Asset Pricing Model and Economic Activity of Firms
PITHAK SRISUKSAI
School of Economics,
Sukhothai Thammathirat Open University,
9/9 Changwattana Road, Pakkret District, Nonthaburi, 11120,
THAILAND
Abstract: - This study derives the asset pricing model by introducing the economic activity of firms in the
business cycle model which explores the expected returns of stocks and sheds light on the equity premium risk.
Such a model follows the discrete-time optimization to come up with the asset pricing model that includes the
economic activity variable. The result shows that the considerable factors affecting the rate of stock returns at a
time
1t
are the rate of time preference, the firm investment at a time
1t
, the stock price, and the growth
rate of private consumption at the time
t
. Therefore, the economic activity of firms influences the expected
returns on stock in a positive direction. In contrast, the growth rate of consumption has the opposite impact on
the expected rate of stock returns.
Key-Words: - Asset Pricing, Optimization, General Equilibrium, Bellman Equation, Euler Equation, Taylor
Approximation
Received: May 4, 2023. Revised: August 5, 2023. Accepted: August 26, 2023. Published: September 26, 2023.
1 Introduction
The correlation between stock price and
macroeconomic variables, especially aggregate
consumption is still challenged for investment
decision-making in the stock markets. Since the
funds to be invested are expected to generate high
returns later, they should be the remaining income
from consumption or the savings from postponing
consumption to the future. If households bring any
funds to invest in the stock market, they expect that
the stock price should be low at the time they buy,
and it will rise at the time they sell. In other words,
the asking price of the stock should be higher than
the bid price of one to generate returns for investors.
Investing in the stock market is important to
households because they want to allocate scarce
resources for smooth consumption over time. That
is, increasing or decreasing in the current
consumption will affect the future consumption.
This is why all stocks have high returns during the
period of extremely volatile consumption. On the
other hand, they have low returns through low and
smooth consumption, for instance, insurance.
Moreover, the investment in the stock market is the
loss of marginal utility from reducing the current
consumption and buying equity stocks at current
prices. It is similar to the expected benefit from the
marginal utility of consumption on the conditional
forecast that the next period’s consumption will
increase from the future sale of the stocks. That's
why each type of stock has different returns, i.e. any
stocks, in good times and high consumption level, or
less marginal utility of consumption, are therefore
less desirable than stocks in bad times and a low
level of consumption, or highly marginal utility as
[1], [2], [3], [4]. Thus, the consumption in each
period regularly affects the stock prices differently.
In addition, there are several products for
consumption in a good time which cause less useful
stocks than ones in a bad time. This situation leads
the stock prices during good times to be lower than
another one. As a consequence, the expected returns
in good times are always higher than the expected
returns in bad times. In summary, the stock prices
have different relations with consumption in each
period. The existing challenge of micro-foundation
of asset pricing is still the relationship between
stock price and consumption in each period.
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Table 1. Annual Equity Premium for Major
Countries
Country
Period
Real
Market
Return
(%)
Relatively
Riskless
Return (%)
Equity
Premiu
m (%)
U.S.
1889-
2005
7.67
1.31
6.36
U.K.
1900-
2005
7.4
1.3
6.1
Japan
1900-
2005
9.3
-0.5
9.8
Germany
1900-
2005
8.2
-0.9
9.1
France
1900-
2005
6.1
-3.2
9.3
Sweden
1900-
2005
10.1
2.1
8.0
Australia
1900-
2005
9.2
0.7
8.5
India
1900-
2004
12.6
1.3
11.3
Source: [3], [5], [6].
Such correlation helps to examine the impact of
aggregate consumption on stock returns.
Furthermore, the current price of stock has an
exactly inverse relationship with the expected return
of a stock. Equally importantly, the effects of
changes in aggregate consumption on changes in
stock returns produce asset pricing and equity
premium model. Table 1 documents the difference
between the annual returns on risky assets and the
annual returns on risk-free assets, which is
particularly known as the equity risk premium. It is
illustrated that the equity risk-premium of annual
returns occurred in eight major stock exchanges
over the last 105 years, which is a comparison
between the annual return on the stock market of
each country and the return on a relatively riskless
security. This turns out that the excess returns to
equity holdings of the Indian capital market were
the highest premium at 11.3 percent, followed by
Japan (9.8), France (9.3), Germany (9.1), Australia
(8.5), Sweden (8.0), the U.S. (6.36), and the UK
(6.1), respectively. Concerning Thailand’s equity
premium, the stock returns have been highly volatile
over the last 25 years. That is, it had positive
monthly returns in some periods; in turn, it showed
negative value in other periods, especially during
2008–2009. Moreover, when comparing the
monthly equity returns with the yields of the one-
month treasury bills between 1995-2019, the equity
premium which turned out to the positive and a few
excess returns was about 0.89 percent per month.
