# Essays On Prospect Theory And Asset Pricing Math

## Content

List of abbreviations

List of symbols

1 Introduction

1.1 Introduction to asset pricing

1.2 Objective of this paper

2 The Capital Asset Pricing Model

2.1 Derivation of the CAPM

2.1.1 Firm-Specific Risk vs. Market Risk

2.1.2 The beta coefficient

2.1.3 The CAPM Equation

2.2 The Security Market Line (SML)

2.3 Assumptions of the CAPM

3 Problems of the CAPM

3.1 Unrealistic assumptions

3.2 Empirical Testing of CAPM

3.2.1 General Testing Problems

3.2.1.1 The Problem of ex ante Data

3.2.1.2 Roll’s critique

3.2.2 Results of empirical Tests

3.2.2.1 Controversy about Beta

3.2.2.2 Empirical support for other risk factors

3.2.3 Conclusion

4 New Developments

4.1 Neoclassical Models – Traditional Asset Pricing

4.1.1 ‘Smaller’ adjustments of the CAPM

4.1.1.1 Zero-Beta CAPM

4.1.1.2 Introducing taxes and transaction costs

4.1.1.3 International Capital Asset Pricing Model

4.1.1.4 Option pricing in CAPM context

4.1.2 Multi-factor Models

4.1.2.1 The Arbitrage Pricing Theory (APT)

4.1.2.2 Fama-French Three-Factor Model

4.1.3 Multi-period models

4.1.3.1 The intertemporal CAPM (ICAPM)

4.1.3.2 The Consumption-Based CAPM (CCAPM)

4.1.3.3 Production-based Asset Pricing Model

4.1.4 General Puzzles of Traditional Asset Pricing Models

4.1.4.1 Equity premium puzzle

4.1.4.2 Risk-Free Rate Puzzle

4.2 Behavioral Finance

4.2.1 Introduction

4.2.2 Evidence contradicting the efficient market

4.2.2.1 Royal-Dutch-Shell shares

4.2.2.2 IPO Palm

4.2.3 Pillars of Behavioral Finance

4.2.3.1 Psychology

4.2.3.2 Limits to arbitrage

4.2.3.3 Summary

4.2.4 Prospect-Theory Model

4.2.4.1 Key Elements

4.2.4.2 Assessment

4.2.5 Habit Formation Models

4.2.6 Models with heterogeneous Agents

4.2.7 Conclusion

4.3 Chaos, synergetic models and neural networks

5 Conclusion

Bibliography

Declaration of Academic Integrity

## List of abbreviations

illustration not visible in this excerpt

## List of symbols

illustration not visible in this excerpt

## 1 Introduction

### 1.1 Introduction to asset pricing

Asset pricing theory tries to explain why some assets pay higher average returns than others. Accordingly, the objective is to understand the prices or values of claims to uncertain payments. (Cochrane, 2005, p. XIII)

The central aspect is the risk-return tradeoff. It is rational that investors demand additional return for an asset incorporating more risk. This relationship can also be empirically examined when looking at the return development of different assets. For example, between 1926 and 1999, small U.S. stocks yielded average returns of almost 19%, while at the same time large stocks yielded 13% and US Treasury-Bills only about 4%. When looking at the risk of the assets, as measured by the standard deviation of the returns, the relationship becomes obvious: small stocks had a standard deviation of almost 40%, while large stocks and U.S. treasury-bills had 20 % and 3%, respectively. (Tuck School of Business, 2003, p. 2)

Problems arise, however, when one tries to determine the relevant risk factors and their expected compensation. The basis for this theory was already laid in the 1950s and 60s with the portfolio selection theory by Markowitz and the Capital Asset Pricing Model (CAPM) by Sharpe, for which he received a Nobel Prize in 1990. (Wilhelm, 2001, p. 15) The CAPM significantly shaped and changed financial management (Užík, 2004, p. VII). Today it is still widely used in practice and plays the centerpiece in the theoretical discussion of asset pricing, although it continues to be sharply criticized (Fama & French, 2004, p. 25). This leads to a variety of adaptations and further developments of the CAPM, but so far no model has been able to sufficiently persuade financial scientists and practitioners (Užík, 2004, p. VII).

