Download Dynamic Asset Pricing Model with Exhibition

Initial results examining the Equity Premium and Volatility puzzles using a
modified asset pricing model
Oren Shmuel
The Graduate Center, City University of New York
Current version: June 2015
Abstract
I examine the ability of a dynamic asset-pricing model to explain the Equity Premium and the
Volatility puzzles. I modify the standard asset pricing model in four aspects. First, I use a time
varying price of risk. Second, I incorporate Duesenberry’s demonstration effect. Third, I include
tax rates in my model to control for any extreme valuation, relative to GDP, caused by tax rates
and not by stock market factors. Fourth, I represent the stock price movement using a right
skewed non-Gaussian model (the Gumbel distribution). My results suggest that the model I use
can help to better understand both puzzles.
I) Introduction
The two above puzzles represent the shortcomings of standard asset pricing. The equity premium
puzzle emanates from the inability of the theoretical model to explain the empirically observed
high equity premium (when the average stock returns so much higher than the average bond
returns). It is based on the fact that in order to reconcile the much higher return on stock
compared to government bonds in the United States, individuals must have very high risk
aversion according to standard economics models. The volatility puzzle manifests itself when
the stock market more volatile relative to the consumption volatility. Shiller (1981) found that
stock market returns are too volatile relative to the volatility of dividends. One type of tests
examines the Euler equation restriction on the product of asset returns with the marginal rate of
substitution of the representative agent. The results from this type of tests show that the stock
market is still too volatile.
When no reasonable parameterizations of the Lucas (1978) asset pricing model, the
equity premium puzzle is created because the theoretical equity premium is as large as the
empirically observed equity premium. However, the size of the equity premium can not be
estimated since there is not enough flexibility in the Lucas model. The Brock (1982) asset
Oren Shmuel, The Graduate Center, City University of New York, 365 Fifth Avenue, New York, NY 10016-4309,
E-mail: [email protected] , Tel: 914-699-4593 .
pricing model can estimate a significant equity premium. Using the Brock’s (1982) asset pricing
model, Akdeniz and Dechert (2007) show that there are parameterizations of the Brock model
that have equity premia that are more consistent with the empirical evidence than the equity
premia that were observed by Mehra and Prescott (1985). Kocherlakota (1996) tries to resolve
the equity premium puzzle and the risk free rate puzzle by reviewing the literature. Kocherlakota
report the papers that try to explain these two puzzles. Campbell and Cochrane (1999) find the
equity premium puzzle using a consumption-based model with external habit formation. They
use a power utility and the Sharpe ratio inequality to explain the equity premium puzzle.
Campbell and Cochrane (1999) use a discrete time model while I am using a continuous time
model. In addition, they use the same utility sensitivity to both the social standard (𝛽) and the
individual consumption (𝛼) while I use different utility sensitivities (𝛼 ≠ 𝛽). Constantinides
(1990) determines that the Equity Premium pazzle is caused by habit persistence.
In this paper, I attempt to modify the standard asset pricing model in four ways. First, I
introduce a time varying price of risk (i.e. time varying excess return per unit risk). Second, I
introduce the Duesenberry’s demonstration effect [Duesenberry(1949)] and define the Habit
formulation. Duesenberry’s demonstration effect is a type of consumption externality where an
individual’s utility depends not only upon his consumption level (Ct ) but also upon the social
average level (or Habit formation) of consumption ( C t ). I incorporate the Duesenberry’s effect
due to its theoretical and empirical attractiveness. Third, I include tax rates in my model to
control for any low valuation or high valuation, relative to GDP, caused by tax rates and not by
stock market factors. Fourth, I represent the stock price movement using a right skewed nonGaussian model (the Gumbel distribution). To better represent a risk averse investor I use a right
skew distribution for the stock price since such a distribution gives higher weight to negative
stock return than positive stock return. Economists James Duesenberry and Robert H. Frank,
claimed that being aware to the consumption habits of others affects the consumption habits of
the individual consumer by emulating the habits of other consumers. Duesenberry (1949) claims
that the demonstration effect causes unhappiness with current levels of consumption, which
affects savings rates and macroeconomic growth. Ragnar Nurkse (1953) claimed that the
exposure to new goods or patterns of living causes unhappiness with normal consumption habits.
Such an "international demonstration effect" in developing countries causes increased access to
new superior goods because people "come into contact with superior goods or superior patterns
of consumption, with new articles or new ways of meeting old wants." Thus, people in such
developing countries are "apt to feel after a while a certain restlessness and dissatisfaction. Their
knowledge is extended, their imagination stimulated; new desires are aroused" (Nurkse quoted in
Kattel et al. 2009, p. 141).
The paper is organized as follows: In section II, the theoretical model is presented.
Section III derives the closed form solution of the theoretical model. Section IV provides
