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Universiteit van Amsterdam

Faculteit Economie en Bedrijfskunde

An application of Behavioral

Portfolio Theory

Author:

Misja Langeler

Supervisor:

Dr. Liang Zou

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Abstract

Behavioral Portfolio Theory postulates that, in practice, fear and hope operate on an investor’s willingness to take risk - where risk is defined as the probability of falling below the threshold that relates to undesired outcomes (for example: disaster or ruin). In this thesis, I develop BPT-efficient portfolios by means of a selection model which allocates assets such that the transformed expectation of the portfolio‘s return is maxi-mized subject to meeting specific transformed loss constraints. Following Lopes (1987), who conducts that fear and hope reside in every individ-ual although not to the same extent, I establish BPT-efficient portfolios for investors that are primary driven by fear and for investors that are primary driven by hope. Hereby, I define hopeful investors as individuals who transform the (assumed as objective) normal distribution of asset returns to a Gumbel distribution, and fearful investors as individuals who transform the normal distribution of asset returns to a reverse Gumbel distribution. Taking into account the presence of two risky assets - an index of stock and an index of commodity - and one risk free asset - at which one can borrow or lend - I find that hopeful investors are typically more exposed to downside risk than fearful investors, but are likely to ob-tain higher returns. A sensitivity analysis suggest that, under unforseen circumstances, BPT-efficient investors may be better off than rational in-vestors.

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Contents

1 Introduction 3

2 Literature review 5

2.1 Modern Portfolio Theory . . . 5

2.2 Fundamentals challenged . . . 6

2.2.1 Prospect Theory‘s weighting function . . . 7

2.2.2 Prospect Theory‘s value function . . . 10

2.3 Behavioral Portfolio Theory . . . 12

2.3.1 Roy‘s Safety First criterion . . . 13

2.3.2 Lopes‘ SP/A theory . . . 14

2.3.3 Formal representation . . . 15

3 Methodology 17 3.1 Data description . . . 18

3.2 Portfolio Selection Model . . . 19

3.2.1 Example and intuition . . . 20

3.2.2 Optimization routine . . . 23 4 Results 24 4.1 The routine . . . 24 4.2 The results . . . 26 5 Sensitivity analysis 30 6 Conclusion 34 7 References 36

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1

Introduction

Modern Portfolio Theory (Markowitz, 1952, 1959) prescribes investors to be mean-variance optimizing. The theory advices investors to maximize a portfolios expected return for a certain level of portfolio risk or, vice versa, minimize portfolio risk for a certain portfolio expected return. Herewith, Modern Portfolio Theory models portfolio expected return as the portfolio mean return and models portfolio risk as the portfolio standard deviation. The theory assumes that an investor perceives standard deviation as risk and maximizes expected absolute utility (or: should maximize expected utility) using either probability beliefs or objective probabilities. I.e.: MPT assumes investors to be rational and averse of volatility.

From the field of behavioral economics however, it is widely believed that investors generally are irrational, and primary averse of losses. Prospect Theory (Tversky and Kahnemann, 1979, 1992) for example, states that individuals tend to violate the rationality assumption by transforming (objective or subjective) probabilities in their minds, meanwhile deriving utility from relative wealth rather than from absolute wealth. Besides, Prospect Theory postulates that individuals tend to treat losses and gains asymmetrically, so that their utility curves (or: value functions) generally show loss aversion rather than strict aversion of volatility.

Following these findings - together with Roy‘s Safety First criterion and Lopes‘ (1987) conclusion that fear and hope operate on each in-vestor‘s willingness to take risk - Shefrin and Statman (2000) introduce an alternative portfolio theory: Behavioral Portfolio Theory. Rather than a prescriptive theory, Behavioral Portfolio Theory is a descriptive theory. The theory reflects how, in practice, economical agents tend to allocate their investments over a portfolio. In particular, Shefrin and Statman es-tablish that individuals tend to choose their investments by maximizing transformed expectation of terminal wealth (Eh(WT)), subject to

meet-ing specific transformed loss constraints (Ph(WT ≤ D) ≤ α) - where

transformed refers to the concept that hopeful individuals calculate ex-pected terminal wealth (Eh(WT)) and probability of undesired outcomes

(Ph(WT ≤ D)) optimistically, and fearful individuals calculate expected

terminal wealth and probability of undesired outcomes pessimistically. The authors conclude that - whereas the cornerstone of MPT is the mean-variance frontier in (µ, σ)-space - the BPT-efficient frontier resides

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in (Eh(WT) , Ph(WT ≤ D) ≤ α)-space. Besides, Shefrin and Statman

find that the BPT-efficient frontier generally does not coincide with the MPT-efficient frontier.

I contribute to the work of Shefrin and Statman by providing a textit-particular (and applicable) solution to a BPT-efficient portfolio. In this particular solution, I define hopeful investors as individuals who transform the normal distribution of asset returns (assumed as equivalent to the true probability distribution of asset returns) to a Gumbel distribution. Like-wise, I define fearful investors as individuals who transform the normal distribution of asset returns to a reverse Gumbel distribution. Hereby, I adhere to Lopes‘ finding that an individual who is primary driven by fear shifts probability weight from the right side of a distribution‘s support to the left, and an individual who is primary driven by hope shifts probability weight from the left side of a distribution‘s support to the right. Through developing BPT-efficient portfolios from the perspective of both hopeful and fearful individuals - based on the aforementioned definitions - I aim at analyzing the downside risk that the separate types of investors face. Are typically fearful investors better protected against losses than typi-cally hopeful investors? And are fearful and hopeful investors more or less exposed to downside risk than a rational investor? Following Campbell et al. (2000), the answers to these questions will be based on a simplified scenario in which three assets are present - two of them being risky.

In the remains of this thesis, firstly a review of all relevant literature concerning Modern Portfolio Theory, Behavioral Portfolio Theory and their underlying assumptions will be provided. Secondly, the methodology of the study will be specified. Thereafter the results will be presented, and finally, the conclusion will be drawn.

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2

Literature review

In this chapter, a short overview of Modern Portfolio Theory will be provided. Special attention will go out to the assumptions that MPT makes about investors‘ preferences and beliefs. Subsequently, alongside the fundamentals of Prospect Theory, these assumptions will be chal-lenged - explaining the anomaly that investors do not tend to choose for the mean-variance efficient solution. This all - together with Roy‘s Safety First criterion and Lopes‘ SP/A theory - leads to the alternative portfolio theory that is at the heart of this thesis: Behavioral Portfolio Theory.

2.1

Modern Portfolio Theory

Modern Portfolio Theory originally relies on Markovitz‘s portfolio the-ory (Markowitz, 1952, 1959), and is a financial thethe-ory in which expected portfolio return is related to expected portfolio risk. The theory generally aims at maximizing portfolio expected return, given a certain level of risk. Vice versa: according to this theory, portfolio risk should be minimized given certain portfolio expected return. As described in the introduction to this thesis, the theory models portfolio expected return as the portfolio mean return and models portfolio risk as the portfolio standard devia-tion. In other words: Markowitz proposes the portfolio selection process as a problem of portfolio mean and portfolio variance only, simplifying the quest of the optimal investment strategy to the formulation of an ef-ficient frontier in (µ, σ)-space from which the investor can choose his or her preferred portfolio, depending on individual preferences (Bodie and Kane, 2011). Markowitz herewith emphasizes that the selection process of securities should not be based on characteristics that are unique to the separate assets. Instead, Markowitz sends out the message that an investor should analyze how each of the available assets co-moves with the other assets. Accounting for these co-movements subsequently, each investor should be able to establish a portfolio that has a higher return than the portfolio that was found to be optimal in the case of ignoring co-movements - without affecting this portfolio‘s variance.