As it might seem on first sight, asset pricing is not only solely important for financial investors, because in reverse this also means that companies have to meet the expected returns of their investors. This falls under the ‘Shareholder Value concept’, which has increased in significance over the past years and is being rigorously proclaimed by many investors. According to this concept, companies have to know the return expectations of the investors in order to include them in their capital costs for investment decisions. (Wallmeier, 1997, p. 1)

### 1.2 Objective of this paper

The objective of this paper is to give an overview of the most important movements of the complex area of asset pricing. This will be tried by logically structuring and building up the topic from its origins, the Capital Asset Pricing Model, and then over its main points of critique, in order to arrive at the different options developed by financial science that try to resolve those problematic aspects.

Due to the complexity of this subject and the limited scope of this paper, obviously it will not be possible to discuss each model or movement in depth. Coherently, the aim is to point out the main thoughts of each aspect discussed. For further information, especially concerning the deeper mathematical backgrounds and derivations of the models, the author would like to refer the reader to the books mentioned in this paper. Many of those works, finance journal publications and the literature on asset pricing in general, set their focus on different parts of this paper, which again underlines the complexity in terms of scientific scope and intellectual and mathematical intricacy of this topic.

## 2 The Capital Asset Pricing Model

As mentioned above, the Capital Asset Pricing Model (CAPM) laid the basis for modeling the risk-return relationship as it is considered “the basic theory that links risk and return for all assets.” (Gitman, 2006, p. 246)

The foundation of this model has to be seen in the portfolio choice model, especially as developed at the beginning of the 1950s by Harry Markowitz. Later, in the middle of the 60s, Sharpe, Lintner and Mossin adapted the basic idea of Markowitz by generalizing the individual decision problem of a single investor to all capital market participants. This step led to the CAPM and other asset pricing models (Wilhelm, 2001, p. 66-67). Accordingly, the CAPM builds upon the model of portfolio choice. Due to the limited scope of this paper the portfolio selection model can not be discussed in detail and is assumed to be known.^{[1]}

### 2.1 Derivation of the CAPM

The initial development of the Capital Asset Pricing Model is generally attributed to William F. Sharpe^{[2]} based on his article in the ‘Journal of Finance’ from 1964 about Capital Asset Prices (Gitman, 2006, p. 246).

In this article he summarizes that:

*“In equilibrium, capital asset prices have adjusted so that the investor, if he follows rational procedures (primarily diversification), is able to attain any desired point along a capital market line. He may obtain a higher expected rate of return on his holdings only by incurring additional risk. In effect, the market presents him with two prices: the price of time, or the pure interest rate […] and the price of risk, the additional expected return per unit of risk borne […].”* (Sharpe, 1964, p. 425)

Accordingly, there are two important aspects of the CAPM. First of all, the investor is compensated for delaying consumption over the planning horizon with the *price of time*. Secondly, the investor is rewarded with the *risk premium* for taking on the risk associated with the investment. Obviously, the latter is the more complex issue. In the following it will be analyzed on what this risk premium depends.

#### 2.1.1 Firm-Specific Risk vs. Market Risk

Sharpe speaks of diversification. By diversifying the portfolio one is able to eliminate firm-specific risks. But assets also contain another type of risk which is called the ‘market risk’. It is attributed to factors that affect all firms and thus cannot be eliminated through diversification. Therefore it is also called ‘nondiversifiable risk’. (Gitman, 2006, p. 247)

Accordingly, the only part of risk that a rational, diversified investor has to focus on is the market risk. Furthermore, in an efficient market only the nondiversifiable risk is compensated. (Weston, Besley & Brigham, 1996, p. 204)

#### 2.1.2 The beta coefficient

But if one will always be stuck with the market risk and the firm specific risk will not play an important role in a diversified portfolio, does that mean that all assets are equally risky? No, because not all assets react the same way on changes in the market. Thus, there are assets that might only be slightly affected by changes in the market and others that depend to a high degree on the market development and will, therefore, react strongly to a changing market. (Weston et al., 1996, p, 201)

This leads us to the central aspect of the CAPM, namely the *beta coefficient*. It is a measure of nondiversifiable risk, so it measures the “degree of an asset’s return in response to a change in the market return.” (Gitman, 2006, p. 247) Although the beta should be forward-looking and compared to the whole market return, in practice it is usually based on historical returns and a common stock index as a proxy for the market return (Gitman, 2006, p. 247).

