THE APPLICATION OF CUMULATIVE PROSPECT THEORY IN BUILDING OPTIMAL PORTFOLIO IN VIETNAMESE STOCK MAKRKET

The major studiesspecializing in portfolio optimization emphasize that i investors are normalStatman, 2005; ii they use S-shape utility function Kahneman and Tversky,1979 that reflects t

THE APPLICATION OF CUMULATIVE PROSPECT THEORY IN BUILDING OPTIMAL PORTFOLIO IN

VIETNAMESE STOCK MAKRKET

Student name: Nguyễn Thị Thu Hằng Student code: 1001020040

Class: A8 Intake: 49 Supervisor: M.Sc Le Thi Thu

Ha Noi, May 2014

Especially, I also thank my close friends Ngo Bach Thien Huong, Le Ngoc Hai,Nguyen Tung Minh, Ngo Thi Thu Huong, Nguyen Lan Anh, Pham Hai Yen andNguyen Phuong Thanh for all their encouragement throughout the completion of thework

Above all, many thanks to mom, dad, and my younger brother and sisters whoalways stimulate and spend all love for me

Nguyen Thi Thu Hang

Class: A8, Faculty of Business Administration, Intake: 49, Foreign Trade University

CONTENT

Stochastic DominanceVietnamese Index of Stock Price

LIST OF TABLES

Table 5.3 Level of knowledge of individual investors in 2000-2007 in Viet Nam 45

Table 7.1 List of joint hypotheses in testing the curvature of value function 57

Table 7.3 List of joint hypotheses in testing the curvature of probability

LIST OF EXHIBITS

Exhibit 2.5 Schematic depiction of the class of probability weighting function 18

CHAPTER 1: INTRODUCTION

1.1. The rationale of the research

Individual wealth management, especially individual optimal portfolio has beenrelatively new but expanding field that attract more and more the concern offinancial researchers Numerous studies regarding this domain are carried out overthe world, including standard finance models and behavioral finance model

The major characteristics of private investors are small capital size, lack oftechnology support and affected by behavioral biases Small scale of investmentsprevents individual investors from selecting many securities for their portfolio Theshortage of supporting high-tech tools poses the problem of how individualpractioners apply optimal portfolio models Lastly, behavioral biases are the in-depth reason for investors’ wrong decisions and mistakes while constitutingportfolio In three above features, individual behaviors is considered as the mosttypical difference, which divides optimal portfolio models into two approaches: onebased on Standard Finance paradigm, and one based on Behavioral Financeparadigm

Standard Finance paradigm proposes Markowitz Portfolio Theory, which isconsidered as the best mathematical model for optimizing portfolio This model ofportfolio optimization bases on the assumption that individual investors areanalytically sophisticated and knowledgeable about markets By assumption, privateinvestors in such these constituted models make optimal decision in a rationalmanner However, MPT is strongly criticized by behavioral finance scholars.According to Bernstein (1998), “evidence reveals repeated patterns of irrationality,inconsistency and incompetency in the ways human being arrive at decisions andchoices when faced with uncertainty” Nofsinger (2001) asserts that assumption ofrationality and unbiasedness of economic participants has been drubbed bypsychologist for a long time

As the mandatory requirement of financial research, behavioral finance researchersadvance substitute models of individual portfolio management The major studiesspecializing in portfolio optimization emphasize that (i) investors are normal(Statman, 2005); (ii) they use S-shape utility function (Kahneman and Tversky,1979) that reflects their attitudes toward risk; (iii) investors are also affected by theiremotions (Lopes, 1987) Derived from these realistic assumptions, a vast number ofresearches regarding individual portfolio have conducted in over the world Theprincipal contribution of individual optimal portfolio is with no doubt CumulativeProspect Theory initiated by Kahneman and Tversky (1992) – the keystone ofBehavioral Finance Theory

Moving focus on Vietnamese stock market, due to the great number of privateinvestors to the financial market, individual wealth management is still a pivotaldomain According to the interview result of Tran Dac Sinh – chairman of HOSE,

by the end of the year 2013, there were 1.3 million trading accounts comprise1,282,071 accounts of domestic individual investors compared with 5,081 accounts

of domestic institutional investors, 13,950 accounts of foreign individual investorsand the 1,631 remaining of foreign institutional investors In addition, during thedevelopment of the Vietnamese stock market, there is an increasing number ofprivate investors picking stock and allocating their portfolio instead of short-termtrading

Nevertheless, in reality, Vietnamese individual investors are not equipped bystrategic models helping them to overcome their emotional and cognitive biases.Many of them simplify portfolio selection process by using heuristics approachbecause they find models of optimal portfolio sophisticate and difficult to apply.Other individual investors designing portfolio based on available models are stillunable to optimize their wealth because of models’ implausible assumption ofrationality This status is one of the reasons causing the speculative bubble crash in

2007, even maybe threatening the sustainability of Vietnamese stock market

Thus, the matter of wrong individual portfolio investment decisions affecting on thesustainability and the enhancement of Vietnamese stock market is the rationale of

my thesis “The application of Cumulative Prospect Theory in building optimal

portfolio for individual investors in Vietnamese stock market”.

