Losses Loom Larger Than Gains and Reference

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Aug 11, 2018 - In a nutshell, Bernoulli's utility function is alive and well. ..... The “q” shown in the original sketch has a squiggle .... a continuation of the concave curve over BP (relative gains). ..... wealth must be less than the reference wealth level for loss aversion to be up- held. Beyond that point, the formula breaks down.

Losses Loom Larger Than Gains and Reference Dependent Preferences in Bernoulli’s Utility Function∗ G. Charles-Cadogan † August 11, 2018 Abstract Some analysts claim that Bernoulli’s utility function is “reference-independent”, so it is not able to generate a loss aversion index, and that the theoretical framework of Prospect Theory (PT) is required to achieve those results. This paper examines that claim and finds that the geometry of Bernoulli’s original utility function specification either explains or implies key elements of PT: reference dependence and a loss aversion index. Theory and evidence show that the loss aversion index constructed from reference wealth in Bernoulli’s utility specification is in the domain of attraction of a stable law. That is, its distribution is a slow varying function with a fat tail that decays like a power law. Additionally, the index can be tested with a modified Fisher z-transform test. Bernoulli‘s utility function also sheds light on why loss aversion may be over-estimated under PT. In a nutshell, Bernoulli’s utility function is alive and well. Keywords: prospect theory, Bernoulli utility, loss aversion, Fisher z-transform test, stable distribution, fat tail JEL Classification Codes: C0, D03, D81

∗ I thank Ivan Moscati for bringing my attention to pertinent literature, Sure Mataramvura, Haim Abraham, Ramaele Moshoshoe, Mark Taylor, Mario Siniscalchi, Gana Pogrebna, Pablo Bra˜nas Garza, Glenn Harrison and Don Ross for their incisive comments and suggestions on earlier drafts. I thank Peter Wakker, Elad Yechiam and participants at Foundations for Utility and Risk Conference 2018 in York for their comments. I am grateful to the Editor (Dan Houser), and two anonymous referees for many helpful comments which greatly improved the paper. I thank RUBEN at University of Cape Town, and IITM at Ryerson University for their financial support. The usual disclaimer applies. This paper is based on Chapter 5 of the author’s PhD dissertation. It replaces earlier drafts circulated under the title “Prospect Theory’s Cognitive Error About Bernoulli’s Utility Function”. † University of Leicester, School of Business, Division of Finance, School of Business, Leicester, LE1 7RH; Institute for Innovation and Technology Management (IITM), Ted Rogers School of Management, Ryerson University, 575 Bay, Toronto, ON M5G 2C5; Tel: +44 (0116) 229 7385; e-mail: [email protected]

Contents 1 Introduction

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2 Prospect theory value function vs Bernoulli utility function 2.1 The reference point in Bernoulli’s utility function . . . . . . . . . . . . . . . . 2.2 Bernoulli’s forgotten reference wealth level and geometric mean return . . . . . 2.3 Bernoulli utility function vs. Kahneman-Tversky skew S-shape value function . 2.3.1 Case (i): Deviations from the reference point . . . . . . . . . . . . . . 2.3.2 Case(ii): Generally concave for gains and commonly convex for losses 2.3.3 Case(iii): Steeper for losses than for gains . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

3 Is there a kink at the reference point?

5 6 6 8 9 9 10 10

4 A loss aversion index formula implied by Bernoulli’s utility function 12 4.1 The loss aversion index, its conjugate, and Fisher’s z-transform . . . . . . . . . . . 14 4.1.1 Example–Fisher z-transform test for utility loss aversion index . . . . . . . 17 4.2 Theory and evidence of α -stable loss aversion index . . . . . . . . . . . . . . . . . 20 5 Conclusion

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6 Appendix

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A Proof of Z-test for Fisher z-transform test for large sample loss aversion index Theo24 rem 4.3 B Proof of Z-test Fisher z-transform test for large sample gain seeking index Corollary 1 25 C Proof of t-test for Fisher z-transform for small sample loss aversion index Theorem 4.4 25 D Proof of t-test for Fisher z-transform for small sample gain seeking index Corollary 2 26 E Gain seeking in Bernoulli’s (1738) implicit conjugate utility function

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References

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List of Figures 1 2 3 4 5 6 7

Reproduction of utility of wealth function sketched in Bernoulli (1738, p. 26). Fishburn-Kochenberger utility with reference point . . . . . . . . . . . . . . Bernoulli Utility with reference wealth . . . . . . . . . . . . . . . . . . . . . Distribution of Fisher z-transform . . . . . . . . . . . . . . . . . . . . . . . Distribution of Loss Aversion Index for Bernoulli Utility . . . . . . . . . . . Fitted Log Pearson 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernoulli’s (1738) Implied Conjugate Utility Function . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

5 11 11 19 19 21 27

List of Tables 1 2

Sample Distribution of Loss Aversion Index for Bernoulli Utility . . . . . . . . . . 17 Diagnostics for Fishurn-Kochenberger loss aversion index data . . . . . . . . . . . 21

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“What has been will be again, what has been done will be done again; there is nothing new under the sun.” Ecclesiastes 1:9

1 Introduction A recent survey by Barberis (2013, p. 173) describes Kahneman and Tversky (1979) original version of prospect theory (OPT), and its amendment, cumulative prospect theory (CPT) (Tversky and Kahneman, 1992) thusly. “Prospect theory is still widely viewed as the best available description of how people evaluate risk in experimental settings”,1 while duly noting that “there are relatively few well-known and broadly accepted applications of prospect theory in economics”. This paper compares Bernoulli’s (1738) model to the Kahneman and Tversky (1979); Tversky and Kahneman (1992) models of prospect theory to see whether concepts like reference point and loss aversion index are explained or implied by Bernoulli’s original utility function. In contrast to Barberis, papers by Birnbaum and Navarrete (1998); Birnbaum (2005, 2008) point out that CPT is unable to explain why certain gambles show systematic violations of stochastic dominance2 –while certain types of rank dependent models can, e.g., configural weighting and transfer exchange (TAX) of weights by Birnbaum and Chavez (1997). Baltussen et al. (2006) found that CPT was unable to explain decision makers choices in mixed gambles with moderate probabilities where stochastic dominance was also violated. Schmidt and Zank (2008) introduced “third generation” CPT to address, inter alia, issues related to the uncertain reference point deficit in CPT. However, Birnbaum’s (2018) staunch criticism of CPT remains unabated as he indicates that configural weigting models still outperform third generation CPT. Meanwhile, theorists like Gilboa and Marinacci (2013) are more circumspect. They believe that no single theory has emerged to replace Expected Utility Theory (EUT). Indeed, experiments conducted by Hey and Orme (1994) found that EUT was upheld as a valid model of decision making for many subjects.3 List (2004) reviewed several experiments where inexperien1 In contrast, Moscati (2016) references Gilboa and Marinacci (2013, p. 232) to note that “it is not clear that a single theory of decision making under uncertainty will replace expected utility theory, and “even if a single paradigm will eventually emerge, it is probably too soon to tell which one it will be.”” 2 If A and B are gambles, and for some outcome x ∈ A and x ∈ B, and threshold x⋆ , Pr{x > x⋆ |A} > Pr{x > x⋆ |B}, then A stochastically dominates B. 3 Other models in the EUT class like regret theory (Loomes and Sugden, 1982; Bell, 1982), disappointment aversion (Gul, 1991), and the more recent weakly separable rank dependent utility theory (Charles-Cadogan, 2016) explain, inter alia, phenomena like

