Comparing risks with reference points: A stochastic dominance approach

Insurance: Mathematics and Economics 70 (2016) 105–116

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Comparing risks with reference points: A stochastic dominance approach Dongmei Guo a , Yi Hu b , Shouyang Wang c , Lin Zhao c,∗ a

School of Economics, Central University of Finance and Economics, Beijing, 100081, China

b

School of Economics and Management, University of Chinese Academy of Sciences, Beijing 100190, China

c

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

article

info

Article history: Received October 2014 Received in revised form April 2016 Accepted 7 May 2016 Available online 16 June 2016 Keywords: Stochastic dominance Reference point Loss aversion Downside risk Allais-type anomalies Endowment effect for risk

abstract This paper develops a stochastic dominance rule for the reference-dependent utility theory proposed by Kőszegi and Rabin (2007). The new ordering captures the effects of loss aversion and can be used as a semiparametric approach in the comparison of risks with reference points. It is analytically amenable and possesses a variety of intuitively appealing properties, including the abilities to identify both ‘‘increase in risk’’ and ‘‘increase in downside risk’’, to resolve the Allais-type anomalies, to capture the violation of translational invariance and scaling invariance, and to accommodate the endowment effect for risk. The generalization to third-order dominance reveals that loss aversion can either reinforce or weaken prudence, depending on the location of the reference point. Potential applications of the new ordering in financial contexts are briefly discussed. © 2016 Published by Elsevier B.V.

1. Introduction Reference points such as benchmarks or targets manifest themselves pervasively in portfolio management. Regardless of being individual or institutional, investors commonly have a benchmark to follow or a target to beat. Reference points usually exert substantial influence on investors’ risk appetites, which further motivate their strategies (Dittmann et al., 2010). In the financial literature, the evaluation of risks with reference points is often performed using risk measures such as fixedtarget lower-partial moments, downside betas, Value-at-Risk and ExpectedShortfall.1 These measures are designed primarily on the basis of computational considerations and lack a choice-theoretic foundation. As response to this shortcoming, the literature has

∗

Corresponding author. E-mail addresses: [email protected] (D. Guo), [email protected] (Y. Hu), [email protected] (S. Wang), [email protected] (L. Zhao). 1 Lower-partial moment models were first introduced by Markowitz (1959). Fishburn (1977) and Price et al. (1982) further developed more general forms of downside risk measures based on fixed-target lower-partial moments. Downside betas were first advocated by Bawa and Lindenberg (1977) and have been adopted by many authors such as Ang et al. (2006). Value-at-Risk and Expected Shortfall are rooted in the safety-first criterion proposed by Roy (1952) and use a reference percentile. http://dx.doi.org/10.1016/j.insmatheco.2016.05.003 0167-6687/© 2016 Published by Elsevier B.V.

suggested a non-parametric approach for evaluating risks based on stochastic dominance rules. These rules are rooted in utility theory and rank prospects with minimal assumptions about investors’ risk attitudes (Levy, 1992). However, to the best of our knowledge, with few exceptions (Baucells and Heukamp, 2006), researchers have not yet developed stochastic dominance rules that are suited for comparing risks with explicit reference points. The behavioral stochastic dominance rules such as ‘‘prospect stochastic dominance’’ and ‘‘Markowitz stochastic dominance’’ studied previously mainly focus on exploring the concavity and convexity of the value function, and thereby cannot provide guidance on how to properly mitigate downside risk. This paper represents a first effort to tailor a stochastic dominance rule suited for comparing risks with reference points. In modeling agents’ preferences with reference points, the referencedependent utility theory developed by Kőszegi and Rabin (2006, 2007) is a popular candidate. To invoke this theory, one needs a von Neumann–Morgenstern utility function to describe the intrinsic taste for outcomes, a parameter to capture the magnitude of loss aversion, and a parameter to measure the weight of gain–loss utility. In this paper, we develop a stochastic dominance rule for Kőszegi and Rabin’s theory to reduce the parametrization. To invoke our stochastic dominance rule, one does not need to know exactly the form of the von Neumann–Morgenstern utility function or the values of the two parameters. What is needed is just a

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D. Guo et al. / Insurance: Mathematics and Economics 70 (2016) 105–116

Fig. 1. The inclusion relationship between SSDr˜ , SSD and TSD. The relationship ˜ 1 SSDw ˜2 SSD ⊂ SSDr˜ (SSD ⊂ TSD) denotes that for any two random variables, w ˜ 2 (w ˜ 1 TSDw ˜ 2 ), but the converse is not true. The relationship (SSDr˜ ∩ implies w ˜ 1 SSDr˜ w TSD)\ SSD ̸= ∅ states that there are pairs of random variables that cannot be ranked by SSD but can be ranked by TSD and SSDr˜ in the same order.

judgment about whether the von Neumann–Morgenstern utility is increasing or concave and whether the two parameters are bigger than some pre-specified lower bounds. As we have to specify the reference point and the lower bounds of the parameters in the piecewise linear value function when employing our dominance rule, we regard it as a semi-parametric approach for comparing risks. In developing the stochastic dominance rule, we follow Kőszegi and Rabin (2006, 2007) to abstract from the S-shaped nature of the value function by assuming a piecewise linear value function. Piecewise linearity captures risk aversion over gains but precludes risk-lovingness over losses, which is consistent with portfolio mangers’ disaster avoidance motive, making our dominance rule suitable for use in practice.2 We use three specifications of the reference point: (i) exogenous and deterministic; (ii) exogenous and stochastic; and (iii) endogenous (stochastic or not). Among the them, (i) provides a basis where the intuitions underlying our stochastic dominance rule will be elaborated. Most of our discussion is confined to the second-order dominance, while a generalization to higher orders will also be explored. Our new dominance rule proves to be a convenient tool for analytically characterizing the effects of reference point and loss aversion on the risk-taking behavior. When the reference point is exogenous (whether deterministic or stochastic), our new ordering exhibits four features that distinguish it from the traditional dominance rules. First, compared with the traditional second-order stochastic dominance (‘‘SSD’’, henceforth), the new ordering admitting a reference point r˜ , denoted by SSDr˜ , shows higher aversion towards the risk spread: in the SSDr˜ ordering, not only all mean-preserving spreads are disliked, but also certain kinds of mean-increasing spreads satisfying the condition specified in Proposition 4 will be disliked. For a spread to be preferred according to SSDr˜ , the average increase in return must be high enough to compensate for the downside risk. Second, the inclusion relationship between SSDr˜ , SSD and thirdorder stochastic dominance (TSD) is found to satisfy Fig. 1. On the one hand, SSDr˜ contains SSD as a sub-ordering and can rank certain risk pairs in the same order as TSD does when risk pairs satisfy the condition specified in Proposition 5. On the other hand, SSDr˜ is definitely a new ordering that is neither sufficient nor necessary for TSD. Therefore, SSDr offers an effective complement to SSD and TSD in identifying the ‘‘increase in risk’’ and the ‘‘increase in downside risk’’ (‘‘more skewed to the left’’). Third, since investors’ sense of loss is influenced by the location and scale of the risk, SSDr˜ is not invariant under either translations or nonnegative scaling of underlying risks. In particular, as we

2 Convexity for losses found in experiments is not very pronounced. See Abdellaoui et al. (2005).

will see in Proposition 6, the original undominated component can become dominated after an upward scaling of a pair of risks, capturing that investors exhibit larger risk aversion in the face of risks with larger scale. Fourth, SSDr˜ offers an analytical characterization of the endowment effect for risk. We show in Proposition 8 that the investor can become less averse to the spread of risk when the reference point becomes more dispersed. In other words, the investor is less risk averse if she expects to face greater risks. This effect was firstly formalized by Kőszegi and Rabin (2007) under the assumption of linear intrinsic utility function (see their Proposition 1 on pg. 1053) and has found an experimental support in Sprenger (2015). Our new ordering generalizes the analysis to arbitrary concave utility functions. Our stochastic dominance rule can easily accommodate endogenous reference points. When reference points are endogenously formed by expectations as in Kőszegi and Rabin (2006, 2007), we show in Proposition 9 that our ordering yields a new resolution of the Allais-type anomalies that includes both the common consequence effect and the common ratio effect. The essence of the resolution is that anticipating more risky lotteries drives investors to become less risk-averse by making their reference points more dispersed. Our stochastic dominance rule can also be generalized to explore the effects of reference points on higher-order preferences. Although the effects of reference points on the second-order preference (risk aversion) are well studied, little is known on how reference points change higher-order preferences. Maier and Rüger (2012) find that if the reference point is endogenously formed by expectations, individuals exhibit even- but never uneven-order risk attitudes. Complementary to this result, our third-order stochastic dominance rule provided in Proposition 10 shows that when the reference point is exogenous, loss aversion can either reinforce or weaken prudence, depending on the location of the reference point. We contribute to the literature on risk theory by offering a new method rooted in utility theory for comparing risks with reference points. It establishes robust predictions based on limited information about the utility function, and can be used in experiments as a guide to design pairs of prospects to examine theories with reference-dependence structure (Baucells and Heukamp, 2006). It also adds to the literature on behavioral portfolio (Shefrin and Statman, 2000; Berkelaar et al., 2004; Jarrow and Zhao, 2006; De Giorgi and Post, 2011; He and Zhou, 2011) by providing a new approach in identifying downside risk. Mathematically, SSDr˜ is a simple extension of that of SSD, as it just introduces a stepwise weighting function to the integrand. This makes the existing statistical approaches for testing SSD such as Davidson and Duclos (2000) readily extendable for testing SSDr˜ . The remainder of this paper is structured as follows. Sections 2 and 3 are devoted to the baseline case where the reference point is non-stochastic and exogenous. We concentrate on second-order preference in these two sections. Sections 4–6 are three important extensions, generalizing the analysis to accommodate stochastic reference points, endogenous reference points, and higher-order preferences, respectively. Section 7 concludes this paper with a discussion of potential applications. All proofs are relegated to Appendix. 2. Definition and characterizations of SSDr We begin with the baseline case where the reference point is constant and exogenous, denoted by r. We use w ˜ to denote a random variable and w its realization. When the random variable w ˜ is discrete, we use

w ˜ = (w1 , p1 ; · · · ; wn , pn )

