As the improvement of economic theory and empirical research, there are a large number of anomalies challenging the classical theories. The Equity Premium Puzzle is one of these anomalies, which was declared by Mehra and Prescott (1985). They found out there is a really high equity premium over 6% in the past 100 years in U.S. stock market. To explain this phenomenon, they used the Capital Asset Pricing Model (CAPM) to analysis the average annual equity premium. The result showed that the classical model cannot explain such a high equity premium under the normal relative risk aversion. If the equity premium must be corresponded with the real data , then investors' relative risk aversion in the model have to reach 30 to 40 (Siegel and Thaler, 1997). This is implausible. This large difference between theory and reality is called "The Equity Premium Puzzle". There are numerous economists tried to solve this question with various explanations, like Survivorship Bias (Brown et al., 1995), Habit Formation (Costantinides, 1990) and so on, but most of these theories still cannot analysis the equity premium puzzle reasonably. Nowadays, a growing number of economists suggest the behavioral finance can explain the large equity premium more efficiently. The behavioral finance explains two main reasons: Myopic Loss Aversion (Benartzi and Thaler, 1995) and Disappointment Aversion (Ang et al., 2005). This paper has been divided into three parts. It begins with the background of the equity premium puzzle and then it will go on to talk about why the traditional approaches such as survivorship bias and habit formation are not accurate in explaining the puzzle, the third part discusses how behavioral finance explains the question.
The equity premium puzzle was pointed out by Mehra and Prescott (1985). They realized that the U.S. sock returns extraordinary larger than one year treasury bills over a hundred years. As table 1 illustrates, in different period of the history in America, the equity premiums existed and were conspicuous, especially in the twentieth century, the premium reached to 8%.
Table . U.S. Return 1802-2000
Mean Real Return (%)
Period
Market Index
Relatively Riskless Security
Risk Premium (%)
1802-1998
7.0
2.9
4.1
1889-2000
7.9
1.0
6.9
1926-2000
8.7
0.7
8.0
1947-2000
8.4
0.6
7.8
Sources: "The Equity Premium: Why Is It a Puzzle?" (Mehra 2003: 55)
Furthermore, as Mehra's (2003) study emphasizes, such anomaly not only exist in U.S. financial markets but also in lots of other countries' markets, which have a similar high level of risk premium. It can seem from table 2 that there were also high premium in British, Japan, Germany and France, particularly in Germany and France, their historical data of equity premium both larger than 6%. As a result, the equity premium puzzle normally exists in other developed countries.
Table . Returns for Selected Countries, 1947-98
Mean Real Return (%)
Country
Period
Market Index
Relative Riskless Security
Risk Premium (%)
Untied Kingdom
1947-99
5.7
1.1
4.6
Japan
1970-99
4.7
1.4
3.3
Germany
1978-97
9.8
3.2
6.6
France
1973-98
9.0
2.7
6.3
Sources: "The Equity Premium: Why Is It a Puzzle?" (Mehra 2003: 55)
As Mehra and Prescott (1985) state, suppose the changes of consumption of investors is a Markfov process and investors' objectives are always maximizing expected utility under a standard general equilibrium model, so the expected utility function of investors can be:
, (1)
Where is per capita consumption level, is subjective time discount factor, describes investors' consumption of time preference, if is low, then means investors have a strong preference in present. Moreover, they confined the utility function (1) as Constant Relative Risk Aversion Utility Function (CRRA):
(2)
Where is the coefficient of relative risk aversion, it directly charge the curvature of the utility function, the reasons they chose CRRA utility function are two: one is the scale is constant, maintains the yield rate stable during the total asset price rise. Another one is CRRA utility function made investors' utility functions have nothing to do with their original wealth distribution. However, this kind of utility function has a limitation, which is it connected risk preferences with time preferences. Then made the relative risk aversion become the reciprocal of elasticity of intertemporal substitution. Basic on the previous formulas, find the derivation of stock and bond's assets expected utility functions separately and come up with:
(3)
Where is the stock return on asset, is risk-free rate, x is consumption increase rate, is variance of . From formula (3), we can calculate the relative risk aversion coefficient . Mehra and Prescott (1985) report the data on the economy in the U.S. between 1889 to 1978 as a sample shows in table 3.
