Unifying the CAPM with Prospect Theory behavior: A theoretical and numerical analysis

Transcription

1 Unifying the CAPM with Prospect Theory behavior: A theoretical and numerical analysis A.P.P. Janssen ANR: Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science, and of the Master s program in Finance at the Tilburg School of Economics and Management JEL classification: G02, G11 Thesis committee: dr. Sebastian Ebert (Supervisor) dr. Alberto Manconi Tilburg University Tilburg School of Economics and Management Tilburg, The Netherlands November 2014

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3 Abstract This thesis provides an analysis of the assumptions and technical arguments that allow for the unification of the Capital Asset Pricing Model by Sharpe (1964), Lintner (1965) and Mossin (1966) and Cumulative Prospect Theory by Tversky & Kahneman (1992). In particular, the proofs provided by Levy, De Giorgi & Hens (2012) and Barberis & Huang (2008) are compared and related. This thesis adds to existing literature by computing and displaying numerical examples of CAPM-equilibria as positions on the capital market line. Its main finding is that investors (still) do not choose reasonable places on the capital market line, either leveraging hugely or by not investing at all Whereas hope was being put in the fact that Cumulative Prospect Theory is a more realistic theory than Expected Utility Theory in describing investor behavior. An empirical example is carried out, with its conclusion being that only investors that are slightly loss averse are willing to invest in the market portfolio proxy slightly loss averse implying a loss aversion parameter well below empirically estimated values. Overall, despite the fact that the CAPM with Prospect Theory behavior should be a model with more power to work in reality, much amendments are still to be made. Acknowledgements Judging on own experience, writing a thesis can be a lone- and tiresome process, but I am grateful for all who have helped pull me through. From a content-related standpoint, I am grateful to my supervisor Sebastian Ebert for more than once helping me in structuring my thoughts and steering me into the right direction. From a personal standpoint, first and foremost, I am grateful to my parents for providing their unconditional support and for checking up with me regularly to ensure I was not drowning in the process. The second person would be Ruben Mak, for providing me with the idea that led to the subject, and later on for extremely helpful comments on my thesis. Ruben, although the process was tedious, I still think the subject could not have been a better fit. The third to be mentioned personally would be Thijs Bongertman, for I am impressed by and grateful for how hard he worked to give me a final round of feedback (and a lot, to be honest...) just before the deadline. Job and Gijs, you two may conclude the list for providing me with a combination of content-related feedback as well as making sure I still relaxed once in a while throughout the whole process I am sorry for the occasional case of bad-temperedness. Of course, this list is incomplete and should incorporate many more friends, Watergeuzen, colleagues and others, who either supported me by showing interest per se or put things in perspective from time to time.

4 Contents 1 Introduction 1 2 Prospect Theory vs. Expected Utility Cumulative Prospect Theory The CAPM model 7 4 Linking CPT and the CAPM The approach of Levy, De Giorgi & Hens (2012) The approach of Barberis & Huang (2008) Equilibria with CPT preferences Optimal holdings with the exponential value function An empirical example Conclusions 26 7 Further research 26 A References B Appendix B.1 Definitions B.2 Proofs and derivations

5 List of Figures 1 The linear, power and exponential value function Karmarkar s and Tversky & Kahneman s weighting functions An example of the CAPM The normal CDF and two weighted normal CDFs The normal PDF and two weighted normal PDFs Equilibrium prices for market portfolios of different volatility Equilibria Sharpe Ratios for heterogeneous weighting functions and power value function Equilibria Sharpe Ratios for heterogeneous weighting functions and the exponential value function Equilibrium prices for different deltas and the power value function Equilibrium prices for different deltas and the exponential value function Three realizations of V (θ) Optimal portfolio holdings around base-line case for varying market volatility Optimal portfolio holdings around base-line case for varying loss aversion Histogram of the Fama-French R M with kernel smoothing density Histogram of the Fama-French R M with normal density Values of θ opt for the empirical example and fixed loss aversion List of Tables 1 Optimal holdings θ opt Values of θ opt for {γ, δ} = {0.61, 0.69}