D. Guo et al. / Insurance: Mathematics and Economics 70 (2016) 105–116

to signify that w ˜ = wi with probability pi , i = 1, . . . , n, i=1 pi = 1. For random variables, capital letters, such as F and G, denote their cumulative distribution functions (cdfs). We build our stochastic dominance rule on the referencedependent utility theory proposed by Kőszegi and Rabin (2006, 2007), which adopts a von Neumann–Morgenstern utility function to describe the intrinsic taste for the outcome and a Kahneman–Tversky value function to describe the sensation of gain or loss due to a deviation from the reference point. Formally, if w ˜ denotes the risky wealth with cdf F and r denotes the reference point, then the investor’s reference-dependent utility function is

n

E [v(w; ˜ r , u)] = E [u(w) ˜ + R η,λ (u(w) ˜ − u(r ))]

 =

u(w) + R η,λ (u(w) − u(r ))dF (w),

ηx, for x ≥ 0, R η,λ (x) = ηλx, for x < 0. 

(2)

Here η > 0 is the relative weight of gain–loss utility, and λ > 1 measures the magnitude of loss aversion. The limiting case of η → +∞ corresponds to the situation where the reference-dependent utility is only composed of a Kahneman–Tversky value function.3 For analytical convenience, we assume that all the random variables to be dealt with have finite supports. This assumption avoids imposing conditions on boundaries when employing the integration by parts. Alternatively, one can abandon it by imposing upper bounds on the utility functions (see Hadar and Russell, 1971). Before proceeding, let us give a short review of the standard stochastic dominance rules. Let U 1 = {u : (−∞, ∞) → R | u′ ≥ 0}

and

U 2 = {u : (−∞, ∞) → R | u′ ≥ 0, u′′ < 0}.

A random variable, w ˜ 1 , is said to first-order (resp. second-order) stochastically dominate another random variable, w ˜ 2 , written w ˜ 1 FSDw ˜ 2 (resp. w ˜ 1 SSDw ˜ 2 ), if E [u(w ˜ 1 )] ≥ E [u(w ˜ 2 )] for all u ∈ U 1 (resp. u ∈ U 2 ). Hadar and Russell (1969) obtain a precise characterization of FSD and SSD:

w ˜ 1 FSDw ˜ 2 if and only if F2 (x) − F1 (x) ≥ 0 for all x,  w ˜ 1 SSDw ˜ 2 if and only if (F2 (t ) − F1 (t ))dt ≥ 0 for all x, (−∞,x]

˜ 1 and w ˜ 2 . The meaning of SSD can where F1 and F2 are the cdfs of w be illustrated graphically by use of the so-called simple spreads. Definition 1. w ˜ 2 is called a simple spread of w ˜ 1 (or equivalently, w ˜1 is called a simple concentration of w ˜ 2 ) if there exists a single crossing point a ∈ R such that F2 (x) ≥ F1 (x) on (−∞, a] and

Definition 2. Consider the reference-dependent utility specified in (1). Assume that η∗ ≥ 0, λ∗ ≥ 1 are pre-specified lower bounds for the parameters η and λ, and the reference point r is given. Let w ˜ 1, w ˜ 2 be two random variables. (i) w ˜ 1 first-order stochastically dominates w ˜ 2 relative to r, written w ˜ 1 FSDr w ˜ 2 , if E [v(w ˜ 1 ; r , u)] ≥ E [v(w ˜ 2 ; r , u)], for all u ∈ U 1 , η ≥ η∗ , λ ≥ λ∗ . (ii) w ˜ 1 second-order stochastically dominates w ˜ 2 relative to r, written w ˜ 1 SSDr w ˜ 2 , if E [v(w ˜ 1 ; r , u)] ≥ E [v(w ˜ 2 ; r , u)],

(1)

where u is a concave utility function, and R η,λ is a universal value function

F2 (x) ≤ F1 (x) on (a, ∞).

In this definition, the term ‘‘spread’’ indicates that w ˜ 2 has more probability mass on the tails of the distribution. Simple spread not only has the virtue of being visually suggestive, but also serves as the most basic form of SSD change. Indeed, when w ˜ 1 and w ˜ 2 have the same mean, Rothschild and Stiglitz (1970) prove that w ˜ 1 SSDw ˜2 if and only if w ˜ 2 can be obtained by a sequence of mean-preserving simple spreads of w ˜ 1. Now we define the stochastic dominance rules for (1).

3 When taking the limit, we scale (1) by 1 such that the weights of the intrinsic 1+η η 1 utility and the gain–loss utility are 1+η and 1+η respectively.

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for all u ∈ U 2 , η ≥ η∗ , λ ≥ λ∗ . In the definition, we conform to tradition by making minimal assumptions about the intrinsic utility u and the parameters η and λ, but impose a rather specific condition that r should be given ex ante. Although the specification of the reference point r limits the generality of stochastic dominance in the sense that the resulting ordering no longer represents the agreement for all investors with different reference points, it is desirable from the practical viewpoint of seeking a convincing risk ordering for a small group of investors sharing a common specific target. Moreover, it will be seen from Proposition 2 that different reference points always lead to disagreement on risk choices. Proposition 1. Let the cdfs of w ˜ 1, w ˜ 2 be F1 , F2 , respectively. Given r, η∗ , λ∗ , and define D

η∗ ,λ∗

 ∗ ∗ 1 + η λ , 1 + η∗ (x; r ) =  1,

if x ≤ r , if x > r .

(i) w ˜ 1 FSDr w ˜ 2 if and only if w ˜ 1FSDw ˜ 2. ∗ ∗ (ii) w ˜ 1 SSDr w ˜ 2 if and only if (−∞,x] (F2 (t ) − F1 (t ))D η ,λ (t ; r )dt ≥ 0 for all x. Proposition 1 offers a precise characterization of FSDr and SSDr . It clarifies that the gain–loss concern with respect to a reference point does not change investor’s preference for the increase in wealth according to FSD, but changes investor’s preference for the spread of risk according to SSD. Proposition 2. Given two different reference points r1 and r2 , there always exist two risks w ˜ 1 and w ˜ 2 such that w ˜ 1 SSDr1 w ˜ 2 , but w ˜ 1 does r2 not SSD w ˜ 2. Proposition 2 confirms that for any r1 ̸= r2 , SSDr1 is neither sufficient nor necessary for SSDr2 . In other words, for any two different reference points, one can always find two risks such that investors with different reference points cannot agree with each other on the risk choice. 3. Properties of SSDr 3.1. The relationship between SSDr and SSD Different from SSD, SSDr exhibits preference for mean-reducing contractions as well as aversion to mean-increasing spreads. The relationship between SSD and SSDr is characterized below. Proposition 3. Let the reference point r be fixed. For any two random variables w ˜ 1 and w ˜ 2, w ˜ 1 SSDw ˜ 2 implies w ˜ 1 SSDr w ˜ 2 , but the converse

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η∗ (λ∗ −1) 1+η∗

is not true. Further, when w ˜ 2 is a mean-preserving spread of w ˜ 1 , we have

(ii) w ˜ 1 SSDr w ˜ 2 if and only if E [w ˜ 1 ] − E [w ˜ 2] + (F2 (t ) − F1 (t ))dt ≥ 0.

(i) Preference for mean-reducing contraction: there exists a random variable w ˜ 3 , which is a mean-reducing contraction of w ˜ 2 such that, w ˜ 3 SSDr w ˜ 2 , but w ˜ 3 does not SSDw ˜ 2; (ii) Aversion to mean-increasing spread: there exists a random variable w ˜ 4 , which is a mean-increasing spread of w ˜ 1 such that, w ˜ 1 SSDr w ˜ 4 , but w ˜ 1 does not SSD w ˜ 4.