1.2. Objectives of the research

Behavioral Finance paradigm is a theoretical and empirical system that includesnumerous sub-theories such as Heuristics, Prospect Theory, Cumulative ProspectTheory, behavioral biases, disposition effect, etc Each relative theory, which has itsmathematic forms, can be a base constructing models of portfolio optimization Due

to limited time and a lack of research capacity, my thesis will only concentrate onCumulative Prospect Theory – the keystone of Behavioral Finance Theory, and themost simple model, which is so-called “Static Portfolio Optimization model”, withone risky asset and one free-risk asset in one-period economy

My thesis aims to answer two key research questions Are hypotheses ofCumulative Prospect Theory compatible with Vietnamese individual investors’characteristics? If the model is suitable for privately applying, are there anyrecommendations to realize the models in practical investments?

1.3. Scope of the research

Individual investors were picked up for the survey since they were more likely tohave limited knowledge about application of the Behavioral Finance or CumulativeProspect Theory in portfolio construction, hence prone to make psychologicalmistakes The influence has primarily analyzed in term of whether behavioralfactors affect the portfolio management behavior of individual investors

1.4. Research methodology

This study follows the methodology of survey research design of which dataprocessing was supported by quantitative approach

As Holme and Solvang (1996), a quantitative method is formalized, structured and

is characterized by selectivity as well as a distance from the source of information.The approach concentrates on numerical observations and attempts to generalize aphenomenon through formalized analysis of observed data where statistic indicators

are indispensable parts On the other hand, a qualitative approach is formalized to alesser extent is directed at testing whether the information is valid The typicalfeature of this method is the use of verbal description instead of purely numericaldata and aims at creating a common understanding of the subject in research.

In my thesis, by using descriptive survey, primary data is collected for quantitativeand qualitative analyses Stochastic Dominance is used to interpret individualdecisions between two options

1.5. Research structure

Except Introduction, Conclusion and Appendices, the thesis is structured as follows:

• Chapter 2: Cumulative Prospect Theory

• Chapter 3: Building Optimal Portfolio for individual investors

Theory

• Chapter 5: Introduction to Vietnamese individual investors in the stock market

• Chapter 7: Empirical results

CHAPTER 2: CUMULATIVE PROSPECT THEORY

2.1. Introduction to Cumulative Prospect Theory

Cumulative Prospect Theory (Kahneman and Tversky, 1992) is one of the mostimportant theories of Behavioral Finance Paradigm CPT has assistance forbehavioral researchers to understand and explain individual decision-makingprocess under uncertainty Hence, CPT has many important implications inconstructing portfolio

Cumulative Prospect Theory is the second version of Prospect Theory (Kahnemanand Tversky, 1979) Both of them are considered as two of the best theories toexplain individual decision under risk In essence, there are many such relativetheories as Expected Value, Expected Utility having great contribution to thefinancial decision-making process under conditions of risk, but each of them has itsown limitations Expected value is calculated by multiplying its payoff with itsprobability This model fails in predicting the final choice because the value was notalways directly related to its precise monetary worth, but rather dependent on otherpsychological factors Daniel Bernoulli (1738) releases works discovering thiscontradiction and advancing an alternative to the expected value notion Throughouthis experiments, Bernoulli recognizes that the value a person attaches to an outcomecan be influenced by such factors as the likelihood of winning, or probability, etc.Expected Utility, however, the notion of Expected Utility also fails in predicting all-loss choices

In 1979, Kahneman and Tversky provided an alternative, empirically supportedtheory of choice, so-called Prospect Theory, one that accurately describes howpeople actually go about making their decision In short, the theory predicts thatindividuals tend to be risk averse in a domain of gains and relatively risk-seeking in

a domain of losses However, there are some theoretical problems in PT The mainproblem is that the functional form of PT violates “stochastic dominance”(Kahneman and Tversky, 1979, pp 283±284) Stochastic dominance requires that ashift of probability mass from bad outcomes to better outcomes leading to an

improved prospect The theoretical problems have recently been resolved in a newversion of PT, called Cumulative Prospect Theory (CPT) that was introduced byTversky and Kahneman (1992); in particular, CPT satisfies stochastic dominance

2.2. Hypotheses of Cumulative Prospect Theory

2.2.1. Three basic hypotheses

According to Kahneman and Tversky in their work in 1992, there are three elementsforming the decision making process of CPT CPT-investor is defined as an investorwho behaves consistently with Cumulative Prospect Theory

Firstly, a CPT –investor will be concerned with the deviation of his final wealthfrom a reference level instead of final value under EU hypothesis Secondly, CPT-investor is more sensitive with losses than gains Lastly, investors do not evaluaterandom outcomes using reasonable probabilities, but base upon distortion byoverestimating low probabilities

For the first hypothesis, Kahneman and Tversky presented in their study in 1979 thefollowing experimental evidence to illustrate that the evaluation of decisionoutcomes has to be reference-dependent (“reference” in this context refers to thecurrent state of wealth), a principle that is incompatible with Expected Utility Inthis empirical work, experimental participants were asked to choose between alottery offering a 25% chance of winning 3000 and a lottery offering a 20% chance

of 4000, 65% of their participants chose the second option (20%; 4000) On thecontrary, when they were asked to choose between a 100% chance of winning 3000and an 80% chance of winning 4000, 80% of them chose the former (100%, 3000)

On contrary to reality, EU predicts that they should not choose different option inboth circumstances as the expected utility in the second choice is always better thanthe first one

To understand this certain situation, consider a gamble:

Where the notation should be understand as “gain with probability with ; where theoutcomes are organized in increasing order, so that for , and where = 0 Forinstance, a lottery offering a 50% chance of winning $333 or losing $111 would beformulated as Under EU, a rational investor valuates the above gamble as:

Where: is the current asset and is the utility function that is increasing andconcave

This formulation demonstrates the four key components of prospect theory: 1)reference – dependence, 2) loss aversion, 3) diminishing sensitivity, and 4)probability weighting

Exhibit 2.1: The value function

(Source: Kahneman and Tversky (1979))

According to Miyamoto (1987) and Kahneman and Tversky (1979), they advance avalue function with a reference point at the outcome, located at zero (see Exhibit1.1) Their findings emphasize on the graphic function of u is S-shape, reflecting theprincipal of “diminishing sensitivity” for the outcome evaluation For example, thesubject strongly discriminates between 0 and 20 rather than between 80 and 100

even though both where these figures regarding gains and losses In other words, adollar is always appreciated less as an investor becomes wealthier

Secondly, it is also found that is more sensitive to losses than for gains (lossaversion) Empirical tests conducted by Kahneman and Tversky (1991) indicate thatlosses are weighted about twice as heavily as gains – losing $1 is about twice aspainful as the pleasure of gaining $1

Last but not least, Kahneman and Tversky find that preferences of subjects can bemodeled by probability weighting that amplifies small probabilities and reduceshigher probabilities Therefore, the weighting function is definitely sensitive tochanges in probability near the final points 0 and 1 but obviously insensitive tochanges in probability in middle region As Kahneman and Tversky (1979),weighting function is an important hypothesis in supporting explanation forinvestors’ decision

2.2.2. Mathematic form

Let W and be the wealth and the reference level at the end of the period

Define the deviation D from the reference level as follows:

2.2.2.1. The value function

According to Kahneman and Tversky (1992), the value functionis defined as

follows:

Where: is random variable D,

It can be referred from definition of the value function that the function and arepositively homogenous, increasing, invertible, and twice differentiable

Parameter and demonstrate risk aversion, parameter illustrate loss-aversion.Moreover investors show the tendency to risk-averse for gains and risk-seeking forlosses, hence it is clear to find that Kahneman and Tversky (1979) suggest that ,and See Exhibit 2.2

Exhibit 2.2: The value function for different values of

(Source: Kahneman and Tversky, 1979)

2.2.2.2. The probability weighting function

The third element forming CPT decision-making process lies in the systematicdistortion of physical probability measure The probabilities distortion may beslightly different in case of gains () and losses ()

The probability distortions (or probability weighting functions) are denoted by and For a random variable D with cumulative distribution function and decumulativedistribution , Tversky and Kahneman (1992) suggest the following two probabilitiesweighting functions with for gains andfor losses:

Where:, and are both less than 1 as if or , there is no distortion in gain domain orloss domain, respectively

It is can be referred from the definition of the probability weighting function andthat they are differentiable Remember that investors show the tendency to risk-averse for gains and risk-seeking for losses, hence it is clear to find that

Kahneman and Tversky (1992) estimated and for a typical investor (See the

Exhibit 2.3)

Exhibit 2.3: The probabilities distortion functions, and

(Source: Kahneman and Tversky, 1992)

Ingersoll (2008) shows that the condition of ( ) ensures that and are increasing.Rieger and Wang (2004) indicate that the probability weighting function is notmonotone for

Prelec (1998) proposes an alternative specification for the weighting function: ,where parameter is similar to the one in the function proposed by Kahneman andTversky.

2.2.2.3. Objective function (Prospect function)

Bernard and Ghossoub (2009) define the objective function of the CPT-investor,denoted by, as:

Or, the CPT-objective function also rewrite as:

Where: and are cumulative distribution function and decumulative distributionfunction, respectively (see page 8), and are probability weighting functions (seepage 10)

In order to ensure that both integrals are finite and computable, the objectivefunction requires that and , where is parameter in the value function (see page 9),are parameters in the probability weighting function (see page 10)

2.2.3. Stochastic Dominance approach to test hypotheses

2.2.3.1. Overview of approaches

In order to test features of Prospect Theory, Kahneman and Tversky (1979) employthe Certainty Effect approach that also supports for Cumulative Prospect Theory Intheir experiments in 1979 and 1992, they rely upon comparison of two outcomes,one certain, one uncertain; hence, probabilities distortion function can beexplainable for their results Wu and Gonalez (1996) also apply the Certainty Effectapproach on their study supporting Tversky and Kahneman’s probability weightingfunction

Even though Certainty Effect approach has many applications and implications indecision theory under uncertainty, it has the well-known drawback, recognized by

Levy and Levy (2001) Many experiments show that the approach is ineffective incase of more than two outcomes and each outcome has the same probability (forinstance, gain $1000 with probability of 25%; loss -$200 with probability of 25%;gain $0 with probability of 25%; gain -$300 with probability of 25%) The problem

of Certainty Effect poses the need of alternative approaches

In their work, Levy and Levy (2001, 2002a) propose to employ StochasticDominance (SD) criteria to analyze decisions and implied preference inexperimental research The prominent advantages of SD approach over CertaintyEffect approach are that SD can compare two uncertain choices with manyoutcomes, which can be all positive, all negative or mixed According to Levy andLevy (2002b), Certainty Effect approach is not explainable for the curvature of thepreference with mixed prospects while SD approach can provide conclusion.Furthermore, based on SD condition, recent studies suggest experimental designsthat can isolate elements of CPT without having to estimate all parameters offunctions

In my thesis, the experiment, testing whether investors’ decisions are consistent toCPT, is conducted with the support of SD approach Firstly, we consider theoreticalframework of Stochastic Dominance Approach

2.2.3.2. Stochastic Dominance approach

It is clear that if you determine the distribution of each () and value function of eachoption, you can predict and explain the portfolio preferred to others.

