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How Endogenization of the Reference Point Affects Loss Aversion: A Study of Portfolio Selection
Abstract
Endogenization of the Reference Point Reduces Loss Aversion
A central idea in behavioral economics is that agents derive utility from gain and losses relative to a certain reference point and that losses loom larger than gains. In “How Endogenization of the Reference Point Affects Loss Aversion: A Study of Portfolio Selection,” He and Strub study the implications of various models of partially endogenous reference points on portfolio selection. In these models, an agent faces a salient exogenous reference point that influences the formation of endogenously determined expectations about the future, through rational expectations, optimal expectations, or mental updating of the reference point. A key finding is that the predictions of these models are identical to a model with an exogenous reference point but a lower degree of loss aversion. This suggests that it is difficult to separately identify an agent’s degree of loss aversion and his or her reference point and may help to explain why experienced and sophisticated agents appear to be less loss averse than expected in some field settings.
... In the context of portfolio choice, Berkelaar et al. (2004) incorporate the effect of exogenous references and find that loss aversion and risk aversion are empirically entangled, while Jin and Zhou (2008) also examine the effect of probability distortion and riskseeking attitude to losses. For a thorough literature review on reference dependence, we refer to the general survey of O' Donoghue and Sprenger (2018), and to He and Strub (2022) for a sharper focus on portfolio choice. ...
... Thus, they propose a partially endogenous expectations model, whereby the endogenous reference is chosen within a neighborhood of an exogenous reference. He and Strub (2022) study reference point endogenization under three models: the partially endogenous expectations of De Giorgi and Post (2011), optimal expectations, and mental updating with partial adaptation. Applying these models to portfolio selection, they establish that reference point endogenization reduces loss aversion across all models. ...
In a complete market, we find optimal portfolios for an investor whose satisfaction stems from both a payoff's intrinsic utility and its comparison with an endogenous reference as modeled by Kőszegi and Rabin. In the regular regime, arising when reference dependence is low, the marginal utility of the optimal payoff is proportional to a twist of the pricing kernel. High reference dependence leads to the anchors regime, whereby investors reduce disappointment by concentrating significant probability in one or few fixed outcomes, or “anchors.” Multiple equilibria arise because anchors may not be unique. If stocks follow geometric Brownian motion, the model implies that investors with longer horizons choose larger stocks holdings.
... For example, Baron et al. (2015) study a newsvendor problem in which consumers are loss-averse with stochastic reference points that represent their beliefs about possible price and product availability. He and Strub (2022) study how the endogenization of stochastic reference points affects the portfolio optimization under loss aversion. Zhang and Li (2021) investigate the implications of loss aversion on firm profit, consumer surplus, and social welfare when firms endogenously make quality disclosure decisions. ...
In many service industries, information disclosure about the product can alleviate customers' loss aversion induced by uncertain product valuation. In this paper, we consider a single-server queueing system in which the manager who privately learns the valuation information discloses the valuation information strategically to loss-averse customers. We investigate the impact of the customers' loss aversion on the system's equilibrium arrival rate and the manager's optimal disclosure policy. We find that loss aversion restrains customers from joining the queue. Surprisingly, we find that there is no one disclosure policy that always prevails over other disclosure policies. Specifically, the full disclosure policy is optimal only when the valuation is large and the degree of loss aversion is moderate. The full non-disclosure policy is optimal when the degree of loss aversion is too large or too small, or the valuation is small. The threshold disclosure policy is optimal when the valuation and the degree of loss aversion are moderate. Furthermore, under the threshold disclosure policy, the increasing degree of loss aversion makes managers be more reluctant to disclose the valuation.
... Among the studies on reference-dependent choice, the updating rule of the reference level is one of the most important factors influencing these models. De Giorgi and Post (2011) investigate a stochastic, state-dependent reference level and develop a choice model in which adjustment costs can exhibit a kink at an exogenous reference level, while He and Strub (2022) extends their model to partially endogenous reference point formations. Strub and Li (2020) explore the influence of a dynamically evolving reference level with recursive and non-recursive updating rules on portfolio selection, examining which leads to more realistic investment behavior. ...
We develop a comprehensive model to derive optimal consumption-investment choices for agents aiming to sustain spending power over an infinite horizon. Our model accounts for agents constrained by costly adjustable spending bounds and evaluates utility based on both current and future minimum allowable consumption levels, as well as actual consumption. Higher minimum levels assure investors that their consumption will not fall below a certain threshold, thereby generating utility from this anticipating security. However, adjusting these minimum levels incurs utility costs, leading to a non-trivial trade-off. The problem is transformed into an optimal switching problem through its conversion to a dual problem. The optimal switching problem is analytically characterized as a two-dimensional double obstacle problem. Our explicit solutions reveal an empirically consistent consumption pattern and U-shaped portfolio choices, implying that investors exhibit loss aversion toward changes in consumption.
... With the rapid development of the social economy, the scale of the financial market is likewise constantly expanding, which in turn promotes economic prosperity, and also creates a positive and profound impact on people's social lives (Li et al., 2018). At present, trading venues represented by the stock market, bond market, fund market, and various other financial asset markets have formed a relatively sound financial system encompassing a diversity of products (He & Strub, 2022). At the same time, the interaction and impact between various financial products constitute a close and complex relationship in the financial market, which puts forward new requirements for the formulation of financial market management policies and the scientific investment behavior of investors. ...