solutions to the equity premium and volatility puzzles. Section V concludes.
2
II) The Model
I assume the utility function for the representative agent in this economy to be of the
following power type:
1
𝑢(𝐶𝑡 , 𝐶𝑡 ) = 𝛼 𝐶𝑡 𝛼 𝐶𝑡
(1)
𝛽
where Ct is the per capita consumption rate and C t is the social average consumption rate at time
t (or habit behavior), 𝛼 is the utility sensitivity to the individual consumption (i.e. elasticity of
consumption substitution or the risk aversion parameter) and 𝛽 is utility sensitivity to the social
standard.
C t (the Habit formation) is defined by:
𝑡
𝐶𝑡 = 𝐶0 𝑒 −𝑎𝑡 + 𝑏 ∫0 𝑒 𝑎(𝑠−𝑡) 𝐶(𝑠)𝑑𝑠
(2)
Thus, the coefficient of absolute risk aversion is:
𝛽
(𝛼 − 1) 𝐶𝑡 𝛼−2 𝐶𝑡
𝑢′′ (𝐶𝑡 , 𝐶𝑡 )
1−𝛼
)
𝐴(𝐶𝑡 = − [
] = −[
]
=
𝛽
𝐶𝑡
𝑢′ (𝐶𝑡 , 𝐶𝑡 )
𝐶𝑡 𝛼−1 𝐶𝑡
The coefficient of relative risk aversion is:
𝑅(𝐶𝑡 ) = 𝐶𝑡 𝐴(𝐶𝑡 ) = 1 − 𝛼
Hence, the utility in equation (1) follows Constant Relative Risk Aversion (CRRA).
For simplicity, I assume that there are only two assets (without loss of generality), one risky
stock and one risk-free bond, to invest in.
To better represent a risk averse investor, I use a right skew distribution for the stock price since
such a distribution gives higher weight to negative stock return than positive stock return.
To represent the stock price motion I use the Gumble probability distribution which is a nonGaussian distribution with an asymmetric right skew. Thus, the formulation of the stock price St
is:
𝑑𝑆𝑡 = [𝜇𝑡 + 𝜀𝜎𝑡 ]𝑆𝑡 𝑑𝑡 +
(3)
Where
𝜋
𝜎
√6 𝑡
𝜋
𝜎 𝑆 𝑑𝑍𝑡
√6 𝑡 𝑡
is the Gumble distribution standard deviation, 𝜇𝑡 + 𝜀𝜎𝑡 is the Gumble distribution
mean, 𝜀 = 0.577216 the Euler’s constant, 𝜎𝑡 is the distribution scale (i.e. the normal distribution
standard deviation) of the price of stock (which is a risky asset), 𝜇𝑡 is the distribution location
(i.e. the normal distribution mean) of the risky rate of return and 𝑑𝑍𝑡 is a Wiener process.
3
The percent change, i.e. the return, of the stock price is described as:
𝑑𝑆𝑡
(4)
𝑆𝑡
= [𝜇𝑡 + 𝜀𝜎𝑡 ]𝑑𝑡 +
𝜋
𝜎 𝑑𝑍𝑡
√6 𝑡
In McGrattan and Prescott (2001) the authors found that the stock market was neither overvalued
nor undervalued in 1962 and 2000. The main reason for the low valuation in 1962 relative to
GDP and the high valuation in 2000 relative to GDP is the tax rate on distributions (𝜏𝑑 ) was
much higher in 1962 than it was in 2000. Thus, I would like to add tax rates to my model to
control for any low valuation or high valuation, relative to GDP, caused by tax rates and not by
stock market factors.
I define 𝑊𝑡 as the wealth before valuation, the tax rate on distributions (𝜏𝑑 ) and the tax rate on
corporate income (𝜏𝑐 ). Risky assets (such as stocks) are taxed at the corporate level and then
again at the distribution level (dividends). Riskless assets (non stock assets) are taxed only at the
distribution level.
The change in the per capita wealth ( W t ) is determined by the after tax return from the risky
asset (the stock) and the after tax return from the riskless asset (the risk-free bond) less
consumption. Hence:
𝑑𝑊𝑡 = (1 − 𝜏𝑑 )(1 − 𝜔𝑡 )𝑊𝑡 𝑟𝑑𝑡 + (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝜔𝑡 𝑊𝑡
(5)
𝑑𝑆𝑡
𝑆𝑡
− 𝐶𝑡 𝑑𝑡
where r is risk-free rate of return, Ct is per capita consumption, 𝜏𝑑 is the tax rate on distributions,
𝜏𝑐 is the tax rate on corporate income, and t is the portion, i.e. weight, of the before tax per
capita wealth ( W t ) that is invested in the risky asset.
Inserting equation (4) into equation (5) yields the equation governing the changes in the before
tax wealth written as follows:
(6)
𝑑𝑊𝑡 = {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝜔𝑡 𝑊𝑡 (𝜇𝑡 − 𝑟) + 𝜀𝜔𝑡 𝑊𝑡 𝜎𝑡 ] − (1 − 𝜏𝑑 )𝜏𝑐 𝜔𝑡 𝑊𝑡 (𝜇𝑡 + 𝜀𝜎𝑡 ) − 𝐶𝑡 }𝑑𝑡 +
𝜋
+ ( 6) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝜔𝑡 𝑊𝑡 𝜎𝑡 𝑑𝑍𝑡
√
where 𝑊𝑡 is the before tax per capita wealth, 𝑟 is risk-free rate of return, 𝜇𝑡 is risky rate of return,
𝑤𝑡 is the proportion of wealth invested in the risky asset, and 𝑑𝑍𝑡 is a Wiener process.
I define 𝑋𝑡 = 𝜔𝑡 𝑊𝑡 𝜎𝑡 which can be interpreted as the level of investment in units of risk, and
𝜇 −𝑟
𝐴𝑡 = 𝑡𝜎 as the price of the risk (or the excess return per unit risk). I set for the price of risk to
𝑡
be stochastic so that the rate of return and the volatility rate of the risky asset vary over time. I
also assume that the change in the price of risk is represented by a Wiener process. For
4
simplicity, the price of the risk process is set to be perfectly correlated with the process of the
risky asset (Φ = 1). Thus, the process for the price of risk is defined as follows:
𝑑𝐴𝑡 = 𝑓(𝐴𝑡 )𝑑𝑡 + 𝑔(𝐴𝑡 )𝑑𝑍𝑡
(7)
Using definitions of X t and At , the law of motion of the wealth, equation (6), becomes:
(8)
𝑑𝑊𝑡 = {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡 + 𝜀𝑋𝑡 ] − (1 − 𝜏𝑑 )𝜏𝑐 [𝐴𝑡 𝑋𝑡 + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡 ] − 𝐶𝑡 }𝑑𝑡 +
𝜋
+ ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 𝑑𝑍𝑡
√6
III) A Closed Form Solution
I define the state valuation function, V (.,.), as a function of the two variables, Wt and At. Thus,
the maximization problem is:
∞
𝑉(𝑊𝑡 , 𝐴𝑡 ) = max 𝐸𝑡 ∫𝑡 𝑒 −𝜌(𝑠−𝑡) 𝑢(𝐶𝑠 , 𝐶𝑠 )𝑑𝑠
(9)
Subject to:
(7)
(8)
𝑑𝐴𝑡 = 𝑓(𝐴𝑡 )𝑑𝑡 + 𝑔(𝐴𝑡 )𝑑𝑍𝑡
𝑑𝑊𝑡 = {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡 + 𝜀𝑋𝑡 ] − (1 − 𝜏𝑑 )𝜏𝑐 [𝐴𝑡 𝑋𝑡 + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡 ] − 𝐶𝑡 }𝑑𝑡 +
𝜋
+ ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 𝑑𝑍𝑡
√6
(10)
𝑑𝑍𝑡 𝑑𝑍𝑡 = Φ 𝑑𝑡 = 1𝑑𝑡
where  represents the constant utility discount rate, Φ is the correlation between the price of
the risk process and the process of the risky asset, and Et represents the operator for expectations
conditional on the information set at time t. The Hamilton-Jacobi-Bellman (HJB) equation is
derived using the principle of optimality with Ito’s lemma.
Ito’s lemma formulation is written as:
V
V
1   2V
dV (Wt , At ) 
dWt 
dAt  
Wt
At
2  Wt 2