From formula (3), Mehra and Prescot (2003) observe the relative risk aversion coefficient is 45.91, but the normal risk aversion coefficient should below 10, meanwhile, this means the margin value of every dollar will rise 45.91% when the consumption of investors drop 1%, this unexplainable high equity premium called " equity premium puzzle" (Mehra and Prescott, 1985).
Table . U.S. Economy Sample Statistics, 1889-1978
Mean risk free rate
1.008
Mean return on equity
1.0698
Mean growth rate of consumption
1.018
Standard deviation of the growth rate of consumption
0.036
Mean equity premium -
0.0618
Sources: "The Equity Premium in Retrospect" (Mehra and Prescott 2003: 25).
Survivorship Bias is one of the explanations of the equity premium puzzle. As Reitz (1988) suggests, the existence of catastrophic events will increase the gap between risk-free rate and equity returns, even though the probability of the catastrophic events is very small. As equity returns are much higher than the risk-free rate, then generate a large equity premium. However, this explanation is hard to test. As Siegel and Thaler (1997) state, the study of Mehra and Prescott (1985) is between 1802 to 2000, there was a catastrophic event: a Great Depression which happened in 1929-1933. It is clear from table 1 there are four time periods: 1802-1998, 1889-2000, 1926-2000 and 1947-2000, the first three periods are contain the Great Depression whereas the last one is not, if survivorship bias is correct, then we can understand why the first three risk premiums are so high, but after several decades of the catastrophe, the risk premium was 7.8, still very large. What's more, Brown et al. (1995) emphasize that the inferior stocks will quit and the best-performing stocks will survive. All the stocks that Mehra and Prescott (1985) measure are best stocks, then the average rate of return will be overvalued. That means the risk of stock is undervalued and then the equity premium should lower. However, the precondition of this view is the effect of stock market after the financial crisis is different from the bond market, but it is obvious that both risk-free investment and stock may be a huge loss during the financial crisis (Mehra and Prescott, 2003). Clearly, survivorship bias cannot explain the equity premium.
Habit formation is seen as another resolution of the equity premium puzzle, but there still no convincing evidence to prove this. Constantinides (1990) states, the utilities of investors are not only deciding by current purchase but also the difference between current and previous consumption (habit), so previous consumption will affect current utility and current consumption will affect later utility. Moreover, as Campbell and Cochrane (1999) note that if the habit is larger, past consumption has a stronger effect on current utility, relatively current purchase brings less utility. According to their research, when the probability of economic recession increases, the risk aversion of investors will rise significantly, hence require a higher risk premium. At the same time, the demand of bonds is increasing cause the higher risk aversion, so risk-free rate will decrease. Although this model conforms to the data of consumption and financial market, the risk aversion of investors probably different from the prediction.
Campbell and Cochrane's (1999) study is from the outside environment to consider the change of investors' risk aversion, they ignored the inside changing of investors. One of the assumptions of the capital asset pricing model (CAPM) is all investors use the same manner of analysis securities and has the same point of view seeing the economic society (Bodie et al., 2011). Constantinides et al. (2002) construct an infinite-horizon model, they separated investors into three groups: youth, middle-aged and old-aged. In general speaking, youths are the poorest, old-aged are the richest and the wealth level of the middle-aged is between them. Their model supposes borrowing is restricted. The consumption source of investors divides into salary income (pension) and stock income. The pension of old-aged is certain, if someone invests in stock, his consumption volatility are all come from stock income changes. So the old-aged risk aversion is the highest, then comes the middle-aged and the risk aversion of youth is the lowest. In addition, as Lascu et al. (1997) point out that men have a lower risk aversion than women, one determine factor is self-esteem, men has higher self-esteem than women, on the other hand, men would like to take more risks because they have high expectations of success. Overall, the risk aversion of investors is difficult to predict, without a regular and predictable risk aversion, habit formation cannot explain the equity premium puzzle as well.
