Behavioral Explanation of the Equity Premium Puzzle

Table of Contents

List of Abbreviations

List of Figures and Tables

1 Introduction

2 The Equity Premium Puzzle
2.1 The Model
2.2 Empirical Observations and the Predictions of the Model
2.3 Approaches to the Puzzle
2.4 Potential Failures of the Model
2.4.1 The Risk Aversion of Investors
2.4.2 Consumption-Based Asset Pricing
2.4.3 Expected Utility Theory
2.4.3.1 The Axioms of Expected Utility Theory
2.4.3.2 Violations of Expected Utility Theory

3 Prospect Theory and Mental Accounting
3.1 Prospect Theory
3.1.1 The Value Function
3.1.2 The Weighting Function
3.1.3 Mathematical Notions of Prospect Theory
3.1.4 Loss Aversion
3.2 Mental Accounting

4 Myopic Loss Aversion and the Equity Premium Puzzle
4.1 Implications from Prospect Theory and Mental Accounting for the Equity Premium
4.2 Myopic Loss Aversion Approaches to the Equity Premium Puzzle
4.2.1 Explanation With Myopic Loss Aversion
4.2.2 Explanation With Consumption, Narrow Framing, and Loss Aversion
4.2.2.1 Investor Preferences
4.2.2.2 Incorporating Historical Consumption and Dividend Correlation
4.2.2.3 Results
4.3 Remarks on Myopic Loss Aversion

5 Regret Theory and the Equity Premium Puzzle
5.1 The Intuition Behind Regret Theory
5.2 Linking Regret Theory, Prospect Theory, and Narrow Framing
5.3 Formal Model of Regret Theory
5.4 Explaining the Equity Premium Puzzle With Regret

6 Conclusion

References

Appendixes

List of Abbreviations

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List of Figures and Tables

Figure 1 Admissible Region of the Equity Premium

Figure 2 Double-Stage Choice Problem

Figure 3 Hypothetical Value Function

Figure 4 Hypothetical Weighting Function

Figure 5 Weighting Functions for Gains and Losses

Figure 6 Prospective Utilities for Different Evaluation Periods of Homogenous Portfolios

Figure 7 Prospective Utilities for Different Evaluation Periods of Mixed Portfolios

Figure 8 Equity Premia for Different Evaluation Periods

Figure 9 Value Function With Respect to Prior Gains and Losses

Table 1 Historical Data (1889-1978)

Table 2 Choice Problem 1 to 6

Table 3 Choice Problem 7 to 8

1 Introduction

Ever since the equity premium puzzle (EEP) was published by Mehra and Prescott (1985), it has become one of the most investigated problems in economics (Mehra, 2003, p. 54). The EEP describes the fact that we cannot link historic stock returns with the volatility of consumption growth (in a sense to be made precise below). Mehra and Prescott call this a puzzle as their consumption-based asset pricing model can not plausibly explain the S&P 500’s annual risk premium of 6.2% over relatively risk-free governmental treasury bills between 1889 and 1978. This model reproduces an equity premium of 6.2% solely by adapting unreasonable estimates of agents’ risk aversion (Mehra & Prescott, 1985, pp. 155-156). In this way, the model also predicts an extreme size of the risk-free rate (Cochrane, 2000, p. 416). Thus, the equity premium is not able to be explained exclusively by the risk of stock price fluctuations.

In general, there are three possible reasons for the puzzle. First, investors could indeed be much more risk-averse than economists generally think. Some arguments below will show that we are safe in disregarding this possibility. Second, the stock performance of the last century was mostly due to luck. Since the study of Mehra and Prescott (1985) was done using US data which was the most successful economy of the 20th century, the results may be biased. Rietz (1988), for instance, explains the EEP by a peso-event which is unobserved in the data. Thus, the EEP should be particular to the US where no extreme event has occurred. In a global study, however, Campbell (1996) shows that the EEP is a general phenomenon. Thus, also this second explanation does not seem to hold. We are, therefore, led to the conclusion that something must be fundamentally wrong with the model. In this regard, two possibilities arise. Either the EEP arises due to time-varying risk premia or expectational errors. In this thesis, the focus will lie on the latter, in particular, on how the (incorrect) use of expected utility maximization leads to biased results. For an overview of the former literature, the reader is referred to the seminal work by Campbell and Cochrane (1999), the recent paper by Verdelhan (2010), as well as references therein.

