An Analysis of Heterogeneous Utility Benchmarks in a Zero Return Environment

8/2/2019 An Analysis of Heterogeneous Utility Benchmarks in a Zero Return Environment

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An Analysis of Heterogeneous Utility Benchmarks in a Zero Return Environment

by

Fred VioleOVVO Financial Systems

[email protected]

And

David NawrockiVillanova University

Villanova School of Business

800 Lancaster AvenueVillanova, PA 19085 USA

[email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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An Analysis of Heterogeneous Utility Benchmarks in a Zero Return Environment

Abstract

The utility of an investor should be based on an acceptable loss in the loss region and a target

return in the gain region of a set of investment opportunities. The level of these benchmarks will

unveil an opportunity cost, break-even effect, or indifference when the return of an investment

equals zero. This condition has been arbitrarily assumed away for continuity and other

simplification purposes over the last few decades. Historical utility functions, those that are von

Neumann-Morgenstern compliant and not, are all constrained via a single target or reference

point. This single target restriction coupled with the arbitrary zero-return assumption has

ignored the important interpretation of this salient point on the utility curve as a proxy for the

investors current wealth.


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I. INTRODUCTION

We propose a utility function comprised of two autonomous functions: one function describing

the utility of losses and one function describing the utility of gains. Each of these autonomous

functions will be represented by a Markowitz Stochastic Dominance (MSD) Reverse-S curve per

Levy and Levy (2002). We expand the notion of a dual target or multiple reference points as

proposed in Lopes SP/A Theory (1987). Historically, with both von Neumann and Morgenstern

compliant functions and not, a commonality is the single reference point or target. Our proposal

better reflects the behavioral aspect of investing by incorporating an acceptable level of loss and

an upside target. Moreover, the dissection of variance into upper (UPM) and lower (LPM)

partial moments allows for the heterogeneous interpretations investors have towards

above-target variances versus below target variances.

1. THE USE OF BENCHMARKS

A valid utility function representing wealth over time should be governed by the simple axiom:

utility is equal to a change in wealth with respect to an initial wealth condition. The formula

representing this can be stated in Equation 1 as:

U(Wt) = U(wt-1) + U(w) (1)

We cannot go back to zero wealth and track utility from the first change in wealth such as a

birthday card from a relative or a summer job as a teen. We have to assume some level of initial

wealth. There is a subjective interpretation that individuals have towards a nominal level of

wealth (w). $1,000,000 may mean two different things to two different people. The amount of


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utility U(1,000,000) will be reflected by how close that wealth is to their Personal Consumption

Satiation level (PCS). Some individuals may feel content with that amount for a lifetime, while

others will simply see it as inadequate to provide satiation. Any differences in interpretations of

a given level of an identical wealth will solely be the result of different PCS benchmarks.

Example, if Investor A has a PCS level of $5,000,000 while Investor B has a PCS level of

$50,000,000 and they both have a level of wealth of $1,000,000, who is going to have a higher

utility U(w)? The $1,000,000 will simply mean more to Investor A as it represents a greater

percentage of the PCS, and thus will feel wealthier.

Harry Markowitz notes in his 1952 work whereby he criticizes Friedman and Savages 1948

hypothesis,

Consider two men with wealth equal to C +1/2(D-C) (i.e., two men who aremidway between C and D). There is nothing these men would prefer, in the wayof a fair bet, rather than one in which the loser would fall to C and the winnerwould rise to D. The amount of the bet would be (DC)/2 half the size of theoptimal lottery prize. At the flip of a coin the loser would become poor; the

winner, rich. Not only would such a fair bet be acceptable to them but nonewould please them more.

We do not observe persons of middle income taking large symmetric bets. Weexpect people to be repelled by such bets. If such a bet were made, it wouldcertainly be considered unusual and probably irrational.