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ced subjects tended to behave according to prospect theory, but more experienced subjects tended to behave in accordance with EUT. Papers by Harrison and Rutstr¨om (2008a); Bruhin et al. (2010) also find a mixture of CPT and EUT types in their experiments. Brandst¨atter et al. (2006) posit that many results attained by the “weighting and summing” feature of EUT and CPT can be obtained by a priority heuristics. However, Gl¨ockner and Betsch (2008) provide evidence in favour of CPT against the priority heuristic. In a critical review paper, Gal and Rucker (2018) noted that Daniel Kahneman stated that loss aversion is perhaps the most useful contribution of prospect theory to behavioral decision theory (Kahneman, 2002). They argue that “most writings on loss aversion appear to accept the assumption that losses do loom larger than gains and deviations from this are aberrations and violations of the norm that do not challenge the basic principle”. However, they find that loss aversion is less robust and universal than assumed, and call for “critical reevaluation of prevailing paradigms” of loss aversion. Higgins and Liberman (2018) and Simonson and Kivetz (2018) were more guarded in their response but by and large concurred with Gal and Rucker’s (2018) challenge to critically reevaluate loss aversion as a behavioural phenomenon. Ert and Erev (2013) question the efficacy of loss aversion as a behavioural phenomenon and highlighted six experimental design patterns that increase the likelihood of loss aversion in subjects. Yechiam (2018) also provides a succinct review of the literature and notes that proponents of loss aversion tend to over-interpret its results. Wu and Markle (2008) found that the strong gain loss separability assumption, often used to facilitate estimation of CPT’s subutility functions as well as a loss aversion index, was not supported by their experiments. Por and Budescu (2013) established the robustness and persistence of strong gain-loss separability violations across multiple elicitation methods. They speculate that decision makers unobserved reference point may play a role in this violation. Charles-Cadogan (2016) introduced a weakly separable rank dependent utility model with reference dependent loss aversion that militates against the strong gain-loss hypothesis. In this paper we show how Bernoulli’s original utility function specification deals with gain-loss separability, and help shed light on why loss aversion may be overestimated. preference reversal, Allais paradox, and loss aversion that are in the solution space of prospect theory.

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To the extent that Bernoulli (1738) utility theory laid a foundation for Von Neumann and Morgenstern (1953) axiomatic EUT, it is interesting to know which elements of prospect theory are explained or implied by Bernoulli’s utility function. This paper undertakes that task. A key motivation for this approach is the relative simplicity of EUT versus the complexity of CPT. Whereas EUT requires a “regular” utility function, and objective probabilities, CPT involves two key transformations. First, a transformation for outcomes that require two different subutility functions called value functions: one over gains, and the other over losses relative to a reference point. Second, a transformation of objective probabilities to probability weighting functions. In the experimental literature this is often accompanied by choice functions to model the stochastic choice of subjects in an experiment. For instance, Stott (2006, p. 102) examined 256 model variants for implementing CPT based on a combination of different specifications for value functions, probability weighting functions, and choice functions popularized in the literature. Prospect theory was proposed in response to purported anomalies from experiments in psychology and behavioral economics which led to revisions of Von Neumann and Morgenstern (1953) expected utility theory (EUT) model, and utility theory more generally. However, this paper shows that Bernoulli’s (1738) original utility function specification, which falls under rubric of EUT, explicitly or implicitly satisfies several innovations attributed to prospect theory’s construct: (1) gain loss asymmetry, i.e., “losses loom larger than gains”,4 (2) reference dependent preferences,5 (3) utility valuation over changes in wealth; and (4) support for a loss aversion index. We also show that under mild assumptions Bernoulli’s utility function accommodates a Fisher ztransformation test for the loss aversion index. Moreover, the index follows an α -stable law. These findings are important in their own right. These phenomena seem to have escaped the attention of analysts in debates about the efficacy of EUT, CPT and other nonexpected utility theories. Of course our findings do not mean that EUT should replace CPT. They simply mean that aspects of behavioural decision theory, hitherto attributed solely to CPT’s theoretical construct, are also implied or explained by EUT construct upon further examination. It should be noted that Bernoulli 4

(See e.g., Kahneman and Tversky, 1979, p. 279). and Rabin (2006, p. 1134) “build on the essential intuitions in Kahneman and Tverskys [1979] prospect theory and subsequent models of reference dependence” but failed to acknowledge reference dependence in Bernoulli’s utility function specification. 5 K˝ oszegi

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model is silent on probability weighting functions and risk seeking behavior. So it does not explain very important probabilistic risk attitudes: something that CPT does well, see e.g., Camerer (2005, p. 130) and Booij et al. (2010). This paper was initially stimulated by Daniel Kahneman Nobel Prize lecture, a significant part of which is devoted to what he deemed “Bernoulli’s error”. He states in relevant part: Perception is reference-dependent: the perceived attributes of a focal stimulus reflect the contrast between that stimulus and a context of prior and concurrent stimuli. ********* [Amos Tversky and I] noted, however, that reference-dependence is incompatible with the standard interpretation of Expected Utility Theory, the prevailing theoretical model in this area. This deficiency can be traced to the brilliant essay that introduced the first version of expected utility theory (Bernoulli, 1738). One of Bernoulli’s aims was to formalize the intuition that it makes sense for the poor to buy insurance and for the rich to sell it. He argued that the increment of utility associated with an increment of wealth is inversely proportional to initial wealth, and from this plausible psychological assumption he derived that the utility function for wealth is logarithmic. He then proposed that a sensible decision rule for choices that involve risk is to maximize the expected utility of wealth (the moral expectation). This proposition accomplished what Bernoulli had set out to do: it explained risk aversion, as well as the different risk attitudes of the rich and of the poor. The theory of expected utility that he introduced is still the dominant model of risky choice. The language of Bernoulli’s essay is prescriptive it speaks of what is sensible or reasonable to do but the theory is also intended to describe the choices of reasonable men (Gigerenzer et al., 1989). As in most modern treatments of decision making, there is no acknowledgment of any tension between prescription and description in Bernoulli’s essay. The idea that decision makers evaluate outcomes by the utility of final asset positions has been retained in economic analyses for almost 300 years. This is rather remarkable, because the idea is easily shown to be wrong; I call it Bernoulli’s error. Bernoulli’s model is flawed because it is reference-independent: it assumes that the value that is assigned to a given state of wealth does not vary with the decision makers initial state of wealth[Footnote in original][What varies with wealth in Bernoulli’s theory is the response to a given change of wealth. This variation is represented by the curvature of the utility function for wealth. Such a function cannot be drawn if the utility of wealth is reference-dependent, because utility then depends not only on current wealth but also on the reference level of wealth.]. This assumption flies against a basic principle of perception, where the effective stimulus is not the new level of stimulation, but the difference between it and the existing adaptation level. The analogy to perception suggests that the carriers of utility are likely to be gains and losses rather than states of wealth, and this suggestion is amply supported by the evidence of both experimental and observational studies of choice (see Kahneman & Tversky, 2000). [Emphasis added]. (Kahneman, 2002, pp. 460-461). 4

Accordingly, this paper provides a critical review of the Bernoulli (1738) model, and compares it to the claims made against it in the Kahneman lecture excerpted above.6 In section 2 we compare the geometry of Bernoulli’s utility function to that of Kahneman-Tversky skew S-shape value function. And we show how loss aversion is implied by Bernoulli’s specification. In Section 3 we compare and contrast the kink at prospect theory’s reference point with the smoothness at the reference point in Bernoulli’s utility function. In section 4 we show how Bernoulli’s specification supports a closed form loss aversion index, that the index is α -stable, and we characterize its relation to Fisher’s z-transformation test. In Section 4.1.1 we provide several heuristic examples of large and small sample Fisher’s z-transformation tests. We conclude in section 5.