#### 2.1.3 The CAPM Equation

The following is the CAPM equation and calculates the required return on asset i:

illustration not visible in this excerpt

Where Abbildung in dieser Leseprobe nicht enthalten= expected return on the market portfolio

Accordingly, the market risk premium is the additional return over the risk-free rate needed to compensate investors for taking on the average amount of risk associated with holding the market portfolio of assets. Thus, it depends on the average degree of risk aversion of investors and is calculated by the difference between the return on the market portfolio and the risk free rate of return. (Weston et al., 1996, p. 207)

If an asset has the same response as the market, its beta must be 1. On the other hand, if it responds stronger than the market, its Abbildung in dieser Leseprobe nicht enthaltenis higher than 1, leading to a higher risk premium. Obviously, for a less responsive asset the beta is smaller than 1 and it therefore has a lower risk premium. (Gitman, 2006, p. 249)

### 2.2 The Security Market Line (SML)

The Security Market Line graphically depicts the risk-return trade-off. Accordingly, it is the visualization of the CAPM and graphs the required return of an asset dependant on each level of beta (and thus its market risk). The interception of the SML with the y axis (*b* = 0) is at the risk-free rate of return. (Gitman, 2006, p. 252)

illustration not visible in this excerpt

The Security Market Line (SML) (Weston et al., 1996, p. 208)

Since the risk-free rate is the investor’s compensation for the ‘price of time’, in other words the delay of consumption over the period, this rate will change with a change in inflationary expectations. Obviously, with growing inflationary expectation the compensation for delaying consumption must also be higher. Thus, inflation shifts the SML curve upwards. (Weston et al., 1996, pp. 209/210)

As discussed, the market risk premium (depicted by the slope of the SML) depends on the risk aversion of investors. Increasing risk aversion will therefore make the SML steeper and vice versa. (Gitman, 2006, pp. 254/255)

### 2.3 Assumptions of the CAPM

Since the preconditions of the CAPM will be of great importance in the following course of this paper, one has to be aware of their rigidity in order to assess the limitations of the models and to discuss new developments in asset pricing theory. The key assumptions of the CAPM were first stated by Michael C. Jensen in 1972 and are as follows: (Brigham & Gapenski, 1996, p. 68 and Jensen, 1972, pp. 358/359)

1. All investors think in terms of a single period.

2. Investors act rational and choose their portfolio solely based on the expected return and its standard deviation over that period, which means that returns have to be normally distributed (Wilhelm, 2001, p. 69).

3. All investors can borrow or lend an unlimited amount of money at a given (and for all equal) risk-free rate of interest.

4. All investors have homogeneous expectations, meaning that they identically estimate expected returns, standard deviations and correlations of returns among all assets.

5. Homogeneous expectations require that all investors have constant and free access to all required information regarding the investment decision. Furthermore, this information has to be analyzed and evaluated equally by all. (Hug, 1993, pp. 131/132)

6. All assets are perfectly divisible and are perfectly marketable at the going price.

7. There are no transaction costs, taxes and restrictions on short sales of any asset.

8. The market is not constricted by any institution (Weber, 1990, p. 72).

9. Investors assume that their own acting will not affect prices (= price takers).

10. The quantities of all assets are given and fixed.

11. Investors are risk averse (Hug, 1993, p. 130).

## 3 Problems of the CAPM

The decisive question is to what extent the CAPM is able to fulfill its high-aimed objective, namely to explain the risk – return relationship of assets. The title of this paper already indicates that the CAPM has its limitations. Actually, most empirical tests of the CAPM have even had trouble to explain the past, let alone to predict the future (Weber, 2006, pp. 83-86). In those tests, empirical research even noticed a few regularities in the divergences (Užík, 2004, p.47). These empirical problems might be the result of theoretical failings, namely many simplifying assumptions (Fama & French, 2004, p. 25). These will be discussed in the next paragraph. Afterwards, problematic aspects of testing the CAPM and a few of the unexplained phenomena will be analyzed.

### 3.1 Unrealistic assumptions

Already when the assumptions of the CAPM were stated in the upper part of the paper, the reader must have been surprised by their rigorousness and lack of reality. Here, only the most important assumptions are to be discussed.