In order to cope with the complex risk environment of the current financial market, achieve portfolio optimization and accurate risk prediction, this paper conducts effective research using SVM algorithm. This article uses stock data as a sample to empirically analyze the risk return and risk prediction performance of investment portfolio strategies based on SVM algorithm. Compared with traditional index fund investment strategies, the risk resistance of investment portfolio strategies is significantly improved, and the risk return is also stable at a high level. In addition, with the support of SVM algorithm, the risk prediction error level in the financial market remains within a relatively low range. From the perspective of practical applications, the financial market investment portfolio selection and risk prediction based on SVM algorithm has strong feasibility.
... This aspect renders them highly potent in modeling real-world economic issues and supporting decision-making processes undertaken by economists and policymakers (Schwerter, 2023). The versatility of evolutionary games extends to various domains, such as economics, management, political science, and social and biological evolution, addressing problems such as pricing wars between businesses, government-business cooperation, market volatility, and competition (He and Strub, 2022). The prospect theory was integrated into an evolutionary game framework to scrutinize the decisionmaking processes of businesses and governments within the realm of a green supply chain. ...
Fostering sustainable development through green supply chains is of paramount significance. Government subsidies emerge as a successful strategy for motivating businesses to actively participate in such eco-friendly practices. This study employs prospect theory and an evolutionary game model to analyze the transition toward carbon peaking and neutrality while promoting the expansion of highly sustainable businesses. By exploring the decision-making processes of businesses and governments regarding sustainability, we develop an evolutionary game-based decision model to assess the impact of government subsidies on businesses engaged in green supply chains. Through numerical simulation obtained via MATLAB, we examine various factors influencing the evolution of the game system between green supply chain businesses and the government. Additionally, we investigate how government incentives impact the decision-making behavior of green supply chain businesses. Our findings indicate that governmental fines can effectively encourage the adoption of green supply chains. Furthermore, moderate government subsidies incentivize enterprises to opt for sustainable supply chains, benefiting both the government and businesses. However, providing hefty government subsidies not only fails to encourage the adoption of green supply chains but also incurs costs for the government, without yielding any positive change in the businesses’ approach. By incorporating evolutionary game theory and prospect theory, this study contributes to the body of knowledge on government-supported green supply chains, offering incentive programs tailored to the real-world conditions faced by businesses while demonstrating practical application values.
... In addition to the works mentioned above, there exists an extensive literature on equilibrium asset pricing and portfolio selection with loss aversion or time-varying risk aversion and thus we do not attempt to summarize it here (e.g., Guo and He (2017);van Bilsen et al. (2020); Strub and Li (2020); Choi et al. (2022); He and Strub (2022); Li et al. (2022), and Tse and Zheng (2023)). 1 ...
We develop a continuous-time pure exchange model with heterogeneous households who differ in their consumption adjustment behavior. One group exhibits standard preferences and adjusts consumption immediately in response to shocks, while the other is loss-averse with respect to consumption changes and adjusts infrequently. This micro-level consumption heterogeneity endogenously generates business cycle dynamics—even when aggregate endowment follows a smooth stochastic trend. The resulting shifts in wealth distribution lead to time-varying effective risk aversion, which drives counter-cyclical patterns in asset pricing moments such as risk premia, Sharpe ratios, and return volatility. Unlike prior studies that rely on habit formation or recursive preferences to explain these dynamics, our model links them directly to household consumption behavior. We establish equilibrium existence under non-monotone and non-convex preferences, derive closed-form asset pricing implications, and show consistency with key empirical observations.
Project leaders are responsible for planning, controlling, and revising projects. As a project unfolds, the leader evaluates the project’s progress by comparing ongoing costs and scope to a baseline plan and considers potential revisions. We offer a general model of managerial mental accounting, which includes loss aversion, reference point updating, and narrow framing, and examine how it impacts downstream decisions. Our model predicts insufficient adjustments of project scope and cost at revision, resulting in reduced financial profit. We show that the choice of measure to quantify the project progress—planned, actual, or earned—affects the updating of reference points, and hence the downstream decisions. Thus, progress measures could be wisely employed to mitigate insufficient adjustments. It turns out that measuring progress via planned scope is often advantageous, whereas utilizing earned value for cost is never advisable.
This paper was accepted by David Simchi-Levi, behavioral economics and decision analysis.
Supplemental Material: The online appendix is available at https://doi.org/10.1287/mnsc.2021.02929 .
We study the impact of a reference point determined by social comparison on wealth growth and inequality. The reference point of each individual investor contains both personal and social components. Whereas the personal component depends on the investor’s own history of wealth, the social component is determined by the wealth level of other investors in her network. In the benchmark case without social interactions and under the assumption of homogeneous preferences, each investor’s expected wealth grows at a common rate, the wealth gaps widens and the Gini coefficient remains constant. On the other hand, if the reference point is determined solely by social interactions, then it is possible that the network simultaneously experiences high wealth growth and a reduction in inequality. Finally, for the general case where the reference point incorporates both personal and social components and some extensions of the model, we numerically show that increasing the degree of social interactions is beneficial for both increasing wealth growth and reducing inequality.