The average rate of stock return of the stock
exchange of Thailand during that period was 1.08
percent per month, while the yield on the one-month
Treasury bills (risk-free rate of return) was 0.19
percent per month. It implies that the risk premium
of Thailand’s capital market is considerably lower
than the significant countries.
Explaining the stock returns and equity
premiums of such stocks has significantly resulted
in the model development of exploring the
relationship between stock prices and aggregate
consumption. Such a relationship is still very
challenging which is based on the concept that
households will postpone their current consumption
for future consumption by bringing the remaining
resources at the present to invest or save. As a
result, they expect that the rewards will later be
obtained in the future. The well-known model is
commonly referred to as the Consumption-based
Capital Asset Pricing Model (C-CAPM).
Nevertheless, the development of the C-CAPM is
still flawed. This is because such a model cannot
account for the equity returns and the equity
premium in the U.S., Taiwan, South Korean, and
Thailand stock markets. This is why the exploration
of the relationship between aggregate consumption
and the stock returns in explaining the equity
premium via the development of the C-CAPM
remains a major challenge for economists who have
motives to shed light on the link between economic
activity and the returns of stocks. As a result, this
assertion, based on the derived model of financial
economics to reveal the rate of stock returns and the
pattern of excess returns to equity holdings that are
related to economic activity, is very valuable for the
asset pricing model.
2 Literature Review
The previous studies related to the equity premium
are mainly theoretical research. Initially, the most
well-known paper of, [7], demonstrates the
correlation between stock price and consumption in
an endowment economy similar to, [8]. The only
difference in both studies is the assumption of
endowment, [7], assumed that the endowment levels
evolved according to a Markov process, but, [8],
assumed that the growth rate of endowment changed
gradually following the Markov process. However,
the remarkable results on the equity premium of the
two models are the same. In other words, [9],
showed that any security with negative covariance
between the stochastic discount factor and the stock
returns led the expected rate of stock returns to be
higher than the rate of returns on risk-free securities.
As shown by, [10], any asset that depended on the
covariance between the growth rate of aggregate
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consumption and the gross rate of return paid off a
higher expected rate of returns than the risk-free rate
for bearing risk. That is, asset payoff co-varies
positively with consumption. Thus, this implied that
an asset return is high if its marginal utility at a time
1t
is low. Conversely, an asset return is low if the
marginal utility at a time
1t
is high. More
importantly, the work of, [8], also found that the
excess returns in the model economy were higher
than the ones in the U.S. economy. In fact, for the
actual data over the period 1889-1978, the risk
compensation from the economic model was 0.35
percent. Unlike the risk premium from the U.S.
stock market, it equals 6.18 percent. Therefore, the
difference between these compensations is called
the “equity premium puzzle”. Table 1 documents
the equity premiums in eight major stock exchanges
in the past 115 years, which are still a puzzle.
Many studies have attempted to develop models
to explain the equity premium puzzle, but there are
no financial economics models to appropriately
account for such premiums, [9], [11], developed an
asset pricing model by changing the standard utility
function to a power utility function. The finding of
model testing with the General Moment Method
(GMM) stated that the pricing model did not fit the
equity premium from 1978 to 1995. In addition, the
C-CAPM pricing model was further derived to shed
light on the risk premium of stocks by combining
the production function with the household utility
function in a general equilibrium model. The study,
[10], derived a financial model by adding a habit
formation and capital adjustment cost into a real
business cycle model. As a result, such a model well
explains the risk compensation and the stock
returns. However, if habit formation or capital
adjustment cost was added, the risk premium of
securities was not explained suitably as before. In
addition, [4], found that taking account of the bid-
ask spread variable in a model of, [7], the equity
premium could be better described than the C-
CAPM of, [8]. In, [1], the authors also explored that
an unexpected idiosyncratic risk was a key factor in
determining stock returns due to an insufficient risk
diversification of securities. Even though most
investors invested in a large number of stocks to
eliminate the unsystematic risk of each stock, the
number of stocks was not enough to completely get
rid of these risks. Moreover, the speculators who
tried to seek an abnormal pricing of stocks faced the
specific risks of the stocks and the unusual events
affecting the stock price.
