Download Aggregate Stock Market

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Aggregate Stock Market
Introduction
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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

Introduction
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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.

Introduction
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After prior gains, he becomes less loss averse.
Conversely, after a prior loss, he becomes more loss averse.
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The Model
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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}, where dividend growth is given by eq. (1).
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The Model
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In this model, agents choose a consumption level Ct
and an allocation to the risky asset St 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(Xt+1, St, zt) is a function of the gain or loss Xt+1, St ,
the value of the investor’s risky asset holdings at time t ,

The Model
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and a state variable zt 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, bt is an exogenous
scaling factor.
 The definition of Xt+1
is given in eq. (14). The
specification of v function and λ (zt ) are in eq. (15)~(17).
Finally, eq. (18) shows the dynamics of the state variable
zt .
The Model
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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).
Numerical results
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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 gC and σC, the mean and standard deviation of log
consumption growth, follow Cecchetti, Lam, and Mark
(1990) who obtain gC =1.84 percent and σC = 3.79 percent
from a time series of annual data from 1889 to 1985.
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Numerical results
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Given the values of gC and σC , equation (23) shows
that γ =1.0 and ρ = 0.98 bring the risk-free interest rate
close to Rf -1 = 3.86 percent.
 Tversky and Kahneman [1992] estimate λ= 2.25.
 For simplicity, the authors set gD = gC = 0.0184. Using
NYSE data from 1926–1995 from CRSP, the authors find
σD = 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.
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Numerical results
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2. Table IV shows the simulated moments of stock
returns for various values of b0 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 (b0 =0).
 The bottom panel in Table IV shows that the results can
be significantly improved by increasing k. For b0 =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.