Notation:

The cumulative distribution function of X:

The complementary or tail distribution function of X:

Density function:

b. Absolute and First-order Stochastic Dominance

Absolute dominance/ almost-sure dominance: Y is absolutely dominant over X if

and there is at least one y such that

First-order stochastic dominance/ simple stochastic dominance: Y is first-order

stochastically dominant over X if for all y, and there is at least one y such that Itcan be understood that Y has more chance than X of being bigger than any givenvalue y When Y is first order stochastic dominant over X, this relationship isdefined as

And, thus, if Y is absolutely dominant over X, then Y is also first-orderstochastically dominant over X

Theorem: Suppose, then portfolio is preferred to portfolio if either w is absolutely

dominant over w or w (See the proof in Appendix B)

Second-order stochastic dominance: Y is second-order stochastically dominant

over X if for all x, and there is at least one x, for which the above inequality isstrict When Y is second-order stochastic dominant over X, this relationship isdefined as

Theorem: Suppose that (implying that investors prefer more to less) and (implying

that investor is risk averse), is preferred to if is second - order stochasticallydominant over, or:

(See the proof in Appendix B)

d. Prospect Stochastic Dominance

Prospect stochastic dominance: Y is prospect stochastically dominant over X

(for , and ) if and only if

With at least one strict inequality

2.2.3.3. Applying Stochastic Dominance approach to test hypotheses of CPTInvestors are assumed to abide by the framework of CPT It implies that investorsare compatible with three basic hypotheses of CPT (See page 6)

The value function is where x is defined as the deviation of wealth in comparisonwith a determined reference point) (See page 9)

The probability function is for non-negative outcomes and for negative outcomes,with, where is the cumulative distribution function of X (See page 10)

Exhibit 2.4: Prospect Theory S-shape function and Reverse S-shape function

(Source: Levy and Levy (2002a))

Denote the set of prospect value functions containing that are convex for andconcave for

Denote the set of inverse prospect value functions containing that are convex forand concave for

Denote the set of prospect value functions containing that convex for all

Denote the set of inverse prospect value functions containing that are concave forall

In essence, to test the value function or the probability weighting function, it isadvised that researchers will test the curvature of graphic curves of both functions

If that convex for and concave for , then is an element of the set of prospect valuefunctions Similarly, if the probability function has the reserve S-shape, then is also

a probability weighting function of Cumulative Prospect Theory

Stochastic Dominance Approach relies on comparison of two outcomes Byanalyzing choice results, scholars are able to predict the shape of the value functionand probability weighting function

1.3.3.1. Applying Stochastic Dominance to test the value function

Consider two prospects X and Y:

Consider the function:

Let and be intervals such that Given prospects X and Y, Then,

, and

Both inequalities hold if and only if X is preferred to Y for all function that areconvex in A and concave in B

X dominates Y according to Prospect theory (See page 14), denoted by, if and only

if, and Remember that if and only if X is preferred to Y according to Prospect Theory or

Denote when X dominates Y according to Inverse Prospect Theory, if and only if,and And, if and only if X is preferred to Y according to Inverse Prospect Theoryfor all

Denote when X dominates Y according to second-order stochastic dominance (seepage 18), if and only if And, if and only if X is preferred to Y according to second-order stochastic dominance for all

Denote when X dominates Y according to inverse second-order stochasticdominance, if and only if And, if and only if X is preferred to Y for all

1.3.3.2. Test the probability weighting function

We assume that the decision maker abides by Cumulative Prospect Theory, wehave:

The empirical researches show that probability weighting function is “shallow inthe open interval and changes abruptly near the end-points where ” (Tversky andKahneman, 1992) More specifically, an inverse S-shape probability weightingfunction will be concave first, then convex

Consider the probability distortion functions that are concave in and convex in , forgiven values of d, c in [0,1] Denote this class by (See Exhibit 2.5)

• If , then the segment between c and d is nearly linear, then the probabilityweighting function is inverse S-shaped and continuous in (0,1) (as desired)

• If , thus c will be the inflection of these inverse S-shaped function

• If, the probability weighting function is unrestricted between d and c It ishard to conclude the shape of the probability weighting function

Exhibit 2.5: Schematic depiction of the class of probability weighting function

(Source: Manel and Franz (2007))

2.3. Typical biases explaining for Cumulative Prospect Theory

This thesis focuses on three typical biases explaining for Cumulative ProspectTheory, namely loss aversion, anchoring - adjustment and herding Loss aversion is

a pivotal part of risk attitude influencing investment choices Analyzed cognitivebias is anchoring – adjustment, which forms the reference lever, stemming fromfaulty reasoning In addition, emotional biases such as herding originating fromimpulsive feelings or intuition, rather than conscious reasoning and are hardlypossible to be adjusted to traditional rationality