Upon investigation of this scenario, it is clear that both individuals along with identical wealth

and risk profiles, will have identical symmetrical benchmarksfor this wager (D, rich and C,

poor). Thus the benchmarks for this wager are a proxy for wealth and risk profiles. Whether the

two individuals are compelled to take such a wager is not relevant. This focuses our attention the

benchmarks used in the computation of U(w).


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Markowitz (1952) centers his utility curve on a level of wealth that he describes as customary

wealth, or current wealth U(w). We do not differ as our y-axis is U(w). Where we do differ is

that our y-axis is the meeting point for both our autonomous loss and gain functions. As

explained in Viole and Nawrocki (2011), an increased subjective wealth interpretation will be

reflected via a relatively smaller upside benchmark (UPM) relative to the UPM of a decreased

feeling of wealth, and a larger acceptable level of loss (LPM) compared to the LPM of decreased

feeling of wealth. Our curve is simply two Markowitz Stochastic Dominance (MSD) Reverse-S

curves that meet at customary wealth (See Levy and Levy, 2002); one curve is centered on an

upside target, and the other curve centered on an acceptable level of loss. Figure 1 illustrates the

MSD utility curve. In the aggregate wealth function, the upside benchmark is the PCS and the

lower benchmark is S, Roys(1952) safety first level of subsistence. Analysis of these

benchmarks associated with an investment U(x) will be a valid proxy for effects on U(w) in the

instance when x = 0 or no return. The MSD utility curve is also self-similar between the

aggregate utility of wealth and the utility of an individual investment. The difference between

aggregate and per investment utility is the x-axis on the aggregate wealth function U(w) is lower

bound by zero, whereby the x-axis on the investment function U(x) is lower bound by the

amount of the investment.


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Figure 1. Utility function consistent with Markowitz Stochastic Dominance

2. A PROPOSED UTILITY FUNCTION USING UPM AND LPM

Our function builds on a utility function by Holthausen (1981) by making a few assumptions:

Assumption 1

The first assumption we make is that the investor has performed all of the necessary probability

computations they require to accept their targets. Whether they are ultimately right, yielding an

optimal solution is not of concern. This avoids placing a weight or probability variable to the

function which would only be additive to the inherent specification error, since that weight or

probability is already reflected in the computation of the targets.


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Assumption 2

Our second assumption is there is no a priori reason to constrain subjective variance parameters.

One major result derived in the paper is that the tmodel is congruent

with a von Neumann-Morgenstern utility function of the form

() ( )() (2)

Where kis a positive constant and, by construction U(t) = 0 and U(t+1) = 1Holthausen (1981).

xis the investment, tis the single target from which Holthausen derives both utilities, kis

the loss aversion parameter and is the curvature of the loss function, andis the

curvature of the gain function.

Equations 3 and 4 representing the n-degree LPM and q-degree UPM:

(,,) 1 [{0,,}

=] (3)

(, ,) 1 [{0,, }

=] (4)

where Rx,zrepresents the returns of the investmentxat timez, nis the degree of the LPM, qis the

degree of the UPM, his the target for computing below target returns, and l is the target for

computing above-target returns.

By relaxing the constriction of Holthausens (1981) andin Equation 2 we do not need a

constant kto generate loss aversion. Rather the ratio of loss aversion exponent nand gain-

seeking exponent qwill be able to replicate the loss aversion notion via n> q. With nand q> 1


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we can also compensate for above-target risk-seeking behavior that eludes Holthausen due to

1. For our model, n< qdictates overall risk seeking while supporting any nonlinear variance

interpretations.

Assumption 3

The third assumption we make is if that target or reference point equals zero, it is a fair bet. In

Markowitzs example of his dual targets, C and D, DC = 0 because they are symmetrical.

Thus the aggregated target gains minus losses equals zero.

In Figure 2 below we have illustrated the complete function using Figure 1, the MSD Reverse-S

function. We have one MSD for gains, centered on an upside target and one MSD for losses,

centered on an acceptable level of loss. This function is for an individual investment, U(x).

