The Evolution of Capital Asset Pricing Models
Po-Jung Chen
Department of Finance
College of Management
National Taiwan University
Taipei, Taiwan
d96723004@ntu.edu.tw
Sheng-Syan Chen
Department of Finance
College of Management
National Taiwan University
Taipei, Taiwan
fnschen@management.ntu.edu.tw
Cheng-Few Lee
Department of Finance
School of Business
Rutgers University
New Brunswick, NJ 08903, USA
lee@business.rutgers.edu
Yi-Cheng Shih
Department of Finance
College of Management
National Taiwan University
Taipei, Taiwan
ycshih@ntu.edu.tw
1
The Evolution of Capital Asset Pricing Models
1. Introduction
This paper surveys the evolution of CAPM from 1964 to 2009. The original
CAPM of Sharpe (1964), Lintner (1965), and Mossin (1966) is developed in a
hypothetical world where the following assumptions are made about investors and the
opportunity set: (1) Investors are risk-averse individuals who maximize the expected
utility of their wealth; (2) Investors are price takers and have homogeneous
expectations about asset returns that have a joint normal distribution; (3) There exists
a risk-free asset such that investors may borrow or lend unlimited amounts at a
risk-free rate; (4) The quantities of assets are fixed and all assets are marketable and
perfectly divisible; (5) Asset markets are frictionless and information is costless and
simultaneously available to all investors; and (6) There are no market imperfections
such as taxes, regulations, or restrictions on short selling (Copeland, Weston and
Shastri, 2005).
After Sharpe (1964), Lintner (1965), and Mossin (1966), many
researchers have tried to develop more general asset pricing models by relaxing those
assumptions of CAPM and tested the empirical implications.
Merton (1973) relaxes
the single-period assumption to develop the intertemporal CAPM model with
stochastic investment opportunities, stating that the expected return on any asset is
deduced from a multi-beta version of CAPM in a continuous-time model.
2
However,
Merton’s intertemporal CAPM with stochastic investment opportunities states that the
expected excess return on any asset is given by a multi-beta version of the CAPM
with the number of betas being equal to one plus the number of state variables needed
to describe the relevant characteristics of the investment opportunity set.
Since all of
those state variables are not easily identified, Breeden (1979) utilizes the same
continuous-time economic framework as that used by Merton, likewise permitting
stochastic investment opportunities. However, it is shown that Merton’s multi-beta
pricing equation can be collapsed into a single-beta equation, where the instantaneous
expected excess return on any security is proportional to its ‘beta’ (or covariance)
with respect to aggregate consumption alone. After Breeden (1979), it is important
to emphasize, as Fama and French (1988a), and others have noted, that, in the context
of intertemporal models, predictability is not necessarily inconsistent with the concept
of market efficiency. Therefore, Balvers, Cosimano, and McDonald (1990) present a
general equilibrium theory relating returns on financial assets to macroeconomic
fluctuations in a context that is consistent with efficient markets in that no
excess-profit opportunities are available. The intuition underlying their theoretical
model arises from consumption smoothing by investors. Consumption opportunities
are linked to output, and, consistent with conventional macroeconomic models, output
is serially correlated and hence predictable. To maximize utility, investors attempt to
3
smooth consumption by adjusting their required rate of return for financial assets.
For example, investors anticipating lower output in the next period will attempt to
transfer wealth to this anticipated period of scarcity and therefore will accept a lower
rate of return in order to smooth consumption over time.
Although Merton (1973)
and Breeden (1979) relaxed the single-period assumption of CAPM, they just
provided the demand-side models without supply-side. Black (1976) examined the
effects of disequilibrating shocks on individual behavior in financial markets and the
effects of such modified behavior on market out-comes.
A short-run dynamic,
multi-period capital asset pricing model is constructed by assuming rational
expectations and adding the supply side to the static model of capital asset pricing.
After Black (1976), Grinols (1984) extended Merton's intertemporal capital asset
pricing model with multiple consumers to include a description of the supply of
traded securities.
The production decisions of firms are described in a model with
stochastic investment opportunities and incomplete markets.
Moreover, Lee , Tsai
and Lee argued that (2009) Black’s theoretically elegant model has never been
empirically tested for its implications in dynamic asset pricing.
theoretically extend Black’s CAPM.
They first
Then they use price, dividend per share and
earnings per share to test the existence of supply effect with U.S. equity data.
find the supply effect is important in U.S. domestic stock markets.
4
They
Besides, Levy, Levy and Benita (2006) relaxed the homogeneous beliefs
assumption of CAPM. CAPM can be derived under various sets of assumptions.
However, one of the fundamental assumptions that all of the above-mentioned models
share, which is considered as the most critical drawback of CAPM, is that all
investors have the homogeneous beliefs regarding the expected returns and the
variance-covariance matrix.
Under the homogeneous beliefs assumption, all
investors invest in the same mix of risky assets, the CAPM holds, and there is a
simple relationship between risk and return well known as the security market line
(SML).
Levy, Levy and Benita (2006) examined the robustness of CAPM to the
relaxation of one of its most problematic assumptions: homogeneous beliefs. They
proved that in a heterogeneous-belief market with an infinite number of investors and
an infinite number of risky assets, CAPM risk-return relationship precisely holds.
Regarding the relaxation of homogeneous investment horizon assumption, Lee, Wu
and Wei (1990) examined the effect of heterogeneous investment horizons on the
functional form of capital asset pricing and proposed a translog model for estimating
the risk-return relationship. In addition, their paper contended that some empirical
findings that are inconsistent with the traditional CAPM have resulted from
misspecification of the CAPM by ignoring the discrepancy between the observed data
periods and the true investment horizons.
5
Concerning the relaxation of taxation assumption, Brennan (1970) first
proposed an extended form of the single period CAPM model that accounted for the
differential taxation of dividends over capital gains. Litzenberger and Ramaswamy
(1979) extended the model of Brennan (1970) to account for restrictions on investors’
borrowing.
Sharpe (1964), Lintner (1965), and Mossin (1966), following the work of
Markowitz (1959), developed the first formulations of the mean-variance capital asset
pricing model (CAPM).
However, many researchers argued that assets pricing
models should subsume the effects of the higher moments.
Lee (1977) first
employed the transformation technique developed by Box and Cox (1964) to
determine the true functional form for testing the risk-return relation and to examine
the possible impact of skewness effect on capital asset pricing.
Then Sears and Wei
(1988) indicated that although the estimated coefficient of co-skewness gives
important information on the marginal rate of substitution between skewness
preference that is independent of the effects of the market risk premium. Furthermore,
Harvey and Siddique (2000) suggested that if asset returns have systematic skewness,
expected returns should include re-wards for accepting this risk. They formalized this
intuition with an asset pricing model that incorporates conditional skewness. Their
results showed that conditional skewness helps to explain the cross-sectional variation
6
of expected returns across assets. Besides the effect of the higher moments on utility
function, traditional asset pricing theorists assume that investors seek to maximize
expected utility. However, proponents of behavioral finance suggest that people
behave more in accordance with a psychologically based theory, such as prospect
theory which was developed by Kahneman and Tversky (1979).
After we discuss these theoretical models to relax the assumptions of traditional
CAPM, we will talk about existence of equilibrium.
Hart (1974) argued that in
deriving these properties of equilibrium prices, it has been assumed that equilibrium
does in fact exist.
Surprisingly, no attempt appears to have been made to establish
the existence of equilibrium in the basic Lintner-Sharpe model or in more general
versions of the model.
Yet, the existence of equilibrium is not implied by any of the
standard existence theorems since these theorems assume that consumption sets are
bounded below, whereas the assumption that investors can hold securities in unlimited
negative amounts implies that consumption sets are unbounded below.
In his paper,
he found the conditions for the existence of equilibrium in a very general version of
the Lintner-Sharpe model.
Moreover, Nielsen (1989) presents simple conditions and
a simple proof of the existence of equilibrium in asset markets where short-selling is
allowed and satiation is possible.
Unlike standard non-satiation assumptions, the
one used here is weak enough to be reasonable in the mean-variance Capital Asset
7
Pricing Model and in asset market models where investors maximize expected utility
and where total returns to individual assets may be negative.
The empirical evidence has led scholars to conclude that the pure theoretical
form of the traditional CAPM does not agree well with reality.
For example Roll’s
critique (1977), he argued “The only legitimate test of the CAPM is whether or not the
market portfolio is mean-variance efficient” and” If performance is measured relative
to an index that is ex post efficient, then from the mathematics of the efficient set no
security will have abnormal performance when measured as a departure from the
security market line.”
Besides, most empirical studies of the static CAPM assume
that betas remain constant over time and that the return on the value-weighted
portfolio of all stocks is a proxy for the return on aggregate wealth.
The general
consensus is that the static CAPM is unable to explain satisfactorily the cross-section
of average returns on stocks. Therefore, Fama and French (1992, 1996) provided the
three-factors model to the cross-section of expected stock returns. So the great factor
debate rages on. Sharpe (1998) said”...I’d be the last to argue that only one factor
drives market correlation. There are not as many factors as some people think, but
there’s certainly more than one …”
2. Intertemporal Models
8
The static capital asset Pricing Model (CAPM) of Sharpe (1964), Lintner
(1965), and Mossin (1966) states that the expected premium on any risky asset is
proportional to the premium on the market as a whole.
It has been criticized for the
requirement of additional assumption, especially homogeneous expectations and the
single-period nature of the world.
An intertemporal model for the capital market is
concluded from the multiperiod setting.
2.1 Merton Model
The theoretical linear relationship on static CAPM model, however, is subject
to criticisms on the premises of a constant opportunity set. Merton (1973)
subsequently relaxes this assumption, and shows that when the opportunity set
fluctuates over time due to changes in the state of the economy, individual securities
are also priced with respect to selected portfolios, providing a hedge against
unanticipated fluctuations.
Merton (1973) first develop the intertemporal CAPM model with stochastic
investment opportunities states that the expected return on any asset is deduced from a
multi-beta version of CAPM in a continuous-time model.
Consider K investors who
are concerned with maximizing the expected utility of wealth at the end of period,
where the utility functions are twice-differentiable concave function.
The k-th
investor is assumed to be able to invest in a riskless asset with instantaneous rate of
9
return r , and in i-th risky asset with instantaneous rate of return is given by the
stochastic differential equation
dPi
  i dt   i dzi ,
Pi
where dz i is the increment to a standard Brownian motion.
(1)
Processes such as (1)
are call Itô processes and while them are continuous and are not differentiable.
Thus
the stochastic process for the investor’s wealth as
n
n

