Almost stochastic dominance for risk averters and risk seeker

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Almost Stochastic Dominance for Risk Averters and Risk Seeker Xu Guo, Wing-Keung Wong, Lixing Zhu PII: DOI: Reference:

S1544-6123(16)30082-4 10.1016/j.frl.2016.05.005 FRL 534

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Finance Research Letters

Received date: Revised date: Accepted date:

4 January 2016 11 April 2016 5 May 2016

Please cite this article as: Xu Guo, Wing-Keung Wong, Lixing Zhu, Almost Stochastic Dominance for Risk Averters and Risk Seeker, Finance Research Letters (2016), doi: 10.1016/j.frl.2016.05.005

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Highlights • Introduce the concept of almost stochastic dominance (ASD) for risk seekers. • Develop properties of the ASD for both risk averters and risk seekers.

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• Discuss the advantages and disadvantages of using ASD and generalized ASD.

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Almost Stochastic Dominance for Risk Averters and

Xu Guo

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Risk Seekers

School of Statistics, Beijing Normal University College of Economics and Management,

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Nanjing University of Aeronautics and Astronautics

Wing-Keung Wong

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Department of Economics, Hong Kong Baptist University

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Lixing Zhu

School of Statistics, Beijing Normal University

May 6, 2016

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Department of Mathematics, Hong Kong Baptist University

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Acknowledgements The authors are grateful to Brian Lucey, the Editor-in-Chief, and anonymous referee for substantive comments that have significantly improved this manuscript. The authors would like to thank Howard E. Thompson for his valuable comments that have significantly improved this manuscript. The second author would also like to thank Robert B. Miller and Howard E. Thompson for their continuous guidance and encouragement. This research has been partially supported by grants from the Natural Science Foundation of Jiangsu Province, China (grant number BK20150732), Hong Kong Baptist University, and the Research Grants Council (RGC) of Hong Kong (project numbers 12502814 and 12500915).

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Almost Stochastic Dominance for Risk Averters and Risk Seeker Abstract In this paper we first extend the theory of almost stochastic dominance (ASD) (for risk averters) to include the ASD for risk-seeking investors. We then study

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the relationship between ASD for risk seekers and ASD for risk averters. Recently, Tsetlin, et al. (2015) develop the theory of generalized ASD (GASD). We then briefly discuss the advantages and disadvantages of ASD and GASD.

Keywords:

Almost stochastic dominance, generalized almost stochastic dominance,

JEL Classification:

Introduction

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1

C00, D81, G11.

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expected-utility maximization, risk averters, risk seekers, moments

Studying the selection rules for risk-averse and risk-seeking behaviors is one of the most

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important issues in behavioral economics. Friedman and Savage (1948) and others observe that investors exhibit not only risk-averse behavior but also risk-seeking behavior.

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Markowitz (1952) and Tobin (1958) introduce the mean-variance (MV) selection rules for both risk averters and risk seekers. Hanoch and Levy (1969) and others develop the

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theory of SD for risk averters. On the other hand, Hammond (1974), Wong and Li (1999), Meyer (1977), Stoyan (1983), Wong (2007), Levy (2015), and many others develop the

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SD rules for risk seekers. To circumvent the limitation that the MV and SD rules could lead to paradoxes

in decision making, Leshno and Levy (2002) introduce the theory of almost stochastic dominance (ASD) for risk averters. Tzeng, Huang, and Shih (2013) modify the definition of the ASD to acquire this property. Nonetheless, Guo, et al. (2013) have constructed some examples to show that the ASD definition modified by Tzeng, Huang, and Shih (2013) does not possess any hierarchy property, while Guo, et al. (2014) establish the necessary conditions for ASD criteria of various orders. 1

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2

The Theory

We assume that random variables X and Y defined on Ω = [a, b] have cumulative distribution functions (CDFs) F and G, probability density functions f and g, and means µX and µY , respectively. For any integer j, we define x

Hj−1 (y) dy and

a

HjR (x)

=

Z

x

b

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Hj (x) =

Z

R (y) dy , Hj−1

(2.1)