We study optimal price discrimination when a monopolist faces a continuum of consumers with reference‐dependent preferences. A consumer's valuation for product quality consists of an intrinsic valuation affected by a private state signal (type) and a gain–loss valuation that depends on deviations of purchased quality from a reference point. Following [Kőszegi and Rabin, 2006], we consider loss‐averse buyers who evaluate gains and losses in terms of changes in the consumption valuation, but in our model each buyer evaluates consumption outcomes relative to his own state‐contingent reference quality level. We capture the process by which reference qualities are formed via a reference consumption plan, and use a generalization of the Mirrlees representation of the indirect utility to fully characterize optimal contracts for loss‐averse consumers. We find that, depending on the reference plan, optimal price discrimination may exhibit (i) downward distortions beyond the standard downward distortions due to screening, (ii) efficiency gains relative to second‐best contracts without loss aversion, and (iii) upward distortions above first‐best quality levels without loss aversion. We consider ex ante and ex post consistent contracts in which quality offers by the firm coincide, in expectations or at every state realization, respectively, with the reference quality levels. We find the firm's unique preferred ex ante and ex post consistent contract menu and specify conditions under which, for the second case, it also constitutes the consumers' preferred menu.
This paper investigates the dynamics of individual portfolios in a unique data set containing the disaggregated wealth of all households in Sweden. Between 1999 and 2002, we observe little aggregate rebalancing in the financial portfolio of participants. These patterns conceal strong household-level evidence of active rebalancing, which on average offsets about one-half of idiosyncratic passive variations in the risky asset share. Wealthy, educated investors with better diversified portfolios tend to rebalance more actively. We find some evidence that households rebalance toward a greater risky share as they become richer. We also study the decisions to trade individual assets. Households are more likely to fully sell directly held stocks if those stocks have performed well, and more likely to exit direct stockholding if their stock portfolios have performed well; but these relationships are much weaker for mutual funds, a pattern that is consistent with previous research on the disposition effect among direct stockholders and performance sensitivity among mutual fund investors. When households continue to hold individual assets, however, they rebalance both stocks and mutual funds to offset about one-sixth of the passive variations in individual asset shares. Households rebalance primarily by adjusting purchases of risky assets if their risky portfolios have performed poorly, and by adjusting both fund purchases and full sales of stocks if their risky portfolios have performed well. Finally, the tendency for households to fully sell winning stocks is weaker for wealthy investors with diversified portfolios of individual stocks. © 2009 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology.
Many investment models in discrete or continuous-time settings boil down to
maximizing an objective of the quantile function of the decision variable. This
quantile optimization problem is known as the quantile formulation of the
original investment problem. Under certain monotonicity assumptions, several
schemes to solve such quantile optimization problems have been proposed in the
literature. In this paper, we propose a change-of-variable and relaxation
method to solve the quantile optimization problems without using the calculus
of variations or making any monotonicity assumptions. The method is
demonstrated through a portfolio choice problem under rank-dependent utility
theory (RDUT). We show that solving a portfolio choice problem under RDUT
reduces to solving a classical Merton's portfolio choice problem under expected
utility theory with the same utility function but a different pricing kernel
explicitly determined by the given pricing kernel and probability weighting
function. With this result, the feasibility, well-posedness, attainability and
uniqueness issues for the portfolio choice problem under RDUT are solved. The
method is applicable to general models with law-invariant preference measures
including portfolio choice models under cumulative prospect theory (CPT) or
RDUT, Yaari's dual model, Lopes' SP/A model, and optimal stopping models under
CPT or RDUT.
A new model of consumer behavior is developed using a hybrid of cognitive psychology and microeconomics. The development of the model starts with the mental coding of combinations of gains and losses using the prospect theory value function. Then the evaluation of purchases is modeled using the new concept of “transaction utility.” The household budgeting process is also incorporated to complete the characterization of mental accounting. Several implications to marketing, particularly in the area of pricing, are developed.
A key open question for theories of reference-dependent preferences is: what determines the reference point? One candidate is expectations: what people expect could affect how they feel about what actually occurs. In a real-effort experiment, we manipulate the rational expectations of subjects and check whether this manipulation influences their effort provision. We find that effort provision is significantly different between treatments in the way predicted by models of expectation-based, reference-dependent preferences: if expectations are high, subjects work longer and earn more money than if expectations are low. (JEL D12, D84, J22)
How is risk-taking affected by prior gains and losses? While normative theory implores decision makers to only consider incremental outcomes, real decision makers are influenced by prior outcomes. We first consider how prior outcomes are combined with the potential payoffs offered by current choices. We propose an editing rule to describe how decision makers frame such problems. We also present data from real money experiments supporting a "house money effect" (increased risk seeking in the presence of a prior gain) and "break-even effects" (in the presence of prior losses, outcomes which offer a chance to break even are especially attractive).
We introduce a modification of the discounted utility model that accounts for both satiation and habit formation in intertemporal choice. Preferences depend on the satiation level and the habitual consumption level. These two state variables, together with the shape of the value function, drive the properties of the model. One unique feature of our model is that it addresses the trade-off between seeking variety and maintaining acquired habits. We examine several properties of our model, such as the nontrivial patterns of desirability (willingness to pay) for an additional unit of consumption, or the effect of abstaining from consumption (craving). We explore the shape of optimal consumption patterns in discrete and continuous choice settings. If subjects underestimate the changes in satiation and habituation levels, as occurs under projection bias, our model explains why people buy more when hungry, or prefer variety in advance of consumption but stay with the same consumption good in actual use.
A growing body of qualitative evidence shows that loss aversion, a phenomenon formalized in prospect theory, can explain a variety of field and experimental data. Quantifications of loss aversion are, however, hindered by the absence of a general preference-based method to elicit the utility for gains and losses simultaneously. This paper proposes such a method and uses it to measure loss aversion in an experimental study without making any parametric assumptions. Thus, it is the first to obtain a parameter-free elicitation of prospect theory's utility function on the whole domain. Our method also provides an efficient way to elicit utility midpoints, which are important in axiomatizations of utility. Several definitions of loss aversion have been put forward in the literature. According to most definitions we find strong evidence of loss aversion, at both the aggregate and the individual level. The degree of loss aversion varies with the definition used, which underlines the need for a commonly accepted definition of loss aversion.