In the aggregate utility of wealth U(w), the upside target is the PCS from which a MSD Reverse-

S function will be centered on. In the loss region, the lower target is Roys Safety first level, or

S,serving as the reference point for that MSD Reverse-S function.


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MSD Reverse-S

centered on upside

target

MSD Reverse-S

centered on

acceptable level of

loss

GAINS

LOSSESUpside target for

investment. PCS level

for aggregate wealth.

Acceptable loss for

investment. Slevel for

aggregate wealth.

Figure 2. Superimposing 2 MSDs onto larger utility axes, revealing final shape of our

function, concave - convexconcaveconvex.

U(w) or customary

wealth (dashed y-axis).


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SINGLE BENCHMARK UPM / LPM UTILITY

Figure 3. Single Benchmark Utility Function.

LPM Conditional Values(LPM(n,h,x)

LPM(n,y,x) + U(0)) If LPM (n,h,x)LPM(n,h,y)>0

( LPM(n,h,y)+ UPM(n,y,x) + U(0)) If LPM (n,h,y)LPM(n,h,x)>0

(LPM(n,h,y)) If LPM (n,h,x) = 0U(x) =

UPM Conditional Values(UPM (q,l,a) LPM (q,a,x) + U(0)) If UPM (q,l,a)UPM (q,l,x)>0

(UPM (q,l,x)+ UPM (q,a,x) + U(0)) If UPM (q,l,x) UPM (q,l,a) >0

(UPM (q,l,a)) If UPM (q,l,x)= 0

UPM ( ) = Upper partial moment (degree, target, investment)LPM ( ) = Lower partial moment (degree, target, investment)n = Investor's loss aversion levelq = Investor's gain-seeking appetiteh = Target for computing LPMl = Target for computing UPMx = Investmenty = Benchmark Y (Acceptable level of loss)a = Benchmark A (Upside target)


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The next section of the paper will place our proposed utility model into the literature. Then we

will explore the implications of the literature for our utility model. A short discussion,

conclusions, and currently observed zero return investor behavior will follow.

II. REVIEW OF THE LITERATURE

It should be noted that a more extensive review of the literature is provided in Appendix A. The

utility formula for an investmentU(x) can be represented by the marginal change in wealth it

produces.

U(x) = U(w) (5)

One would assume a zero return on an investment should produce no change in the aggregate

utility of wealth.

U(wt) = U(wt-1) + U(x)

U(wt) = U(wt-1) + U(0)

U(wt) = U(wt-1) (6)

But, if we do not assume U(0) = 0 and isolate U(0) we have Equation 7 which illustrates that

U(0) can be zero, positive or negative.

U(wt) = U(wt-1) + U(0)

U(0) = U(wt)U(wt-1) (7)


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Using Allaisparadox (Allais, 1953), we can show that U(0) does not equal zero when there is a

certain outcome and can be zero, negative, or positive when there is no certain outcome.

EXPERIMENT 1 (Most participants choose A)

SITUATION A SITUATION B

$1m with certainty 89% chance of $1m; 10% chance of $5m; 1% chance of nothing

U($1 M) > 0.89U($1 M) + 0.01U($0 M) + 0.1U($5 M)

-0.01U($0 M) > 0.1U($5 M)0.11U($1 M)

U($0 M) < 0

EXPERIMENT 2 (Most participants choose B)

SITUATION A SITUATION B

89% chance of nothing; 11% chance of $1m 90% chance of nothing; 10% chance of $5m

0.89U($0 M) + 0.11U($1 M) < 0.9U($0 M) + 0.1U($5 M)

-0.01U($0 M) < 0.1U($5 M)0.11U($1 M)

U($0 M) 0


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What Allais decisionexamples isolate is the effect of the certainty equivalence and how it can

rationally distort any expected value calculation. Post et al. (2008) note the Dutch contestants

take a higher discount to expected value when the offer is a larger multiple of their annual

income in an analysis of the gameshow Deal or No Deal. This observed behavior is completely

consistent with risk aversion and a larger upside target of their local MSD for gains utility

decisions.