2 Prospect theory value function vs Bernoulli utility function In this section we emphasize the geometric properties of Bernoulli (1738) utility function, identify its reference point, and contrasts it to the qualitative and geometric properties of Kahneman and Tversky (1979) value function. Figure 1: Reproduction of utility of wealth function sketched in Bernoulli (1738, p. 26).

6 It should be noted that the descriptive validity of Bernoulli’s theory was questioned at least three decades before Daniel Kahneman’s lecture (see e.g. Samuelson, 1952).

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2.1 The reference point in Bernoulli’s utility function We begin with Bernoulli (1738, pg. 26) description of the geometry of his utility of wealth function reproduced in Figure 1: “[L]et AB represent the quantity of goods initially possessed. Then after extending AB, a curve BGLS must be constructed, whose ordinates CG, DH, EL, FM, etc., designate utilities corresponding to the abscissas BC, BD, BE, BF, etc., designating gains in wealth. Further, let m, n, p, q, etc., be the numbers which indicate the number of ways in which gains in wealth BC, BD, BE, BF, etc., can occur”. [Emphasis added]7 Undeniably, the point B in Figure 1 is Daniel Bernoulli’s reference point. Furthermore, Bernoulli (1738, pg. 29) states: First, it appears that in many games, even those that are absolutely fair, both of the players may expect to suffer a loss; indeed this is Nature’s admonition to avoid the dice altogether . . . This follows from the concavity of curve sBS to BR. For in making the stake, Bp, equal to the expected gain, BP, it is clear that the disutility po which results from a loss will always exceed the expected gain in utility, PO. [Emphasis added] It is indisputable that the italicized text emphasized in Bernoulli’s analysis above involves gains and losses relative to the reference point B. Furthermore, he compared “utility” of expected gain BP to the “disutility” of a loss of an equal amount Bp, and plainly concludes that “loss will always exceed the expected gain in utility”. In other words, Bernoulli noticed that “losses loom larger than gains” (Kahneman and Tversky, 1979, p. 279) in his utility function specification. Nonetheless, Kahneman and Tversky (1979, pg. 276) state: “[Markowitz (1952)] was the first to propose that utility be defined on gains and losses rather than on final asset positions, an assumption which has been implicitly accepted in most experimental measurements of utility”. 2.2 Bernoulli’s forgotten reference wealth level and geometric mean return Given Bernoulli’s log-concave specification, the reference wealth level can only cut the horizontal axis at x = 1. In other words, Bernoulli normalized wealth levels so that a given wealth level Wx , say, is numeraire–the reference wealth. In which case, any other wealth level, say Wz , is represented 7A

referee pointed out that points “m” and “q” do not appear on the sketch. The “q” shown in the original sketch has a squiggle so it appears to be different from the q which refers to the number of ways in which gains in wealth can occur. Similarly, “m is not shown in Bernoulli’s sketch because it is a number.

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by

Wz Wx .

Thus, the points in his graph are changes in wealth relative to the reference wealth. This

fact may have been obscured by his use of “analytic geometry” as opposed to “algebraic geometry” to represent the geometric mean. Stigler (1950, pg. 374, fn. 118) also analyzed Bernoulli’s utility function by introducing the notion of a “subsistence level at c” where for some “constant” (call it a) we have U (c) = k ln(c) + a = 0 ⇒ a = −k ln(c) ⇒ U (x) = k ln

x c

(2.1)

Using Stigler’s interpretation, U (c) = 0 at precisely where “subsistence wealth level” x = c and relative wealth

x c

= 1.8 More on point, Bernoulli (1738, pg. 28) writes:

b log

AC AE AF + nb log AD AP mb log AB AB + pb log AB + qb log AB + . . . = AB m + n + p + q + ...

(2.2)

That equation can be rewritten as AP = AB

"

AC AB

!m

AD AB

!n

AE AB

!p

AF AB

!q

...

# m + n + p1+ q + ...

(2.3)

which is a weighted geometric mean relative to the reference wealth level AB (see e.g., Stearns, 2000, p. 221)(“Bernoulli taught us how to measure risk with the geometric mean”). Since AP < AB in Figure 1,

AP AB

< 1 if and only if at least one or all of the fractions on the right hand side in (2.3)

is smaller than 1. Let {WP ,WB ,WC ,WD ,WE ,WF , . . .} be a ranking of nominal wealth where the subscripts coincide with the corresponding letters in Bernoulli’s model in Figure 1. Thus, we have the strict partial preference order WP ≺ WB ≺ WC ≺ . . .. Choose WB as reference wealth so that relative wealth has the following correspondence: WB WC WD WP ∼ AP, = 1 ∼ AB, ∼ AC, ∼ AD, WB WB WB WB WF WE ∼ AE, ∼ AF, . . . WB WB 8 There

(2.4)

is evidence that those at “subsistence levels” of income are more prone to purchasing lottery tickets (Friedman and Savage, 1948; Light, 1977; Beckert and Lutter, 2013; Scott and Barr, 2012).

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Let N = m + n + p + q + . . . and

WC WB

D = (1 + rC ), W WB = (1 + rD ) and so on. We rewrite (2.3) as

h i p q m n 1 + rP = (1 + rC ) N (1 + rD ) N (1 + rE ) N (1 + rF ) N . . . where

m n N, N,...

(2.5)

are relative frequencies. For relatively small r j we get the following law of large

numbers for rP = exp



ln(1 + r j )

j∈{C,D,E,F,...} k∈{m,n,p,q,...}

rN =

1 N

1 ∑ k(r j − 2 r2j ), j∈{C,D,E,F,...}

k N

!

1 − 1 = exp N



j∈{C,D,E,F,...} k∈{m,n,p,q,...}

rP ≈ lim rN = r⋆ = µr − N→∞

!

ln(1 + r j )k − 1

σr2 WB we

have Wx /WB = 1 + ∆Wx /WB = 1 + x where 0 < x < 1. For Wx < WB we have Wx /WB = 1 − x. Thus, Bernoulli’s specification for relative wealth is of type

u(x) =

  ug (x) = ln(1 + x) > 0 for gain  u (x) = ln(1 − x) < 0 ℓ

(2.8)

for loss

In the sequel ug (x) = ln(1 + x) for concave part of function over gains, and uℓ (x) = ln(1 − x) for concave part of function over losses. Note that for 0 < x < 1, uℓ (x) = ln(1 − x) < 0 in loss domain. Analytically, u′g (x) = (1 + x)−1 > 0 and u′′g (x) = −(1 + x)−2 < 0 implies ug (x) is concave for 0 < x < 1 since it is growing at a decreasing rate. Similarly, u′ℓ (x) = (1 − x)−1 > 0 for uℓ (x) < 0,

and u′′ℓ (x) = (1 − x)−2 > 0 implies uℓ (x) is growing at an increasing rate in the loss quadrant.

