Prospect Theory’s Cognitive Error About Bernoulli’s Utility Function∗ G. Charles-Cadogan
†
December 6, 2014 Abstract We claim that analysts misconception of Bernoulli’s utility function lead them to believe that it is “reference-independent” and hence not able to generate a loss aversion index. We prove that the geometry of Bernoulli’s original utility function specification accommodates reference dependence and a loss aversion index. In fact, it provides a solution to the open problem of closed form global loss aversion index formula in prospect theory as well as a Fisher z-transform test for the utility loss aversion index. Moreover, higher order approximation of Bernoulli’s specification of preferences predicts that an otherwise risk averse but temperate agent would prefer a high kurtosis gamble to certainty equivalent if the payoff is sufficiently high. So it admits convexity and gain seeking over gain domains. In contrast, prospect theory’s strong gain-loss separability hypothesis predicts that no agent would prefer that gamble in gain domain. In a nutshell, Bernoulli’s utility function is alive and well. It subsumes prospect theory’s skewed S-shape value function for decision making under risk and uncertainty. Keywords: prospect theory, cognitive bias, Bernoulli utility, loss aversion, value function JEL Classification Codes: D81
∗
I thank Haim Abraham and Ramaele Moshoshoe for their incisive comments on an earlier draft. The usual disclaimer applies. † University of Cape Town, School of Economics, Research Unit in Behavioral Economics and Neuroeconomics (RUBEN), Private Bag 5-3b, Rondebosch 7701. e-mail:
[email protected]; Tel: (786) 329-5469.
Contents 1 Introduction
1
2 Prospect theory value function vs Bernoulli utility function 2.1 The reference point in Bernoulli’s utility function . . . . . . . . . . . . . . . 2.2 Bernoulli utility function vs. Kahneman-Tversky skew S-shape value function 2.2.1 Case (i): Deviations from the reference point . . . . . . . . . . . . . . 2.2.2 Case(ii): Generally concave for gains and commonly convex for losses 2.2.3 Case(iii): Steeper for losses than for gains . . . . . . . . . . . . . . . 2.3 Fishburn and Kochenberger (1979) two piece analysis . . . . . . . . . . . . .
3 3 4 5 7 10 12
3 A global loss aversion index formula from Bernoulli utility 3.1 Bernoulli’s forgotten numeraire wealth level . . . . . . . . . . . . . . . . . . 3.2 Global loss aversion index, its conjugate, and Fisher’s z-transform . . . . . .
15 17 19
4 Conclusion
23
5 Appendix
24
A Proof of Fisher z-transform test for loss aversion Theorem 3.5
24
References
24
List of Figures 1 2 3 4 5 6 7 8 9
Bernoulli Utility of Wealth Function . . . . . . . . . . . . . . . Prospect theory’s value function . . . . . . . . . . . . . . . . . . Bernoulli Utility Function With Reference Point . . . . . . . . . Bernoulli vs. Kahnemen-Tversky Value Function . . . . . . . . . Fishburn-Kochenberger utility with reference point . . . . . . . Bernoulli Utility with reference wealth . . . . . . . . . . . . . . Distribution of Fisher z-transform . . . . . . . . . . . . . . . . . Distribution of Global Loss Aversion Index for Bernoulli Utility Conjugate Bernoulli Utility Function . . . . . . . . . . . . . . .
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3 5 6 9 12 12 22 22 22
Hierarchical risk attitudes: Prospect value vs. Bernoulli utility . . . . . . . Sample Distribution of Loss Aversion Index for Bernoulli Utility . . . . . . .
8 21
List of Tables 1 2
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”, while duly noting that “there are relatively few well-known and broadly accepted applications of prospect theory in economics”. 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. In this paper we argue that Bernoulli (1738) “original” utility function, which falls under rubric of EUT, explicitly satisfies several characteristics of prospect theory’s skew S-shape value function. We claim “cognitive error”1 as the cause of analysts misperception of Bernoulli’s utility function specification. This paper is motivated 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 1
Also known as a cognitive bias, a cognitive error is a pattern of deviation in judgment that occurs in particular situations, which may sometimes lead to perceptual distortion, inaccurate judgment, illogical interpretation, or what is broadly called irrationality. See e.g., http://en.wikipedia.org/wiki/Cognitive bias Last visited 11/15/2014.
1
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). This paper provides a critical review of the (Bernoulli, 1738) model, and compares it to the claims made against it in the Kahneman lecture above. In section 2 we compare the geometry of Bernoulli’s utility function to that of Kahneman-Tversky skew S-shape value function. We show how loss aversion is latent in Bernoulli’s specification, and how it accomodates higher order risk attitudes. In section 3 we show how Bernoulli’s specification supports a closed form global loss aversion index, and we characterize its relation to Fisher’s z-transformation test. We conclude in section 4.