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7 1 Introduction Numerous models in finance coexist whilst being derived from fundamentally different assumptions. Three researchers in particular have taken it upon themselves to unify two such theories: the Capital Asset Pricing Model (CAPM), and probably the most popular theory from behavioral finance: Cumulative Prospect Theory (CPT). The CAPM stems from an era in which economists viewed people as part of the homo economicus, in the sense that they were assumed to behave rationally meaning that they obey a certain, very specific set of rules as described by Expected Utility Theory (EUT). CPT is essentially a revision of these rules, but backed by psychological experiments. Despite the fact that the empirical evidence provided by CPT does not support the underlying assumptions of the CAPM, the CAPM is still considered to be of huge importance in finance as a field of research. One reason is historically determined and stems from the pricing possibilities that equilibria models (such as the CAPM) provided before arbitrage pricing theories were formulated. A second reason stems from the close link between the CAPM and the Sharpe Ratio, which is still considered to be a useful performance measure in the industry. The apparent contradiction between CPT and the assumptions underlying the CAPM forms the roots for the inspiration of Haim Levy, Enrico de Giorgi and Thorsten Hens. They have tried to answer the fundamental question: How to unify these two theories? Already in 2003, they provided the arguments to link these theories in a working paper, but it took until 2012 to publish these in a serious journal. Their main insight is that the mean-variance frontier from the CAPM has an equivalence to (first-order) stochastic dominance. Stochastic dominance in a mean-variance setting implies that for equal variance, the portfolio with the highest mean is also best in terms of (first-order) stochastic dominance. Concurrently, other researches and spin-offs have been published as well: the most notable being a paper by Nicholas Barberis and Ming Huang published in 2008, in which they provide a different way of reasoning. This thesis provides an analysis of the assumptions necessary to allow both theories, and it briefly summarizes and compares the arguments from Levy, de Giorgi & Hens (2012) as well as Barberis & Huang (2008). The main contribution is that numerical examples of CAPM equilibria are provided in the framework of Barberis & Huang (2008) and Levy et al. (2012), with one important addition: the reconsideration of the piecewise-power value function as a function suitable to describe investors utility, and replacing it by the exponential value function. The numerical approximations show large variety in investors positions on the capital market line, with most of them being infeasible in practice: the amount of leverage is either huge, or investors do not invest at all in risky assets, thereby not clearing the market. An empirical example based on the Mkt-Rf and Rf factors from the Kenneth R. French library supports the observation that positions on the capital market line seem to be on either ends of the spectrum, with no market clearing as the result. The structure of this thesis is as follows: Section 2 compares Expected Utility Theory to Prospect Theory (PT) and provides the arguments necessary to understand the introduction of CPT as its revision. Section 3 introduces the CAPM. Section 4 compares and relates the arguments in both Levy et al. (2012) and Barberis & Huang (2008). Section 5 provides the additional arguments to actually compute equilibria, as well as the results of the numerical approximations. Section 6 concludes. 2 Prospect Theory vs. Expected Utility Numerous different theories of investment behavior have been proposed in (academic) economic literature, but it might be said undoubtedly that none of these theories have gained as much 1

8 track as Prospect Theory, defined first by Daniel Kahneman and Amos Tversky in Kahneman & Tversky (1979). Prospect Theory has been proposed as an alternative to the classical idea of the homo economicus: an investor behaving according to the rules of Expected Utility Theory. In contrast to EUT, Prospect Theory is backed by psychological experiments rather than theory. To be more precise, Kahneman & Tversky (1979) analyzed the results of these experiments to formulate a number of situations in which EUT is contradicted. From this respect, their work is barely unique: two major papers on this field are Allais (1953) and Ellsberg (1961), which both even earned their own-name paradox, the Allais and Ellsberg s paradox respectively. However, the two main pillars of PT are still considered to be major insights in the field of behavioral finance. Kahneman & Tversky (1979) provide, almost as briefly as is done here, a framework for EUT as follows: 1. EUT satisfies the concepts of (probabilistic) expectation: U(x 1, p 1 ;...; x n, p n ) = n p i U(x i ), or the overall utility derived from participating in a contract with payoffs {x 1,..., x n } with probabilities {p 1,..., p n } is the (probability-)weighted average of all outcomes; 2. In EUT, adding an asset with payoffs {x 1,..., x n } with probabilities {p 1,..., p n } to a current wealth level w is acceptable if and only if U(w + x 1, p 1 ;...; w + x n, p n ) > U(w); 3. The agent s utility function expresses risk aversion by being concave: U ( ) < 0. The function U( ) above is a utility function: a function that assigns a certain level of utility from a monetary level or outcome, such as an endowment w or an endowment increased or decreased by gain x > 0 or loss x 0. Kahneman & Tversky (1979) identify four major issues on which EUT is contradicted: the Certainty Effect, the Reflection Effect, Probabilistic Insurance and the Isolation Effect. The Certainty Effect relates to the relatively big difference in perceiving a near-certain chance (i.e. 99.9%) versus a certain chance (i.e. 100%). The Reflection Effect relates to the difference in perceiving gains and losses. Probabilistic Insurance relates to taking precautionary measures such as a fire alarm rather than a full insurance: a fire alarm provides an uncertain relief, whereas a full insurance provides a relief with certainty. The Isolation Effect relates to differences in presenting a bet (also called framing ), and that two equivalent bets might be preferred differently due to their presentation. For exact descriptions and examples, I refer to the original paper. Most importantly, the aforementioned effects are not explained by EUT but are by PT. The key insight here is that EUT as described above is an objective theory: the rules above do not leave much room for interpretation, but are rather strict. However, the examples used to describe the aforementioned effects show that decisions between two bets are made, in general, based on a more complicated set of rules, including subjective interpretations made by the decision maker. The valuation phase as described by Kahneman & Tversky (1979) boils down to applying two functions: a value function v and a weighting function π to the (investor-edited 1 ) problems 1 Kahneman & Tversky (1979) also explain that, before the valuation phase, an investor goes through an editing phase in which, to put it briefly, the choice problem is simplified or put into perspective so it becomes prepped for valuation. i 2