Proposition 4 makes a clear distinction between SSD and SSDr . First, the inequality shown in assertion (ii) collapses to E [w ˜ 1] − E [w ˜ 2 ] ≥ 0, which is an equivalent condition for SSD, in the case that η∗ = 0 or λ∗ = 1. This fact confirms that SSDr will reduce to SSD when the loss aversion relative to the reference point is removed. Second, in the case that η∗ > 0 and λ∗ > 1, the condition E [w ˜ 1] − E [w ˜ 2 ] ≥ 0 becomes only sufficient but not necessary for SSDr . As shown in Example 1, another sufficient condition for SSDr is that E [w ˜ 1 ] < E [w ˜ 2 ] and in the meanwhile the inequality in assertion (ii) holds true. In this situation, E [w ˜ 2 ] − E [w ˜ 1] > 0 measures the incremental expected return from choosing w ˜2 η∗ (λ∗ −1)  instead of w ˜ 1 , while 1+η∗ (−∞,r ] (F2 (t )− F1 (t ))dt > 0 measures the incremental losses relative to the reference point brought about by w ˜ 2 . Investors will favor the mean-reducing contraction w ˜ 1 and avoid the mean-increasing spread w ˜ 2 , as long as the incremental return from w ˜ 2 is not high enough to compensate for the incremental losses caused by it.

The above relationship can be summarized as follows:

k5 w˜ 3 kkk k k FSD kk kkk kkk k k kk SSD w ˜ 1 SSS SSS SSS SSS SSS SSDr , not SSD SSS ) w ˜4

SSS SSS SSS SSDr , not SSD SSS SSS SSS ) k/5 w˜ 2 k k k kkk kkk k k FSD kkk kkk

(3)

Proposition 3 highlights that SSD is merely a sufficient but not necessary condition for SSDr . To get an intuitive recognition of this proposition, let us look at a concrete example. Example 1. Suppose that r = 0,

w ˜ 1 ≡ 0,

w ˜ 2 = (ε, 50%; − ε, 50%) ,

ε > 0.

w ˜ 2 is a mean-preserving spread of w ˜ 1 . For assertion (i), let w ˜3 ≡ −

η∗ (λ∗ − 1) ε. 2(1 + η∗ λ∗ )

It is easy to show that w ˜ 3 SSDr w ˜ 2 , but w ˜ 3 does not SSD w ˜ 2 . The intuition is as follows. Because w ˜ 3 gives a certain outcome, its dispersion is smaller than that of w ˜ 2 . However, w ˜ 3 has a mean smaller than that of w ˜ 2 . Thus, for investors whose risk aversion is small enough (i.e., investors who are proximately risk-neutral), w ˜ 3 is less appealing than w ˜ 2 , since w ˜ 2 gives a relatively high risk compensation for switching from w ˜ 3 to w ˜ 2 . This is why w ˜ 3 does not SSD w ˜ 2 . In contrast, when taking into account the loss aversion inherent in SSDr , the compensation of w ˜ 2 becomes insufficient, and w ˜ 3 becomes dominant as it gives a smaller downside risk than w ˜ 2. This implies that for SSDr , not only a mean-preserving contraction of risk is preferred, but also some mean-reducing contraction will be preferred. Assertion (ii) is in the same spirit. Let

w ˜4 =



1 + η∗ λ∗ 1 + η∗



ε, 50%; − ε, 50% .

Because w ˜ 4 gives a risky outcome, its dispersion is bigger than that of w ˜ 1 . However, since w ˜ 4 has a mean bigger than that of w ˜ 1, it is more attractive for investors whose risk aversion is small, and hence w ˜ 1 does not SSD w ˜ 4 . In contrast, according to SSDr , the compensation of w ˜ 4 becomes insufficient due to loss aversion and ˜ 4 . In words, for SSDr , not only a mean-preserving thus w ˜ 1 SSDr w spread of risk is disliked, but also some mean-increasing spread will be disliked. The above example provides the intuition for the effect of loss aversion relative to a reference point on the risk ordering. This effect can be formulated more precisely in terms of simple spreads. Proposition 4. Assume that w ˜ 2 is a simple spread of w ˜ 1 . Then, we have: (i) w ˜ 1 SSDw ˜ 2 if and only if E [w ˜ 1 ] − E [w ˜ 2 ] ≥ 0;



(−∞,r ]

3.2. The relationship between SSDr and TSD Because SSDr captures investors’ aversion to downside risk, it is highly desirable to see the similarities and differences between SSDr and third-order stochastic dominance (TSD), which is another ordering that shows downside risk aversion. We will prove that although SSDr is neither sufficient nor necessary for TSD, for some kinds of downside risk spreads that can be identified by TSD, SSDr is also able to identify and give the same ordering as TSD. To illustrate that TSD captures downside risk aversion that is missing from SSD, let us consider an example provided by Menezes et al. (1980). Example 2. Suppose

w ˜ 1 = (1, 75%; 3, 25%) ,

w ˜ 2 = (0, 25%; 2, 75%) .

(4)

w ˜ 1 and w ˜ 2 have equal mean and variance, and neither of them dominates the other in the SSD sense. As illustrated in Panel A of Fig. 2, w ˜ 2 can be obtained from w ˜ 1 through a transfer of the dispersion from the right to the left.4 This transfer increases the downside risk in the sense that it decreases the likelihood of extreme positives but increases the chance of large negatives. Menezes et al. (1980) show that w ˜ 1 TSDw ˜ 2 . For w ˜ 1 and w ˜ 2 , SSDr can 1+η∗ λ∗ yield the same ranking as TSD. In fact, taking 1+η∗ = 2 + s (0 ≤ 1 s ≤ 1) and r ∈ 1+ , 2+3s , one can easily verify that w ˜ 1 SSDr w ˜ 2 .5 s 2+2s To generalize the regularity behind this example, we introduce the notion of spread-contraction transformation.





Definition 3. We say that w ˜ 2 is a spread-contraction of w ˜ 1 , if there exist two crossing points a < b such that F2 (x) ≥ F1 (x) on (−∞, a],

F2 (x) ≤ F1 (x) on (a, b],

F2 (x) ≥ F1 (x) on (b, ∞).

4 w ˜ 2 can be obtained from w ˜ 1 by first performing a mean-preserving contraction to transfer the event ‘‘gaining 3 with probability 14 and gaining 1 with probability 14 ’’ to ‘‘gaining 2 with probability 21 ’’ and then performing a mean-preserving spread to transfer the event ‘‘gaining 1 with probability

1 ’’ 2

to ‘‘gaining 0 with probability

1 4

and gaining 2 with probability 41 ’’. 5 A very extensive literature confirms that the loss aversion parameter elicited either through experiments or empirical studies lies between 1 and 3 (e.g., Tversky and Kahneman, 1992; Berkelaar et al., 2004).

D. Guo et al. / Insurance: Mathematics and Economics 70 (2016) 105–116

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Fig. 2. The cdfs of w ˜ 1 and w ˜ 2 , with w ˜ 1 TSDw ˜ 2 . The left panel depicts the cumulative distributions of w ˜ 1 and w ˜ 2 , which are denoted by Fw˜ 1 and Fw˜ 2 , respectively. The ‘‘+’’ area is that where Fw˜ 2 > Fw˜ 1 , and the ‘‘−’’ area is that where the opposite holds. w ˜ 1 TSDw ˜ 2 if and only if (−∞,x] (−∞,t ] (Fw˜ 2 (s) − Fw˜ 1 (s))dsdt ≥ 0 for all x. The right panel is an illustration of a spread-contraction, which resembles what we observe in the left panel.

The spread-contraction transformation offers a graphic illustration of the transfer of the dispersion from the high-wealth range to the low-wealth range, and serves as the most basic form of TSD change. Indeed, when w ˜ 1 and w ˜ 2 have the same mean, Menezes et al. (1980) prove that w ˜ 1 TSDw ˜ 2 if and only if w ˜ 2 can be obtained by a sequence of spread-contraction transformations of w ˜ 1. Proposition 5. Assume that w ˜ 2 is a spread-contraction of w ˜ 1 with E [w ˜ 1 ] = E [w ˜ 2 ]. Denote the crossing points of their cdfs F1 and F2 by a, b (a < b). Then, we have: (i) w ˜ 1 TSDw ˜ 2 if and only if (−∞,b] (F2 (t ) − F1 (t ))(b − t )dt ≥ 0;  ∗ ∗ r (ii) w ˜ 1 SSD w ˜ 2 if and only if (−∞,b] (F2 (t ) − F1 (t ))D η ,λ (t ; r )dt ≥ 0.