Bernoulli’s utility function does not capture diminishing sensitivity over losses (i.e., risk seeking behaviour) because u′′ℓ (x) = (1 − x)−2 > 0, whereas the CPT value function specification does. So it is concave over gains and losses. In contrast, CPT’s data driven value function is commonly concave for gains and commonly convex over losses.

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2.3.3

Case(iii): Steeper for losses than for gains

From (2.8) the slope of uℓ (x) over losses is u′ℓ (x) = (1 − x)−1 > 0 for uℓ (x) < 0 and the slope for ug (x) over gains is u′g (x) = (1 + x)−1 > 0. However, for 0 < x < 1 we have u′ℓ (x) = (1 − x)−1 > u′g (x) = (1 + x)−1 . So the slope of uℓ (x) over losses is steeper than the slope of ug (x) over gains.

3 Is there a kink at the reference point? With respect to CPT’s bifurcated value function, Tversky and Kahneman (1992, p. 309) proffered the following specification:

v(x) =

  xα ,

x>0

 −λ (−x)β ,

(3.1) x 0.

For gains (losses), the power function is concave if α < 1 (β > 1), linear if α = 1 (β = 1), and convex if α > 1 (β < 1). For the power family the loss aversion coefficient λ is defined as U (−1) − ”. The convex piece of Figure 2 (or Bs′ over BA in Figure 3) is shown in the sketch. U (1) However, there is no kink in Bernoulli’s concave-concave specification, and utility over loss is not as pronounced for Bs as it is for Bs′ in the convex-concave specification popularized by Tversky

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and Kahneman since VℓB < VℓKT .10 More on point, Kahneman and Tversky (1979) referenced Fishburn and Kochenberger (1979) metastudy to support their reference point and strong gain-loss separability hypotheses. However, in order to arrive at a reference point, Fishburn and Kochenberger transformed very noisy data they collected from eyeballing published graphs in some cases. And their reference Figure 3: Bernoulli Utility with reference wealth

Figure 2: Fishburn-Kochenberger utility with reference point

Q

Q’ S

L A

ܸ௚

L’ B

ll

ܸκ஻

ܸκ௄்

s’

s

Figure 2 is a reproduction of Fishburn and Kochenberger (1979, Fig. 1, p. 504) juxtaposed to Figure 3 for convenience and comparison. t is a reference point called “target utility” in the Fishburn and Kochenberger (1979) metastudy based on “changes in wealth or return on investment”. They fitted separate utility functions in loss and gain domains using linear, power and exponential specifications. Most of their fits were “convex-concave” type in Figure 2 (see also Bs and Bs′ in Figure 3) and they concluded that the convex portion was steeper than the concave portion. But some of their fits were of the “concave-concave” type depicted in Figure 2. However, they noticed that for both “concave-concave” and “convex-concave” the “below-target utility ... is almost always steeper than above-target utility”. This is consistent with Bs being steeper than BS in Bernoulli’s specification.

point(s) or “target utility” was obtained by transforming data in many instances (see e.g., Fishburn and Kochenberger, 1979, Table 1B). Additionally, they fitted the power utility specification a1 xa2 based on strong gain-loss separability assumptions, and patched the value function over gain to the value function over loss to get the combination of shapes in Figure 2. From a geometry perspective, 10 Bernoulli’s

(1738) utility function in (2.8) can be modified with a so called power log-utility u(η |γ ) =

1 losses (Kale, 2006) so that VℓB > VℓKT and limγ →0 (1 − η )γ = − ln(1 − η ) based on L’Hospital’s rule. γ

11

1 (1 − η )γ , γ < 0 over γ

the convex segment of the curve over loss domain is equivalent to rotating the concave portion of the curves over loss domain in Figure 2 and Figure 3.11 Fishburn and Kochenberger (1979, p. 510) found that 46% (13/28) of their fitted curves were convex-concave and only 10.7% (3/28) were concave-concave. Tversky and Kahneman (1992) found that the median value for α and β was 0.88. So 50% (the median shape) for CPT’s value function was convex-concave. Abdellaoui et al. (2008) found that 40% (19/47) of the curves in their study were concave-concave, and 30% (14/47) were convex-concave. So the iconic convex-concave value function attributed to CPT is a common but not universal shape for decision making under risk and uncertainty over gain and loss domains. A concave-concave fit supports Bernoulli’s EUT specification and it has no kink as shown in Figure 3.12 Research by Harrison and Rutstr¨om (2008a) and Bruhin et al. (2010) find that preferences reflect a mixture of EUT and CPT. This writer is unaware of any behavioral dynamical system theory that predicts bifurcation or kink of the value function at the reference point in decision making under risk and uncertainty. The kink at the reference point is a manifestation of patching subutility functions derived from CPT’s strong gain-loss separability assumption. However, that assumption is rejected by the Wu and Markle (2008) study designed to test it. Also, Charles-Cadogan’s (2016) weakly separable rank dependent utility model produces a reference dependent loss aversion index with no kink at the reference point. So there is theoretical and empirical evidence against strong gain-loss separability and a kink at the reference point.

4 A loss aversion index formula implied by Bernoulli’s utility function In this section, we use the geometry of Bernoulli’s utility function to derive a utility loss aversion index and establish its relation to Fisher’s z-transform statistic.13 Cursory inspection of Figure 3 h i cos(θ ) − sin(θ ) b x = R(θ )xx where R(θ ) = sin(θ ) cos(θ ) is a rotation matrix and x = [ xx12 ]. When θ = π we generate a 180◦ rotation  1 operation so that b x = R(π )xx = −x −x2 . 12 This is verified analytically by using a ratio of slopes approach favored by K¨ obberling and Wakker (2005) to get λ B (0) =   u′ℓ (η ) 1+η limη →0 ′ = 1 at the reference point. = limη →0 ug (η ) 1−η 13 Cohen (2014, Fig. 1, p. 9) introduced a “state dependent loss aversion Bernoulli utility” function that is kinked at the reference point. However, he used a K˝oszegi and Rabin (2006) type specification to characterize his loss aversion index. 11 Let

12

shows that the value VℓKT for Kahneman and Tversky’s skew is such that VℓKT > VℓB > Vg ⇒

VB VℓKT > ℓ >1 Vg Vg

(4.1)

Thus, the “disutility” of loss in either case is such that it is greater than the “utility” of an equal nominal gain. However, the impact of loss in Tversky and Kahneman CPT is greater than the impact of loss in Bernoulli’s (1738) utility model. Let AV KT , AVgKT , AV B , AVgB be the areas under ℓ



the gain-loss sub-utility functions for impact of an incremental change in wealth η under the Kahneman-Tversky (KT) and Bernoulli (B) value functions. Thus, from (2.8) and (3.1), for a symmetric deviation η from the “reference point” we get the impacts: AV KT ℓ

AVgKT AV B ℓ

AVgB

Z 0

n (−η )β +1 o (−λ (−x) )dx = −λ , η >0 = β +1 −η Z η η α +1 = xα dx = α +1 0 Z x dy = ln x − ln(x − η ) = x−η y Z x+η dy = ln(x + η ) − ln x = y x β