dW   wi  i  r   r Wdt   wiW i dzi   y  c dt
i 1
 i 1

(2)
where wi is the fraction of the wealth invested in the i-th asset, y is the wage income,
and c is the consumption.
For computational simplicity, they assume that investors drive all their income
from capital gains and for notational simplicity, they introduce the state-variable
vector, X, whose m elements, xi, denote the current levels of P,  , and  . The
dynamics for X are written as the vector Itô processes,
dX  F ( X )dt  G ( X )dQ ,
(3)
where F is the vector [f1, f2... fm], G is a diagonal matrix with diagonal elements [g1,
g2... gm], dQ is the vector Wiener process [dq1, dq 2... dq m],  ij is the instantaneous
correlation coefficient between dqi and dzj, and vij is the instantaneous correlation
coefficient between dqi and dqj.
The necessary optimality conditions for an investor
who act according to in choosing his consumption-investment program for using Ito’s
10
lemma, at each point of time,


 n
 m





U
c
,
t

J

J
w


r

r
W

c




t
W   i
i
   J i fi 
1
1






0  max
c , w 
n
n
m n
m m
.
1
1
 J W W  wi w j ijW 2   J iW w jWg i j ij   J ij g i g j vij 
2 1 1
1
1
1
1
 2

(4)
The n+1 first-order conditions derived from (4) are
0  U c (c, t )  J W (W , t , X )
(5)
and
n
m
1
1
0  J W (  i  r )  J W W  w jW ij   J jW g j i ji
(6)
where c  c(W , t , X ) , wi  wi (W , t , X ) are the optimum consumption and portfolio
rules as functions of the state variables and  ij are the instantaneous covariance
between the returns on the i-th and j-th asset.
Solving explicitly for these functions
by matrix inversion,
n
m
n
1
1
1
wiW  A vij (  j  r )   H k  j g k jk vij ,
(7)
where the vij are the elements of the inverse of the instantaneous variance
covariance matrix of returns,   [ ij ] , A   JW / JW W , and H k   J kW / J W W .
n
The demand function of (7) has two component: The first term, A vij (  j  r ) , is
1
the usual demand function for a risky asset by s single-period mean-variance
maximizer, where A is proportional to the reciprocal of the investor’s absolute risk
aversion; the second term,
m
n
1
1
 H 
k
j
g k jk vij , reflects the demand for the asset as a
vehicle to hedge against shifts in the investment opportunity set.
11
Merton (1973) shows that if investment opportunities are varying over time,
then long-term investors generally care about shocks to investment opportunities－the
productivity of wealth－and not just about wealth itself.
They may seek to hedge
their exposures to wealth productivity shocks, and this gives rise to intertemporal
hedging demands for financial assets.
2.2 Consumption-based Models
The intertemporal CAPM model of Merton (1973) with stochastic investment
opportunities states that the expected return on assets is derived from a multi-beta
version of the CAPM with numbers of betas being equal to one plus the number of
state variables needed to describe the relevant characteristics of the investment
opportunity set.
However, Breeden (1979) utilizes the same continuous-time
economic framework as that used by Merton, likewise permitting stochastic
investment opportunities, shows that the expected return on any asset is proportional
to its beta with respect to aggregate consumption alone.
He argues that a single beta
relative to a specific variable, given certain stationary assumptions on the joint
distributions of rates of return and aggregate consumption, make the model easier to
test and implement.
Therefore, Breeden (1979) is the main theory of
consumption-based capital asset pricing model explaining return variation across
assets from an optimizing intertemporal perspective under the assumption that if
12
agents have time-additive utility, that is, locally quadratic, then the expected return of
assets are linear with their aggregate consumption.
An investor k’s portfolio holdings are founded in terms of his indirect utility
function for wealth,
J w, t, x , and equilibrium expected asset returns are
correspondingly found in terms of aggregate wealth and the returns on assets that are
perfectly correlated with changes in the various state variables, that is, x, if they exist.
The individual’s direct utility function for consumption, U c, t  , in the analysis of
equilibrium expected returns on assets. Then the optimal portfolio demands in terms
of individual k’s optimal consumption function that J W  U c , which implies the
J W x  U cc C x and J W W  U cc CW , where subscripts of U, J and c denote partial
derivatives. Define T to be individual k’s absolute risk tolerance: T  U c / U cc .
Then the optimal portfolio may be written as