2
Figure 1: Bernoulli Utility of Wealth Function
2
Prospect theory value function vs Bernoulli utility function
In this section we emphasize the geometric properties of Bernoulli’s utility function as described in Bernoulli (1738) and contrasts it to the qualitative and geometric properties of Kahneman and Tversky (1979) value function. 2.1
The reference point in Bernoulli’s utility function We begin this subsection with (Bernoulli, 1738, pg. 26) description of the geometry of
his utility 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, F M, 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] 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 3
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, P O. [Emphasis added] It is indisputable that the italicized text 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, “losses loom larger than gains” (Kahneman and Tversky, 1979, p. 279) in Bernoulli’s utility function specification. Nonetheless, (Kahneman and Tversky, 1979, pg. 276) states: “[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 utility function vs. Kahneman-Tversky latent skew S-shaped utility function with loss aversion Next we examine the shape of (Kahneman and Tversky, 1979, pg. 279) value function
sketched in Figure 2, and the specification in Tversky and Kahneman (1992) and its implication for the ubiquitous loss aversion index. According to (Kahneman and Tversky, 1979, pg. 279) (KT79) In summary, we have proposed that the value function is (i) defined on deviations from the reference point; (ii) generally concave for gains and commonly convex for losses; (iii) steeper for losses than for gains. A value function which satisfies these properties is displayed in Figure 3. Note that the proposed S-shaped value function is steepest at the reference point, in marked contrast to the utility function postulated by Markowitz which is relatively shallow in that region. [Emphasis added] We will go through each one of those points above ((i) − (iii)) in seriatim to see if Bernoulli’s utility construct satisfies them. We will show that Bernoulli utility function satisfies all the conditions in (i) − (iii). 4
Figure 2: Prospect theory’s value function
V ( x) value function
Vg ( x)
Reference point Neigbourood
UaTK
x < 0 Loss
Gains
x>0
x>0
V ( x)
2.2.1
Case (i): Deviations from the reference point
From the outset, we note that Figure 3 depicts Bernoulli’s utility function with reference point B. The (Bernoulli, 1738, pg. 29) quote above satisfies KT79 condition (i). Bernoulli conducted his analysis over positive values AB and beyond in Figure 3. However, if we shift (i.e. translate) Bernoulli’s axis Qq in Figure 1 so that it passes through B, then his loss Bq is negative while his gain BQ is positive. Ironically, this change of axes involves a translation2 of relative wealth by −1 so that the reference point B is now 0–the same as Kahneman and Tversky’s reference point–as shown in Figure 3. To see this, note that according to (Stigler, 1950, pg. 374) Bernoulli’s specification is of type
u(x) =
2
ln(1 + x)
ln(1 − x)
for gain
(2.1)
for loss
(Eeckhoudt et al., 1995, Fig. 1, p. 335) shows that Bernoulli type utility is a concave function of translation in wealth.
5
Figure 3: Bernoulli Utility Function With Reference Point
Q
Q’ S
L _
A
ܸ
L’ B ܸκ
ܸκ்
s’
s
Replacing x by X = x − 1 generates u(X) = ln(X) for gains. However, that change of axis induces undefined concepts like the logarithm of negative terms when X < 0 for losses. So in Bernoulli’s model, a “pure utility” of loss is undefined under the translation scheme and the reflection effect is unobservable under that transformation. Kahneman and Tversky (1979). Tversky and Kahneman (1992) “resolved” this problem by writing −X when X < 0 and they “hardcoded” a loss aversion index λ to account for the skew. However, the utility of loss can be recovered in Bernoulli’s specification without resort to “hardcoding” as evidenced by the following approximations when return on wealth x > 0: x2 x3 x4 + − , for gains 2 3 4 x2 x3 x4 uℓ (x) = ln(1 − x) ≈ −x + − + , for loss 2 3 4
ug (x) = ln(1 + x) ≈ x −
(2.2) (2.3)
In each case, ug (0) = uℓ (0) = 0 when the reference point is zero. This is semantics because under the original specification ln(1 + 0) = 0. However, we introduce (2.2) and (2.3) for comparison with Tversky and Kahneman (1992) model of loss aversion. Note that the curve is smooth at the reference point 0–there is no kink–contrary to Kahneman and Tver-
6