The classical theories discuss above both lack of reasonable effectiveness, the following paper will describe more reasonable explanations of the equity premium puzzle basic on the point of view of behavioral finance. There are two main explanations: myopic loss aversion (Benartzi and Thaler, 1995) and disappointment aversion (Ang et al., 2005). First, myopic loss aversion, it contains of two thoughts: loss aversion and frequent periodic performance evaluation. Kahneman and Tversky (1979) give an example: suppose there are two choices, one is loss $7500, another is 75% of probability loss $10000, 25% of probability no loss. Their research found out that most of people choose the latter one. This is loss aversion and its utility function is a quasi linear piecewise function:
(4)
Here, we can construct an example to explain. Suppose an investor take part in an investment, the cash flow as the follow table 4:
Table . The cash flow of the investment in future two years
Year 1
Year 2
Season
1
2
3
4
Total
1
2
3
4
Total
Cash flow
2000
-1000
1500
--800
1700
-1000
1500
3000
-500
3000
If the investor use seasonal performance evaluation method, the total utilities after two years is:
However, if the investor change to evaluate once a year, then total utilities will change:
It is clear that the latter method of evaluation generates higher utilities than the former. Although this is the same investment, the investor only changes the evaluate cycle, then the result is totally different, thus it can be seen that evaluate cycle is very important under the loss aversion. Additionally, Siegel and Thaler (1997) emphasize that investors are more sensitive to losses than gains, if investors facing the same level of losses and gains, the pain of losses is much more than the happiness of gains. Back to the equity premium puzzle, the high risk premium in U.S. stock market caused by both investors' loss aversion and frequent periodic performance evaluation. Because investors maybe exist loss aversion to their investment portfolios, they will pay more attention to the security of the portfolios. Such an attention makes them evaluate the performance frequently, so that they can charge the changes of the value of the securities they have. As Barberis and Huang (2001) state that investors' utilities not only decided by consumption but also the volatility of the financial asset they hold. It is known that one of the characteristics of stocks is volatility, the probability of stocks' temporary losses always higher than the probability of bonds' losses (Ruslan and Andrey, 2009). As a result, investors can feel much more about the loss in the stock market and the utilities are getting lower, so the attraction of stocks to investors will decrease. Therefore, only if the long-term average profit of the stock maintain at a high level, stock and bond can be seen as alternative for investors. In other words, under the condition of myopic loss aversion, we need the high risk premium to help balance the stock and bond.
In terms of disappointment aversion, Ang et al. (2005) explain the high risk premium in U.S. market based on this theory. Disappointment aversion is very similar to myopic loss aversion, they both explain the equity premium puzzle from the behavior of investors. It was first pointed out by Gul (1991), the theory indicates that there is an expected value which computed by utility function is the standard of measure gain or loss, if investors' final profits lower (higher) than this value, it will cause disappoint (satisfy). The traditional utility function is:
(5)
(6)
represents the relative risk aversion coefficient of investors, it is indirect decides the level of investors' risk aversion, W is wealth, is initial wealth and a is the proportion of risky assets in initial wealth. Actually under the classical theory, investors' assets are determined by three factors: risky asset income (y), risk-free rate (r) and relative risk aversion of investors. Since risky asset income and risk-free rate are both decide by the objective market, so relative risk aversion is the only subjective factor. However, this kind of explanation is too simple and cannot resolve the risk premium puzzle. Under the traditional analytical framework, Ang et al. (2005) revised the utility function, added disappointment factor:
(7)
A is the coefficient of disappointment aversion and A, is certainty equivalent. We can see from the formula (7), there are two parts of the utility: (a) When the wealth lower than , investors feel disappoint and present by . (b) When the wealth higher than investors feel satisfy and present by . Moreover, when 0 means increasing utility from satisfy is less than decreasing utility from equivalent disappoint. Therefore, investors have disappointment aversion. And the determined factors on final portfolio up to five: beside the first three factors, disappointment aversion coefficient A and reference variable were added. Thus, the subjective factors became to three: relative risk aversion coefficient , disappointment aversion coefficient A and the deviation between wealth and reference level . The dimension of A decides the different utilities during disappoint and satisfy (Ang et al., 2005). The lower A, the disappointment of investors is higher, similar to loss aversion, the utility loss of disappointment is much more than the utility increase of satisfaction. Again, the volatility of the stock market is large, so the present gains are easily deviating the reference, the larger negative deviation, investors feel more disappointed about stocks. As a consequence, investors will hold less stocks. To balance the stock and bond, stock have to maintain at a high level of returns. And the high returns come with high risks (Campbell, 1996). This explains the high equity premium in the U.S. market.