2.3.1. Loss Aversion

Kahneman and Tversky (1979, 1992) advanced Prospect Theory and CumulativeProspect Theory that describe how decision-makers actually behave whenconfronted with choice under uncertainty The value function shows the asymmetrybetween the values people treat between gains and losses This theory hypothesizesthat prior losses increase risk-seeking, while prior gains reduce it This asymmetry

is called loss aversion

Empirical tests conducted by Kahneman and Tversky (1991) indicate that losses areweighted about twice as heavily as gains – losing $1 is about twice as painful as thepleasure of gaining $1 In other words, people tend to hold on losses in the hope thatprices will eventually go back up It can be explained on the basis of the CumulativeProspect Theory, that value function is upward sloping for wealth levers under eachindividual’s reference point In additional, investors are predicted to be risk averse

in gains Shefrin and Statman (1985) called this occurrence, stemmed from lossaversion, of “selling winners too early and riding losers too long” as dispositioneffect

Loss aversion is one of three components of risk attitude under the lenses ofbehavioral finance Numerous studies resolving the problem of portfoliooptimization derive from the base assumption of risk, particularly, loss aversion,there can be Static portfolio optimization model and Multi-stock portfoliooptimization under Prospect Theory for instances

2.3.2. Anchoring and Adjustment

As proposed by Tversky and Kahneman (1974), Anchoring and Adjustmentheuristic is one strategy for estimating unknown magnitude by starting frominformation that is adjusted to yield the acceptable value A vast number of studiesdemonstrate that regardless of how the initial anchors were selected, people havethe tendency to adjust their anchors inefficiently, leaving final estimates too close tothe original anchor, consequently, irrationally In other words, people are generallybetter at relative comparison than absolute numbers.

In his survey carried out in 2006, Pompian require participants to estimate a goodbuy price for a share Investors are likely to start by using an initial value as ananchor which can be the 52-week price of stock for instance People, then adjusttheir information by using their analysis and interpretation which are indicated asinefficient approach It is undeniable that investors anchor their thoughts to alogically irrelevant reference point while making portfolio investment decision Andersen (2010) presents the involvement of anchoring in investment decision ofmarket participants by using an existing arbitraging algorithm He applies thealgorithm for practical date of Dow Jones Industrial average, providing evidencethat anchoring plays an indispensable part in the weekly price fixing of the DowJones Industrial Index

Anchoring and Adjustment bias shapes a reference point in investors’ mind whenthey designing portfolios People, basing on their experience with their ownanchors, select securities for their portfolio Furthermore, reference level of price isalso forming in their invisible cognition The reference point is the central point ofthe S-shape function of the Cumulative Prospect Theory

2.3.3. Herding Bias

“Herding behavior is an alternative explanation of the way that investment choicesare made by investors” (Demirer and Kutan, 2006, Ferruz at al., 2008) Hirshleiferand Teoh (2003) define herding in financial markets as mutual imitation leading to aconvergence of action In other words, herding is a fundamental tendency of humansociety that people follow the investment decisions taken by majority That is why

people tend to alter their “wrong” answer when they are confronted with thejudgment of large group of people

Popular analysts have considerable influence on private investors‘ decisions.However, even completely rational professionals can deal with herding bias whenthey take into account other’s viewpoints, even if they know people react in a herdlike manner One reason is originating from the past when our ancestors used to livesociably and generally tend to seek the allowance from the crowd rather than being

a stand-out Furthermore, they believe when a large number of people areunanimous in its judgments, they are certainly right due to their illusion that thecrowd may know something they do not

Word of mouth is a pivotal importance of herding Investors generally trust theirrelatives, colleagues, friends instead of credible institutions or media (printednewspaper, television, radio) Talking to others seems rapid and effectiveinformation - spreading approach that no means of communication can surpass Intheir study, Shiller and Pound (1986b) with their intensive survey in investor’sbehavior, only six percent of the respondents specified newspapers and periodicals The existence of herding may have implications for asset-pricing models because itsbehavioral affects on stock price movement The assumption of EMH is totallyincorrect because in the real world, people, instead be rationally valuate the stockprice, they react in herd-like manner

In spite of the fact that herding bias is not a component of CPT-investor, thisimportant bias provides a proof that it is not plausible to apply models of portfoliooptimization of which assumption is investors’ rationality and independence

CHAPTER 3: BUILDING OPTIMAL PORTFOLIO FOR INDIVIDUAL

INVESTORS

3.1. Individual investors

An individual investor is a person who buys and sells securities for their personalaccount, and not for another company or organization Private investors play anindispensable part in stock market from the developed stock markets such as USA’s

to the emerging financial market such as Viet Nam’s

Standard Finance Paradigm assumes that individual investors are analyticallysophisticated and knowledgeable about markets By assumption, private investors insuch these constituted models make optimal decision in a rational manner.However, numerous studies criticizing the notion of rationality, pointing out thatindividual investors are affected by irrational nature of buying and sellingbehaviors According to Bernstein (1998), “evidence reveals repeated patterns ofirrationality, inconsistency and incompetency in the ways human being arrive atdecisions and choices when faced with uncertainty” Nofsinger (2001) asserts thatassumption of rationality and unbiasedness of economic participants has beendrubbed by psychologist for a long time

The irrationality of individual investors is discovered during the decision-makingprocess because this process is a cognitive process resulting in the choice of acourse of action among several alternatives In this process, the emphasis is onthinking based on weighting the outcomes and alternative prior to the last decision.During this process, individual investors are under the influence of numerous biasesthat drive them to wrong decisions and mistakes

Regularly, individual investors are irrational while they have to make buying andselling decisions in the stock market In reality, private investors are under the lack

of abilities, knowledge and technology, therefore, they decide to manage their assetthrough investing in an investment trust The investment trust is just cognizedinside, not the rational entity in the market Pompian (2006) lists more than twenty

biases appearing in the decision-making process, which alter and motivate alldecisions of individual investors.