 2
dWt 2  1   V

2  At 2

5

 2
dAt 2    V
 W A

t
t



dWt dAt 

By dividing both side of the equation by dt I get:
(11)
dV  V

dt  Wt
 dWt   V

  
 dt   At
 dAt  1   2V

  
2
 dt  2  Wt
 dWt 2

 dt

 1   2V
 
 2  A 2

 t
 dAt 2

 dt

   2V

  W A
t
t
 
 dWt dAt 

dt

Solving the expressions:
𝜋
(12)
𝑑𝑊𝑡 𝑑𝐴𝑡 = ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 𝑔(𝐴𝑡 )𝑑𝑡 (see Appendix 1)
(13)
(𝑑𝑊𝑡 )2 = ( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 (𝑋𝑡 )2 𝑑𝑡
6
(14)
(𝑑𝐴𝑡 )2 = [ 𝑔(𝐴𝑡 )]2 𝑑𝑡
√6
𝜋2
Thus Ito’s lemma formulation becomes:
(15) ρ𝑉(𝑊𝑡 , 𝐴𝑡 ) = 𝑉𝑊 {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡 + 𝜀𝑋𝑡 ] − 𝜏𝑐 (1 − 𝜏𝑑 )[𝐴𝑡 𝑋𝑡 + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡 ] − 𝐶𝑡 } +
1
𝜋2
𝑉𝑊𝑊 ( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 (𝑋𝑡 )2 +
2
6
1
𝜋
+ 𝑉𝐴𝐴 [ 𝑔(𝐴𝑡 )]2 + 𝑉𝑊𝐴 𝑔(𝐴𝑡 ) ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡
2
√6
+𝑉𝐴 𝑓(𝐴𝑡 ) +
Utilizing the principle of optimality, the Hamilton-Jacobi-Bellman (HJB) equation becomes:
(16) ρ𝑉(𝑊𝑡 , 𝐴𝑡 ) = max {𝑢(𝐶𝑡 , 𝐶𝑡 ) + 𝑉𝑊 {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡 + 𝜀𝑋𝑡 ] − 𝜏𝑐 (1 − 𝜏𝑑 )[𝐴𝑡 𝑋𝑡 +
𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡 ] − 𝐶𝑡 } + 𝑉𝐴 𝑓(𝐴𝑡 ) +
𝜋
1
𝜋2
1
𝑉 ( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 (𝑋𝑡 )2 + 2 𝑉𝐴𝐴 [ 𝑔(𝐴𝑡 )]2 +
2 𝑊𝑊 6
𝑉𝑊𝐴 𝑔(𝐴𝑡 ) ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 }
√6
From the above HJB equation, I can find the two first order conditions for optimality:
(17)
(18)
𝑢𝑐 (𝐶𝑡 , 𝐶𝑡 ) − 𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 0
𝜋2
𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 )(1 − 𝜏𝑐 )(1 − 𝜏𝑑 )(𝐴𝑡 + 𝜀) + 𝑉𝑊𝑊 (𝑊𝑡 , 𝐴𝑡 ) ( 6 ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 𝑋𝑡 +
𝜋
+𝑉𝑊𝐴 𝑔(𝐴𝑡 ) ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 ) = 0
√6
6
If Ct* and X t* satisfy the above first order conditions the HJB equation becomes:
(19)
ρ𝑉(𝑊𝑡 , 𝐴𝑡 ) = 𝑢(𝐶𝑡∗ , 𝐶𝑡 )
+ 𝑉𝑊 {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡∗ + 𝜀𝑋𝑡∗ ] − 𝜏𝑐 (1 − 𝜏𝑑 )[𝐴𝑡 𝑋𝑡∗ + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡∗ ] − 𝐶𝑡∗ }
1
𝜋2
1
+ 𝑉𝐴 𝑓(𝐴𝑡 ) + 𝑉𝑊𝑊 ( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 (𝑋𝑡∗ )2 + 𝑉𝐴𝐴 [ 𝑔(𝐴𝑡 )]2
2
6
2
𝜋
+ 𝑉𝑊𝐴 𝑔(𝐴𝑡 ) ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡∗
√6
remember that Ct* and X t* (the optimal choice variables), are functions of the state variables, W
and A.
To calculate a closed form solution, I make a simplifying assumption: The price of risk follows a
stationary lognormal process, and is perfectly correlated with the market risk, thus
(20)
𝑑𝐴𝑡 = 𝑓0 𝐴𝑡 𝑑𝑡 + 𝑔0 𝐴𝑡 𝑑𝑍𝑡
Using equations (1) and (20), I verify whether the model allows a closed form solution where
both the consumption rate and the risky investment rate are linear function in wealth. Thus, I
conjecture the following:
(21)
𝐶𝑡 = 𝐶0 (𝐴𝑡 )𝑊𝑡
(22)
𝑋𝑡 = 𝑋0 (𝐴𝑡 )𝑊𝑡
Next, I use the conjecture and verification method to find the form of state valuation function.
Using the form of the utility function in equation (1), I conjecture the equation for a state value
function to be:
(23)
𝑉(𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝑊𝑡 𝛼
I will assert the functional form for  ( A) (the discount factor) later.
To match the empirical results with the financial data I must use several propositions.
Proposition 1: finding 𝐶0 (𝐴𝑡 ) (see proof in Appendix 2)
−𝛽
(24)
1
𝐶0 (𝐴𝑡 ) = (𝐶𝑡 )𝛼−1 [𝛾𝛿(𝐴𝑡 )𝛼]𝛼−1
7
Next, I will formulate the different volatilities in my model. I set  c2 to denote the variance rate
of consumption and  W2 denote the variance rate of wealth.
Proposition 2: finding 𝜎𝑐 (see proof in Appendix 3)
𝐴𝑡 +𝜀 √6
( )
1− 𝛼 𝜋
𝜎𝑐 =
(27)
𝜋
or 𝐴𝑡 = 𝜎𝑐 (1 − 𝛼) ( ) − 𝜀
√6
Proposition 3: finding 𝜎𝑊 (see proof in Appendix 4)
(28)
1
√6