Proposition 5 compares the equivalent conditions for TSD and SSDr . It is obvious that these two conditions are different and consequently SSDr is neither sufficient or necessary for TSD in general. However, there exist some common features between the two conditions. First, they both rely on an integral over the range ∗ ∗ (−∞, b]. Second, both (b − t ) and D η ,λ (t ; r ) in the integrands are positive and monotone decreasing in t ∈ (−∞, b]. Due to such similarity, there exist pairs of risks satisfying both TSD and SSDr , as we have seen in Example 2. Two remarks on SSDr are in order. First, in contrast to SSD and TSD that represent the agreement for all individuals with any possible reference point, SSDr is sensitive to the value of r. Continuing with the example in (4), one can verify that for r ≤ 0, w ˜ 2 is no longer a SSDr deterioration of w ˜ 1 , whatever the lower bound of the loss aversion parameter takes. This result is reasonable, as when the reference point is below zero, both w ˜1 and w ˜ 2 are ‘‘gains’’, and their downside risks are zero. Similarly, for r ≥ 3, w ˜ 2 is also no longer a SSDr deterioration of w ˜ 1 . This is also intuitive, as for reference points with large value, both w ˜1 and w ˜ 2 are ‘‘losses’’, and their downside risks are noncomparable because they have the same dispersion measured by variance. As we will see below, the dependence of SSDr on r is at the core of many novel implications. Second, although SSDr˜ contains SSD as a sub-ordering and can rank certain risk pairs in the same order as TSD does when risk pairs satisfy the condition specified in Proposition 5, it is definitely a new ordering that effectively complements SSD and TSD. For example, both TSD and SSD cannot capture the downside risks in meanincreasing spreads, but SSDr can do this according to Proposition 4. To see it, suppose we compare w ˜ 1 = 0 and w ˜ 2 = (2, 50%; − 1, 50%). Neither SSD nor TSD produces an unambiguous ranking for 1+η∗ λ∗ this pair of risks. In contrast, taking r = 0 and 1+η∗ = 2, we ob-

tain w ˜ 1 SSDr w ˜ 2 . This ordering is consistent with the conventional wisdom that the downside risk of w ˜ 1 is smaller than that of w ˜ 2 because the former has a smaller lower-partial moment (Price et al., 1982).

3.3. The ‘‘invariance’’ property of SSDr As pointed out by Hadar and Russell (1971), the SSD ordering is preserved under a nonnegative scale change of the risks or a translation transform of underlying risks. This assertion does not generally hold when the effect of downside loss aversion is taken into account. Example 3. Let [a, b] ⊂ (0, ∞) be the support of w ˜ i (i = 1, 2) and the reference point r > 0. Assume that w ˜ 1 SSDr w ˜ 2 , but w ˜ 1 does not ˜ i (i = 1, 2) becomes SSD w ˜ 2 . If β > ar or β < br , the support of β w isolated from r, i.e., β w ˜ i (i = 1, 2) becomes pure gains or pure losses. It follows immediately that β w ˜ 1 does not SSDr β w ˜ 2 because r in this case, SSD reduces to SSD. Similarly, if the constant k satisfies k > r − a or k < r − b, the support of w ˜ i + k (i = 1, 2) also becomes isolated from r, and thus w ˜ 1 + k does not SSDr (w ˜ 2 + k). When w ˜ 2 is a simple spread of w ˜ 1 with crossing point 0, the sense of loss relative to the reference is influenced by the scale of lotteries. This leads us to the following result. Proposition 6. Given a reference point r ̸= 0, let w ˜ 2 be a simple r spread of w ˜ ˜ ˜ 2 . If E [w ˜1 + 1 with crossing point 0 such that w 1 SSD w ∗ ∗ ∗ ∗ R η ,λ (w ˜ 1 − r )] = E [w ˜ 2 + R η ,λ (w ˜ 2 − r )], then β w ˜ 1 SSDr β w ˜ 2 if and only if β ≥ 1. A concrete example of Proposition 6 is as follows. Suppose that investors with reference r = 1 are required to make a choice between the two risks:

w ˜ 1 ≡ 0,   1 + η ∗ λ∗ ε, 50% , w ˜ 2 = −1 − ε, 50%; 1 + 1 + η∗

ε > 0.

According to the SSDr rule, it can be easily checked that

βw ˜ 1 SSDr β w ˜ 2 if and only if β ≥ 1. That is, if investors take w ˜ 2 as a SSDr deterioration of w ˜ 1 , they ˜ 1 when β > 1. would also take β w ˜ 2 as a SSDr deterioration of β w When β < 1, i.e., after a downsizing scale, investors may no longer take β w ˜ 2 as a SSDr deterioration of β w ˜ 1 . This result stands in contrast with the scaling invariance of SSD, and is reminiscent of the observation that perceived risk increases if all outcomes of a lottery are multiplied by a positive constant greater than one (Coombs and Meyer, 1969; Jia et al., 1999). 4. Extension to stochastic reference points In the basic model (1), we have assumed that the reference point is single and deterministic. This assumption is appropriate when the model is used to describe the preference of portfolio managers who track a unique constant benchmark. In real-life investment activities, casual observations and psychological research suggest

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that investors usually compare their performance with several reference points, e.g., indices of different markets, most popular stocks in the news, portfolios of their friends, performances in different regimes, and so on (Koop and Johnson, 2010; Dodonova, 2012). One way to model such variety of comparison is by means of stochastic reference points. This implies that the reference point itself is a lottery. In this section, we extend the ordering SSDr to accommodate such stochastic reference points. Following Kőszegi and Rabin (2006, 2007); Dodonova (2012), we assume that the investor compares her wealth with several reference points separately and then aggregates her feelings in the utility. It captures a well documented psychological regularity that people can feel positively relative to one reference point and at the same time feel negatively relative to another reference point (Ordóñez et al., 2000). In symbols, assume that the investor has m reference points, rj , j = 1, . . . , m, and each reference point has a weight qj , q1 + · · · + qm = 1. Denote the cdf of the risky wealth w ˜ by F and the cdf of the reference points r˜ = (r1 , q1 ; · · · ; rm , qm ) by G. Then the investor’s reference-dependent utility writes as E [v(w; ˜ r˜ , u)] = E [u(w) ˜ + R η,λ (u(w) ˜ − u(˜r ))]

 =

u(w) + R η,λ (u(w) − u(r ))dF (w)dG(r ).

Notice that m can be infinity, r˜ can be continuous, and w ˜ is assumed to be stochastically independent of r˜ . We define SSDr˜ in the same way as in Definition 2. Definition 4. Consider the reference-dependent utility specified in (1). Assume that η∗ ≥ 0, λ∗ ≥ 1 are pre-specified lower bounds for the parameters η and λ, and the stochastic reference point r˜ is given. Let w ˜ 1, w ˜ 2 be two random variables. w ˜ 1 second-order stochastically dominates w ˜ 2 relative to r˜ , written w ˜ 1 SSDr˜ w ˜ 2 , if E [v(w ˜ 1 ; r˜ , u)] ≥ E [v(w ˜ 2 ; r˜ , u)], for all u ∈ U 2 , η ≥ η∗ , λ ≥ λ∗ . Parallel to Proposition 1, the mathematical characterization of SSDr˜ is as follows. Proposition 7. Let the cdfs of w ˜ 1, w ˜ 2 be F1 , F2 , respectively, and the cdf of r˜ be G. Define 1 + η∗ λ∗ η∗ (λ∗ − 1) ∗ ∗ − G(x). D η ,λ (x; r˜ ) = 1 + η∗ 1 + η∗ Then w ˜ 1 SSDr˜ w ˜ 2 if and only if ≥ 0 for all x.

(−∞,x] (F2 (t )



− F1 (t ))D η

(5) ∗ ,λ∗

(t ; r˜ )dt

Proposition 7 is a generalization of Proposition 1. It shows that the introduction of multiple reference points only changes the ∗ ∗ specification of the stepwise weighting function D η ,λ (x; r˜ ). Ber cause most properties for SSD derived with a single reference ∗ ∗ point hinge only upon the fact that D η ,λ (x; r ) is a decreasing ∗ ∗ function of x and this fact remains true for D η ,λ (x; r˜ ), Propositions 2–6 appearing in Section 3 remain valid when the reference point r is replaced by a stochastic reference r˜ . ∗ ∗ The weighting function D η ,λ (x; r˜ ) in (5) is a linear function of G(x), which is the cdf of the stochastic reference point r˜ . A basic question arising here is whether it is possible to introduce an average reference point for investors so that the extension to stochastic multiple reference points can be reduced to a single expected non-stochastic reference point. Mathematically, this amounts to asking whether there exists a deterministic rˆ such that

 (−∞,x]

(F2 (t ) − F1 (t ))D η

 = (−∞,x]

∗ ,λ∗

(t ; r˜ )dt

(F2 (t ) − F1 (t ))D η

∗ ,λ∗

(t ; rˆ )dt

for all x. Unfortunately, such equivalent single reference point is not generally available. For example, if we take r˜ = (1, 50%; − 1, 50%), then we have



(F2 (t ) − F1 (t ))D η ,λ (t ; r˜ )dt (−∞,x]  1 + η∗ λ∗   (F2 (t ) − F1 (t ))dt , if x < −1,   1 + η∗  (−∞,x]         1 + η∗ λ∗ 2 + η∗ + η∗ λ∗   +  1 + η∗ 2(1 + η∗ ) (−∞,−1] (−1,x] = × (F2 (t ) − F1 (t ))dt , if − 1 ≤ x < 1,           2 + η∗ + η∗ λ∗ 1 + η∗ λ∗   + +   1 + η∗  2(1 + η∗ ) (−∞,−1] (−1,1] (1,x]  × (F2 (t ) − F1 (t ))dt , if x ≥ 1, ∗

∗

and



(F2 (t ) − F1 (t ))D η ,λ (t ; rˆ )dt (−∞,x]  1 + η∗ λ∗   (F2 (t ) − F1 (t ))dt , if x < rˆ ,   1 + η∗ (−∞,x]    =  1 + η∗ λ∗    + (F2 (t ) − F1 (t ))dt , 1 + η∗ (−∞,ˆr ] (ˆr ,x] ∗

∗

if x ≥ rˆ .