R b where H = F or G and H0 = H0R = h with h = f or g, Fn (x) − Gn (x) = a Fn (x) − R b R R Gn (x) dx, FnR (x) − GR n (x) = a Fn (x) − Gn (x) dx, Sn (F, G) = {x ∈ [a, b] : Gn (x) <

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R Fn (x)}, SnR (F, G) = {x ∈ [a, b] : FnR (x) < GR n (x)} for n = 1, 2, 3. Hi and Hi can be used

to develop the SD theory for risk averters and risk seekers, respectively. Levy (2015) calls the SD theory for risk seekers the risk-seeking SD theory, denoted as RSSD. We call Hi (HiR ) the ith -order SD (SDR ) integral.

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Leshno and Levy (2002) construct the following example to show the limitation of the

Example 2.1

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traditional SD rules (for risk averters):

Suppose that an investor considers two mutually exclusive prospects A

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and B which involve the same initial investment. Prospect A yields $900 with a probability

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of 1/2 and $100,000 with a probability of 1/2. Prospect B yields $1,000 with certainty. In this example, it is clear that most people will prefer Prospect A to Prospect B. However,

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Prospect A does not dominate Prospect B under traditional SD rules. To circumvent the limitation, Leshno and Levy (2002) introduce the concept of almost SD (ASD) rule (for risk averters), while Tzeng, et al. (2013) modify the rule. We rewrite the rule as follows:1 Definition 2.1

For any two random variables X and Y with their respective DFs F

and G and for 0 <  < 1/2, X (F ) is at least as large as Y (G) in the sense of: almost()

1. -almost F SD or -AF SD, denoted by X 1  G1 (x) dx ≤  F1 (x) − G1 (x) ,

1

Y if and only if

R  S1

F1 (x) −

We have modified their notations to distinguish them from the notations used for risk seekers.

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almost()

2. -almost SSD or -ASSD, denoted by X 2  G2 (x) dx ≤  F2 (x) − G2 (x) and µX ≥ µY , and

Y if and only if

R  F2 (x) − S2

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R  almost() 3. -almost T SD or -AT SD, denoted by X 3 Y if and only if S3 F3 (x) −  G3 (x) dx ≤  F3 (x) − G3 (x) and Gn (b) ≥ Fn (b) for n = 2, 3, where Sn (F, G) and Fn (x) − Gn (x) for n = 1, 2, 3 are defined in (2.1), and -almost F SD, SSD, and T SD stand for -almost first-, second-, and third-order SD, respectively, for risk averters.

In Example 2.1, the ASD rule (for risk averters) developed by Leshno and Levy (2002)

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and others could draw preference of Prospect A over Prospect B for most risk averters. However, the rule may not be able to draw any preference for risk seekers. To complete the ASD theory, we have to introduce the ASD concept for risk seekers. We first construct

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the following example to illustrate the motivation: Example 2.2

Prospect A yields $1 with a probability of 1/2 and $100,000 with a prob-

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ability of 1/2 and Prospect B yields $99,999 with certainty.

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In this example, it is clear that most people will prefer Prospect B to Prospect A but the traditional SD rules for risk seekers could not be used to determine any preference for

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A or B. To circumvent the limitation, in this paper we introduce the concept of the ASD

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for risk seekers as shown in the following definition: Definition 2.2

For any two random variables X and Y with their respective DFs F

and G and for 0 <  < 1/2, X (F ) is at least as large as Y (G) in the sense of: almost()

almost()

almost()

almost()

1. -almost F SDR or -AF SDR , denoted by X 1R Y or F 1R R  R  R R only if S R GR 1 (x) − F1 (x) dx ≤  F1 (x) − G1 (x) ,

G, if and

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2. -almost SSDR or -ASSDR , denoted by X 2R Y or F 2R G, if and R  R  R R and µX ≥ µY , and only if S R GR 2 (x) − F2 (x) dx ≤  F2 (x) − G2 (x) 2

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almost()

almost()

3. -almost T SDR or -AT SDR , denoted by X 3R Y or F 3R G, if and  R  R (x) and GR only if S R GR (x) − F3R (x) dx ≤  F3R (x) − GR n (a) ≤ Fn (a) for 3 3 3 for n = 1, 2, 3 are defined in n = 2, 3, where SnR (F, G) and FnR (x) − GR n (x) (2.1), and -almost F SDR , SSDR , and T SDR stand for almost first-, second-, and