HISTORICAL UTILITY FUNCTIONS

An examination of the historical utility functions that are compliant with von Neumann and

Morgenstern (1947) as well as Prospect Theory (Kahneman and Tversky, 1979) will reveal what

the utility interpretation is for a zero-return situation. Table 2 summarizes these results derived

in Appendix A. One consistency to note is that there is a single target or reference point, and it

may take any value as per Fishburn and Kochenberger (1979):

This target point (Fishburn, 1977 and Libby and Fishburn, 1977) or referencepoint (Kahneman and Tversky, 1979) is often the zero-gain point, as inMarkowitzs (1952) case or in Swalms(1966) data, but it may be in the lossregion (for some wildcatters as in Grayson, 1960) or the gain region (Green,1963).


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a. FISHBURN (1977)

U(x) = x for all x t

U(x) = xk (tx) for all x t

where x is the investment, t is the target, k is the loss aversion parameter and is thecurvature of the loss function.

b. FISHBURN AND KOCHENBERGER (1979)

U(x0) = u0+ u(k1(x0t))/k2 for all x0 t

U(x0) = u0u(k3(tx0))/k4 for all x0 t

where x0is the investment, u0 is the utility function in the original data format, t is thetarget, k1and k2 are the gain-seeking parameters, k3 and k4 are the loss aversionparameters.

c. HOLTHAUSEN (1981)

U(x) = (xt) for all x t

U(x) =k(tx) for all x t

where x is the investment, t is the target, k is the loss aversion parameter and is thecurvature of the loss function, and is the curvature of the gain function.


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d. PROSPECT THEORYKAHNEMAN AND TVERSKY (1979)

(|) ( ) ( ) >

where x is the investment, RP is the reference point, is the loss aversion parameter and is both the curvature of the loss function and the gain function.

Table 2. Comparison of U(0) and benchmark scenarios

(*) von Neumann-Morgenstern (1947) compliant() U(0) = 0 forced assumption

U(0)Fishburn*

Fishburn and

Kochenberger*Holthausen*

Prospect

Theory

Positive

Benchmark

Negative 0 Negative Negative

Negative

Benchmark

0 0 Positive Positive

Zero

Benchmark

(Fair Bet)

0 0 0 0


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III. U(0) INTERPRETATIONS AND FURTHER DEVELOPMENT

One possible explanation for the negative utility generated from a zero return is opportunity cost.

However, isnt the first goal of risk aversion not to lose? Furthermore, if a risk-averse investor's

first goal is to not lose, doesn't a zero return satisfy that goal? Then why would an investor

realize a negative utility from such an event? Opportunity cost would be more aligned with a

risk-seeking investor. A risk-averse investor should exhibit a form of relief or positive utility

from a zero return, consistent with a break-even effect. Markowitz (2010) notes this notion,

Because not buying the ticket has a utility U(0) = 1. The logic behind this observation is

provided by Markowitz (1959):

"In general, any outcome can be assigned utility equal to zero, any better outcomecan be assigned utility equal to one. On a utility curve, such as that in Figure 1for example, the level of return with zero utility and the (higher) level of returnwith unit utility can be chosen arbitrarily. The zero and unit of the utility scale,therefore, is a matter of convention and convenience.It is shown in the footnote that, once the zero and unit of the utility scale arechosen, the rest of the utility curve follows from the individual's preferences. Nomore points may be arbitrarily selected without affecting the preferencesrepresented by the curve."

The single target or reference point is one major hindrance in reflecting a level of wealth. By

identifying an acceptable level of loss and an upside target (akin to C and D per Markowitz,

1952, or something as temporal as a stop loss and a limit order investors typically use), we avoid

this hindrance and are able to let the individual preferences of the function generate both

negative and positive levels of utility for a zero return under different wealth benchmark

configurations, rather than arbitrarily set the wealth benchmark at zero or one.