Apart from the crucial premise that investors perceive portfolio vari-ance as portfolio risk, the theory relies on the fundamental assumption that investors are (or: should be) rational. In his 1959 book, Markowitz

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emphasizes that rational decision making, in his eyes, is characterized by conditions which imply that one should maximize expected absolute util-ity, using probability beliefs where objective probabilities are not known. One can thus say that, in part, Markowitz adheres to the concept of Sub-jective Expected Utility - a proved result for each individual that acts in compliance with Savage seven axioms (or: postulates) of rationality (Camerer and Harless, 1994). Nevertheless: the axioms concerning ra-tional decision making under uncertainty that Markowitz proposes in his 1959 book, slightly differ from the axioms as presented in Savage‘s 1954 book. In particular, the axioms as presented by Markowitz (1959) are simpler than those stated by Savage (1954). This can, in the first place, be explained by the fact that Markowitz assumes only a finite number of possible hypotheses that may be true - whereas Savage accounts for an infinite number of hypotheses (Markowitz, 2012). In the second place, Markowitz‘s axioms are simpler because Savage does not include any con-sideration of objective probability (Markowitz, 2014). In essence, Savage finds that a rational decision maker should always maximize expected utility using probability beliefs. Markowitz, on the other hand, concludes that a rational decision maker only maximizes expected utility using prob-ability beliefs, when objective probprob-ability is not known. I.e.: according to Markowitz the decision maker, in theory, can rank alternatives that involve the possibility of objective probability. One should notice here however, that Markowitz does not imply that objective probability does exist.

2.2

Fundamentals challenged

Since its introduction, Modern Portfolio Theory has received a persistent amount of criticism, which at first was directed at the communis opinio that the theory simplifies reality by only taking into account the mean re-turn and standard deviation of portfolios (Elton and Gruber, 1997). Tobin (1969) for example emphasizes that, for a utility maximizing investor to be interested in the mean-variance solution, one must be sure that the combination of this investor‘s utility and the distribution of asset returns leads the maximization of expected utility to be equivalent to the mean-variance optimization problem. In particular, Tobin finds that a utility maximizing and volatility disliking investor can only act in compliance

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with the mean-variance solution if either 1.) the utility curve of the in-vestor is quadratic, 2.) asset returns are normally distributed, or 3.) both. In practice however, both utility and asset returns may depend on higher moments than only the mean and variance of returns (e.g.: Mandelbrot, 1963). Following these findings, some authors developed alternative port-folio selection models that, apart from a portport-folio‘s mean and variance, accounted for higher portfolio moments such as portfolio Skewness and portfolio Kurtosis (for instance, see Kraus and Litzenberger, 1976, and Lee, 1977).

In later literature, the more fundamental assumptions (hence, the premises of rationality and aversion of risk - where risk is identified as volatility) underlying Modern Portfolio Theory are challenged - especially from the field of behavioral economics and psychology. Taking into ac-count such literature, the remainder of this section seeks to appoint that (and why) investors, in practice, behave differently than assumed by Mod-ern Portfolio Theory. Alongside (Cumulative) Prospect Theory‘s value function and weighting function, I explain that individuals tend to maxi-mize the product of transformed probability and reference dependent util-ity (or: value) summed over all possible states, instead of being rational, i.e.: instead of maximizing the product of (subjective) probability and expectation of absolute wealth). Besides, I appoint how th summed over all states. shape of the value function implies that individuals cannot be perceived as strictly averse of volatility, but should rather be perceived as averse of losses.

2.2.1 Prospect Theory‘s weighting function

Where rationality implies that individuals take into account either prob-ability beliefs or objective probprob-ability, Prospect Theory postulates that individuals tend to make use of so-called transformed probabilities (or: decision weights) - a concept that is known as probability weighting (Tver-sky and Kahnemann, 1979, 1992). One should notice here first of all that, rather than to erroneous beliefs, probability weighting refers to the transformation of probabilities (Barberis, 2012). For an illustration of probability weighting, one could consider the example of an individual that participates in a lottery, even though he or she knows that buying a lottery ticket - taking into account the true probability of winning the

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lottery - is an unbeneficial investment from a monetary point of view. The question then rises: why does this individual participate? The con-cept of probability weighting provides an answer to this question through establishing that individuals tend to reweight the likelihood of events in their minds, therewith transforming probability. In the case of buying a lottery ticket, the participant then might overweight the probability corresponding to the state of the world in which the jackpot is won and underweight the probability corresponding to the states of the world in which someone else wins the jackpot - such that the ticket is perceived as a profitable investment. Likewise, through assuming that the probability corresponding to disastrous states of the world are generally overweighted, probability weighting can explain the phenomenon that individuals tend to buy insurance.

In scientific literature, the concept of probability weighting has been an issue of interest since the fifties of the past century. Following the introduction of Allais‘ (1953) common ratio violation - which can be seen as a consequence of probability weighting (Barberis, 2012) - both theoret-ical and empirtheoret-ical literature have addressed the existence of probability weighting. Loomes (1991), Harless and Camerer (1994) and Hey and Orme (1994), for example, find evidence in the support of probability weighting by means of experimental studies in which the participants are faced with strictly nonnegative monetary payoffs. More recently, Holt and Laury (2004) and Mason et al. (2005) conducted comparable experiments, in which the participants were provided a positive real cash balance at the start of the survey - so that negative as well as positive monetary outcome payoffs were made possible. Also these two studies provide evidence in support of probability weighting.

Answering to this (and other, earlier) empirical evidence, probabil-ity weighting has been included as a key assumption in several theories of choice under risk (Barberis, 2012), such as Prospect Theory (Kahne-man and Tversky,1979), Rank Dependent Expected Utility Theory (Quig-gin,1993), and Cumulative Prospect Theory (Tversky and Kahneman, 1992). Besides, especially in the past two decades, several authors have tried to quantify the concept of probability weighting both parametrically and non-parametrically. These authors‘ main concern was to estimate a weighting function w(.) whose argument is the objective probability

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Figure 1: Tversky and Kahneman‘s weighting function

P . An example of such a weighting function, provided by Tversky and Kahneman (1992), is shown in figure 1. In this figure, the vertical axis represents the transformed probability w(P ) and the horizontal axis rep-resents the objective probability P . The parametric relationship between P and w(P ) is given by w(P ) = Pδ/(Pδ+ (1 + Pδ))1/δ. Note that the solid line can be obtained by filling in 0.65; the value for δ that Tversky and Kahneman estimated based on experimental data. The dotted line is obtained by filling in δ = 1, and corresponds to the situation in which no probability weighting occurs, hence: to the situation in which w(P ) = P . As, in Tversky and Kahneman‘s weighting function, typically for small probabilities w(P ) > P and for large probabilities w(P ) < P , one could conclude that, indeed, individuals tend to overweight small probabilities and underweight large probabilities (Barberis, 2012), as was established in the example of the lottery ticket.

