1. Market-Wide Anomalies

2. Outline Puzzles in the Aggregate Market-wide Data
Equity Premium Puzzle
Excessive Volatility Puzzle
Behavioral Modeling
Preference-based Models: Barberis, Huang, and Santos (2001)
Belief-based Models: BSV(1998), DHS(1998),
Hong and Stein (1990)

3. Puzzles in the Aggregate Market-wide Data
The standard framework for thinking about aggregate stock market behavior has been the consumption-based approach. However, this framework encounters difficulties.
Equity premium puzzle: high historical average return in excess of risk-free return
Volatility puzzle: excess volatility
Predictability puzzle

4. Puzzles in the Aggregate Market-wide Data
Consumption-based model and puzzles
There is a continuum of identical infinitely lived agents in the economy, with a total mass of one, and two assets: a risk-free asset in zero net supply, paying a gross interest rate of Rf,t between time t and t +1; and one unit of a risky asset, paying a gross return of Rt+1between time t and t +1. The risky asset—stock—is a claim to a stream of perishable output represented by the dividend sequence {Dt}. The framework of the model is as follows.

5. Puzzles in the Aggregate Market-wide Data
Max =

6. Puzzles in the Aggregate Market-wide Data
In this model, agents choose a consumption level Ct and an allocation to the risky asset St to maximize the intertemporal total expected utility.
There exists an equilibrium for this economy which is demonstrated in Appendix.
A constant risk-free rate shown in eq. (18).
The stock’s price-dividend ratio f(.) determined by eq. (19).
After the price-dividend ratio f(.) is solved, the stock return can be obtained by eq.(20).

7. Puzzles in the Aggregate Market-wide Data
Equity premium puzzle
The puzzle was initiated by Mehra and Prescott (1985). The average historical premium on the US stock market, 6.2 percent in 1889–1978, was too high given what had been theoretically predicted based on changes in aggregate consumption.
The model used by Mehra and Prescott was based on the standard utility function. It also assumed agent representativeness and stable level of risk aversion.

8. Puzzles in the Aggregate Market-wide Data
In order to account for the high-risk premium observed historically, investors would have to display unrealistically high coefficients of relative risk aversion. It follows that very high risk-aversion coefficients would mean that investors should try to even out the consumption level intertemporally.
However, this attitude should result in high real interest rates. In practice, real historical returns on risk-free assets are surprisingly low (Weil, 1989).

9. Puzzles in the Aggregate Market-wide Data
Numerous theoretical attempts to explain the equity premium puzzle questioned the assumptions adopted by Mehra and Prescott.
Mankiw and Zeldes (1991) and Haliassos and Bertaut (1995) suggest that the reason the equity premium might seem too high compared to fluctuations of consumption is because market segmentation has not been factored in.

10. Puzzles in the Aggregate Market-wide Data
Mehra and Prescott (1985) used aggregated data on American consumption forgetting that almost three quarters of Americans do not invest in shares at all.
Such caveats solve the problem only partially. Even with market segmentation taken into account, equity premium still seems too elevated.
Another strand of literature focuses on ways to modify the utility function used to evaluate changes in the consumption level.

11. Puzzles in the Aggregate Market-wide Data
For example, Abel (1990) and Campbell and Cochrane (1999) claim current consumption should be evaluated against the present consumption of other market players with whom the decision maker compares his own situation.
The marginal utility of future consumption rises as it is assumed that others will also consume more in the future. Thus investors are less likely to borrow from future consumption to the benefit of current one. This can explain low real returns on safe instruments without much change of degree of risk aversion.

12. Puzzles in the Aggregate Market-wide Data
Fama and French (2002a) tried to explain the puzzle looking at changes in fundamental values. They verified whether the historically observed equity premium is justified by a relevant increase in dividends or profits. It turned out that the equity premium stemming from an increase in fundamental factors in the second half of the twentieth century should be 4.3 percent, still significantly below the actual empirical values in that period.

13. Puzzles in the Aggregate Market-wide Data
Behavioral finance offers its own explanation of the equity puzzle.
Benartzi and Thaler (1995) point out that mental accounting coexists with strong loss aversion. Being myopic in their attitude, investors frame and mentally account returns of each period separately. Hence, the high equity premium is a reward for taking the risk of short-term fluctuations and, as such, is necessary to overcome short-term loss aversion investors commonly experience.

14. Puzzles in the Aggregate Market-wide Data
Barberis, Huang, and Santos (2001) propose a formal model to explain the equity puzzle by myopic loss aversion.
Drawing on the prospect theory, they assume that investors evaluate utility not only on the basis of the total level of consumption, but also changes in wealth, and that they are more sensitive to losses than gains.
Max

15. Puzzles in the Aggregate Market-wide Data
1. The first term in this specification, utility over consumption, is a standard feature of asset pricing models.
2. The second term represents utility from fluctuations in the value of financial wealth.
v(Xt+1) is a function of the gain or loss Xt+1, St , the value of the investor’s risky asset holdings at time t and a state variable zt which measures the investor’s gains or losses prior to time t.