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third-order SDR , respectively, for risk seekers. We note that Levy (2015) calls the SD for risk seekers risk-seeking SD and denotes it by RSD. Following Levy (2015), we refer ASD for risk seekers risk-seeking ASD and denote it by SDR . In addition, we first specify different types of utility functions as follows:

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Definition 2.3 For n = 1, 2, and 3,   Un∗ () = u ∈ Un : (−1)n+1 u(n) (x) ≤ inf{(−1)n+1 u(n) (x)}[1/ − 1] ∀x and UnR∗ () = u ∈ UnR : u(n) (x) ≤ inf{u(n) (x)}[1/ − 1] ∀x .

We call investors the nth -order -risk averters (seekers) if their utility functions u ∈ Un∗ ()

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(UnR∗ ()). We note that one could easily extend the theory to any order n. We note that

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Tsetlin, et al. (2015) use a different  for different order utility functions and one could follow their approach to define Un∗ (n ) and UnR∗ (n ) in Definition 2.3.

Fundamental Theorem

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2.1

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Tzeng, et al. (2013) modify the almost SD rule so that the almost SD rule for risk averters possesses the property of expected-utility maximization. In this paper we will show

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that the almost SD rule for risk seekers also possesses the property of expected-utility maximization. Here, we state both results in the following: Theorem 2.1

almost()

For n = 1, 2, and 3,2 X n

almost()

[nR

]Y if and only if E[u(X)] ≥

E[u(Y )] for any u ∈ Un∗ () [UnR∗ ()]. Readers may refer to Tzeng, Huang, and Shih (2013) for the proof of the almost SD rule for risk averters and refer to the appendix for the proof of the almost SD rule for risk seekers in Theorem 2.1. 2

We note that one could easily extend our work to n > 3.

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2.2

Preferences Between Risk Averters and Risk Seekers

Now, we turn to examining whether there is any relationship between the almost SD rule and the almost SDR rule. We first show in the following theorem that almost SD and SDR could be a dual problem: almost()

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Theorem 2.2

For any random variables X and Y and for n =1,2, and 3, X n almost()

if and only if −Y nR

−X.

Y

Sometimes the preference for assets using almost SD could move in the same direction

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as that using almost SDR but sometimes they move in the opposite direction. We first show in the following theorem for the first order that they move in the same direction: Theorem 2.3 almost()

Y.

Y if and only if

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X 1R

almost()

For any random variables X and Y , X 1

It is well known (Levy, 2015) that if prospects X and Y have the same mean, the

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preferences of risk averters and risk seekers could be opposite. We show that this is true

Theorem 2.4

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for almost SD in the following theorem: almost()

If µX = µY , then X 2

almost()

Y if and only if Y 2R

X.

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The proofs of Theorems 2.2 to 2.4 are straightforward and thus we skip reporting their

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proofs. We find that the preference of the third-order SD and SDR between prospects X and Y could be the same, as shown in the following theorem: Theorem 2.5 almost()

X 3R

almost()

If µX = µY and F3 (b) = G3 (b), then X 3

Y if and only if

Y.

We exhibt the proof of Theorem 2.5 in the appendix. We note that the risk-averse and risk-seeking ASD relationship are related in the situations mentioned in Theorems 2.1 to 2.5, but, in general, their relationship is not predictable.

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2.3

Hierarchy Property

We now examine the hierarchy property. To do so, we first discuss whether for  > 0, ∗ R∗ we have Uj+1 () ⊂ Uj∗ () and that Uj+1 () ⊂ UjR∗ (). To answer this question, first, the

’s in these two sets may be different. Second, U2∗ () only places constraints on u00 , while

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U1∗ () focuses on u0 . As we know, higher order derivatives cannot determine lower order ones. Thus, even if we have −u00 (x) ≤ inf{−u00 (x)}[1/ − 1] ∀x, we still do not have u0 (x) ≤ inf{u0 (x)}[1/ − 1] ∀x. We give the following example to illustrate this problem: Example 2.3

Consider u(x) = 2x − x2 , x ∈ [0, 1], we have u0 (x) = 2 − 2x and u00 (x) =

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−2. Clearly, u ∈ U2∗ (), while it does not belong to U1∗ () since inf{u0 (x)} = 0.