There exists no rˆ such that the above two integrals can become identical for all x. The reason behind the non-availability of an equivalent single reference point is that, in contrast to the disappointment-aversion models of Bell (1985), Loomes and Sugden (1986) and Gul (1991), Kőszegi and Rabin (2007) do not specify the reference point as a lottery’s certainty equivalent, but rather assume that investors compare outcomes and reference points across states. The cross-state comparison makes investors’ gain–loss sensations dependent on the entire distribution of the reference point rather than on some kinds of certainty equivalents. As pointed out by Sprenger (2015, pg. 1457), ‘‘The modeling convention of using either the certainty equivalent or the entire distribution of expected outcomes as a reference is more than a minor nuance. A central distinction is the existence of a so-called endowment effect for risk, present in Kőszegi and Rabin (2007)’s model that is absent in disappointment-aversion models’’. This argument suggests that the reliance of SSDr˜ on the distribution function of r˜ plays a key role in generating the endowment effect for risk. The endowment effect for risk is a relatively new notion in the literature of behavioral economics. Kőszegi and Rabin (2007, pg. 1053) use it to refer to the phenomenon ‘‘a person is less risk averse in eliminating a risk she expected to face than in taking on the same risk if she did not expect it’’, while Sprenger (2015, pg. 1456) generalizes it to be ‘‘risk attitudes differ when reference points change from certain to stochastic’’. Sprenger (2015) supports the existence of the endowment effect for risk by conducting an experiment, but he provides no analytical insights. Our ordering SSDr˜ can be used to analytically formulate the endowment effect for risk. A simple example is offered below. Example 4. Suppose that investors are asked to choose between

w ˜ 1 ≡ 0,

w ˜2 =



1 + η∗ λ∗ 1 + η∗



ε, 50%; − ε, 50% .

When r˜ ≡ 0, there is w ˜ 1 SSDr˜ w ˜ 2 . That is, with a deterministic reference point, investors dislike the mean-increasing spread. However, when r˜ becomes (ε, 50%; − ε, 50%), the ordering w ˜ 1 SSDr˜ w ˜ 2 no longer holds true. In other words, when r˜ becomes more dispersed, investors become less averse to the spread of risk and can take the spread w ˜ 2 more appealing.

D. Guo et al. / Insurance: Mathematics and Economics 70 (2016) 105–116

Generalizing Example 4, we can show that when the risks to be compared are simple spreads, the ordering is preserved only when r˜ becomes more concentrated. When r˜ becomes more dispersed, the average weight assigned to losses relative to that assigned to gains in the utility evaluation becomes smaller. Therefore, investors become less risk-averse and the original ordering can be reversed. A formal statement of the endowment effect for risk is included in Proposition 8. Proposition 8. Let r˜1 and r˜2 be two references. Assume that w ˜ 2 is ˜ 2. a simple spread of w ˜ 1 with crossing point w0 such that w ˜ 1 SSDr˜1 w If r˜2 is a simple contraction of r˜1 with the same crossing point, then ˜ 2 . On the contrary, if r˜2 is a simple spread of r˜1 with respect w ˜ 1 SSDr˜2 w ˜ 2 may become false. to w0 , the ordering w ˜ 1 SSDr˜2 w

(A1) (A2) (A3) (A4)

Proposition 9. Assume that x˜ 1 , x˜ 2 , y˜ 1 , y˜ 2 satisfy (A1)–(A4). Let r˜1 and s˜1 be independent of x˜ i , y˜ i (i = 1, 2) such that r˜1 =d x˜ 1 and s˜1 =d y˜ 1 . If ∗ ∗ ∗ ∗ E [˜y1 + R η ,λ (˜y1 − s˜1 )] − E [˜y2 + R η ,λ (˜y2 − s˜1 )]

< 0 < E [˜x1 + R η

Definition 5. Suppose that an investor needs to choose between two lotteries w ˜ 1 and w ˜ 2 . Choosing w ˜ 1 is a second-order stochastic dominant personal equilibrium (SSD-PE), if and only if

w ˜ 1 SSDr˜ w ˜ 2,

with r˜ =d w ˜ 1.

x˜ 1 FSDy˜ 1 , x˜ 2 FSDy˜ 2 . F2 − F1 ≡ k(G2 − G1 ) for some k > 0. x˜ 2 is a simple spread of x˜ 1 with crossing point x∗ . F1 ≡ G1 on [x∗ , ∞).

Under these assumptions, the Allais-type behavior refers to preferring x˜ 1 over x˜ 2 and yet preferring y˜ 2 over y˜ 1 in the pairwise choices. Notice that x˜ i and y˜ i (i = 1, 2) in Examples 5 and 6 satisfy the assumptions (A1)–(A4) with k = 1 and 4 respectively.

5. Extension to endogenous reference points So far, the reference point r or r˜ is exogenously given. Kőszegi and Rabin (2006, 2007) endogenize the reference point by introducing the concept of ‘‘personal equilibrium’’. Roughly speaking, the personal equilibrium is defined as the situation where the stochastic outcome implied by the optimal choice conditional on reference coincides with the reference itself. In terms of stochastic dominance, we can define personal equilibrium in the following way.

(6)

Condition (6) ensures that when investors take w ˜ 1 as the reference point, they will not change their idea to choose w ˜ 2 . The reference point is identical in distribution to the optimal choice. By virtue of SSD-PE, we can show that endogenous reference points offer a new resolution to the Allais-type anomalies, which appears in the following choice problems (Quiggin, 1993). Example 5. Common consequence effect. Scenario 1. Choose between x˜ 1 ≡ 2400 and x˜ 2 = (0, 1%; 2400, 66%; 2500, 33%); Scenario 2. Choose between y˜ 1 = (0, 66%; 2400, 34%) and y˜ 2 = (0, 67%; 2500, 33%). The choice set (˜y1 , y˜ 2 ) can be obtained by moving away the ‘‘common consequence’’ of ‘‘winning 2400 with probability 0.66’’ from the choice set (˜x1 , x˜ 2 ). According to the expected utility theory, choosing x˜ 1 in Scenario 1 implies choosing y˜ 1 in Scenario 2. However, repeatedly confirmed experiments show that most subjects choose x˜ 1 in Scenario 1 and y˜ 2 in Scenario 2. Example 6. Common ratio effect. Scenario 1. Choose between x˜ 1 ≡ 3000 and x˜ 2 = (0, 20%; 4000, 80%); Scenario 2. Choose between y˜ 1 = (0, 75%; 3000, 25%) and y˜ 2 = (0, 80%; 4000, 20%). The ratio of the winning probabilities is the same for both choice sets. Similar to Problem 1, choosing x˜ 1 in Scenario 1 implies choosing y˜ 1 in Scenario 2 according to the expected utility theory. However, in experiments, most subjects choose x˜ 1 in Scenario 1 and y˜ 2 in Scenario 2. Machina (1982) and Segal (1987) generalize these two problems in a unified framework. Let F1 , F2 and G1 , G2 be the cdfs of x˜ 1 , x˜ 2 and y˜ 1 , y˜ 2 respectively. Assume that

111

∗ ,λ∗

(˜x1 − r˜1 )] − E [˜x2 + R η

∗ ,λ∗

(˜x2 − r˜1 )],

(7)

then x˜ 1 is a SSD-PE of the choice between x˜ 1 and x˜ 2 , while y˜ 1 is not a SSD-PE of the choice between y˜ 1 and y˜ 2 . In other words, all referencedependent risk-averse investors will choose x˜ 1 in personal equilibrium in the face of x˜ 1 and x˜ 2 , but may choose y˜ 2 in personal equilibrium in the face of y˜ 1 and y˜ 2 . Proposition 9 justifies the existence of Allais-type behavior in personal equilibrium. The essential spirit of this resolution is that because y˜ i is more risky than x˜ i (by Assumption A1), investors with endogenous reference points will become less risk-averse in the face of y˜ i (by Proposition 8), and thus they may prefer the more dispersed choice y˜ 2 . For Examples 5 and 6, condition (7) holds true as long as η∗ = 0.6 and λ∗ = 2. Capturing the Allais-type anomalies using endogenous reference points has an important advantage over the alternative of introducing nonlinear probability weighting functions (Machina, 1982): it greatly improves the tractability of the decision model to cope with dynamic environments that are commonly adopted in game theoretical contexts. 6. Extension to higher-order dominance Up to now, we have seen that for second-order stochastic dominance, loss aversion with respect to a reference point is in effect equivalent to a reference-driving (stepwise) weighting function associated to the integrand. This property is in fact preserved for higher-order stochastic dominance, making stochastic dominance a very convenient approach for studying how loss aversion impacts higher-order risk preferences. This section explores how loss aversion interacts with prudence. Since the ordering preferred by all prudent investors is captured by TSD, we turn to study how loss aversion alters TSD. Formally, given the lower bounds η∗ , λ∗ of η and λ, we say w ˜ 1 thirdorder stochastically dominates w ˜ 2 relative to a (deterministic or stochastic) reference point r˜ , written w ˜ 1 TSDr˜ w ˜ 2 , if E [v(w ˜ 1 ; r , u)] ≥ E [v(w ˜ 2 ; r , u)] for all u ∈ U 3 = {u : (−∞, ∞) → R | u′ ≥ 0, u′′ < 0, u′′′ > 0},

η ≥ η ∗ , λ ≥ λ∗ . The mathematical characterization of TSDr˜ is as follows. Proposition 10. Let the cdfs of w ˜ 1, w ˜ 2 be F1 , F2 , respectively,  and the cdf of r˜ be G. Then w ˜ 1 TSDr˜ w ˜ 2 if and only if (F2 (t ) − ∗ ∗ F1 (t ))D η ,λ (t ; r˜ )dt ≥ 0 and



 (−∞,x]

(−∞,y]

(F2 (t ) − F1 (t ))D η

∗ ∗ where D η ,λ is specified by (5).