sky’s specification, infra. The translation does not affect the [absolute] magnitude (x) of Bernoulli’s relative gains or losses in wealth. But it does change the orientation of his curve from the solid concave portion Bs to the dotted convex portion Bs′ as shown in Figure 3 to accommodate the fact that now points to the left of B are negative, and so the curve is in the negative quadrant with a longer tail because of the asymmetric response to gains and losses. This point is explained in detail later in the paper. For the purpose of illustration, we provide a numerical example in Table 1 which brings us to Case (ii). 2.2.2
Case(ii): Generally concave for gains and commonly convex for losses
The asymptotic expansion of Bernoulli’s utility function in (2.2) and (2.3) to higher moments introduces risk attitude concepts like prudence, temperance, etc (Eeckhoudt et al., 1995). For example, u′′′′ g (x) < 0 implies that a decision maker (DM) with Bernoulli utility over gains is temperate (Eeckhoudt et al., 1995, Cor. 2). Applying an expectation operator E to (2.2) and (2.3) implies the following higher risk attitudes (Noussair et al., 2014, p. 326, fn.1). Here ⊕ is treated as an abstract conjoint operation. For example, the higher order terms are analogized to measures of dispersion. Hierarchical higher order risk attitudes
• E[Y1 (x)] ≡ mean =⇒ “risk neutrality”; • E[Y2 (x)] ≡ mean ⊕ variance; =⇒ “risk aversion”; • E[Y3 (x)] ≡ mean ⊕ variance ⊕ skewness =⇒ “prudence”; • E[Y4 (x)] ≡ mean ⊕ variance ⊕ skewness ⊕ kurtosis =⇒ “temperance” Table 1 contains the distribution of those values for equally spaced values of x. The underlying premise is that the range of wealth is normalized by its maximum value. A plot of Bernoulli’s value function with hierarchical higher order risk attitudes, and the KahnemanTversky value function (in red) is superimposed in Figure 4. 7
Table 1: Hierarchical risk attitudes: Prospect value vs. Bernoulli utility x -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Y1 (x) -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Y2 (x) -1.500 -1.305 -1.120 -0.945 -0.780 -0.625 -0.480 -0.345 -0.220 -0.105 0 0.095 0.180 0.255 0.320 0.375 0.420 0.455 0.480 0.495 0.500
Y3 (x) -1.833 -1.548 -1.291 -1.059 -0.852 -0.667 -0.501 -0.354 -0.223 -0.105 0 0.095 0.183 0.264 0.341 0.417 0.492 0.569 0.651 0.738 0.833
Y4 (x) -2.083 -1.712 -1.393 -1.119 -0.884 -0.682 -0.508 -0.356 -0.223 -0.105 0 0.095 0.183 0.266 0.348 0.432 0.524 0.629 0.753 0.902 1.083
YP T (x) -2.250 -2.051 -1.849 -1.644 -1.435 -1.223 -1.005 -0.780 -0.546 -0.297 0 0.132 0.243 0.347 0.446 0.543 0.638 0.731 0.822 0.911 1.000
Prospect theory’s artificial kink at the origin is due to experimenter bias
Kahneman-Tversky value function is a power function xα which is concave over gains and convex over losses. So Figure 4 is a qualitative comparison. Each of the curves in Table 1 represent hierarchical risk attitudes, and they are subsets of (2.2) and (2.3). The “loss tail” for Bernoulli’s construct is depicted by VℓB (x) in Figure 4. YP T = x0.88 is the Kahneman-Tversky value function. The value function over losses, VℓP T (x), is obtained by pre-multiplying the loss component by λ = 2.25–the median value of the loss aversion index reported in (Tversky and Kahneman, 1992). That introduces an artificial kink at the origin where there is no kink in the Bernoulli construct. Undeniably, for x < 0 we have E[x] > VℓB (E[x]). Thus, a DM would prefer to take risk over losses rather than receive the sure payment (Machina, 1987, p. 538). This implies that VℓB should be convex over losses despite its appearance in
8
Figure 4: Bernoulli vs. Kahnemen-Tversky Value Function 1.5
ࢂࡼࢀ ࢍ ሺ࢞ሻ
1
ࢂ ࢍ ሺ࢞ሻ
0.5 0 -1.5
-1
-0.5
0
0.5
-0.5
1
Y_1(x) 1.5
Y_2(x) Y_3(x) Y_4(x)
-1
Y_PT(x)
-1.5
ࢂ κ ሺ࢞ሻ
ࢂκࡼࢀ ሺ࢞ሻ
-2 -2.5
the sketch. This is depicted by Bs′ in Figure 3. Because Kahneman-Tversky hard coded −2.25 or −λ if you will in their tail specification their model is confounded. Furthermore, in gain domains Bernoulli preferences are strictly risk averse with the exception of Y4 (with kurtosis) which exceeds the expected value near the tail, i.e., for 0.9 < x ≤ 1 as indicated in the last row of Table 1 and the black curve in Figure 4. This suggests the existence of a convex segment of the curve consistent with gambling over gain domain. So an otherwise risk averse subject faced with the prospect of doubling her wealth is willing to take risks if she has kurtosis prefereces. According to Menezes and Wang (2005) that subject takes probability mass from the mean and transfers it to the tail of the distribution. Recent experiments by Ebert and Wiesen (2012) confirms the existence of the behaviours characterized above for higher moments. In particular, Bernoulli’s subjects are risk seeking over gain and loss domains in this veritable “preference for skewness” setting that has implications for asset pricing outside the scope of this paper (Kraus and Litzenberger, 1976; Harvey and Siddique, 2000; Dittmar, 2002; Polkovnichenko and Zhao, 2013). 9