One of them is obviously the single period time horizon of the model. This means that investors are only concerned with the wealth their portfolio produces at the end of the current period. Investors in the real world have the intention of securing their lifetime consumption level by the means of investing. Making optimal investment decisions by considering returns over the next period only (single period model), is just achievable under further assumptions. (Armitage, 2005, p. 52)

Another problem is that the model should only be based on forward-looking data, e.g. the expected rate of return and the expected beta. Obviously, these cannot be estimated with precision and are therefore often historically based. (Brigham & Gapenski, 1996, p. 85)

Other assumptions that do not comply with reality are the lack of free and instantly available information (information market efficiency) and the exclusion of taxes and transaction costs (Hug, 1993, pp. 151-162). Furthermore, in reality a risk-free asset does not exist. Even government bonds, which play this role in the practical usage of the CAPM actually contain risk as well. (Užík, 2004, p. 53)

It is also quite unrealistic that all investors have homogeneous expectations and that they all act rationally, based on the expected return and the standard deviation (Shefrin, 2005, pp. 1-5).

### 3.2 Empirical Testing of CAPM

Obviously, models will always be a simplification of the real world in order to create a processible model. Pierre-Yves Moix states that “Models have proven to be very successful in the field of engineering since they can adequately capture the relationships in the physical world. Economic models on the other hand involve a formalization of human (economic) behavior, which is definitely, and fortunately, too rich to be fully described in quantitative terms.” (Moix, 2001, p. 5) Accordingly, the important part is that the models are good approximations of reality. This can only be decided by empirical tests. The following part will discuss problems involved with testing the CAPM and show results of some empirical tests.

#### 3.2.1 General Testing Problems

##### 3.2.1.1 The Problem of ex ante Data

An important problem when testing the CAPM is that the model should be based on *expected*, forward-looking data. Actually though, it is based entirely on historical data. The expected return is an example and “there is no reason to believe that realized rates of return over the past holding periods are necessarily equal to the expected rates of return.” (Brigham & Gapenski, 1996, p. 84) The same holds true for the beta values. Accordingly, one has to be careful with the result of the tests. (Brigham & Gapenski, 1996, p. 84 and Q3, pp. 133/134)

##### 3.2.1.2 Roll’s critique

Roll argued in 1977 that the CAPM cannot be tested. It can neither be proven wrong nor true. (Armitage, 2005, p. 51) The general problem is that the CAPM is based on the market portfolio. Alongside stocks, the market portfolio actually contains real estate and other risky assets, even human capital. Human capital actually seems quite important since, for example, it is worth about 2/3 of the US-GDP. (Jahnke, 2006, pp. 45/46)

Obviously, it is impossible to hold the market portfolio. Thus, a proxy to the market portfolio has to be found. Roll argues that ex post, the possibility is given to create an efficient portfolio. But this proxy portfolio does not necessarily have to represent the real market portfolio. (Weber, 2006, p. 84) On the other hand, if a portfolio that ex post does not indicate a relation between the mean return and the beta, all one can infer for sure is that the market-proxy portfolio is not efficient ex post. (Armitage, 2005, pp. 51/52)

Thus, nobody so far has neither been able to definitively deny nor prove the CAPM, because it has never ‘really’ been tested (Fama & French, 2004, p. 41).

#### 3.2.2 Results of empirical Tests

Roll’s critique did not stop science from trying to test the CAPM. This paragraph is dedicated to the results that were found, independent from Roll’s critique, and to their conclusions.

##### 3.2.2.1 Controversy about Beta

In 1980 Wallace wrote an article about the CAPM entitled “Is Beta Dead?” (Weber, 2006, p. 84). In an article in 1992, Fama and French came to a similar conclusion, stating that their research did not show that there is any necessary relationship between the average stock return and its betas (Weber, 2006, p. 85).

But others found a correlation and in turn criticized the assumptions made by Fama and French and the database they used (Weber, 2006, p. 85). One of the most important and often cited studies that support the CAPM was carried out by Black, Jensen and Scholes in 1972. The study came to the conclusion that a positive relationship between the beta and the average return exists. Unfortunately, the slope of the security market line was too flat to explain the empirical values. Furthermore, the empirical analysis indicated a risk-free rate that was higher than the actual one. This conclusion was supported by many other studies, e.g. Blume and Friend (1973) and Fama and MacBeth (1973). (Fama & French, 2004, p. 32)

Accordingly, the CAPM approach of trying to explaining all risk with the market risk represented by *β*, seems not to be the perfect choice. A natural reaction to this is the search for other factors that influence risk. A few of the factors that science came up with will be discussed in the next paragraph.

**[...]**

^{[1]} For information on portfolio selection and risk and return fundamentals please refer to standard finance textbooks, e.g. Gitman (2006), pp. 224 ff.