In light of the above discussion, individual investors are irrational and biased duringthe process of making decision or process of buying and selling Private investorsmanage their assets on the basis of investment trusts instead of investment analyses.Thus, it is advisable that individual investors should have different investmentinstruments from institutional investors

3.2. Optimal portfolio

3.2.1. Introduction

Wealth management and especially the portfolio choice, one important bloc of thefinancial literature, have developed substantially over several decades, utilizing theenormous advancement under power of mathematics and calculus science

For the start, my thesis provides definitions of portfolio and optimal portfolio Aportfolio is defined as a grouping of financial assets such as stocks, bonds and cashequivalents, as well as their mutual, exchange – traded and closed-fundcounterparts Optimal portfolio is a set of portfolios that offers the highest expectedreturn rate for a particular investor’s acceptable level of risk or the lowest risk for agiven level of expected return

Portfolio construction is designed based on underlying principle of the notion thatrisk can be diversified by adding other assets that allow the portfolio to achieve abetter outcome per each risk unit From an investor’s perspective, portfolios are to

be constructed taking into account risk return preference of investors with optimalportfolios lying on the efficient frontier With each intensive objective eitherminimizing risk or maximizing return, more models of portfolio choice areproposed

The optimal portfolio literature can be reviewed as being in two major partsaccording to the approaches The first is Markowitz mean – variance model which

is well-known as “Modern portfolio theory”, which developed on the premise ofexpected utility theory by Markowitz (1952b, 1959) and Tobin (1958, 1965) Each

security is modeled by two parameters: mean and variance of its return rate.Parameter “mean” is representative for expected return concept, while “variance” isrepresentative for risk concept The key insight of the model is the expected return

is combination weighted average return of each individual security, but variance ofportfolio is not Thus, rational investors focus on the subset of portfolios lying on

“efficient frontier” which achieve the maximum value for a given variance or theminimum risk for each expected return rate The investor’s ultimate decision is onthe basis of their preference along the efficient frontier

Although the mean-variance model seems attractive and useful, there is a variety ofproblems for practitioners As Michaud (1989), the principal problems arestemming from optimization procedure that leads to concentrated portfolios, cornersolutions, the shortage of robustness and especially requirement of much input data,hence it is unsuitable for private investors The model is also strongly criticized bypsychologists because it is built in terms of investors’ rationality The underlyingassumption of Modern portfolio theory prevents MPT from applying in reality.The second approach is developed under the advancement of behavioral finance thatproposes better understanding of portfolio management behavior as well asdecision-making process When people confront with risk and gain, they areaffected of invisible biases deriving from psychology (Kahneman and Tversky,

1979, 1991) and emotions (Lopes 1987) Furthermore, investors have more accuratesecurity assessment in long-time rather than within one year

Portfolio construction based on behavioral finance assumes that investors areirrational Each behavioral model introduced in this domain concentrates on severalmajor psychological concepts such as risk asymmetry, emotions, behavioral biases,the prospect theory, mental accounting etc Within this thesis, my concentration is

on the model designed on Cumulative Prospect Theory base

3.2.2. Approaches of portfolio optimization

The expected utility theory, developed by Von Neumann and Morgenstern (1947),originates from early working paper of Bernoulli (1738), providing an idealized,

normative economic model of rational decision under uncertainty Complying withthe theory, investors maximize their utility through aggregating the weightedoutcomes Utilities, formulated in a utility function, are graphed nonlinearly related

to monetary amounts

In their study in 2005, Copeland, Weston and Shastri witness that expected utilityrests on a set of axioms, such as comparability or completeness, transitivity andinvariance Comparability means that agents know exactly their preference, hencecan select the most desired outcomes Transitivity implies that people haveconsistent preferences that are unable to be altered Invariance can be understoodthat preferences are framed independently Based on these assumptions, models ofportfolio optimization, asset allocation and valuation are constructed The expectedutility model, as with all theoretical models, is not without its limitations One isthat the theory considers uncertainty as objective risk It is obviously unacceptable

to plan for probabilities of events

Despite its limitations, EU assumption is irreplaceable until the occurrence ofpsychological concept in finance Each behavioral concept, such as risk asymmetry,emotions, behavioral biases, prospect theory, mental accounting, can be added into amodel of portfolio optimization Nevertheless, Prospect Theory and CumulativeProspect Theory are two most popular approaches to resolve the problem ofoptimization under lenses of behavioral finance

The Prospect Theory is similar in character to that of utility function, but the majordifference between two theories is the reference point While EU is the keybackground for construction of mean-variance portfolio theory, PT is an essentialpremise for models of portfolio choice in behavioral finance However, manyresearches show the limitation of PT that the theory can be only applied to gambleswith at most two nonzero outcomes; it predicts that people sometimes choosedominated gambles

In modified version published in 1992, the theory known as “Cumulative ProspectTheory” is popularly accepted and typically used in both academic and practical

Create a policy statement

Develop an investment strategy

Portfolio selection

Asset allocation

Monitor and update portfolios

worlds Empirical research has been testing CPT and providing evidence of itsrelevance for models of investment decision than original version The theoreticaland empirical studies, proposed by Tversky and Kahneman (1979), are a strikingproof in support of the CPT when CPT is accessible to resolve some limitations ofProspect Theory

To conclude, different assumptions shape different models of portfolio optimization.The Expected Utility Theory is the key motivation for traditional mathematicalmodel while PT and CPT are indispensable parts of the enhancement of optimalportfolio models accounting for behavioral biases