𝜎𝑊 = ( 𝜋 ) (𝛼−1)(1−𝜏
𝑐 )(1−𝜏𝑑
[(𝛼 − 1)𝜎𝑐 −
)
𝑔(𝐴𝑡 )𝛿 ′ (𝐴𝑡 )
]
𝛿(𝐴𝑡 )
I conjecture the state valuation function to be of the form as in equation (23). To find an
expression for the function  ( A) , I use the above equations to derive a differential equation for
 ( A) . The solution for this differential equation will give the correct closed form solution. Since
even the simplest differential equation frequently has multiple solutions, my solution cannot be
guaranteed to be unique.
Proposition 4: finding 𝑋0 (𝐴𝑡 ) and verify my conjecture (see proof in Appendix 5)
I define above and in equation (22) that 𝑋𝑡 = 𝜔𝑡 𝑊𝑡 𝜎𝑡 = 𝑋0 (𝐴𝑡 )𝑊𝑡 that is
𝑋0 (𝐴𝑡 ) = 𝜔𝑡 𝜎𝑡 . The proportion of wealth invested in the risky asset, 𝜔𝑡 , is proportional to the
inverse of the market variance rate,  t . Thus 𝑋0 (𝐴𝑡 ) = 𝑋0 , i.e. 𝑋0 (𝐴𝑡 ) does no depend on 𝐴𝑡 . I
choose one solution for  ( A) that can create these conditions with parametric restrictions, thus
for
𝛿(𝐴𝑡 ) = 𝛿0 𝐴𝑡 𝛼−1
(30)
I derive
1
1
1
0 = 𝛼 (𝛼−1) 𝛾 (𝛼−1) 𝛿0 (𝛼−1) 𝐶𝑡
𝛽
(
)
1−𝛼
+ 𝛼(1 − 𝜏𝑑 )𝑋0 (𝐴𝑡 ) − 𝛼𝜏𝑐 (1 − 𝜏𝑑 )𝑋0 (𝐴𝑡 ) −
−𝛼𝐶𝑡
[𝛼
𝑋0 (𝐴𝑡 ) =
(
𝛽
(
)
1−𝛼
1
[𝛾𝛿0 𝛼]𝛼−1
𝛽
2−𝛼
1
)
)
(
) (
𝛼−1 (𝛿0 𝛾) 𝛼−1 𝐶𝑡 1−𝛼 (𝛼
− 1)]
(1 − 𝜏𝑐 )(1 − 𝜏𝑑 )
solving for 𝑋0 (𝐴𝑡 ) yields an expression that does not depend on 𝐴𝑡 as per my conjecture. Hence,
my conjecture that 𝑋0 (𝐴𝑡 ) = 𝜎𝑡 𝜔𝑡 and 𝑋𝑡 = 𝑋0 (𝐴𝑡 )𝑊𝑡 is verified.
8
Corollary 5: deriving 𝐶0 (𝐴𝑡 ) using the definition for 𝛿(𝐴𝑡 )
𝐶0 (𝐴𝑡 ) = (𝐶𝑡 )
(32)
(
𝛽
)
1−𝛼
1
[𝛾𝛿0 𝛼](𝛼−1) 𝐴𝑡
Proof: Insert equation (30) into (24) will yield equation (32)
𝛿(𝐴𝑡 ) = 𝛿0 𝐴𝑡 𝛼−1
(30)
𝐶0 (𝐴𝑡 ) = (𝐶𝑡 )
(24)
(
−𝛽
)
𝛼−1
1
[𝛾𝛿(𝐴𝑡 )𝛼](𝛼−1)
Corollary 6: deriving 𝜎𝑊 using the definition for 𝛿(𝐴𝑡 )
(33)
𝜎𝑊 =
1
√6
( ) [𝜎𝑐
(1−𝜏𝑐 )(1−𝜏𝑑 ) 𝜋
− 𝑔0 ]
Proof: Insert equation (30) into (28) to derive equation (33)
𝛿(𝐴𝑡 ) = 𝛿0 𝐴𝑡 𝛼−1
(30)
(28)
√6
1
𝜎𝑊 = ( 𝜋 ) (𝛼−1)(1−𝜏
𝑐 )(1−𝜏𝑑 )
[(𝛼 − 1)𝜎𝑐 −
𝑔(𝐴𝑡 )𝛿 ′ (𝐴𝑡 )
]
𝛿(𝐴𝑡 )
IV) Examining the Equity Premium Puzzle and Volatility Puzzle
The first step in investigating the puzzles requires that I match the risk-free rate of return
𝑑𝐶
(r) and the consumption growth rate ( 𝐶 𝑡 ) with real data. In this paper I assumed that the risk𝑡
free rate of return (r) is constant over time. Thus, I can set the constant risk-free rate of return to
match the estimate of real average risk-free rate. Using equation (32), I can determine the
consumption values and  0 to fit my estimate of consumption growth rate while taking into
account the estimate of the growth of wealth and the price of risk.
A) Examining the Equity Premium Puzzle:
The formulation for the equity premium derived is: (see proof in Appendix 6)
(34)
𝜋2
𝜋
𝜇𝑡 − 𝑟 = 𝜎𝑡 [−𝜀 + (1 − 𝛼)(1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝜎𝑡 𝜔𝑡 ( 6 ) + (1 − 𝛼)𝑔0 ( )]
√6
From equation (34), I can have a high equity premium (𝜇𝑡 − 𝑟 > 0) without assuming an
unreasonable value of the risk aversion parameter,  .
9
Since 𝜎𝑡 > 0 , to create a scenario of the equity premium puzzle the condition should be:
𝜋2
𝜋
(1
)(1
)𝜎
−𝜀 + − 𝛼)(1 − 𝜏𝑐
− 𝜏𝑑 𝑡 𝜔𝑡 ( ) + (1 − 𝛼)𝑔0 ( ) > 0
6
√6
Hence
𝑔0 >
𝜋2
𝜀 − (1 − 𝛼)(1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝜎𝑡 𝜔𝑡 ( 6 )
𝜋
(1 − 𝛼) ( )
√6
𝜀
𝜋
= [
] − ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝜎𝑡 𝜔𝑡
𝜋
√6
(1 − 𝛼) ( )
√6
Since 𝜎𝑡 > 0 , 𝜔𝑡 > 0 , 𝜏𝑐 < 1 , 𝜏𝑑 < 1 then
𝜀
𝑚𝑎𝑥(𝑔0 ) = [
]
𝜋
(1 − 𝛼) ( )
√6
In addition, given g0 and 𝜎𝑡 , the equity premium (𝜇𝑡 − 𝑟 ) moves with the variance rate of market
portfolio (𝜎𝑡 2 ) in the same direction.
B) Examining the Volatility Puzzle:
From equation (27), I can choose the value of elasticity of consumption substitution (𝛼) together
with my estimate of the price of risk (𝐴𝑡 ). This allows me to find the average consumption