The shape of the weighting function has been the subject of scientific research by other authors as well. According to Prelec (1998), for instance, empirical estimates indicate that w(P ) is regressive (first w(P ) > P , there-after w(P ) < P ), inversely S−shaped (first concave, therethere-after convex), and asymmetrical (intersecting the diagonal before P = 0.5). Other para-metric estimates of the weighting function are found by Hey and Orne (1994) and Goldstein and Einhorn (1987) - all of them verifying the typ-ical inverse S−shape as shown in figure 1. More recently, several studies

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have tried to estimate the weighting function by means of more sophisti-cated, non-parametric methods. These studies include Gonzalez and Wu (1999), Abdellaoui (2000) and Bruhin et al. (2010), and generally provide strong evidence for the phenomenon of Probability Weighting - confirming the inverse S-shape of the weighting function. Drawback of these type of studies is, that they require very much information and need a detailed measurement of utility before probability weighting can be examined and proved (Van de Kuilen, Wakker and Zou, 2006).

2.2.2 Prospect Theory‘s value function

As stated in the first paragraph of this chapter, an investor is required to be averse of variance in order to be interested in Markowitz‘s mean-variance optimization problem. According to Prospect Theory, however, individuals generally dislike variance over gains only. Over losses on the other hand, individuals tend to be rather seeking for variance. A striking example of this anomaly, is provided by the combination of problems 11 and 12 from Kahneman and Tversky‘s 1979 paper:

• Experiment 11:

– In addition to whatever you own, you have been given 1000. You are now asked to choose between A (1000, 0.5) and B (500)

• Experiment 12:

– In addition to whatever you own, you have been given 2000. You are now asked to choose between C (-1000, 0.5) and D (-500)

In these problems, the subjects of the experiment generally preferred op-tion B over opop-tion A, meanwhile preferring opop-tion C over opop-tion D. Hence: the participants preferred a certain gain of 500 over a 50 percent chance of winning 1000, but simultaneously preferred a 50 percent chance of losing 1000 over a certain loss of 500, i.e.: in the loss region the subjects accepted the highly volatile gamble, whereas in the gain region they played the safe card - making them seeking for variance in the loss region and averse of variance in the gain region. As a result, the preferences of the subjects show diminishing sensitivity (Barberis, 2012). That is, each gained dollar contributes less to utility than the previous gained dollar, and each lost dollar (negatively) impacts utility by less than the previous lost dollar. In the questionnaire, one should notice as well that, in terms of absolute

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wealth, the problems are completely similar. By both option B and op-tion D, a terminal posiop-tion of 1500 is achieved, whereas opop-tions A and C each provide a 50 percent chance of obtaining terminal position 2000 and a 50 percent chance of obtaining terminal position 1000. Neverthe-less, the majority of the subjects in the experiment preferred option B in experiment 11 while they preferred option C in experiment 12 indicating that individuals do not care about absolute wealth but instead care about relative wealth. This concept of Prospect Theory is known as reference dependence (Barberis, 2012).

Since the introduction of Prospect Theory, especially the phenomenon of reference dependence has been subject of extensive empirical research, as it plays a role in many day-to-day applications, especially in the field of finance. Core et al. (2001), for example, find that reference points matter in the context of exercising stock option. In particular, they find that individuals are much more likely to exercise their options when stock prices exceed a maximum price that was set sometime during the previous year. In the context of mergers and acquisitions, the reference point has also empirically been found to be of importance. Baker et al. (2012) for example find that peak prices can serve as a reference point in negotia-tions concerning mergers and acquisinegotia-tions. The target may then consider the most recent (52 weeks) peak price as an anchor for negotiating. Baker et al. (2012) propose however, that this anchor is also clear to the po-tential acquirer/bidder. Also in theoretical studies, the reference point has been of interest. Koszegi and Rabin (2006) for instance developed a model of reference-dependent preferences in which the reference point is determined endogenously by the economic environment, being equal to rational expectations that were held in the recent past about outcomes -determined in a personal equilibrium.

In order to address the issues of reference dependence and diminish-ing sensitivity, Tversky and Kahneman parametrically estimated a value function in their 1992 paper. This value function can be seen as Prospect Theorys equivalent to the classical families of utility functions (Barberis, 2012). The general formula for the value function as estimated by Tver-sky and Kahneman is given by v(x) = xα for x > 0 and v(x) = −(−x)α for x < 0, where x represents a unit gain or loss. Based on experimen-tal data, the authors estimate α at 0.88 and λ at 2.25 - the values for

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Figure 2: Tversky and Kahneman‘s value function

which v(x) is plotted in figure 2. By the fact that the argument of the value function (hence: x) is a relative variable, it follows that, accord-ing the value function, individuals are indeed assumed to derive utility from gains and losses, measured relative to some reference point. I.e.: the value function accounts for the finding that individuals tend to be refer-ence dependent. Besides, the obvious convex shape in the loss region and the concave shape in the gain region infers what was established at the beginning of this section: individuals are diminishingly sensitive through being seeking for volatility in the loss region and being averse of volatility in the gain region. Therewith, the value function contradicts the strict aversion of volatility that is assumed by Modern Portfolio Theory. Instead of strict aversion of volatility however, one can recognize the concept of loss aversion in the graph. That is, the value function is obviously steeper in the loss region than in the gain region, indicating that a one unit loss generally hurts more than a one unit gain contributes.

2.3

Behavioral Portfolio Theory

Answering to the analyses of Tversky and Kahneman, Shefrin and Stat-man (2000) introduce an alternative theory to Markovitz‘s Modern Port-folio Theory: Behavioral PortPort-folio Theory. Doing so, Shefrin and Strat-man assume that 1.) instead of taking into account probability beliefs or

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objective probability (as follows from Markowitz‘s definition of rational-ity), investors take into account transformed probability (as is shown by Tversky and Kahneman‘s weighting function), and 2.) instead of variance aversion (as implied by Modern Portfolio Theory), investors show loss aversion (as is shown by Tversky and Kahnemans value function). Be-havioral Portfolio Theory however, proposes a specific type of probability weighting, based on Lopes‘ SP/A theory, and accounts for a specific type of loss aversion, built on Roy‘s Safety First criterion.

2.3.1 Roy‘s Safety First criterion

Although the stylized fact of loss aversion is generally assigned to Prospect Theory, Andrew Donald Roy proposed a portfolio selection model which accounted for the assumption that investors like to avoid losses long be-fore. Already in 1952 - shortly following the introduction of Markovitz‘s portfolio theory - Roy published his article Safety First and the Holding of Assets, in which he assumes that investors generally adopt the principle of Safety First instead of being mean-variance efficient.

