Cumulative prospect theory and its application in vietnamese security market

FOREIGN TRADE UNIVERSITY FACULTY OF BANKING AND FINANCE FINAL DRAFT -Cumulative prospect theory and applications to GRADUATION THESIS portfolio construction in Vietnamese security market Major: International Finance By Nguyen Duc Dung CUMULATIVE PROSPECT THEORY AND Advisor:TO Dr.PORTFOLIO Phan Tran Trung Dung APPLICATIONS CONSTRUCTION Faculty of Finance and Banking IN VIETNAMESE SECURITY MARKET Student’s Full Name: Nguyen Duc Dung Student ID: 1001030090 Intake: 48 – A9 Supervisor: Phan Tran Trung Dung, PhD Hanoi, May 2014 ii CONTENT CONTENT ii LIST OF TABLES iii LIST OF FIGURES iii INTRODUCTION CHAPTER CUMULATIVE PROSPECT THEORY AND BEHAVIORAL PORTFOLIO THEORY 1.1 Original Prospect Theory and Cumulative Prospect The ory 1.1.1 Phenomena of choice in laboratory settings 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 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 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 RESEARCH FINDINGS 48 CHAPTER SUGGESTIONS TO IMPROVE THE RESEARCH 51 iii CONCLUSION 54 REFERENCE 55 LIST OF TABLES Tables Page number Table 1.1: Forms of value functions 13 Table 1.2: Forms of weighting functions 15 Table 2.2: Questions asked to determine the utility for gains and the utility for losses 37 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 x i being medians of the 20 classes in table 3.1 and probabilities being frequencies of those classes 42 Table 3.3: Former estimations of CPT parameters 46 Table 3.4: Value of 48 for each set of estimation LIST OF FIGURES Figures Page number Figure 1.1: The Prospect Theory Value Function Figure 1.2: The probability weighting function Figure 1.γ: The value function v(x) with different value of α, and λ 12 Figure 1.4: Distorted cumulative probabilities w - (p) and w+ (p) with = 61 and δ = 69, estimated by Kahneman and Tversky (1992) 14 Figure 1.5: Possible portfolios comprised of assets 19 Figure 1.6: How an optimal portfolio can be determined under Safety First Theory 23 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 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 reallife 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 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 demonstrations that, in laboratory 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 five 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 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 Firstly, 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 ; …; x , p ; …; x n , p n ) (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 x i < x j for i < j, and where x = In this notation, p -m , p -m+1 , … and p n are corresponding probabilities of the outcomes Under the Expected Utility Theory, an individual would evaluate the above gamble equal to: ∑ =− + where W is his or her current wealth and U(.) is an increasing and concave utility function Under the Original Prospect Theory, by contrast, the game would be evaluated as: ∑ � =− where v(.) is called “value function” being an increasing function with v(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 + xi Kahneman and Tversky motivated this assumption, known as “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 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 46 This problem statement becomes very difficult to solve numerically Therefore, in this study, we choose to find an approximate solution of it by using Solver add-in of MS Excel Note that the solution, the optimal allocation produced, depends on assumed parameters of value function and weighted probability function 3.2.6 Empirical estimates of parameters In this study, the estimates of and δ is fixed at the estimates of Tversky and Kahneman (1992), that is, (δ, ) = (0.69, 0.61) while the estimates of α, and λ will vary among estimates of several author’s work as follow: Authors α λ Tversky and Kahneman (1992) 88 88 2.25 Abdelloui at al (2007) 72 73 2.54 Tu (2005) 68 74 3.2 Andersen et al (2006) 81 80 1.07 Abdelaoui et al (2008) 86 1.06 2.61 Fennema and van Assen (1998) 39 84 - Abdelaoui (2000) 89 92 - Abdelaoui et al (2005) 91 96 - Harrison and Rutstrom (2009) 71 72 1.38 Fehr-Duda et al (2006) 1.01 1.05 - Table 3.3: Former estimations of CPT parameters 47 For the former estimations that not contain estimations of λ we test by varying λ within (1; 3) linearly 3.2.7 CPT’s portfolio matching market portfolio In respond to a set of parametric estimates, an optimal allocation is produced The set of estimates will be tested by measuring degree of match between the produced allocation and the real allocation We will measure it by means of where � � = ∑ = � − � � is the weight of stock D in the real portfolio The less the more suitable the set of estimates would be is, the 48 CHAPTER RESEARCH FINDINGS The following table shows the results which describe the matching between theoretical optimal portfolios correspondence to sets of parameters of Cumulative Prospect Theory and the market portfolio: Parameter estimates Authors (α, ) λ (0.88, 0.88) 2.25 0.417175 Abdelloui at al (2007) (0.72, 0.73) 2.54 0.821946 Tu (2005) (0.68, 0.74) 3.2 0.772824 Andersen et al (2006) (0.81, 0.80) 1.07 0.087167 Abdelaoui et al (2008) (0.86, 1.06) 2.61 0.058379 Fehr-Duda et al (2006) (1.01, 1.05) 1.38 0.371269 1.25 0.056115 1.50 0.056094 1.75 0.056075 2.00 0.055971 2.25 0.055991 2.50 0.056013 2.75 0.056034 3.00 0.056054 1.25 0.077723 1.50 0.081939 1.75 0.108046 2.00 0.120823 Tversky and Kahneman (1992) Fennema and van Assen (1998) Abdelaoui (2000) (0.39, 0.84) (0.89, 0.92) 49 Abdelaoui et al (2005) Harrison and Rutstrom (2009) 2.25 0.12975 2.50 0.15368 