We note that Theorem 2.1 shows that the almost SD and the SDR rules both possess the property of expected-utility maximization. It is well-known that SD rules for both risk

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averters and risk seekers possess the hierarchy property (Levy, 1992). Guo, et al. (2013) find that the almost SD defined in Definition 2.1 does not possess the hierarchy property

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such that almost F SD does not imply almost SSD, which also does not imply almost T SD. Similarly, one could easily show that the almost SDR defined in Definition 2.2 does

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not possess the hierarchy property such that almost F SDR does not imply almost SSDR ,

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which also does not imply almost T SDR . To illustrate the non-hierarchy of almost SDR , we give the following example to show that almost F SDR does not imply almost SSDR :

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Example 2.4

Prospect A yields $1 with a probability of 1/2 and $5 with a probability

of 1/2 and Prospect B yields $3.33 with certainty. It’s easy to know that Prospect B dominates A by almost F SDR . Let the distributions of A and B be G(x) and F (x), respectively. Note that F2R (x)−GR 2 (x) = 0.83−0.5x, if 1 ≤ x ≤ 3.33, while if 3.33 < x ≤ 5, F2R (x) − GR 2 (x) = 0.5(x − 5). Then according to Definition 2.2, we can conclude that Prospect B does not dominate A by almost SSDR .

2.4

Almost Stochastic Dominance and Moments 6

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Ma and Wong (2012), Fong (2016), and the references therein have established some relationships between SD and mean-risk models. Guo, et al. (2014) develop the relationship between moments and ASD for risk averters. In this paper, we extend their results to include the relationship between moments and ASD for risk seekers. We summarize the

risk averters and risk seekers: Theorem 2.6

For any pair of random variables X and Y with their corresponding i

moments miX = E(X i ) and miY = E(Y i ), we have

Y , then mkX > (<)mkY for the smallest k ≤ j for which mkX 6= mkY if

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almost()

1. if X j

k is odd (even), and almost()

2. X jR

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results in the following theorem for the relationship between moments and ASD for both

Y =⇒ mkX > mkY for the smallest k ≤ j for which mkX 6= mkY .

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Readers may refer to Guo, et al. (2014) for the proof of Part (1) of Theorem 2.6. We

Discussion and Concluding Remarks

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state the proof of Part (2) of Theorem 2.6 in the appendix.

One of the main advantages of using GASD is that GASD possess the hierarchy property

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(Levy, 1992, 1998). Readers may refer to Guo, et al. (2013) for the discussion of the importance of the hierarchy property. We discuss some of the advantages of using ASD

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in the following:

The first advantage of using ASD over GASD is that ASD is simple and easy to

understand and interpret. In addition, the beauty of the ASD theory is that the definitions of ASD and the corresponding definitions for the classes of utilities are simple and yet the ASD theory still contains the beauty of the SD theory: its definitions (Definitions 2.1 and 2.2) do not use any information on utility, while the order of ASD among assets is the same as the order of expected utilities for decision makers (refer to Theorem 2.1) for both risk averters and risk seekers. Another advantage of using ASD is that there is a very 7

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nice relationship between the ASD theory for risk averters and risk seekers, as discussed in Section 2.2: when one obtains the ASD relationship on that pair of prospects for risk averters, one could draw inference on the ASD relationship for risk seekers on the pair of prospects and vice versa. Because of the complicated integral conditions for GASD,

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we are not sure whether it could have the simply relationships between GASD for risk averters and risk seekers.

The third advantage of using ASD is that from Definitions 2.1 and 2.2, one can easily observe that the ASD for risk averters (seekers) is defined by using the nagging violation

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R area where Fn is above Gn ( GR n is above Fn ) to be a small fraction () of the total R absolute area difference between Fn and Gn (GR n and Fn ). Since the definition of GASD

is complicated, it is not easy for GASD to get this nice results as the results of ASD we got in this paper.

