions of A and B in Task I

However, we use these 3 choices to check for consistency of the decisions. For example, if in choice 4 the subject prefers option B and in choice 6 the subjects write the minimum value z = $400, these two choices are inconsistent because the minimum z should be $350 or less. Similarly, if in choice 4 the subject selects B and in choice 5, the subject selects also B we consider these choices inconsistent (see Table 1). We have about 8% inconsistent choices. All the cases of inconsistent choices were eliminated, as we suspect these subjects did not give serious consideration to the choices, and filled out the questionnaire just to get rid of it. In problem 6, which is the heart of Task I of the experiment, the subjects had to write the minimum value $z such that B is preferred over A. From this minimum value we can learn about the subjects’ preferences and in particular about the relationship between FSD and FSD∗ and therefore between U1 and U1 . Figure 2 demonstrates the cdf of A and B for a value $z of $1000 corresponding to decision number 6. We have to distinguish between various ranges of z where different conclusions regarding U1 can be drawn. First note that B can never dominate A by FSD. However, by the same token it is obvious that for a very large value z, e.g., $1 million, in practice all investors would prefer B. Thus, though B does not dominate A in U1 such a dominance may hold for the bounded set U1 i.e. there is FSD∗ dominance. Finally note that the selected value by the ith subject, zi , determine her allowed area violation εi , and for the highest selected zi we have the smallest value εi (see Fig. 2). Also, once zi is determined, we have that the actual and allowed area violation are identical, as F and G are determined by the selection of zi . Yet, by taking z = max zi , we know that if B is preferred over A for zi it is a fortiori preferred for a larger value of z. Thus, z = max zi determine the FSD∗ of B over A, because with z all subjects prefer B to A. Let us illustrate by means of Fig. 2, the notion of “economically irrelevant” preferences, the concepts of ε1 and the difference between FSD and FSD∗ implying the difference between U1 and U1 . Suppose that we have many subjects and each subject selects her value z. Furthermore, suppose that the highest value z across all subjects is z = 1000 (see Fig. 2). It is easy to show that the difference in expected utility  ≡ EB u(x) − EA u(x) is given by (see Fig. 2)  =

∞

−∞

[FA (x) − FB (x)]u (x)dx

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165

where FA and FB are the two cumulative distributions of A and B, respectively (see Levy 2006, p. 57). In our specific example with z = 1000,  can be rewritten as follows:  = −50u¯1 + 800u¯2 where u¯1 is the average u in the range 50 ≤ X ≤ 100 and u¯2 is the average u in the range 200 ≤ X ≤ 1000. If for z = 1000 all subjects prefer B over A with max zi = z = 1000, we have FSD∗ of B over A, even though we have no FSD. Thus,  > 0 implies that  = −50u¯1 + 800u¯2 > 0,

or

u¯1 /u¯2 < 16 must hold for all subjects. Thus, though U1 includes also all preferences with u¯1 /u¯2 > 16, from U1 such preferences are eliminated. Thus, the subjects reveal that such a reduction in the marginal utility by shifting from the range 50 ≤ X ≤ 100 to the range 200 ≤ X ≤ 1000, though theoretically is allowed, simply does not reflect their preferences. Thus, any u ∈ U1 but u ∈ U1 (e.g. those functions with u¯1 /u¯2 > 16) is “economically irrelevant”. To sum up, though B does not dominate A by FSD regardless of the selected value z, with z ≥ 1000, B dominates A by FSD∗ . With z = 1000 the value ε1 = 50/(50 + 800) = 1/17 (see Fig. 2) is the allowed area violation by all subjects. This area violation is allowed, simply because the subjects’ choices reveal that u does not decreases dramatically, hence for z ≥ 1000 B is preferred over A despite the fact that there is no FSD. As we have a sample of subjects and not the whole population, ε1 is estimated experimentally. Of course, we may have a subject with an extreme reduction in u who is not included in our sample. Hence, we say that B dominates A by “Almost FSD”, meaning that some pathological preference may exist showing a preference for A. Therefore, the SD∗ rules are investment decision rules for almost all investors but not for all investors. Moreover, if our sample is a representative sample, we can determine the p% FSD, e.g. p = 95%, indicating the B dominates A by 95% of the investors where ε1 is chosen such that there are 5% higher εi . This has practical implication as the investment consultant knows that she recommends the right choice for 95% of the inventors, though there are no FSD. The results of Task I Out of the 196 subjects 16 subjects (i.e. about 8.2%) are characterized by inconsistent choices and in particular violated FSD dominance.10 Eliminating the inconsistent choices we are left with 180 subjects. Table 2a reports the results regarding the selected value z and Table 2b reports the implied allowed relative area violation ε1 by the various subjects, such that we have a preference of option B over option A. Table 2b focuses on the 180 relevant subjects and use the same information of Table 2a but the percentage figure are calculated based on n = 180 subjects. As we can see from Table 2b about 23% of the subjects selected z = 250, which can be explained by risk neutrality preferences. For all other choices with z > 250, risk-seeking preferences can not explain the choices, yet also there is no FSD or SSD dominance (see Fig. 2). Moreover, as we see from Table 2b, many subjects required a substantially higher value z than $250, indicating that we may have FSD∗ or SSD∗ of B over A. Thus, in practice as we increase z more subjects prefer B over A, and for z = $1,000, 100% of the subjects prefer B over A, hence we find experimentally that in our population for this value we have FSD∗ of B over A. Thus though 10 For example, subject who select z < 200 violates FSD as for all z < 200, A dominates B by FSD.

