Cumulative prospect theory and its application in vietnamese security market

Figure 1.2: The probability weighting function 9 Figure 1.γ: The value function vx with different value of α, Figure 1.4: Distorted cumulative probabilities w- p and w+ p with = .61 and

FINAL DRAFT -

Cumulative prospect theory and applications to portfolio construction in Vietnamese security market

By Nguyen Duc Dung Advisor: Dr Phan Tran Trung Dung Faculty of Finance and Banking

FACULTY OF BANKING AND FINANCE

GRADUATION THESIS

Major: International Finance

CUMULATIVE PROSPECT THEORY AND

APPLICATIONS TO PORTFOLIO CONSTRUCTION

IN VIETNAMESE SECURITY MARKET

Student ID: 1001030090 Intake: 48 – A9

Supervisor: Phan Tran Trung Dung, PhD

Hanoi, May 2014

CONTENT

CONTENT ii

LIST OF TABLES iii

LIST OF FIGURES iii

INTRODUCTION 1

CHAPTER 1 CUMULATIVE PROSPECT THEORY AND BEHAVIORAL PORTFOLIO THEORY 4

1.1 Original Prospect Theory and Cumulative Prospect The ory 4

1.1.1 Phenomena of choice in laboratory settings 4

1.1.2 The ideas of Original Prospect Theory 6

1.1.3 Cumulative Prospect Theory 9

1.2 Rational Portfolio Theory versus Behavioral Portfolio Theory 15

1.2.1 Modern Portfolio Theory 15

1.2.2 Challenges from psychologists 19

1.2.3 Behavioral Portfolio Theories 22

1.3 Applications of Cumulative Prospect Theory to Portfolio Theory 26

1.3.1 Challenges in applying Cumulative Prospect Theory 26

1.3.2 Portfolio Construction under Cumulative Prospect Theory 27

CHAPTER 2 ESTIMATIONS OF CUMULATIVE PROSPECT THOERY’S PARAMETERS 30

2.1 Tversky and Kahneman (1992) 30

2.2 The trade-off method of Fennema and Van Assen (1998) 31

2.3 The traceable method of Abdellaoui (2007) 34

CHAPTER 3 DATA 40

3.1 Data selection 40

3.2 Guideline of data processing 40

3.2.1 Equity return 40

3.2.2 Value function 43

3.2.3 Weighting function 43

3.2.4 Objective function 44

3.2.5 Optimal portfolio problem 45

3.2.6 Empirical estimates of parameters 46

3.2.7 CPT’s portfolio matching market portfolio 47

CHAPTER 4 RESEARCH FINDINGS 48

CHAPTER 5 SUGGESTIONS TO IMPROVE THE RESEARCH 51

CONCLUSION 54 REFERENCE 55

LIST OF TABLES

number

Table 1.2: Forms of weighting functions 15 Table 2.2: Questions asked to determine the utility for gains and

Table 3.1: Defining the return classification of the stock AAA

in which return spread of AAA is divided into 20 classes 41 Table 3.2: Describe the prospect of AAA with outcomes xi

being medians of the 20 classes in table 3.1 and probabilities

being frequencies of those classes

Figure 1.2: The probability weighting function 9

Figure 1.γ: The value function v(x) with different value of α,

Figure 1.4: Distorted cumulative probabilities w- (p) and w+ (p)

with = 61 and δ = 69, estimated by Kahneman and Tversky

INTRODUCTION

Portfolio allocation problem is at heart of investment theories , with the

central premise being that individual’s portfolio and market portfolio are

derived from utility maximizing agents During the last century there are two paradigms being the contrast foundations for the development of portfolio theory:

“The rational paradigm”: assuming investors rationally behave by

optimizing their object being the expected utility function (which is smooth, concave and based on their final wealth) Thereby, the paradigm argues that markets are efficient and each investor needs one optim al portfolio

“The behavioral paradigm”: here investors’ decision making process is

susceptible to psychological biases compared to that of rational investors Utility is based on deviation from a reference point and can displays concave

or convex areas Therefore, unlike rational investors, behavioral investors do not have one efficient portfolio but fragmented portfolios, markets display trends and reverting patterns instead of efficiency

The struggle between the two paradigms was most pronounced in the twentieth century Success shifted between the two multiple times, and the most prominent economists and psychologists were involved Now it seems widely accepted that investors display major bias from rational behavior In addition, patterns with an information content actually exists, and even out of sample studies show on past data the possibility to beat some markets

