1. Aggregate Stock Market 1

2. Introduction 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 2

3. Introduction This paper provides an alternative way of thinking about the aggregate stock market. In the model presented below, the investor derives direct utility not only from consumption but also from changes in the value of his financial wealth.  The specification of this additional source of utility captures two ideas. First, the investor is much more sensitive to reductions in his financial wealth than to increases, a feature sometimes known as loss aversion. Second, how loss averse the investor is, depends on his prior investment performance. 3

4. Introduction  After prior gains, he becomes less loss averse. Conversely, after a prior loss, he becomes more loss averse. 4

5. The Model 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 R f,t between time t and t +1; and one unit of a risky asset, paying a gross return of R t+1 between time t and t +1. The risky asset—stock—is a claim to a stream of perishable output represented by the dividend sequence {Dt}, where dividend growth is given by eq. (1). 5

6. The Model In this model, agents choose a consumption level C t and an allocation to the risky asset S t to maximize eq.(13). 1. The first term in this preference 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(X t+1, S t, z t ) is a function of the gain or loss X t+1, S t, the value of the investor’s risky asset holdings at time t, 6

7. The Model and a state variable z t which measures the investor’s gains or losses prior to time t. The authors allow the investor’s prior investment performance to affect the way subsequent losses are experienced, and hence his willingness to take risk. Finally, b t is an exogenous scaling factor.  The definition of X t+1 is given in eq. (14). The specification of v function and λ (z t ) are in eq. (15)~(17). Finally, eq. (18) shows the dynamics of the state variable z t. 7

8. The Model 3. Consider the economy in which consumption and dividends follow distinct process (eq. (29)~(31)). Then 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). 8

9. Numerical results This study presents price-dividend ratio f(.) that solves eq. (34), and then creates a long time series of simulated data and uses it to compute various moments of asset returns which can be compared with historical numbers. 1. Parameter values. Table III summarizes the choice of parameter values.  For g C and σ C, the mean and standard deviation of log consumption growth, follow Cecchetti, Lam, and Mark (1990) who obtain g C =1.84 percent and σ C = 3.79 percent from a time series of annual data from 1889 to 1985. 9

10. Numerical results  Given the values of g C and σ C, equation (23) shows that γ =1.0 and ρ = 0.98 bring the risk-free interest rate close to R f -1 = 3.86 percent.  Tversky and Kahneman [1992] estimate λ= 2.25.  For simplicity, the authors set g D = g C = 0.0184. Using NYSE data from 1926–1995 from CRSP, the authors find σ D = 0.12. Campbell (2000) estimates in a time series of U. S. data spanning the past century, and based on his results, the authors set ω= 0.15. 10

11. Numerical results 2. Table IV shows the simulated moments of stock returns for various values of b 0 and k.  The top panel keeps k fixed at 3. The equity premium and the volatility of returns are much higher than what can be obtained under the traditional model (b 0 =0).  The bottom panel in Table IV shows that the results can be significantly improved by increasing k. For b 0 =2, a k of 10 is enough to give a premium of 5.02 percent and a volatility of 23.84 percent; note also that this corresponds to an average loss aversion of only 3.5. 11