Mehra and Prescott’s (1985) model turns out to have significant shortcomings due to its underlying assumptions. Subsequent attempts to solve the puzzle by attenuating those assumptions are not sufficient as well to explain the full magnitude of the equity premium. This raises the question if the EPP exists simply due to the failure of classical theory. Apparently, another approach is needed to explain the EPP. At this point it is important to mention that advances in psychological concepts, like mental accounting and prospect theory, pave the way for behavioral approaches to the EPP. These new approaches include myopic loss aversion and implications from regret theory. On the one hand, these approaches examine the problem from a different perspective than it had been investigated initially. On the other hand, however, they provide much needed explanations for the equity risk premium. Thus, behavioral theory might be able to capture the subtleties which traditional theory neglects.

This thesis will examine the EPP from a behavioral perspective. The major research question to be pursued is this: How do behavioral approaches explain the equity premium puzzle? In order to answer this question, a variety of subtasks must be addressed. This includes the investigation of the initial model of Mehra and Prescott (1985) as well as its underlying assumptions. That is, in particular, needed because several well-established classical assumptions must be dropped to set up descriptive behavioral models. In addition, implications from psychology and behavioral economics must be introduced to answer the overall question of this thesis. Hence, the thesis will focus on the notions of loss aversion, narrow framing, and regret theory in an effort to explain the EPP.

The EPP was the subject of a large research effort after its initial publication by Mehra and Prescott (1985). For the context of this thesis the approaches of Abel (1990), Constantinides (1990), and Epstein and Zin (1991) are of particular interest. The reason for this is that they provide more realistic models by overcoming certain classical assumptions of the initial approach. Epstein and Zin (1991) break up the connection between risk aversion and the elasticity of inter-temporal substitution. In addition to this, habit formation, as discussed by Constantinides (1990) and further developed by Abel (1990), assumes that an agents’ current utility is, amongst other things, dependent on past consumption. With respect to the general applicability of consumption-based asset pricing, Campbell and Cochrane (2000) best illustrate poor performance compared to other pricing models. Furthermore, Campbell (1998) and Mankiw and Zeldes (1991) provide historical evidence of weak correlations between consumption growth and asset returns.

Prospect theory and mental accounting are the cornerstones of behavioral approaches to the EPP. The observations regarding violations of expected utility theory led to the introduction of prospect theory by Kahneman and Tversky (1979). Subsequent to this, extensive research of Kahneman and Tversky (1984, 1991, & 1992) helped to advance this field. Moreover, the work of Kahneman and Lovallo (1993); Kahneman and Tversky (1981 & 1986); Loewenstein, Rabin, and Read (1999); and Thaler (1980 & 1999) explain the concept of mental accounting and particularly of narrow framing. This paves the way for behavioral approaches to discussing the puzzle. Initially it were Benartzi and Thaler (1985) who advance that area by combining implications from prospect theory and mental accounting to the concept of myopic loss aversion. Barberis, Huang, and Santos (2001) as well as Barberis and Huang (2008) extend the behavioral notions of Benartzi and Thaler. Moreover, they combine utility from consumption and behavioral traits into one model. Another behavioral concept that explains preferences under uncertainty as observed by Kahneman and Tversky (1979) is regret theory, which was established by Bell (1982) and Loomes and Sugden (1982).