w W  T / cW Vjj-1  j  r   Vjj-1Vjx cx / cW ,
(8)
where (  j  r ) is the vector of instantaneous expected excess return on assets, Vjj
is their variance-covariance matrix on assets, and Vjx is the J×X matrix of covariances
of asset returns with changes in the state variables.
Pre-multiplying (8) by cW Vjj
and rearranging terms gives
T  j  r   VjWcW  Vjx cx ,
where VjW is the vector of covariance of asset returns with wealth change.
13
(9)
Since
k’s optimal consumption is a function, cW , t , x  , of his wealth, time , and the state
variables.
Ito’s Lemma implies that the local covariance of asset returns with
changes in k’s consumption rate are given by Vjc  VjWcW  Vjx cx .
Intuitively,
Equation (9) can be seen by
Vjc  T  j  r  .
(10)
The Equation (10) holds for each individual k and can be aggregated by summing
over all individuals, with defining the aggregate consumption rate to be C, and
defining the aggregate risk tolerance to be T M  k T k , it follows that the expected
excess returns on assets in equilibrium will be proportional to their covariance with
changes in aggregate consumption,
 j  r  T M  VjC
1

 T M /C

1
(11)
Vj,l n C.
For any portfolio M with weights w M gives
M  r /  M ,lnC  T M / C 1
(12)
and pre-multiplying Equation (11) by Equation (12) gets
 j  r  Vj,lnC /  M ,ln C  M  r 
  jC /  MC  M  r  ,
(13)
where  jC and  MC are the consumption-betas of asset returns and of portfolio M’s
return.
If there exists a security whose return is perfectly correlated with changes in
14
aggregate consumption over the next instant, then the risk － return relation of
Equation (13) can be written in terms of assets’ betas measured relative to that
security’s return,  C , and the expected excess return on this security
 j  r  C C*  r .
(14)
Therefore, Breeden (1979) derives a single－beta asset pricing model in a
multi-good, continuous-time model with uncertain consumption-goods prices and
uncertain investment opportunities.
To understand the equity premium puzzle, consider the intertemporal choice
problem of an investor k who can trade in some asset i and can obtain a gross simple
rate of return (1  Ri ,t 1 ) on the asset held from time t to time t+1.
If the investor
consumes C k ,t at time t and has time-separable utility with discount factor  and
period utility U (Ck ,t ) , then the first-order condition of utility function is
U (Ck ,t )   Et 1  Ri ,t 1 U (Ck ,t 1 ) .
(15)
The left-hand side of Equation (15) is the marginal utility cost of consuming one real
dollar less at time t ; the right-hand side is the expected marginal utility benefit from
investing the dollar in asset i at time t, sell it at time t+1, and consuming the proceeds.
The investor equates marginal cost and marginal benefit, so Equation (15) must
describe the optimum.
Dividing Equation (15) by U (C k ,t ) yields

U (Ck ,t 1 ) 
1  Et (1  Ri ,t 1 ) 
  Et 1  Ri ,t 1  M k ,t 1  ,
U (Ck ,t ) 

15
(16)
where M k ,t 1   U (Ck ,t 1 ) / U (Ck ,t ) is the intertemporal marginal rate of substitution
of the investor, also known as the stochastic discount factor.
Equation (16) assumes
the existence of an investor maximizing a time-separate utility function.
The
existence of a positive stochastic discount factor is guaranteed by the absence of
arbitrage in markets in which non-satiated investors can trade freely without
transaction costs.
In general, different investors k whose marginal utilities follow
different stochastic process will have different M k ,t 1 , but each stochastic discount
factor must satisfy Equation (16).
It is common practice to drop the subscript k from
the equation and rewrite
1  Et 1  Ri ,t 1  M t 1  .
(17)
In complete markets the stochastic discount factor M t 1 is unique because investors
can trade with one another to eliminate any idiosyncratic variation in their marginal
utilities.
Equation (17) is written as the expectation of the product equals the
product of expectations plus the covariance
Et 1  Ri ,t 1  M t 1   Et 1  Ri ,t 1  Et M t 1   Covt Ri ,t 1 , M t 1  .
(18)
Substituting Equation (17) into Equation (18) and rearranging gets
1  Et Ri ,t 1  
1  Covt Ri ,t 1 , M t 1 
.
Et M t 1 
(19)
Equation (19) must hold for any asset, including a riskless asset whose gross simple
return is 1  R f ,t 1 .
Since the simple riskless return has zero covariance with the
16
stochastic discount factor, that is
1  R f ,t 1 
1
.
Et M t 1 
(20)
Equation (20) can be used to rewrite Equation (19) as
1  Et Ri ,t 1   1  R f ,t 1  1  Covt Ri ,t 1 , M t 1  .
(21)
Because the intertemporal marginal rate of substitution of the investor, also
known as the stochastic discount factor is M k ,t 1   U (Ck ,t 1 ) / U (Ck ,t ) . Under the
assumption on the form of utility function, we can get the stochastic discount factor
from the first order conditions of maximizing investors’ utilities.
Then, according to
Equation (21), we can describe any expected excess return on risky assets over the
riskless rate.
Campbell and Cochrane (1999) present a habit persistence model to explain
the dynamic pricing phenomena, that is, using lagged consumption as the state
variable.
They replace the utility function U (Ct ) with U (Ct  X t ) .
The
identical agents maximize the utility function

E t
t 1
(Ct  X t )1  1
,
1 
(22)
where X t denotes the level of habit and  belongs to the time discount factor.
They use surplus consumption ratio St  (Ct  X t ) / Ct to capture the relation between
consumption and habit.
The surplus consumption ratio increases with consumption:
S t  0 means a bad state in which consumption is equal to habit; St  1 as
17
consumption rises relative to habit.
Under the assumption that the habit is the
external specification and the habit is determined by the history of aggregate
consumption rather than the history of individual consumption.
Therefore the
marginal utility is