16. Puzzles in the Aggregate Market-wide Data
The authors allow the investor’s prior investment performance to affect the way subsequent losses are experienced, and hence his willingness to take risk.
Consequently, the concept is similar to the one noticed by Thaler and Johnson (1990) that the degree of risk aversion depends on previous investor’s experience.

17. Puzzles in the Aggregate Market-wide Data
Consider the economy in which consumption and dividends follow distinct process (eq. (29)~(31)). There exists an equilibrium for this economy which is demonstrated in Proposition 2.
A constant risk-free rate shown in eq. (33).
The stock’s price-dividend ratio f(.) given by eq. (34).
After the price-dividend ratio f(.) is solved, the stock return can be obtained by eq. (32).

18. Puzzles in the Aggregate Market-wide Data
This model can explain a major part of the actually observed real premium, which, at the time, stood at about 6 percent per annum.
Another psychological phenomenon that may make investors expect higher equity premiums is ambiguity aversion whereby people are reluctant to take part in lotteries with undefined probability distribution (Olsen and Troughton, 2000).

19. Puzzles in the Aggregate Market-wide Data
Excessive Volatility Puzzle
Another aggregate market phenomenon the classical theory of finance struggles to explain is excessive volatility of prices compared to observed fluctuations of fundamental values, such as earnings or dividends, or changes in expected consumption.
LeRoy and Porter (1981) and Shiller (1981) find that price variance is much higher than what might have been justified by changes in dividend levels.

20. Puzzles in the Aggregate Market-wide Data
If the variance of the returns is higher than the variance of dividends, it is obvious that there must be fluctuations of the price-to-dividend ratio (P/D).
The theoretical explanation for the fluctuations of P/D: changing expectations of future dividends or changes in discount rates
Changes in discount rates, in turn, might stem from changing expectations related to returns on risk-free assets, changes in the expected level of risk, or changing risk aversion.

21. Puzzles in the Aggregate Market-wide Data
Campbell and Shiller (1988), Campbell (1991), and Fama and French (2002a) document that historical P/D levels do not translate into real dividend growth rates. Neither does the P/D ratio explain changes in the risk-free rate or fluctuations of risk levels in historical time series.
Therefore, the only factor that potentially can influence fluctuations of the P/D ratio and, consequently, explain higher variance for prices compared to dividends is the changing degree of risk aversion.

22. Puzzles in the Aggregate Market-wide Data
Campbell and Cochrane (1999) propose a model in which people are relatively slow to get used to a specific level of consumption.
If current consumption is higher than what consumers are accustomed to, risk aversion will decrease. Conversely, when current consumption is likely to fall below the habit level, risk aversion will increase.
Changes in the attitude to risk are reflected in the level of discount rates and, as a consequence, cause fluctuations in the P/D ratio.

23. Puzzles in the Aggregate Market-wide Data
In Barberis, Huang, and Santos (2001) people are less risk averse after prior gains and more risk averse after prior losses.
According to this model, a positive fundamental signal will generate a high return which lowers the investors’ risk aversion. Therefore, investors apply a lower discount rate to the future dividend stream, giving stock prices an extra push upward. A similar mechanism holds for a bad fundamental signal.

24. Puzzles in the Aggregate Market-wide Data
One result of this effect is that stock returns are much more volatile than dividend changes. Normally, this pattern could be viewed as exhibiting market overreaction to initial good/bad news. Stock returns are made up of two “justified” components: one due to fundamental signals and the other to a change in risk aversion.
BHS(2001) demonstrate that their model fits well with several empirical observations. The equity premium is justified because loss-averse investors require a high reward for holding a risky or excessively volatile asset.

25. Puzzles in the Aggregate Market-wide Data
Behavioral finance offers a number of potential explanations that could be roughly divided into two groups: those related to irrational investor beliefs and those based on instability of preferences.
1. The first group includes a representativeness bias in the form of the short-series error, which leads to the extrapolation bias and a belief in trends. Another mistake that may contribute to high price volatility is overconfidence combined with selective attribution.

26. Puzzles in the Aggregate Market-wide Data
Representativeness bias: having observed a firm with a consecutive series of higher-than-expected earnings, investors underestimate the probability that the situation might be random and become mistakenly convinced that a permanent change has taken place and the observed growth will continue in the future. These expectations are reflected in current prices causing irrational fluctuations in the P/D ratio.
Overconfidence. Attaching too much importance to

27. Puzzles in the Aggregate Market-wide Data
privately acquired knowledge to the detriment of public information triggers price fluctuations that are not justified by real changes in earnings or dividends (Odean (1998b) and Daniel, Hirshleifer, and Subrahmanyam (1998, 2001)).
2. The second group includes irrational instability of investor preferences and shifts in the degree of risk aversion.

28. Puzzles in the Aggregate Market-wide Data
BHS(2001) suggest investors will change their preferences depending on the results of their former choices.
Following previous positive experiences, falling risk aversion should be observed and vice versa—after adverse events, investors will be more sensitive to risk. Shifts in the degree of risk aversion will result in applying different discount rates to forecasted cash flows resulting in periodical under- and overvaluation.