Review of Equity Risk Premium Puzzle

WU Zhi-lin， SONG Hui

（1.Durham University，Durham Howland，AlderDH13DE； 2.Shandong University，Jinan，Shandong 250000）

1 Introduction

It is well-known that stock returns are on average higher than bond returns.One possible interpretation is from traditional asset pricing models（like CAPM） based on the fact that stocks are riskier assets than bonds.However， in reality， stocks returns during historical periods in the U.S.are too high to be captured by these models considering investors'risk aversion，implying other possible mechanisms at work in stock returns.

This puzzle（equity risk premium puzzle） has raised attention of many economists.Mehra and Presccot[1]research uses traditional CAPM model to analyze the American stock market from 1889 to 1994.Weil[2]employs a non-expected utility function.Epstein and Zin[3]use a generalized expected utility function model（GEU） .Constantinides[4]introduces habit formation to cut off the direct correlation between the elasticity of intertemporal substitution and a risk aversion coefficient.Barberis et，al[5]first consider including countercyclical risk aversion into an intertemporal consumption model.Fama and French[6]uses Gorden growth model tocalculate the expected stock return.Kartashova[7]finds that whether the private ERP is reasonable depends on economic prosperityand stock type.

Although many theories have been proposed，no one can present a satisfactory answer for the ERP puzzle.It is useful and necessary to review the theoretical development.This paper is a summary of the extensive studies developed to re-examine the existence of（ERP）puzzle or provide more reasonable explanations.

2 The proposal of puzzles

Mehra and Presccot[1]empirically researched American stock market data from 1889 to 1978.By using S&P 500 to replace the market gains and the fact that treasury rates are seen as risk-free rate，they found that during the period，there is 6.18%equity premium more than which can be explained by using traditional asset pricing model.Mehraand Presccot[1]named this phenomenon as the equity risk premium（ERP） puzzle.The ERP is caused by the fact that stock has much higher risk than bond， therefore， investors would request additional risk compensation for stock.In their research，under the CAPM mode，the corresponding relative risk aversion should reach 26 which is beyond the normal range.Because the highest risk aversion factor is only 10[7]，Campbell etc[8].also repeat the research of Mehra and Presccot[1].Based on the data from 1889 to 1994 in American stock market，they found that by using traditional CAPMmodel to explain，this would require a risk aversion factor of 19，which is still an incredible large number，

Weil[2]is the first one who uses the non-expected utility function preference to adjust investors expected utility theory.In Weil's model，investors'elasticity ofintertemporal substitution is inversely related to the relative risk aversion coefficient.When the relative risk aversion coefficient increases， the elasticity of intertemporal substitution would be lower，which means investors is not willing to postpone the present wealth to future consumption.When there is a high risk premium，it means that investors have a high relative risk aversion coefficient， which requests a low elasticity of intertemporal substitution corresponding to a higher risk-free rates.Therefore，the high risk premium is incompatible with a lowrisk-free rate and cannot explain the equity premium puzzle.Kandel and Stambaugh[9]also documented that risk aversion is higher than conventional acknowledgement.The risk-free interest rate calculated in this theory is much larger than actual risk-free rate.This model can not explain the mystery of the equity premium，and also raises another problem——“mystery ofthe risk-free interest rate”puzzle.

2017-11-21

伍之琳，1994年生，女，湖南怀化人，杜伦大学硕士生，研究方向：金融理论、金融市场；

Until now，the study of ERP theory has not been unanimous.There are mainly two directions of the（ERP） puzzle research.One is based on the traditional model by introducing market friction combined with the reality. Though improving the old model and re-innovation，researchers try to find a new model that can explain the excess ERP.Another is to improve the empirical research method based on historical data，such as using bonds with different structured maturities to estimate the true risk-free interest rate and determine the market earnings.

3 Theoretical explanation development

The first category of explanations includes model's modifications and innovations.

3.1 Explanation ofefficient market hypothesis theory

In the 1920s of US，there has been been a lot of study about the equity premium.At first，Markowitz[10]combined risk and benefit to research assets pricing.After that，a comprehensive and classic CAPM model was contributed by several scholars including Sharpe，Lintner and TreynorMossin.This model was the backbone of the modern financial capital asset pricing model and also the basic model to study the equity premium.After that，most models are improved based on CAPM.

