Excerpt

## Table of contents

1 Introduction

2 The Equity Premium Puzzle

2.1 Investigation from a rational perspective

2.2 Investigation from a behavioral perspective

3 Empirical evidence

4 Discussion

5 Conclusion

References

## 1 Introduction

As Siegel and Tahler stated in 1997, Equity Premium Puzzle (the puzzle) is a rare bird among economic anomalies. Indeed, the puzzle has challenged the adequacy of explanation provided by rational as well as behavioral economics. For the time being, no com- plete solution for the puzzle has been accepted, and as we shall see in this work, it covers a wide range of puzzling and unanswered questions.

The discrepancy between the return of stocks and a risk-free investment, also known as the risk premium compensating investors for bearing risk, has been roughly 6% over the ninety-year period 1889-1978. While the average return on the risk-free investment has been less than one percent, the average return on stocks has been 7%.

The question why the average return on stocks is so high, and the average return on risk-free investments is so low, leading to an equity premium of 6%, has resulted in the puzzle of the same name, and leads to the following research question:

*Can behavioral economics contribute the explanation of the Equity Premium Puzzle, and which solution(s) do rational economics suggest?*

As the research question implies, this work will examine solutions of the puzzle sug- gested from both the field of rational economics as well as behavioral economics. Due to a large number of various approaches to explain the puzzle over the past 30 years, this work will concentrate on a selected number of suggestions from various econo- mists. I will examine assumptions and conditions of the solutions suggested, and ana- lyze key theories applied in the resolutions. The quantitative foundation of models ana- lyzed in this work will not be examined, as this would require detailed statistical and mathematical considerations.

The work is structured as follows. I will to begin with outline the initial puzzle stated by Mehra and Prescott in 1985 and analyze key assumptions of their equilibrium model. Different resolutions of the puzzle suggested by rational and behavioral economics will then be examined, followed by an investigation of the empirical evidence behind the proposed solutions. A discussion will follow, and the author will reflect on the robust- ness and relevance of the puzzle. A conclusion will sum up the key elements of the work.

## 2 The Equity Premium Puzzle

This section will examine the initial equilibrium model to calculate return on equity and risk-free investments suggested by Mehra and Prescott (1985). Key assumptions behind and the robustness of the model will be analyzed, and the subsections 2.1 and 2.2 will examine solutions proposed by rational and behavioral economics respectively.

In 1985, economists R. Mehra and E.C. Prescott suggested an equilibrium model with the purpose to calculate the return of equity and risk-free investments observed in data over a ninety-year period. Table 1 shows key data from the original research examined (Mehra and Prescott, 1985), which the model was supposed to replicate:

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Table 1: Average return of risky and risk-free securities, risk premium and consumption growth rate, 1889-1978. Mehra and Prescott, 1985, page 147

For the computation of their model, Mehra and Prescott considered three key assump- tions.

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The first assumption captures the consumption of the single representative household in the frictionless economy considered in the model. Preferences are ordered over random consumptions paths captured in (1), with the term *ct* representing per capita consump- tion and ß is the subjective discount factor. Consumers are assumed to discount future values by ß, and the parameter expresses the utility of future consumption and hence time preferences of consumers. The utility represented by the concave function *U* ap- plies for the representative agent, hence consumers share and will seek to maximize the same utility function.

The following process applies to the utility function:

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The parameter *a* captures the constant relative risk aversion (CRRA) of the consumer, and measures the curvature of the utility function. Mehra and Prescott argue that *a* can be restricted to assume values between one and 10, and should approximately be one, meaning consumers display neutral risk preferences and will prefer to smooth consump- tion between time periods. This restriction of *a* will be discussed later in this work.

Consumers are assumed to be risk averse and will want consumption to be similar in different states and at different dates. Hence, consumers dislike risk, which implies time- and state-separability of preferences, meaning that consumption in one state and/or at one date does not depend on consumption in previous states and or/dates. This assumption will be discussed later in the work.

Finally, markets are assumed to be complete and there is no transaction costs associated with buying either stocks or bonds. To sum up, the representative agents of the friction- less economy gain utility by preferences given by (1) and the economy in which agents act (buy and sell investments) is assumed to be complete.

Mehra and Prescott (1985) analyze a two-state theoretical framework, where the economy is either in a good or in a bad state. In good states with high consumption and thus low marginal utility (that is, the consumer will not experience a great gain on a further consumption unit), assets that covary positively with consumption will yield a high premium and thus have a destabilizing effect on consumption (Mehra, 2003). In bad states, the opposite will apply, and assets will yield a low premium.