3.2.3. Processes of portfolio management

Exhibit 3.1: The process of portfolio management

(Source: CFA Institute, 2014, CFA Level1Book 1, Portfolio management)

3.2.3.1. Create a policy statement

Policy statement is a commitment of investors about goals and constraints as itrelates to their investment This step is judged as the most important of all stages inportfolio management process

It is requisite for an investor to understand his true financial needs both in short-runand long-run Based on this good understanding, the investor will manage his

portfolio to meet his needs When there is market volatility or a change in hisprivate needs, the policy statement will guide him to make necessary adjustments in

decision-3.2.3.2. Develop an investment strategy

Strategic investment plan is the strategy combining investors’ goals and objectiveswith current financial market and economic conditions

Before investment decision, investors should spend time on researching andanalyzing the macroeconomic situation There is no one denying the dependencebetween the development of stock market upon the sustainability of nationaleconomy and the stability of manufacturing environment

Actual experiences show that stock and other asset prices are important parts of thedriving forces to economic growth For example, the rise of stock prices haspositive effects to the increased investment of enterprises (excluding too highspeculation and imperfect information).Stock prices also have effects to the wealth

of the households and their spending

3.2.3.3. Select securities

Portfolio selection is the process that investors decide to pick securities for theirportfolio In this stage, the investor will choose securities including foreignexchange, gold, stocks, bonds, etc Portfolio selection is an indispensable step in

portfolio management process Based on policy statement, investment strategy andprivate screening systems, assets are add in to list of portfolio

There are many approaches to pick stock or securities Many fundamental investorsprefer huge companies that generate more profit with sustainable growth rates.While some individual investors select technical tools to pick growing stocks based

on market performance rather than the company’s fundamental factors Types ofpreference lead to the different selection of stocks, bond or cash

3.2.3.4. Allocate assets

After having a list of securities, investors jump into the next stage, allocation Themajor objective of this step is to distribute total original monetary wealth intodifferent investments In other words, they have to answer the following questions:how proportion of cash should an investor maintain? How much proportion of asset

X should be purchased? In my thesis, the model Static Portfolio Optimizationmodel, designed on the basis of Cumulative Prospect Theory, aims to help investors

to answer these questions in real world

In reality, with the support of such models as Capital Asset Pricing Model, FamaFrench Three Factors, Discounted Cash Flow model, Dividend Discounted Model,etc, investors are equipped with many screening system allowing to shortenchoosing process

3.2.3.5. Monitor and update portfolios

The last stage of portfolio management process requires investors to adjust whenboth markets and investors’ needs change It is necessary for investors to monitorfor these changes as they occur and update the plan as soon as the market changeshas big influence on portfolio performance in the foreseeable future

3.2.4. Optimization constraints

3.2.4.1. Regulation and taxes

Regulation and taxes are constraints imposed on the optimization process Investorsmay be forbidden by law to hold some assets because in some cases, unconstrainedportfolio optimization would lead to short-selling of some assets while short-sellingcan be forbidden in several countries Additionally, it is impractical to hold an assetdue to too high associated tax cost

3.2.4.2. Transaction costs

Transaction costs are the costs of trading in order to change the portfolio weights.Since the optimal portfolio changes with time, there is a financial incentive tooptimize again frequently However, too frequent trading will lead to too-frequenttransactions costs; hence, the optimal strategy is to find the frequency of re-optimization and trading that balance between transaction costs and up-to-dateoptimal portfolios

3.3. Designing optimal portfolio for individual investors

Each individual investor is affected by different types of cognitive and emotionalbiases These biases influence on purchasing and selling decisions, hence have greatimpacts on selection and allocation stages (See Exhibit 3.1)

Selection stage is the stage when individual investors choose stocks, bonds and cashfor their portfolio The choices can deviate from the initial investment policy andstrategy due to herding and anchoring for instances Therefore, investors shouldtake advantage of screening system to isolate them from the craziness of the stockmarket

Allocation is the process of optimization, thus it is of critical importance to applysuitable models to allocate portfolios For example, if one of the most obviousbiases is loss aversion, the investor should apply the model of portfolio optimizationbased on loss aversion index or loss aversion function Another examples, when aninvestor abide by Cumulative Prospect Theory, he should use models based ontheoretical framework of CPT

To summarize, designing individual portfolio is different from financial institutionsbecause of investors’ irrationality Each investor requires a private model benefitingthem in order to optimize their portfolio These models should be based on the mostclear biases affecting on them

CHAPTER 4: MODEL OF STATIC PORTFOLIO OPTIMIZATION

UNDER CUMULATIVE PROSPECT THEORY

4.1. Introduction

Cumulative Prospect Theory has been emerging as the best financial premises forconstructing optimal portfolio in comparison with Expected Utility Hypothesis andProspect Theory; hence, some theoretical optimization models have been designedunder CPT

Pirvu and Schulze (2012), in their working paper No 742, propose advance model

of multi-stock portfolio optimization under CPT The model is developed on thebasis of Static portfolio optimization of Bernard and Ghossoub (2009) Theyconsider a CPT-investor in one-period economy with one riskless bond and multiplerisky stocks, which follow a multivariate elliptical distribution The keycontribution of their work is a a two-separation between the riskless bond and amean-variance-portfolio Based on their finding, they resolve the optimizationproblem by imposing a regulatory risk constraint