According to Roy, investors are primary focused at the worst case sce-nario of an investment. This assumption originally comes from the believe that, in general, investors view the outcome of economic investments as a stochastic process in which each investment or combination of investments has a probability distribution from where best case scenarios and worst case scenarios can be extracted. Roy states that consequently, a large amount of economical agents have the idea that at each moment of time some disaster may occur. In particular, Safety First investors are assumed to be interested in avoiding worst case scenarios. From this point of view, Roy provides a framework for establishing portfolios so that the probabil-ity of these worst case scenarios is minimized. Herewith, he assumes that investors have a certain threshold, D, that they associate with the event of disaster. If the payoff of the investment(s) turns out to be lower than this threshold, disaster occurs. If the payoff of the investment(s) turns out to be higher than the threshold, no disaster occurs. Formally, Roy reckons that Safety First investors solve the following constrained optimization problem:

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max

w E (WT)

s.t. P (WT ≤ D) ≤ α

(1)

where the set w refers the respective probability weights (i.e.: the propor-tion of each available asset that is to be included into the portfolio), (WT)

denotes the level of wealth at the end of the investment horizon (assumed equivalent to the terminal portfolio value) and α gives an indication of the probability by which the investor likes to avoid disastrous outcomes. From equation 1), Behavioral Portfolio Theory adopts both the concept of wealth maximization and the principle of safety first. Nevertheless, Shefrin and Statman argue that, firstly, investors transform probabilities, or: weight probabilities.

2.3.2 Lopes‘ SP/A theory

Shefrin and Statman propose that investors do not only tend to overweight unlikely events at the expense of likely events, as follows from Prospect Theory‘s weighting function, but as well adopt the specific type of proba-bility weighting that SP/A theory - a psychological model of choice under uncertainty in which S stands for security and P stands for potential (Lopes, 1987) - propagates. In SP/A theory, Lopes labels individuals that have high need for potential as hopeful, and individuals that have high need for security as fearful. Lopes conducts that fear and hope re-side in every individual, although not to the same extent. Moreover: she concludes that, in case of a relatively fearful investor, transformation of probability occurs through an overweighting of the worst case outcomes at the expense of the best case outcomes and, in case of a relatively hopeful individual, transformation of probability occurs through overweighting of the best case outcomes at the expense of the worst case outcomes.

In the particular case of fearful individuals, Lopes computes Eh(WT),

hence the expectation of wealth at termination, by means of the decu-mulative function hs(D) - in which the subscript s stands for security.

The decumulative function attaches disproportionately great weight to higher values of D, leading Lopes to the conclusion that a typically fear-ful investor computes Eh(WT) in a pessimistic way, effectively shifting

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left side. In the hopeful case, Lopes computes Eh(WT) by means of the

decumulative function hp(D), in which the subscript p stands for

poten-tial. This decumulative function attaches disproportionate great weights to lower values of D. Lopes therefore concludes that a typically fearful investor computes Eh(WT) in an optimistic way, effectively shifting

prob-ability weight from the left side of the distribution‘s support to the right side.

2.3.3 Formal representation

Based on the fundamentals of Prospect Theory, Shefrin and Statman con-clude that investors generally are not mean-variance optimizing, because they are not likely to be strictly averse of variance (i.e.: they are not likely to perceive variance as risk) and not likely to show rational behavior (i.e.: they are not likely to take into account probability beliefs or objective probability). Instead, based on the principles of Roy‘s portfolio theory, Behavioral Portfolio Theory assumes that investors perceive the probabil-ity of falling below a threshold that relates to disaster as risk, leading them to apply the principle of Safety first while constructing their portfolios. Furthermore, based on the principles of SP/A theory, Behavioral Portfolio Theory postulates that investors take into account transformed probabil-ity instead of objective probabilprobabil-ity or probabilprobabil-ity beliefs - where the trans-formation of probability primary occurs through either an overweighting of worst case outcomes at the expense of the best case outcomes or an overweighting of the best case outcomes at the expense of the worst case outcomes. Shefrin and Statman subsequently emphasize that - whereas the cornerstone of MPT is the mean-variance frontier in (µ, σ)-space - the BPT-efficient frontier resides in (Eh(WT) , Ph(WT ≤ D) ≤ α)-space.

More specifically, Behavioral Portfolio Theory reckons that investors, in practice, maximize transformed returns, subject to avoiding disaster at a certain level of confidence, leading to the following formal representation:

max

w Eh(WT)

s.t. Ph(WT ≤ D) ≤ α

(2)

where the set w denotes the respective probability weights (i.e.: the pro-portion of each asset that is to be included into the portfolio), Eh(WT)

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refers to the transformed (or: perceived) expectation of terminal wealth (hence, the expectation of final wealth that is obtained through either overweighting the best case outcomes at the expense of the worst case outcomes or overweighting the worst case outcomes at the expense of the best case outcomes), and Ph(WT< A) refers to the (likewise) transformed

probability of falling below the threshold that relates to disaster. Finally, α gives an indication of the probability by which the investor wants to avoid disastrous outcomes.

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3

Methodology

The methodology of this thesis will have a structure similar to the study as originally published by Campbell, Huisman and Koedijk (2001), in which the authors construct examples of single-period Value-at-Risk-efficient portfolios. In particular, Campbell et al. develop a portfolio selection model in which stock, bond and cash are allocated over a portfolio such that the expected return of the portfolio is maximized, subject to the con-straint that the expected maximal loss does not exceed the Value-at-Risk limits as set by the manager. The applicant portfolios are developed based on the empirical distribution of the asset returns, and thereafter evaluated at the normal distribution and the students t -distribution. Campbell et al. present results for the 0.95, 0.96, 0.97, 0.98 and 0.99 confidence levels respectively.

However, whereas Campbell et al. construct Value-at-Risk-efficient portfolios, I will construct BPT-efficient portfolios. In particular, I de-velop a portfolio selection model in which the available assets are allocated over a portfolio such that transformed expectation is maximized subject to meeting specific, transformed, loss constraints. More specifically, I develop a portfolio selection model that complies with the constrained optimiza-tion problem as presented in equaoptimiza-tion (2), in which I consider two cases of the BPT-efficient portfolio: one case that is appropriate for a typically hopeful investor, and one case that is appropriate for a typically fearful investor. In the hopeful case, Eh(WT) and Ph(WT ≤ D) are computed

optimistically (hence, computed through overweighting the best case out-comes at the expense of the worst case outout-comes - referring to relatively hopeful behavior in Lopes‘ SP/A theory). In the fearful case Eh(WT) and

Ph(WT ≤ D) are computed pessimistically (hence, through

overweight-ing the worst case at the expense of the best case outcomes - referroverweight-ing to relatively fearful behavior in Lopes‘ SP/A theory). I furthermore assume that the reference point is equal to initial wealth, following empirical re-search that has identified the status quo or lagged status quo as reference point (Kahneman, Knetsch and Thaler, 1990; Odean, 1998; Genesove and Mayer, 2001). Furthermore, the investment period comprises one year, the investors are assumed to fully invest initial wealth into their portfo-lios and, following Campbell et al., the investors are prohibited to sell short risky assets.