In addition, the equity risk premium is still
challenged concerning the financial economics
model. In, [2], the study examined Lucas's C-CAPM
to shed light on the equity premium in Taiwan and
South Korea’s capital markets. The result
demonstrated that such a model could not explain
the stock returns and the equity premium. In, [12],
the study attempted to test the C-CAPM with a
Thailand data set for the period 1980-1989. The
findings illustrated that the risk-free rate of returns
based on the derived model was more than the one
based on the actual data set. This led the study to
conclude that the C-CAPM may not be correct.
Consistent with the findings of, [13], there was no
equity premium puzzle in the Stock Exchange of
Thailand over the period 1986 – 1996. Not
surprisingly, [14], took into account the asset
pricing model with the money supply variable;
however, it did not fully describe the risk
compensation of the Thai stock market.
3 Research Methodology
The asset pricing model is derived from a pricing
model related to the economic activity of the firm
under the real business cycle model to describe the
rate of stock returns and compensation for bearing
the risk of stocks. This paper carries out the research
by using mathematical methods and discrete time
optimization to develop an asset pricing model
within a general equilibrium analysis under an
imperfect competition market. In other words, this is
a model set up to determine the price of stocks with
economic activity variables in the stock market. by
applying the Lagrangian equation, and Bellman
equation and calculating Euler’s equation and the
Envelope condition before calculating market
equilibrium and its application to stock price.
4 The Model
The economic environment based on this model set-
up consists of representatives of two economic
sectors as follows: 1) the Infinitely-lived
homogenous households and 2) the Infinitely-lived
heterogeneous firms in the economic system.
Furthermore, there is only one type of investment
stock in this economy, namely common stock.
Hence, an infinitely representative household
maximizes the expected lifetime utility subject to
periodic budgetary constraints at each time, and
firms with different characteristics (the infinitely
heterogeneous firms) maximize the present value of
expected cash flows subject to their budget
constraints. Both households and firms carry out all
economic activities in a perfectly competitive
market, thus all prices are taken as given. The
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homogeneous households must decide how much to
consume at each period, and how much to invest in
stocks at each period. The households will receive
money from labor wages, common stocks, and
dividend payments at any time
t
. At the same time,
those firms must decide on the amount of dividends
to be paid to households, the number of workers to
be hired to work, and decide on the investment
amount of the firm to allocate funds from which the
firms are financed by debts and the sale of produce.
Therefore, all agents in both sectors are
optimizations together which leads to effective
resource allocation under the general equilibrium in
this economy.
4.1 Household
The economic model is an extension of the work of,
[15]. There are infinitely-lived identical households
that exist forever. Hence, a model describing the
economic behavior of all households can be
represented by a single agent. Moreover, under the
limited time of the household, it is divided into
leisure time
t
l
and working time,
t
h
. For simplicity,
1lh
tt
Therefore, the utility function of a
representative household can be defined over
stochastic sequences of consumption and leisure as
the following equation.
0
,1
t
t t t
t
E U c h
;
01
(1)
Where
t
E
is the expectation operator
conditional on information available at time
t
.
t
c
stands for the consumption at the time
t
.
t
h
represents the hours worked at the time
t
.
denotes the subjective discount factor.
The household utility function is a curved
function derived from the change in consumption.
and changes in work. This implies that the first and
second partial derivatives of the utility function with
respect to both arguments are as follows:
0, 0, 0, 0
c h cc hh
U U U U
and
20
cc hh ch
U U U
. Considering the budget
constraints, a representative household receives
income from wages, stock selling, and dividend
payment at the time
t
He or she will allocate for
consumption, investment, and payment of lump-sum
taxes. The investment in this economy is the only
type of investing in equity stocks at the time
1t
.
Therefore, the equation expressing the household
budget constraint can be written as follows:
1t t it it it it it it t t
i i i
w h b s d p s p c T
(2)
Denote
i
as firm
i
.
t
w
is the wage rate at the
time
t
.
it
p
represent the price of equity stock
i
at
time
t
.
it
d
represents the dividend payment received
from stock
i
at the time
t
.
it
s
represents the equity
stocks for the firm
i
at the time
t
.
i
T
are lump-
sum taxes financing the tax benefits received by
firms.