2.75 0.20041 3.00 0.26723 1.25 0.081104 1.50 0.069133 1.75 0.065331 2.00 0.068732 2.25 0.092138 2.50 0.117832 2.75 0.163245 3.00 0.210304 1.25 0.057932 1.50 0.069134 1.75 0.130862 2.00 0.244529 2.25 0.378323 2.50 0.452309 2.75 0.587496 3.00 0.746301 (0.91, 0.96) (.71, 0.72) Table 4.1: Value of for each set of estimation The table 3.4 shows that in comparison to other estimations the estimation of (α, ) = (0,γ9; 0,84), which is experimentally measured in Fennema and van Assen (1998), best describes the CPT behavior of investor in Vietnamese security market In this estimation, the value of α and are both smaller than 50 This reinforces the idea of diminishing sensitivity, a component of Prospect Theory The estimation also display the biggest difference between α and (α is much smaller than ) This proves the diminishing sensitivity of gain side is much larger than that of loss side In other words, this implies that when investors shift from gain side to loss side, they dramatically change their attitude towards risk, from risk-aversion to risk-seeking In terms of loss aversion, the λ is estimated at β.00 This value of the λ proves the loss aversion phenomenon in Vietnamese investors’ behaviors 51 CHAPTER SUGGESTIONS TO IMPROVE THE RESEARCH This study has developed a framework to apply Cumulative Prospect Theory to portfolio construction in Vietnamese security market The framework includes fully assumptions of the theory and adds some assumptions which amend the ability to apply the theory and are qualitatively suitable for Vietnamese market setting In this study, for each set of parameters of Cumulative Prospect Theory, through process of constructing optimal portfolio under the theory’s assumptions, a theoretical portfolio would be produced After that, we implement a test of the match between the theoretical portfolio and the real market portfolio by using Ordinary Least Sum of Squares criterion This test is to measure the extent to which a set of parameters fits with nature of the market According to the result of the calculation, this study finds that the power indices of CPT’s utility function, which best match the nature of Vietnamese security market, are respectively α = 0,γ9 and = 0,84 The values of the parameters imply the existence of the psychological phenomena which are foundations of Cumulative Prospect Theory, i.e diminishing sensitivity, risk aversion for gains and risk-seeking for losses The study also finds the value of the multiplier λ equal to 2.00, greater than This proves the existence of loss aversion in behavior of investors in Vietnamese market As the capability and the time for this thesis are limited , the scope of this study merely includes the testing for the former estimations of CPT’s parameter and find the best out of them, but does not estimate the parameters exactly describing the CPT behavior of investors in Vietnamese market In addition, this study only focuses on the behavior of inve stors in the security market, but does not cover other assets such as real estate, gold, foreign currency In order to improve the result of this study, we suggest the following research methodologies: - Use data of a longer time period and divide the period into several shorter stages The extent to which parameters fit with the market should be 52 tested on each of these stage This method can describe the change in behavior of the investors overtime - For better measurement of loss aversion, we can use the ide a of equity premium puzzle and its solution, i.e myopic loss aversion, suggested by Mehra and Prescott (1985) This analysis not merely finds out the loss aversion degree but also allows to measure the implied investment horizon A good measurement of the implied investment horizon allows to structure the data well and make the testing become more accurate - To find a numerical solution for the optimal portfolio problem under Cumulative Prospect Theory, we can chose the approach of Eaves (1971) and carry it out by an adapted procedure, in multi-start framework This approach allows to solve optimal problems with non-smooth objective like the objective function of Cumulative Prospect Theory 53 54 CONCLUSION This paper has proposed a new and none-experimental method to measure utility and loss aversion under Cumulative Prospect Theory The method is conducted through applying the assumptions pf Cumulative Prospect Theory in constructing an optimal portfolio Therefore, it is easy to implement in empirical data and analysis Analyzing the data by solving the optimal problem backward induces the coefficients for the utility function We hope that the provision of such a non-experimental method will foster the use of Cumulative Prospect Theory in applications We proposed a procedure to process data in an attempt to use in construct an optimal portfolio under the Cumulative Prospect Theory assumptions We also proposed some assumptions which are suitable for nature of Vietnamese market These assumptions make the components of Cumulative Prospect Theory easy to apply Our results were broadly consistent with the assumptions of Cumulative Prospect Theory We found concave utility for gains, convex for losses and strong evidence for loss aversion The power parameters of utility function we found are also nearest to the results of Fennema and Van Assen (1998) We hope that the new approach to test the existent of Cumulative Prospect Theory’s assumptions and estimate its 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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 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... � = � − ℎ[ ] 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