This study investigates reference-dependent choice with a stochastic, state-dependent reference point. The optimal reference-dependent solution equals the optimal consumption solution (no loss aversion) if the reference point is selected fully endogenously. Given that loss aversion is widespread, we conclude that the reference point generally includes an important exogenously fixed component. We develop a choice model in which adjustment costs can cause stickiness relative to an initial, exogenous reference point. Using historical U.S. investment benchmark data, we show that this model is consistent with diversification across bonds and stocks for a wide range of evaluation horizons, despite the historically high-risk premium of stocks compared to bonds.
This paper was accepted by Peter Wakker, decision analysis.
This study examines individual behavior in two well-functioning marketplaces to investigate whether market experience eliminates
the endowment effect. Field evidence from both markets suggests that individual behavior converges to the neoclassical prediction
as market experience increases. In an experimental test of whether these observations are due to treatment (market experience)
or selection (e.g., static preferences), I find that market experience plays a significant role in eliminating the endowment
effect. I also find that these results are robust to institutional change and extend beyond the two marketplaces studied.
Overall, this study provides strong evidence that market experience eliminates an important market anomaly.
The equity premium puzzle refers to the empirical fact that stocks have outperformed bonds over the last century by a surprisingly
large margin. We offer a new explanation based on two behavioral concepts. First, investors are assumed to be “loss averse,”
meaning that they are distinctly more sensitive to losses than to gains. Second, even long-term investors are assumed to evaluate
their portfolios frequently. We dub this combination “myopic loss aversion.” Using simulations, we find that the size of the
equity premium is consistent with the previously estimated parameters of prospect theory if investors evaluate their portfolios
annually.
We introduce the concept of forward rank‐dependent performance criteria, extending the original notion to forward criteria that incorporate probability distortions. A fundamental challenge is how to reconcile the time‐consistent nature of forward performance criteria with the time‐inconsistency stemming from probability distortions. For this, we first propose two distinct definitions, one based on the preservation of performance value and the other on the time‐consistency of policies and, in turn, establish their equivalence. We then fully characterize the viable class of probability distortion processes and provide the following dichotomy: it is either the case that the probability distortions are degenerate in the sense that the investor would never invest in the risky assets, or the marginal probability distortion equals to a normalized power of the quantile function of the pricing kernel. We also characterize the optimal wealth process, whose structure motivates the introduction of a new, ‘distorted’ measure and a related market. We then build a striking correspondence between the forward rank‐dependent criteria in the original market and forward criteria without probability distortions in the auxiliary market. This connection also provides a direct construction method for forward rank‐dependent criteria. Finally, our results on forward rank‐dependent performance criteria motivate us to revisit the classical (backward) setting. We follow the so‐called dynamic utility approach and derive conditions for existence and a construction of dynamic rank‐dependent utility processes.
We explicitly derive and explore the optimal consumption and portfolio policies of a loss-averse individual who endogenously updates his or her reference level over time. We find that the individual protects current consumption by delaying painful reductions in consumption after a drop in wealth, and increasingly so with higher degrees of endogeneity. The incentive to protect current consumption is stronger with a medium wealth level than with a high or low wealth level. Furthermore, this individual adopts a conservative investment strategy in normal states and typically a more aggressive strategy in good and bad states. Endogeneity of the reference level increases overall risk-taking and generates an incentive to reduce risk exposure with age even without human capital. The welfare loss that this individual would suffer under the conventional constant relative risk aversion (CRRA) consumption and portfolio policies easily exceeds 10%.
This paper was accepted by Tyler Shumway, finance.
In a dynamic setting, decision makers update their reference point as a function of previous decision and outcomes. In “Failing to Foresee the Updating of the Reference Point Leads to Time-Inconsistent Investment,” Strub and Li investigate the influence of reference point updating on decision making and in particular, address whether a decision maker foresees the updating of the reference point in the context of discrete time portfolio optimization. By deriving and comparing optimal trading strategies under various frameworks and reference point updating rules and then, simulating how typical investment behavior would look like in each setting, they come to the following conclusion: a loss-averse decision maker with a time-varying reference point exhibiting realistic investment behavior fails to foresee the future updating of the reference point, and this failure leads to time-inconsistent investment.
Koszegi and Rabin (2006, 2007) develop a model of expectations-based reference-dependent preferences in which the agent experiences “gain-loss utility” by comparing his actual consumption outcome to all of his previously expected consumption outcomes. Koszegi and Rabin (2009) then generalize the static model to a dynamic setting by assuming that the agent experiences both contemporaneous gain-loss utility over present consumption and “prospective” gain-loss utility by comparing his updated to his previous expectations about future consumption. Moreover, Koszegi and Rabin (2009) generalize the outcome-wise “static comparison” of gain-loss utility to a percentile-wise “ordered comparison,” in which the agent compares expected consumption outcomes at each percentile. This paper generalizes the static comparison slightly differently, to what I call a “separated comparison.” In the separated comparison, the agent compares each expected consumption outcome but experiences gain-loss utility only over uncertainty that has been realized, by effectively separating it from remaining future uncertainty. This paper shows that the separated comparison yields simple, tractable, and well-behaved equilibria in a broad class of economic models. Moreover, the equilibria are easy to solve for because the separated comparison preserves an outcome-wise linear structure.