Cumulative Prospect Theory (CPT) traces its origin to Prospect Theory (Kahneman and Tversky 1979), which straightforwardly replaces the Expected Utility Theory by assuming that people make decision based on gains or loss, probabilities are replaced by decision weights (which tends to overweight small probabilities in tails and underweight bigger probabilities) Also, value function is concave for gain side (risk aversion) and convex for loss side (loss aversion)

This replacement of EUT with Prospect Theory and its advanced version

– CPT proposed a new approach to the processes and objectives of investors

constructing their optimal portfolio In addition, after the invention of Prospect Theory and its advanced version – Cumulative Prospect Theory,

there are several empirical studies that attempted to estimate the parameters

of the theory (e.g degree of loss aversion, probabilities preference) In the studies, based on models of weighted probability function and value function, the parametric estimations were produced through analysis of experimental data The explaining portfolio choice with respect to objective function of Cumulative Prospect Theory proposed the way choice patterns display in real-life investment decision and provided a new non-experimental estimation of the parameters of Cumulative Prospect Theory by backward induction from market portfolio

This thesis has three objectives: 1) Modeling: establish behavioral optimal allocation model incorporating the three key elements of CPT 2) Solution: under the model, for each set of parameters, which were estimated by former studies, derive objective functions for the stocks in the statistical data and then produce an optimal portfolio 3) Testing: examine the match of each produced portfolio to assess the best parametric estimation for Vietnamese market

The remainder of this thesis consists of the following sections The first one is to review the invention of Cumulative Prospect Theory as well empirical evidences on it and studies on applications of CPT to investors’

portfolios The second section provides the approach that have been employed

to estimate Cumulative Prospect Theory’s parameters and the results given by

the methods In the next section, we present the data resources and the guideline of data processing in attempt to construct an optimal portfolio under assumptions of CPT The third section is to present findings of the optimal portfolio in Vietnamese security market and the degree of matching between each estimated set of parameters of former studies and that of the market And

the last one consists of the conclusion, implications of this study and suggestions for later studies on this topic

CHAPTER 1 CUMULATIVE PROSPECT THEORY AND BEHAVIORAL PORTFOLIO THEORY

In 1979, Daniel Kahneman and Amos Tversky, who were already well known due to their study on judgment heuristics, published a paper in the journal Econometrica entitled “Prospect Theory: An Analysis of Decision under Risk” The paper achieved two things Firstly, it collected in one place

-a series of convinced demonstr-ations th-at, in l-abor-atory setting, systematically, people did not make their choice as the predict ions of expected

utility theory, economists’ general accepted model of decision-making under

uncertainty It also presented a new model of risk -return attitudes called

“prospect theory,” which captured the experimental evidence on risk-taking,

including the documented violations of expected utility theory

Over more than 30 years, Prospect Theory is currently considered as the theory that best describes how people evaluate a risky choice in laboratory settings Actually, so many years after the publication of the paper, there were few broadly known applications of the Prospect Theory in economics One of the reasons might be that, even if Prospect Theory is an excellent description

of behavior in experimental settings, it is not effective in real life Howev er, over 1990s and 2000s, researchers in the behavioral finance area have put

many ideas into how Prospect Theory’s insights can be applied in economic

settings

This chapter discusses the insights of Prospect Theory and latter research findings improving its capability of applying to economic settings, specifically, to portfolio construction in investment

1.1 Original Prospect Theory and Cumulative Prospect Theory

1.1.1 Phenomena of choice in laboratory settings

Inducing from the experimental results, there are fiv e major phenomena

of choice, which violate the standard model and set a minimal challenge that must be met by any adequate descriptive theory of choice All these findings

have been confirmed in a number of experiments, with both real and hypothetical payoffs

Framing effects: The rational theory of choice advocates description invariance: equivalent formulations of a choice puzzle should give rise to the same preference order (Arrow, 1982) In contrast to this assumption, there are many evidences that variations in the framing of options (e.g., in terms of gains or losses) derive systematically different preferences (Tversky and Kahneman, 1986)

Nonlinear preferences: According to the expectation principle, the utility

of a risky prospect is linear in outcome probabilities Allais's (1953) famous paradox challenged this principle by showing that the difference betw een probabilities of 0.99 and 1.00 has more impact on preferences than the difference between 0.10 and 0.11 More recent studies observed nonlin ear preferences in choices that do not involve sure things (Camerer and Ho, 1991) Source dependence: People's willingness to bet on an uncertain event depends not only on the degree of uncertainty but also on its source Ellsberg (1961) observed that people prefer to bet on a box containing equal numbers

of red and green balls, rather than on a box that contains red and green balls

in uncertain proportions which is called Ellsberg’s paradox More recent

evidence indicates that people often prefer a bet on an event in their area of competence over a bet on a matched chance event, although the former probability is vague and the latter is clear ( Heath and Tversky, 1991)