3.2 Modifyingpreferences

Some researchers improved the traditional equilibrium model.According to the problem caused in Weil[2]，if there is no directly correlation between elasticity of intertemporal substitution and relative risk aversion，there might be a solution for those puzzles.Epstein-Zin[3]came up with the generalized expected utilityfunction model（GEU） .He adopted the recursive preference models to establish a relative independent and separate relationship between the elasticity of intertemporal substitution and relative risk aversion.This model separates elasticity of intertemporal substitution and relative risk aversion coefficient.Risk aversion coefficient represents investors'will of increasing the risk offuture investment byreducing the current consumption.The elasticity of intertemporal substitution represents investors'will of transferring their wealth into different period.Thefavoriteoutcomeof thismodifying is individual investors has different attitude towards the risk and their preference consumption style also calculated under independent parameter.In addition，Epstein-Zin model uses market rate of return to replace the return required by investors.However，Kandel and Stambaugh[9]documented that Epstein-Zin model can't absolutely addressthispuzzle，aseven underthismodel，investors'relative risk aversion was still toohigh.

3.3 Habit persistent

Another waytodeal with this problemis to introduce habit formation to cut off the direct correlation between the elasticity of intertemporal substitution and a risk aversion coefficient.Constantinides[4]andCampbell and Coehrane[11]both get accustomed to a internally lifestyle that is habit persistence.Habit formation theory claimed that with the increasing of consumption level and wealth investors would has less risk averse.Constantinides[4]found the differences in per capita consumption growth rate.He thinks that the current level of consumption was affected by earlier consumption， therefore， investors'elasticity of intertemporal substitution cannot directly relate to relative risk aversion coefficient.After including habit formation，Constantinides's model seems to explain the equity premium puzzle and the mystery of low risk-free interest rate.Although this model theoretically explains that the risk premium and low risk-free interest rate can be compatible，this model requires risk aversion coefficient has inverse relationship consumption volatility which is contrarytothe American consumer data.

Campbell and Coehrane[11]also investigated a consumption-based model.Thismodelclaimed an independently consumption growth process and personnel habit.They think the expectation of economic depression would decrease consumption level and increase investors'risk aversion.Therefore，the limited consumption would stimulate people's fanaticism of purchasing bonds and decrease risk free rate.This improvement partially addresses equity premium puzzle，but the estimate of risk aversion coefficient under this model is still higher than reality.

Abel[12]did the further research after Constantinides[4]andCampbell and Coehrane[12].A famous word“Keeping up with Joneses”is from his relative consumption theory.Different fromConstantinides[4]internal habit，Abel came up with the external habit formation model.He set up the model as a power function of the consumption ratio and habit，and calculated the utility by comparing the consumption level ofpresent and past.

3.4 Heterogeneous consumers and Market incomplete market

The improvement of assets pricing model under incomplete market will bring the theory closer to reality.Telmer[13]presented a model that takes heterogeneity of economic man into account.Under market constraint，the endowment of economic man is different，however，the reality market cannot reach the true risk-free rate.But as long as existing a note discount market，brokers can buy and sell bonds to balance their current consumption levels.Therefore，generally there is indifference between the incomplete market and the complete market.Market segmentation also rendered it difficultto find the properly proxy ofindividual consumption growth[14].From the research，they found almost 3/4 investors didn't hold stock and the market segmentation made separate and independent investors facing different conditions and attitudes to trade their assets.Therefore，there is enormous covaries among individual investors.But，the single market segmentation theory is totally not enough to explain equity premium puzzle.

Heaton and Lucas[15]also studied the ERP puzzle under incomplete market.They claimed that when risk diversification is inefficient under transaction friction since agents are difficult to effectively allocate assets，agents will require a corresponding risk compensation for the return of the investment.This is because the market friction and the risk dispersion are real.However，there is a dispute on transaction cost which would lead high stock return.As Aiyagari and Gertler[16]claimed that if using transaction cost to understand this puzzle，the difference between bonds transaction cost and stock transaction cost is the only way could lead high equity premium.Because according to standard financial theory this premiumis compensation for higher transaction cost.

The following literature such as Constantinides and Duffle[17]， Constantinideset， al[18]also studied the explanation of incomplete market for equity premium.Constantinides and Duffle[17]considerate the heterogeneity of consumers since every per individual has different consumer level.This model claimed that when investors suffer the risk of unemployment，as there is no insurance to eliminate the idiosyncratic of income，investors would not hold stocks only if the risk premium is satisfying and permanent. Therefore， heterogeneity is a strong supplement toexplain this puzzle.

A famous announcement by Constantinideset，al[18]claimed that“ Junior can't borrow” combined heterogeneity and incomplete market and built up a three period overlapping generations model.They introduced the life cycle intomodel that believes youngagents'ability of investment is constrained by their wage as their income is uncertain.But those people have the largest desire for holding stock，because their total amount in equity is limited，a portion of equity in the portfolio is a good way to hedge of their risk of unemployment.The middle-age people have relatively stable income，and the earnings from equity can be seen as an additional income for one household，so they have lower cast of investment.Bansal and Yaron[19]assertthatwhen introducing EZW preference and catastrophic events in the meantime，it might properly explain equity premium puzzle.This gives us another view to extend equity premium puzzle researches but it is largely not enough to properly reconcile the existence ofequitypremiumpuzzle.