U c (Ct , X t )  (Ct  X t )   Ct St .
(23)
In this model, the intertemporal marginal rate of substitution depends on change in the
ratio of consumption to habit as well as on consumption growth,
M t 1
Lettau
and
C 
U (C , X )
  c t 1 t 1    t 1 
U c (Ct , X t )
 Ct 
Ludvigson
(2001b)
is
the


 S t 1 

 .
 St 
first
reexamination
(24)
of
a
consumption-based factor model, the first recent paper that finds some success in
pricing the value premium from a macro-based model.
They examine a conditional
version of the linear consumption-based CAPM model with time-varying coefficients,
the stochastic discount factor is
M t 1  at  (b0  b1 zt )  ct 1
(25)
The innovation is to allow the slope coefficient b, which acts as the risk-aversion
coefficient in the model, to vary over time; ct 1 is consumption growth, the single
factor in the asset pricing model.
However, the risk parameters at and bt will
depend on risk premium, thus they seek a scaling variable, log consumption-wealth
ratio, to measure z t [as in Lettau and Ludvigson (2001a)].
18
In the condition version
of CAPM models, Jagannathan and Wang (1996) argue that the CAPM holds in a
conditional sense, that is, betas and the market premium vary over time.
They add
the labor income to explain the cross-section asset returns. Furthermore, Kumar et al.
(2008) examine the information-dependent conditional CAPM with adding the
innovations in market volatility, oil prices, exchanges rates, and dispersion of
analysts’ forecasts to explain the cross section of stock returns.
Balvers and Huang (2009) add money to the standard consumption-based
CAPM of Breeden (1979).
Balvers and Huang (2009) consider asset pricing in a
monetary economy that the consumption CAPM driving from real money growth as
an additional factor to determine the asset return.
They argue that the availability of
money as a source of liquidity improves transactions and affects the marginal value of
wealth in generating consumption.
A representative consumer/investor in an
endowment economy maximizes expected life time utility subject to a budget
constraint and given that transaction costs to purchase consumption goods are
mitigated by money holdings.
V wt , mt , xt   Max uct   Et V wt 1 , mt 1 , xt 1 
ct , t
(26)
subject to
wt 1  Rt 1 wt  ct  T ct , mt   mt  t 
mt 1  t  zt 1  /  t 1
(27)
(28)
where wt and mt represent respectively real financial wealth (excluding money
19
holdings) and real money holdings; xt indicates a set of state variables, assumed
exogenous to the consumer-investor, that is sufficient to represent changes in the
investment opportunity set.
T c, m represents the real transaction cost of
purchasing the current level of consumption and the agent in period t chooses current
consumption ct . The choice of real money balances at the current price level to be
used in the next period,  t , differs from the real balances used in the next period at
the then-relevant price level, due to the gross inflation rate  t 1  pt 1 / pt and due to
an
assumed
transfer
zt 1  M t 1  M t  / pt .
of
governance
revenues
from
money
creation,
To isolate the contribution of the money factor, Balvers and
Huang (2009) exclude Merton (1973) factors by assuming that there are no changes
over time in the exogenous dividend processes, ruling out shifts in the investment
opportunities set and conclude that real money growth as an additional factor
determine asset returns.
However, Campbell (1993) substitutes consumption out of the model to get a
discrete-time version of the intertrmporal CAPM of Merton (1973).
The
representative agent’s dynamic budget constraint can be written as
Wt 1  Rm,t 1 (Wt  Ct ) ,
(29)
where Wt is total wealth, C t is consumption at period t, Rm ,t 1 represents gross
return on the portfolio of all invested wealth from period t to period t+1.
20
The
loglinear approximation begins by dividing Equation (29) by Wt to obtain
Wt 1
C
 Rm ,t 1 (1  t ) ,
Wt
Wt
(30)
or in logs

1
wt 1  rm,t 1  k  1  ct  wt  ,
 
(31)
indicating that if consumption to aggregate wealth ratio is stationary, the budget
constraint maybe be approximated by taking a first-order Taylor expansion of the
equation.
In Equation (31), when the log consumption-wealth ratio is constant, then
 can be interpreted as (W-C)/W and k is a constant. Solving Equation (31) by
assuming that lim i  i ct i  wt i   0 , the log consumption-wealth ratio may be
written

ct  wt    i (rm ,t i  ct i ) 
i 1
k
.
1 
(32)
Taking the conditional expectations of both side of Equation (32) to obtain

ct  wt  Et   i (rm ,t i  ct i ) 
i 1
k
1 
(33)
Equation (33) shows that, if the aggregate consumption-wealth ratio is not constant, it
must forecast changing returns to the market portfolio or changing consumption
growth. That is the consumption-wealth ratio can only vary if consumption growth
or returns or both predictable.
Building on the work of Kreps and Porteus (1978), Epstein and Zin (1989,
1991) and Weil (1989) develop a more flexible version of the basic power utility
21
model. The objective function is
1

U t  1   Ct    EtU t11



1





1
.
(34)
Here  is the coefficient of relative risk aversion,   1    /1  1 /  , and  is
the elasticity of intertemporal substitution.
The Euler equation for asset i’s return,
Ri ,t 1 can be written as
1
1 /  


 



C
1


t

1
.
  


1  Et  
1

R
i ,t 1
  Ct    1  Rm ,t 1 




(35)
For the market portfolio itself, this takes the simpler form

1 / 

 


C

1  Et   t 1  Rm ,t 1   .
   Ct 
 


(36)
When asset returns and consumption are jointly conditional homoscedastic and
lognormally distributed, the Euler equations (35) and (36) can be written in log form.
The log version of Equation (36) takes the form


1   
2 2
2
0   log   Et ct 1  Et rm,t 1    Vcc   Vmm 
Vcm 

2   


2
(37)
Equation (37) implies that there is a linear relationship between expected
consumption growth and the expected return on the market portfolio, with the slope
coefficient equal to the intertemporal elasticity of substitution  . The relationship is
Et ct 1   m  Et rm,t 1
where

1   
 m   log    Vcc  Vmm  2Vcm 
2   

22
(38)
  log  
1 
 Vart ct 1  rm,t 1  .
2  
The intercept term  m is related to the variance of the error term in the ex post
version of the liner relationship (38), that is the degree of uncertainty about
assumption growth relative to the return on the market.
The log version of general Euler Equation (35) can be used for cross-sectional
asset pricing.
It takes the more complicated form
0   log  

E c    1Et rm,t 1  Et ri ,t 1 
 t t 1
2

1   
2
2
2
(  1)(Vcm )  Vci  2(  1)Vim 
  Vcc  (  1) Vmm  Vii 
2   



. (39)
When the asset under consideration is a risk-free real return r f ,t 1 , this expression
simplifies because some variance and covariance drop out, then
Et ri ,t 1  rf ,t 1  
Vii
V
  ic  (1   )Vim .
2