volatility (𝜎𝑐 ). I derive the following (see proof in Appendix 7)
(39) 𝜎𝑊 − 𝜎𝑐 =
1
√6
( ) [𝜎𝑐
(1−𝜏𝑐 )(1−𝜏𝑑 ) 𝜋
− 𝑔0 ] − 𝜎𝑐
I can to create a scenario of the volatility puzzle by requiring:
𝜎𝑊 − 𝜎𝑐 =
1
√6
( ) [𝜎𝑐
(1−𝜏𝑐 )(1−𝜏𝑑 ) 𝜋
− 𝑔0 ] − 𝜎𝑐 > 0
Hence
𝜎𝑐 [1 − (
𝜋
) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 ) ] > 𝑔0
√6
If the 𝑔0 requirement is met then there is more volatile stock market relative to the consumption
volatility and a Volatility Puzzle scenario exists.
10
C) Examining both the Equity puzzle and the Volatility Puzzle simultaneously:
Using the results from parts A and B above, the required condition for a simultaneous
scenario for both puzzles to exist is defined by:
𝜋
𝜀
𝜋
𝜎𝑐 [1 − ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 ) ] > 𝑔0 > [
] − ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝜎𝑡 𝜔𝑡
𝜋
√6
√6
(1 − 𝛼) ( )
√6
V) Conclusion
In this paper I try to theoretically determine conditions for two puzzles. The Equity
Premium puzzle occurs when the average of stock returns are so much higher than the average of
bond returns. The Volatility puzzle occurs when the stock market is much more volatile relative
to the consumption volatility.
I have designed a methodology to help understanding, at least theoretically, these two
puzzles by improving the standard asset pricing model. First, I use a time varying price of risk
(i.e. time varying excess return per unit risk). Second, I implement the Duesenberry’s
demonstration effect and define the Habit formulation. Duesenberry’s demonstration effect is a
type of consumption externality where an individual’s utility depends not only upon his
consumption level but also upon the social average level (or Habit formation) of consumption.
Third, I include tax rates in my model to control for any low valuation or high valuation, relative
to GDP, caused by tax rates and not by stock market factors. Fourth, I represent the stock price
movement using a right skewed non-Gaussian model (the Gumbel distribution). The results
suggest that the model I use can help to better understand both the equity premium and the
volatility puzzles.
11
Appendices:
Appendix 1: deriving equation (12)
(12) 𝑑𝑊𝑡 𝑑𝐴𝑡 = [{(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡 + 𝜀𝑋𝑡 ] − (1 − 𝜏𝑑 )𝜏𝑐 [𝐴𝑡 𝑋𝑡 + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡 ] − 𝐶𝑡 }𝑑𝑡 +
𝜋
+ ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 𝑑𝑍𝑡 ] {𝑓(𝐴𝑡 )𝑑𝑡 + 𝑔(𝐴𝑡 )𝑑𝑍𝑡 } =
√6
= {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡 + 𝜀𝑋𝑡 ] − (1 − 𝜏𝑑 )𝜏𝑐 [𝐴𝑡 𝑋𝑡 + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡 ] − 𝐶𝑡 } 𝑓(𝐴𝑡 )(𝑑𝑡)2 +
+{(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡 + 𝜀𝑋𝑡 ] − (1 − 𝜏𝑑 )𝜏𝑐 [𝐴𝑡 𝑋𝑡 + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡 ] − 𝐶𝑡 } 𝑔(𝐴𝑡 )𝑑𝑡𝑑𝑍𝑡 +
𝜋
+ ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 𝑓(𝐴𝑡 )𝑑𝑡𝑑𝑍𝑡 +
√6
𝜋
+ ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 𝑔(𝐴𝑡 )𝑑𝑍𝑡 𝑑𝑍𝑡 =
√6
𝜋
= ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡 𝑔(𝐴𝑡 )𝑑𝑡
√6
Since dt   dtdzt  0 and dzt dzt  dt
2
Appendix 2: Proof of Proposition 1
Insert equations (1), (21) and (23) into (17).
(17)
𝑢𝑐 (𝐶𝑡 , 𝐶𝑡 ) − 𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 0
where
(25)
𝑢𝑐 (𝐶𝑡 , 𝐶𝑡 ) =
1
𝛼
𝛼𝐶𝑡 𝛼−1 𝐶𝑡
𝛽
and
(26)
𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝛼𝑊𝑡 𝛼−1
Inserting all into equation (17) I get
𝛽
[𝐶0 (𝐴𝑡 )𝑊𝑡 ]𝛼−1 𝐶𝑡 = 𝛾𝛿(𝐴𝑡 )𝛼𝑊𝑡 𝛼−1
Solving for 𝐶0 (𝐴𝑡 ) will derive equation (24)
12
Appendix 3: Proof of Proposition 2
The law of motion of the percent change in consumption is:
dCt
 dC 
2
  c dt   c dz t  var 
   c dt
Ct
 C 
The differential on consumption is:
dc  CW (W , A)dW  C A (W , A)dA
Inserting the consumption differential formulation into the variance yields:
1
 dC 
1