3.5 Alternative explanations

In the recent years，the research of equity premium puzzle increasinglycombined with realityand blamingthe puzzle is caused by financial data，hence the empirical research suffers the problem of non-normality.This is reasonable because this requests a relatively long period data that underwent financial crisis，civil wars or bank panics which is hard tofind in reality.

Rietz[20]found that serious catastrophic events like wars and natural disasters would increase the gap between risk-free interest rates and stock returns.This is similar to peso problem byKrasker[21]，when take economic catastrophe into account that serious depression would exterminate all stock value，investors would discount the true value of equity.However，the reality data is extremely limited.After the financial crisis of 2008，there is another flurry of empirical research of ERP puzzle usingthis theory.

Although the smallprobability eventcannot completely solve ERP puzzle，it provides another perspective to do research，such as introducing survivor bias to explain the equity premium puzzle.In empirical study，historical data is generally used to verify the accuracy of the model.When we use the asset pricing model to predict stock returns，the bankrupted and delisted company shares were ignored on the calculation of average market yield.They think that American financial crisis would eliminate the poor-performance companies and companies that has grim future.The remaining survivors will improve the average performance of the sample period stock.Therefore，survivorship bias may lead to a long-term risk premium for stocks on the data.Li H and Xu Y[22]also proposed the same opinion.Bansal and Coleman[23]calculated a monetary model that differentiates money from other assets.This model claimed that Treasury bonds which is used to measure the risk free rates do not have the convenience of currency trading.Under this model，a very low risk-free rate and a portion of the liquidity premium may result in a high return on equity.But the current deposit rate used to measure the liquidity premium has changed every second in realityand keeps zerofor a longtime in America.

Based on the model present by Bansal and Yaron[19]，Collin et，al[24]revised the measurement of equity premium after studying the relationship between the observed credit spreads and the implied credit spreads of the model.

3.6 Explanation ofbehavioral finance theory

The explanation of behavioral finance for equity risk premium mainly includes prospect theory[25-26]，shortsighted risk aversion and disappointment aversion theory.Different fromclassical function[27]behavior finance theory shows us a new perspective to unravel equity premium puzzle.Based on the selection of different reference points，behavior finance calculated the utility by losses and gains rather than the absolute wealth.Barberis and Thaler[28]believe the correlation of reference points based on the prospect theoryand ambiguityaversion can explain investors'extra worries ofstocks.

Kahneman and Tversky[25]documented the prospect theory.This theory believes agent's utility function would change when it is close to the reference point.The negative feelings from loss is overwhelmingly larger than the pleasant from profitability.Therefore，only a satisfied stock return rate would attract investors bearingadditional risk on stock.Campbell et，al[29]claimed that under the theoryof ambiguity aversion，the asset pricing model is a biased，so investors will demand a higher risk premium tocompensate the ambiguityaversion return on equity.

Based on two hypotheses that prospect theory of risk aversion and mental accounts，Benartzi and Thaler[26]proposed the short-term risk aversion（myopic loss aversion，MLA） .Benartzi and Thaler[26]are alsothe first combining behavior finance with equity premium puzzle.This theory believes long-term investors would frequently revaluate asset portfolio within a short time，and then choose the one with the highest expected utility.In a short term，the stock always fluctuatesmore violence comparing to bond.Therefore，there is a larger risk premium on stock investment in the long term.Zervoudi and Spyrou[30]employed several value functions compared with the MLA，and found that the investors would be sensitive toassets allocation duringfinancial crisis.

Barberiset，al[5]gave the first try to consider including countercyclical risk aversion into an intertemporal consumption model which is closer to reality. Barberiset， al[5]constructed a dynamic equilibrium model（BHS） also based on the prospect theory.This model extends their investment period from single-period to multiple periods.Considering wealth change from the previous investment，investors'risk aversion coefficient would followthis change.That means when the wealth increases，this investor's risk aversion would decrease， butwheneverthere are losses，investors'risk aversion coefficient will sharplyincrease.

Among the following researches，behavior finance keeps the mainstream to explain the equity premium puzzle.There is a lot other correction for the model，such as Ang et，al[31]arguing that an independent and isolated risk aversion coefficientisunsufficientto explain investors expectations of the stock profit，so they also added disappointment aversion coefficient correction model.Gollier[32]introduced a model of vague aversion，showing that investor aversion to uncertainty would require a high risk premium as compensation.Jouini and Napp[33]claimed that the extremely pessimism attitude render investors extra afraid of stocks which would lead to a high REP.