Risk seeking: Risk aversion is generally assumed in economic analyses of decision under uncertainty However, risk-seeking choices are consistently observed in two classes of decision problems First ly, people tend to prefer a small probability of winning a large amount over the expected value of that prospect Second, risk seeking is prevailing when people must choose between

a certain loss and a substantial probability of a bigger loss

Loss' aversion: One of the basic phenomena of choice under both risk and uncertainty is that losses loom larger than gains (Kahneman and Tversky,

1984; Tversky and Kahneman, 1991) The observed asymmetry between gains and losses is far too extreme to be explained by income effects or by decreasing risk aversion

1.1.2 The ideas of Original Prospect Theory

In the traditional theory, the utility of a risky prospect i s the sum of the utilities of the outcomes and each outcome weighted by itself probability To explain the phenomena mentioned above, Kahneman and Tversky (1979) suggested the Original Prospect Theory which modifies the former theory in two points: (1) the carriers of the value of outcomes are their deviations relative to a benchmark (gains or losses) instead of final wealth and (2) the value of each outcome is multiplied by a decision weight, not by an additive probability Specifically, consider a gamble:

(x-m, p-m; x-m+1, p-m+1; …; x0, p0; …; xn, pn) (1) where x-m, x-m+1,… and xn are gains for the outcomes of the gamble These

gains are arranged in increasing sequence, so xi < xj for i < j, and where x0 =

0 In this notation, p-m, p-m+1, … and pn are corresponding probabilities of the outcomes Under the Expected Utility Theory, an individual would evaluate the above gamble equal to:

∑ �

=−

where v(.) is called “value function” being an increasing function with v(0) =

0 and where πi are the “decision weights” This formula illustrates the four

key elements of Original Prospect Theory: 1) Reference – dependence, 2) loss

aversion, 3) diminishing sensitivity, and 4) probability weighting

Firstly, in Original Prospect Theory, people assess utility through gains and losses, which are measured relatively to a reference point, rather than through an absolute level of wealth, so the argument of v(.) is xi, but not W +

“reference-dependence”, not only by documented experimental evidences, but also by

noting that people’s perceptual system works in a similar way: we usually

focus more on changes in attributes such as light, temperature, noise than we are to the absolute magnitudes of them

Secondly, the value function captures “loss aversion” phenomenon, the

idea argues that people are more sensitive to losses than gains of the same size This phenomenon is presented in the value function through the loss region of the value function being steeper than its gain region This expression can be seen in Figure 1, which plots a typical value function ; the horizontal axis represents the gain or loss, and the vertical axis, the value v(x) assigned

to that gain or loss Apparently, the value placed on a $40 gain, v(40), is smaller in absolute magnitude than v(-40), the value placed on a $40 loss Kahneman and Tversky infer loss aversion from the fact that most people turn down the gamble ($40, 0.5; $-50, 0.5) As Rabin (2000) shows, it is hard to understand this fact in the expected utility framework: the dollar amounts are very small relative to typical wealth levels that, under expected utility, the gamble is evaluated in a risk-neutral way; given its positive expected value,

it is therefore attractive For a loss averse individual, however, the gamble is unappealing: the pain of losing $40 far outweighs the pleasure of winning $50 Thirdly, as Figure 1 shows, the value function is concave for the region

of gains but convex for the loss region This component of prospect theory is called diminishing sensitivity because it implies that, while replacing a $10 gain (or loss) by a $20 gain (or loss) has a substantial utility impact, replacing

a $200 gain (or loss) by a $210 gain (or loss) has a smaller impact The concavity over gains captures the finding that people tend to be risk averse over moderate probability gains: they typically prefer a certain gain of $100

to a 50 percent chance of $200 In contrast, people tend to be risk-seeking

over losses: they prefer a 50 percent chance of losing $100 to losing $50 for sure This is shown as the loss region is concave

Figure 1.1: The Prospect Theory Value Function The fourth and last component of Original Prospect Theory is probability weighting Under the theory, people do not weight chance of an outcome happening by its probability, but rather, by the decision weight on the outcome