^{[2]} Almost at the same time the model was also independently developed by John Lintner (1965) and Jan Mossin (1966) (Užík, 2004, p. 27). But the main credit is given to Sharpe (Gitman, 2006, p. 246).

Home

## Prospect Theory

## Cumulative Prospect Theory Calculators

## Introduction

The most cited paper ever to appear in *Econometrica*, the prestigious academic journal of economics, was written by the two psychologists Kahneman and Tversky (1979). They present a critique of expected utility theory as a descriptive model of decision making under risk and develop an alternative model, which they call *prospect theory*. Kahneman and Tversky found empirically that people underweight outcomes that are merely probable in comparison with outcomes that are obtained with certainty; also that people generally discard components that are shared by all prospects under consideration. Under prospect theory, value is assigned to gains and losses rather than to final assets; also probabilities are replaced by decision weights. The value function is defined on deviations from a reference point and is normally concave for gains (implying risk aversion), commonly convex for losses (risk seeking) and is generally steeper for losses than for gains (loss aversion) (see Figure 1). Decision weights are generally lower than the corresponding probabilities, except in the range of low probabilities (see Figure 2).

Figure 1: A hypothetical value function

Figure 2: Weighting functions for gains (w+) and losses (w-) based on median estimates of parameters

## Prospect theory vs Expected utility theory

There are two fundamental reasons why prospect theory (which calculates value) is inconsistent with expected utility theory. Firstly, whilst utility is necessarily linear in the probabilities, value is not. Secondly, whereas utility is dependent on final wealth, value is defined in terms of gains and losses (deviations from current wealth).

## Quotes

‘Kahneman & Tversky’s (1979) “Prospect Theory: An analysis of decision under risk” is the second most cited paper in economics during the period, 1975-2000 (Coupe, in press; Laibson & Zeckhauser, 1998).’

Wu, Zhang and Gonzalez (2004)

‘Prospect theory, developed in the early 1980s by the psychologists Daniel Kahneman and Amos Tversky, represents an important milestone in this context. Kahneman and Tversky proved in numerous experiments that the day-to-day reality of decision makers varies from the assumptions held by economists.’

Goldberg and von Nitzsch (2001, p. 12)

‘Prospect theory was developed by Kahneman and Tversky (1979). In its original form, it is concerned with behavior of decision makers who face a choice between two alternatives. The definition in the original text is: “Decision making under risk can be viewed as a choice between prospects or gambles." Decisions subject to risk are deemed to signify a choice between alternative actions, which are associated with particular probabilities (prospects) or gambles. The model was later elaborated and modified.’

Goldberg and von Nitzsch (2001, p. 62)

‘Prospect theory has probably done more to bring psychology into the heart of economic analysis than any other approach. Many economists still reach for the expected utility theory paradigm when dealing with problems, however, prospect theory has gained much ground in recent years, and now certainly occupies second place on the research agenda for even some mainstream economists. Unlike much psychology, prospect theory has a solid mathematical basis — making it comfortable for economists to play with. However, unlike expected utility theory which concerns itself with how decisions under uncertainty **should** be made (a prescriptive approach), prospect theory concerns itself with how decisions are **actually** made (a descriptive approach).

Prospect theory was created by two psychologists, Kahneman and Tversky, who wanted to build a parsimonious theory to fit a number of violations of classical rationality that they (and others) had uncovered in empirical work. Prospect theory bears more than a passing resemblance to expected utility theory.’

Montier (2002, p. 20)

Short description: ‘We have an irrational tendency to be less willing to gamble with profits than with losses..’

Tvede (1999, p. 94)

‘*Prospect theory*, which was developed by Kahneman and Tversky (1979), is one of the most often quoted and best-documented phenomena in economic psychology. The theory states that we have an irrational tendency to be less willing to gamble with profits than with losses.’

Tvede (1999, p. 166)

*Prospect theory*. We have an irrational tendency to be less willing to gamble with profits than with losses. This means selling quickly when we earn profits but not selling if we are running losses’

Tvede (1999, p. 169)

‘In their landmark work on *prospect theory*, a descriptive framework for the way people make choices in the face of risk and uncertainty...’

Shefrin (2000, p. 24)

‘...get-evenitis is central to prospect theory...’