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1. SYMMETRICAL BENCHMARKS

For a zero return x = 0, this situation would infer that there would be no UPM or LPM.

When

LPM(n,h,x) = 0 and UPM (q,l,x) = 0

Then our utility function reduces to,

U(0) =LPM(n,h,y) + UPM(q,l,a).

Ify(acceptable level of loss) = a(upside return target), h(target to compute lower partial

moment from) = l(target to compute upper partial moment from) and n(loss aversion exponent)

= q(risk-seeking exponent) (symmetrical benchmarks from the same target[does not have to be

0 as for nominal gains and losses]; indifference loss exponent ratio) then, U(0) = 0.

Figure 4. Upside target = acceptable level of loss. Symmetrical benchmarks.

-80

-60

-40

-20

0

20

40

60

80

-8

-6

-4

-2 0 2 4 6 8

Lower Partial

Moment

Upper PartialMoment

Upside

Target

Acceptable

level of loss

LOSSES GAINS

TILIT

Y


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The symmetrical inflection points at benchmarksyand a(C and D respectively per Markowitz

1952), intersects the y-axis, utility, at 0. This symmetrical benchmark setup represents a fair bet

to the investor. The function is continuous when U(0) = 0.

The local investment U(0) = 0 or no change in wealth from the investment, will not affect the

aggregate level of overall utility. This makes sense considering a zero return and indifference of

the investor.

But what about an opportunity cost?


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2. ASYMMETRICAL BENCHMARKS: INCREASED WEALTH

Ify> a, h= land n= q(asymmetrical benchmarks from the same target; indifference loss

exponent ratio) then

U(0) =LPM(n,h, y) + UPM(q,l,a)

U(0) = negative or U(0) < 0

The LPM will be larger than the UPM generating a negative utility from a zero return reflecting

the opportunity cost. This is a function of the asymmetrical benchmarks identified with an

increased wealth applied to an investment.

No change in wealth from a zero return lowers the aggregate utility of overall wealth due to the

opportunity cost of that investment as illustrated in Figure 5.

Figure 5. Upside target < acceptable level of loss. Asymmetrical benchmarks reflecting

increased wealth.

-80

-60

-40

-20

0

20

40

60

80

100

-8

-6

-4

-2 0 2 4 6 8

Lower PartialMoment

Upper Partial

Moment

Upside

Target

Acceptable

level of loss

LOSSES GAINS

TILITY


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This opportunity cost can be viewed as the intersection of the y-axis at a negative utility point for

the loss function. The amount of the discontinuity at U(0) equals opportunity cost when the

acceptable level of loss is greater than the upside target. The opportunity cost is also additive, it

is experienced for all losses where x < 0.


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3. ASYMMETRICAL BENCHMARKS: DECREASED WEALTH

If a>y, h= l, and n= q(asymmetrical benchmarks from the same target; indifference loss

exponent ratio) then,

U(0) =LPM(n,h,y) + UPM(q,l,a)

U(0) = positive or U(0) > 0

The UPM will be larger than the LPM generating a positive utility from a zero return reflecting

the relief a less wealthy individual experiences from a break-even effect as shown in a decreased

UPM target and an increased LPM acceptable level of loss in Figure 6.

A no change in wealth raises the aggregate utility of overall wealth due to the break-even effect

of that investment.

Figure 6. Upside target > acceptable level of loss. Asymmetrical benchmarks reflecting

decreased wealth.

-100

-80

-60

-40

-20

0

20

40

60

80

-8

-6

-4

-2 0 2 4 6 8

Lower Partial

Moment

Upper Partial

Moment

Acceptable

level of loss

Upside

Target

TILITY

LOSSES GAINS


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This break-even effect can be viewed as the intersection of the y-axis at a positive utility point

for the gain function. The amount of the discontinuity at U(0) = break-even effect when the

acceptable level of loss is less than the upside target. The break-even effect is also additive, it is

experienced for all gains where x > 0.