2.2.3
Case(iii): Steeper for losses than for gains
´ According to Hospital’s rule (Apostol, 1967, pp. 292-293) Bernoulli’s loss aversion index at the origin is u′ℓ (−x) =1 x→0 u′ (x) g
λB = lim
(2.4)
When we impose Tversky and Kahneman (1992) reported median loss aversion index estimate of 2.25 we introduce a kink at the “reference point” where there was none before. The tails are now longer and the loss aversion index is a lot larger than it is under Bernoulli’s specification. Perhaps most important, the Tversky and Kahneman (1992) “interference” induced a kink at 0 so local loss aversion is now undefined there. Instead of λB above, we now get K¨obberling and Wakker (2005) estimate λKW =
vℓ′ (0− ) vg′ (0+ )
(2.5)
which spawned a whole literature around estimation of the loss aversion index. See Wakker (2010). In the sequel we show how a simple global loss aversion index can be derived and provide some estimates. More on point, in Figure 3 Bernoulli’s analysis is invariant to the orientation of Bs or Bs′ since wealth levels are not affected by the orientation. In a nutshell, the curve Bs′ is latent in Bernoulli’s analysis because he did not place his reference point for changes in wealth at the origin. Ironically, (Tversky and Kahneman, 1992, p. 309) proffered the following specification for their value function:
v(x) =
xα ,
x>0
−λ(−x)β , x < 0
(2.6)
There, α and β are shape parameters and λ is the celebrated loss aversion index that captures
10
asymmetric responses to symmetric gain and loss. More on point, if x < 0 then −x > 0. So the value function segment (−x)β is positive and concave for 0 < β < 1 (β = 0.5 is Gabriel Cramer’s specification) and functionally equivalent to Bernoulli’s Bs. Thus, Bernoulli satisfied KT79 condition (ii). By pre-multiplying the concave segment (Bs) by −λ (for λ > 1), Tversky and Kahneman in effect reoriented Bs so that it is convex (say Bs′ ). In the context of behavioural operator theory, that is a “spin”–no pun intended. Thus, but for the power law specification, the value function in (2.6) is qualitatively the same as Bernoulli’s utility function. In that case, if a sufficiently small gain and loss are equidistant from the reference point B, i.e., BL′ and BL in Figure 3, we would expect to find α ≈ β. Lemma 2.1 (Equality of shape parameters). For a sufficiently small symmetric gain and loss equidistant from Bernoulli’s reference point B shape parameters are approximately equal. In that case, one should not be surprised that (Tversky and Kahneman, 1992, pg. 311) upholds Lemma 2.1 when they report: [I]t is common to assume a parametric form (e.g., a power utility function), but this approach confounds the general test of the theory with that of the specific parametric form. For this reason, we focused here on the qualitative properties of the data rather than onparameter estimates and measures of fit. However, in order to obtain a parsimonious description of the present data, we used a nonlinear regression procedure to estimate the parameters of equations (5) and (6), separately for each subject. The median exponent of the value function was 0.88 for both gains and losses, in accord with diminishing sensitivity. [Emphasis added] Perhaps more important, one need not premultiply the curve by −λ to capture the effect of the loss aversion parameter λ since the reoriented curve represented by the dotted line Bs′ already captures asymmetric skew evidenced by VℓB (−x) > VgB (x), where the former is Bernoulli’s value for loss (ℓ) and the later is value for symmetric gain (g). That is, λ is a latent parameter in Bernoulli’s model. Thereby, causing Bernoulii’s latent analysis of Bs′ under auspice of Bs to satisfy KT79 (iii). In fact, a cursory inspection of Table 1 shows that 11
Figure 5: Fishburn-Kochenberger utility with reference point
Figure 6: Bernoulli Utility with reference wealth
Q
Q’ S
L _
A
ܸ
L’ B ܸκ
ܸκ்
s’
s
at points equidistant from the origin 0, Bernoulli utility is larger in loss domain compared to gain domain. This is evident in Figure 2 where Vℓ (−x) > Vg (x). By virtue of satisfying (i) − (iii) in (Kahneman and Tversky, 1979, pg. 279) we proved Proposition 2.2 (Bernoulli’s value function).