This paper finds optimal portfolios for the reference-dependent preferences by Kőszegi and Rabin with piecewise linear gain–loss utility in a one-period model with a safe and a risky asset. If the return of the risky asset is highly dispersed relative to its potential gains, two personal equilibria arise, one of them including risky investments and the other one only safe holdings. In the same circumstances, the risky personal equilibrium entails market participation that decreases with loss aversion and gain–loss sensitivity, whereas the preferred personal equilibrium is sensitive to market and preference parameters. Relevant market parameters are not the expected return and standard deviation, but rather the ratio of expected gains to losses and the Gini index of the return.
This book presents the definitive exposition of ‘prospect theory’, a compelling alternative to the classical utility theory of choice. Building on the 1982 volume, Judgement Under Uncertainty, this book brings together seminal papers on prospect theory from economists, decision theorists, and psychologists, including the work of the late Amos Tversky, whose contributions are collected here for the first time. While remaining within a rational choice framework, prospect theory delivers more accurate, empirically verified predictions in key test cases, as well as helping to explain many complex, real-world puzzles. In this volume, it is brought to bear on phenomena as diverse as the principles of legal compensation, the equity premium puzzle in financial markets, and the number of hours that New York cab drivers choose to drive on rainy days. Theoretically elegant and empirically robust, this volume shows how prospect theory has matured into a new science of decision making.
In this chapter, we present theories and applications of reference-dependent preferences. We provide some historical perspective, but also move quickly to the current research frontier, focusing on developments in reference dependence over the last 20 years. We present a number of worked examples to highlight the broad applicability of reference dependence. While our primary focus is gain–loss utility, we also provide a short treatment of probability weighting and its links to reference dependence.
Experimental studies have demonstrated that a typical investor derives utility of the gain and loss, relative to certain reference point, realized at each sale of a stock, and that the reference point adapts asymmetrically to the stock's prior gains and losses in that the adaptation to a gain is more substantial than to a comparable loss. This empirical finding motivates us to consider a dynamic trading problem in which an agent decides when to sequentially sell and buy a stock in order to maximize her utility of terminal wealth and realized gains and losses based on a reference point that adapts asymmetrically to the stock's prior gains and losses. We show that this agent is more reluctant to sell the stock at a loss than another agent whose reference point does not adapt to the stock gain or loss at all, leading to a higher percentage of gains realized (PGR) and a lower percentage of losses realized (PLR). We also show that when the agent weighs the utility of realized gains and losses more heavily in her investment criteria, she trades the stock more frequently, the PGR becomes lower, and the PLR becomes higher.
Recent evidence suggests that investors are inattentive to their portfolios and hire expensive portfolio managers. This paper develops a life-cycle portfolio-choice model in which the investor experiences loss-averse utility over news and can ignore his portfolio. In such a model, the investor prefers to ignore and not rebalance his portfolio most of the time because he dislikes bad news more than he likes good news such that expected news causes a first-order decrease in utility. Consequently, the investor has a first-order willingness to pay a portfolio manager who rebalances actively on his behalf. Moreover, the investor can diversify over time and his consumption aligns with predictions of mental accounting. I structurally estimate the preference parameters by matching stock shares and stock-market non-participation over the life cycle. My parameter estimates are in line with the literature, generate reasonable intervals of inattention, and simultaneously explain consumption and wealth accumulation over the life cycle. Here, it matters that news utility preserves first-order risk aversion even in the presence of stochastic labor income, which also causes stock shares to rise in wealth.
Experimental studies show that people's risk preferences depend nonlinearly on probabilities, but relatively little is known about how probability weighting inuences investment decisions. In this paper we analyze the portfolio choice problem of investors who maximize rank-dependent utility in a single-period complete market. We prove that investors with a less risk averse preference relation in general choose a more risky final wealth distribution, receiving a risk premium in return for accepting conditional-mean-zero noise (more risk). We also propose a new scenario-based notion of less risk taking that can be applied when state probabilities are unknown or not agreed upon.
Theories of expectations-based reference-dependent preferences have provided a critical modeling innovation, incorporating a structured theory of the formation of reference points. An important prediction of these models is a monotone response in behavior to changes in expectations. To test such models we conduct a real-effort experiment manipulating expectations and examining consequences on effort provision. In contrast to the theory, we document substantial nonmonotonicities in the effort response to changing expectations. Our results provide some evidence on the limitations of expectations-based reference dependence. (JEL: D81, D84, D12, D03)
We examine the reference-dependent risk preferences of Ko″szegi and Rabin (2007), focusing on their choice-acclimating personal equilibria. Although their model has only a trivial intersection (expected utility) with other reference-dependent models, it has very strong connections with models that rely on different psychological intuitions. We prove that the intersection of rank-dependent utility and quadratic utility, two well-known generalizations of expected utility, is exactly monotone linear gain-loss choice-acclimating personal equilibria. We use these relationships to identify parameters of the model, discuss loss and risk aversion, and demonstrate new applications. (JEL D11, D81).
We provide conditions on a one-period-two-date pure exchange economy with rank-dependent utility agents under which Arrow-Debreu equilibria exist. When such an equilibrium exists, we show that the state-price density is a weighted marginal rate of intertemporal substitution of a representative agent, where the weight depends on the differential of the probability weighting function. Based on the result, we find that asset prices depend upon agents' subjective beliefs regarding overall consumption growth, and we offer a direction for possible resolution of the equity premium puzzle.