4. EVIDENCE OF REVERSE-S UTILITY CURVES

One need only identify the notion of risk seeking with gains to provide an argument for

convexity of gain functions, originally proposed by Markowitz (1952). In order to not violate

the St. Petersburg Paradox, this convexity was ultimately concave to put a boundary on the y-

axis. We avoid the paradox by putting a boundary on the x-axis, by assuming there is some finite

level of wealth that can be ascribed to an asset. On the upside, the ultimate utility boundary is

the total amount of monetary assets available in circulation, while the downside boundary is zero

wealth or bankruptcy. Since utils are subjective units of measure, convexity of gains is less of a

concern than forcing concavity in a bid to address an absurd prior assumption of infinite

monetary resources.

Concavity of gains is also used to reinforce the notion of risk aversion. However, we are also

able to demonstrate overall risk aversion, not by concavity, but multiple other methods. First, the

ratio of the loss-seeking exponent versus the gain-seeking exponent (n:q) will be greater than one

for a risk-averse investor. Second, the associated benchmarks as described earlier will be shifted

to reflect this aversion, as noted in Figure 7 (in Appendix A) from Viole and Nawrocki (2011).

This flexibility and asymmetry is far superior to the rigidity of historical functions when being

applied to complex human behavior, such as greed in a risk-averse manner.


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Another example of human behavior that confounds historical functions is philanthropy. In this

sense, the utility derived from helping others is intangible. This intangible utility is also better

governed by an economies-of-scale convex function rather than the assumed concavity to

represent the positive feedback experienced.

To the first point, the evidence we reviewed is quite supportive: Happier peoplegive more and giving makes people happier, such that happiness and giving mayoperate in a positive feedback loop (with happier people giving more, gettinghappier, and giving even more) (Anik, Aknin, Norton and Dunn, 2009).

5.

DISCONTINUITY OF THE UTILITY FUNCTION

We note how the function is discontinuous under any asymmetry of benchmarks. This

discontinuity is critical in avoiding a major philosophical discrepancy arising from shifting a

continuous curve along the y-axis. To properly reflect a break-even effect (positive utility for a

zero return); the continuous function would have to be shifted upwards to intersect the y-axis at

this positive point. The unintended consequence from this action is positive utility readings from

losses, amplified by the amount of break-even effect and underlying curve characteristics.

Under no circumstances could a nominal loss be viewed as a positive utility generator in our

function. The discontinuity at one specific point, zero, ensures this critical notion is upheld with

our loss function upper bound by zero and our gain function lower bound by zero.

Fishburn and Kochenberger (1979) note a similar discontinuity at U(0) with their two-piece von

Neumann-Morgenstern utility function,

Finally, we note that P+and E+have a limiting form that is discontinuous at theorigin with U(0) = 0 and U(x) = k for all x > 0. This arises from P+when a1 = kand a2goes to zero, and from E

+when b1= k and b2goes to infinity.


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IV. CONCLUSIONS

How can a zero change in wealth offer different utility values? The intransitive nature of a utility

observation at time tmeans that even if Xt= Xt-1they can have wildly different interpretations

due to the shift in either a UPM or LPM benchmark. Post et al. (2008) note this,

The specification of the subjective reference point (or the underlying set ofexpectations) and how it varies during the game is crucial for our analysis, as itdetermines whether outcomes enter as a gain or a loss in the value function andwith what magnitude.

Fishburn and Kochenberger (1979) also note the influence of a target point,

First, there is usually a point t on the abscissa at which something unusualhappens to the individuals utility function. We shall refer to t as the target pointalthough we do not know with certainty that it served as a conscious target levelfor the individuals concerned.

We have identified how the heterogeneous benchmarks drive U(w) and we have noted how

they are influenced by the individuals subjective wealth interpretation. In a zero return scenario

we can examine how the benchmarks reflect this investment result, either as indifferent, an

opportunity cost or a relief.