Shefrin (2000, p. 108)

‘A theory that incorporates such framing effects has been proposed by Kahneman and Tversky (1979). Termed *prospect theory*, it has been extraordinarily influential. It is based on the idea that people evaluate gains or losses in prospect theory from some neutral or status quo point, an assumption consistent with the adaptation-level findings that occur not just in perception but in virtually all experience. That is, we adapt to a constant level of virtually any psychological dimension and find it to be neutral. In a similar way, we adapt to the reduced light in a movie theater when we enter it—finding it not particularly dark after a few seconds—and then readapt to the much brighter light outside when we leave the theater—finding it not to be unusually bright after a few seconds. But since choice varies by framing it as a gain or a loss, it cannot reveal underlying preferences.’

Dawes (2001, p. 195)

‘Not very long after expected utility theory was formulated by von Neumann and Morgenstern (1944) questions were raised about its value as a descriptive model (Allais, 1953). Recently Kahneman and Tversky (1979) have proposed an alternative descriptive model of economic behavior that they call prospect theory.’

Thaler (1980)

First, individuals do not assess risky gambles following the precepts of von Neumann-Morgenstern rationality. Rather, in assessing such gambles, people look not at the levels of final wealth they can attain but at gains and losses relative to some reference point, which may vary from situation to situation, and display loss aversion—a loss function that is steeper than a gain function. Such preferences—first described and modeled by Kahneman and Tversky (1979) in their ‘Prospect Theory’—are helpful for thinking about a number of problems in finance. One of them is the notorious reluctance of investors to sell stocks that lose value, which comes out of loss aversion (Odean 1998). Another is investors' aversion to holding stocks more generally, known as the equity premium puzzle (Mehra and Prescott 1985, Benartzi and Thaler 1995).’

Shleifer (2000, pp. 10–11)

‘If Richard Thaler's concept of mental accounting is one of two pillars upon which the whole of behavioral economics rests, then prospect theory is the other.’

Belsky and Gilovich (1999, p. 52)

‘...prospect theory deals with the way we frame decisions, the different ways we label—or code—outcomes, and how they affect our attitude toward risk. Indeed, we might just as easily have constructed this book as one long essay explaining prospect theory and all of the ideas that flow from it—that’s how influential and seminal the ideas discussed in Kahneman and Tversky's paper are.’

Belsky and Gilovich (1999, p. 52)

‘Prospect theory [...] helps explain how loss aversion, and an inability to ignore sunk costs, leads people to take actions that are not in their best interest. The sting of losing money, for example, often leads investors to pull money out of the stock market unwisely when prices dip.’

Belsky and Gilovich (1999)

‘In a nutshell, prospect theory assumes that investors' utility functions depend on *changes* in the value of their portfolios rather than the value of the portfolio. Put another way, utility comes from returns, not from the value of assets.’

Cornell (1999, p. 148)

‘Prospect theory was developed by Daniel Kahneman and Amos Tversky (1979), and it differs from expected utility theory in a number of important respects.

First, it replaces the notion of “utility” with “value.” Whereas utility is usually defined only in terms of net wealth, value is defined in terms of gains and losses (deviations from a reference point). Moreover, the value function for losses is different than the value function for gains. [...] the value function for losses (the curve lying below the horizontal axis) is convex and relatively steep. In contrast, the value function for gains (above the horizontal axis) is concave and not quite so steep. These differences lead to several noteworthy results.

Because the value function for losses is steeper than that for gains, losses “loom larger” than gains. For instance, a loss of $500 is felt more than a gain of $500.’

Plous (1993, p. 95–96)

‘Unlike expected utility theory, prospect theory predicts that preferences will depend on how a problem is framed. If the reference point is defined such that an outcome is viewed as a gain, then the resulting value function will be concave and decision makers will tend to be risk averse. On the other hand, if the reference point is defined such that an outcome is viewed as a loss, then the value function will be convex and decision makers will be risk seeking.’

Plous (1993, p. 97)

‘Prospect theory also differs from expected utility theory in the way it handles the probabilities attached to particular outcomes. Classical utility theory assumes that decision makers value a 50 percent chance of winning as exactly that: a 50 percent chance of winning. In contrast, prospect theory treats preferences as a function of “decision weights,” and it assumes that these weights do not always correspond to probabilities. Specifically, prospect theory postulates that decision weights tend to overweight small probabilities and underweight moderate and high probabilities.’

Plous (1993, p. 98)

‘Prospect theory represents a great improvement over classical expected utility theory. Indeed, many violations of expected utility theory are explicitly predicted by prospect theory.’