πi The decision weights are calculated by t he weighting function w(.), depending on the objective probability The Figure 2 shows a typical weighting function raised by Kahneman and Tversky (1992) In this figure, the inverse S-shape curve presents the weighting function of Original Prospect Theory, whereas the 450 line presents the probability assessment of a rational person, which is exactly equal to objective probability Apparently, the inverse S-shape curve explains that people tend to overweight small probability and underweight big probabilit y

Figure 1.2: The probability weighting function Apart from explaining the above empirical phenomena, Original Prospect Theory also emphasized that the process of decision making consists of two phases: framing and valuation In the framing phase, peopl e constructs representation of the acts, contingencies and outcomes that are relevant to the decision In the valuation phase, the decision makers evaluate the value of each prospect and choose accordingly

1.1.3 Cumulative Prospect Theory

1.1.3.1 Drawbacks of Original Prospect Theory

The Original Prospect Theory provides an idea of weighting scheme in the valuation phase of decision making process This scheme in the original version is a monotonic transformation of outcome probability This scheme encounter a problem that it violates the first-order stochastic dominance Specifically, due to enhance every small probability, under Original Prospect Theory, value of a dominated prospect is enhanced and the prospect can be chosen The problem of this kind of decision weights, with respect to the generalization to many different outcome prospects, can be described by the following example

Consider a gamble with many different outcomes as follows:

(-20, 0.04; -10, 0.04; 0, 0.04; 10, 0.04; …; β10; 0.04; ββ0, 0.04)

Under the Original Prospect Theory, if probabilities of all the outcomes

prospect will be valued higher than its expected value $100 for sure It is obviously implausible that people would prefer a prospect to its expected value for sure This anomaly is the result of overweighting every outcomes of the prospect This phenomenon leads to the violation of stochastic dominance

of Original Prospect Theory

In addition, in the original version of prospect theory, prospects mentioned include not more than two outcomes This limits the ability to apply Original Prospect Theory in real financial markets in which people have make decision by discriminating among prospect with many outcomes

These problems can be solved by transparently eliminating dominated prospects in framing phase of decision making process and by normalizing the weights so that the sum of them equals 1 These issues also can be solve by rank-dependent expected utility (cumulative functional), invented by Quiggin (1982) for decision under risk, and by Schmeidler (1989) for decision under uncertainty Kahneman and Tversky (1992) revise their original work with a new version, Cumulative Prospect Theory, a combination of the concepts of loss aversion and a nonlinear rank dependent weighting of probabi lity assessment Instead of transformation probability of each outcome separately, the new model provides transformation the entire cumulative distribution function In addition, the new model allows different probability transforming for gain side than for loss side The development extends the original version

to uncertainty and risky prospects with continuous set of outcomes while it preserves most of key features of the original theory

1.1.3.2 The new model

In 1992, Tversky and Kahneman announced a new versi on of prospect theory called cumulative prospect theory Keeping most of essential theoretical characteristics of prospect theory, this new theory incorporates a

value function and a probability distortion function The value function, like prospect theory, presents reference dependence, diminishing sensitivity, and

loss aversion However, cumulative prospect theory modifies prospect theory

by applying the probability distortions to the cumulative probabilities as opposed to the individual probabilities, in an attempt to includ e nonlinear preferences This formulation is called rank -dependence and was first proposed by Quiggin (198β) The advantage being that it satisfies stochastic

dominance, and has the ability to be applied to prospects with large number

of outcomes Cumulative prospect theory also allows for different distortion functions while in the gain region or in loss region Consider the gamble (1),

in Cumulative Prospect Theory the value of a prospect, V(x, p) is given by the following calculation:

=

=−

Where v(x) is the monotonic value function as in Original Prospect Theory

The notations π+(p) and π-(p) are subjective distorted probability derived

from weighting cumulative function Particularly,

π+(pn) = w+(pn) and π+(pi) = w+(pi + pi+1+ … + pn) – w+(pi+1 + pi+2 + … + pn)

π+(p-m) = w-(p-m) and π-(pi) = w-(p-m + p-m+1+ … + pi) – w-(p-m + p-m+1+ … + pi-1) where w+(p) is the weighting function for probability within gain side and w -(p) is the weighting function for probability within loss side

The advantage of Cumulative Prospect Theory is that it satisfies the

stochastic dominance which is violated by Original Prospect Theory The Cumulative Prospect Theory is generally accepted as a n ormative and descriptive model of decision making under uncertainty, and have broadly applied in economics to explain the inconsistency with standard economic rationality, e.g equity premium puzzle, the asset allocation puzzle, etc However, it has received some criticism as well For instance, Wu (1994) has

pointed out the ordinal independent violation of cumulative prospect theory Other literatures have pointed out various other violations as well