He and Zhou (2011) resolve the static problem in the presence of n risky choices,corresponding to a multi-stock financial market They introduce a new measure ofloss aversion for large payoffs, known as large-loss aversion degree (the LLAD),which is proved to be a pivotal determinant of the model The problem ofmaximizing the prospect value is explicitly demonstrated for the cases when thereference level is the risk-free return and when it is not They compose the LLAD,the reference point and the curvature of of the probability distortion within theirstatics of optimal risky portfolio

Gomes (2003) in “Portfolio Choice and Trading Volume with Loss-AverseInvestors” presents a model of portfolio selection and security trading volume incase of loss aversion bias The demand function of model is discontinuous and non-monotonic risky assets Loss-averse investors complying with disposition effect willnot hold stocks unless the equity return rate is quite high Gomes provides the

cogent proof of that elasticity of the aggregate demand curve fluctuate considerably,depending upon the distribution of wealth.

Within the thesis, my principal objective is to introduce Static portfolio optimizationmodel holding in a risky asset and a risk- free asset under Cumulative prospectTheory, in a one-period economy This model valid in case of the assumption ofCPT-investor is reasonable

4.2. Static Portfolio Choice under Cumulative Prospect Theory

4.2.1. Background

Consider the portfolio choice problem in case of one-period economy with one free asset (return rate p over the period) and one risky asset (return rate q overperiod)

risk-Denote to be the investor’s initial wealth An amount is invested in the risky assetand the remaining is invested in the risk-free asset Assume that short-selling isforbidden

The final wealth at the end of the period is given by:

Define as the excess return rate on the risky asset over the risk-free rate:

Define , the reference level of wealth at the end of period as:

is the amount the individual would have receive at the end of period if he investedall his initial in the risk-free asset (for example: bank account or Treasury bills)

It is clear to see that:

The deviation from the reference level is defined as:

4.2.2. Content of Static Portfolio Optimization model

The objective function of the CPT – investor, , is given by:

(See page 11) and is the cumulative distribution functions and decumulative distributionfunctions of risky asset and risk-free asset, respectively

Letting, in order to and

Thus, and Then, obtain:

To simplify, rewrite the formulation as follows:

Where

Thus, portfolio optimization holds if

Let denote by the ratio of to We have:

CASE 1: Firstly, consider the situation where only borrowing is allowed, so

Problem 1: Given where is parameter in the value function (see page 9), are

parameters in the probability weighting function (see page 10), suppose that selling is prohibited and investors are allowed to borrow in order to invest in therisky asset We resolve the optimization problem of maximizing the prospect value

short-of (See page 31)

If , then we can write , then we consider 3 cases as follows:

- If , any holding in the risky asset is optimal The prospect value is constantand equal to 0

- If , the borrowing finite amount to invest in the risky asset optimize theportfolio The prospect value is equal to

- If , the optimal amount to invest in the risky asset is equal to 0

If , the maximum prospect value holds when:

According to Bernard and Ghossoub (2009), the equality yields the only root:

In order to is the optimal point for the equality, it leads to the requirement asfollows:

When , then (as desired) Thus, is the optimal allocation when borrowing isallowed

CASE 2: Consider the condition where both short-selling and borrowing constraints

are imposed This leads to

Problem 2: Given , where is parameter in the value function (see page 9), are

parameters in the probability weighting function (see page 10), suppose that bothshort-selling and borrowing are not allowed, we resolve the optimization problem ofmaximizing the prospect value of (See page 31)

If , then we can write , then we consider 3 cases as follows:

- If , any holding in the risky asset is optimal The prospect value is constantand equal to 0

- If , It is optimal to invest in the risky asset optimize the portfolio

- If , the optimal amount K to invest in the risky asset is equal to 0

If , the maximum prospect value holds when:

Thus, in this section we have just indicate the optimal portfolio allocation rate Clearly, the optimal holding is dependent upon As pointed out by Bernard andGhossoub, is the key to resolve the problem of portfolio allocation

As you can see the result of both cases, has key contribution to the final allocation

K Thus, to find the optimal allocation rate K, it is essential to find the optimal value

of

The higher, the higher the optimal allocation in the risky asset:

is also called CPT-ratio This ratio quantifies the risky asset‘s upside and downsidemeasured by and , respectively

In their work, Bernard and Ghossoub include their finding of the maximum value of

as follows:

To summarize: (See the detailed proof in page 32, 33 when )

Case 1: consider the situation where only borrowing is allowed, so

Case 2: Consider the condition where both short-selling and borrowing constraintsare imposed This leads to

Cumulative Prospect Theory, hence it is applicable for investors who are

loss-averse, anchoring in the reference level and overweighting small probabilities Furthermore, this model is easy to understand and apply in practical world

To academics, Bernard and Ghossoub introduce a new approach to resolve the problem of optimization under the lens of CPT Based on their working paper, more researches are conducted with more complex and sophisticated scenarios

4.3.2. Disadvantages

4.3.2.1. Violate Loss Aversion Index

Risk attitude consists of three components: (i) the basis utility; (ii) probabilitydistortion, and (iii) loss aversion known as “behavioral concept” measured through

LA index Numerous academics point out that there are many different alternativemeasures of behavioral criterion of loss aversion in the literature with their ownadvantages and disadvantages As Kobberling and Wakker, loss aversion isillustrated by an index defined as follows:

The gist of the formulation is to consider foundations of risk attitude outsidemarginal utility by using a “probabilistic risk attitude” resulting from model of