  var  CW (W , A)dW  C A (W , A)dA  
C
 C 
C

 c2 dt  var 
2
2
2
C 
C 
1
  W  var( dW )   A  var( dA)  2  CW C ACov( dW , dA) 
C 
 C 
 C 
2
1
𝜋
= { [𝐶𝑊 ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑋 + 𝑔(𝐴)𝐶𝐴 ]} 𝑑𝑡
𝐶
√6
Therefore,
𝜎𝑐 =
1
𝐶
𝜋
[𝐶𝑊 ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑋 + 𝑔(𝐴)𝐶𝐴 ]
√6

Next, I calculate the partial derivatives with respect to W and A on equation (17). Thus,
VW W (W , A)  u (C , C )CW ,
VW A(W , A)  u (C , C )C A .
and
Now, inserting the above two equations and equation (17) into (18) will result in the following:
(22)
𝜋2
𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 )(1 − 𝜏𝑐 )(1 − 𝜏𝑑 )(𝐴𝑡 + 𝜀) + 𝑉𝑊𝑊 (𝑊𝑡 , 𝐴𝑡 ) ( 6 ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 𝑋𝑡 +
+𝑉𝑊𝐴 (
𝜋
) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑔(𝐴𝑡 ) = 0
√6
𝜋2
𝑢′ (𝐶𝑡 , 𝐶𝑡 )(1 − 𝜏𝑐 )(1 − 𝜏𝑑 )(𝐴𝑡 + 𝜀) + 𝑢′′ (𝐶𝑡 , 𝐶𝑡 )𝐶𝑊 ( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 𝑋𝑡 +
6
+𝑢′′ (𝐶𝑡 , 𝐶𝑡 )𝐶𝐴 (
𝜋
) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑔(𝐴𝑡 ) = 0
√6
13
thus
𝑢′ (𝐶𝑡 , 𝐶𝑡 )(𝐴𝑡 + 𝜀) + 𝑢′′ (𝐶𝑡 , 𝐶𝑡 ) {𝐶𝑊 (
𝜋2
𝜋
) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑋𝑡 + 𝐶𝐴 ( ) 𝑔(𝐴𝑡 )} = 0
6
√6
Dividing both sides by 𝑢′ (𝐶𝑡 , 𝐶𝑡 ), and using the expression for  c and the coefficient of
Absolute risk Aversion found above I have the following:
(𝐴𝑡 + 𝜀) +
𝑢′′ (𝐶𝑡 , 𝐶𝑡 )
𝜋2
𝜋
{𝐶𝑊 ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑋𝑡 + 𝐶𝐴 ( ) 𝑔(𝐴𝑡 )} = 0
6
𝑢′ (𝐶𝑡 , 𝐶𝑡 )
√6
hence
(𝐴𝑡 + 𝜀) +
𝛼−1 𝜋
( ) 𝜎𝑐 𝐶𝑡 = 0
𝐶𝑡
√6
Solving for 𝜎𝑐 I find equation (27).
Appendix 4: Proof of Proposition 3
Insert equations (23) into (18)
(18)
𝜋2
𝑉𝑊 (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )(𝐴𝑡 + 𝜀) + 𝑉𝑊𝑊 ( 6 ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 𝑋𝑡 +
𝜋
+𝑉𝑊𝐴 ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑔(𝐴𝑡 ) = 0
√6
and from equation (20)
𝑔(𝐴𝑡 ) = 𝑔0 𝐴𝑡
since from equation (23)
(23)
𝑉(𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝑊𝑡 𝛼
the partial derivatives are:
𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝛼𝑊𝑡 𝛼−1
𝑉𝑊𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝛼(𝛼 − 1)𝑊𝑡 𝛼−2
𝑉𝑊𝐴 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿 ′ (𝐴𝑡 )𝛼𝑊𝑡 𝛼−1
14
I insert the above partial derivatives and equation (27) into equation (18) and derive
𝑋
6
1
(29) 𝑊𝑡 = (√𝜋 ) (𝛼−1)(1−𝜏
𝑡
𝑐 )(1−𝜏𝑑 )
[−𝐴𝑡 −
𝑔(𝐴𝑡 )𝛿 ′(𝐴𝑡 )
]
𝛿(𝐴𝑡 )
√6
1
= ( 𝜋 ) (𝛼−1)(1−𝜏
𝑐 )(1−𝜏𝑑 )
[(𝛼 − 1)𝜎𝑐 −
𝑔(𝐴𝑡 )𝛿 ′ (𝐴𝑡 )
]
𝛿(𝐴𝑡 )
Remember
 t2  1  t 2  c2  t 2  W2  2(1  t )t c W 
where
 t is the standard deviation of At ,which is the price of risk.
 W is the standard deviation of the stock market.
 c is the standard deviation of consumption.
 t is the portion of the portfolio that is invested in risky assets (i.e. the stock market).
From the above equation I can see that when t  1 then  t   W .
Since I defined X t   ttWt thus when t  1 and  t   W I get X t   W 1Wt
Noting that  W 
X
and 𝜎𝑐 =
W
𝐴𝑡
1− 𝛼
from proposition 2, (29) becomes (28).
Appendix 5: Proof of Proposition 4
Insert equations (1), (20)-(24), (27), (28) and (30) into (19). Then I can derive a differential