1.1.3.3 Value function

Based on results of their experiments, K ahneman and Tversky (1992) raised the forms of the value function and weighting function as follows: The formula of the value function:

= {− −+ − � < �

Where u+: R+ → R+ and u-: R+ → R+ satisfy:

(i) v(0) = v+(0) = v-(0) = 0

(iii) v+(x) = xα, with 0 < α < 1 and x 0

(iv) v-(x) = λx , with α, < 1, λ > 1 and x 0

Figure 1.γ: The value function v(x) with different value of α, and λ

The form above of the value function is of power functional form which

is the most widely used value functions are the power functional form Stevens (1957) suggested the power function form in his psychophysical model He argued that with power forms the psychological magnitudes were interpreted better than the Fechnerian method which uses the logarithmic function This

x

λ =1 α , =0.2 , =0.4

λ =2 α , =0.6 , =0.9

λ =2.25 α =0.80 =0.88

value (utility) function was used by Kahneman and Tversky (1992), Luce (1991) and Wakker and Tversky (1993)

In addition to power form, the exponential form was also cited frequently (e.g Camerer and Ho (1994); Fishburn and Kochenberger (1979)) Luce and Fishburn (1991) showed that the exponential form is applicable under reasonable condition Similar, Wakker and Tversky (1993) showed that this functional form can be cited if preferences are invariable under the addition

of a positive constant to outcomes De Giorgi and Hens (2006) proposed that the piecewise exponential function should be used rather than the piecewise power function They asserted that the piecewise power function has the limitation of unboundedness which limits the ability to apply to finance

Nevertheless, many authors noted some drawbacks of this function such as for large outcome, the exponential value function exhibits more curvature and hence the function discourages extreme risk taking

The last function studied is quadratic form which plays an important role

in finance The advantage of this form is that the value of prospects can be priced through their mean and variance which is commonly used in finance

This type of value function together with power value function and exponential value function has been examined in Stott (2006) In the study, Stott has conducted research on the complete pattern of 256 model variants including different types of weighting functions, value functions and stochastic choice functions

Form of value function Equation

Assuming 0.28 < and 0.β8 < δ to ensures that the two weighting

functions are increasing

Figure 1.4: Distorted cumulative probabilities w- (p) and w+ (p) with = 61

and δ = 69, estimated by Kahneman and Tversky (199β)

The invers S-shape of weighting function implies that people tend to overweight extreme outcomes at which cumulative probability is near to 0 or

1 and underweight near-median outcomes

Besides the formula above, there are some other forms of weighting function which display invers S-shaped curves These form were suggested in Karmakar (1979), Prelec (1998), Goldstein and Einhorn (1987), Currim and

0.4 0.3 0.2 0.1

0 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Distribution Function w+: =0.61

w−:δ=0.69

Sarin (1989), Lattimore, Baker and Witte (1992) These function together with

Kahneman and Tversky’s function are listed in Table 1.2 They were also

evaluated in number of papers (e.g Camerer and Ho (1994), Tversky and Fox (1995), Wu and Gonzalez (1996), Gonzalez and Wu (1999), Abdellaoui (2000) Bleichrodt and Pinto (2000), Kilka and Weber (2001), Etchart -Vincent (2004), and Abdellaoui, Vossmann and Weber (2005)

Karmarkar (1979) has suggested a weighting function, that we call Karmarkar form, as shown in Table 1.2 Karmarkar weighting function crosses the point (0,5; 0,5) for all value of ∈ [ , ] Prelec (1998) proposed the

compound invariant form of weighting function, that we call Prelec form, as listed in Table 1.2 This function allows us to explain common ratio effect more thoroughly Prelec (1998) built this risky weighting function based on a system of behavioral axioms The function always cross the 450 line at a fixed

point p = 1/e When is decreasing the function is more concave to the left of

the point, and more convex to the right of the point

Form of weighting function Equation

Kahneman & Tversky

Table 1.2: Forms of weighting functions

1.2 Rational Portfolio Theory versus Behavioral Portfolio Theory

1.2.1 Modern Portfolio Theory

1.2.1.1 Normative arguments

Markowitz (1952) had introduced a mathematical framework to construct

an efficient portfolio This idea, called “Modern Portfolio Theory”, is as

simple as powerful Markowitz pointed out the two most important criteria for

an investment being the return and the risk In this theory, the preference should be given to a portfolio with higher return for the same risk or a portfolio with lower risk given a similar return If we would plot all possible portfolio with their risk and return, there would be an upper boundary for all

portfolio This boundary, called “Efficient Frontier”, can be characterized by

that for each portfolio lying on the frontier, there is no portfolio with the same risk offering a higher return