The remainder of this thesis is organized as follows: Chapter 2 investigates the EPP and its predictions. This chapter considers potential failures of the model, especially the violations of expected utility theory which may lead to the puzzling results. Chapter 3 focuses on the behavioral concepts of prospect theory and mental accounting. Based on those concepts, chapter 4 deals with myopic loss aversion in an effort to explain the EPP. Chapter 5 discusses regret theory as another behavioral concept. This chapter also proposes an explanation of the EPP from a regret perspective. Chapter 6 concludes and discusses potential directions for future research.

2 The Equity Premium Puzzle

2.1 The Model

This section deducts the reconsidered EPP by Mehra and Prescott (2003). In addition, the key assumptions of Mehra and Prescott are examined. This gives, on the one hand, a reasonable overview of the source of the puzzle, and, on the other hand, provides a knowledge base for upcoming argumentation concerning behavioral explanations of the puzzle.

Mehra and Prescott’s (2003) model assumes consumption-based asset pricing. The idea behind consumption-based asset pricing can be derived from the capital asset pricing model (CAPM). The CAPM predicts a linear relationship between the expected return of an asset and its movement with the market captured through the asset’s beta. High beta assets imply returns above the market return if the market return is positive. The consumption CAPM uses assets to smooth consumption. Thus, if the economy is running well and agents typically consume a lot, assets that pay off in booming phases provide less utility than those assets whose payoff occurs during periods of weak consumption. Therefore, assets that pay off when consumption is high anyway must convince investors with a high risk premium (Mehra & Prescott, 2003, p. 899).

Investors face the possibility of consuming now or in a future period. Due to time preferences, consuming a good in the present is valued differently than consuming that good in the future. Therefore, future consumption is discounted by a subjective time discount factor [illustration not visible in this excerpt]. Cumulated expected consumption at the present time then amounts to

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where [illustration not visible in this excerpt] denotes the expectation at time [illustration not visible in this excerpt], and [illustration not visible in this excerpt] represents the household’s utility function depending on per capita consumption [illustration not visible in this excerpt] (Mehra & Prescott, 2003, p. 900).^[1] Assuming the present time [illustration not visible in this excerpt] and the subsequent period [illustration not visible in this excerpt], agents are indifferent between present and future consumption if the utility from present consumption equals the expected utility from consumption in [illustration not visible in this excerpt], discounted by the subjective time discount factor. This formally means [illustration not visible in this excerpt]. However, if the discount factor decreases, the agent prefers immediate consumption. Therefore, [illustration not visible in this excerpt] captures the agent’s impatience for early consumption over random future consumption (Cochrane, 2000, pp. 14-15). Regarding the utility function in equation (1), Mehra and Prescott (2003) assume constant relative risk aversion (CRRA) through power utility

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This function is concave, where [illustration not visible in this excerpt] measures the curvature of the function. Thus, a higher [illustration not visible in this excerpt] implies higher risk aversion of investors. Since [illustration not visible in this excerpt], [illustration not visible in this excerpt] is the coefficient of relative risk aversion (see Appendix 1). The power utility function of equation (2) follows expected utility theory, which assumes rational investors. Furthermore, [illustration not visible in this excerpt] is the reciprocal of the inter-temporal substitution elasticity (see Appendix 2). Thus, the model implies that investors prefer smooth consumption across different states of nature while at the same time a smooth consumption over time. Those are major characteristics of the model which are challenged in subsequent sections.