4 The empirical research

Another literature analyzes ERP puzzle from the perspective of empirical study.There is a difference between ex-post and ex-ante based on the method of calculating the the return rate.Most studies like Mehra and Presccot[1]， Fase[34]， Dimson et al[35].use the historical data.But increasing scholars found the realized return and expected return are different.The ex-ante studies compute the return of stock by dividend yield，growth and the fluctuation in valuation.

4.1 Ex-post

Campbell[36]claimed that the sample period choose does not effect the excess equity premium.As from using the post-war period data，to reaching the calibration in this model，large excess premium and relatively small consumption variance request an incredible higher risk aversion.

Fase[34]investigated among international countries.By comparing US data with European countries in the post-war period，he found the level of risk aversion need to be higher in France， Germany， Belgium，Netherland and UK than in US.This is also confirmed by Campbell[36]and Dimson et al[35].Campbell and Yogo[37]extended the study among 12 developed countries，which shows most areas'market exists ERP puzzle，but Australia and Italy only present 1.486%and 0.797%equity premium.Salomons and Grootveld[38]found ERP does not only exist in developed countries but also in emergingmarket.

Kartashova[7]specializedly analyzed the private equitypremiumpuzzle.He found that the the private ERP is reasonable during the economic prosperity period 1990s but after the financial recession 2008，the discrepancy between public and private equities are largely shrunken.There is obviously difference between private stock and public stock performance.Therefore，there is necessary to investigate ERP puzzle independent on stock type.

Bellelaha，et.al[39]empirically examined the ERP of France stock market in six sector after the financial crisis.In most literature，the high systematic risk is the major element of the explanation of excess premium.Based on the data of 2003 to 2012，they found that the excessive volatility of market would continuously and increasingly add idiosyncratic risk during the post-war period.Therefore，the excess return of stock is kind of a compensation for the enormous loss in financial crisis.

4.2 Ex-ante

Considerating the dividend yield，Fama and French[6]use Gorden growth model to calculate the expected stock return.This model produced an unbiased estimate of the long term expected stock return according to the hypothesis of relevant stochastic processes.Because when the research covers a relative long time，the processes of the dividend growth rate and ratio of capital gains would cointegrate.In addition，theyclaimed that in the pre-war era the expected return on equity is approaching to the actual return on equity.In their research from 1871 to 1945 the expected and actual equity premium is around 3.5%.But in the post-war period the actual equity premium has reached 6，the expected equity premium is still 3.7%.Fama and French[6]explained it as the unexpected gains in the post war period.

Salomons[40]repeated the Fama and French[6]'s research and extended to the sample countries.They confirmed that the volatility of expected return of stock is much higher than the counterpart of actual return which means from the long term perspective the estimation of return on equity based on historical data is more precise.This is because the world war tremendously changed the expected and actual price ofequity.

Salomons[41]compared other major stock market with American stock market in a long sample period.It is obviously that the US stock investors are most profitable above expectation.Instead of using historical growth data，this model applied the predictable valuation ratios as an indication of expected stock return.He concluded that the ERP is a long period notion where the intrinsic risk is always hidden behind.Algaba and Boudt[42]also assert this method which use generalized financial ratios to forecast stock return. They use Generalized Price-Dividend Ratio（GPDR） method to predict ERP.In their theory，the changes of market structure and macroeconomic condition would influent the distribution ofdividend and equity premium.Therefore，the statisticaland economicalbeneficialcan improve accuracyofREP prediction.

5 Conclusion and Forecasting

Since the propose of ERP puzzle，there are a lot of literatures explaining this puzzle with their innovated models and theories，or empirically examining the robust of ERP puzzle，but we still didn't get consensus on the conclusion.Decades years past，the research of ERP puzzle is still in progress and still earns our special attention.This is not just because there is no satisfactory theory and model to properly explain the puzzle，but the research of ERP puzzle helps us understand that stock price is a comprehensive present of the financial market guided by assets pricing theory， macroeconomics，psychology and demographics[41].This essay extended the previous literature reviews on ERP puzzle and introduced an unfamiliar perspective in empirical research which gradually became a mainstream of ERP puzzle study.The examination of ERP is based on a framework of valuation of market and stock which improved the long-term prediction of stock pricing.There are still a lot of questions around ERP puzzle to explore.Which asset pricing model is the most precise one?If behavior theory can properly explain all investors behavior including ERP?IfERP puzzle exists or just exists in some market?

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