9 v(x) 10 v lin 15 v pow 20 v e x Figure 1: The linear, power and exponential value function represented by the blue line, the green line and the red line respectively, for β = 1, α 1 = α 2 = 0.88, α = 0.2, λ = 2.25, λ + = 6.57 and λ = 14.7, such that λ /λ The reference point is set to 0 for all three value functions. and/or bets. The value function can be seen as a special type of utility function which was also used in EUT. Three examples of such value functions are plotted in Figure 1. However, it would be more precise to say that the value function is an a-typical utility function in the sense that it has been altered to display three essential differences from the classical, concave utility function: 1. The value function displays loss aversion: in short, a loss of size x can not be offset by a gain of similar size 2 ; 2. The value function displays risk-averseness over gains and risk-seekingness over losses. In technical terms, it is concave over gains and convex over losses; 3. The value of an outcome is set relative to a reference point, for instance current wealth or a risk-free rate. The three value functions displayed in Figure 1 are stated furtheron. Equations (1), (2) and (3) are known as the linear, power and exponential value functions respectively. Equation (1) is by far the most simplistic value function, but has nonetheless found its use in the literature (e.g. Barberis, Huang & Santos (2001)) It has, however, a major drawback since it does not display diminishing sensitivity (see Tversky & Kahneman (1992), p. 303). Equation (2) is by far the most well-known and is a generalization of the piecewise-power value function as proposed by Tversky & Kahneman (1992) (see Tversky & Kahneman (1992), Equation (5)), and is used by Barberis & Huang (2008). Equation (3) is proposed by De Giorgi & Hens (2006) to solve a range of problems occuring in finance (see De Giorgi & Hens (2006), p. 340). 2 A formal definition of loss aversion does not exist, although some papers describe it as v(x) + v( x) > v(y) + v( y) for x < y. 3

10 v lin (x) = { βx, x 0 λβx, x < 0 ; (1) { v pow (x) = βx α1, x 0 λβ( x) α2, x < 0 ; (2) { v e (x) = λ + exp( αx) + λ + x 0 λ exp(αx) + λ x < 0. (3) The specific choice(s) of a value function (value functions) will be postponed towards Section 5. The rest of this section will introduce the weighting function, first introduced in PT, and provide the arguments to understand why the weighting function was revised in CPT. The weighting function π( ) is an investor s way to subjectively interpret a set of actual probabilities {p 1,..., p n } 3 such that lottery-like small probabilities are overweighted (p < π(p) for small probabilities p) and vice versa (p > π(p) for probabilities p close to 1). Still, the weighting function respects the boundaries of probabilities since π(0) = 0 and π(1) = 1. Moreover, the weighting function π is such that it underweights medium 4 probabilities. Difficulties might arise when the probabilities are weighted in such a way that the set {π(p 1 ),..., π(p n )} is not a proper set of probabilities, despite π(0) = 0 and π(1) = 1, and it is exactly the underweighting of medium probabilities that might lead to the set {π(p 1 ),...π(p n )} not representing a proper distribution. For instance, if it is such that p = 0.5 > π(0.5), then the sum of the two does not add up to 1, so: π(0.5) + π(0.5) < 1. Kahneman & Tversky (1979) define this phenomenon as subcertainty. Why it may cause problems will be discussed in Section 2.1. Generally speaking, subcertainty has been considered a major hurdle in accepting Prospect Theory, and it even lead to the publishing of Tversky & Kahneman (1986) in which the authors commented on the criticism, next to expanding the evidence supporting Prospect Theory even more. Nonetheless, Prospect Theory was updated a few years later in Tversky & Kahneman (1992), which will be the subject of Section Cumulative Prospect Theory The allowance for subcertainty in Prospect Theory had multiple problematic implications, and although Kahneman & Tversky (1979) already acknowledged a number of these implications, a revision of Prospect Theory turned out to be necessary according to Tversky & Kahneman (1992): First, [Prospect Theory] does not always satisfy stochastic dominance, an assumption that many theorists are reluctant to give up. Second, it is not readily extended to prospects with a large number of outcomes. These problems can be handled by assuming that transparently dominated prospects are eliminated in the editing phase, and by normalizing the weights so that they add to unity. Alternatively, both problems can be solved by the rank-dependent or cumulative functional, first proposed by Quiggin (1982) (...) and Schmeidler (1989) (...). (Tversky & Kahneman (1992), p.299) 3 Although it might seem that this initial form of Prospect Theory is suitable for problems with n > 2 non-zero outcomes, the authors argue that this formulation is only suitable for problems with at most 2 non-zero outcomes. Some years after the publication, Prospect Theory was expanded for a wider range of bets. 4 A precise definition of medium will not be given here, but it is small enough to be underweighted and not as small that it is lottery-like, so that it will be overweighted. 4

