Preferences In Prospect Theory

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Preferences In Prospect Theory

An essential ingredient of any model trying to understand asset prices or trading behaviour is an assumption about investor preferences, or about how investors evaluate risky gambles. The vast majority of models assume that investors evaluate gambles according to the expected utility framework (EU). The theoretical motivation for this goes back to Von Neumann and Morgenstern (1944) (VNM), who show that if preferences satisfy a number of plausible axioms – completeness, transitivity, continuity, and independence (see the next slide) – then they can be represented by the expectation of a utility function. Saturday 22 April 2023 09:01 AM

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Completeness entire; having everything that is needed.

Transitivity a relation between three elements such that if it holds between the first and second and it also holds between the second and third it must necessarily hold between the first and third. For example a>b, b>c ⇒ a>c.

Continuity uninterrupted; the opposing concept is discreteness.

Independence the occurrence of one event makes it neither more nor less probable that the other occurs.

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Unfortunately, experimental work in the decades after VNM has shown that people systematically violate EU theory when choosing among risky gambles. For an historical perspective, tracing its roots to the Swiss mathematician Nikolaus Bernoulli (September 1713), see Szpiro (2013).

Economics: Value judgements - Szpiro - Nature 2013

In response to this, there has been an explosion of work on so-called non-EU theories, all of them trying to do a better job of matching the experimental evidence.

http://www.nature.com/nature/journal/v500/n7464/full/500521a.html

http://www.nature.com/nature/journal/v500/n7464/full/500521a.html

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Some of the better known models include

Weighted-utility theory (Chew and MacCrimmon 1979, Chew 1983),

Implicit expected utility (Chew 1989, Dekel 1986), Disappointment aversion (Gul 1991), Regret theory (Bell 1982, Loomes and Sugden 1982), Rank-dependent utility theories (Quiggin 1982, Segal 1987, 1989, Yaari 1987), Prospect theory (Kahneman and Tversky 1979, Tversky and Kahneman 1992).

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Should financial experts be interested in any of these alternatives to expected utility?

It may be that EU theory is a good approximation to how people evaluate a risky gamble like the stock market, even if it does not explain attitudes to the kinds of gambles studied in experimental settings.

“Essentially, all models are wrong, but some are useful.” ‘Empirical Model-Building and Response Surfaces’ 1987, George E. P. Box and Norman R. Draper, p. 424, ISBN 0471810339

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On the other hand, the difficulty the EU approach has encountered in trying to explain basic facts about the stock market suggests that it may be worth taking a closer look at the experimental evidence.

Indeed, recent work in behavioural finance has argued that some of the lessons we learn from violations of EU are central to understanding a number of financial phenomena.

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Guidelines

Normative a model of how people ought to choose

Descriptive a practical aid to choice

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Of all the non-EU theories, prospect theory may be the most promising for financial applications, and we discuss it in detail.

The reason we focus on this theory is, quite simply, that it is the most successful at capturing the experimental results.

In a way, this is not surprising. Most of the other non-EU models are what might be called quasi-normative, in that they try to capture some of the anomalous experimental evidence by slightly weakening the VNM axioms.

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Normative refers to value judgments as to "what ought to be”, in contrast to descriptive which is about "what is”.

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The difficulty with such models is that in trying to achieve two goals

normativedescriptive

they end up doing an unsatisfactory job at both.

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The normative theory is concerned with prescribing courses of action that conform most closely to the decision maker’s beliefs and values.

Describing these beliefs and values and the manner in which individuals incorporate them into their decisions is the aim of descriptive decision theory.

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In contrast, prospect theory has no aspirations as a normative theory, it simply tries to capture people’s attitudes to risky gambles as parsimoniously as possible.

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Indeed, Tversky and Kahneman (1986) argue convincingly that normative approaches are doomed to failure, because people routinely make choices that are simply impossible to justify on normative grounds, in that they violate dominance or invariance.

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Kahneman and Tversky (1979) (KT) lay out the original version of prospect theory, designed for gambles with at most two non-zero outcomes. They propose that when offered a gamble

(x,p; y,q)

to be read as “get outcome x with probability p, outcome y with probability q”, where x≤0≤y or y≤0≤x.

So a win and a loss.

People assign it a value …

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People assign it a value of

π(p)υ(x)+π(q)υ(y)

Where υ and π are shown in the figure. When choosing between different gambles, they pick the one with the highest value.

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Kahneman and Tversky’s 1979 proposed value function υ and probability weighting function π.

When choosing between different gambles, they pick the one with the highest value.

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This formulation has a number of important features.

First, utility is defined over gains and losses rather than over final wealth positions, an idea first proposed by Markowitz (1952).