His argument is to reject that investor would only focus on maximizing return because this hypothesis leads to non-diversified portfolios He also suggest to use variance of return on portfolios as a measure of risk He wrote:

“We next consider the rule that people should consider expected return a desirable thing and variance of return an undesirable thing” and next showed

the mean-variance criterion

Further he stated that variance “is commonly used measure for dispersion”

and even when one uses another measure of risk such as standard deviation or

coefficient of variance (σ/R) then one would find the same optimal portfolios

The essence of this theory is that it is unacceptable that a portfolio selection method would not lead to diversified portfolios He proposed a solution that meets diversification requirement and argues that it is wise to apply his

criteria since they are better than “speculative behavior”

1.2.1.2 A mathematical formulation of mean-variance criterion

Firstly, the standard deviation of a random variable X is defined as:

= √�∑ − ̅

=

where { = ∫−∞+∞

distribution and for discrete distribution

Consider a portfolio comprising N possible risky assets This portfolio can be referred to through the vector of weights among the different assets:

with � is the covariance between asset i and asset j

Using the properties of the assets’ returns, we can define the properties of a

portfolio p with a vector of weights w as follows:

The mea variance criterion developed by Markowitz is now reduce to the following problem:

It is possible to solve the N+2 equations in N+2 variables

Figure 1.5: Mean-variance analysis for possible portfolios comprised of 3

assets The mean-variance model of Markowitz not merely gives a powerful and clear application in investment but establishes a foundation for later

inventions Particularly, the “Sharpe-Linter-Mossin mean-variance equilibrium model of exchange”, or Capital Asset Pricing Model, which was

proposed independently by Treynor (1961), Treynor (1962), Sharpe (1964) and Mossin (1966) and currently a widely applied pricing model, was built on the earlier work of Markowitz

1.2.2 Challenges from psychologists

an instance or occurrence In other words, people tend to enhance frequency

of an event which happened recently in comparison to that of an event which

occurred a long time ago Also, when people are asked to make a decision, they are inclined to use their available knowledge rather than other alternatives, which might be more relevant

A year later, in addition to Available Heuristic, Kahneman and Tversky continued to introduce two more heuristics that are used in making decisions with uncertain outcomes Representativeness is a bias occurring when people are asked to judge the probability of class or process A containing an object

or event B, of which probability is known, so the probability of A would be evaluated by the degree of which B is representati ve of A

In terms of “anchoring and adjustment”, when an anchor available, people

adhere their mind to its initial value, which may be inferred from formulation

of the problem or can be the result of partial computations In those case, the adjustment people make are small, and their decisions are strongly affected

by the initial anchor

1.2.2.2 Framing effects

This is a bias derived from Kahneman and Tversky’s (1979) prospect

theory Beside heuristics or rules of thumb, framing effects are other factors that challenge modern portfolio theory as well as traditional assumptions Kahneman and Tversky showed that when one same problem is framed differently, decision makers would change their decision predictably

An implication of framing effects can be observed in prospect theory’s

assumption When people evaluate a prospect, they tend to use the direct consequences of the prospect (e.g gain and loss in their money) Therefore, they assess prospects in terms of mental account, which includes only gain or loss from the prospect but not their total wealth Another direct implication

of framing effects is that people change their views on their portfolio based

on information and data that may be not relevant to the portfolio fundamentals As a result, they ignore covariance among assets they hold and therefore select a dominated portfolios which lie below the efficient frontier

1.2.2.3 The exceed volatility puzzle

It has been known long time ago that security prices usually undergo dramatic fluctuations that do not fit with the fundamental changes of the security such as prospective earnings, dividends or variables involved the discounted factors like interest rate This fact contrasts to the traditional pricing models in which every change in stock price should be attributed to news relating to future dividend stream or discounted factors A big question, which was raised by Robert Shiller (1981) and Leroy and Porter (1981) , is weather the puzzle appears systematically or in individual episodes Shiller,

in his papers, observed that if stock price is equal to the expectation of sum

of discounted dividends, then stock price volatility should have an upper bound being the volatility of ex-post rational stock price, considered as actual summed discounted dividends Shiller points out that ex-post rational price ∗