Bernoulli (1738) value function is qualita-
tively equivalent to Kahneman and Tversky (1979) value function. Proposition
2.2
is
depicted
in
Figure 5
and
Figure 6
which
juxtaposes
(Fishburn and Kochenberger, 1979, Fig. 1, pg. 504), and an annotated sketch of (Bernoulli, 1738, pg. 26). 2.3
Fishburn and Kochenberger (1979) two piece analysis
The Fishburn and Kochenberger (1979) (FK) paper was selected for critical review here because it was referenced by (Kahneman and Tversky, 1979, pg. 268) as empirical support for their skewed S-shaped value function with a kink at the origin. However, FK conducted a “mini-metastudy” comprised of 30 observations from “empirically assessed utility functions from five sources”. Because their data is a “convenience sample” drawn from a very het12
erogenous population, and they were gleaned from plots depicted from the “five [published papers] sources”, it contains a lot of noise. Therefore, the analysis that follows focuses on the qualitative results of the FK paper with parenthetical remarks about methodology. That said, FK used a minimum mean squared error (MMSE) procedure to fit the subject value functions. That procedure is popular among Bayesians like Fishburn for point estimates, and it is used when the analyst has some prior information about the parameter [s]he wants to estimate (Shao, 2003, pp. 122-123). The stated objectives of the FK study are (1) to re-examine published data on utility function estimation over “significant changes in individual or corporate wealth”, and (2) to see how well “simple functional forms” fit the data. The functional forms selected were linear, power (a functional equivalent of CRRA) utility, and an exponential form reminiscent of “one switch utility” (Bell and Fishburn, 2000). In a nutshell, they found that power functions gave the best fit. That finding is confirmed in a recent and more extensive metastudy by Stott (2006). Several comments are in order. First, the qualitative shapes of the skewed S-shaped value functions they estimated were known a priori to be in the subject data. Therefore, FK did not test prospect theory per se. Instead, their study is more akin to seeking congruence among the selected papers. They did not state why they selected the subject five papers from the universe of papers available for inclusion in the study. So their results are open to criticism for selection bias. Further, their paper is silent on whether there was any accompanying probability weighting functions since the curvature of probability weighting could dilute findings of curvature in corresponding value and or utility functions.3 (Harrison and Rutstr¨om, 2008; Holt and Laury, 2014). Second, (Fishburn and Kochenberger, 1979, pg. 505) transformed the data so that it had a common “reference point” of zero. They conducted translation and rotation operations on the data to achieve that goal. This is in contrast to Bernoulli (1738) reference wealth model analyzed above which achieves the same qualitative results without such strong trans3 (Luce, 2000, p. 221) reports that in those studies “utility functions for gains and losses had been separately estimated using expected utility [W (p) = p]”.
13
formation. The authors knew about Bernoulli’s utility function because it was referenced in the second sentence of their paper. However, they chose not to include that specification which may have provided the best fit of all. Third, as if to highlight our observation above, (Fishburn and Kochenberger, 1979, pg. 511) state: The obvious implications of these observations is that the below-target functions tend to exhibit more curvature than the above-target functions. Put otherwise, the above-target functions tend to be closer to linear functions than the below-target functions. [Emphasis added]. Inspection of Figure 4 show that almost all the curves are “closer to linear” Y1 (x) in gain domains with the exception of Y4 (x) which crosses Y1 (x) near the end point x = 1. Specifically, Y1 (x) represents “risk neutrality” because E[Y1 (x)] = E[x] is the mean relative wealth under Bernoulli’s model. Similarly, E[Y2 (x)] includes mean and variance; E[Y3 (x)] includes mean, variance and skewness; and E[Y4 (x)] includes mean, variance, skewness and kurtosis. The increased curvature in “below-target” functions shows a “preference for skewness” or measures of dispersion over loss domain. Interestingly, E[Y4 (x)] implies that a subject presented with a high kurtosis gamble would be gain seeking in gain domains for a sufficiently large increase in relative wealth. [S]he is less temperate when given an opportunity to double her wealth.4 Fourth, (Fishburn and Kochenberger, 1979, pg. 523) state, in pertinent part: “power functions tend to give better fits to flat data, whereas exponential functions tend to give better fits to steep data. At present, we do not have a reasonable theoretical explanation for this finding” (emphasis). By “flat data” the authors mean data consistent with the skewed Sshape value function. “Steep data” refers to convexity over gain domains and concavity over loss domains. We have a theoretical explanation for the phenomenon–again in the context of Bernoulli’s utility function. The “flat data” observation was already explained at the end of our third point above. The “steep data” observation rests on the conjugate Bernoulli 4
(Luce, 2000, pg. 221) provides a distribution of studies which shows that the shape of utility functions are affected by framing effects, and the method of estimation–certainty equivalent or probability equivalent.