Plous (1993, p. 105)

‘The assumption of a diminishing marginal returns utility function relating dollar gains to utilities has been a cliché in economic theorizing and most reseach shows that our evaluations of gains show a negatively accelerating, diminishing returns pattern. In 1979, Daniel Kahneman and Amos Tvversky proposed what they termed prospect theory as a descriptive theory of decision behavior. A basic tenant of this theory is that the law of diminishing returns applies to *good and bad* quantitative consequences of decisions.’

Hastie and Dawes (2001, p. 216)

- ‘An individual views monetary consequences in terms of changes from a
*reference level*, which is usually the individual's status quo. The values of the outcomes for both positive and negative consequences of the choice then have the diminishing returns characteristic. - The resulting value function is steeper for losses than for gains. This implies
*loss aversion*; equal-magnitude gains and losses do*not*have symmetric impacts on the decision. Losses hurt more than gains satisfy; most empirical estimates conclude that losses are about twice as painful as gains are pleasurable. - The curve is concave for gains and convex for losses, impying that decision makers will be
*risk averse when choosing between gains and risk seeking when choosing between losses*.’

Hastie and Dawes (2001, p. 216)

‘The addition of a moveable reference level is the major difference between prospect theory and traditional economic utility theories.’

Hastie and Dawes (2001, p. 216)

‘v(x) = x^{α} if x > 0

v(x) = -λ(-x^{α}) if x < 0

(with a typical α = 0.88 and λ = 2.25)

This process has three major characteristics:

- Reference level dependence: An individual views consequences (monetary or other) in terms of changes from the
*reference level*, which is usually that individual's*status quo*. - Gain and loss satiation: The values of the outcomes for both positive and negative consequences of the choice have the diminishing returns characteristic. The α term in the value function equation captures the marginally decreasing aspect of the function. Empirical studies estimate that α is typically equal to approximately .88 and always less than 1.00. When the exponent α < 1.00, the curve will accelerate negatively (if α = 1.00, the function would be linear; and if α > 1.00, if would accelerate positively).
- Loss aversion: The resulting value function is steeper for losses than for gains; losing $100 produces more pain than gaining $100 produces pleasure. The coefficient λ indexes the difference in slopes of the positive and negative arms of the value function. A typical estimate of λ is 2.25, indicating that losses are approximately twice as painful and gains are pleasurable. (If λ = 1.00, the gains and losses would have equal slopes; if λ < 1.00, gains would weigh more heavily than losses.)’

Hastie and Dawes (2001, p. 294)

‘Prospect theory is the best comprehensive description we can give of the decision process. It summarizes several centuries' worth of findings and insights concerning human decision behavior. Moreover, it has produced an unmatched yield of new insights and predictions of human behavior in decision making.’

Hastie and Dawes (2001, p. 310)

‘Some behaviors observed in economics, like the disposition effect or the reversing of risk aversion/risk seeking in case of gains or losses (termed the reflection effect), can be explained referring to the prospect theory.’

## Articles via Google Scholar

All

Articles published since 2000|2001|2002|2003|2004|2005|2006|2007|2008|2009|2010|2011|2012

### Top 3 Papers

- Kahneman and Tversky, 1979.

Abstract: ‘This paper presents a critique of expected utility theory as a descriptive model of decision making under risk, and develops an alternative model, called prospect theory. Choices among risky prospects exhibit several pervasive effects that are inconsistent with the basic tenets of utility theory. In particular, people underweight outcomes that are merely probable in comparison with outcomes that are obtained with certainty. This tendency, called the certainty effect, contributes to risk aversion in choices involving sure gains and to risk seeking in choices involving sure losses. In addition, people generally discard components that are shared by all prospects under consideration. This tendency, called the isolation effect, leads to inconsistent preferences when the same choice is presented in different forms. An alternative theory of choice is developed, in which value is assigned to gains and losses rather than to final assets and in which probabilities are replaced by decision weights. The value function is normally concave for gains, commonly convex for losses, and is generally steeper for losses than for gains. Decision weights are generally lower than the corresponding probabilities, except in the range of low probabilities. Overweighting of low probabilities may contribute to the attractiveness of both insurance and gambling.’ - Tversky and Kahneman, 1992.