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In the remainder of this chapter, I specify how the specific form of probability weighting that is propagated by Behavioral Portfolio Theory (which as stated - does not only account for the anomaly that investors tend to overweight unlikely events at the expense of likely events, but also reflects that left tail events are generally overweighed at the expense of right tail events or vice versa) will be modeled. Doing so, I will make spe-cific assumptions about the real distributions of asset returns and the per-ceived distribution of asset returns. However, firstly, I provide an overview of the set of assets that the separate types of BPT-efficient portfolios may consist of.

3.1

Data description

Following Campbell et al., I assume presence of three different assets: a stock, a commodity and a risk-free asset. The historical returns and standard deviations concerning stock, and commodity, as well as their correlation coefficient, are made quantitative in table 1. The historical data on the stock and the commodity come from crbtrader and comprise the period 1959-2014. Note that historical returns are computed following the method used by Gorton (2004), who constructed a so-called equally-weighted index of commodity futures and equity returns over the timespan between 1959 and 2004. I only present the first two moments of the assets‘ historical distribution because, as will be explained in the next paragraph, I assume that only these two moments play a role in the true (and perceived) probability distribution of the separate assets. Following Campbell et al., I assume the risk free rate to be equal to 0.03. The risk free rate, at which the investors can lend or borrow, carries zero risk and thus has zero volatility.

ρ = -0.25 historical return historical volatility stock 0.1158 0.1643

commodity 0.0976 0.1125 Table 1: Historical returns

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3.2

Portfolio Selection Model

Where Campbell et al. take into account the empirical, normal and stu-dent’s t distributions for establishing and testing their Value-at-Risk ef-ficient portfolios, I make use of the normal distribution, the Gumbel dis-tribution and the reverse Gumbel disdis-tribution. Specifically, my BPT-efficient portfolios are constructed based on the Gumbel distribution and the reverse Gumbel distribution, and thereafter evaluated at the normal distribution. Herewith I implicitly assume that, according to objective probability, asset returns are normally distributed. Specifically, following the information as provided in table 1, I assume that the true distribution of the stocks return is defined by mean µ = 0.1158 and standard deviation σ = 0.1643. Likewise, I assume that the true distribution of the commod-itys return is defined by mean µ = 0.0976 and standard deviation σ = 0.1125. I assume this information to be publicly known.

A typically fearful investor however, is assumed to transform this nor-mal distribution of returns (even though it is public information) in his or her mind to a Gumbel distribution - therewith overweighting the worst case scenarios at the expense of the best case scenarios. Likewise, a typ-ically hopeful investor transforms the normal distribution of assets to a reverse Gumbel distribution - therewith overestimating the best case out-comes at the expense of the worst case outout-comes. Or formally:

Definition 1 A fearful investor is an individual that perceives a normally distributed return with mean µ and standard deviation σ as a return that follows a reverse Gumbel distribution with location parameter µ and scale parameter σ.

Definition 2 A hopeful investor is an individual that perceives a nor-mally distributed return with mean µ and standard deviation σ as a return that follows a Gumbel distribution with location parameter µ and scale pa-rameter σ.

Although the Gumbel distribution is generally used in order to analyze minimum and maximum values, it has also been found to be appropriate for modeling skewed events such as population strength (many strength populations show few weak units in the left tail but a high number of strong units in the right tail) or endurance of products that experience very quick wear-out after reaching a certain age (Beirlant, Goegebuer,

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Segers and Tengels, 2004). Besides, in economics, the Gumbel distribu-tions have been found useful for modeling Leontieff producdistribu-tions funcdistribu-tions. I consider the Gumbel distribution appropriate for this study, because it replicates exactly the type of probability weighting that is proposed by Behavioral Portfolio Theory. That is: starting from a normal distribu-tion with mean µ and standard deviadistribu-tion σ, a Gumbel distribudistribu-tion with location parameter µ and scale parameter σ 1.) shifts probability weight from the top of the distribution to the tails of the distribution, and 2.) shifts probability weight from the left side of the distribution‘s support to the right side of the distribution‘s support (Kinney, 1997). Likewise, a reverse Gumbel distribution 1.) shifts probability weight from the top of the distribution to the tails of the distribution, and 2.) shifts probability weight from the right side of the distribution‘s support to the left side of the distribution‘s support. One should also notice here that, while shift-ing probability weight, the transformation of a normal distribution with mean µ and standard deviation σ to a (reverse) Gumbel distribution with location parameter µ and scale parameter σ does not affect the location of the distribution‘s top unlike for example the skew-normal distribution. (For an illustration of this concept, please refer to figure 3, in which I present the three distributions of an example portfolio in one graph.)

3.2.1 Example and intuition

For illustrating the specific form of probability weighting that is implied by the transformation of a return‘s normal distribution to the (reverse) Gumbel distribution, one can take into account the following example. Consider a portfolio that consists for 25 percent of stock, for 25 percent of commodity and for 50 percent of the risk free asset. It is straight-forward that this portfolio, just as its increments, follows a normal dis-tribution, since basic probability theory prescribes that a linear combi-nation of assets is normally distributed as well, meaning that the dis-tribution of this portfolio is completely defined by its mean (equal to 0.5 ∗ 0.03 + 0.25 ∗ 0.1158 + 0.25 ∗ 0.0976 = 0.0684) and its standard devia-tion ( equal to (0.252∗ 0.16432+ 0.252∗ 0.11432+ 2 ∗ (−0.25) ∗ 0.25 ∗ 0.25 ∗

0.1643 ∗ 0.1143)1/2= 0.0438). Please refer to figure 3 (obtained via MATLAB) for the graphical representation of this particular distribution -denoted by the red dotted line. One should notice here, that the

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objec-Figure 3: The normal, Gumbel and reverse Gumbel distribution

tive probability that someone who invests full wealth into this particular portfolio will be faced with a disastrous outcome at the end of an in-vestment period of one year (i.e.: the probability that, after one year, WT ≤ W0), is equal to the probability that a normally distributed

vari-able with µ = 0.0684 and standard deviation σ = 0.0438 turns out to be smaller than or equal to zero. This probability can be computed by taking the integral from −∞ to 0 of the distribution curve as presented in figure 3 and is equal to 0.0592. (I use the MATLAB-command normcdf in order to derive this probability. In particular, I compute the probability that a random variable with mean µ = 0.0684 and standard deviation σ = 0.0438 is smaller than or equal to zero is by means of the command normcdf (0, 0.0684, 0.0438).)

A fearful investor however, following definition 1, transforms the prob-ability distribution of the aforementioned portfolio to a reverse Gumbel distribution with location parameter µ = 0.0684 and scale parameter σ = 0.0438. (Please refer to the blue line in figure 3 for the graphi-cal representation of this particular distribution.) More specifigraphi-cally, the fearful investor perceives the probability that investing full wealth into this portfolio leads to a disastrous outcome (hence, the probability that, after one year, WT ≤ W0), assuming that full wealth was invested

into the portfolio) as equal to the probability that a reverse-Gumbel dis-tributed variable with µ = 0.0684 and σ = 0.0438 turns out to be smaller

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than or equal to zero. This probability can be computed by taking the integral from −∞ to 0 of the distribution curve as presented in figure 3, and is equal to 0.1892. (In order to compute this integra, I use the MATLAB-command evcdf. In particular, I compute the probability that a reverse-Gumbel distributed variable with location parameter µ = 0.0684 and scale parameter σ = 0.0438 is smaller than or equal to zero by means of the command evcdf (0, 0.0684, 0.0438).) Besides, the expected value of a reverse-Gumbel distributed variable with scale parameter µ = 0.0684 and location parameter σ = 0.0438 is equal to 0.0684 − γ ∗ 0.0438 = 0.0431 , where γ represents Euler‘s constant (Kinney, 1997).