According to Mehra and Prescott’s (2003) model, investors have the opportunity to purchase a risky asset characterized by the stock of a company producing output that is of value [illustration not visible in this excerpt]. Output is considered as a dividend claim of the company’s asset, which is worth [illustration not visible in this excerpt]. Buying or selling that asset does not entail transaction costs. Thus, as agents maximize their utility, an inter-temporal equilibrium occurs if the utility related with purchasing an asset (i.e., [illustration not visible in this excerpt]) equals the expected utility of the asset in the next period (i.e., [illustration not visible in this excerpt]). Hence, we yield an equilibrium if

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By means of this expression, the one period returns on equity and risk-free assets can be derived. At this point, it is convenient to define the stochastic discount factor

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which comprises all adjustments that are due to risk associated with random future consumption (Cochrane, 2000, pp. 16-17). By expressing the gross return on equity as [illustration not visible in this excerpt], equation (3) is simplified to

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Equivalently, the following expression contains the risk-free rate which is known at time [illustration not visible in this excerpt]:

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By means of equation (5) and (6) one yields an expression for the equity premium, as shown in Appendix 3:

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(Mehra and Prescott, 2003, p. 902). Accordingly, the equity premium is determined by the covariance of the marginal utility of consumption with the return on equity. Hence, a premium is required if the covariance of consumption and the return on equity is positive. Intuitively, this is because positively covarying assets do not pay off if consumption is weak. These assets are, thus, not useful in an effort to smooth consumption. Contrariwise, positive covariance may lead to higher consumption fluctuation. Therefore, a premium is needed for investors to hold those assets (Mehra, 2003, p. 57). However, the question of whether or not covariance can explain the magnitude of the equity premium has not been answered yet.

In their updated model of the EPP, Mehra and Prescott (2003) add some assumptions which are not included in their model from 1985. They presuppose i.i.d. consumption growth (i.e., [illustration not visible in this excerpt]) as well as dividend growth (i.e., [illustration not visible in this excerpt]). Furthermore, [illustration not visible in this excerpt] and [illustration not visible in this excerpt] are jointly log-normally distributed (Mehra, 2003, p. 58). Due to this, [illustration not visible in this excerpt] is i.i.d. as well. Utilizing those definitions and the fact that [illustration not visible in this excerpt] , equation (3) can be rewritten as the price of the company’s stock:

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With some algebra we yield expressions for the gross return on equity

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and the gross return on risk-free assets

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(see Appendix 4 for intermediate steps). Since it is assumed that [illustration not visible in this excerpt] and [illustration not visible in this excerpt], where [illustration not visible in this excerpt] and [illustration not visible in this excerpt] denote the mean and the variance of the log-normal distribution, equation (9) can be transformed to

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as shown in Appendix 5. Analogously, one can derive the log gross return of a risk-free asset from equation (10):

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Hence, by subtracting equation (12) from equation (11), one yields

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According to the assumption that agents consume the dividends of every period (i.e., [illustration not visible in this excerpt]), consumption growth and dividend growth are perfectly correlated, and, thus, equation (13) can be rewritten as

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As a consequence, the log equity premium depends on the coefficient of risk aversion times the covariance of consumption growth and dividend growth, or, alternatively, the coefficient of risk aversion times the variance of the growth rate of consumption. These expressions are sufficient to obtain a risk premium through the model with empirical data. In addition, the expressions allow an examination of risk aversion that is needed to reproduce the historical equity premium.

2.2 Empirical Observations and the Predictions of the Model

This section shows why Mehra and Prescott (1985) were confronted with a puzzle. Mehra and Prescott compare empirical observations of the equity premium and the prediction of their model. Therefore, they use historical data from the S&P 500 and US T-bills from 1889 to 1978 as shown in Table 1.

Table 1 – Historical Data (1889-1978)

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Adapted from Mehra and Prescott, 2003, p. 905