We present a model of portfolio choice and stock trading volume with loss-averse investors. The demand function for risky assets is discontinuous and nonmonotonic: As wealth rises beyond a threshold, investors follow a generalized portfolio insurance strategy, which is consistent with the disposition effect. In addition, loss-averse investors hold no stocks unless the equity premium is quite high. The elasticity of the aggregate demand curve changes substantially, depending on the distribution of wealth across investors. In an equilibrium setting, the model generates positive correlation between trading volume and stock return volatility but suggests that this relationship is nonlinear.
We formulate and study three multi-period behavioral portfolio selection models under cumulative prospect theory: (i) S-shaped utility maximization without probability weighting in a market with one risky asset; (ii) S-shaped utility maximization without probability weighting in a market with multiple risky assets which follow a joint elliptical distribution; and (iii) S-shaped utility maximization with inverse-S-shaped probability weighting in a market with one risky asset. For the first two time consistent models, we identify the well-posedness conditions and derive the semi-analytical optimal policies. For the third time inconsistent model, we assume that the investor is aware of the time inconsistency but is unable to commit to his initial plan of action. Then, we reformulate the model into an intrapersonal game model and derive the semi-analytical subgame perfect Nash equilibrium (time consistent) policy under well-posedness condition. All the three policies take a piecewise linear feedback form. Our analysis of the three models not only partially explains the well documented phenomena of non-participation puzzle and horizon effect, but also extends the two fund separation theorem into multi-period S-shaped utility setting and pushes forward the study on time inconsistency issue incurred by probability weighting.
We formalize the reference point adaptation process by relating it to a way people perceive prior gains and losses. We then develop a dynamic trading model with reference point adaptation and loss aversion, and derive its semi-analytical solution. The derived optimal stock holding has an asymmetric V-shaped form with respect to prior outcomes, and the related sensitivities are directly determined by the sensitivities of reference point shifts with respect to the outcomes. We also find that the effects of reference point adaptation can be used to shed light on some well documented trading patterns, e.g., house money, break even, and disposition effects.
We use the portfolio selection model presented in He and Zhou [Manage. Sci., 2011, 57, 315–331] and the NYSE equity and US treasury bond returns for the period 1926–1990 to revisit Benartzi and Thaler’s myopic loss aversion theory. Through an extensive empirical study, we find that in addition to the agent’s loss aversion and evaluation period, his reference point also has a significant effect on optimal asset allocation. We demonstrate that the agent’s optimal allocation to equities is consistent with market observation when he has reasonable values of degree of loss aversion, evaluation period and reference point. We also find that the optimal allocation to equities is sensitive to these parameters. We then examine the implications of money illusion for asset allocation. Finally, we extend the model to a dynamic setting.
A hallmark result within behavioral economics is that individuals' choices are affected by current endowments. A recent theory due to Kőszegi and Rabin (, Quarterly Journal of Economics, 121, 1133–1165) explains such endowment effect with a model of expectations-based reference-dependent preferences. Departing from past work, we conduct complementary experiments to disentangle expectations—verified probabilistic beliefs held by subjects—from other features of endowment—such as “assignment” to a good—hence allowing us to compare the effect of expectations with that of other variations. While mere assignment can affect choices, we do not find a large role in the effect for Kőszegi–Rabin expectations.
We study how a behavioral agent allocates her portfolio. We consider a cumulative prospect theory investor in a single period setting with one riskless bond and multiple risky stocks, which follow a multivariate elliptical distribution. Our main result is a two-fund separation between the riskless bond and a mean–variance-portfolio, up to an exogenous benchmark portfolio. The mean–variance-portfolio, which we derive explicitly, is the same for all agents. Individual risk preferences are mirrored only in the participation in this portfolio. This dependence is illustrated by considering empirical returns. Furthermore we solve ill-posed optimization problems by imposing a regulatory risk constraint. Finally we address specific parameterizations of the value function by studying power, linear, and exponential utility.
We examine whether investing experience can dampen the disposition effect, that is, the fact that investors seem to hold on to their losing stocks to a greater extent than they hold on to their winning stocks. To do so, we devise a computer program that simulates the stock market. We use the program in an experiment with two groups of subjects, namely experienced investors and undergraduate students (the inexperienced investors). As a control procedure, we consider random trade decisions made by robot subjects. We find that though both human subjects show the disposition effect, the more experienced investors are less affected.
EVIDENCE EXISTS THAT PEOPLE DO NOT ALWAYS MAKE DECISIONS INVOLVING UNCERTAIN MONETARY REWARDS AS IF THEY WERE MAXIMIZING EXPECTED UTILITY OF FINAL ASSETS. EXPLANATIONS FOR THIS BEHAVIOR POSTULATE THAT THE COGNITIVE DEMANDS OF CONSISTENCY TO SUCH A THEORY ARE TOO GREAT. HOWEVER, SITUATIONS EXIST IN WHICH MORE THAN MENTAL SHORTCUTS ARE INVOLVED AND THESE ANOMALIES RAISE EQUATIONS ABOUT EXPECTED UTILITY THEORY AS A GUIDE TO BEHAVIOR. THIS STUDY EXPLORES THE POSSIBILITY THAT EXPECTED UTILITY THEORY APPEARS TO FAIL BECAUSE THE SINGLE OUTCOME DESCRIPTOR - MONEY - IS NOT SUFFICIENT. AFTER MAKING A DECISION UNDER UNCERTAINTY, A PERSON MAY DISCOVER, ON LEARNING THE RELEVANT OUTCOMES, THAT ANOTHER ALTERNATIVE WOULD HAVEBEEN PREFERABLE. THIS KNOWLEDGE MAY IMPART A SENSE OF LOSS,OR REGRET. THE DECISION MAKER WHO IS PREPARED TO TRADEOFF FINANCIAL RETURN IN ORDER TO AVOID REGRET WILL EXHIBIT SOME OF THE BEHAVIORAL PARADOXES OF DECISION THEORY.