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Ann Oper Res (2010) 176: 153–178

Table 2a The distribution of the 196 choices including those with FSD violation. Task I (Experiment 1)

Selected z

Number of choices

1

1

50

1

100

7

200

6

201

1

250

41

251

10

255

1

275

1

300

63

301

1

325

3

350

9

400

27

401

1

500

15

600

2

750

1

800

1

1000

4

Total

196

Table 2b The Distribution of the selected value z and implied FSD* ε1 of the consistent 180 choices.a (Task I, Experiment 1) Selected z

Number of choices

In percent

The implied FSD violation, ε1

250

41

22.8

251–275

12

6.7

0.500

300

63

35.0

301–350

13

7.2

400

27

15.0

0.20

401–800

20

11.1

0.077–0.199

1,000

4

2.2

Total

180

0.400–0.495 0.333 0.250–0.331

0.059

100

a We obtain in this experiment that max z = $1,000 ⇒ min ε  = 5.9% i 1,i

B never dominates A by FSD (regardless of the selected z), experimentally we find that for z = $1,000, 100% of the population participating in the experiment prefer B over A. It is possible that with another sample of subjects the minimum value of z, which reveals 100% preference for B will be above or below z = $1,000. Yet, it is clear that the same type of results would be obtained in any sample of subjects: namely, there is some finite value z for which 100% of the subjects would prefer B over A despite the fact that B does not dominant A by FSD, and this is the main justification for the introduction of SD∗ criteria. Finally note

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167

that with z = 1000 we have a preference for B over A by 100% of the subjects. This means for all subjects that −50u¯1 + 800u¯2 ≥ 0 or u¯1 /u¯2 ≤ 16. Thus there is a bound on the average reduction in u by shifting from the range 0 ≤ z ≤ 50 to the range 200 ≤ z ≤ 1000. Let us illustrate how the results reported in Table 2b are calculated. Suppose that indeed our 180 subjects represent the whole population, i.e., we would have the same percentage choice of z (and ε1 ) in the population as in our experiment. The subjects who selected, say, z = $300 allow an area violation of up to 33.3% of the area enclosed between FA and FB , (see ε1 reported in Table 2b), and hence prefer B over A. For z = 300, the 33% is given by the formula  $25 S1 [F (x) − G(x)]dx   = 33.3%. = $25 + $50 [F (x) − G(x)]dx + [G(x) − F (x)]dx S1 S1 By the same way all other εi are calculated. The most severe group contains those who selected z = $1,000, i.e., allowing no more than ε1 = 5.9% area violation.11 Of course, U1 (ε1 = 20%) ⊂ U1 (ε1 = 5.9%) as those who would allow 20% area violation in the cdf to prefer B over A would a fortiori prefer B with a lower violation ε1 = 5.9% in FSD.12 In short, those subjects who required z = $400 will be more than happy to receive z = $1,000. Therefore, in our experiment, for z = $1,000 all investors would prefer B to A. With z = $800 (or ε1 = 7.7%) (see Table 2b) we can say that B dominates A for 97.8% of the population, and in this respect, B dominates A for “almost all” but not “all” investors. For z = $1,000, B dominates A by all investors (in our sample with u ∈ U1 ) though such dominance is not intact for all u ∈ U1 . If our subjects represent the population of all investors we have that all those preferences eliminated from U1 are economically irrelevant though mathematically these preferences are included in U1 . Thus, in the specific example given in Table 1, decision 6, 5.9% area violation in the FSD is allowed by all subjects. Moreover, this 5.9% area violation must be allowed to avoid the deficiency of FSD induced by the fact that 100% of the subjects prefer B over A, yet there is no FSD (see Table 2b). The bounded set U1 represents the sets of all economically relevant preference, while the set U1 contains the set of all mathematically relevant preferences. Therefore, though FSD rule is mathematically correct, FSD∗ rule is economically more relevant in practice: on the one hand it applies to all investors and on the other hand it avoids paradoxes. Let us now clarify the relationship between actual (ε1 ) and allowed area violation (ε1 ). Suppose that ε1 = 5.9%, as we found in the experiment and further suppose that this result is approximately robust for other similar experiments with different set of choices. Now if we find with two actual prospects F and G, such that ε1 < ε1 , say, ε1 = 1%, we can conclude that F dominance G by FSD∗ . The reason is that all investors allow ε1 = 5.9% or more area violation in order to prefer F over G, and such preference is a fortiori intact for a lower  is the minimum allowed area actual area violation of ε1 = 1%. Thus, ε1 = 5.9% = mini ε1,i violation such that 100% of the subjects prefer one option over the other, and for any actual smaller area violation FSD∗ is intact. Of course for ε1 = 0 we have FSD for all u ∈ U1 regardless of the allowed relative area violation ε1 . 11 Note that since in the FSD experiment we have a left tail violation, the SSD and FSD ε parameters are