11 To give some background on the above: Quiggin (1982) considered the case in which the probabilities of possible events are known, whereas Schmeidler (1989) considered the case where probabilities were unknown but still a belief could be formulated on the basis of considering (sub)sets of events. In particular, Quiggin (1982) pointed out a number of problems associated with the weighting function π( ) suggested by Kahneman & Tversky (1979). One intuitive problem comes with a bet between payouts {x 1,..., x n } with probabilities {p 1,..., p n }, but where the probabilities are proportional: for instance, each probability equals 1/n or some equal a multiple k/n. In the idealized example, with all p i = 1/n, i {1,..., n}, it becomes difficult to accept that people would still under- or overestimate the probabilities, since in that case it is immediately clear that they are violating the rules of a proper probability distribution. That is to say, it is so obvious that someone is violating those rules, that this person is likely to recall his interpretation instantly whereas in less idealized cases, people might not be so aware. In that idealized setting, the weighting function should be such that π(1/n) = 1/n. The example above, as Quiggin (1982) shows, is a violation of stochastic dominance, which was already admitted in Kahneman & Tversky (1979), in which they also argue that this violation might lead to intransivity of choices. A definition of stochastic dominance can be found in Appendix B.1. Although the solution proposed in Tversky & Kahneman (1992) to normalize the sum of all the probabilities might be practical, this is not ideal from a theoretical point of view. As a direct example where the weighting function as proposed in Kahneman & Tversky (1979) violates stochastic dominance, consider Example 2.1 below: Example 2.1 (A violation of stochastic dominance). Consider gaining an amount W for sure and betting on payouts {W +x 1,..., W +x n } with probabilities {1/n,..., 1/n}. Let the x i, i {1,..., n} be small, that is to say 0 x i ɛ. Denoting the bet by the random variable W X and the sure gain by a random variable W with distribution functions F WX and F W respectively. It is straightforward to see that W X stochastically dominates W, since: F WX (x) = 0 = F W (x) x < W, F WX F W = 1 x W. If the probabilities are such that they will be underweighted, then at least one situation exists such that: n n v(w ) > π(1/n)v(w + x i ) = π(1/n) v(w + x i ), which violates the principles of stochastic dominance. (Quiggin (1982), p.325) i Although Quiggin s paper does provide a basis for the publication of Tversky & Kahneman (1992), his paper is not an attempt to repair Prospect Theory: Quiggin merely proposes an alteration to utility theory which he names anticipated utility (AU) 5. Despite the fact that Quiggin (1982) did not attempt deliberately to do so, a new version of Prospect Theory was introduced by 5 In its essence, his paper is an axiomatization of Karmarkar (1978) A paper written by Uday S. Karmarkar, an Indian engineer that around the time of obtaining his Ph.D. visited a conference in Jerusalem and came into contact with an early version of Prospect Theory presented by Daniel Kahneman and Amos Tversky, at that time dubbed Value Theory. There is some irony here: Inspired by the concepts proposed in Value Theory, Karmarkar proposed an alteration to the classic utility theory named Subjective Weight Utility (SWU), where odds were interpreted subjectively in a manner that looks strikingly similar to the weighting function by Kahneman & Tversky (1979) But even more similar to the cumulative weighting function as proposed in Cumulative Prospect Theory by Tversky & Kahneman (1992). The weighting function w as proposed by Karmarkar was: i 5

12 Tversky & Kahneman (1992), dubbed Cumulative Prospect Theory (CPT) since the weighting function was to be applied over cumulative probabilities Or, in terms of Schmeidler (1989), partitions of events rather than individual events. The weighting function proposed by Tversky & Kahneman (1992) is: p γ π + (p) =, π (p γ + (1 p) γ ) 1 (p) =, (4) γ (p δ + (1 p) δ ) 1 δ where π + denotes the weighting function for gains and π for losses. Some examples are plotted in Figure 2, along with an early version of a cumulative weighting function proposed by Karmarkar (1978) (see Footnote 5 for additional background). p δ ω(p) π + (p) π (p) ω(p), π(p) p Figure 2: The subjective weighting function of Karmarkar (1978) ω(p) and two possible weighting functions following Tversky & Kahneman (1992) π (p) and π +(p) as the blue line and the green and red line respectively with α = 0.61, γ = 0.61 and δ = So far two versions of Prospect Theory are considered: PT as proposed by Kahneman & Tversky (1979) and CPT as proposed by Tversky & Kahneman (1992). The only difference between PT and CPT concerns the weighting function: in CPT, the way probabilities are (subjectively) interpreted by investors is such that stochastic dominance 6 is satisfied. As will be p α π(p) = p α + (1 p) α. Apart from Karmarkar s suggestion that a weighting function should be applied to the cumulative probabilities, the suggested function shows striking similarities as is displayed in Figure 2. In short, it seems that Uday Karmarkar already solved the problems of Prospect Theory before Prospect Theory was officially published whilst being inspired by an early draft of Prospect Theory. Hence, the irony. 6 Technically, CPT satisfies stochastic dominance of the first-order. Other types of stochastic dominance are considered in Section 4.1 and