should satisfy the relation:

where is discounted factor which is assumed to be constant From the formula above, a time series of ∗ can be constructed by backward induction,

given an initial condition Shiller made graphs of ∗ and and showed that

the graph of ∗ was much smoother than that of This discrepancy improves

the excessive volatility

1.2.2.4 Equity premium puzzle

The puzzle traces its origin to the article “The equity premium – a puzzle”

by Mehra & Prescott (1985) The article showed how the observed return relationships between asset classes contradicted the prediction of several traditional asset pricing model such as CAPM, CCAPM, which were based on expected utility theory The traditional model predicted an equity premium based on the higher levels of risk when trading equities (relative to risk -free assets) however the theoretically explained premium was far below the empirically observed premium for the period This implied that investors had other aversions than risk aversion as former theories explained, that is, they required a higher reward for the additional risk of holding equities

1.2.3 Behavioral Portfolio Theories

1.2.3.1 Roy’s Safety First Portfolio Theory

In 1952, Roy argued that investors tend to minimize the probability of portfolio return falling below a subjectively designated disaster level This behavioral maxim is referred to as the Safety First Principle

Note the return of the portfolio Rp and the minimal desired return Rm (the returns that would bring the wealth down to the subsistence level Ws, then the Safety First Principle is equivalent to:

If returns are normally distributed, then Roy’s Safety First Principle can

be reduced to maximizing the “Safety First Ratio” (SF-ratio henceforth)

If we focus on the case where no risk free asset exists (i.e σp > 0: ∀p ∈

P), where P is the set of all acceptable portfolios for a certain investor and/or investment problem Also Roy focused on the case where no risk free asset exist, and it is a quite reasonable assumption: there are no investments that have a zero variance over time horizons that are relevant for investments (multiple years to decades)

Figure 1.6: How an optimal portfolio can be determined under Safety First

Theory

In order to find the iso-SF curves in the (µp, σp) − plane we rewrite the

principle equation for a certain SF level SF0

So, people will notice that the iso-SF-curves are straight lines in the

, � – a plane that intersects the y-axis in Rm

After the introduction of Safety First Portfolio Theory, there were a number of generalization of it:

(Tesler 1955) generalized the Safety First portfolio theory by introducing

a desired probability level (α) connected to the violation of the minimal return

So, an investor will choose a portfolio that maximizes expected wealth (E[W]), subject to the constraint P(W ≤ Ws) ≤ α

(Arzac and Bawa 1977) extend Tesler’s model by allowing the probability

level α to vary In that case, the expected utility function becomes

with c is a scalar, and DR(WS) the decumulative distribution function Markowitz (1959) agreed that this was the only functional form that was consistent with the expected utility hypothesis

(Elton and Gruber 1996) discuss some generalizations, a.o the extension presented from Kataoka Kataoka proposes that it is an investors aim to maximize the subsistence level subject to the constraint that the probability that wealth falls below the subsistence level does not exceed a predetermined α

to reach to a higher level of wealth (this concept that is absent in SF-theory)

In all Safety First models risk is represented by the probability of falling below a certain subsidence wealth level, Ws This probability is best expressed by the complementary cumulative distribution function (or a decumulative probability function), D, defined as D(x) = P(W ≥ x)

Assume the discrete setting, so that after a certain horizon an investment can have M different states, that occur with a probability pi so that pi = P(Wi)

∀i ∈ {1,2, ,M} Assume without loss of generality that the wealth levels are

ordered so that W1 ≤ Wβ ≤ ··· ≤ WM Then one will notice that the expected

wealth after the experiment equals

Fear is the human need for security and will lead to make calculatio ns not with pi but with modified probabilities that overweight the probabilities of bad outcomes (so assume that the probability of low gains (low i) corresponds

to higher values than the actual probabilities This is the concern for security and it can be formulated as follows

where the s refers to security and with qs > 0

Hope for potential will result on the contrary in overweighting positive outcomes (so the higher i) This hope for potential will result in SP/A theory

in another distortion of the cumulative distribution function

where the p refers to potential and with qp > 0 One will notice that because

of the overweighting of the higher results, hp(D) will stochastically dominate

ℎ + − ℎ So, here SP/A theory differs from EUT by using a rank dependent formulation (i.e replacing DX by h(DX)

Furthermore Lopez postulates that decisions when outcome is uncertain are based on two functions

 the modified expected wealth, Eh[W]