14
function which we present in Corollary 3.4, infra. Bernoulli’s conjugate utility function is exponential increasing over gain domains but exponential decreasing over loss domains. See Figure 9, infra. We note in passing that (Fishburn and Kochenberger, 1979, pg. 516) also state “If u is convex-concave and steeper in losses than in gains, then the individual will never purchase a lottery ticket for a large prize. This assumes of course that the cost of the ticket exceeds its expected winnings” (emphasis added). Again, that claim is refuted by admissible utility curve Y4 (x) where high kurtosis gambles that offer sufficiently large prizes are more attractive than the mean! In closing, even though prospect theory was in its infancy when FK wrote their paper, Bernoulli’s utility function was already in existence for over 300-years. Our interpretation of Bernoulli’s utility function provides a theoretical explanation of the K two piece functions. Minimum Mean Square Error analysis is a valid diagnostic statistical procedure. However, our comparative analysis rests on FK qualitative assessments the subject data in their analysis were simply too noisy, and the sample size too small to give substantive weight to numerical estimates. Cf. Abdellaoui et al. (2007).
3
A global loss aversion index formula from Bernoulli utility
Cursory inspection of Figure 6 shows that the value VℓKT for Kahneman and Tversky’s skew is such that VℓKT > VℓB > Vg ⇒
VℓKT VB > ℓ >1 Vg Vg
(3.1)
Thus, the “disutility” of loss in either case is such that it is greater than the “utility” of an equal nominal gain. Let AVℓKT , AVgKT , AVℓB , AVgB be the impact of an incremental change in wealth η under the Kahneman-Tversky (KT) and Bernoulli (B) value functions. Thus, from
15
(2.1) and (2.6), for a symmetric deviation η from the “reference point” we get the impacts: AVℓKT = AVgKT = AVℓB = AVgB =
Z
0
(−λ(−x)β )dx = −λ
−η Z η
Z
0
x
xα dx =
β+1
, η<0
(3.2) (3.3)
dy dy = ln x − ln(x − η) y
x−η Z x+η x
η α+1 α+1
n (−η)β+1 o
(3.4)
dy dy = ln(x + η) − ln x y
(3.5)
Inasmuch as Lemma 2.1 suggests that shape parameters are equal for sufficiently small symmetric gain and loss, the loss aversion index λKT derived from the impact of a change in wealth in (3.2) and (3.3); and λB derived from the same impact in (3.4) and (3.5) are given by β+1
λ
KT
=
λB (x, η) =
| | − λ (−η) β+1 ηα+1 α+1
=λ
α + 1 β−α |η| ≈ λ, β+1
when α ≈ β
ln x − ln(x − η) ln(x + η) − ln x
(3.6) (3.7)
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). Nonetheless, it is a valid measure as shown by (3.6) (Wakker, 2010, p. 268). Undeniably, the loss aversion index in (3.7) depends on the reference wealth level x (depicted by B in Figure 3) and the amount of loss η (depicted by BL). For internal consistency of (Tversky and Kahneman, 1992) specification the relation in (3.6) holds. (Wakker, 2010, p. 268) addresses the effect on local utility loss aversion index measures when α 6= β. Thus we proved Theorem 3.1 (A reference dependent loss aversion index for Bernoulli). The loss aversion index λB (x, η) in (3.7) computed from Bernoulli’s value function is reference dependent. In particular, λB (x, η) is a global loss aversion index over the distribution of wealth x and loss η. 16
3.1
Bernoulli’s forgotten numeraire wealth level
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 by
Wz . Wx
Thus, the points in his graph are changes in wealth rel-
ative to the numeraire. 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) also analyzed Bernoulli’s utility function by introducing the notion of a “subsistence level at c” where U(c) = k ln(c) + a = 0 ⇒ a = −k ln(c) ⇒ U(x) = k ln
x c
(3.8)
Using Stigler’s interpretation, U(c) = 0 at precisely where “subsistence wealth level” x = c and relative wealth
x c
= 1. Ironically, there 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, 2012; Scott and Barr, 2012). More on point, (Bernoulli, 1738, pg. 28) writes: AF AC + nb log AD + pb log AE + qb log AB + ... mb log AB AP AB AB b log = AB m + n + p + q + ...