(40)
Substituting Equation (38) into Equation (33) to obtain

ct  wt  1    Et   i (rm ,t i ) 
i 1
 k   m 
.
1 
(41)
Equation (41) means the log consumption-wealth ratio is a constant, plus 1   
times the discounted value of expected future returns on invested wealth.
Instead of
substituting the wealth return out of the Epstein-Zin-Weil model, Compbell (1993)
substitutes consumption out of the model to get a discrete-time version of the
intertemporal CAPM of Merton (1973).
The innovation in consumption is
23

ct 1  Et ct 1  rm,t 1  Et rm,t 1  1   Et 1  Et   i rm,t 1i .
(42)
i 1
Thus the covariance of any asset return with consumption growth must satisfy
Covri ,t 1 , ct 1   Vic  Vim  1   Vih ,
(43)
where Vih denotes the covariance of asset return i with revisions in expected future
returns on wealth:



Vih  Cov ri ,t 1  Et ri ,t 1 , ( Et 1  Et )  i rm,t 1i  .
i 1


(44)
The letter h here is used as a mnemonic for hedging demand [Merton (1973)], a term
commonly used in the finance literature to describe the component of asset demand
that is determined by investors’ responses to changing investment opportunities.
Substituting Equation (43) into Equation (40) and using the definition
  1    /1  1 /  ,
Et ri ,t 1  r f ,t 1  
Vii
 Vim  (  1)Vih
2
(45)
The only parameter of the utility function that enters Equation (45) is the coefficient
of relative risk aversion  . The elasticity of intertemporal substitution  does not
appear once consumption has been substituted out of the model.
Intuitively, this
result comes from the fact that  plays two roles in the theory. A low value of 
reduces anticipated fluctuations in consumption [Equation (38)], but it also increase
the risk premium required to compensate for any contribution to the fluctuations
[Equation (40)].
These offsetting effects lead  to cancel out of the pricing
24
Equation (45).
A large number of models amount the discount factor generalizes the
power-utility case by adding another state variable [Cochrane (2005)].
It is very
danger in such models that they often work well for short run returns, but not in the
long run. The trouble is that the marginal utility of consumption depends not only
on consumption but also on an additional variable.
Even if they assume the function
form of the utility function, there is no general agreement on how to measure the
marginal utility of the consumption: any combination of state variable may have a
quantitatively significant impact on marginal utility [Balvers and Huang (2007)].
Lewellen and Nagel (2006) have also criticized consumption model on the argument
that the low covariance between the risk premium and the betas. The covariance
between consumption (market) betas and the consumption (market) risk premium
obtain from a series of estimates over small time windows is too small to support the
importance of any conditional variable.
2.3 Production-based Models
After Breeden (1979), it is important to emphasize, as Fama and French
(1988a), and others have noted, that, in the context of intertemporal models,
predictability is not necessarily inconsistent with the concept of market efficiency.
Therefore, Balvers, Cosimano and McDonald (1990) present a general equilibrium
25
theory relating returns on financial assets to macroeconomic fluctuations in a context
that is consistent with efficient markets in that no excess-profit opportunities are
available.
Balvers, Cosimano, and McDonald (1990) argue that aggregate output is
equal or proportionate to aggregate consumption and that one can evaluate the
marginal utility of consumption at the observed level of output so that aggregate
output growth becomes the key asset pricing factor.
The advantage to this approach
is that output growth is likely measured more accurately than consumption growth.
Balvers, Cosimano, and McDonald (1990) indicate that changes in aggregate output
lead to attempts by agents to smooth consumption, which affect the required rate of
return on financial assets.
Output yt at time t and at that time the firm divides the
output into dividends, dt and investment, it. The investment becomes productive as
capital one period after, kt+1, and leads to production yt+1 after the random
productivity shock θt is revealed. The dividends, dt+1, are paid to the investors and,
together with changes in share prices pt and pt+1, determine the gross realized return,
Rt+1, on stock held in the period from t to t+1.
The representative firm determines
its level of investment each period to maximize shareholder wealth

E0
 R 
t 0
t
1
i 0
i

d t
(46)
subject to
dt  yt  it  yt  kt 1
26
(47)
yt  AB t t kt
(48)
where A and B are positive constant,  is less than one, and Rt is one plus the
appropriate discount rate.
Substituting Equation (47) and (48) into (46) and
differentiating with respect to kt+1 yields the stochastic Euler condition:


Et Rt 1    yt 1 / kt 1   1
1
(49)
That is, the expectation of the marginal product of investment must equal the one unit
of the consumption good sacrificed in favor of investment.
The consumer
maximizes the present value of time-additive utility function from consumption of all
goods. The utility function indicates concave one-period consumption, ct and  as
the consumer’s discount factor for utility. The consumer maximizes

  uc 
(50)
ct  pt  yt st 1   pt  yt   d t  yt st
(51)
E0
t
t
t 0
subject to
In the budget constraint, d t  yt  represents the dividends per share, paid at the
beginning of the period; pt  yt  is the price per share in state yt immediately after
dividends d t are paid; and st is the number of shares held at the beginning of
period t. Maximization by the representative consumer yields the following Euler
equation with respect to the choice variable st 1 :
pt  yt uct   Et  pt 1  yt 1   dt 1  yt 1 uct 1 
(52)
The Euler equation relates the price and return of a share to the cost of delaying
27
consumption, where they define
Rt 1  yt 1 , yt    pt 1  yt 1   dt 1  yt 1  / pt  yt 
(53)
as the realized holding period return on shares. Equation (52) represents that the
consumer will choose current consumption such that the utility of current
consumption equals the expected discount return of buying a stock times the marginal
utility of consumption when the stock is sold in the next period.
However, returns
will vary since consumption caries due to the randomness in aggregate output.
Solving Equation (52) forward yields a general expression for ex-dividend share
prices

pt  Et   i u ct i  / u ct d t i .
(54)
i 1
Under the assumption of logarithmic utility function to solve the general equilibrium
model,
uct   a ln ct  .
(55)
Using the fact that uct   a / ct and the market-clearing condition ct  dt , the
demand for consumption goods equals the net supply to the consumers of the
multi-purpose good, the share pricing Equation (54) yields

pt  Et   i d t   / 1   d t .
(56)
i 1
Under the logarithmic form of the utility function, pt does not depend on future
dividends. From Equation (53) for share prices and (56) to define returns,
28
Rt 1  1 /  d t 1 / d t 
(57)
Equation (47), (48), (49) and (57) characterize the dynamic path for d t , kt , yt and Rt .
Where investment is proportional to output then yields kt 1  yt , which implies
from Equation (47) that dt  1    yt .
Therefore, Equation (44) becomes
Rt 1  1 /   yt 1 / yt 
(58)
then substitution Equation (58) into (49) verifies that the posited solution is correct.
Since future output depends on current investment, it can be predicted from current
observations. The solution kt 1  yt implies that
yt 1   t 1 t yt
where   AB  .