 D(A) = P(W > A), or the probability that the outcome will exceed a

certain aspiration level

Hence the function to evaluate possible outcomes

1.3 Applications of Cumulative Prospect Theory to Portfolio Theory

1.3.1 Challenges in applying Cumulative Prospect Theory

The central idea in Cumulative Prospect Theory is that people realize

utility from “gains” and “losses” measured relatively to a reference point But

it is often unclear how to define precisely how much a gain or loss is beca use Kahneman and Tversky provided relatively few guidance on how to determine the reference point

The following example in investment can help to make this difficulty more concrete Suppose that we want to find out what characteristics of portfolio

an investor under prospect theory preferences would hold Firstly, we need to determine the “gains” and “losses” the investor is thinking of Are they

undergo gains and losses in his overall wealth, in the value of his entire portfolio, or in the value of each particular stocks? If the investor focuses on gains and losses in the value of the entire portfolio, is a “gain” realized by the

investor simply that the return on the portfolio was positive? Or does it mean that the return comes greater than the risk-free rate, or the return the investor expected to receive? And does the investor think about daily gains and losses, monthly or a longer horizontal?

One important attempt to clarify how people think about gains and losses

is the work of Koszegi and Rabin (2006, 2007, 2009) In these papers, the authors proposed a framework for applying prospect theory in economics that they argued is both restricted and flexible across different contexts Their

framework has several elements, but the most key idea is the idea that the reference point people use to compute gains and losses is their expectations,

or “beliefs… held in the recent past about outcomes.” Particularly, they

proposed that people derive utility from the difference between consumption and expected consumption, where the utility function displays loss aversion and diminishing sensitivity To close the model, t hey also assume that expectations are rational, in that they match the di stribution of outcomes that people will face if they 8 follow the plan of action that is optimal, given their expectations

Koszegi and Rabin (2006) also emphasized, as other authors did, that the problem is not whether we should replace traditional model s with models in which people derive utility only from gains and losses, but rather whether it

is useful to consider models in which people derive utility from both gains and losses and, as in traditional analysis, from consumption levels After all, even if gains and losses must be taken into account, consumption levels surely matter too, it would be a mistake to ignore them In some models based on prospect theory, people derive utility only from gains and losses However, this modeling choice simply reflects a desire for tractability, not a belief that rejects the relevance of consumption levels

1.3.2 Portfolio Construction under Cumulative Prospect Theory

The towards acceptance of Behavioral Finance and the advent of Cumulative Prospect Theory are incentives for researchers to conduct new

studies on applying Cumulative Prospect Theory’s assumption in explaining

individual behavior on their portfolio

Some research have been done on the optimal portfolio choice under Cumulative Prospect Theory Berkelaar et al (2000) has solved portfolio choice problem for loss-averse investor by assuming that investor aims to maximize probability that terminal wealth exceeds his aspiration level when facing with losses However, in this study, the probability distortion was n ot

applied Gomes (2003) conducts research on asset allocation behavior of loss averse investor and its implications for trading volume This research also finds that there exists a certain threshold over which the surplus wealth becomes, investors therefore sell a large part of their stock holdings and follow a portfolio insurance rule in order to protect themselves from loss De Giorgi and Hens (2004) analyze the consistency between Cumulative Prospect Theory (Kahneman and Tversky 1992) and Capital Asset Pricing Model under

-an assumption of normally distributed return They find that the Security Market Line holds in every financial market equilibrium Unfortunately, the piecewise-power utility function, suggested by Kahneman and Tversky, under the infinite short-selling condition, leads to financial markets with no equilibrium They suggest an alternative utility function which is consistent with experimental results of Kahneman and Tversky as well as the existent of equilibria

Carole Bernard and Mario Ghossoub (2009) is another significant finding

In this study, the authors introduced the Omega performance, which is the ratio of gain-side part of value of a prospect over its loss -side part and does not depend on portfolio weights And they show that there is a strong relationship between Omega performance measures and the optimal holding

of an investor under CPT with piecewise-power value function with different shape parameters, under a suitable choice of the status quo They studied the properties of this optimal holding and discuss how sensitive an investor behaving according to CPT is to skewness, asymmetric distributions and curvature of the value function

Finally, note that this study uses the piecewise-power utility function suggested by Kahneman and Tversky (1992) and also takes into account the effect of probability distortion This study does not focus on the existence of market equilibrium but follows the procedures of a CPT -investor in investment decision making Thereby, we attempt to find an optimal security portfolio under assumptions of the theory In addition, applying Cumulative Prospect Theory to finding an optimal portfolio provides a non -experimental