identical. Thus S1 and S2 are the same set. 12 We obtained robust results in the estimation of ε  and ε  . However, more experiments are called for with a 1 2 violation area located in various places and may be with more than one violation area, where the size of ε1 is kept constant.

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Ann Oper Res (2010) 176: 153–178

Finally, in this study we stress that we need a decision rule e.g. FSD∗ which corresponds to 100% of the subjects. However, an investment consultant may decide to consider less sever restrictions on U1 , and therefore will be satisfied with a rule corresponding to p% of the population, e.g. p = 95%. Appendix A analyzes this case. Let us turn to the SSD∗ experimental results. 3.1.2 Task II: Almost SSD (SSD∗ ) In Task II we examine the relationship between SSD and SSD∗ which reflects the relationship between U2 and U2 . Table 3 presents the five decisions of Task II. Once again, decisions 1–4 are used as the check for consistency of the decision process. For example, selecting B in decision 1 and B in decision 2 reflects inconsistency. Similarly, if one chooses B in decision 3 and writes z = $500 in decision 5, it would be considered inconsistent behavior. We learn on the relationship between SSD and SSD∗ from decision 5, where z is the minimum value such that A is preferred (see Table 3). Figure 3 presents the cdf of A and B, for a hypothetical value z > $300. There is no FSD and no SSD dominance of either B over A or A over B. Moreover, even if we select z to be  200 very large, e.g., z = $10,000, there is no SSD of A over B because 100 [FB (x)−FA (x)]dx < Table 3 The choices in Task II: decision choose between A and B in the following five decisions (Experiment 1)

Decision

1

2

3

4

Option A

Option B

Probability

Outcome

Probability

Outcome

1 3 1 3 1 3

$100

1 3 1 3 1 3

$125

1 3 1 3 1 3

$100

1 3 1 3 1 3

$125

1 3 1 3 1 3

$100

$200 $300

1 3 1 3 1 3

$125

1 3 1 3 1 3

$100

1 3 1 3 1 3

$125

$150 $325

$200 $300

$200 $300

$150 $300

$200 $300

$150 $400

$150 $400

What is the minimum value z for which you prefer A over B? Please write z = $_ Decision