This fits naturally with the way gambles are often presented and discussed in everyday life.

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More generally, it is consistent with the way people perceive attributes such as brightness, loudness, or temperature relative to earlier levels, rather than in absolute terms.

Kahneman and Tversky (1979) also offer the following violation of EU as evidence that people focus on gains and losses.

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Subjects are asked:

In addition to what ever you own, you have been given 1000. Now choose between

A = (1000,0.5) Win 1000 with a probability of 0.5 so 1000 or

2000

B = (500,1) Win 500 with certainty so 1500

What is your choice?B was the more popular choice.

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The same subjects were then asked:

In addition to whatever you own, you have been given 2000. Now choose between

C = (-1000,0.5) Lose 1000 with probability 0.5 so 1000 or 2000

D = (-500,1) Lose 500 with certainty so 1500

What is your choice?This time, C was more popular.

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Note that the two problems are identical in terms of their final wealth positions and yet people choose differently. (Choices B: 1500 and C: 1000 or 2000 while A=C and B=D)

The subjects are apparently focusing only on gains and losses.

Indeed, when they are not given any information about prior winnings, they choose B (B = (500,1)) over A and C (C = (-1000,0.5)) over D.

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The second important feature is the shape of the value function υ, namely its concavity in the domain of gains and convexity in the domain of losses.

Put simply, people are risk averse over gains, and risk seeking over losses.

Simple evidence for this comes from the fact just mentioned, namely the case in which there was no information about prior winnings.

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G1►G2 should be read as “a statistically significant fraction of subjects preferred G1 to G2.”

In the previous example

B ►A, C ►D

The υ function also has a kink at the origin, indicating a greater sensitivity to losses than to gains, a feature known as loss aversion.

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Loss aversion is introduced to capture aversion to bets of the form:

E = (110, ½ ;-100, ½).

Recall (x,p; y,q)

It may seem surprising that we need to depart from the expected utility framework in order to understand attitudes to gambles as simple as E, but it is none the less true.

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In a remarkable paper, Rabin (2000) shows that if an expected utility maximizer rejects gamble E at all wealth levels, then he will also reject

(20000000, ½;-1000, ½),

an utterly implausible prediction.

The intuition is simple: if a smooth, increasing, and concave utility function defined over final wealth has sufficient local curvature to reject E over a wide range of wealth levels. It must be an extraordinarily concave function, making the investor extremely risk averse over large stake gambles.

http://www.jstor.org/stable/pdfplus/2999450.pdf?acceptTC=true&jpdConfirm=true

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The final piece of prospect theory is the non-linear probability transformation. Small probabilities are over weighted, so that π(p) > p. This is deduced from Kahneman and Tversky’s finding that

(5000, 0.001) ►(5, 1),

and

(-5, 1) ►(-5000, 0.001),

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Together with the earlier assumption that υ is concave (convex) in the domain of gains (losses). More over, people are more sensitive to differences in probabilities at higher probability levels. For example, the following pair of choices,

(3000, 1) ►(4000, 0.8; 0, 0.2), That is υ(3000) π(1) > υ(4000) π(0.8)

(4000, 0.2; 0, 0.8) ►(3000, 0.25),That is υ(4000) π(0.2) > υ(3000) π(0.25)

which violate EU theory (transitivity).

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Preferences In Prospect Theoryυ(3000) π(1) > υ(4000) π(0.8)

υ(4000) π(0.2) > υ(3000) π(0.25)

Imply

that is

π(0.25)

<

υ(4000

)<

π(1)

π(0.2)υ(3000

)π(0.8)

π(0.25)<

π(1)

π(0.2) π(0.8)

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The intuition is that the 20% jump in probability from 0.8 to 1 is more striking to people than the 20% jump from 0.2 to 0.25.

In particular, people place much more weight on outcomes that are certain relative to outcomes that are merely probable, a feature sometimes known as the “certainty effect”.

π(0.25)<

π(1)

π(0.2) π(0.8)

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Along with capturing experimental evidence, prospect theory also simultaneously explains preferences for insurance and for buying lottery tickets.

Although the concavity of υ in the region of gains generally produces risk aversion, for lotteries which offer a small chance of a large gain, the over weighting of small probabilities dominates, leading to risk-seeking.

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Along the same lines, while the convexity of υ in the region of losses typically leads to risk seeking, the same over weighting of small probabilities induces risk aversion over gambles, which have a small chance of a large loss.

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Gollier and Muermann (2010) apply their decision criterion to the simple portfolio/insurance decision problem, They show that individuals are more reluctant to take on risk than predicted by the expected utility model and that this reluctance is increasing in the degree of optimism.