(4.2) (4.3) (4.4) (4.5)

The loss aversion index λ KT , derived from the impact of a percentage change η in wealth in (4.2) and (4.3); and λ B derived from the same change in (4.4) and (4.5) are given by

λ KT (η ) = λ B (x, η ) =

|−λ

(−η )β +1 β +1 | η α +1 α +1



ln x − ln(x − η ) ln(x + η ) − ln x

α + 1 β −α |η | β +1

(4.6) (4.7)

Our ratio-of-areas approach to deriving the loss aversion index differs from that in the literature which favors first derivatives and ratio of utilities (Wakker, 2010, p. 239). The formula in (4.6) takes the power utility curvature parameters α and β , and distance from the reference point η into account. Notice that when α = β in (4.6) the formula for λ KT collapses to λ .14 So λ KT (η ) is an admissible measure of loss aversion index. Harrison and Rutstr¨om (2008a, Fig. 14, p. 97) (2010, pp. 268, 271) addresses the effect on local utility loss aversion index measures when α 6= β by proposing a rescaling of the input for the utility function, or alternate specification to avoid the problem. 14 Wakker

13

impose the restriction α = β in their structural CPT model estimation of reference dependent λ (s) over a range of assumed reference points s. Even though they did not specify an explicit formula for λ (s) in Harrison and Rutstr¨om (2008a), they referenced a working paper version of Harrison and Rutstr¨om (2008b) where a reference dependent utility specification of type U (s, x) = (s + x)r was used. There, s is an endowment or reference point and x is the payoff from a lottery in their experiment. The foregoing arguments support the following Theorem 4.1 (Reference dependent loss aversion index implied by Bernoulli original utility function). The loss aversion index λ B (x, η ) in (4.7) computed from Bernoulli’s value function is reference dependent. In particular, λ B (x, η ) is a loss aversion index over the distribution of change in reference wealth x and relative change in wealth η . 4.1 The loss aversion index, its conjugate, and Fisher’s z-transform Without loss of generality, in the sequel we replace reference wealth x in (4.4) and (4.5) with x = 1 and the corresponding relative gain and loss amount with η .15 Given the relationships in (4.1), (4.6) and (4.7) we have reference point driven loss aversion index relationships:

λ KT > λ B > 1 ⇒ λ B =

ln 1 − ln(1 − η ) − ln(1 − η ) = >1 ln(1 + η ) − ln 1 ln(1 + η )

(4.8)

⇒ ln(1 − η 2 ) < 0 ⇒ |η | < 1

(4.9)

Since η > 0, the operational inequality is 0 < η < 1. That inequality implies that the absolute nominal change in wealth must be less than the reference wealth level for loss aversion to be upheld. Beyond that point, the formula breaks down. There is no bound on Tversky and Kahneman’s (1992) loss aversion index λ KT (η ) in (4.6) for changes in wealth relative to the reference point. In contrast, the loss aversion index λ B (η ) in (4.8) is reference dependent and responsive to all changes in wealth less than the reference wealth level. Thus, we proved Theorem 4.2 (A Loss Aversion Index Formula implied by Bernoulli’s (1738) utility function). A loss aversion index formula for a loss η (expressed as a percent change in wealth relative to a 15 In the context of Harrison and Rutstr¨ om (2008a) utility setup U(s, x) = (s + x)r so 0 ≤ η ≤ 1.

14

s+x s

= 1 + xs . In our case, η =

x s

such that

reference wealth level), when utility is log concave, is given by

λ B (η ) = − where 0 < η < 1,

ln(1 − η ) ln(1 + η )

0 ≤ λ B ≤ ∞.



The literature shows that K˝oszegi and Rabin (2006); K˝osegi and Rabin (2007) formulated stylized models of reference dependent preferences, based on concave utility functions, but failed to proffer a closed form loss aversion index formula. We posit that the loss aversion index is robust to criticism against the application of log concave utility to all wealth levels,16 because a log concave utility function is a special case of the abstract concave utility in K˝oszegi and Rabin (2006). Perhaps more important, assuming bivariate normality between wealth levels, Theorem 4.2 suggests that λ B (η ) is related to Fisher’s z-transformation17 if we treat (η ) as a pseudo correlation coefficient, i.e., |η |≤ 1.18 That is, it transforms the (truncated) interval, normalized by “reference

wealth”, from [0, 1] for η to [0, ∞], so that 0 ≤ λ B ≤ ∞. We summarize this observation in the following19

Theorem 4.3 (Large sample Fisher z-transform test for loss aversion index). Assume that W1 , . . .,Wn are rank ordered independent identically distributed wealth levels for n > 3. Let Wr be a reference wealth level 1 < r < n. Assume that W j ,Wr are iid bivariate lognormal and that Wj Wr

b 2j − 1 be a random variable, z be Fisher’s z-transform, and b j , j 6= r. Let η e = 1n ∑nj=1 η = 1+η

e |< 1 be a given truncated symmetric gain and loss relative to Wr . It is known that if E[η e ] = 0, |η

e is such that z is normally distributed with then the population parameter η with sample estimate η       1+η 1+η 1 1 −1 −1 (Anderson, 2003, mean 2 ln 1−η and variance (n − 3) , i.e., z ∼ N 2 ln 1−η , (n − 3) 16 Technically,

one could assume that Bernoulii’s utility function u is “least concave” by separating the preferences u of a DM from her attitude f toward risk so that we get a concave utility function representation U(x) = ( f ◦ u)(x). See e.g., Debreu (1976); Kannai (1977).   17 Fisher’s z-transformation is a variance stabilizing transformation for correlation |r|< 1. It is z = 1 ln 1 + r = tanh−1 (r) 2 1−r √ where z ∼ ln F where F is the F-distribution. Refer to Cram´er (1962, p. 241) for details on derivation. 18 Tversky and Kahneman (1992, Table 5) (explained on page 307 of the paper) provides means of the correlations between high and low probability gains and losses after transformation to Fisher’s z statistic. 19 This is motivated by ongoing research on direct statistical tests for the loss aversion index set forth in Charles-Cadogan (2018a), Charles-Cadogan (2018b).

15

p. 134). Then from Theorem 4.2 1 b z = (1 + b λ B ) ln(1 + η ) 2

λ B and η with test statistic where b z is the sample Fisher z-transform for sample estimates b Zb =

!  1 + η  √n − 3 λ B ) ln(1 + η ) − ln ∼ N(0, 1) (1 + b 1−η 2

Proof. See Appendix A. Remark 4.1. Berry and Mielke Jr (2000) show that the Fisher z-transform test would be valid here only when E[η ] = 0. That condition is satisfied here a fortiori. Corollary 1 (Large sample Fisher z-transform test for gain seeking index). The large sample test B statistic for gain seeking index estimate b λ ∗ is:

"

B λ∗ ) 1 − 12 ln (b Zb = ln B 1 + 1 ln (b λ∗ )

2

!

− ln

#  1 + η  √n − 3 1−η

2

∼ N(0, 1),

B e−2 < b λ∗ < 1

(4.10)

Proof. See Appendix B.