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The fourth advantage of the ASD theory is that there is a very nice relationship between ASD and moments, as stated in Section 2.4: when one obtains the ASD relationship

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for any pair of prospects, one could draw inference on the relationship of moments for the pair of prospects and vice versa. We note that Theorem 2.6 shows that there exist

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simple moment conditions for ASD for both risk averters and risk-seekers. We also note that, interestingly, SD has a similar moment condition as stated in the following theorem

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by Fishburn (1980):

For any pair of random variables X and Y with their corresponding i

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Theorem 3.1

moments miX = E(X i ) and miY = E(Y i ), we have if X j Y , then 1. mkX > mkY for the smallest k ≤ j for which mkX 6= mkY if k is odd, and 2. mkX < mkY for the smallest k ≤ j for which mkX 6= mkY if k is even. We note that one could modify Theorem 3.1 to obtain the relationship between SD for risk seekers and moments. Comparing Theorem 2.6 with Theorem 3.1, we can clearly see that to obtain the moment conditions, it is not necessary to require a strong concept of SD but using ASD is sufficient. 8

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The fifth advantage of using ASD is that the concept of nth -degree risk developed by Ekern (1980), extended by Eeckhoudt, et al. (2009), and modified by, Tsetlin, et al. (2015) could be defined in a way consistent with ASD. Here, we rewrite their results as follows:

F

(k)

For 0 ≤ n ≤ 1/2, G has more n -AnR than F if and only if (k)

(b) = G (b), k = 2, · · · , n,

Z

Sn

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Theorem 3.2

[Fn (x) − Gn (x)]dx ≤ n ||Fn (x) − Gn (x)||.

From Tsetlin et al. (2015), one can also observe that ASD can be characterized in terms of

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allowing some relaxation of the conditions on probability shifts for SD. ASD also satisfies a preference for combining good with bad. For details, kindly refer to Tsetlin et al. (2015). Last, we note that some academics may believe that there are only risk averters in the markets. However, it is well known that the market could have other types of investors,

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see, for example, Friedman and Savage, 1948, Markowitz, 1952, Thaler and Johnson, 1990, and Egozcue, et al. 2011 for more discussion. In addition, there are many empirical

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findings which show that investors could be risk seeking. For example, Fong, et al. (2005) find that risk averters will prefer to invest in winners, whereas Sriboonchitta, et al. (2009)

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find that risk seekers will prefer to invest in losers in the momentum profits. Qiao, et al.

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(2012) find that risk averters prefer to buy stocks, while risk seekers are attracted to long index futures. Recently, Hoang, et al. (2015) show that in general, risk averters prefer

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not to include gold in their stock-bond portfolios while risk seekers prefer to include gold in their portfolios, especially in crisis periods. In addition, they find that risk-seekers prefer including gold in an equal-weighted portfolio while risk-averters prefer including gold in efficient portfolios. On the other hand, Bai, et al. (2015) extend the SD test for risk averters to get the SD test for risk seekers. Recently, Guo, et al. (2015) develop a ASD test for risk averters. We note that it is not difficult to extend Guo, et al. (2015) to develop a ASD test for risk seekers.

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Appendix Appendix A. Proof of Theorem 2.1: We only prove Part 2 of Theorem 2.1. We first prove the case for n = 2.

Z

S2R



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(1) If part: We will show that if R  R R and µX ≥ µY , GR (x) − F (x) dx ≤  F (x) − G (x) 2 2 2 2

then EF (u) ≥ EG (u) for every u ∈ U2R∗ (). Note that H2R (x) =

x

H1R (y)dy = b − x −

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H2 (b) + H2 (x). Thus, we have

Rb

F2 (x) − G2 (x) = F2R (x) − GR 2 (x) + F2 (b) − G2 (b). Applying this equation, we obtain 0

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EF (u) − EG (u) = u (b)(G2 (b) − F2 (b)) +

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= u0 (a)(G2 (b) − F2 (b)) +

Z

b

a

Z

a

(A.1)

(A.2)

(F2 − G2 )u00 dx

b

00 (F2R − GR 2 )u dx.