13 shown later in this thesis in Section 4, satisfying stochastic dominance is essential in linking (C)PT to the CAPM. Therefore, the remainder of this thesis will only concern CPT. Although in the rest of this thesis the distinction between PT and CPT might not always be made clearly, this is for notational convenience only, since the difference between PT and CPT is fundamental. So far, PT and CPT are introduced as well as some intuitive arguments on why CPT is required. It is still left to show the link of stochastic dominance and the CAPM, as will be done in Section 3. 3 The CAPM model The fundamentals of the Capital Asset Pricing Model (CAPM) are traditionally led back to the 1960s with Sharpe (1964), Lintner (1965) and Mossin (1966) seen as three fundamental papers, although this list might be incomplete 7. Sharpe (1964) was the first to display the line tangent to the mean-variance frontier and to show that a systematic market risk was present in the CAPMmodel. He also presented an early version of the famous Sharpe ratio, although it can only be σ found in a footnote and looks as the peculiar expression 8 µ R f in slightly updated notation. Sharpe worked with variance-mean diagrams which explains the reversal of the numerator and denominator. Lintner (1965) forms an optimization problem explicitly excluding the case where investors might take short positions in assets. Lintner does this without referring to Sharpe (1964), thereby suggesting that he independently performed his analysis. Lintner also considers a case in which only two risky assets are available, one representing a single asset and one representing all other assets and shows that there should be a pricing relation between the two, hinting towards an earlier version of the capital asset line: Lintner provides a theorem that explicitly states that excess return is proportional to the aggregate risk. This is an early description of the contribution of a stock towards the systematic risk, nowadays better known as its β. Mossin (1966) builds upon Sharpe (1964), by formulating the CAPM as an optimization problem and explicitly solved the Lagrangian, thereby noting that in equilibrium every investor must buy the same percentage of the total outstanding stock of all risky assets; thereby touching upon the notion of a market portfolio that consists of equal pieces of the aggregate of all marketed assets. Moreover, Mossin (1966) presents a formulation for the price of risk as a function of standard deviation. Nonetheless, the popularity of the CAPM allows for a complete derivation to be found in numerous textbook, for example Luenberger (1998). Since the objective of this section is to introduce the reader to the CAPM and provide a springboard towards the next section (which aims to unify CPT and the CAPM), it is sufficient to stick to Luenberger (1998). The derivation of the CAPM commences by assuming a market containing multiple risky assets of which the returns are Normal distributed, so that the returns are fully specified by their respective means and variances and, thus, can be drawn in a mean-variance diagram. Furthermore, all investors on the market must agree on the distributions of these assets. If these risky assets are not perfectly correlated 9, then diversification is valuable in the sense that their combined expectation behaves linearly, but their combined volatility does not. That is to say, suppose two risky assets x 1 N(µ 1, σ1) 2 and x 2 N(µ 2, σ2) 2 with correlation coefficient ρ < 1, 7 Some sources also state work by Jack Treynor and James Tobin, although Sharpe, Lintner and Mossin were supposedly the first to view the CAPM as an equilibrium model. 8 σ To be more literal, Rg, see p. 438 of Sharpe (1964). E Rg P 9 In a Normal distribution framework, correlation is always defined in terms of Pearson s linear correlation coefficient ρ. 7

14 then a combination θx 1 + (1 θ)x 2 yields return: µ 1 θµ 1 + (1 θ)µ 2 µ 2 (assuming µ 1 µ 2 ), but for a properly chosen θ it is possible to obtain as volatility: σ (θx1+(1 θ)x 2) = σ(θx 2 < θσ 1+(1 θ)x 2) 1 + (1 θ)σ In other words, diversification pays out for imperfectly correlated assets if done properly (see Luenberger (1998), p ). This has a specific implication of the shape of the set of allowed combinations of risky assets: if one constructs a portfolio out of risky assets only, i.e. it takes a certain weight in a certain combination of assets such that the sum of all these weights is equal to 1, then the feasible region will be convex to the left (Luenberger (1998), p. 156). Although it might require some imagination in some cases, such a shape can be best compared by a bullet or the left side of an ellipsoid: the shape is even known as the Markowitz bullet. The left boundary of this ellipsoid is then referred to as the minimum-variance set: the set that, for a given return, provides the minimum variance. Obtaining this minimum-variance set can be done by formulating the minimization problem that minimizes total variance subject to the constraint that all combinations of risky assets should be allowed, see p. 158 in Luenberger (1998). This yields the (global) minimum-variance point; for any other point on the minimum-variance set, add a constraint that fixes the return. A Theorem known as the Two-fund Theorem even provides extra convenience: knowing two points of the minimum-variance set, then any allowed combination of these portfolios will yield another point on the minimum-variance set mean return µ return volatility σ Figure 3: An example of the minimum-variance frontier formed by the available assets (displayed as colored stars) and the capital market line. The yellow circle represents the minimum-variance point, the green circle represents the tangency portfolio. The risk-free rate is located outside the graph at coordinates (σ, µ) = (0, 3.6%) For ρ 1, the <-sign becomes a -sign. 8

15 So far, only combinations of risky assets are considered. However, when a risk-free asset is available, the efficient set reduces to one portfolio only: the portfolio that is tangent to the line drawn from the risk-free asset and the minimum-variance set. Equivalently stated, the line drawn from the risk-free rate of return R f and the return on the tangency portfolio R t. For a portfolio that is not the tangency portfolio, this line will intersect the minimum-variance set: therefore, a portion of the minimum variance set will always be above the line. Hence, intersecting the minimum-variance set is not optimal, since the portion above the intersection line represents opportunities of equal variance but with higher expected return. The line drawn from the risk-free asset R f and the tangency portfolio R t is known as the capital market line (Luenberger (1998), p. 176). At this point, the term capital market line might seem to be coming out of the blue. However, when it is considered that the line between the risk-free rate and the tangency portfolio is always on or above the minimum-variance set, it becomes clear that every investor will perform optimally if it buys a portion of the tangency portfolio and the risk-free asset: hence, the tangency portfolio is also known as the market portfolio, the portfolio that is demanded by every investor in the market. The only assumptions needed to form such a line is that anyone is able to lend and borrow against the same risk-free rate. To ensure that the tangency portfolio is efficient, it is also necessary to assume that the set of allowed portfolio combinations also includes short positions in individual assets. Luenberger (1998) uses this line of reasoning to conclude with the capital asset pricing model: Definition 3.1 (Capital Asset Pricing Model (CAPM)). If the market portfolio M is efficient, the expected return E(r i ) of any asset satisfies: E(r i ) R f = β i (E(r M ) R f ), (5) with β i = σ i,m the covariance of r σm 2 i and r M divided by the variance of r M. (Luenberger (1998), p. 177) The proof is stated in Luenberger (1998), p It is now possible to more precisely define what linking the CAPM with CPT entails. The link between these two theories will be established as soon as it is possible to show that Equation (5) holds, so that CPT investors just like EUT investors will eventually pick a portfolio on the capital market line. The introduction above is a general derivation with as little use of mathematical equations and statements as possible, and assumes that any reader of this thesis has a basic level of acquaintance with the CAPM. For the remainder of this thesis, recalling Definition 3.1 is sufficient. Before continuing with Section 4, it is necessary to restate explicitly the observation that the portfolio with the highest expected return µ given a certain level of variance dominates any other portfolio of equal variance. Hence, every portfolio on the capital market line dominates any other portfolio available in the market, provided the variance is equal. It turns out that, for Normal distributed returns, this description of dominance is equivalent to stochastic dominance of the first-order, the main shortcoming of the original Prospect Theory and the major improvement of Cumulative Prospect Theory. A definition of first-order stochastic dominance is given in Appendix B.1, a proof along with a graph is stated in Appendix B.2. 4 Linking CPT and the CAPM In this Section, both Levy et al. (2012) as well as Barberis & Huang (2008) will be discussed, since both provide (independently) proofs of the holding of the capital market line when investors 11 Source: Figure 3 is of own production, originally as part of a group assignment for the course Introduction to Finance (course code 35B112). 9