Theorem 4.4 (Small sample t-test for Fisher z-transform for loss aversion index). For small samples (n ≤ 20) the t-test statistic with n − 2 degrees of freedom for Fisher z-transform for loss aversion index is given by: T =η

"

n−2

2

b B)

(1 + η )(1−λ

Proof. See Appendix C.

#1

(4.11)

Remark 4.2. Fouladi and Steiger (2008) provides details on modifications of Fisher’s z-transform for small sample tests. Corollary 2 (Small sample t-test for Fisher z-transform for gain seeking index). For small samples (n ≤ 20) the t-test statistic with n − 2 degress of freedom for Fisher z-transform for gain seeking

16

index is given by: 

B  1 T = − ln (b λ∗ ) 2

Proof. See Appendix D. 4.1.1

1 2



n−2

1 − − 12 ln (b λ ∗B )

 2  ,

B λ∗ < 1 e−2 < b

(4.12)

Example–Fisher z-transform test for utility loss aversion index

We provide a few heuristic examples for how one could use the statistical tests for loss aversion index and gain seeking index obtained from the theorems and corollaries above. Table 1: Sample Distribution of Loss Aversion Index for Bernoulli Utility Loss η 0 0.00001 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.99 0.99999 1.0 Mean STDev

Loss Aver. Index λ B (η ) 0 1.000010000 1.051303993 1.105448714 1.223901086 1.359464654 1.518180605 1.709511291 1.949539695 2.268957226 2.738132742 3.587397603 6.692251671 16.60976029 ∞ 2.18367494* 1.617554778

Conj. Loss Aver. Index ∗ λ B (η ) 1 0.999980 0.904837 0.818731 0.67032 0.548812 0.449329 0.367879 0.301194 0.246597 0.201897 0.165299 0.138069 0.135338 0.135335 0.433408308 0.317314522

* This value is for 0 < λ B (η ) ≤ 0.99. Loss column η represents relative loss as a fraction of Bernoulli’s reference wealth level. λ B (η ) is the loss aversion index we would expect to find for a subject with ⋆ Bernoulli preferences. η B (η ) is the corresponding gain seeking index for the implied convex utility function for a subject. See Theorem E.1 in Appendix E.

17

Large sample loss aversion index

According to Theorem 4.3 we would reject an observed loss aversion index value b λ , at η (relative) percentage points from the reference point, if it was statistically significant different from the

λ B expected in Table 1. So if n = 30, η = 20%(0.2) and b λ B = 2.25 we get from Theorem 4.3

Zb = 0.494967027 and p ≈ 0.35. So we would accept the loss aversion index value of 2.25 at a

distance about 20% away from the reference point. In contrast, if η = 90%(0.9) we get Z = −2.27

and p ≈ 0.01. So we would reject the loss aversion index value as being too small at a distance 90% away from the reference point. In other words, a subject faced with the prospect of losing 90% of her wealth should have a much larger loss aversion index than 2.25. For example, Table 1 shows that λ B should be about 3.58 at 90% distance from the reference point. Large sample gain seeking index

λ ∗B = 0.9 In this case, for sample size n = 30, if we observed a gain seeking index estimate b located at η = 20%(0.2) distance from the reference point, then the test statistic from Corollary 1 is Zb ≈ −0.4799 with p = 0.32. So we would accept a gain seeking index value of b λ ∗B = 0.9 at

20% distance from reference point. In contrast, for η = 90%(0.9) from the reference point, we get

Zb = −7.1974 and p = 0.00. So we would reject the gain seeking value as being too large at that

distance from the reference point. For example, Table 1 tells us that at about 90% relative distance from reference point we should expect to see a gain seeking value of about 0.165. Small sample loss aversion index

To illustrate the small sample test for loss aversion index, we choose n = 10, η = 20%(0.2)

λ B = 2.25 as before. In this case, the test statistic from Theorem 4.4 is T = 0.6340 with and b p = 0.31 for n − 2 = 8 degrees of freedom. So we would accept a loss aversion index value of

2.25 at about 20% relative distance from the reference point. However, when η = 0.9 we find that T = 3.80 and p = 0.004. So we reject λ B = 2.25 as being too small at a relative distance of 90% away from the reference point.

18

Small sample gain seeking index

To illustrate the small sample test for gain seeking index, we choose n = 10, η = 20%(0.2) and

λ ∗B = 0.9 as before. Here, the test statistic from Corollary 2 is T = 0.1492 with p = 0.38 for n − 2 = 8 degrees of freedom. However, when η = 0.9 we get the same values. This test is different from the other tests in that it is not sensitive to relative distance from reference point. It is only sensitive to degrees of freedom. However, if we choose λ ∗B = 0.2 we find that T = 3.8340 and p = 0.004 for 8-degrees of freedom. It should be noted in passing that research has shown that for “small and moderate losses”, i.e. relative distance η , loss aversion does not emerge. It is only beyond a certain distance from the reference point, say η ⋆ , that losses loom larger than gains when η > η ⋆ (e.g., Wang and Johnson, 2012). This observation resonates in Table 1. For example, loss aversion index of 2.0 or more is observed at about 0.6 or 60% of the distance from the reference point. The 2.25 number popularized by Tversky and Kahneman (1992) is observed about 70% from the reference point.

Figure 5: Distribution of Loss Aversion Index for Bernoulli Utility

Figure 4: Distribution of Fisher z-transform 18

6

16 4

14 12

2

10 λ

0 -1

-0.5

0

0.5

λ_B

8

1

λ_B* 6

-2

4 -4

2 0

-6

-0.2 η

Fisher-z

0

0.2

0.4

0.6

0.8

1

1.2

Amt of loss (η)

In Figure 4 |η | ↑ 1 ⇒ Fisher |z|↑ ∞. In Figure 5 η → 1 ⇒ λ B → ∞. Intuitively, this implies that as an agent approaches losing all her wealth, loss aversion becomes quite ∗ high and explosive. In contrast λ B exhibits gain seeking. The smaller the number, the higher the gain seeking. So gain seeking is highest when an agent wants to win a ∗ lost endowment, and it approaches the limiting value λ B = 0.135335 for relative wealth in Table 1. If reference wealth approaches infinity, i.e. it is quite large, then for ∗ yr → ∞, we get λ B → 0. See Appendix E equation (E.1).

One shortcoming of the statistical test in Theorem 4.3 is it applies only to loss aversion 19

index values greater than or equal to 1. So it does not test for gain seeking. This is because there is no diminishing sensitivity over losses in Bernoulli’s specification. To address this shortcoming we constructed the antilog of Bernoulli’s (1738) specification to generate an artificial convex utility function that exhibits diminishing sensitivity over losses, and a gain seeking index we refer to as a conjugate loss aversion index in Table 1. The details are included in Appendix E. Figure 4 is an unscaled plot of Fisher z-transform for the loss data η in Table 1. The ztransform is approximately linear for −0.5 < z < 0.5 and it steepens fairly rapidly after that. It has

asymptotes at η = ±1. Table 1 provides a sample distribution for λ B (η ) and its conjugate gain ∗

seeking index λ B (η ) (seeAppendix E for derivation) based on equally spaced intervals between

0.1 and 0.9. The points 0.00001, 0.05, 0.99, 0.99999 were inserted to highlight the behavior of the distribution near the edge. A plot of λ B (η ) is depicted in Figure 5. In Figure 7 in Appendix E a conjugate Bernoulli function is plotted to depict risk seeking behaviour. 4.2 Theory and evidence of α -stable loss aversion index In this subsection we prove that the statistical distribution of the loss aversion index is fat tailed. This distribution is independent of the power law specification for utility (cf. Gabaix, 2009, §4.4). Definition 4.1 (Regularly varying.). De Haan and Ferreira (2007, B.1.1, p. 362). f (η ) is regularly varying (RV) with index α if lim

t→∞

f (t η ) = ηα f (η )

(4.13)

We use the notation f ∈ RVα to represent this phenomenon. The inverse function f −1 ∈ RV1\α for 1

η α.