∗ ,λ∗

(t ; r˜ )dtdy ≥ 0 for all x,

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dominance rules for expected utility to accommodate downside loss aversion and explicit reference points. We show that the new dominance rule has four attractive features as a risk ordering.

Fig. 3. The inclusion relationship between SSDr˜ , TSDr˜ , SSD and TSD. The notation in this figure follows from Fig. 1.

In contrast to second-order stochastic dominance, TSD is neither sufficient nor necessary for TSDr˜ . That TSDr˜ does not imply TSD can be verified using the result in Section 3.2, which shows that SSDr ̸⊂ TSD. Because SSDr ⊂ TSDr , we have TSDr ̸⊂ TSD. To see that ˜ 1 and w ˜ 2 therein, TSD ̸⊂ TSDr˜ , let us go back to Example 2. For w w ˜ 1 TSDr w ˜ 2 holds true only when r ≤ 32 or r ≥ 3. When 32 < r < 3, w ˜ 1 TSDr w ˜ 2 does not hold any longer. The intuition is as follows. Recall that w ˜ 1 is obtained by adding ε˜ = (−1, 50%; 1, 50%) to the state of winning 1 in the lottery z˜ = (1, 50%; 2, 50%), and w ˜ 2 is obtained by adding ε˜ to the state of winning 2 in z˜ . Without reference points, all prudent investors prefer adding ε˜ to the better state of winning 2. However, when the reference point is close to 2, investors will care more about the risk surrounding winning 2, and thereby may prefer adding ε˜ to the state of winning 1. Following the terminology of Eeckhoudta and Schlesinger (2006), prudence represents a type of ‘‘location preference’’ for the harms within a lottery. Our example clarifies that loss aversion with respect to a reference point usually attenuates this location preference. The inclusion relationship between SSDr˜ , TSDr˜ , SSD and TSD is shown in Fig. 3, which is an extension of Fig. 1. From Fig. 3, we see that TSDr˜ strengthens SSDr˜ , which in turn strengthens SSD; however, neither of them strengthens TSD. We end this section with a remark that one can incorporate loss aversion and reference point into the jth-order stochastic dominance with j = 4, 5, . . . in the same spirit. To be specific, given the lower bounds η∗ , λ∗ of η and λ, we say w ˜ 1 SDr˜ w ˜ 2 at order j, if E [v(w ˜ 1 ; r , u)] ≥ E [v(w ˜ 2 ; r , u)] for all u∈Uj = 

(−1)



u : (−∞, ∞) → R|u′ > 0, −u′′ > 0, u′′′ > 0, . . . ,

  j

j −2 2 +1

η ≥ η∗ ,





u(j) > 0 ,

j

j

where ⌈ 2 ⌉ denotes the maximal integer not exceeding 2 . Then the mathematical characterization of the jth-order SDr˜ turns out to be

 ···

(−∞,x3 ]



(−∞,xj−1 ]

· · · dx3 ≥ 0; 

(F2 (t ) − F1 (t ))D η

 ···

(−∞,x1 ]

(−∞,x2 ]

(−∞,xj−1 ]

From the theoretic perspective, SSDr˜ offers a precise characterization of ‘‘increasing risk’’ for the reference-dependent preference formulated by Kőszegi and Rabin (2006, 2007). As stochastic dominance rules are useful in many economic contexts such as portfolio analysis, inequality of income distribution, investment and saving, and option evaluation (Levy, 1992), we expect that SSDr˜ will be helpful for comparative static analysis with a referencedependence structure in the future. Mathematically, the characterization of SSDr˜ simply introduces a reference-driving (stepwise) weighting function to the integrand, which makes the existing statistical approaches for testing SSD proposed in Anderson (1996), Davidson and Duclos (2000), Barrett and Donald (2003), Linton et al. (2003) and Post (2008) readily extendable for testing SSDr˜ . An extension based on Davidson and Duclos (2000) can be found in Appendix B. Finally, in financial contexts, our stochastic dominance rule can be employed in several ways. For example, we can develop a linear programming test for SSDr˜ in the same spirit of Post (2003) to test for the stochastic dominance efficiency of a given portfolio with a prescribed reference point; we can also use SSDr˜ to evaluate the performance of securities with a return target in a similar way of Fong et al. (2005) or Li and Linton (2007). Acknowledgments

λ ≥ λ∗ ,



(1) It is consistent with portfolio mangers’ disaster avoidance motive as it captures risk aversion over gains but precludes risk-lovingness over losses. (2) It offers an effective complement to SSD and TSD in ordering risks. It contains SSD as a sub-ordering, coincides with TSD under certain specific conditions, and can identify the downside risks in mean-increasing spreads that neither TSD nor SSD could identify. (3) It is a convenient tool for analytically characterizing the effects of reference points on risk choices. It rationalizes the violation of translational invariance and scaling invariance, accommodates the endowment effect for risk, and resolves the Allais-type anomalies. (4) It is can be easily generalized to study how loss aversion with respect to the reference point interacts with higher-order preferences.

∗ ,λ∗

(t ; r˜ )dtdxj−1

(F2 (t ) − F1 (t ))D η

∗ ,λ∗

(t ; r˜ )dtdxj−1

· · · dx2 ≥ 0 for all x1 . The jth-order SDr˜ provides an analytical characterization of the influence of loss aversion on the jth order risk preference. These characterizations may be helpful in the studies of higher-order effects of reference-dependent preferences (Maier and Rüger, 2012).

The authors appreciate the helps of the managing editor Rob Kaas and the valuable comments of an anonymous referee. Lin Zhao acknowledges financial support from the research grant of National Science Foundation of China (NSFC) (No. 71301161 and No. 71532013). Yi Hu acknowledges financial support from the research grant of National Science Foundation of China (NSFC) (No. 71301160). Dongmei Guo acknowledges financial support from the research grant of National Science Foundation of China (NSFC) (No. 71301173) and the support from the Program for Innovation Research in Central University of Finance and Economics. Appendix A All the integrals appearing in the following are calculated in the Lebesgue–Stieltjes sense. Lemma A.1. Let φ1 , φ2 be right-continuous functions satisfying lim φ1 (x) = lim φ2 (x) = 0,

7. Concluding remarks

x→−∞

Building on the reference-dependent utility theory proposed by Kőszegi and Rabin (2006, 2007), we extend the stochastic

x→+∞

x→−∞

lim φ1 (x) = lim φ2 (x) < +∞.

Then we have

x→+∞

D. Guo et al. / Insurance: Mathematics and Economics 70 (2016) 105–116

113

u(x)dφ1 (x) ≥ u(x)dφ2 (x) for all u ∈ U 1 iff φ2 (x)−φ1 (x) ≥ 0 for all x.    (ii) u(x)dφ1 (x) ≥ u(x)dφ2 (x) for all u ∈ U 2 iff (−∞,x] (φ2 (t ) − φ1 (t ))dt ≥ 0 for all x. (i)





Proof. Notice that integration by parts can be used when φ has a jump, say at a,



u(x)dφ(x)

 = (−∞,a]

u(x)dφ(x) + u(a)[φ(a) − φ(a−)]

 + (a,∞)

u(x)dφ(x)

= u(a)φ(a−) −



φ(x)du(x) + u(a)[φ(a) − φ(a−)]  + u(∞)φ(∞) − u(a)φ(a) − φ(x)du(x) (a,∞)  = u(∞)φ(∞) − φ(x)du(x). (−∞,a]

Fig. A.1. A graphic illustration of F1 and F2 constructed in Proposition 2. This figure depicts the cdfs F1 and F2 . The ‘‘+’’ area is that where F2 > F1 , and the ‘‘−’’ area is that where the opposite holds.

there is

 (−∞,x]

′ Therefore,  ′′  u(x)d(φ1 (x) − φ2 (x)) = (φ2 (x) − φ1 (x))u (x)dx = − u (x) (−∞,x] (φ1 (t ) − φ2 (t ))dtdx. Then the results follow from Hadar and Russell (1971) straightforwardly. 





Lemma A.2. Suppose (−∞,x] A(t )dt ≥ 0 for all x. If B ≥ 0 is a  nonincreasing function, then (−∞,x] A(t )B(t )dt ≥ 0 for all x.