Taking all prices as given, the representative
household will choose the consumption at the time
t
, the number of working hours at the time
t
, and
investing in common stocks at the time
1t
to
maximize the expected discount utility function
subject to budget constraints. This leads to the
optimal choices of the first-order conditions and the
solution for the optimization problem is the Euler
Equations as follows:
1
,, 0
max ,1
t t it
t
t t t
c h s t
E U c h
;
01
(3)
subject to
1t t it it it it it it t t
i i i
w h b s d p s p c T
Euler Equations are as follows:
,1
,1
h t t t
c t t
U c h w
U c h
(4)
11
11
,1 ,1
it it
t c t t c t t
it
dp
E U c h U c h
p
(5)
Denote
1
s
it
R
as the returns on stock
i
at
time
1t
, then it can be define as
11
1
sit it
it
it
pd
Rp
(6)
Substituting Equation 6 into Equation 5, then
the Euler Equation becomes
11
1
,1 1
,1
c t t s
t it
c t t
U c h
ER
U c h
(7)
Equation 4 shows that the wage rate is equal to
the expected value of the proportion of marginal
utility of working hours and the marginal utility of
consumption. or the wage rate equals the marginal
rate of substitution between working hours and
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household consumption at the time
t
. Importantly,
Equation 7 expresses that the expected value of the
marginal rate of intertemporal substitution between
the next period consumption and the consumption at
time
t
equals the inverse of the stock return.
4.2 Firms
In this economy, there are also infinitely
heterogeneous firms that produce a large amount of
consumption goods through their production
function. They take labor
it
h
and capital
it
k
as
factors of production. The capital depreciation rate is
. Additionally, all firms face idiosyncratically
stochastic productivity,
it
, according to, [16].
Therefore, the production function is the following.
1
,,
it it it it it it
F k h k h
(8)
it
k
represents the capital for the firm
i
at
time
t
.
it
h
is the labor for the firm
i
at the time
t
.
it
is idiosyncratically stochastic risk of the firm
i
at
the time
t
.
is a capital share. Such stochastic risk
is assumed further to follow a first-order
autoregressive Makov process.
1it it it
;
2
0,
it N
(9)
;
01
it
is independently and identically
distributed for the firm
i
at a time
t
with mean zero
and constant variance, i.e.
2
0,
it N
. Firms
i
accumulate capital through investment as follows.
11
it it it
k k I
(10)
it
I
is the investment of the firm
i
at time
t
. A firm
that has an adjustment cost is equal to
it it
it
Ik
k
,
whose function is characterized by a decreasing
return to scale in capital. For simplicity, this study is
defined
as a deterministic function in which
technology shocks can be incorporated into the
model. When each firm carries out its business by
maximizing the value of the firm that is equal to the
present value of future cash flows. Consequently, the
maximization problem of firms can be written in the
form of a recursive equation in which a firm will
maximize its market value as follows:
0 0 0
,L 0
, max
it it
i t it
It
V k E M D
(11)
t
M
denotes the stochastic factor.
it
D
defines as
dividend payment of a firm
i
at time
t
for holding
equity stock. Then,
it
it it it t it
it
I
D Y k w h
k
(12)
Defined
tt
WW
as the process of equilibrium
wage. Therefore, given
, , ,
it it t t
kH
as the state
variable. Let’s denote
t
H
as the summary of the
next period information.
1
,,
it it it
I k h
are the control
variables. Hence, the Bellman Equation can be
written as the following.
1
,L
111
, max
tt
it
t it it it it it t it
Iit
t
t t it
t
I
V k k h k w h
k
M
E V k q
M
(13)
subject to
11
it it it
k k I
(14)
1it it it
(15)
The first-order conditions are computed to find the
optimality. Moreover, Euler Equations and Envelope
conditions are solved to get the producer
equilibrium. Thus,
1
*111
it
it t
t k t it
it t
IM
E V k
kM
(16)
*11
1
1
t it
it t
t
it t it
Vk
IM
E
k M k
(17)
The production in this economy also assumed
that the outputs come from constant returns to scale
of production function and investment technologies
following the Q-theory of Investment as, [17], [18].