Based on the above analysis, myopic loss aversion and disappointment aversion are similar, but as Fielding and Stracca (2007) remark, disappointment aversion can explain the risk premium puzzle better than loss aversion in that it can adjust various time range. As figure 1 shows, there including two sample periods 1881-2001 and 1926-2001, the overlap part was used by Benartzi and Thaler (1995) and the rest part was from Fielding and Stracca (2007), beside they did not use the same methodology too.
Figure . Loss aversion and time horizon
Source: "Myopic loss aversion, disappointment aversion, and the equity premium puzzle" (Fielding and Stracca 2007:262)
The results of loss aversion both turned out to very near 2.25 when h=1, but with longer time horizons , loss aversion parameter continues to rise, especially the last 5 years it ascended rapidly, after 10 years, it was about 25 or even over 25. In fact, the parameter only reasonable in the first 3 year horizon, as well as Kahnneman and Tversky (2000) suggest that loss aversion is a small number. So the economists have come out formula (4), another simple example to understand, use the data of table 4 again, this time we have changed the loss aversion parameter to 25, use seasonal performance evaluation method again, the total utilities after two years is:
It is absurd. Thus, loss aversion can only solve the equity premium puzzle in a short time, it is myopic.
Figure . Disappointment aversion and time horizon
Source: "Myopic loss aversion, disappointment aversion, and the equity premium puzzle" (Fielding and Stracca 2007:263)
In figure 2, we analyze disappointment aversion in the same way, by contrast, the results in both sample periods are near 1 in short time, and steadily grow to about 2 in the long time. It is more reasonable. Therefore, both two behavioral explanations can solve the puzzle, but consider the time horizon, disappointment aversion is the prefect choice.
In conclusion, the equity premium puzzle as an anomaly have been bothering economists for a long time, the rational relative risk aversion coefficient cannot explain the high equity premium phenomenon in U.S. stock market during the past 100 years. Although there are a large number of traditional theories try to explain and solve the puzzle, still no one is satisfied. Such as survivorship bias (Brown et al., 1995) and habit formation (Constantinides, 1990). Survivorship bias is not correct because both risk-free investment and stock may be a huge loss during the financial crisis (Mehra and Prescott, 2003), also the risk aversion of investors is difficult to predict, without a regular and predictable risk aversion, habit formation cannot explain the equity premium puzzle as well. However, behavioral finance brings a new dawn to the equity premium puzzle. The main two behavioral finance: myopic loss aversion (Benartzi and Thaler, 1995) and disappointment aversion (Ang et al., 2005) explain the high premium reasonably and effectively from different aspects. High risk premiums can attract investors to hold stocks and keep the balance between stocks and bonds. Besides, based on loss aversion and disappointment aversion, Fielding and Stracca (2007) have an empirical analysis on time horizon in the U.S. stock market, the result shows that disappointment aversion is the perfect explanation.