Proof. After an integration by parts, we obtain (−∞,x] A(t )B(t )dt =  



B(x) (−∞,x] A(t )dt − (−∞,x]







(−∞,t ]

A(s)ds dB(t ) ≥ 0.



Proof of Proposition 1. According to (1), we have E [v(w; ˜ r , u)]

 =

u(x)dF (x) + ηλ

+η 

 (−∞,r ]

[u(x) − u(r )]dF (x)



[u(x) − u(r )]dF (x)  = u(x)dF (x) + ηλ u(x)dF (x) − ηλu(r )F (r ) (−∞,r ]  +η u(x)dF (x) − ηu(r )(1 − F (r )) (r ,∞)  = u(x)d[F (x)D η,λ (x; r )] − ηu(r ). (r ,∞)

Then, for w ˜ 1 and w ˜ 2 with cdfs F1 and F2 , there is E [v(w ˜ 1 ; r , u)] − E [v(w ˜ 2 ; r , u)]

 =

u(x)d[(F1 (x) − F2 (x))D

η,λ

(x; r )].

By Lemma A.1, given η and λ, the equivalent conditions (i) and (ii) of E [v(w ˜ 1 ; r , u)] ≥ E [v(w ˜ 2 ; r , u)] for all u ∈ Ui , i = 1, 2, are F2 (x) ≥ F1 (x) for all x, and (−∞,x] (F2 (t ) − F1 (t ))D η,λ (t ; r )dt ≥ 0 for all x, respectively. For all η ≥ η∗ , λ ≥ λ∗ , B(t , r ) =

  

1+ηλ 1+η 1+η∗ λ∗ 1+η∗

 =

(F2 (t ) − F1 (t ))D η,λ (t ; r )dt

(F2 (t ) − F1 (t ))D η  (−∞,x]  A(t ,r )

(t ; r ) B(t , r )dt ≥ 0, 

for all x.

This proves that ‘‘ (−∞,x] (F2 (t ) − F1 (t ))D η,λ (t ; r )dt ≥ 0 for all x  and all η ≥ η∗ , λ ≥ λ∗ ’’ is in fact equivalent to ‘‘ (−∞,x] (F2 (t ) −



∗ ∗ F1 (t ))D η ,λ (t ; r )dt ≥ 0 for all x’’.



Proof of Proposition 2. There is no doubt that many pairs of w ˜1 and w ˜ 2 satisfying the condition in Proposition 2 exist. To make our example as simple as possible, assume without loss of generality that 0 < r1 < r2 and choose an interval [x0 , x1 ] lying strictly ∗ ∗ ∗ ∗ inside (r1 , r2 ). Then it follows D η ,λ (x; r1 ) < D η ,λ (x; r2 ) for x ∈ [x0 , x1 ]. Let the support of w ˜ 1 and w ˜ 2 be [0, b] with b > r2 . Choose a constant a ∈ (0, r1 ) and two continuous cdfs F1 and F2 such that F1 (x) = F2 (x) on [0, b] \ {(0, a) ∪ (x0 , x1 )}, F1 (x) < F2 (x) on (0, a), F1 (x) > F2 (x) on (x0 , x1 ),



(F2 (x) − F1 (x))D η ,λ (x; r1 )dx [0,a]  ∗ ∗ = (F1 (x) − F2 (x))D η ,λ (x; r1 )dx, ∗

∗

[x0 ,x1 ]

as illustrated in Fig. A.1. Then the random variables w ˜ 1 and w ˜ 2 with the cdfs given by F1 and F2 satisfy the desired properties.  Proof of Proposition 3. The proof that w ˜ 1 SSDw ˜ 2 implies w ˜ 1 SSDr w ˜ 2 follows straightforwardly from Lemma A.2. The remainder of the proof is devoted to the construction of w ˜ 3 and w ˜ 4 . Because the mean-preserving spread can be derived as the limit of a sequence of simple (single-crossing) spreads, we at first assume that the cdfs of w ˜ 1 and w ˜ 2 , denoted by F1 (x) and F2 (x), intersect only once at some point x = x∗ (see Fig. A.2, panels A and B). We have

 ≥ 1,

∗ ,λ∗

if t ≤ r ,

(−∞,x∗ ]

(F2 (t ) − F1 (t ))dt =

 (x∗ ,∞)

(F1 (t ) − F2 (t ))dt > 0.

is nonincreasing in t. By Lemma A.2, we thus have when

Because D η ,λ (x; G) is nonincreasing in x, we can always shift the curve of F1 (x) upward to obtain F3 (x) under the constraints F3 (−∞) = F1 (−∞), F3 (∞) = F1 (∞), F3 (x∗ ) = F1 (x∗ ) such that





 

(−∞,x]

1,

(F2 (t ) − F1 (t ))D

∗

if t > r ,

η∗ ,λ∗

(t ; r )dt ≥ 0,

for all x,

(−∞,x∗ ]

∗

(F2 (t ) − F3 (t ))D η

∗ ,λ∗

(t ; r )dt

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D. Guo et al. / Insurance: Mathematics and Economics 70 (2016) 105–116

Fig. A.2. The constructions of F3 and F4 . This figure depicts the cdfs F1 , F2 , F3 and F4 . The ‘‘+’’ area is that where F2 > F1 , and the ‘‘−’’ area is that where the opposite holds.

 = (x∗ ,∞)

(F3 (t ) − F2 (t ))D

η∗ ,λ∗

  η∗ (λ∗ − 1) (F2 (t ) − F1 (t ))dt + (F2 (t ) − F1 (t ))dt = 1 + η∗ (−∞,r ]  η∗ (λ∗ − 1) = (F2 (t ) − F1 (t ))dt + E [w ˜ 1 ] − E [w ˜ 2 ] ≥ 0, 1 + η∗ (−∞,r ]

(t ; r )dt > 0

(see panel A of Fig. A.2). It is easily checked that the coined random variable w ˜ 3 , whose cdf is F3 , satisfies the relationship (3). Moreover, E [w ˜ 3 ] < E [w ˜ 1 ] = E [w ˜ 2 ].

which proves assertion (ii).

In a similar way, we can alternatively shift the curve of F2 (x) downward to obtain F4 (x) under the constraints F4 (−∞) = F2 (−∞), F4 (∞) = F2 (∞), F4 (x∗ ) = F2 (x∗ ) such that

 (−∞,x∗ ]

(F4 (t ) − F1 (t ))D η

 = (x∗ ,∞)

∗ ,λ∗

(F1 (t ) − F4 (t ))D η

(t ; r )dt ∗ ,λ∗

(t ; r )dt > 0

(see panel B of Fig. A.2). The coined random variable w ˜ 4 with cdf F4 satisfies (3) and E [w ˜ 4 ] > E [w ˜ 2 ] = E [w ˜ 1 ]. For the general mean-preserving spread with multiple crossing properties, we can construct w ˜ 3 and w ˜ 4 in the same spirit; see panels C and D of Fig. A.2 for an illustration.  Proof of Proposition 4. Since w ˜ 2 is a simple spread of w ˜ 1, w ˜ 1 SSD w ˜ 2 if and only if





Proof of Proposition 5. According to Whitmore (1970), w ˜ 1 TSD

w ˜ 2 under the condition E [w ˜ 1 ] = E [w ˜ 2 ] if and only if   (F2 (t ) − F1 (t ))dtdy (−∞,x] (−∞,y]  = (x − t )(F2 (t ) − F1 (t ))dt ≥ 0 (−∞,x]

for all x. Since w ˜ 2 is spread-contraction of w ˜ 1 , the above inequality holds true for all x if and only if it holds true for x = b. This proves assertion (i) Similar argument can be used to prove assertion (ii).  Proof of Proposition 6. Denote the cdfs of w ˜ 1, w ˜ 2 by F1 , F2 , η∗ ,λ∗ ( w ˜ ˜2+ respectively. Then the condition E [w ˜ 1 +R 1 − r )] = E [w ∗ ∗ R η ,λ (w ˜ 2 − r )] is equivalent to

 (−∞,0]

(F2 (x) − F1 (x))D η

 =

(F2 (t ) − F1 (t ))dt = E [w ˜ 1 ] − E [w ˜ 2 ] ≥ 0,

(0,∞)

∗ ,λ∗

(F1 (x) − F2 (x))D η

(x; r )dx ∗ ,λ∗

(x; r )dx.

which proves assertion (i). Similarly, w ˜ 1 SSDr w ˜ 2 if and only if

The cdf of β w ˜ i is Fˆi (x) = Fi (x/β), i = 1, 2. Hence





(F2 (t ) − F1 (t ))D η ,λ (t ; r )dt  1 + η∗ λ∗ = (F2 (t ) − F1 (t ))dt 1 + η∗ (−∞,r ]  + (F2 (t ) − F1 (t ))dt ∗

(r ,+∞)

∗

(Fˆ2 (x) − Fˆ1 (x))D η ,λ (x; r )dx (−∞,0]  ∗ ∗ − (Fˆ1 (x) − Fˆ2 (x))D η ,λ (x; r )dx (0,∞)  ∗ ∗ =β (F2 (x) − F1 (x))D η ,λ (β x; r )dx ∗

(−∞,0]