(3.9)
That equation can be rewritten as AP = AB
"
AC AB
!m
AD AB
!n
AE AB
!p
AF AB
!q
...
# m + n + 1p + q + ...
(3.10)
which is a weighted geometric mean relative to the reference wealth level AB. 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
(3.10) is much smaller than 1 (≪ 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
17
model. So that we have the strict partial preference order WP ≺ WB ≺ WC ≺ . . .. Choose WB as numeraire so that WB WC WD WP ∼ AP, = 1 ∼ AB, ∼ AC, ∼ AD, WB WB WB WB WE WF ∼ AE, ∼ AF, . . . WB WB
(3.11)
This is an implicit assumption in Bernoulii’s model. Let N = m + n + p + q + . . . we rewrite (3.10) as h i N1 1 + rP = (1 + rC )m (1 + rD )n (1 + rD )p (1 + rE )p (1 + rF )q . . . ! X 1 ln(1 + rj )k − 1 ⇒ rP = rN = exp N
(3.12) (3.13)
j∈{C,D,E,F,...} k∈{m,n,p,q,...}
≈
1 N
X
krj ⇒ rP = lim rN = r ⋆ < ∞ N →∞
j∈{C,D,E,F,...} k∈{m,n,p,q,...}
(3.14)
up to a first order approximation. Lemma 3.2 (Bernoulli’s reference dependent change in wealth). Bernoulli normalized his wealth levels with a numeraire so that reference wealth x = 1 where his log-concave utility cuts the horizontal and all other points on the axis represent percent changes in wealth relative to x = 1. So that points to the left of x = 1 correspond to percent loss in wealth and points to the right correspond to percent gain in wealth.
Remark 3.1. This seemingly over looked result has implications for asset pricing models. See (Campbell et al., 1997, §1.4).
18
3.2
Global loss aversion index, its conjugate, and Fisher’s z-transform
Given the relationships in (3.1), (3.6) and (3.7) we have reference point driven loss aversion index relationships: λKT > λB > 1 ⇒ λB =
(3.15)
− ln(1 − η) ln 1 − ln(1 − η) = >1 ln(1 + η) − ln 1 ln(1 + η)
⇒ ln(1 − η 2 ) < 0 ⇒ |η| < 1
(3.16) (3.17)
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. It is an open question as to what happens when wealth increases by more than a factor of 1. Perhaps most important, Kahneman and Tversky’s loss aversion index λKT in (3.6) is uniformly distributed over their value function for all changes in wealth. So in theory, there is no bound on the magnitude of changes relative to the reference point–it is irrelevant. By contrast, the loss aversion index λB in (3.7) and (3.16) is reference dependent and responsive to all changes in wealth less than the reference wealth level. To wit, it is “global”. Thus, we proved Theorem 3.3 (A Global Loss Aversion Index Formula). A global loss aversion index formula for a loss η (expressed as a percent change in wealth relative to a numeraire), when utility is log concave, is given by λB (η) = − where 0 < η < 1,
ln(1 − η) ln(1 + η)
0 ≤ λB ≤ ∞.
The conjugate loss aversion index formula is derived when utility is not logconcave but when its the “antilog”. That is Bernoulli’s logconcave function is now transformed to U(x) = exp(x). In which case for a nominal symmetric gain\loss η in a neighbourhood of wealth 19
level x, the conjugate loss aversion index formula is B
λ∗ (η) =
exp(x − η) = exp(−2η) exp(x + η)
(3.18)
B
Here, 0 < λ∗ ≤ 1 for 0 ≤ η < ∞. In this case, for relative wealth levels x = 1 we have Corollary 3.4 (Conjugate global loss aversion index formula). The conjugate global loss aversion index formula for a loss η (expressed as a percent change in wealth relative to a numeraire), when utility is convex exponential is given by B
λ∗ (η) = where 0 < η < 1,
exp(1 − η) = exp(−2η) exp(1 + η)
0 ≤ λB ≤ ∞.