Given the assumptions of the model and the estimation of consumption processes (two- state), the largest premium obtainable when varying the preference parameters *a* and *ß* within the given restrictions is 0.35% (Mehra and Prescott, 1985). The result of the model is inconsistent with the observed premium of roughly 6%, and thus the model does not deliver a consistent result under the restrictions and assumptions applicable. The robustness of the puzzle is confirmed by accounting for different possible errors, increasing the equity premium to 0.39%, 0.4 pp. higher than the initially calculated premium. The calculation of the premium is further concluded to show insensitivity to persistence in the consumption growth rate, inflation as well as implications from taxation (Mehra and Prescott, 1985). Furthermore, the premium showed insensitivity to measurement of higher moments of the risk distribution.

Though the Equity Premium Puzzle is proven to show strong robustness, other econo- mists both within the rational as well as behavioral economics have suggested solutions to the puzzle, and suggestions will be examined in the following.

### 2.1 Investigation from a rational perspective

Due to the Mehra and Prescott (1985) two-state specification, the economy is in either a good or a bad state, not accounting for disaster-like states such as an economic market crash. However, taking low-probability, disaster-like states into account in the suggest- ed equilibrium model is assumed to solve the Equity Premium Puzzle (Rietz, 1987). The effects of a market crash are captured in this third state, allowing the same condi- 3 tions and restrictions of the model suggested by Mehra and Prescott to apply. Rietz (1987) argues that the occurrence of a probable (but unlikely) market crash will affect all future consumption levels of the consumer, and that consumers will demand a high return on risky investments in order to compensate for losses that may occur during a crash. Whether an economic collapse did occur in the data examined over the period 1889-1978 and could be considered a disaster-like state will be discussed later in this work. For the case that the output in the economy considered should decrease by 50% of its normal value during a crash, the model considering a third state yields a risk premium near to the observed. The results of the model calibrations thus yield a risk premium of 6.15%, a risk-free rate of 0.89% and preference parameters of *a* = 5.30 and *ß* = 0.98. The results comply with the restrictions on the parameters *a* and *ß* imposed by Mehra and Prescott. The incorporation of the disaster-like third state as described by Rietz (1987) does not alter the original model presented by Mehra and Prescott, but ra- ther adds a further feature to the model.

Another approach to resolve the puzzle has been allowing modifications of the preference structure of the representative agent, thereby relaxing the assumption of stationari- ty and time- and state-separability of risk preferences.

Epstein and Zin (1990) argue that a model containing preference parameters exhibiting first-order-risk aversion can solve the puzzle, and further show how this preference structure contradicts the expected utility theory. In their model, the representative agent acts in the same economic environment as considered in the model of Mehra and Pres- cott, but the risk premium is assumed to be proportional to the standard deviation of the gamble facing consumers (in this case, consumption). In an expected utility model, the risk premium would be proportional to the variance of a gamble, thus implying second- order risk aversion. Hence, Epstein and Zin assume agents will maximize a non- expected utility function, with risk preferences exhibiting first-order risk aversion (i.e. sensitivity to the standard deviation). The standard deviation is larger than the variance for small risks, which implies that the equity premium observed in data can be rational- ized when consumers exhibit first-order risk aversion in their risk preferences. Calibra- tion of preferences yield an equity premium of 2%, roughly 4 p.p. lower that the premium observed in data, but however larger than the maximum premium calibrated apply- ing the Mehra and Prescott model. The modification of utility and risk preferences as- sumed in the original equilibrium model thus yields a result closer to the observed data. The introduction of non-expected utility preferences as described above as a suggestion to resolve the puzzle was further specified by Weil (1989). Besides arguing in favor of the relaxation of the expected, time-additive utility restriction (by incorporating non- expected, nonlinear utility functions), Weil suggests independent parameterization of the CRRA and the coefficient of intertemporal substitution can partly explain the low level of the risk-free rate and thus the discrepancy between the risk-free rate and the return on equity (Weil, 1989). This suggestion allows the separate modeling of consumer attitudes towards risk and towards growth. In the Mehra and Prescott model, the elas- ticity of the intertemporal substitution is limited be the inverse of the CRRA. This as- sumption implies that a high level of CRRA is synonymous with a low value for the elasticity of intertemporal substitution. The latter would suggest a high level of the riskfree rate, which is inconsistent with the data. Hence, splitting the two parameters and then calibrating the model yields a return of the risk-free rate of 0.85% and a risk premium of 5.72% (Weil, 1989). However, this result assumes a CRRA of 45 and a coeffi- cient of intertemporal substitution of 0.10 (note the latter is not the inverse of the CRRA, since this value would have been 1/45 = *0.022).* The high level of risk aversion a low value of intertemporal substitution thus leads to a puzzle why the level of the risk-free rate is so low. The emerging puzzle, also known as the Risk-Free Rate Puzzle, will briefly be discussed later in this work.