Contrary to the empirical observations, the actual model implies a much smaller equity premium. As equation (14) reveals, the emergence of the puzzle depends significantly on the coefficient of risk aversion as this is the most critical input factor. Based on previous research the coefficient of risk aversion should be approximately 1 and definitely smaller than 10 (Mehra & Prescott, 1985, p. 154). Based on those values for [illustration not visible in this excerpt], equation (14) yields a much lower equity premium than provided in the empirical data (Mehra & Prescott, 1985, p. 156). Mehra and Prescott (1985) calculate a maximum equity premium of 0.35% which can be explained through the model. This is tremendously below the real average equity premium of 6.18%. As a result, the model cannot explain the historical risk premium without assuming implausible values for [illustration not visible in this excerpt]. Figure 1 illustrates the admissible region for the equity premium predicted by the model with respect to the risk-free rate. This reveals a considerable increase in the risk-free rate in obtaining a high equity premium. Accordingly, the graph shows that even an equity premium of less than 0.5% requires a risk-free rate of 4%. This phenomenon was dubbed the risk-free rate puzzle (Kocherlakota, 1996, pp. 50-51), which is obviously linked to the EPP. Some further evaluations of the risk-free rate and the coefficient of risk aversion will be provided in section 2.4.1.

2.3 Approaches to the Puzzle

There have been numerous attempts to solve the EPP by changing or relaxing the assumptions of Mehra and Prescott (1985). However, none of these attempts have explained the entire magnitude of the equity premium reasonably (Mehra, 2003, p. 60). Reviewing these non-behavioral approaches would clearly go beyond the scope of this thesis. Nevertheless, it is noteworthy that some works reveal considerable improvements to the work of Mehra and Prescott by accounting for more realistic characteristics of economic agents. In particular, Epstein and Zin (1991) correct the equity premium model by breaking up the connection between risk aversion of changes in consumption at different states and of consumption over time (i.e., the coefficient of risk aversion does not necessarily equal the reciprocal of the elasticity of inter-temporal substitution). Furthermore, the approach of habit formation argues that, contrary to the model of Mehra and Prescott, utility from consumption in previous periods does affect the utility from consumption in the current period. Agents that used to consume at a certain level in the past do not want to fall below this level in the present (Constantinides, 1990, p. 521). Thus, agents develop habits. This idea was combined with the assumption that individuals try to “catch up with the Joneses” (i.e., keep consumption as high as peers do) (Abel, 1990, p. 38). Despite the fact that those approaches still rely on classical theory and that they cannot entirely solve the puzzle (Abel, 1990, p. 41), it is noteworthy that they enhanced the initial model by implementing more realistic features of preferences.

There are many more attempts to solve the puzzle; some even arguing that the equity premium is not a puzzle. For instance, investors may be as risk-averse as predicted by Mehra and Prescott (1985) because small negative consumption shocks are the consequence of recessions which agents excessively fear (Fama, 1991, p. 1596). Examining this thought in the absence of any other arguments makes it seem somewhat reasonable. However, it does not make sense when confronted with facts denying such high levels of risk aversion. Likewise, every attempt to solve the puzzle might appear meaningful in its own sense. However, the following sections will show that classical theory fails in modelling investor preferences which attenuates the credibility of all approaches that rely on classical theory in any way. This follows the guess of Mehra and Prescott (1988) that “perhaps the introduction of some other preference structure will do the job” (p. 133).

2.4 Potential Failures of the Model

After having assessed some approaches to the EPP, we evaluate the major constraints and pitfalls of the model proposed by Mehra and Prescott (1985). As the model is subject to several strong assumptions and previous research findings, it is questionable by how far these assumptions reflect preferences under uncertainty. Therefore, the following sections deal with the assumptions on an agent’s risk aversion as well as the applicability of consumption-based asset pricing to explain the EPP. Furthermore, the axioms of expected utility theory will be identified and assessed.