16 are CPT instead of EUT investors. 4.1 The approach of Levy, De Giorgi & Hens (2012) Based on a working paper already made available in 2003 (Levy, De Giorgi & Hens (2003)), Levy, De Giorgi & Hens (2012) describe the first attempts to connect CPT and the CAPM model, driven by the notion that both form corner stones of financial theory but that their underlying assumptions are substantially different. Levy et al. (2012) mainly focus on the equivalence between the capital market line 12 and first-order stochastic dominance, and investigate whether or not any step in CPT might violate stochastic dominance. In other words, they consider every difference between CPT and EUT and check if each difference might lead to a violation of the capital market line representing all dominant portfolios in the mean-variance framework. If none of these differences will violate stochastic dominance, then a CPT investor will choose, eventually, a portfolio on the capital market line as well: the key insight here is that, if first-order stochastic dominance holds, this implies a specific preference when an investor s value function satisfies certain properties (see e.g. Levy (1992), Equation (4)). This definition is formally stated in Appendix B.1. Considering this, it is immediately clear why Prospect Theory as proposed by Kahneman & Tversky (1979) is impossible to link with the CAPM. Knowing that the original Prospect Theory might violate stochastic dominance, it might occur that a (stochastically) non-dominating portfolio is preferred over a (stochastically) dominating portfolio. The non-dominating is not located on the capital market line, which contradicts the CAPM. Levy et al. (2012) consider several main differences between CPT and EUT for which they separately show that, given that a certain portfolio A is dominant in the classical CAPM framework, it will also be dominant after a CPT investor has been given the choice between A and a different portfolio A : 1. Investors value opportunities over their difference with respect to a reference point rather than the total final wealth; 2. In the CAPM, agents preferences are assumed to be homogeneous, whereas in CPT agents are often considered heterogeneous by either differences in their reference point and different curvatures in the weighting function; 3. In the CAPM, asset returns are considered to be Normal distributed. Although the CPT can still allow for Normal distributed returns ex ante, after applying the weighting function the returns are, generally, not perceived to be Normal distributed. This technical argument is displayed in Figure 4; 4. In the CAPM, agents preferences are driven by a risk-averse utility function from EUT whereas in CPT, agents preferences are driven by a S-shaped value function. Given that any point on the capital market line stochastically dominates any other portfolio of equal variance, the arguments to tackle each of the differences above are relatively straightforward. First, consider the valuation of opportunities as differences of wealth rather than total, final wealth. This relates to thinking about a change of wealth x rather than using total wealth w + x as input. Given that stochastic dominance holds, by definition 13, for the full domain of a distribution function, this change of variable does not affect the original preference ordering 12 Although they use the term security market line. 13 See the definition of stochastic dominance as stated in Appendix B.1. 10