Abstract: ‘We develop a new version of prospect theory that employs cumulative rather than separable decision weights and extends the theory in several respects. This version, called cumulative prospect theory, applies to uncertain as well as to risky prospects with any number of outcomes, and it allows different weighting functions for gains and for losses. Two principles, diminishing sensitivity and loss aversion, are invoked to explain the characteristic curvature of the value function and the weighting functions. A review of the experimental evidence and the results of a new experiment confirm a distinctive fourfold pattern of risk: risk aversion for gains and risk seeking for losses of high probability; risk seeking for gains and risk aversion for losses of low probability.’ - Barberis, Huang and Santos, 2001.

Abstract: ‘We study asset prices in an economy where investors derive direct utility not only from consumption but also from fluctuations in the value of their financial wealth. They are loss averse over these fluctuations, and the degree of loss aversion depends on their prior investment performance. We find that our framework can help explain the high mean, excess volatility, and predictability of stock returns, as well as their low correlation with consumption growth. The design of our model is influenced by prospect theory and by experimental evidence on how prior outcomes affect risky choice.’

## Book

- KAHNEMAN, Daniel and Amos TVERSKY (editors), 2000. Choices, Values, and Frames, Cambridge University Press. [Cited by 1110] (185.44/year)
- WAKKER, Peter P., 2010. Prospect Theory: For Risk and Ambiguity. Cambridge: Cambridge University Press.

## Links

## Bibliography

- BARBERIS, Nicholas, Ming HUANG, and Tano SANTOS, 2001. Prospect Theory and Asset Prices.
*The Quarterly Journal of Economics*,**116**(1), 1–53. - BELSKY, Gary, and Thomas GILOVICH, 1999.
*Why Smart People Make Big Money Mistakes—And How To Correct Them: Lessons From The New Science Of Behavioral Economics*. New York: Simon & Schuster. - CORNELL, Bradford, 1999.
*The Equity Risk Premium: The Long-Run Future of the Stock Market*. New York: Wiley. - DAWES, Robyn M., 2001.
*Everyday Irrationality: How Pseudo-Scientists, Lunatics, and the Rest of Us Systematically Fail to Think Rationally*. Boulder, CO: Westview Press. - GOLDBERG, Joachim, and Rüdiger von NITZSCH, 2001.
*Behavioral Finance*. Chichester: Wiley. First published in German under the title*Behavioral Finance*by FinanzBuch Verlag GmbH. Translated from German by Adriana Morris. - HASTIE, Reid, and Robyn M. DAWES, 2001.
*Rational Choice in an Uncertain World: The Psychology of Judgment and Decision Making*. Thousand Oaks: Sage Publications. - KAHNEMAN, Daniel, and Amos TVERSKY, 1979. Prospect Theory: An Analysis of Decision under Risk.
*Econometrica*,**47**(2), 263–292. - KAHNEMAN, Daniel, and Amos TVERSKY, 2000.
*Choices, Values, and Frames*. Cambridge: Cambridge University Press. - MONTIER, James, 2002. Darwin’s Mind: The Evolutionary Foundations of Heuristics and Biases. Dresdner KleinwortWasserstein – Global Equity Strategy.
- PLOUS, Scott, 1993.
*The Psychology of Judgment and Decision Making*. New York: McGraw-Hill. - SHEFRIN, Hersh, 2000.
*Beyond Greed and Fear: Understanding Behavioral Finance and the Psychology of Investing*. Financial Management Association Survey and Synthesis Series. Boston, MA: Harvard Business School Press. - SHLEIFER, Andrei, 2000.
*Inefficient Markets: A Introduction to Behavioral Finance*. Oxford: Oxford University Press. - THALER, Richard, 1980. Toward a Positive Theory of Consumer Choice.
*Journal of Economic Behavior & Organization*,**1**(1), 39–60. - TVEDE, Lars, 1999.
*The Psychology of Finance*. Chichester: Wiley. First edition published by Norwegian University Press in hardcover in 1990. - TVERSKY, Amos, and Daniel KAHNEMAN, 1992. Advances in Prospect Theory: Cumulative Representation of Uncertainty. Journal of Risk and Uncertainty,
**5**(4), 297–323. - WU, George, Jiao ZHANG, and Richard GONZALEZ, 2004. Decision Under Risk.
*In*: Derek J. KOEHLER and Nigel HARVEY, eds.*Blackwell Handbook of Judgment & Decision Making*, Handbooks of Experimental Psychology. Malden, MA: Blackwell Publishing, Chapter 20, pp. 399–423.