The literature shows that K˝oszegi and Rabin (2006, 2007) formulated stylized models of reference dependent preferences but failed to proffer a closed form loss aversion index formula even though one is evidently available from careful analysis of Bernoulli’s original ideas. Thus, the index is robust to criticism against the application of logconcave utility to all wealth levels. Perhaps more important, Theorem 3.3 suggests that λB is related to Fisher’s z-transformation5 of symmetric gain and loss (η). That is, it transforms the normalized “reference wealth” interval from [0, 1] for η to [0, ∞], so that 0 ≤ λB ≤ ∞. We summarize this artefact in the following Theorem 3.5 (Fisher z-transform test for loss aversion). 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 so that
Wj Wr
= 1 + ηbj , j 6= r and ηbj ∼ iid N(0, 1). Let z be Fisher’s
z-transform and |b η| < 1 be a given symmetric gain and loss relative to Wr , and z is normally 1+η 1+η and variance (n − 3)−1 , i.e., z ∼ N 12 ln 1−η , (n − 3)−1 distributed with mean 12 ln 1−η 5
Refer to (Cram´er, 1962, p. 241)
20
(Anderson, 2003, p. 134). Then zb =
1 bB ln(1 + ηb) 1+λ 2
bB and ηb. where zb is the sample Fisher z-transform for sample estimates λ
Proof. See Appendix A.
Table 2: 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 λB (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 ∗ λB (η) 0 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
* This value is for 0 < λB (η) ≤ 0.99.
Figure 7 is an unscaled plot of Fisher z-transform for the loss data η in Table 2. 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 2 provides a sample distribution for λB (η) and ∗
its conjugate λB (η) 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
21
Figure 7: transform
Figure 8: Distribution of Global Loss Aversion Index for Bernoulli Utility
Distribution of Fisher z6
18 16
4
14 2
12 10
0 -1
-0.5
0
0.5
1
λ
λ_B
8
λ_B* -2
6 4
-4
2 0
-6 η
-0.2
Fisher-z
0
0.2
0.4
0.6
Amt of loss (η)
Figure 9: Conjugate Bernoulli Utility Function ࢁሺࣁሻ
ࢁሺࣁሻ ൌ ࢋ࢞ሺ ࣁሻ ࢁሺሻ
࢞ൌ η=-1
22
࢞ൌ ࣁ
0.8
1
1.2
the edge. A plot of λB (η) is depicted in Figure 8. In Figure 9 a conjugate Bernoulli function is plotted to depict convex utility over gains. The plot shows that η → 1 ⇒ λB (η) → ∞ ∗
and that λB (η) is slow varying as η approaches 1. Curiously, λB (η) + λB (η) ≈ 2.0, 0 < η ≤ 0.5. Perhaps more important, the mean value for λB (η) ≈ 2.18,
0 < η < 0.99 is
consistent with (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 Bernoulli loss aversion index λB (η) is monotone increasing in the amount of loss η. B
The conjugate loss aversion λ∗ (η) in Corollary 3.4 depicts the case when the slope of the curve at points above x = 1 is greater than the slope at points less than x = 1. That is, when the utility curve is convex–the conjugate of concave–over gains. This is an important result because Kahneman and Tversky (1979) only accounts for those cases when the utility curve is concave over x > 1 and convex when x < 1. However, laboratory results often report estimates 0 < λ < 1 which connotes gain seeking (Wakker, 2010, p. 239) and convexity for relative wealth x > 1, and relative concavity for x < 1. See (Harrison and Rutstr¨om, 2008, Fig. 14, pg. 97). It implies that low income are more prone to gamble since their utility function is convex near reference wealth (Scott and Barr, 2012; Beckert and Lutter, 2012; Light, 1977).
4
Conclusion
Close inspection of the geometry of (Bernoulli, 1738) original utility function specification show that it accommodates gains and losses relative a reference point. Furthermore, it has a reference dependent feature that supports a global loss aversion index that is monotone increasing in the magnitude of loss. This implies that many of the results explained by prospect theory’s skew S-shaped value function are explicable with Bernoulli’s incipient utility function specification.
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5
Appendix
A
Proof of Fisher z-transform test for loss aversion Theorem 3.5
P Proof. By hypothesis ηbj ∼ iidN(0, 1). By abuse of notation let n1 nj=1 ηbj2 = 1 + ηb. h P i Thus E n1 nj=1 ηbj2 = 1 =⇒ E[b η ] = 0. In which case, according to (Cram´er, 1962, 1 + ηb eq(18.3.3), p. 242) we can write e2bz = . However from Theorem 3.3 we get 1 − ηb − ln(1 − ηb) = b λB ln(1 + ηb). After taking log of e2bz and substituting the foregoing expression
for ln(1 − ηb) in the formula we get the desired result.
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