Their decision criterion can thus help explain the equity premium puzzle and the preference for low deductibles (the amount of expenses that must be paid out of your pocket ) in insurance contracts.

Their revisited model provides a suitable framework to think of simultaneous demand for insurance and lotteries, a puzzle pointed out by Friedman and Savage (1948). Their model can explain the coexistence of insurance and lottery demand with the fear of disappointment (after the event) and the desire to savour (prior to the event).

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Jouini et al. (2014) when considering savouring (prior to the event) and disappointment (after the event) explain the popularity of lottery games (Thaler and Ziemba 1988) despite their negative expected returns and the underperformance of lottery-type stocks (Kumar 2009, Bali et al. 2011) (i.e., gambling enables agents to dream). This taste for lottery-type stocks and extreme values is also a possible explanation for portfolio under diversification (Mitton and Vorkink 2007).

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Earlier in this section, we saw how prospect theory could explain why people made different choices in situations with identical final wealth levels (choices A,B,C,D) . This illustrates an important feature of the theory, namely that it can accommodate the effects of problem description, or of framing.

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Such effects are powerful. There are numerous demonstrations of a 30% to 40% shift in preferences depending on the wording of a problem (Page 20 Barberis and Thaler 2005).

The wording of a problem induces a shift in the reference point and leads to different choices (Bottom and Studt, 1993; van Schie and van der Plight, 1995). Lopes (1987) points out that reference points can come from a variety of sources, such as a person's sense of what outcome is reasonable or the presence of other alternatives in the choice set.

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No normative theory of choice can accommodate such behaviour since a first principle of rational choice is that choices should be independent of the problem description or representation.

Framing (4.1) refers to the way a problem is posed for the decision maker. In many actual choice contexts the decision maker also has flexibility in how to think about the problem.

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For example, suppose that a gambler goes to the racetrack and wins $200 in his first bet, but then losses $50 on his second bet.

Does he code the outcome of the second bet as a loss of $50 or as a reduction in his recently won gain of $200?

In other words, is the utility of the second loss υ(‑ 50) or (υ(150) - υ(200))?

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The process by which people formulate such problems for themselves is called mental accounting (Thaler 2000). Mental accounting matters because in prospect theory, υ is non-linear.

One important feature of mental accounting is narrow framing, which is the tendency to treat individual gambles separately from other portions of wealth. In other words, when offered a gamble, people often evaluate it as if it is the only gamble they face in the world, rather than merging it with pre-existing bets to see if the new bet is a worthwhile addition.

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Redelmeier and Tversky(1992) provide a simple illustration, based on the gamble

F = (2000, ½ ;-500, ½ ).

Subjects in their experiment were asked whether they were willing to take this bet; 43% said they would (so 57 % would not).

They were then asked whether they would prefer to play F five times or six times; 70 % preferred the six-fold gamble.

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Finally they were asked:Suppose that you have played F five times but you don’t yet know your wins and losses. Would you play the gamble a sixth time?

60 % rejected the opportunity to play a sixth time, reversing their preference from the earlier question.

This suggests that some subjects are framing the sixth gamble narrowly, segregating it from the other gambles. Indeed, the 60 % rejection level is very similar to the 57 % rejection level for the one-off play of F.

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Hoffmann et al. (2013) examined the importance of aspirations as reference points in a multi-period decision-making context. After stating their personal aspiration level, 172 individuals made six sequential decisions among risky prospects as part of a choice experiment. The results show that individuals make different risky-choices in a multi-period compared to a single-period setting. In particular, individuals' aspiration level is their main reference point during the early stages of decision-making, while their starting status (wealth level at the start of the experiment) becomes the central reference point during the later stages of their multi-period decision-making.

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First, they demonstrate that a multi-period feature is important to better understand and predict individuals’ risk-behaviour.

Second, the results indicate that aspiration levels act as main reference point during the early stages of decision-making, whereas the starting status becomes an important reference point during later stages.

Third, the results show how prior outcomes influence subsequent choices (Hoffmann et al. 2013) .

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Lady Luck is fickle, but many of us believe we can read her mood. A new study of one year's worth of bets made via an online betting site shows that gamblers' attempts to predict when their luck will turn has some unexpected consequences (Xu and Harvey 2014).

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A common error in judging probabilities is known as the Gambler's Fallacy. This is the belief that independent chance events have an obligation to 'even themselves out' over the short term, so that a run of wins makes a loss more likely, and vice versa. An opposite error is the belief that a run of good luck predicts more good luck – when a basketball player succeeds in a number of successive shots, they are said to have a 'Hot Hand', meaning a better chance of succeeding with their next shot. While the hot hand might be possible in games of skill, it is a logical impossibility for truly chance events (Xu and Harvey 2014).

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