The following representation theorem implies that λ B (η ) is α -stable. It is a restatement of Karamata’s Theorem in De Haan and Ferreira (2007, Thm. B.1.6). Theorem 4.5 (α -stable loss aversion). If λ B ∈ RV1\α , then there exist measurable functions a :

20

R+ → R and c : R+ → R with lim c(t) = c∗ , 0 < c∗ < ∞, and lim a(t) = α

t→∞

t→∞

(4.14)

and t0 ∈ R+ such that for t > t0

λ (t) = c(t) exp B

Z

t

t0

a(s) ds s



(4.15)

Conversely, if (4.15) holds with a and c satisfying (4.14), then λ B ∈ RV1\α . Proof. See De Haan and Ferreira (2007, Thm. B.1.6, p. 365) or Feller (1970, p. 282). Table 2: Diagnostics for Fishurn-Kochenberger loss aversion index data Statistic Sample Size Range Mean Variance Std. Deviation Coef. of Variation Std. Error Skewness Excess Kurtosis

Value 30 164.4 12.34 876.65 29.608 2.3994 5.4057 5.0684 26.809

Percentile Min 5% 10% 25% (Q1) 50% (Median) 75% (Q3) 90% 95% Max

Figure 6: Fitted Log Pearson 3

Value 0.8 1.295 1.83 2.825 4.85 7.725 22.96 87.925 165.2

The descriptive statistics in Table 2 are for the loss aversion index estimates for the ratio of slopes for the two-piece linear (L− L+ ) local utility function in Fishburn and Kochenberger (1979, Tales 1A, 1B, pp. 508-509). They show that the index is leptokurtic and right skewed. The data were fitted to a Log Pearson Type III distribution (p = 0.32514 for Anderson-Darling goodness of fit statistic) plotted in Figure 6. Fishburn and Kochenberger (1979) also reported extreme values for the loss aversion index, i.e., λ = 3300, λ = ∞ for a two-piece exponential (E − E +) local utility function. However, they provided no ratios for the power utility function(s).

We use sample data for the loss aversion indexes in Fishburn and Kochenberger (1979) metastudy to test the efficacy of Theorem 4.5. Table 2 presents the descriptive statistics. It shows that the underlying distribution has excess kurtosis (right skew) with large variance, relatively narrow interquartile range Q1 − Q3 and a long tail between Q3 − Q4. Figure 6 depicts a Log Pearson Type III (Log Gamma) distribution which was fitted to the data. Refer to Kleiber and Kotz (2003, 21

§5.3 Log–Gamma Distribution) for details on Log Pearson Type III distributions. The specific parameterization fitted via maximum likelihood estimation (MLE) is

p(λ ) =

1 λ |β | Γ(α )





ln λ − γ β



   ln λ − γ exp − , β exp(γ ) < λ < ∞,



0 < λ < exp(γ ),

β < 0; (4.16)

β >0



The fitted values are α = 2, β = 1, and γ = 0 where α and β are shape parameters. The distribution is characterized by its mode which has the form exp((α − 1)\(β + 1)). Perhaps most important, if we put (4.16) in correspondence with (4.15) in Theorem 4.5 we find that   lnt − γ 1 c(t) ≡ t|β |Γ(α ) β Z t     a(s) lnt − γ exp ds ≡ exp − , β t0 s

(4.17) 1 a(s) = − , ln(t0) = γ β

(4.18)

Thus, empirical evidence shows that the probability distribution is α -stable.20 Vizly, for η ∈ [0, 1],

the composite function (p ◦ λ )(η ) ∈ [0, 1] implies (p ◦ λ )(η ) = η ∈ [0, 1], and λ (η ) = p−1 (η ).21

So that p is an inverse function in the class RV1\α and it follows that λ ∈ RV1/α as predicted by Theorem 4.5. ∗

Curiously, λ B (η ) + λ B (η ) ≈ 2.0, 0 < η ≤ 0.5. Perhaps more important, the mean value

of λ B (η ) ≈ 2.18, 0 < η < 0.99 in Table 1 is close to the median value in Tversky and Kahneman

(1992, pg. 311)(“[t]he median λ was 2.25, indicating pronounced loss aversion”). Unlike Tversky and Kahneman (1992) who posit a constant local loss aversion index λ , the loss aversion index implied by Bernoulli, i.e., λ B (η ), is monotone increasing in the amount of loss η . One major restriction of the Fisher z-transform test is that it does not accommodate |η |> 1. So it only addresses gains or loss which size is less that one hundred percent (100%) of the numeraire reference level. B

The gain seeking index λ ∗ (η ) in Theorem E.1 in Appendix E depicts the case when the slope of the curve in Figure 7 at points y > 1 is greater than the slope at points y < 1. That is, 20 Charles-Cadogan (2016) proves that the loss aversion index is unmeasurable under CPT. That result is consistent with the loss aversion index belonging to an α -stable distribution like the generalized Cauchy distribution. 21 The implicit assumption here is that the family of functions p(·) is equicontinuous, i.e., it satisfies a common Lipschitz condition. This facilitates pointwise convergence in [0, 1] and application of the Arzela-Ascoli Theorem (Klambauer, 2005, p. 263) for points in [0, 1].

22

when the utility curve exhibits risk seeking behaviour. This is a useful result because experiments often produce loss aversion index estimates 0 < λ < 1 which connotes gain seeking (Wakker, 2010, p. 239) and convexity over gains and concavity over losses.22 It implies that low income are more prone to gamble because their utility function maybe convex near reference wealth. See, e.g. Markowitz (1952) but compare Light (1977); Scott and Barr (2012); Beckert and Lutter (2013).

5 Conclusion Contrary to popular belief among some behavioral economists, we prove that (1) Bernoulli’s incipient utility function specification is reference dependent, and (2) a loss aversion index is implied by Bernoulli’s specification. We are unaware of any behavioral decision theory of phase transition that predicts a kink or bifurcation at a reference point for utility functions. In fact, Markowitz (1952) utility function–a precursor of prospect theory’s value function–has no kink at the origin. And data show that the kink-at-reference-point in cumulative prospect theory’s (CPT’s) convexconcave value function stems from patching separate subutility functions, estimated over gains and losses, based on a strong gain-loss separability assumption that is rejected empirically. Cursory inspection of the geometry of Bernoulli’s (1738) original concave utility function specification shows that it accommodates gains and losses relative to a reference point where there is no kink. Even though Bernoulli did not contemplate a loss aversion index, we show how his function supports a reference dependent loss aversion index that is monotone increasing in the magnitude of relative loss. We introduce theory and evidence which show that this loss aversion index is slowly varying and α -stable. So we extend the solution space for loss aversion to α -stable laws. Additionally, we show how Fisher’s z-transform test can be used in statistical tests of the loss aversion index. In a nutshell, Bernoulli’s utility function is alive and well. It is sufficient to imply or explain reference dependence and loss aversion index attributed to the theoretical framework of prospect theory. However, the findings in this paper does not imply that prospect theory should be replaced by Bernoulli’s original model. For example, one major deficiency in Bernoulli’s model is it does not anticipate probability weighting functions–something that CPT does well. Another deficiency 22 Refer

to Harrison and Rutstr¨om (2008b, Fig. 14, pg. 97) for a a distribution of λ with values greater than and less than 1.