∗

D. Guo et al. / Insurance: Mathematics and Economics 70 (2016) 105–116

 − (0,∞)

(F1 (x) − F2 (x))D η

∗ ,λ∗

(β x; r )dx



:= β H (β). ∗ ∗ ∂ D η ,λ (β x; r ) > 0 Because H (1) = 0, and for β > 0, there are ∂β

∂ when x < 0 and ∂β D η ,λ (β x; r ) < 0 when x > 0, it follows that for β > 0, H (β) ≥ 0 if and only if β ≥ 1.  ∗

∗

Proof of Proposition 9. We only need to prove that assumptions (A1)–(A4) are compatible with (7), since (7) implies ‘‘x˜ 1 SSDr˜1 x˜ 2 but y˜ 1 does not SSDs˜1 y˜ 2 ’’ straightforwardly due to the single-crossing condition (A3). By (A2) and (A3), y˜ 2 is a simple spread of y˜ 1 with respect to x∗ . By (A2) and (A4), we obtain ∗ ∗ ∗ ∗ E [˜y1  + R η ,λ (˜y1 − s˜1 )] − E [˜y2 + R η ,λ (˜y2 − s˜1 )]

=

Proof of Proposition 7. We first restructure E [v(w; ˜ r˜ , u)] in the following convenient way6

(−∞,x∗ ]

(G2 (x) − G1 (x))D η

 − (x∗ ,∞)

E [v(w; ˜ r˜ , u)]

= E [u(w) ˜ + η u( w ˜ ∨ r˜ ) + ηλu(w ˜ ∧ r˜ ) − η(1 + λ)u(˜r )]   = u(s)dP (w ˜ ≤ s) + η u(s)dP (w ˜ ∨ r˜ ≤ s)   +ηλ u(s)dP (w ˜ ∧ r˜ ≤ s) − η(1 + λ) u(s)dP (˜r ≤ s)   = u(s)dF (s) + η u(s)d[F (s)G(s)]  +ηλ u(s)d[(F (s) + G(s) − F (s)G(s))]  − η(1 + λ) u(s)dG(s)   = u(s)d[F (s) (1 + ηλ − η(λ − 1)G(s))] − η u(s)dG(s).    (1+η)D η,λ (s;˜r )

115

=

1

(G1 (x) − G2 (x))D η



k

∗ ,λ∗

− (x∗ ,∞)

∗ ,λ∗

(F2 (x) − F1 (x))D η

(x; s˜1 )dx

∗ ,λ∗

(x; s˜1 )dx  ∗ ∗ (F1 (x) − F2 (x))D η ,λ (x; r˜1 )dx .

(−∞,x∗ ]



(x; s˜1 )dx

By (A1), G1 > F1 on (−∞, x∗ ], following which we obtain



(G2 (x) − G1 (x))D η ,λ (x; s˜1 )dx  ∗ ∗ − (G1 (x) − G2 (x))D η ,λ (x; s˜1 )dx ∗ (x ,∞)  1 ∗ ∗ < (F2 (x) − F1 (x))D η ,λ (x; r˜1 )dx k  (−∞,x∗ ]  ∗ ∗ − (F1 (x) − F2 (x))D η ,λ (x; r˜1 )dx , ∗

∗

(−∞,x∗ ]

(x∗ ,∞)

∗ ∗ ∗ ∗ i.e., E [˜y1 + R η ,λ (˜y1 − s˜1 )] − E [˜y2 + R η ,λ (˜y2 − s˜1 )] <

η∗ ,λ∗

η∗ ,λ∗

1 k

(E [˜x1 +

For w ˜ 1 and w ˜ 2 with cdfs F1 and F2 , there is

R

E [v(w ˜ 1 ; r˜ , u)] − E [v(w ˜ 2 ; r˜ , u)]

Proof of Proposition 10. The result follows straightforwardly from the equation

= (1 + η)



u(s)d[(F1 (s) − F2 (s))D η,λ (s; r˜ )].

The equivalent condition of E [v(w ˜ 1 ; r˜ , u)] ≥ E [v(w ˜ 1 ; r˜ , u)] for all u ∈ U2 , all η ≥ η∗ , λ ≥ λ∗ follows in a similar manner of the proof for Proposition 1.  Proof of Proposition 8. Let the cdfs of w ˜ i and r˜i be Fi and Gi , i = 1, 2. Assume without loss of generality that

 (−∞,w0 ]

(F2 (x) − F1 (x))D η

 = (w0 ,∞)

∗ ,λ∗

(F1 (x) − F2 (x))D

(x; r˜1 )dx

η∗ ,λ∗

(x; r˜1 )dx.

If r˜2 is a simple contraction of r˜1 with respect to w0 , then ∗ ∗ ∗ ∗ D η ,λ (x; r˜2 ) ≥ D η ,λ (x; r˜1 )

D

η∗ ,λ∗

(x; r˜2 ) ≤ D

η∗ ,λ∗

on x ≤ w0 ,

(x; r˜1 ) on x ≥ w0 ,

which leads to

 (−∞,w0 ]

(F2 (x) − F1 (x))D η

 ≥ (w0 ,∞)

∗ ,λ∗

(x; r˜2 )dx

(˜x1 − r˜1 )] − E [˜x2 + R

(˜x2 − r˜1 )]).



E [v(w ˜ 1 ; r˜ , u)] − E [v(w ˜ 2 ; r˜ , u)]

= (1 + η)



u(s)d[(F1 (s) − F2 (s))D η,λ (s; r˜ )]

and the procedure based on integration-by-parts shown in Whitmore (1970). 

Appendix B This appendix extends the approach proposed by Davidson and Duclos (2000) (DD, for short) to testing SSDr˜ . For simplicity, we assume that the reference point is single and constant, denoted by r. All of the following can be generalized to the case of multiple discrete reference points. Let F 1 and F 2 be the cdfs of two risks x˜ 1 and x˜ 2 , and fi1 and fi2 be the corresponding observations of realized values respectively. For a grid of preselected points x1 , x2 · · · xk , the second order DD statistic T (x) is given by T (x) =

 F2 − F1 , √  V

where

(F1 (x) − F2 (x))D

η∗ ,λ∗

(x; r˜2 )dx.

That is, w ˜ 2 SSDr˜2 w ˜ 1 . On the contrary, if r˜2 is a simple spread of r˜1 with respect to w0 , the reverse of the inequality is true, violating the condition for w ˜ 2 SSDr˜2 w ˜ 1. 

6 We use the notation a ∨ b = max{a, b} and a ∧ b = min{a, b}.

 N 1   j   F = (x − fij )+ , j = 1, 2,    N i =1      N   1 1   j 2 j 2  VF j = (x − f )+ − (F ) ,

j = 1, 2, i N N i =1     N   1 1   1 2 12    VF 1 F 2 = (x − fi )+ (x − fi )+ − F F ,    N N i=1    V = VF 1 +  VF 2 − 2 VF 1 F 2 .

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Let the 1 − α percentile of the Studentized Maximum Modulus k distribution with k and infinite degrees of freedom be M∞,α . According to Stoline and Ury (1979), we take k = 10, α = 0.05 and k M∞,α = 3.254. The DD test accepts the null hypothesis ‘‘x˜ 1 SSDx˜ 2 ’’ k at significance level α , if and only if −T (xi ) < M∞,α for all i and k Tj (xi ) > M∞,α for some xi . By definition, x˜ 1 SSDr x˜ 2 if and only if

 2 1    (−∞,x] (F (t ) − F (t ))dt ≥ 0, for all x ≤ r ,     (F 2 (t ) − F 1 (t ))dt  (−∞, x ]     η∗ (1 − λ∗ )   ≥ (F 2 (t ) − F 1 (t ))dt , for all x > r , 1 + η∗ (−∞,r ] which further equates to

   (x − t )(f 2 (t ) − f 1 (t ))dt ≥ 0, for all x ≤ r ,    (−∞, x ]   (x − t )(f 2 (t ) − f 1 (t ))dt   (−∞, x ]    η∗ (1 − λ∗ )    ≥ (r − t )(f 2 (t ) − f 1 (t ))dt , for all x > r . 1 + η∗ (−∞,r ] Discretizing the above, one can follow the DD approach to test SSDr k with the mirror modification of replacing M∞,α with

 k M∞,α , if xi ≤ r ,    η∗ (1 − λ∗ ) 1  k  M∞,α + √ k,r N (1 + η∗ )  M∞,α (xi ) = V  N    2  [(r − fi )+ − (r − fi1 )+ ],  ×

if xi > r .

i=1

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Further reading Barberis, N., Huang, M., Thaler, R.H., 2006. Individual preferences, monetary gambles, and stock market participation: A case for narrow framing. Amer. Econ. Rev. 96, 1069–1090. Baucells, M., Hwang, W., 2013. A model of mental accounting and reference price adaptation. Working Paper. Benartzi, S., Thaler, R.H., 1995. Myopic loss aversion and the equity premium puzzle. Quart. J. Econ. 110, 75–92. Maheu, J.M., McCurdy, T.H., Song, Y., 2012. Components of bull and bear markets: Bull corrections and bear rallies. J. Bus. Econom. Statist. 30, 391–403.