5

Option A

Option B

Probability

Outcome

Probability

Outcome

1 3

$125

1 3

$100

1 3

$150

1 3

$200

1 3

$z

1 3

$300

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169

Fig. 3 The cumulative distribution functions of A and B in Task II

 125 0. Similarly, B does not dominate A because 100 [FA (x) − FB (x)]dx < 0. Thus, in U2 both A and B are in the efficient set. However, for a very large z, most if not all investors would select A because there is SSD∗ of A over B. Namely, the negative SSD violation area (see Fig. 3, area “N”) which is the reason for the no SSD dominance has smaller utility weight relative to the positive large area denoted by “+”. Thus, unless u changes very dramatically, A would be preferred over B. How far should z be to the right to overcome the effect of the negative area over the range 175 < x < 200? What is the value z such that we have SSD∗ for all investors though there is no theoretical SSD? This is an experimental question to which we turn next. The results of Task II Table 3a presents the value z chosen by the subjects and the corresponding frequency distribution. We have relatively large frequency at z = $300, $325, $350, $400 and $500 with the mode at z = $400 with 58 out of the 196 subjects selecting this value. Table 3b focuses on the 180 subjects who did not violate the consistency test.13 Assuming risk-aversion, Table 3b reveals that if one selects z = $1,000 with corresponding ε2 = 0.032 (i.e., relatively very small area violation) than all risk-averse investors, i.e., 100% of the subjects prefer A over B. In such a case U2 ⊂ U2 and the utility functions eliminated from U2 are economically irrelevant. Hence, we have SSD∗ rule which on the one hand corresponds to all relevant risk-averse investors and on the other hand avoids the deficiency of SSD rule. The most important lesson we learn from Task II is that some SSD area violation may be irrelevant in practice as there is some value z, not astronomically large such that A dominates B by SSD∗ for all risk-averse investors in U2 . Namely, the negative utility due to the range where the cdf of A is above the cdf of B (and where SSD is violated, see area “N” in Fig. 3) is much smaller than the positive utility induces from the fact that the cdf of A is below B. And if this does not occur, one can always increase z until it occurs. It is found in this experiment 13 It incidently occurred that in the SSD test like in the FSD test exactly 180 subjects did not violate the

consistency test.

170 Table 3a The choices in Task II including the FSD violations (Experiment 1)

Table 3b Choices with no SSD violation and the implied ε2 in Task II (Experiment 1)

Ann Oper Res (2010) 176: 153–178 Selected z

Number of choices

100

5

150

2

175

4

200

5

250

6

275

1

300

16

310

1

325

26

326

7

350

36

362

1

375

1

400

58

425

1

430

1

450

3

451

1

500

14

600

2

1000

5

Total

196

Selected value z

Number of choices

In percent

The implied ε2

250

6

3.3

275

1

0.56

0.50

300

16

8.9

0.33

310

1

325

26

0.56 14.4

1.0

0.294 0.250

326

7

3.9

0.248

350

36

20.0

0.20

362

1

0.56

375

1

0.56

400

58

425

1

0.56

0.125

430

1

0.56

0.122

450

3

1.67

0.111

451

1

0.56

0.111

500

14

7.78

0.091

600

2

1.11

0.067

1000

5

2.78

0.032

32.2

0.182 0.167 0.143

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171

that with z = $500, the SSD∗ covers 96.1% of the subjects (as only 1.11% + 2.78% required a higher value z, see Table 3) and with z = 600 the SSD∗ covers 97.2% of the subjects.14 Therefore, in practice the u of most subjects does not change dramatically over the various ranges of x. If u of the subjects would change dramatically, then we may need to have a very large z to have SSD∗ which avoids the paradox. So far, we obtain ε1 and ε2 which rely on some specific choices. How robust are all the results? How these values changes with charges in the magnitude of the financial outcomes? How do they change with changes of the structure of the prospects under consideration? To these issues we devotes experiment 2 and 3 given below. 3.2 Experiment 2 To see whether ε1 and ε2 depend on the magnitude of the payoffs, we conducted a similar experiment for FSD∗ and SSD∗ where all outcomes are multiplied by 100. Table 4 and Table 5 provide the choices and the results with n = 88 subjects. We find that the dominance area violation ε1 and ε2 corresponding to FSD∗ and SSD∗ , respectively, are quite invariant to the size of the payoffs, at least for the range given in Tables 3 and 4. Table 4 What is the minimum value z for which you prefer A over B? Please write z = $ (n = 88 subjects)

Decision 5

Prospect A

Prospect B

Probability

Outcome

Probability

Outcome

1 3 1 3 1 3

$12,500

1 3 1 3 1 3

$10,000

$15,000 $z

$20,000 $30,000

Table 5 Choices with no violation and to implied ε1 and ε2 (Experiment 2) Selected value z

Number of choices

In percent

The implied ε1

The implied ε2

30,000

7

8.05

0.667

0.333

32,500

25

28.74

0.500

0.250

32,501

3

3.45

0.500

0.250

32,510

1

1.15

0.499

0.250

33,000

3

3.45

0.476

0.238

33,250

1

1.15

0.465

0.233

34,000

1

1.15

0.435

0.217

35,000

19

21.84

0.400

0.200

36,000

1

1.15

0.370

0.185

37,500

1

1.15

0.333

0.167

40,000

12

13.79

0.286

0.143

42,500

2

2.30

0.250

0.125

45,000

1

1.15

0.222

0.111

100,000

1

1.15

0.065

0.032

14 We define in Appendix A, SSD∗ which corresponds to p% of the subjects. However, in the text we define SSD∗ as the rule which corresponds to p = 100% of the subjects.