17 1 0.9 N(0,1) distribution π distribution 4 x N(0,1) density π density Φ (x), π (Φ(x)), π + (Φ(x)) π + distribution φ(x), π (φ(x)), π + (φ(x)) π + density x x Figure 4: The normal CDF as the blue line and the weighted normal CDFs as the green and red lines for the weighting function for losses with δ = 0.69 and for gains with γ = 0.61 respectively. Figure 5: The normal PDF as the blue line and the weighted normal PDFs as the green and red line representing weighted normal PDFs corresponding to the weighting function for losses π with δ = 0.69 and to the weighting function for gains π + with γ = 0.61 respectively. implied by stochastic dominance. The same argument holds when investors have heterogeneous reference points: still, stochastic dominance of the first order is preserved. Levy et al. (2012) do, however, note that equilibrium prices might change under such a change of variable. The argument for heterogeneity in the weighting function, as well, comes down to straightforwardly preserving stochastic dominance: as long as the weighting function is a monotonic function, the ordering of preferences implied by stochastic dominance is preserved. However, whether or not the weighting function is monotonic depends on the value of the parameters γ or δ. This point is not explicitly stressed by Levy et al. (2012), but is a basic property of CPT and bounds for these parameters are available in the literature. Ingersoll (2008) shows that any parameter value within the interval (0.28, 1) is safe. The argument is similar for the fact that after applying the weighting function, the returns are not anymore Normal distributed: as long as γ and δ are properly chosen, the weighting functions carry out a monotonic transformation which preserves the preference ordering implied by stochastic dominance. Hence, the capital market line will still be dominating any portfolio of equal variance that is not on the capital market line. However, the chosen position on the capital market line might become different, as do Levy et al. (2012) admit. Showing that the shift from a concave utility function towards a convex-concave value function still preserves the ordering implied by stochastic dominance is less straightforward. Levy et al. (2012) give a sufficient as well as a necessary condition to show that even an S-shaped value function will leave the implied preference ordering intact, as long as stochastic dominance is satisfied in the classical CAPM setting. The sufficient condition is that the S-shaped value function is monotonic, so that stochastic dominance of the first-order as proposed by Levy (1992) can be applied (see Levy (1992), Equation (4) or Definition B.2 in Appendix B.1). The necessary condition is relatively unknown in the literature and involves a different kind of stochastic dominance Prospect Stochastic Dominance as described by Levy & Wiener (1998). Due to its peculiarity it is stated below: Definition 4.1 (Prospect Stochastic Dominance). Let U p denote the class of all S-shaped utility functions: S-shaped means that it has a convex (U 0, U 0; x < 0) left part which corresponds to risk-seekingness for losses; and a concave (U 0, U 0; x 0) right part 11

18 which corresponds to risk-averseness for gains. Consider two distributions F and G. Then it is said that F dominates G in Prospect Stochastic Dominance equivalently: F PSD G if and only if y x G(z) F (z)dz 0 for all utility functions U U p. x < 0, y > 0 E F U( ) E G U( ) Essentially, Definition 4.1 is the concluding argument for Levy et al. (2012) in showing that the capital market line even holds when investors have CPT rather than EUT preferences. As a final remark, note that it is easily seen that stochastic dominance of the first order implies prospect stochastic dominance, but not vice versa. However, what makes Definition 4.1 peculiar is that it is derived from a different type of stochastic dominance: stochastic dominance of the second-order 14. What is necessary to realize is that stochastic dominance, in a general sense, implies a certain preference when certain conditions are met. To distinguish between which conditions should be met, there is differentiation between order of stochastic dominance. To be more specific, first-order stochastic dominance describes the conditions for a certain preference ordering when a certain utility function is considered, whereas second-order stochastic dominance describes different conditions when a different type of utility function is considered. Levy et al. (2012) do not go any deeper into relating second-order stochastic dominance with Definition 4.1. However, Barberis & Huang (2008) do establish a link with second order stochastic dominance in their derivation, although they use it differently. 4.2 The approach of Barberis & Huang (2008) Barberis & Huang (2008) use a more analytical approach in the sense that they derive equilibrium conditions in order to show that the capital market line is intact, explicitly deriving Equation (5) from these equilibrium conditions. Next to the fact that their approach is more analytical, deriving the holding of the CAPM is not the main subject of their paper: the main topic is to show that, since small probabilites are overweighted in CPT, CPT investors do prefer a lotterytype stock in a well-diversified (mean-variance) portfolio, such as a position on the capital market line. This skewness preference in the small is a whole different topic (see, e.g. Ebert & Strack (2012)). Barberis & Huang (2008) assume homogeneous investors using a piecewise-power value function (before-stated as Equation (2)) with parameters as estimated by Tversky & Kahneman (1992). They assume that each investor has the goal function: V (W net ) = 0 0 v(w )dπ + (1 F (W )) + v(w )dπ (F (W )), (6) where F ( ) is the distribution function of W net, the end wealth relative to the initial wealth accumulated at the risk-free rate. For mathematical tractability they rewrite the goal function by integration of parts to: V (W net ) = 0 0 π + (1 F (x))dv(x) π (F (x))dv(x). (7) 14 To be more precise, one needs second-order stochastic dominance which by definition applies to concave utility functions as well as its converse: second-order stochastic dominance for convex functions. The original second-order stochastic dominance covers the left side of the area under the integral in Definition 4.1, whereas the alternative version covers the right side-part. Derivations and additional details of the two are given in Appendix B.2, mostly based on Levy & Wiener (1998) and Meyer (1977). 12