A hopeful investor on the other hand, following definition 2, transforms the probability distribution of the aforementioned portfolio to a Gum-bel distribution with location parameter µ = 0.0684 and scale parameter σ = 0.0438. (Please refer to the green dotted line in figure 3 for the graph-ical representation of this particular distribution.) More specifgraph-ically, the hopeful investor perceives the probability that investing full wealth into this portfolio leads to a disastrous outcome (hence, the probability that, after one year, WT ≤ W0)) as equivalent to the probability that a

Gum-bel distributed variable with µ = 0.0684 and σ = 0.0438 turns out to be smaller than or equal to zero. Again, this probability can be computed by taking the integral from −∞ to 0 of the distribution curve as presented in figure 3, which is equal to 0.0085. (I compute the probability that a Gumbel distributed variable with location parameter µ = 0.0684 and scale parameter σ = 0.0438 is smaller than or equal to zero is by means of the command 1 − evcdf (0, −0.0684, 0.0438).) Besides, the expected value of a reverse-Gumbel distributed variable with scale parameter µ = 0.0684 and location parameter σ = 0.0438 is equal to 0.0684 + γ ∗ 0.0438 = 0.0937 , where γ represents Euler‘s constant (Kinney, 1997).

Summarizing, one can say that a portfolio that consists for 25 percent of stock, for 25 percent of commodity and for 50 per cent of the risk free rate, faces a probability of 5.92 percent of becoming worth less than its initial value after one year. Hence, in reality, P (WT ≤ D) = 0.0592.

Be-sides, the true expectation of terminal wealth (normalizing initial wealth at 1) of such portfolio is equal to 1 ∗ (1 + 0.0684) = 1.0684. Hence, accord-ing to objective probability, Eh(WT) = 1.0684. In the perceptions of the

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port-folio loses worth is equal to 0.85 percent and 18.92 percent respectively. Hence, in the hopeful case Ph(WT ≤ D) = 0.0085, and in the fearful case

Ph(WT ≤ D) = 0.1892. Likewise, in the perception of the hopeful and

fearful investor, Eh(WT) = 1.0937 and Eh(WT) = 1.0431 respectively.

3.2.2 Optimization routine

For finding the BPT-efficient portfolios, one should not only transform the probability distribution of the portfolio that is considered in the pre-vious example. Instead, the distribution of each feasible portfolio (hence, of every portfolio that consists of stock and/or commodity and/or the risk free asset that one can think of) should be transformed to a (reverse) Gumbel distribution. Subsequently, all possibly BPT-efficient portfolios are selected by removing the portfolios from the set that do not fulfill the constraint. From the remaining set, the optimal portfolio is found by se-lecting the particular portfolio that has the highest transformed expected return.

Because, in the solution to the BPT-efficient portfolio that I propose, the inequality constraint of the applicant maximization problem is non-linear in that it involves the cumulative distribution function of the (re-verse) Gumbel distribution, an analytic solution is not feasible. Therefore, I run the maximization routine by means of the optimization application in MATLAB - an application that can account for both linear and non-linear (in)equality constraints. Following Campbell et al., results are presented for the 0.95, 0.96, 0.97, 0.98 and 0.99 confidence levels respectively (i.e.: for α = 0.01, 0.02, 0.03, 0.04 and 0.05).

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4

Results

4.1

The routine

As stated in the previous chapter, I find the BPT-efficient portfolios by means of the non-linear consrained optimization solver that is included in MATLAB. This means that I have to construct:

• An objective function, for declaring the function that has to be max-imized.

• A constraint function, for declaring the non-linear and/or linear (in)equality constraint(s).

Obviously, in the general format of BPT, the objective function is as in the formal representation, which was obtained in the literature review. In other words: the objective function, taking into account BPT‘s general form, is represented by:

max

w Eh(WT) (3)

Likewise, the constraint function, taking into account BPT‘s general form, is represented by

Ph(WT ≤ D) ≤ α (4)

As stated, a rational investor takes into account objective probability when available. Therefore, such investor would solve the aforementioned problem through imposing that the returns of the available risky assets (hence, rs and rc) are normally distributed. In other words: a rational

investor considers terminal wealth (which - because it is assumed that full initial wealth - normalized at one - is invested in the portfolio at the start of the investment period - I define equal to terminal portfolio value, hence, to: WT = ws ∗ (1 + rs) + wc∗ (1 + rc) + wf ∗ (1 + rf),

where ws+ wc+ wf = 1) as a normally distributed variable with mean

ws∗ (1 + µs) + wc∗ (1 + µc) + wf∗ (1.03) and standard deviation (ws2∗ σ2s+

wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2, Consequently, a rational investor

would rewrite the objective function as:

max

ws,wc,wf

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where the values for µ are as obtained in the previous chapter. Meanwhile, the rational investor defines the constraint function as:

FWT(1) ≤ α,

ws ≥ 0,

wc ≥ 0,

ws+ wc+ wf = 1

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where FWT(.) denotes the cumulative distribution function of the normal

distribution. Hence, in this particular case, FWT(1) refers to the

proba-bility that a normally distributed variable with mean ws∗ (1 + µs) + wc∗

(1 + µc) + wf∗ (1.03) and standard deviation (ws2∗ σs2+ wc2∗ σc2+ 2 ∗ ws∗

wc∗ ρ ∗ σs∗ σc)1/2is smaller than or equal to one, in which the values for

µ, σ and ρ are as found in the previous chapter. Notice that via ws ≥ 0,

wc ≥ 0,, the restriction on short-selling of risky assets is imposed.

A fearful investor on the other hand, solves the aforementioned prob-lem through imposing that the portfolio follows a reverse Gumbel distri-bution with location parameter ws∗ (1 + µs) + wc∗ (1 + µc) + wf∗ (1.03)

and scale parameter (ws2∗ σ2s+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2.

As a result, the fearful investor writes the objective function as:

max

ws,wc,wf

ws∗ (1 + µs) + wc∗ (1 + µc) + wf∗ (1.03) − γ∗

(ws2∗ σ2s+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2

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where the values for µ, σ and ρ are as obtained in the previous chapter and γ denotes Euler‘s constant. Furthermore, the fearful investor writes the constraint function as:

FWT(1) ≤ α,

ws ≥ 0,

wc ≥ 0,

ws+ wc+ wf = 1

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where FWT(.) denotes the cumulative distribution function of the reverse

Gumbel distribution. Hence, in this particular case, FWT(1) refers to

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parameter ws∗ (1 + µs) + wc∗ (1 + µc) + wf∗ (1.03) and scale parameter

(ws2∗ σ2s+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2is smaller than or equal

to one. Notice that via ws ≥ 0, wc ≥ 0, the restriction on short-selling

of risky assets is imposed.