172 Table 6 The three choices in Experiment 3

Ann Oper Res (2010) 176: 153–178 Task I: What is the minimal value z such that will make you prefer B over A? Please write z = $_ Prospect A

Prospect B

Probability

Outcome

Probability

Outcome

0.5

$1,000

0.5

$-500

0.5

$2,000

0.5

$z

Task II: What is the minimum value z for which you prefer A over B? Please write z = $_ Prospect A

Prospect B

Probability

Outcome

Probability

Outcome

1 3 1 3 1 3

$−750

1 3 1 3 1 3

$−1,000

$1,500 $z

$2,000 $3,000

Task III: What is the minimum value z for which you prefer A over B? Please write z = $_ Prospect A

Prospect B

Probability

Outcome

Probability

Outcome

1 3 1 3 1 3

$−7,500

1 3 1 3 1 3

$−10,000

$15,000 $z

$20,000 $30,000

For example, to get FSD∗ for 100% of the subjects we find with small bets that ε1 = 5.9% (see Table 2b), while with larger bets we have ε1 = 6.5%. For SSD∗ corresponding to 100% of the population we find that ε2 = 3.2% in both, small and large bets. 3.3 Experiment 3 In experiment 3 we have 132 subjects composed of heterogenous groups. Table 6 shows the three choices in experiment 3. As can be seen all choices contain negative as well as positive outcomes. Also, the magnitude of the outcomes differ across the three tasks. In Task I we have FSD area violation and in Tasks II and III we have SSD area violation. Table 7 provides the main result, i.e. the maximum z value across all subjects and the corresponding ε1 and ε2 values. From Table 7 we observe a large difference between groups, in particular with regard to ε1 . Yet, as what is important to us is the minimum-ε1 across all 132 subjects, we see that it is ε1 = 3.03%, little smaller then ε1 = 5.9% obtained in experiment 1. Regarding SSD∗ we obtain ε2 = 3.23% rather than ε2 = 3.2% obtained in experiment 2. Thus the results are very similar across experiments. 4 Practical investment rules which are consistent with SD∗ (or MV∗ ) A common rule of thumb employed among practitioners is that the proportion of equity in the portfolio should be 100% less the investor’s age, well known as “100 less your age rule”.

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173

Table 7 The results of Experiment 3 (N = 132) Group

N

Task I max z

Task II ε1

max z

Task III ε2

max z

ε2

Laboratory Employee

19

30,000

5.08%

30,000

0.90%

50,000

9.09%

MBA Graduate Student

17

50,000

3.03%

10,000

3.23%

100,000

3.23%

Accounting Students (Second year)

29

6,000

27.27%

5,500

7.69%

55,000

7.69%

Accounting Students (Third year)