(A.3)

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R Rb 00 R Since u0 (a) > 0 and u00 > 0, we get EF (u) − EG (u) ≥ a (F2R − GR 2 )u dx = S2R (F2 − R R 00 R R 00 ¯R GR 2 )u dx+ S¯2R (F2 −G2 )u dx. where S denotes the complement of S in [a, b]. Denoting R inf x∈[a,b] u00 (x) = θ and supx∈[a,b] u00 (x) = θ, we obtain EF (u) − EG (u) ≥ θ S R (F2R − 2 R R R R R R R R R G2 )dx + θ S¯R (F2 − G2 )dx = (θ + θ) S R (F2 − G2 )dx + θ F2 − G2 . Since u ∈ U2R∗ (), 2

2

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by definition, we have θ ≤ θ(1/ − 1), i.e.  ≤ θ/(θ + θ) < 1/2. In addition, using our  R  θ R R R . Consequently, we assumption, we get S R GR (x) − F (x) dx ≤ F (x) − G (x) 2 2 2 2 θ+θ 2

can have EF (u) ≥ EG (u) ∀u ∈ U2R∗ ().

(2) Only if part: Here, we will show that if Z

S2R



R  R R GR 2 (x) − F2 (x) dx >  F2 (x) − G2 (x)

(A.4)

or µX ≥ µY , then there exists a u ∈ U2R∗ () such that EF (u) < EG (u). We first show the assertion when (A.4) holds. We assume S2R = [c, d] where a ≤ c ≤

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d ≤ b and define the following marginal utility function:    θ(x − a) if a ≤ x ≤ c   u0 (x) = θ(c − a) + θ(x − c) if c ≤ x ≤ d     θ(c − a) + θ(d − c) + θ(x − d) if d ≤ x ≤ b

.

(A.5)

from (A.4), we conclude that EF (u) < EG (u).

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Then, it is obvious that u ∈ U2R∗ () and  = θ/(θ + θ) < 1/2. Because u0 (a) = 0, from Rb Rd R R 00 R R (A.3), we obtain EF (u) − EG (u) = a (F2R − GR 2 )u dx = θ c (F2 − G2 )dx + θ S¯2R (F2 − R Rd R R R GR 2 )dx = (θ + θ) c (F2 − G2 )dx + θ F2 − G2 . In addition, since  = θ/(θ + θ) < 1/2,

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Next, we turn to considering the situation with µX ≥ µY . We define the following marginal utility function:

  c + η1 x if x ≤ x0 u0 (x) =  c − (η − η )x + η x if x ≤ x 2 1 0 2 0

(A.6)

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where x0 ∈ [a, b] and c, η1 , and η2 are positive constants such that c + η1 x0 > 0 and η2 > η1 . We can easily obtain the result that u ∈ U2R () with  = η1 /(η1 + η2 ).

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Rx Moreover, from (A.3), we obtain EF (u)−EG (u) = [c+η1 a](G2 (b)−F2 (b))+η1 a 0 (F2R − R Rb R R R )dx + η )dx ≤ [c + η a](µ − µ ) + η F − G GR (F − G 2 1 F G 2 2 2 2 . When we choose c 2 2 x0 η2 F R −GR such that c > −η1 a + µG2−µF 2 , we can get EF (u) < EG (u), and thus, the assertion of

Part 2 of Theorem 2.1 holds for n = 2. We turn to prove Part 2 with n = 3 of Theorem

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2.1 as follows:

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Rb 00 Again, from (A.3), we get EF (u)−EG (u) = u0 (a)(G2 (b)−F2 (b))+ a (F2R −GR 2 )u dx = Rb R 0 R R 00 R R u0 (a)(G2 (b)−F2 (b))+ a u00 d(GR 3 −F3 )) = u (a)(F2 (a)−G2 (a))+u (a)(F3 (a)−G3 (a))+ Rb R 000 R R R (F3 − GR 3 )u dx. Since u ∈ U3 () and Gn (a) ≤ Fn (a) for n = 2, 3, it is known a  R  that the first two terms are non-negative. Furthermore, if S R GR (x) − F3R (x) dx ≤ 3 3 R R  F3 (x) − G3 (x) , then using the same argument as in the proof of Theorem 2.1, we Rb 000 obtain a (F3R − GR 3 )u dx ≥ 0. Thus, the sufficiency part of the assertion holds. R  R  R R , one To prove the necessity part, if S R GR (x) − F (x) dx >  F (x) − G (x) 3 3 3 3 3

could easily construct a utility function u ∈ U3R∗ () and use an argument similar to that in the proof of Theorem 2.1 for n = 2 to show that EF (u) < EG (u). Utility u satisfies the 11