(59)
Equation (58) and (59) together explain the return
predictability in our model.
Therefore, Balvers, Cosimano, and McDonald (1990)
develop an intertemporal equilibrium model relates financial asset returns to
movements in aggregate output.
Cochrane (1991, 1996) is an attempt to extend the production-based ideas to
describe the asset returns.
A bit similar to the consumption-based model, the
production-based model ties asset returns to marginal rates of transformation, which
are inferred from the investment through a production function.
It is derived from
the producer’s first order condition for optimal intertemporal investment demand.
The discount factor of the form is
29
M t 1  b0  zt b1   f t 1 ,
(60)
that describe the discount factor might be a linear combination of factors f with
weights that vary as a vector of instrument z varies across different information set.
Scaling the factors f by the instrument z achieves the same result. So Equation (60)
is equivalent to
M t 1  b0 f t 1  b1  f t 1  zt 1  ,
(61)
here f t 1 denote the investment return, functions of investment and capital only, that
is f t 1  f ( I ti1 / K ti1 , I ti / K ti ) .
Therefore, given the choice of instruments, performing
the GMM estimation and testing with scaled factors is a completely general test of a
dynamic, conditional factor pricing model based on the instruments.
Balvers and Huang (2007) they derive the stochastic discount factor that the
productivity shock is the single factor in asset pricing model.
The social planner
maximizes
V kt ,t   Maxuct , n  nt    Et V kt 1 ,t 1 
(62)
 t 1  H  t ,  t 1 
(63)
nt ,kt 1
subject to
ct  f  t , nt , kt   kt 1
(64)
The lifetime utility of the representative consumer is maximized subject to a standard
production function f  t , nt , kt  , where the inputs are labor nt and capital k t , and
an exogenous technology level  t . The productivity level is assumed to follow
30
Markov process as given by Equation (63), with  t representing the zero-mean
white noise productivity shock. Per-capita consumption is given in Equation (64) as
the part of capita production that is not invested. Through Equation (62) to Equation
(64) and the stochastic discount factor M t 1   uc ct 1 , n  nt 1  / uc ct , n  nt  to
obtain
M t 1 
with
 t 1
f k  t 1 , nt 1 , kt 1 
,
(65)
t 1  Vk kt 1 ,  t 1  / Et Vk kt 1 ,  t 1  . The t 1 represents a fundamental,
nondiversifiable risk inherent in capital accumulation that cannot be ignored and
should be needed for asset pricing.
Therefore, Balvers and Huang (2007) relates
t 1 to shocks in the marginal value of capital to obtain an explicit production-based
expression for the asset pricing model.
3. Supply-Side Models
Although Merton (1973) and Breeden (1979) relaxed the single-period
assumption of CAPM, they just provided the demand-side models without supply-side
effect.
Black (1976) examined the effects of disequilibrating shocks on individual
behavior in financial markets and the effects of such modified behavior on market
outcomes.
A short-run dynamic, multi-period capital asset pricing model is
constructed by assuming rational expectations and adding a supply side to the static
31
model of capital asset pricing.
After Black (1976), Grinols (1984) extended
Merton's intertemporal capital asset pricing model with multiple consumers to include
a description of the supply of traded securities.
The production decisions of firms
are described in a model with stochastic investment opportunities and incomplete
markets.
Moreover, Lee, Tsai and Lee (2009) argued that Black’s theoretically
elegant model has never been empirically tested for its implications in dynamic asset
pricing.
They first theoretically extend Black’s CAPM.
Then they use price,
dividend per share and earnings per share to test the existence of supply effect with
U.S. equity data.
markets.
They find the supply effect is important in U.S. domestic stock
Lee, Tsai and Lee (2009) theoretically extend the dynamic, simultaneous
CAPM model of Black (1976) to the existence of the supply effect in the asset pricing
process.
3.1 Demand function of capital assets
The demand equation for the assets is derived under the standard assumptions
of the CAPM.
An investor’s objective is to maximize the expected utility in terms of
the negative exponential function of wealth:
U  a  h  e{bWt 1}
(66)
where the terminal wealth Wt+1 =Wt(1+ Rt); Wt is initial wealth; and Rt is the rate of
return on the portfolio. The parameters, a, b and h, are assumed to be constants.
32
The dollar returns on N marketable risky securities can be represented by:
X j ,t 1  Pj ,t 1  Pj ,t  D j ,t 1 ,
j = 1,…,N,
(67)
where Pj, t+1 = (random) price of security j at time t+1, Pj, t = price of security j at time
t, amd Dj, t+1 = (random) dividend or coupon on security at time t+1.
variables are assumed to be jointly normal distributed.
These three
After taking the expected
value of Equation (67) at time t, the expected returns for each security, xj, t+1, can be
rewritten as:
x j ,t 1  Et X j ,t 1  Et Pj ,t 1  Pj ,t  Et D j ,t 1 ,
j = 1,…,N,
(68)
where Et Pj ,t 1  E Pj ,t 1 t  ; Et D j ,t 1  E D j ,t 1 t ; Et X j ,t 1  EX j ,t 1 t  ; t is
Then, a typical investor’s expected value
the given information available at time t.
of end-of-period wealth is
wt 1  EtWt 1  Wt  r  Wt  qt1 Pt   qt1 xt 1 ,
(69)


where Pt  P1,t , P2,t , P3,t ,..., PN ,t  ; xt 1  x1,t 1 , x2,t 1 , x3,t 1 ,..., xN ,t   Et Pt 1  Pt  Et Dt 1 ;

qt 1  q1,t 1 , q2,t 1 , q3,t 1 ,..., q N ,t  ;
reconstruction of his portfolio;
q j ,t 1  number of units of security j after
r* = risk-free rate.
In Equation (69), the first term
on the right hand side is the initial wealth, the second term is the return on the
risk-free investment, and the last term is the return on the portfolio of risky securities.
The variance of Wt 1 can be written as:

V Wt 1   E Wt 1  wt 1 Wt 1  wt 1   qt1S q ,t 1 ,
33
(70)