46

10,000

15.79%

5,000

9.09%

50,000

9.09%

Practitioners

21

10,000

15.79%

10,000

3.23%

100,000

3.23%

N —Number of choices

This rule conforms with the observed growth in the number of life cycle mutual funds. The basic idea of these funds is that young people who save for retirement should invest relatively large proportion their investment in equity, while older people who save for retirement should invest relatively small proportion of their portfolio in equity. For example, the Life Cycle Mutual Fund “Fidelity Freedom 2030” features 80% stocks, 15% bonds and 5% cash. While Fidelity Freedom 2010 features 50% stocks, 35% bonds and 15% cash (as of September 2005). Thus, if you retire in year 2030 it is recommended that you invest 80% in stocks, but if you retire in year 2010 only 50% should be invested in stocks (see Woodard 2008). How these practical investment policies related to SD∗ (or MV∗ ) and to the concept of area violation? Indeed, it perfectly fits our theoretical and experimental finding. To see this recall that for one period (say one year) stocks have in general high mean and high standard deviation relative to bonds. The two empirical calculated distributions of stocks and bonds returns intersect with a relatively large FSD and SSD actual area violation. However, if one looks at rates of return corresponding to a larger horizon, say, a distribution of bi-annual rate of return, the distribution of stocks shifts to the right faster than the distribution of bonds and actual area violation decreases. Thus as the investment horizon increases the actual area violation decreases. Therefore, for a given allowed area violation we obtain a dominance SD∗ (or MV∗ ) of stocks over bonds as the investment horizon increases. Indeed, Bali et al. (2008) show empirically that this is the case: The longer the investment horizon the smaller the observed actual area violation. Alternatively, the larger the horizon the larger the chance that a portfolio with a high proportion of stock will dominate by SD∗ a portfolio with a relatively small proportion of stocks. Thus, life cycle mutual funds and “100% less your age rule” fits nicely the SD∗ and MV∗ rules: the higher the investment horizon, the smaller the actual area violation εi , and the better the chance that stocks will dominate bonds by SD∗ . We describe above how the concept of Almost SD is used by practitioners. However, the use of this rule is also employed in other areas, as derivative option bound (see Huang 2007), for ranking stock market indexes (see Gasbarro et al. 2007), and for explaining wealth distribution (see Levy 2003). Benitez et al. (2006), use Almost SD to determine the minimal conservation payment required to guarantee that the environmentally-preferred use of land dominates other less environmentally-preferred alternatives (for more details see Levy (forthcoming). Thus, eliminating economically irrelevant utility functions and use SD∗ or MV∗ rules play an important role in the work of both practitioners and academics.

5 Concluding remarks Allais’ paradox and other similar paradoxes were resolved in the CPT or RDEU paradigms by employing decision weights. However, even after taking into account these decision

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Ann Oper Res (2010) 176: 153–178

weights, the commonly employed investment decision rules in the expected utility paradigm as well as in the CPT and RDEU paradigms suffer from a severe drawback: they may be unable to rank two prospects when in practice 100% of the investors prefer one prospect over the other. Thus, the investment efficient set implied by these rules may be too large. For example, the Mean-Variance (MV) rule and Stochastic Dominance (SD) rules cannot rank x ($1 with probability 0.1 and $106 with probability 0.99) and y ($2 with certainty), while in practice we suspect that all investors would choose x. The same deficiency of MV and SD rules exists with less extreme and more realistic cases. Moreover, this deficiency exists regardless of whether the probabilities are objective or subjective (decision weights). The reason that the existing decision rules are unable to rank such a pair of options is because they correspond to all preferences in a given utility class (e.g., all risk-averse functions), including utility functions with derivatives (u or u ), which change wildly over various regions of the financial outcome and which, in practice do not correspond to any investor. We show that though these preferences are mathematically legitimate they are economically irrelevant. If one bounds the variability of u or of u , or both, one reduces the class of preferences to which the decision rules correspond by eliminating preferences which induce the deficiency of MV and SD rules as the one given above. To overcome the deficiency of MV and SD rules, Leshno and Levy (2002) suggested new decision rules called Almost MV, Almost FSD and Almost SSD denoted by MV∗ , FSD∗ and SSD∗ , respectively, corresponding to the sets of economically relevant preferences. The advantage of these new rules is that they allow ranking prospects like x and y given above, and enable us to decrease the investment efficient set. The paper by Leshno and Levy (2002) is a purely theoretical. Here we experimentally analyze the set of preferences that should be considered as economically relevant. This paper’s contributions are as follows: a) We prove SSD∗ when only constraint on u (and not on u as done by Leshno and Levy (2002)) are imposed. b) We design an experimental study by which we are able to determine the set of economically irrelevant utility functions. c) We introduce the concept of “allowed area violation” (in contrast to actual area violation given in Leshno and Levy (2002)) and analyze the magnitude of FSD, SSD and MV “allowed area violation”. d) We show that practitioners implicitly employ the SD∗ or MV∗ rules in their investment recommendation and in particular in selling the Life Cycle Mutual Funds. What restrictions on preference are legitimate? What is considered as a reasonable class of economically relevant preferences? In this study we answer these questions experimentally by conducting three experiments with various groups of subjects, students as well as practitioners, with and without monetary payoff. We provide 416 subjects with a few tasks of choosing from two prospects, where one prospect has an obvious superiority over the other, yet due to some “area violation”, ε, the common SD and MV rules do not reveal a dominance relationship. By the choice of each subject we measure the allowed area violation εi (i = 1, 2, . . . , n), which reflects the preference of the ith subject. The calculated minimum allowed “area violation” across all subjects, mini εi , is determined experimentally, and this, in turn dictates what preferences are economically irrelevant, though mathematically they are relevant. We find that with FSD∗ the minimum allowed area violation across all subjects is ε1 = 5.9%, while the corresponding figure for SSD∗ is ε2 = 3.2%. FSD is a rule for all u1 ∈ U1 as long as u ≥ 0. We determine experimentally a new set U1 such that U1 ⊂ U1 , when some bound is imposed on the volatility of u . This bound allows us to create FSD∗