The suggestion that modifications of the preference structures, thereby incorporating time-nonseparable utility functions, can contribute to the resolution of the puzzle was further examined by Constantinides (1987). Relaxation of time-separability, also known as habit formation, suggests that present consumption (partly) depends on previous con- sumption, hence preferences in different consumptions periods are non-separable. The incorporation of habit formation provides independent parameterization of the CRRA and intertemporal substitution, as also suggested by Weil (1989).

Allowing time-non-separability yields a fair replication of the first moments (mean and variance) of the data observed, producing an average consumption growth rate of 1.8% (for comparison, see table one page two) at a CRRA level of 2.81. However, in order to explain the mean premium, past consumption must generate about 80% of the present level of consumption, and the model thus implies strict time non-separability of utility preferences.

Kocherlakota (1996) suggests alternative solutions of the Equity Premium Puzzle based on the rationales that consumers are either highly risk averse to consumption risk or perceive the trading of stocks significantly more costly than trading bonds. Firstly, a modification of the preference structures (as previously seen from Weil 1989; Constan- tinides, 1987; Epstein and Zin 1990), disentangling the CRRA and elasticity of intertemporal substitution is assumed to be more suitable for consumer preferences, allowing utility functions to be complementary (non-separable). This assumption abandons the expected utility restriction imposed on preferences in the Mehra and Prescott model. In extension to time non-separability, the demand of a high premium can be explained as a result of relative consumption, assuming that the individual utility function is not a function of individual consumption but also of societal consumption (Kocherlakota, 1996; Abel, 1990). The individual will thus not only derive his attitude toward con- sumption risk from his own consumption, but also from the societal attitude towards variability in consumption.

The initial model assumes completeness of markets, where consumers can diversify away unsystematic risk. However, realistic market conditions restrain consumers from insuring completely against consumption risk, implying incomplete markets (Kocherla- kota, 1996). The Mehra and Prescott model further assumes that there are no market frictions, and individuals are not constrained by trading and transactions costs in respect to trading securities. However, accounting for market frictions such as costs associated with trading securities is considered a plausible explanation of the premium given consumers perceive the costs of trading stocks significantly higher than risk-free investments.

Finally, allowing consumer preferences to not be represented by a “representative agent” by introducing heterogenic preferences and utility functions is suggested as a possible solution of the puzzle (Weil, 1989).

### 2.2 Investigation from a behavioral perspective

As described in the previous subsection, rational economics have suggested several ap- proaches to resolve the Equity Premium Puzzle. However, different approaches from behavioral economics suggest alternative explanations. This section will examine se- lected approaches, concentrating on myopia and concepts from the Prospect Theory.

In Prospect Theory, utility is defined over gains and losses rather than wealth in absolute terms (Kahneman and Tversky, 1979). This leads to a redefinition of the utility function of the following form:

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As the function suggests, xa applies for prospects yielding a gain, whereas *—X(* —x) applies for prospects which yield a loss to the consumer. The coefficient of loss aversion is parameterized by Ä. The coefficient implies that consumers will be risk averse in the domain of losses, and the behavior in the respective domains are captured in the value function:

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Figure 1: The value function. Kahneman and Tversky 1979, page 279 As visually shown, the utility function is concave in the domain of gains and convex in the domain of losses, suggesting that losses are more painful for consumers that gains are pleasurable (Kahnemann and Tversky, 1979). The non-linearity of the utility function thereby violates the assumption of linear expected utility preferences. Rather than defining utility over different levels of wealth, Prospect Theory suggests that consumers code prospects as gains or losses relative to a reference point, and then choose the highest/lowest expected value (for gains/losses) based on the coding and associated value of the prospect. Value is thus attached to changes rather than absolute states. It is to be mentioned that prospects are divided into riskless and risky compo- nents, where the evaluation of the risky component is found relevant to the evaluation of the risk premium. The derived utility is then given based on the position serving as a reference point (e.g. current wealth) and the magnitude of change from this point (e.g. loss of 10% of current wealth).

Furthermore, the value of prospects is perceived due to the weight attached to the out- come, not (necessarily) to the probability. Hence, the assumption that utility can be de- rived from expected value of a prospect based on the probability of the outcome is vio- lated in the present theory. The incorporation of a decision weight denoted by result- ing in the weighted probability 'n(p) is therefore not a probability measure, but rather the (subjective) weight assigned to the risky component of a prospect. The weighted utility of a prospect is relevant when consumers are assumed to not obey the rationality axioms given by expected utility, resulting in irrational behavior.

**[...]**

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- Anonymous, 2018, Behavioral Economics and the Equity Premium Puzzle. Which Solutions Do Rational Economist Suggest?, Munich, GRIN Verlag, https://www.grin.com/document/1010174

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