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following conditions: (1) u00 (x) is a piecewise linear function; and (2) u0 (a) = u00 (a) = 0. On the other hand, if F (2) (a) − G(2) (a) < 0, then, using an argument similar to that in the proof of Theorem 2.1 for n = 2, one could easily construct a utility function u ∈ U3R∗ () such that EF (u) < EG (u). The constructed utility function satisfies the

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following conditions: (1) u00 (x) is a piecewise linear function, (2) u00 (a) = 0, (3) u0 (a) is relatively large, and (4) for all x ∈ [a, b], u000 (x) is small enough. Thus, the case with F (3) (a) − G(3) (a) < 0 can be proved similarly and the assertion of Theorem 2.1 holds.

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Appendix B. Proof of Theorem 2.5: Rb Rb Rb Since H3R (x) = x H2R (y)dy = x (b − y − H2 (b) + H2 (y))dy = x (b − y)dy − H2 (b)(b − x) + H3 (b) − H3 (x), we have

F3R (x) − GR 3 (x) = F3 (b) − G3 (b) − F3 (x) + G3 (x) + (b − x)(G2 (b) − F2 (b)).

(B.1)

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In addition, if Fn (b) = Gn (b) for n = 2, 3, then we obtain F3R (x) − GR 3 (x) = G3 (x) − F3 (x) and, furthermore, we get S3 = {x ∈ [a, b] : G3 (x) < F3 (x)} = {x ∈ [a, b] : F3R (x) <

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R GR 3 (x)} = S3 . Conducting simple computations, the assertion of Theorem 2.5 holds.

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Appendix C. Proof of Theorem 2.6:

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We only prove Part (2) of Theorem 2.6. First, we note that R Hj+1 (x)

1 = j!

Z

b

x

(t − x)j dH(t).

(C.1)

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If j = 0, equation (C.1) holds because 0! = 1. Assuming that equation (C.1) holds for j, we now show that it is true for j + 1. To do so, we first apply Fubini’s theorem   R b R b R b R t 1 1 R j−1 j−1 to obtain Hj+1 (x) = (j−1)! x η (t − η) dH(t) dη = (j−1)! x x (t − η) dη dH(t). Evaluating the integral with respect to η, we show that equation (C.1) holds for j + 1. Z

1 R   R Fj (x) − GR Gj (x) − FjR (x) dx ≤  FjR (x) − GR j (x) < j (x) 2 SjR Z Z  R   R  1 1 R ⇐⇒ Gj (x) − Fj (x) dx < Fj (x) − GR j (x) dx 2 SjR 2 S¯jR Z b  R  R ⇐⇒ Gj (x) − FjR (x) dx < 0 ⇐⇒ GR j+1 (a) < Fj+1 (a). a

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Thus, we can obtain that       j j X X j j R   (−a)j−i miX . (C.2)   X i (−a)j−i  = j!Fj+1 (a) = E(X − a)j = E  i i i=0 i=0

R For -AF SDR , from the above equation and the fact that GR 2 (a) < F2 (a), one can easily

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obtain that m1X > m1Y . Now, we turn to considering the general case with j ≥ 2. Assume that k ≤ j is the smallest value with mkX 6= mkY . Then, we can conclude that mlX = mlY R R R for l = 1, · · · , k − 1. For -AjSDR , we obtain GR j+1 (a) < Fj+1 (a) and Gm (a) ≤ Fm (a) for R m = 2, · · · , j. Specifically, we get GR k+1 (a) ≤ Fk+1 (a). According to our assumption, we

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then have k!(F (k+1) (a) − G(k+1) (a)) = mkX − mkY . As a result, we can get that mkX > mkY . 

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