Behavioral finance argues that some financial phenomena can plausibly be understood using models in which some agents are not fully rational. The field has two building blocks: limits to arbitrage, which argues that it can be difficult for rational traders to undo the dislocations caused by less rational traders; and psychology, which catalogues the kinds of deviations from full rationality we might expect to see. We discuss these two topics, and then present a number of behavioral finance applications: to the aggregate stock market, to the cross-section of average returns, to individual trading behavior, and to corporate finance. We close by assessing progress in the field and speculating about its future course.
There has been recent debate about whether prospect theory can explain the disposition effect. Using both theory and simulation, this paper shows that prospect theory often predicts the disposition effect when lagged expected final wealth is the reference point under the principle of preferred personal equilibrium, regardless of whether the reference point is updated or not. When initial wealth is the reference point, however, there is often no disposition effect. Models that use a reference point with no lag under the principle of preferred personal equilibrium or that determine the reference point using the principle of disappointment aversion cannot explain why the investor bought a stock in the first place. Reference point adjustment weakens the disposition effect, leads to more aggressive initial stock purchase strategies, and predicts history dependence in stock holding.
This paper was accepted by John List, behavioral economics.
We study equilibrium firm-level stock returns in two economies: one in which investors are loss averse over the fluctuations of their stock portfolio, and another in which they are loss averse over the fluctuations of individual stocks that they own. Both approaches can shed light on empirical phenomena, but we find the second approach to be more successful: In that economy, the typical individual stock return has a high mean and excess volatility, and there is a large value premium in the cross section which can, to some extent, be captured by a commonly used multifactor model.
A portfolio choice model in continuous time is formulated for both complete and incomplete markets, where the quantile function of the terminal cash flow, instead of the cash flow itself, is taken as the decision variable. This formulation covers a wide body of existing and new models with law-invariant preference measures, including expected utility maximization, mean–variance, goal reaching, Yaari's dual model, Lopes' SP/A model, behavioral model under prospect theory, as well as those explicitly involving VaR and CVaR in objectives and/or constraints. A solution scheme to this quantile model is proposed, and then demonstrated by solving analytically the goal-reaching model and Yaari's dual model. A general property derived for the quantile model is that the optimal terminal payment is anticomonotonic with the pricing kernel (or with the minimal pricing kernel in the case of an incomplete market if the investment opportunity set is deterministic). As a consequence, the mutual fund theorem still holds in a market where rational and irrational agents co-exist.
We develop a rational dynamic model in which people are loss averse over changes in beliefs about present and future consumption. Because changes in wealth are news about future consumption, preferences over money are reference-dependent. If news resonates more when about imminent consumption than when about future consumption, a decision maker might (to generate pleasant surprises) overconsume early relative to the optimal committed plan, increase immediate consumption following surprise wealth increases, and delay decreasing consumption following surprise losses. Since higher wealth mitigates the effect of bad news, people exhibit an unambiguous first-order precautionary-savings motive. (JEL D14, D81, D83, D91)
We derive the optimal portfolio choice for an investor who behaves according to Cumulative Prospect Theory (CPT). The study
is done in a one-period economy with one risk-free asset and one risky asset, and the reference point corresponds to the terminal
wealth arising when the entire initial wealth is invested into the risk-free asset. When it exists, the optimal holding is
a function of a generalized Omega measure of the distribution of the excess return on the risky asset over the risk-free rate. It conceptually resembles Merton’s optimal
holding for a CRRA expected-utility maximizer. We derive some properties of the optimal holding and illustrate our results
using a simple example where the excess return has a skew-normal distribution. In particular, we show how a CPT investor is
highly sensitive to the skewness of the excess return on the risky asset. In the model we adopt, with a piecewise-power value function with different shape
parameters, loss aversion might be violated for reasons that are now well-understood in the literature. Nevertheless, we argue
that this violation is acceptable.
KeywordsCumulative Prospect Theory-Portfolio choice-Omega measure
JEL ClassificationD81-G11-D03
This paper examines the ideas underlying the Kahneman and Tversky (1979) decision-choice model, Prospect Theory, and presetns an extensive chronological review of the literature. The literature review centers on leading articles that either examine aspects of Prospect Theory itself or use Prospect Theory as a basis for other areas of research. The paper concludes with a brief look at the potential for using Prospect Theory in financial research. © 1995 JAI PRESS Inc., All Rights of reproduction in any form reserved.
According to prospect theory [Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk, Econometrica, 47, 263–292], gains and losses are measured from a reference point. We attempted to ascertain to what extent the reference point shifts following gains or losses. In questionnaire studies, we asked subjects what stock price today will generate the same utility as a previous change in a stock price. From participants’ responses, we calculated the magnitude of reference point adaptation, which was significantly greater following a gain than following a loss of equivalent size. We also found the asymmetric adaptation of gains and losses persisted when a stock was included within a portfolio rather than being considered individually. In studies using financial incentives within the BDM procedure [Becker, G. M., DeGroot, M. H., & Marschak, J. (1964). Measuring utility by a single-response sequential method. Behavioral Science, 9(3), 226–232], we again noted faster adaptation of the reference point to gains than losses. We related our findings to several aspects of asset pricing and investor behavior.