19 Instead of W net, other inputs can be used such as excess market return µ e, which essentially is a special case of W net with initial wealth W 0 = 1. Barberis & Huang (2008) then state the conditions necessary for an equilibrium: 1. V (R M,net ) = 0: at market equilibrium, the value of the goal function is 0; 2. V (R subopt ) < 0: the value of the goal function at any suboptimal portfolio return is less than 0. To see that the above conditions are correct, recall that any position on the capital market line is characterized by a portfolio θr M + (1 θ)r f, θ 0, so that using this portfolio as an input, the goal function simplifies to (see Barberis & Huang (2008), Equation (28)): V (θr M + (1 θ)r f R f ) = V (θr e M ) = θ α V (R e M ), where RM e is the excess market return. The key insight here is that any investor will do one of three things: 1. if V (RM e ) < 0, no investor will be willing to invest in anything else but the risk-free asset. Although this is feasible, it is a situation that holds zero interest in considering, since either the market will not be cleared or there will be no market at all; 2. if V (RM e ) = 0, an investor will be willing to invest in the market portfolio; 3. if V (RM e ) > 0, the market portfolio will be so in demand that it pays out to infinitely leverage the position in the market asset. An infinite run on the risk-free asset contradicts any notion of equilibrium. By means of the reasoning above, Barberis & Huang (2008) implicitly define a market equilibrium as an equilibrium of supply and demand, but do not make any conjectures about prices. This issue will be covered explicitly in Section 5. So far, the conditions to which a market equilibrium should oblige are defined. However, it has not (yet) been guaranteed that investors are inclined to look for a combination that offers a tradeoff between expected return and variance. Barberis & Huang (2008) prove in their Appendix that this is the case. They split up the proof in three parts: Proposition A1, Proposition A2 and Proposition A3. In Proposition A1, Barberis & Huang (2008) consider two portfolios with a known distribution function but with unequal mean. They then show analytically that the portfolio with highest mean will be preferred over the other portfolio. In other words: they explicitly show for the goal function (Equation (7)) that, for given portfolios A 1 and A 2 with distribution functions F 1 and F 2 such that F 1 (x) < F 2 (x), x, this implies V (A 1 ) > V (A 2 ). Seeing this is rather straightforward: if F 1 < F 2, then (1 F 1 ) > (1 F 2 ) and F 1 > F 2. Monotonicity of the weighting functions respects this ordering. In Proposition A2, Barberis & Huang (2008) consider two portfolios of which the distribution functions share a single-crossing property; in other words, one is a mean-preserving spread of the other. They then continue to show that for a symmetric distribution (such as the Normal distribution) the portfolio with less variation is preferred over the other. Single-crossing properties and mean-preserving spread also relate to stochastic dominance, but stochastic dominance of the second-order. This is more general than only comparing two portfolios with Normal distributed returns of equal mean and different variance, but the implication is that less variance is preferred over more variance. Hence, the goal function is a decreasing function in variance. Proposition A3 13

20 then combines Proposition A1 and A2: it states that it must be that the goal function is an increasing function of the mean, but a decreasing function of the variance when returns are Normal distributed and are, hence, fully specified by mean µ and variance σ 2. Barberis & Huang (2008) now refer to their initial assumptions, namely that they consider homogeneous investors. Since investors are homogeneous and the CAPM is an equilibrium model that considers market clearing of a positive supply of assets, θ > 0 should hold. Clearly, since the goal function is increasing in the mean, this only occurs if the tangency portfolio R t has a mean return higher than the risk-free rate: E(R t ) > R f > 0. Moreover, in equilibrium the aggregate demand of the tangency portfolio must equate the aggregate supply of assets. The aggregate supply of assets is essentially the market portfolio, so that the returns of the tangency and the market portfolio show equality: R t = R M. That no investor is able to find a portfolio with a higher Sharpe Ratio then follows from the fact that stochastic dominance in the first- as well as the second-order hold. Despite now having considered the outlines of two different approaches to linking the CAPM with CPT, a number of issues still exist and numerous articles have appeared that point out small shortcoming in either Levy et al. (2012) or Barberis & Huang (2008), or that point out practical issues that still need to be solved in order to fully integrate the CAPM and CPT. This will be the subject of Section 5. 5 Equilibria with CPT preferences So far, an equilibrium condition in terms of Equation (6) has been stated, but the exact specifics of a market equilibrium are still undefined. Although Levy et al. (2012) do not explicitly define what they view as a market equilibrium, other papers by (partially) the same authors do, see e.g. De Giorgi, Hens & Rieger (2010) and De Giorgi, Hens & Levy (2011). Their notion of market equilibrium consists of a price vector q and agents portfolio holdings θ such that the market clears and no investor violates his/her own budget: Definition 5.1 (Market Equilibrium as in De Giorgi, Hens & Rieger (2010)). A price vector q and agents portfolio holdings vector θ define a market equilibrium if: θ i = arg max V ((θ i ) T R), (θ i ) T q (θ0) i T q θ i for all investors i I, and the market clears: ι T θ = ι T θ 0, where θ i 0 represents the initial endowment of investor i in available assets. R represents a vector of returns for all available assets including the risk-free asset. The key takeaway is that Definition 5.1 explains an equilibrium as the situation in which demand matches supply of assets, given a certain, appropriate price. The argument provided by Barberis & Huang (2008) leading to R t = R M is essentially based on the same notion of equilibrium, though it is not stated explicitly. None of these papers further restrict the pricing vector q, with an exception being made for the price of the risk-free rate (often denoted as q 0 > 0). Even when the risk-free rate is priced strictly positive, the market portfolio itself can have a negative price (Del Vigna (2011)) when it is volatile enough, or when the loss aversion parameter takes on certain values. For volatility, this is plotted in Figure 6, which is based on Figure 2 of Del Vigna (2011). Del Vigna (2011) estimated that σ (0, 2.19) is a suitable range to obtain positive prices. Del Vigna (2011) also shows that the loss aversion parameter λ should be in a suitable range (λ (1, 2.77)) to ensure nonnegative prices. Here, a different boundary value is obtained of σ M 1.61 for reasons unclear, but the essence of what Del Vigna (2011) 14