equation for the function  ( A) . That is,
(21)
𝐶𝑡 = 𝐶0 (𝐴𝑡 )𝑊𝑡
(22)
𝑋𝑡 = 𝑋0 (𝐴𝑡 )𝑊𝑡
(23)
(24)
(20)
𝑉(𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝑊𝑡 𝛼
𝐶0 (𝐴𝑡 ) = (𝐶𝑡 )
−𝛽
𝛼−1
1
[𝛾𝛿(𝐴𝑡 )𝛼]𝛼−1
𝑑𝐴𝑡 = 𝑓0 𝐴𝑡 𝑑𝑡 + 𝑔0 𝐴𝑡 𝑑𝑍𝑡
That is 𝑓(𝐴𝑡 ) = 𝑓0 𝐴𝑡 and 𝑔(𝐴𝑡 ) = 𝑔0 𝐴𝑡
(1)
1
𝑢(𝐶𝑡 , 𝐶𝑡 ) = 𝛼 𝐶𝑡 𝛼 𝐶𝑡
15
𝛽
Using equations (1), (21) and (24) I get:
𝑢(𝐶𝑡∗ , 𝐶𝑡 )
𝛽
𝛼
1 ∗ 𝛼
𝛽
1
𝛽
1
(
)
1−𝛼
𝛼
𝛼
𝛼−1
(𝑊𝑡 ) (𝐶𝑡 )
= (𝐶𝑡 ) (𝐶𝑡 ) = [𝐶0 (𝐴𝑡 )𝑊𝑡 ] (𝐶𝑡 ) = [𝛾𝛿(𝐴𝑡 )𝛼]
𝛼
𝛼
𝛼
From (23) the partial derivatives are calculated:
𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝛼𝑊𝑡 𝛼−1
𝑉𝑊𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝛼(𝛼 − 1)𝑊𝑡 𝛼−2
𝑉𝑊𝐴 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿 ′ (𝐴𝑡 )𝛼𝑊𝑡 𝛼−1
𝑉𝐴 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿 ′ (𝐴𝑡 ) 𝑊𝑡 𝛼
𝑉𝐴𝐴 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿 ′′ (𝐴𝑡 )𝑊𝑡 𝛼
using equation (30)
(30)
𝛿(𝐴𝑡 ) = 𝛿0 𝐴𝑡 𝛼−1
thus
𝛼−1
𝛿 ′ (𝐴𝑡 ) = 𝛿0 (𝛼 − 1)𝐴𝑡 𝛼−2 = 𝛿(𝐴𝑡 ) (
)
𝐴𝑡
𝛿′′ (𝐴𝑡 ) = 𝛿0 (𝛼 − 1)(𝛼 − 2)𝐴𝑡
𝛼−3
= 𝛿(𝐴𝑡 )
(𝛼 − 1)(𝛼 − 2)
𝐴2𝑡
Now, insert the above equations into (19)
(19)
ρ𝑉(𝑊𝑡 , 𝐴𝑡 ) = 𝑢(𝐶𝑡∗ , 𝐶𝑡 )
+ 𝑉𝑊 {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋𝑡∗ + 𝜀𝑋𝑡∗ ] − 𝜏𝑐 (1 − 𝜏𝑑 )[𝐴𝑡 𝑋𝑡∗ + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋𝑡∗ ] − 𝐶𝑡∗ }
1
𝜋2
1
)
+ 𝑉𝐴 𝑓(𝐴𝑡 + 𝑉𝑊𝑊 ( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 (𝑋𝑡∗ )2 + 𝑉𝐴𝐴 [ 𝑔(𝐴𝑡 )]2
2
6
2
𝜋
+ 𝑉𝑊𝐴 𝑔(𝐴𝑡 ) ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋𝑡∗
√6
16
thus I get:
(31)
ρ𝛾𝛿(𝐴𝑡 )𝑊𝑡 𝛼 =
1
𝛽
[𝐶0 (𝐴𝑡 )𝑊𝑡 ]𝛼 (𝐶𝑡 ) +
𝛼
+𝛾𝛿(𝐴𝑡 )𝛼𝑊𝑡 𝛼−1 {(1 − 𝜏𝑑 )[𝑟𝑊𝑡 + 𝐴𝑡 𝑋0 (𝐴𝑡 )𝑊𝑡 + 𝜀𝑋0 (𝐴𝑡 )𝑊𝑡 ]
− 𝜏𝑐 (1 − 𝜏𝑑 )[𝐴𝑡 𝑋0 (𝐴𝑡 )𝑊𝑡 + 𝜔𝑡 𝑊𝑡 𝑟 + 𝜀𝑋0 (𝐴𝑡 )𝑊𝑡 ] − 𝐶0 (𝐴𝑡 )𝑊𝑡 } +
2
𝛼−1
1
𝛼−2 𝜋
+𝛾𝛿(𝐴𝑡 ) (
) 𝑊𝑡 𝛼 𝑓0 𝐴𝑡 + 𝛾𝛿(𝐴𝑡 )𝛼(𝛼 − 1)𝑊𝑡
( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 [𝑋0 (𝐴𝑡 )𝑊𝑡 ]2
𝐴𝑡
2
6
(𝛼 − 1)(𝛼 − 2) 𝛼
1
+ 𝛾𝛿(𝐴𝑡 )
𝑊𝑡 [ 𝑔0 𝐴𝑡 ]2 +
2
𝐴2𝑡
+ 𝛾𝛿(𝐴𝑡 ) (
𝛼−1
𝜋
) 𝛼𝑊𝑡 𝛼−1 𝑔0 𝐴𝑡 ( ) (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋0 (𝐴𝑡 )𝑊𝑡
𝐴𝑡
√6
I insert equations (24) and (30) into equation (31)
−𝛽
1
(24)
𝐶0 (𝐴𝑡 ) = (𝐶𝑡 )𝛼−1 [𝛾𝛿(𝐴𝑡 )𝛼]𝛼−1
(30)
𝛿(𝐴𝑡 ) = 𝛿0 𝐴𝑡 𝛼−1
and arrive at
1
1
1
(31) 0 = −ρ + 𝐴𝑡 {𝛼 (𝛼−1) 𝛾 (𝛼−1) 𝛿0 (𝛼−1) 𝐶𝑡
−𝛼𝐶𝑡
(
𝛽
)
1−𝛼
(
𝛽
)
1−𝛼
+ 𝛼(1 − 𝜏𝑑 )𝑋0 (𝐴𝑡 ) − 𝛼𝜏𝑐 (1 − 𝜏𝑑 )𝑋0 (𝐴𝑡 ) −
1
[𝛾𝛿0 𝛼]𝛼−1 } +
+𝛼 {(1 − 𝜏𝑑 )[𝑟 + 𝜀𝑋0 (𝐴𝑡 )] − 𝜏𝑐 (1 − 𝜏𝑑 )[𝜔𝑡 𝑟 + 𝜀𝑋0 (𝐴𝑡 )]} + (𝛼 − 1)𝑓0
1
𝜋2
2
2
2
+ 𝛼(𝛼 − 1)(1 − 𝜏𝑑 ) (1 − 𝜏𝑐 ) [𝑋0 (𝐴𝑡 )] ( ) +
2
6
1
𝜋
+ (𝛼 − 1)(𝛼 − 2)[ 𝑔0 ]2 + 𝛼(𝛼 − 1)𝑔0 (1 − 𝜏𝑑 )(1 − 𝜏𝑐 )𝑋0 (𝐴𝑡 ) ( )
2
√6
I would like to verify that 𝑋0 (𝐴𝑡 ) does not depend on 𝐴𝑡 .
17
Thus, I solve equation (31) for 𝑋0 (𝐴𝑡 ) . After equating the coefficient of 𝐴𝑡 to zero (to eliminate
the direct effect of 𝐴𝑡 on 𝑋0 (𝐴𝑡 ) ), I am left with
1
1
1
0 = 𝛼 (𝛼−1) 𝛾 (𝛼−1) 𝛿0 (𝛼−1) 𝐶𝑡
𝛽
(
)
1−𝛼
+ 𝛼(1 − 𝜏𝑑 )𝑋0 (𝐴𝑡 ) − 𝛼𝜏𝑐 (1 − 𝜏𝑑 )𝑋0 (𝐴𝑡 ) −
−𝛼𝐶𝑡
𝛽
(
)
1−𝛼
1
[𝛾𝛿0 𝛼]𝛼−1
And solving for 𝑋0 (𝐴𝑡 ) yields a function that does not depend on 𝐴𝑡 as per my conjecture.
𝑋0 (𝐴𝑡 ) =
(𝛼)
(
𝛽
2−𝛼
1
)
(
) (1−𝛼)
𝛼−1 (𝛾𝛿0 ) 𝛼−1 𝐶𝑡
(𝛼
− 1)
(1 − 𝜏𝑑 )(1 − 𝜏𝑐 )
Appendix 6: deriving equation (34)
Insert 𝐴𝑡 =
𝜇𝑡 − 𝑟
𝜎𝑡
,
𝑋𝑡 = 𝜎𝑡 𝜔𝑡 𝑊𝑡 ,
𝑔(𝐴𝑡 ) = 𝐴𝑡 𝑔0
equations (30) and (23) into equation (18)
(18)
𝜋2