One important distinction is that the lower upside benchmark (UPM) / larger acceptable level of

loss (LPM) can capture risk seeking from a less wealthy individual, essentially emulating a

wealthier individuals preferences. Risk-averse investors can become risk-seeking investors by

emulating targets and acceptable levels of loss from someone wealthier, forcing them to be

bigger risk seekers than their wealth level would rationally infer. Thus, knowing the initial

wealth level cannot provide the insight to the risk profile that the benchmarks reveal.


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Only in the instance of a fair bet does U(0) = 0. Under any asymmetry, the utility realized from a

zero return will be negative (representing an opportunity cost) or positive (representing a break-

even effect). This information cannot be derived from a level of wealth (w), it is only under an

inspection of the benchmarks associated with that level of wealth and a zero return that we can

better understand U(w). Thus,

U(wt-1) U(w)Opportunity Cost

LPM Conditional Values(LPM(n,h,x) LPM(n,y,x) + U(0)) If LPM (n,h,x)LPM(n,h,y)>0

( LPM(n,h,y) + UPM(n,y,x) + U(0)) If LPM (n,h,y)LPM(n,h,x)>0

(LPM(n,h,y)) If LPM (n,h,x) = 0U(w) = Break-Even Effect

UPM Conditional Values(UPM (q,l,a) LPM (q,a,x) + U(0)) If UPM (q,l,a)UPM (q,l,x)>0

(UPM (q,l,x) + UPM (q,a,x) + U(0)) If UPM (q,l,x) UPM (q,l,a) >0

(UPM (q,l,a)) If UPM (q,l,x)= 0

Anecdotally, we are currently witnessing this break-even effect phenomenon with individual

participant cash deposits at banks yielding zero. Furthermore, we are witnessing the same

actions by the banks themselves with their record deposits of excess reserves at the Federal

Reserve yielding zero. Thus it can be inferred that the aggregated target or reference point these

participants are using for this investment is negative, generating a positive utility;since their

actions are clearly maximizing their expected utility. This is fully consistent with a risk-averse,

decreased feeling of wealth for an investment generating the positive utility break-even effect for

a zero investment return.


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Levy, Moshe and Haim Levy. (2002). Prospect Theory: Much Ado About Nothing.Management Science,v48(10), 1334-1349.

Libby, R., and P.C. Fishburn. (1977). Behavioral Models of Risk Taking in Business Decisions:A Survey and Evaluation. Journal of Accounting Research, v15(Autumn 1977), 272-292.

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Experimental Social Psychology, 20, 255-295.

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Markowitz, H. (2010). Porfolio Theory: As I Still See It.Annual Review of FinancialEconomics, v2, 1-41.

Post, T., M. Van den Assem, G. Baltussen, and R. Thaler. (2008). Deal or No Deal? DecisionMaking under Risk in a Large-Payoff Game Show. American Economic Review, v98(1),(March 2008), 38-71.

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http://ssrn.com/abstract=1444831http://ssrn.com/abstract=1444831http://ssrn.com/abstract=1444831


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Swalm, R. D. (1966). Utility Theory Insights into Risk Taking. Harvard Business Review,v47 (November-December 1966), 123-136.

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Von Neumann, J., and O. Morgenstern. (1947). Theory of Games and Economic Behavior,Second Edition. Princeton, NJ: Princeton University Press.


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Fishburn 0 Return and a 0 Target




U(0) = 0


U(0) = 0k (00)

U(0) = 0

If t= 0 and x = 0 then both utilities are valid. Thus U(0) = 0. t = 0 however, infers a fair bet thatwould be unusual and probably irrational for the investor.