The hopeful investor finally, solves the constrained optimiziation rou-tine through imposing that the portfolio follows a Gumbel distribution with location parameter ws∗ (1 + µs) + wc∗ (1 + µc) + wf ∗ (1.03) and

scale parameter (ws2∗ σs2+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2. As a

result, the fearful investor writes the objective function as:

max

ws,wc,wf

ws∗ (1 + µs) + wc∗ (1 + µc) + wf∗ (1.03) + γ∗

(ws2∗ σ2s+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2

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where the values for µ, σ and ρ are as obtained in the previous chapter and γ denotes Euler‘s constant. Furthermore, the hopeful investor writes the constraint function as:

FWT(1) ≤ α,

ws ≥ 0,

wc ≥ 0,

ws+ wc+ wf = 1

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where FWT(.) denotes the cumulative distribution function of the

Gum-bel distribution. Hence, in this particular case, FWT(1) refers to the

probability that a Gumbel distributed variable with location parame-ter ws ∗ (1 + µs) + wc ∗ (1 + µc) + wf ∗ (1.03) and scale parameter

(ws2∗ σ2s+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2 is smaller than (or

equal to) one. Notice that via ws ≥ 0, wc ≥ 0, the restriction on

short-selling of risky assets is imposed.

4.2

The results

Taking into account the objective and constraint functions as specified in

the previous section, I obtain BPT-efficient portfolios for α = 0.01, 0.02, 0.03, 0.04, 0.05 in both the hopeful case and the fearful case. Besides, I obtain the

effi-cient portfolio for a rational investor that prefers to restrict its downside risk equivalently to the BPT-efficient case, hence: through adopting the

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principle of Safety First. Please refer to table 2, the key table of this the-sis, for the exact weights of the separate assets that the efficient portfolios consist of.

α Stock (%) Commodity(%) risk free(%) fearful 0.01 3,88531448 6,017597425 90,0970881 0.02 4,792112556 7,42204174 87,7858457 0.03 5,815348426 8,236319688 85,94833189 0.04 6,265241958 9,703608451 84,03114959 0.05 6,879309882 10,87884561 82,2418445 hopeful 0.01 23,24113816 35,99584988 40,76301196 0.02 31,59703493 48,93743465 19,46553042 0.03 41,63875063 64,49004694 -6,128797566 0.04 55,41711736 85,82996777 -41,24708514 0.05 76,72475407 118,831221 -95,55597507 rational 0.01 10,12429989 15,6805038 74,1951963 0.02 12,53801148 19,41889509 68,04309343 0.03 14,7725508 22,87971368 62,34773552 0.04 17,05972277 26,42207853 56,5181987 0.05 19,51776545 30,22909272 50,25314183

Table 2: BPT-efficient portfolio weights

From table 3 furthermore (the parameters in this table are computed as outlined in the example portfolio that was provided in the methodology section), it is immediately obvious, that the hopeful investor never fulfills the constraint in terms of objective probability. For all applicant levels of α, the hopeful investor perceives the probability that his or her portfolio falls below the threshold A, as lower than the actual probability (assum-ing normal distribution) that corresponds to this scenario. It seems that,

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α perceived P(ruin) real P(ruin) perceived E(V T) real E(V T) fearful 0.01 0,01 2,11094E-06 1,032708573 1,037401496 0.02 0,02 4,77127E-05 1,033340728 1,039128933 0.03 0,03 0,000240278 1,033852301 1,040557321 0.04 0,04 0,000690641 1,034367686 1,041935217 0.05 0,05 0,001488053 1,034850474 1,043256548 hopeful 0.01 0,01 0,063358176 1,102346103 1,074274091 0.02 0,02 0,086275186 1,128356709 1,090191962 0.03 0,03 0,104805664 1,159615047 1,10932132 0.04 0,04 0,121195334 1,202504994 1,135568945 0.05 0,05 0,136279456 1,268832413 1,176159744 rational 0.01 0,01 0,01 1,04928667 1,04928667 0.02 0,02 0,02 1,053884787 1,053884787 0.03 0,03 0,03 1,058141535 1,058141535 0.04 0,04 0,04 1,062498567 1,062498567 0.05 0,05 0,05 1,067181109 1,067181109

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in the first place, hope operates through an underestimation of the prob-ability of ruin and, therewith, the hopeful investor continuously violates the constraint. It can therefore be concluded that the emotion of hope leads an investor to excessive risk-taking behavior. Notice in this context, that the hopeful investor is the only investor that borrows money at the risk free rate (for certain levels of confidence).

The fearful investor on the other hand never violates the constraint. Because, for all applicant levels of α, the fearful investor perceives the probability of falling below threshold related to disaster, as lower than the actual probability (assuming normal distribution) corresponding to this scenario, it can be concluded that fear operates most importantly through an underestimation of the probability of ruin or disaster. Therewith, the fearful investor never violates the constraint. Hence, the emotion of fear leads an investor to excessive risk-avoiding behavior. Notice in this context, that the fearful investor never borrows money. Instead, of all three types of investors, the fearful one lends the highest amount of money.

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5

Sensitivity analysis

Scientific literature over the past decades has shown that correlation be-tween stock and commodity is highly volatile. Gorton and Rouwenhorst (2004), for example, state that the correlation coefficient between stock and commodity is strongly subject to fluctuation, and may even show positive instead of negative values over certain time spans. Therefore, it might be of interest to analyze the performance of the portfolios as ob-tained in the previous section under unexpectedly high or low correlation. How, ceteris paribus, do the obtained portfolios perform in case that the correlation coefficient between stock and commodity (derived at ρ = 0.25) turns out to be stronger than expected (i.e.: when ρ moves further from zero)? And how, ceteris paribus, do the portfolios perform in case that correlation becomes weaker (i.e.: when ρ shifts towards zero)? In order to address this issue, I recalculate, for each portfolio, the actual probability of ruin (or: disaster), taking into account correlation coefficients that are higher or lower than the perceived as true correlation coefficient.

For an illustration of such recalculation, consider the hopeful BPT-efficient portfolio at α = 0.01. At the (previously considered as) true probability measure, this portfolio faces a probability of disaster equal to 0.0633. Recall here that this true probability was computed using FWT(1), where FWT(.) refers to the cumulative distribution function of

the normal distribution. I.e.: the probability that a normally distributed variable with mean ws∗ (1 + µs) + wc∗ (1 + µc) + wf∗ (1.03) and standard

deviation (ws2∗ σs2+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ ρ ∗ σs∗ σc)1/2(where the

portfolio weights are as in table 2 and the values for µ, σ and ρ are as in table 1 ) ends up at or below 1, was computed at 0.0633.

In the new situation however, I assume that correlation turns out to be different than the value that was reported in table 1 (i.e.: I assume that ρ is higher or lower than -0.25). The new true probability of ruin then is computed by FWT(1), however, taking into account the shift in

ρ. In particular, I rederive FWT(1) as the probability that a normally

distributed variable with mean ws∗(1+µs)+wc∗(1+µc)+wf∗(1.03) and

standard deviation (ws2∗ σs2+ wc2∗ σc2+ 2 ∗ ws∗ wc∗ (ρ + δρ) ∗ σs∗ σc)1/2

ends up at or below 1 - where the portfolio weights are as in table 2, the values for µ, σ and ρ are as in table 1 and δρ can adopt the values −0.25, −0.2, −0.15, −0.1, −0.05, 0.05, 0.1, 0, 15, 0, 20, 0, 25. Please refer to

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tables 4 and 5 for the results (reported for all portfolios from table 2) corresponding to this routine.