A reference-dependent generalisation of subjective expected utility theory is presented. In this theory, preferences between acts depend both on final outcomes and on reference points (which may be uncertain acts). It is characterised by a set of axioms in a Savage-style framework. A restricted form of the theory separates attitudes to end states (encoded in a ‘satisfaction function’) from attitudes to gains and losses of satisfaction. Given weak additional assumptions, the restricted theory excludes cycles of choice, explains observed disparities between willingness-to-pay and willingness-to-accept valuations of lotteries, and predicts preference reversal.
Restrictions that a class of general equilibrium models place upon the average returns of equity and Treasury bills are found to be strongly violated by the U.S. data in the 1889–1978 period. This result is robust to model specification and measurement problems. We conclude that, most likely, an equilibrium model which is not an Arrow-Debreu economy will be the one that simultaneously rationalizes both historically observed large average equity return and the small average risk-free return.
We use mutual fund manager data from the technology bubble to examine the hypothesis that inexperienced investors play a role in the formation of asset price bubbles. Using age as a proxy for managers’ investment experience, we find that around the peak of the technology bubble, mutual funds run by younger managers are more heavily invested in technology stocks, relative to their style benchmarks, than their older colleagues. Furthermore, young managers, but not old managers, exhibit trend-chasing behavior in their technology stock investments. As a result, young managers increase their technology holdings during the run-up, and decrease them during the downturn. Both results are in line with the behavior of inexperienced investors in experimental asset markets. The economic significance of young managers’ actions is amplified by large inflows into their funds prior to the peak in technology stock prices.
This paper analyzes the trading records of a major discount brokerage house to investigate the disposition effect, the tendency to sell stocks that have appreciated in price (winners) sooner than stocks that trade below the purchase price (losers). In contrast to previous research that has demonstrated the disposition effect by aggregating across investors, our main objective is to identify differences in the disposition bias across individuals and explain this in terms of underlying investor characteristics. Building on the findings in experimental economics and social psychology, we hypothesize that differences in investor literacy about financial markets and trading frequency are responsible in part for the variation in individual disposition effect. Using demographic and socioeconomic variables as proxies for investor literacy, we find empirical evidence that wealthier individuals and individuals employed in professional occupations exhibit a lower disposition effect. Consistent with experimental economics, trading frequency also tends to reduce the disposition effect. We provide guidelines for investment advisors, regulators, and investment communities to utilize our findings and help investors make better decisions.
Life-cycle models of labor supply predict a positive relationship between hours supplied and transitory changes in wages.
We tested this prediction using three samples of wages and hours of New York City cabdrivers, whose wages are correlated within
days but uncorrelated between days. Estimated wage elasticities are significantly negative in two out of three samples. Elasticities
of inexperienced drivers average approximately −1 and are less than zero in all three samples (and significantly less than
for experienced drivers in two of three samples). Our interpretation of these findings is that cabdrivers (at least inexperienced
ones): (i) make labor supply decisions “one day at a time” instead of intertemporally substituting labor and leisure across
multiple days, and (ii) set a loose daily income target and quit working once they reach that target.
We examined reference point adaptation following gains or losses in security trading using participants from China, Korea, and the US. In both questionnaire studies and trading experiments with real money incentives, reference point adaptation was larger for Asians than for Americans. Subjects in all countries adapted their reference points more after a gain than after an equal-sized loss. When we introduced a forced sale intervention that is designed to close the mental account for a prior outcome, Americans showed greater adaptation toward the new price than their Asian counterparts. We offer possible explanations both for the cross-cultural similarities and the cross-cultural differences.
If options are correctly priced in the market, it should not be possible to make sure profits by creating portfolios of long and short positions in options and their underlying stocks. Using this principle, a theoretical valuation formula for options is derived. Since almost all corporate liabilities can be viewed as combinations of options, the formula and the analysis that led to it are also applicable to corporate liabilities such as common stock, corporate bonds, and warrants. In particular, the formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default.
We introduce adaptive learning behavior into a general-equilibrium life-cycle economy with capital accumulation. Agents form forecasts of the rate of return to capital assets using least-squares autoregressions on past data. We show that, in contrast to the perfect-foresight dynamics, the dynamical system under learning possesses equilibria that are characterized by persistent excess volatility in returns to capital. We explore a quantitative case for theselearning equilibria. We use an evolutionary search algorithm to calibrate a version of the system under learning and show that this system can generate data that matches some features of the time-series data for U.S. stock returns and per-capita consumption. We argue that this finding provides support for the hypothesis that the observed excess volatility of asset returns can be explained by changes in investor expectations against a background of relatively small changes in fundamental factors.
We use Koszegi and Rabin's (2006) model of reference-dependent utility, and an extension of it that applies to decisions with delayed consequences, to study preferences over monetary risk. Because our theory equates the reference point with recent probabilistic beliefs about outcomes, it predicts specific ways in which the environment influences attitudes toward modest-scale risk. It replicates "classical" prospect theory—including the prediction of distaste for insuring losses—when exposure to risk is a surprise, but implies first-order risk aversion when a risk, and the possibility of insuring it, are anticipated. A prior expectation to take on risk decreases aversion to both the anticipated and additional risk. For large-scale risk, the model allows for standard "consumption utility" to dominate reference-dependent "gain-loss utility," generating nearly identical risk aversion across situations. (JEL D81)