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A.2 FISHBURN AND KOCHENBERGER (1979)

U(x0) = u0+ u(k1(x0t))/k2 for all x0 t

U(x0) = u0u(k3(tx0))/k4 for all x0 t

Fishburn and Kochenberger (1979) force U(0) = 0 and explain the rationale as follows:

The restriction of U(0) = 0, or U(t) = u0, limits the goodness of fit, since it forcesthe function to pass through the indicated point. Better overall fits could beobtained by not requiring U(0) to equal zerofor example, by using cx + dinstead of cx for the linear fitsbut the U(0) = 0 constraint for all functions aboveand below target ensures the continuity of each two-piece utility function at thetarget. Continuity at t could also be ensured without forcing u(0) to equal zero,but best two-piece fits with the forms used here would require a simultaneousbelow-above fit. One might also fit a single function through all the data points

(e.g., a function that can have both convex and concave segments). We did not dothis, however, since we felt that the separately fit pieces were adequate toexamine the issues raised in the introduction.


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A.3 HOLTHAUSEN (1981)



Holthausen 0 Return below a POSITIVE Target

U(x) =k(t-x) for all x t

U(0) =k(t0)

U(0) =k(t)

A 0 return will then generate a negative utility with a more negative result for the more risk-averse investor (higher k). U(0) < 0 for all t > 0. This is consistent with Fishburn (1977) andHolthausens goal of consistency (1981), The purpose of this paper is to present a risk-returnmodel that has many of the same attributes as Fishburns model, but one in which the utilityfunction for above-target outcomes need not be linear.

Holthausen 0 Return above a NEGATIVE Target


U(0) = (0t)

U(0) = ((t))

A 0 return will generate a positive utility for all levels of. U(0) > 0 for all t < 0.


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Holthausen 0 Return and a 0 Target




U(0) = (00)

U(0) = 0


U(0) =k(00)

U(0) = 0

If t= 0 and x = 0 then both utilities are valid. Thus U(0) = 0. t = 0, however, infers a fair bet thatwould be unusual and probably irrational for the investor.

Holthausen (1981) handles U(0) only when x = t, not in the context of a U(0) when x t.

One major result derived in the paper is that the tmodel is congruent

with a von Neumann-Morgenstern utility function of the form

() ( )() Where k is a positive constant and, by construction U(t) = 0 and U(t+1) = 1.


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A.4 PROSPECT THEORYKahneman and Tversky (1979)

(|) ( ) ( ) >

Prospect Theory 0 Return below a POSITIVE Reference Point

v(x|RP) = -(RP x) for x RPv(0|RP) = -(RP 0)

v(0|RP) = -(RP)

A 0 return will then generate a negative utility with a more negative result for the more risk-averse investor (higher ). U(0) < 0 for all RP > 0.

Prospect Theory 0 Return above a NEGATIVE Reference Point

v(x|RP) = (x RP) for x > RPv(0|RP) = (0 RP)

v(0|RP) = ( (RP))

A 0 return will then generate a positive utility where = 0.88.

Prospect Theory 0 Return and a 0 Reference Point

v(x|RP) = (RP x) for x RP

v(x|RP) = (RP x) for x RPv(0|0) = (0 0)v(0|0) = 0

Only in the loss function can x = RP = 0. Thus U(0) = 0 under a fair bet scenario for ProspectTheory as well.


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WEALTH EFFECT ON SINGULARITY UTILITY

Figure 7. Increased wealth effect on singularity utility model

As overall wealth increases (blue function), LPM and UPM will shift left representing an

increased ability / propensity to gamble house money effect, reducing the certainty

equivalence for UPM. Downside tolerances are also increased highlighting the effect of

accumulated wealth on the investor's core subsistence level.

A decreased overall wealth (red function) will shift LPM and UPM to the right, illustrating the

increased effect of any losses to a poorer individual as it is closer to encroaching upon their basic

necessities (Roy's, 1952 subsistence level S) as well as the increased amount to reach personal

consumption satiation (PCS).

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An Analysis of Heterogeneous Utility Benchmarks in a Zero Return Environment

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