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situation (with area violation) though there is no FSD (due to the area violation). The FSD∗ rule allows ranking of prospects like x and y given above. The same is true for risk aversion: (SSD∗ , U2 ) replaces (SSD, U2 ) and (MV∗ , U2 ) replaces (MV, U2 ) in the cases of normal distributions. Thus, the size of the sets U1 and U2 is determined experimentally. The SD∗ and MV∗ rules are implicity employed in practice. The investment rule asserting that investors should put in equity the proportion of wealth which is equal to “100% less your age” and the rapid growth in Life Cycle Mutual Funds perfect fits our results: The longer the investment horizon, the lower the actual area violation, and for a given allowed area violation stock dominates bond. Hence, the longer the investment horizon it is recommended to increase the equity proportion in the portfolio. Acknowledgements The authors acknowledge the helpful comments of two anonymous referees of this journal, Moshe Levy and the participants in the CEMMAP conference on Stochastic Dominance in London. The first author gratefully acknowledges the financial support provided by the Krueger Foundation at the School of Business Administration, the Hebrew University, Jerusalem.

Appendix A: The p% Almost FSD (FSD∗ ) From Task 6 in experiment 1, it is obvious that for a very large z, say, z = $106 all investors would prefer B over A. However, an investment consultant may be interested to serve his clients also with actual prospects with lower and more realistic value z. If there is a prospect with, say, z = $800, she may recommend to her clients to invest in B and not in A, though B dominates A by FSD∗ only for 97.8% of the population (see Table 2b). Thus, there is a trade off between the minimum required value z and the percentage of the population for which the FSD∗ is intact. To illustrate these relationships suppose that indeed our sample represents the whole population of decision makers. We can define a decision rule corresponding to 95% of the population but not all of them. This rule will correspond to ε1,95%  and the corresponding bounded utility class is denoted by U1,95% . Similarly, we may decide that covering p = 90% of the population is sufficient. In this case we will have two values  and ε1,90% corresponding to p = 90% of the decision makers. Obviously, the higher U1,90%  . Let us demonstrate this idea p, the lower the corresponding ε1 and the larger the class U1,p with the results of Table 2b. For simplicity, and without loss of generality we illustrate this idea with a piece-wise linear utility function, denoted by u, given in Fig. 4. We denote the set of all utilities showing a preference of B over A by U1 . First note that the utility u1 (see Fig. 4) does not belong to U1 as regardless of the selected value z, B can not be preferred to A (see Table 1) with such a utility function.15 Thus, there is no 0 < ε  < 0.5 that reveals a preference for B with u1 and therefore u1 ∈ U1 . Suppose now that we have a utility function like u2 in Fig. 4 where the slope of u2 corresponding to segment c is a, 0 < a < 1. In this case Eu(B) > Eu(A) if the following holds for z > $200: 1 1 1 1 (50) + [200 + a(z − 200)] > 100 + 200 2 2 2 2 Namely, a(z − 200) > 50 15 Namely, with u we have 1 u(100) + 1 u(200) > 1 u(50) + 1 u(z) for any selected value z. 1 2 2 2 2

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Fig. 4 The cumulative distribution functions of A and B in Task II