α δρ = 0, 05 δρ = 0, 10 δρ = 0, 15 δρ = 0, 20 δρ = 0, 25 fearful

0.01 3,5105E-06 6,52767E-06 1,13467E-05 1,86331E-05 2,9152E-05 0.02 6,91762E-05 0,00010885 0,000163092 0,000234472 0,000325433 0.03 0,000328785 0,000474836 0,000659167 0,000885003 0,001154878 0.04 0,000891848 0,001218797 0,001610857 0,002069768 0,00259617 0.05 0,001854824 0,002434799 0,003104915 0,003863742 0,004708691 hopeful 0.01 0,067898805 0,073923266 0,079790039 0,085493202 0,091030378 0.02 0,091381255 0,098074707 0,104512278 0,110701176 0,116650326 0.03 0,110207558 0,117237113 0,123946961 0,130354217 0,136476058 0.04 0,126765404 0,133975468 0,140819999 0,147323989 0,153511048 0.05 0,141939883 0,149236481 0,15613343 0,162662131 0,168851346 textitrational 0.01 0,011537613 0,013745141 0,016076854 0,018513961 0,021039233 0.02 0,022431648 0,025817462 0,029282893 0,032804616 0,036362858 0.03 0,033104519 0,037352587 0,041623178 0,045894176 0,050147855 0.04 0,043639652 0,048560089 0,053445314 0,058276954 0,063041189 0.05 0,054075585 0,059534918 0,064904009 0,070169458 0,075322161

Table 4: BPT-portfolio real downside risk under shifted correlation

It is immediate that stronger correlation (i.e.: occurring when δ ρ < 0) has a diminishing effect on the probability of ruin. The stronger (hence, more negative) the correlation, the lower the probability of disaster. Vice versa: the weaker the correlation (hence, less negative), the higher the probability of ruin - for all investors at all levels of confidence. Note here, that this result does not come with surprise, because stronger correlation comes with diversification benefits, and so, ceteris paribus, it diminishes portfolio volatility - making the portfolio return distributions narrower.

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α δρ = −0, 25 δρ = −0, 20 δρ = −0, 15 δρ = −0, 10 δρ = −0, 05 fearful

0.01 6,74882E-09 3,0444E-08 1,07275E-07 3,12456E-07 7,83348E-07 0.02 7,26525E-07 2,17303E-06 5,43514E-06 1,18424E-05 2,31454E-05 0.03 8,22224E-06 1,99492E-05 4,18911E-05 7,86914E-05 0,000135397 0.04 3,92198E-05 8,29667E-05 0,000155435 0,000265134 0,000420017 0.05 0,000123067 0,000235585 0,00040615 0,000645774 0,000963247 hopeful 0.01 0,00967746 0,036081574 0,042556313 0,049023337 0,055426754 0.02 0,016076295 0,054134145 0,062042299 0,06974616 0,077213566 0.03 0,028684397 0,069811583 0,078608802 0,087049607 0,095126149 0.04 0,039459109 0,084347691 0,093754835 0,10268212 0,111144465 0.05 0,04976288 0,098209736 0,108046373 0,117300993 0,126009328 rational 0.01 0,002037736 0,003081123 0,004361102 0,005865422 0,007576604 0.02 0,005611284 0,007801651 0,010294911 0,01304628 0,016012279 0.03 0,01011243 0,013399628 0,016984855 0,020801532 0,024791733 0.04 0,015327953 0,019639982 0,024205308 0,028945451 0,033796673 0.05 0,021135453 0,026394941 0,031838885 0,037384845 0,04296952

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Or in other words: when the correlation coefficient between stock and bond becomes more negative, each portfolio’s true probability of ruin de-creases. One should notice from the two tables as well that a decrease in the correlation coefficient between stock and commodity is especially beneficial for the hopeful investor. Through the unexpected shift in cor-relation, this investor may suddenly satisfy the constraint at some levels of confidence. For example, the hopeful BPT-efficient portfolio at the 0.01 level has a true probability of ruin below 0.01 when δ ρ adopts -0.25, i.e.: when the actual correlation coefficient equals -0.5. Likewise, at other values for α the hopeful BPT-efficient portfolio satisfies the constraint as well - provided that δ ρ is low enough. From the perspective of the fearful investor, one should note that the BPT-efficient portfolio provides per-sistent downside protection. That is, even under weaker correlation than expected (hence, in case that the correlation coefficient between stock and commodity is close or equal to zero), the downside protection of the BPT-efficient portfolio is sufficient. Note as well, that this cannot be said about the rational investor.

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6

Conclusion

Behavioral Portfolio Theory postulates that, in practice, fear and hope operate on an investor’s willingness to take risk - where risk is defined as the probability of falling below the threshold that relates to undesired outcomes (for example: disaster or ruin). In this thesis, I developed BPT-efficient portfolios by means of a selection model which allocates assets such that the transformed expectation of the portfolio‘s return is maxi-mized subject to meeting specific transformed loss constraints. Following Lopes (1987), who conducts that fear and hope reside in every individual, although not to the same extent, I established BPT-efficient portfolios for investors that are primary driven by fear and for investors that are primary driven by hope. Hereby, I defined hopeful investors as individu-als who transform the (assumed as objective) normal distribution of asset returns to a Gumbel distribution, and fearful investors as individuals who transform the normal distribution of asset returns to a reverse Gumbel distribution.

Taking into account the presence of two risky assets - an index of stock and an index of commodity - and one risk free asset - at which one can borrow or lend - I found that hopeful investors are typically more exposed to downside risk than fearful investors. In particular, I found that, under the true probability distribution, a hopeful investor always faces more risk than he or she would like to face, and a fearful investor never faces more risk than he or she would like to face. However, the hopeful investor is likely to obtain higher returns than the fearful investor. Besides, the hopeful investor is likely to benefit the most from a negative fluctuation in the correlation coefficient. That is, without affecting the typically high return of the hopeful BPT-efficient portfolio, a negative shock in the correlation coefficient between stock and commodity can lead the hopeful investor to be protected against downside risk as he or she was not under the initial situation.

I would like to emphasize, that the conclusions of this study have to be interpreted with care, because the modeling of the psychologic and behav-ioral foundations of Behavbehav-ioral Portfolio Theory require a certain amount of assumptions that may not hold in reality. The assumption of normally distributed asset returns for example, has already been challenged heavily from the field of quantitative economics. Besides, the model as outlined in

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this thesis has as drawback that it does not allow investors to re-allocate their portfolios at intermediate points.

Regarding future research, it may be of interest to test the BPT-portfolios, as I established them, at different distributions, such as the lognormal distribution or Students t distribution. Moreover, taking into account the insight that - under some (unexpected) circumstances - it may be beneficial to be hopeful or fearful, one could even test the model empirically.

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7

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