2.4.1 The Risk Aversion of Investors

The large difference between empirically observed and mathematically predicted equity premium is due to restricting the parameter [illustration not visible in this excerpt] to less than 10. Despite comprehensive research on risk aversion, it is not yet clear why a high [illustration not visible in this excerpt] is not applicable. Equation (14) facilitates the examination of [illustration not visible in this excerpt]. This equation implies that the equity premium rises with an increase in the coefficient of risk aversion. With respect to the empirical data in Table 1, this equation suggests an [illustration not visible in this excerpt] of 45.9. With that being said, we proceed by transforming equation (10) to

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To make the impact of [illustration not visible in this excerpt] on the risk-free rate more vivid we can use an example. Presuming a moderate gross consumption growth of 1.01 and a subjective time preference discount factor of 0.99 as suggested by Mehra and Prescott (2003), equation (15) yields a risk-free rate of 59.5% with the respective [illustration not visible in this excerpt] of 45.9. On the other hand, everything else equal, a consumption decline by 1% results in a risk-free rate of -36.3%. It is obvious that stronger inter-temporal variations in consumption imply even more unrealistic results for [illustration not visible in this excerpt]. In addition, a lower [illustration not visible in this excerpt] would boost the results to even more extremes in the domain of positive consumption growth. Intuitively, no agent would be willing to pay 36.3% if consumption declines by 1%. Even an [illustration not visible in this excerpt] of 10 leads to conspicuously extreme risk-free rates. Apart from its effect on the risk-free rate, a high [illustration not visible in this excerpt] suggests intuitively implausible behavior. For instance, an [illustration not visible in this excerpt] of 10 predicts that an agent is indifferent between a sure $53,991 and a gamble with equally likely chances to either win $50,000 or $100,000. With an [illustration not visible in this excerpt] above 40, the sure gain shrinks to below $51,000 in the case of indifference (Mankiw & Zeldes, 1991, p. 9). Naturally, people are not that risk-averse and an amount of $70,711 for the provided example, which represents an [illustration not visible in this excerpt] of 1, appears more reasonable. Consequently, we can assume that agents’ risk aversion is associated with a risk aversion coefficient between 1 and 10 as initially assumed in the model (Mehra & Prescott, 1985, p. 154).

We should also regard the risk-free rate from the perspective of inter-temporal substitution. As shown in Appendix 2, the elasticity of inter-temporal substitution is the reciprocal of the coefficient of relative risk aversion. Accordingly, a level of risk aversion that explains the equity premium in Mehra and Prescott’s (1985) model implies a very low elasticity of inter-temporal substitution. Hence, risk-averse investors prefer stable instead of growing consumption over time. Because consumption is growing investors demand a high risk-free rate what is demonstrated by equation (15). Thus, the fact that historical interest rates are remarkably low is puzzling in this context (Weil, 1989, p. 406).

As high risk aversion coefficients may not provide a great explanation to investors’ characteristics, other assumptions might assist us in the EPP. Without applying deeper analysis, the restrictions on the parameter [illustration not visible in this excerpt] appear realistic. Accordingly, Mehra and Prescott (1985) may have been right in their input on the risk aversion of investors. Nevertheless, the aim of the model is to reproduce consumption and investment behavior. Therefore, an implausibly high risk aversion and risk-free rate to capture the historical equity premium simply argues for an inability of consumption-based asset pricing to accurately model reality. The next section will examine this issue in more detail.

2.4.2 Consumption-Based Asset Pricing

In the equity premium model asset pricing is based on consumption (Mehra & Prescott, 1985, p. 150) as depicted in section 2.1. Although consumption-based asset pricing may be intuitively appealing, it is still justified to ask if it does indeed apply to reality. According to Campbell and Cochrane (2000), the power utility consumption-based model is weak in its ability to capture asset prices. In comparison to the traditional CAPM, the consumption-based approach is clearly outperformed. Contrary to the assumption of consumption-based asset pricing, real world consumption growth covaries poorly with asset returns. A large equity premium can be explained by raising the coefficient of risk aversion in equation (13). However, offsetting a low covariance through an increase of [illustration not visible in this excerpt] might be a misleading approach to match historical data (Grüne & Semmler, 2008, p. 3256).

[...]

^[1] Throughout this thesis the same notations will be used. For simplicity, the notations are taken over from references as far as consistency is ensured in this thesis. However, upper and lower cases may be assigned to different quantities.