𝑉𝑊 (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )(𝐴𝑡 + 𝜀) + 𝑉𝑊𝑊 ( 6 ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 𝑋𝑡 +
+𝑉𝑊𝐴 (
𝜋
) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑔(𝐴𝑡 ) = 0
√6
given
𝑉(𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝑊𝑡 𝛼
(23)
𝛿(𝐴𝑡 ) = 𝛿0 𝐴𝑡 𝛼−1
(30)
The partial derivatives are:
𝑉𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝛼𝑊𝑡 𝛼−1
𝑉𝑊𝑊 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿(𝐴𝑡 )𝛼(𝛼 − 1)𝑊𝑡 𝛼−2
𝛼−1
𝑉𝑊𝐴 (𝑊𝑡 , 𝐴𝑡 ) = 𝛾𝛿 ′ (𝐴𝑡 )𝛼𝑊𝑡 𝛼−1 = 𝛾𝛿(𝐴𝑡 ) (
) 𝛼𝑊𝑡 𝛼−1
𝐴𝑡
18
Inserting the above into equation (18)
𝛾𝛿(𝐴𝑡 )𝛼𝑊𝑡
𝛼−1
(1 − 𝜏𝑐 )(1 − 𝜏𝑑 )(𝐴𝑡 + 𝜀) + 𝛾𝛿(𝐴𝑡 )𝛼(𝛼 − 1)𝑊𝑡
𝛼−2
𝜋2
( ) (1 − 𝜏𝑑 )2 (1 − 𝜏𝑐 )2 𝜎𝑡 𝜔𝑡 𝑊𝑡 +
6
𝛼−1
𝜋
+𝛾𝛿(𝐴𝑡 ) (
) 𝛼𝑊𝑡 𝛼−1 ( ) (1 − 𝜏𝑐 )(1 − 𝜏𝑑 )𝑔0 𝐴𝑡 = 0
𝐴𝑡
√6
Inserting 𝐴𝑡 =
𝜇𝑡 −𝑟
𝜎𝑡
and solving for   r will yield equation (34).
Appendix 7: deriving equation (39)
using equation (33) I can calculate 𝜎𝑊 − 𝜎𝑐
(33)
𝜎𝑊 =
1
√6
( ) [𝜎𝑐
(1−𝜏𝑐 )(1−𝜏𝑑 ) 𝜋
− 𝑔0 ]
Inserting equation (33) into 𝜎𝑊 − 𝜎𝑐 I derive equation (39).
19
References
AKDENIZ, L. and W.D. DECHERT (2007), “The Equity Premium in Brock’s Asset Pricing
Model,” Journal of Economic Dynamics and Control, vol. 31, pp. 2263-2292
BROCK, W. A. (1982), “Asset Prices in a Production Economy,” in The Economics of
Information and Uncertainty, ed. by J. J. McCall, pp.1-46. The University of Chicago Press,
Chicago.
BORLAND, L. and BOUCHAUD, J.P. (2004), “A non-Gaussian option pricing model with
skew”, Quantitative Finance, Volume 4, 499-514.
CAMPBELL, J.Y. and COCHRANE, J.H. (1999), “By Force of Habit: A ConsumptionBased Explanation of Aggregate Stock Market Behavior”, The Journal of Political Economy,
Volume 107, Issue 2, 205-251.
CHOI, S., GIANNIKOS, C., FRANCIS, J. C., and PETSAS, I. (2013), “Dynamic Asset
Pricing Model with Demonstration Effect”, A working paper.
CONSTANTINIDES, G. M. (1990), “Habit Formation: A resolution of the Equity Premium
Puzzle,” Journal of Political Economy 98 (3): 519-543.
DUESENBERRY, J.S. (1949), Income, Saving and the Theory of Consumer Behavior,
Harvard University Press, Cambridge.
FRIEND, I. and M.E. BLUME, (1975), The demand for risky assets, American Economic
Review 65, 900-922.
KATTEL, R., KREGEL, J.A, and REINER, E.S. 2009, Radgnar Nurkse: Trade and
Development, Anthem, London. Edited Collection of Nurkse's key works.
KOCHERLAKOTA, N. R. (1996), “The Equity Premium: It's Still a Puzzle,” Journal of
Economic Literature 34 (1): 42-71.
LUCAS, R. E. (1978), “Asset Prices in an Exchange Economy,” Econometrica, 46, 14291445.
MEHRA, R., and E.C. PRESCOTT (1985), “The Equity Premium: A Puzzle”. Journal of
Monetary Economics 15: 145-161.
MCGRATTAN, E. R., and E.C. PRESCOTT (2001), “Taxes, Regulations, and asset Prices”.
Working paper 610, Federal Reserve Bank of Minneapolis.
NURKSE, R.. 1953, Problems of Capital Formation in Underdeveloped Countries,
Blackwell, Oxford.
20
SHILLER, R. J, (1981), “Do Stock Prices Move Too Much to be Justified by Subsequent
Changes in Dividends? “American Economic Review, 71 (3): 421-436.
VEBLEN, T. 2005, Conspicuous Consumption, Penguin Books, London.
21