or 50 z − 200 Thus, the slope of segment c (see Fig. 4) which guarantees dominance of B over A is a function of the selected z as calculated from the above inequality (and of the value ε1 which is determined also by z). Also in order to have a < 1 we must have that z ≥ 250. Table 8 presents the selected value z, the corresponding slope a and the percentage of the subjects selecting value z or less. We see from Table 8 that if we want all investors (p = 100%) to prefer B over A we need to have a ≥ 0.06. Suppose that the slope of line c is 0.06 (see Fig. 4). Then all utility   ⊂ U1 as U1,p=100% functions with a slope below 0.06 are not allowed. Hence U1,p=100% excludes all utility functions enclosed between u1 and u2 (see Fig. 4), and U1 contains all these functions because u ≥ 0 as required. With this respect we say that U1 is a bounded set, which in this specific example the restriction is that all utility functions with a slope of less than 0.06 are not included in U1 . Now suppose we settle for a decision rule corresponding to 97.8% of the population.  , i.e., suitable for Namely, we would like to have a decision rule (FSD∗ ) suitable for U1,97.8% 97.8% of the population. Then, if in option B we have z = $800 we would assert that B dominates A by FSD∗ , which is suitable for 97.8% of the population. By the same token,  and p = 86.7% if p = 86.7%, we would say that B is preferred over A for all u ∈ U1,86.7% 16 corresponds to z = $400. By employing the two options A and B with z = $1000 or more it is obvious that all (i.e. 100%) investors participating in our experience would prefer B over A. The subjects’ behavior in our experiment lends support to the hypothesis that, in our specific example of piecewise preferences that utility functions with a slope of segment c lower than 0.06, although mathematically allowed, are irrelevant economically because they do not conform to any of the investors’ preferences. Therefore, in the above specific example we allow risk aversion but we do not allow an extreme risk aversion as illustrated by utility function, u1 or any utility function with a slope given by the shaded area in Fig. 4. Thus, with FSD we allow all functions u ∈ U1 and with FSD∗ corresponding to our experiment, a>

16 Note that 11.1% + 2.2% = 13.3% of the subjects selected z > 400, hence for z = $400, B dominates A by 100% − 13.3% = 86.7% of the population (see Table 2b).

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Table 8 The Value z, a, α where p% of the subjects prefer B over A in Task 1a Selected z

The minimum slope of the

The minimum value α

Percentage of subjects

utility, a (piece-wise

α with (u(x) = xα )

selecting z or (less)

linear utility)

(see Table 2b) (in %)

250

1

–

22.8

275

0.67

0.76

29.5

300

0.50

0.60

64.5

350

0.33

0.43

71.7

400

0.25

0.33

86.7

800

0.08

0.10

97.8

1000

0.06

0.07

100.0

a Let us explain how the values α given in Table 8 are calculated. Given ε  and z we have that in range [50, z] 1  sup u (x) = 50α−1 and inf u (x) = zα−1 . Thus, we have that (α − 1) log(50) = (α − 1) log(z) + log ε1 − 1 1



or α = 1 +



log 1 −1 ε1

z ) log( 50

. Thus, for a given z, ε1 is determined, hence α can be calculated

we disallow functions with a slope smaller than u2 (see Fig. 4). If we would like to have a decision rule for say p% of the population we disallow more functions, hence we obtain that all functions with a slope below line p are disallowed. The smaller the percentage of the population, p, for which there is a dominance, the higher the slope of line p (see Fig. 4), and more utility functions are disallowed. Therefore, we have the following relationship:   ⊂ U1,p ⊂ U1 U1,p 2 1

for 100% > p1 % > p2 %.

Thus, the smaller the allowed ε1 by switching from FSD to FSD∗ , the closer U1 to U1 . We demonstrated the relationship between p, U1 (p) with the above piecewise linear utility function. Obviously, the same relationship holds for any smooth utility function, e.g., u(x) = xα . In this case the smaller α the more risk-averse are the investors, hence it is possible, that α for p = 95%, say values of α ≤ 0.01 are not allowed and for p = 90%, values of α ≤ 0.1 are not allowed. The same arguments given with the piecewise linear utility holds also in the α case u(x) = xα or for that matter any other utility functions. Table 8 presents the minimum α value α for which B is preferred over A for the utility function u(x) = xα .17 References Allais, M. (1953). Le comportement de l’homme rationnel devant le risque: Critique de posttulats et axiomes de l’ecole americaine. Econometrica, 21, 503–546. Bali, T. G., Demirtas, K. O., Levy, H., & Wolf, A. (2008). Asset allocation to small and big stocks. Working Paper, CUNY, Baruch College. Barrett, G. F., & Donald, S. G. (2003). Consistent tests for stochastic dominance. Econometrica, 71, 71–104. Baumol, W. J. (1963). An expected gain in confidence limit criterion for portfolio selection. Management Science, 10, 174–182. 17 As FSD corresponds to concave as well as convex functions, it can be easy shown that also some convex functions are eliminated from U1 to avoid paradoxes. Thus U1 ⊂ U1 where some concave as well as convex

functions which may create paradoxes are eliminated.

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