23

is it assumes decision makers are risk averse over gains and losses, i.e., it does not accommodate diminishing sensitivity to loss or risk seeking behaviour. Furthermore, we did not address issues like Allais’s (2000) paradox and common ratio effect. Thus, the results in this paper simply mean that we should be more guarded in our praise song for prospect theory vis-a-vis Bernoulli’s EUT model of decision making in the presence of risk and uncertainty.

6 Appendix A

Proof of Z-test for Fisher z-transform test for large sample loss aversion index Theorem 4.3

e is a truncated random variable Proof. By hypothesis if W j , Wr are bivariate iid lognormal, then η

(Greene, 2003, p. 757). To see this, let W j = exp(X j ) and Wr = exp(Xr ) where X j , Xr are normally   j b b j ) so that ln W b distributed. Write W j = Wr (1 + η Wr = X j − Xr = ln(1 + η j ) ≈ η j , and the latter

b j ] = E[X j ] − E[Xr ] = 0. By hypotapproximate difference is iid normally distributed such that E[η i h e ] = 0. Recall that |η e . Thus E 1n ∑nj=1 η b 2j = 1 + η b 2j = 1 =⇒ E[η e |< 1 is a truncated hesis, n1 ∑nj=1 η

random variable by hypothesis. In which case, according to Cram´er (1962, eq(18.3.3), p. 242) 1+η we can write the z-transform e2z = which implies 2z = ln(1 + η ) − ln(1 − η ) for the po1−η e . However from Theorem 4.2 we get − ln(1 − η ) = pulation parameter η with sample estimate η

b λ B ln(1 + η ). Now plug in the value for − ln(1 − η ) in the value for 2z to get the desired result     1 η −1 (Anderson, 2003, p. 134), we have λ B ) ln(1 + η ). Since z ∼ N 21 ln 1+ , (n − 3) z = (1 + b 1−η 2   p η 1 + 1 Zb = z − ln / (n − 3)−1 ∼ N(0, 1). 2 1−η Remark A.1. Because Fisher’s z-trasformation is a variance stabilizing transformation (Winter-

bottom, 1979) we did not employ the delta method in our analysis since it is implied by the variance stabilizing transform. Instead, we use a method of moments type approach (Shao, 2003, p. 207) by replacing population parameters with sample estimates. The price we pay is a possibly less efficient estimator Cram´er (1962, p. 498) so our statistical tests are biased in favor of the null. However, Gayen (1951) proved that statistical tests involving Fisher’s z are fairly robust to violations of underlying assumptions in many cases.

24

B Proof of Z-test Fisher z-transform test for large sample gain seeking index Corollary 1   1 B−2 ∗ . Substitution of = exp(−2η ). So that η = ln λ Proof. From Theorem E.1 we get     B η −1 (Anderson, 2003, p. 134) λ ∗ in expression for η in z ∼ N 12 ln 1+ , (n−3) sample estimate b 1−η B λ∗

gives us the sample statistic

  √ − 21   ∗B b 1 + ln (λ 1+η  n−3 ) b   Z = ln ∼ N(0, 1) − ln − 21 1 − η 2 b ∗B 1 − ln (λ )  

(B.1)

Since (1 − 12 ln(λ ∗B ) > 0 we have λ ∗B > e−2 and λ ∗B < 1 for gain seeking.

C

Proof of t-test for Fisher z-transform for small sample loss aversion index Theorem 4.4

Proof. Samiuddin (1970, p. 462) introduced the following t-test statistic, with n − 2 degrees of

freedom, for Fisher z-transform test for small samples under the assumption H0 : ρ = 0 which e ] = 0: corresponds to our assumption that E[η

√ η n−2 T=p 1 − η2

(C.1)

h i bB λ B ln(1 + η ) + ln(1 − η ) = 0. So that ln (1 + η )λ (1 − λ ) = 0 ⇒ From Theorem 4.2 we get b bB

(1 − η 2 ) = (1 + η )(1−λ ) . Substitution in Equation (D.1) gives us T =η

"

n−2

2

b B)

(1 + η )(1−λ

25

#1

(C.2)

D

Proof of t-test for Fisher z-transform for small sample gain seeking index Corollary 2

Proof. Samiuddin (1970, p. 462) introduced the following t-test statistic, with n − 2 degrees of

freedom, for Fisher z-transform test for small samples under the assumption H0 : ρ = 0 which e ] = 0: corresponds to our assumption that E[η

√ η n−2 T=p 1 − η2

From Theorem E.1 we get

B λ∗

  1 B−2 ∗ . Substitution of sample = exp(−2η ). So that η = ln λ

estimate b λ ∗ in the expression for η in Equation (D.1) gives us B

(D.1)



 1 ∗B − 2  T = ln λ   

1

 n−2  2   1  B− 1 − ln λ ∗ 2

26

2

(D.2)

E Gain seeking in Bernoulli’s (1738) implicit conjugate utility function Figure 7: Bernoulli’s (1738) Implied Conjugate Utility Function

0

Conjugate utility function U ∗ (y) = exp(y) for risk seeking is the inverse of Bernoulli utility function U (y) = ln(y) for risk aversion. For numeraire wealth yr , relative wealth is y\yr = 1 ± η , 0 ≤ η ≤ 1. Without loss of generality, in Figure 7 we assume y is relative wealth. A change of axes to “reference point” (1, e) induces Y = y − 1, b ∗ (Y ) = U ∗ (y) − e for gains Y > 0, U b ∗ (Y ) = e −U ∗ (y) for losses Y < 0. So and U ∗ ∗ ∗ b (−η ) = e −U ∗ (1 − η ) for losses with no b (η ) = U (1 + η ) − e for gains and U U kink at the reference point (1, e) ∼ (0, 0).

The conjugate loss aversion index formula is derived when utility is not logconcave but when it is the “antilog”. That is Bernoulli’s log-concave function is now transformed to U (y) = exp(y). See Figure 7. Recall that if yr is numeraire wealth then y = (1 ± η )yr . In which case for a nominal

symmetric gain\loss η in a neighbourhood of reference wealth level yr , the conjugate loss aversion index formula is B

λ ∗ (η ) =

exp(1 − η )yr = exp(−2η yr ) exp(1 + η )yr

(E.1)

B

Here, 0 < λ ∗ ≤ 1 for 0 ≤ yr < ∞. In this case, for Bernoulli’s canonical reference point 1 we have Theorem E.1 (Gain seeking or conjugate loss aversion index formula). The gain seeking or conjugate loss aversion index formula for a loss η (expressed as a percent

27

change in wealth relative to a reference wealth) is given by B

λ ∗ (η ) =

exp(1 − η ) = exp(−2η ) exp(1 + η )

where 0 ≤ η ≤ 1.

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