This means that the marginal
q
is equal to the
average. Therefore,
t it t it
it it
V k V k
kk
(18)
Denote that
111
t
t t t it
t
M
p E V k
M
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Thus,
*
1
it it
it it
Ip
kk
(19)
Equation 19 reveals that the ratio between the
optimal investment rate of a firm and its marginal q.
That is, it is an example of the relationship between
the economic activity of the firm and its stock price.
In addition, this means that the investment
adjustment costs of a firm are significant for an asset
pricing model to account for the empirically
plausible volatility of stock returns. If
it it
it it
II
kk
so, then the unit price of capital equals one.
Therefore,
1it it
pk
(20)
4.3 Equilibrium
The modeled economy derives from the household’s
resource allocation, the firms’ resource allocation,
and market-clearing conditions. In terms of the
product market, the equilibrium in the product
market can be displayed as
t t t
C I Y
(21)
1
11
t it it it it it
ii
C k k k h
(22)
where
it t
i
kk
it t
i
hh
The stock market:
1
it
i
s
(23)
In a competitive market, all prices are taken as
given as follows: the stock prices
()
t
p
, wage rates
()
t
w
, investment allocation at the time
t
, working
hours at the time
t
, capital at the time
1t
,
consumption at the time
t
, investment in the stock
market at the time
1t
,
11
0
, , , ,
t t t t t t
I h k c s
.
Thus, the household’s decision and firm’s decision
satisfy the optimal condition, the stochastic discount
factor equals the intertemporal marginal rate of
substitution between consumption at time
1t
and
consumption at time
t
.
4.4 Asset Price Implication
An asset pricing model can be derived from the
Euler Equation 7 which is the standard asset pricing
model. To simplify the model of stock returns with
economic activity, the utility function with constant
elasticity of the substitution function is defined as
follows:
11
( ) ,0
1
t
tc
Uc
(24)
Where
is the relative Risk Aversion parameter.
Define gross return on stock as
11
1
sit it
it
it
pd
Rp
Then, Equation 7 can be rearranged as the following:
1 1 1
1t it it
t
t it
c p d
Ecp
(25)
Once
1it it
pk
; hence, Equation 25 can be written
as follows:
11 1 1
t
t it it it
t
c
E p d k
c
(26)
Rearranging the Equation 26, we get
11
1s
t c it it it
E g R p k
(27)
Denote
1
1
, so
11
(1 ) 1 s
it t c it it
k E g R p
(28)
Where
is the rate of time preference.
As a result, Equation 28 can be written in
the form of the log-linearized equation of expected
stock returns as follows. Define
x
as any variables,
then
1t
x
t
x
gx
.
ˆt
x
stands for a deviation from the
steady state of
x
at the time
t
, such that
ˆt
txx
xx
. Equation 28 is approximated by
applying the method of Taylor’s Approximation;
hence, expected stock returns becomes
11
ˆ
ˆˆˆ
12
s
t it it c it
E R k g p
(29)
As can be seen, Equation 28 and Equation
29 represent the factors that affect the stock returns,
namely the rate of time preference, the investment at
the time
1t
, the stock price at the time
t
, and the
growth rate of aggregate consumption. This implies
that the economic activities of firms have a positive
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impact on the future return on stocks, and the effect
of consumption growth rate on stock returns is a
positive direction. In contrast, the influence of
current stock prices on stock returns is negative.
5 Conclusion
The objective of this study is to examine the asset
pricing model with the economic activities derived
from the business cycle model for describing the
rate of return and risk premium of common stocks.
Such a model shows the relationship between the
economic activities and the stock returns in forms of
nonlinearity and linearity. This comes up with the
new asset pricing model. In the modeled economy,
there are infinitely-lived homogeneous households
that maximize utility function subject to budget
constraint. and infinitely-lived heterogeneous firms
that maximize the present value of future cash flows
subject to budget constraints. After that, the general
equilibrium of this economy is computed. As a
result, this study solves for the asset pricing model
which noticeably included the economic activity of
that firm. The main findings demonstrate that the
rate of time preference, the investment in the next
period, the stock price at the time
t
, and the growth
rate of aggregate consumption have significant
impacts on how much the stock prices change.
Therefore, this conclusion is considerably different
from previous studies, especially the investment of
the firm. As a result, the role of economic activity of
firms should be examined in future research to shed
light on why the equity premium is still a puzzle.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2023.22.75
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