where S  E  X t 1  xt 1  X t 1  xt 1  = the covariance matrix of returns of risky
securities. Maximization of the expected utility of Wt 1 is equivalent to:
Max wt 1 
b
V(Wt 1 ) ,
2
(71)
By substituting Equation (69) and (70) into Equation (71), Equation (71) can be
rewritten as:
b
Max 1  r * Wt  qt1  xt 1  r  Pt    qt1 S qt 1 .
2
(72)
Differentiating Equation (72), one can solve the optimal portfolio as:
qt 1  b 1S 1 xt 1  r  Pt  .
(73)
Under the assumption of homogeneous expectation, or by assuming that all the
investors have the same probability belief about future return, the aggregate demand
for risky securities can be summed as:
m
Qt 1   qtk1  cS 1 Et Pt 1  (1  r*) Pt  Et Dt 1  ,
(74)
k 1
where c = Σ (bk)-1.
In the standard CAPM, the supply of securities is fixed, denoted as Q*.
Then,
Equation (74) can be rearranged as Pt  1 / r * xt 1  c 1SQ *  , where c-1 is the market
price of risk.
In fact, this equation is similar to the Lintner’s (1965) well-known
equation in capital asset pricing.
3.2 Supply function of securities
It is assumed that there exists a solution to the optimal capital structure and
34
that the firm has to determine the optimal level of additional investment.
The
one-period objective of the firm is to achieve the minimum cost of capital vector with
adjustment costs involved in changing the quantity vector, Qi ,t 1 :
1
Min Et Di ,t 1Qi ,t 1    Qi,t 1 Ai Qi ,t 1 
2
subject to
Pi ,t Qi ,t 1  0 ,
(75)
where Ai is a n i × n i positive define matrix of coefficients measuring the assumed
quadratic costs of adjustment.
If the costs are high enough, firms tend to stop
seeking raise new funds or retire old securities.
The solution to equation (75) is
Qi ,t 1  Ai1 i Pi ,t  Et Di ,t 1 ,
(76)
where i is the scalar Lagrangian multiplier.
Aggregating equation (76) over N firms, the supply function is given by
Qt 1  Ai1 BPt  Et Dt 1  ,
(77)
 Q1 
1 I

 A11

Q 




1
2 I
A

1
2


, B 
where A  
, and Q   2  .
  






 



1 
N I 
AN 
QN 


Equation (77) implies that a lower price for a security will increase the amount
retired of that security.
In other words, the amount of each security newly issued is
positively related to its own price and is negatively related to its required return and
the prices of other securities.
3.3 Multiperiod Equilibrium Models
35
The aggregate demand for risky securities presented by Equation (74) can be
seen as a difference equation.
The prices of risky securities are determined in a
multiperiod framework It is also clear that the aggregate supply schedule has similar
structure. As a result, the model can be summarized by the following equations for
demand and supply, respectively:



Qt 1  cS 1 Et Pt 1  1  r * Pt  Et Dt 1

(74)
Qt 1  Ai1 BPt  Et Dt 1 
(77)
Differencing Equation (74) for period t and t+1 and equating the result with Equation
(77), a new equation relating demand and supply for securities is



cS 1 Et Pt 1  Et 1 Pt  1  r * Pt  Pt 1   Et Dt 1  Et 1 Dt

 A 1 BPt  Et Dt 1   Vt (78)
where Vt is included to take into account the possible discrepancies in the system.
Here, Vt is assumed to be random disturbance with zero expected value and no
autocorrelation.
Obviously, Equation (78) is a second-order system of stochastic differential
equation in Pt, and conditional expectations Et-1Pt and Et-1Dt.
By taking the
conditional expectation at time t-1 in Equation (78), and because of the properties of
Et-1[Et Pt+1] = Et-1Pt+1 and Et 1 E Vt   0 , Equation (78) becomes



cS 1 Et 1 Pt 1  Et 1 Pt  1  r * Et 1 Pt  Pt 1   Et 1 Dt 1  Et 1 Dt
 A1 BE t 1 Pt  Et 1 Dt 1 .
Subtracting Equation (78′) from Equation (78),
36

(78′)
1  r cS
*
1

 A 1 B Pt  Et 1 Pt  


cS 1 Et Pt 1  Et 1 Pt 1   cS 1  A 1 Et Dt 1  Et 1 Dt 1   Vt
.
(79)
Equation (79) shows that prediction errors in prices (the left hand side) due to
unexpected disturbance are a function of expectation adjustments in price (first term
on the right hand side) and dividends (the second term on the right hand side) two
periods ahead.
This equation can be seen as a generalized capital asset pricing
model.
Lee, Tsai and Lee (2009) first theoretically extend Black’s CAPM.
Then
they use price, dividend per share and earnings per share to test the existence of
supply effect with U.S. equity data.
They find the supply effect is important in U.S.
domestic stock markets and the existence of the supply effect in the asset pricing.
4. Equilibrium Models with Heterogeneity
One of the fundamental assumptions that all of the above-mentioned models
share, which is considered as the most critical drawback of CAPM, is that all
investors have the homogeneous beliefs regarding the expected returns and the
variance-covariance matrix.
Also, in the consumption-based CAPM need to assume
that assets can be priced if there is a representative agent who consumes aggregate
consumption.
Therefore, in this section we will introduce the model of
heterogeneous beliefs, agents, and investment horizon.
37
4.1 Heterogeneous Beliefs
Basak (2005) provide a continuous-time pure-exchange framework to study
asset pricing implication of the present of heterogeneous beliefs, within a rational
Bayesian setting. Equilibrium is determined in terms of a representative investor’s
utility function with stochastic weighting driven by the investor’s disagreement about
the aggregate growth.
In addition, Levy, Levy and Benita (2006) relaxed the
homogeneous beliefs assumption of CAPM.
sets of assumptions.
CAPM can be derived under various
Under the homogeneous beliefs assumption, all investors invest
in the same mix of risky assets, the CAPM holds, and there is a simple relationship
between risk and return well known as the security market line (SML).
They
employ the mathematical analysis and numerical simulations to study the effect of the
introduction of heterogeneity of beliefs on asset prices.
They prove that in an
infinite market when number of investors and assets approach to infinite, with
unbiased heterogeneous beliefs and bounded variance, the CAPM linear risk-return
relationship precisely holds.
However, it is also possible that utility-maximizing stock market investors are
heterogeneous in important ways.
If investors are subject to large idiosyncratic risks
in their labor income and can share these risks only indirectly by trading a few assets
such as stocks and Treasury bills, their individual consumption paths may be much
38
more volatile than aggregate consumption.
Constantinides and Duffie (1996)
assume an economy in which heterogeneous investors k have different consumption
levels ckt.
The cross-sectional distribution of individual consumption is lognormal,
and the change from time t to time t+1 in individual log consumption is
cross-sectional uncorrelated with the level of individual log consumption at time t.
All investors have the same power utility function with time discount factor  and
coefficient of relative risk aversion  .
In this economy each investor's own
intertemporal marginal rate of substitution is a valid stochastic discount factor.
Hence the cross-sectional average of investors' intertemporal marginal rates of
substitution, M t*1 , is a valid stochastic discount factor.
However the marginal
utility of the cross-sectional average of investors' consumption, M tRA
1 , is not a valid
stochastic discount factor when marginal utility is nonlinear.
This false stochastic
discount factor would be used incorrectly by an economist who ignores the
aggregation problem in the economy.
They state the Euler equation of consumption
of consumer k-th for security j:



   1 * 
  Ct 1 

 exp 
Et  
Vart 1   j ,t 1   1 .
C
 2



  t 

(80)
Here Vart*1 denotes a cross-sectional variance of individual log consumption growth
taken after aggregates at time t+1. Equation (80) adds the exponential term to the
standard consumption-based asset pricing equation.
39
The difference between the logs
of these two stochastic discount factors is
M t*1  M tRA
1 
 (  1)
2
Vart*1ck ,t 1 .
(81)
Therefore, we can construct a discount factor to represent any asset pricing anomalies.
Another heterogeneous investors model in Constantinides (1982) argue that if
a complete set of markets exists that enables households to insure against
idiosyncratic income shocks, then the heterogeneous households are able to equalize,
state by state, their marginal rates of substitution.
Therefore, the equilibrium of a
heterogeneous-household, full information economy is in its pricing implications to
the equilibrium of a representative-household, full-information economy.
Under the full consumption insurance assumption implies that heterogeneous
consumers are able to equalize their marginal rate of substitution state by state.
However, Brav, Constantinides, and Geczy (1999)’s finding is based on the set of
Euler equation of household consumption rather than the per capita consumption.
Since the individual consumption data are reported with substantial error and are
difficult to test.
Therefore, they test the hypothesis that the stochastic discount factor
given by the equally weighted sum of the household’s marginal rates of substitution is
a valid stochastic discount factor.
4.2 Heterogeneous investment horizon
Lee (1976) derives two function forms for the CAPM, which will explicitly
40
include the investment horizon parameter to improve the explanatory power of the
CAPM.
To allow the investment horizon parameter to be explicitly included in the
CAPM, it will be assumed that all investors have identical horizons.
The main
contributions of Lee (1976) are to prove the observed function form of CAPM can
become nonlinear and to show that wither the likelihood ratio method or constant
elasticity of substitution function methods can employed to improve the explanatory
power of CAPM.
Levhari and Levy (1977) investigate the empirical implications of
heterogeneous investment horizons. While they have substantial contributed to the
understanding of multi-period investments, but they do not provided a generalized
asset pricing model for the equilibrium risk-return relationship under heterogeneous
investment horizons.
Lee, Wu, and Wei (1990) examine the effect of heterogeneous
investment horizons on the functional form of capital asset pricing and suggest a
translog model for estimating the relation between risk and return. They argue that
some empirical findings that are inconsistence with traditional CAPM from
misspecification of the CAPM by ignoring the inconsistency between observed data
period and the true investment horizon. They assume that the holding period returns
of securities are serially independent and the return distributions are stationary.
Under these two assumptions, the expected returns and variance are identical over
41
time, and the covariance of returns between two periods equals zero.
The translog
model is a suitable function for estimating the relationship between risk and expected
return.
5. Taxation Effect Models
Brennan (1970) first proposed an extended form of the single period CAPM
model that accounted for the differential taxation of dividends over capital gains.
Litzenberger and Ramaswamy (1979) extend the model of Brennan (1970) to account
for restrictions on investors’ borrowing.
Both these models assume that dividends
and interest are taxed as ordinary income and capital gains are taxed at more favorable
rates. Brennan's model is under the assumption of proportional individual tax rate
(not a function of income), certain dividends, and unlimited borrowing at the riskless
rate of interest as
~
E ( Ri )  rf  b*i   (di  rf )
(82)
~
where E( Ri ) is the before tax expected return to security i,  i is the covariance of
asset i 's return with the market divided by the variance of the market,


b*  E( Rm )  rf   (dm  rf ) is the after-tax excess return of the market portfolio,
r f is the return on a riskless asset, di and d m is the dividend yield on security i and
the market portfolio, and  represents a tax differential between dividends and
42
capital gains.
Litzenberger and Ramaswamy (1979) extend this model by assuming both
margin and income constraints on borrowing.
Their model is
~
E ( Ri )  rf  a  bi  c(di  rf )
(83)
~
~
~
where a  E ( Rz* )  rf , b  E ( Rm )  E ( Rz* )  c(d m  rf ) and c is a tax differential
reduced by a shadow price that reflects increases in investors' ability to borrow from
~
an additional dollar of dividends and E ( Rz* ) is the expected return on a zero-beta
portfolio with a dividend yield equal to the taxable riskless rate.
These models are
the standard two-parameter pricing models adjusted for differential taxation of
dividends and interest income relative to capital gains.
6. Skewness Effect Models
Sharpe (1964), Lintner (1965), and Mossin (1966), following the work of
Markowitz (1959), developed the first formulations of the mean-variance capital asset
pricing model (CAPM).
However, many researchers argued that assets pricing
models should subsume the effects of the higher moments. Lee (1977) first employed
the transformation technique developed by Box and Cox (1964) to determine the true
functional form for testing the risk-return relation and to examine the possible impact
of skewness effect on capital asset pricing. Then Sears and Wei (1988) indicated that
43
although the estimated coefficient of co-sknewness gives important information on
the marginal rate of substitution between skewness preferences that is independent of
the effects of the market risk premium. Furthermore, Harvey and Siddique (2000)
suggested that if asset returns have systematic skewness, expected returns should
include re-wards for accepting this risk. They formalized this intuition with an asset
pricing model that incorporates conditional skewness. Their results showed that
conditional skewness helps to explain the cross-sectional variation of expected returns
across assets and is significant even when factors based on size and book-to-market
are included
7. Behavioral Finance
The feature distinguishes the behavioral approach and traditional approach to
asset pricing is the assumption of expected utility.
Traditional asset pricing theorists
assume that investors seek to maximize expected utility.
However, proponents of
behavioral finance suggest that people behave more in accordance with a
psychologically based theory, such as prospect theory which was developed by
Kahneman and Tversky (1979).
Preference-based behavioral models often work with the prospect theory of
Kahneman and Tversky (1979).
According to this theory, people do not judge
44
outcomes on an absolute scale but compare outcomes with an initial reference point.
Their objective function has a kink at the reference point, so risk aversion is locally
infinite at that point.
The objective function is concave for gains (outcomes above
the reference point) but is convex for losses (outcomes below the reference point)
Tversky and Kahneman (1992) modifies their prospect theory by using a
cumulative distribution function for the domain of gains and a cumulative distribution
function for the domain of losses rather than separate decisions called Cumulative
Prospect Theory.
losses.
The value function is a utility function defined over gains and
Investor maximizes a value function of the form,
 x
V ( x)  

  ( x )
if x  0
if x  0
(84)
where 0<α<1, 0<β<1, λ>1 and x is the change of wealth rather than total wealth.
Investor employs decision weights estimated by the formula,
w ( P ) 
w ( P ) 
P
P
P

 (1  P)
P

 (1  P)

1/ 

1/ 
,
(85)
where P stands for cumulate probability, and w- and w+ denote the transformed
cumulative probability in the negative and positive domain, respectively.
Although we know that the mean-variance analysis and the CAPM are based
on the expected utility theory framework. However, Tversky and Kahneman
45
(1979,1992) show that the investor is not always a rational as assumed by expected
utility theory economists. Levy (2009) establishes a very interesting result; the
prospect theory investor will choose a portfolio that is mean-variance efficient.
He
also suggests that a modified version of mean-variance analysis and the traditional
CAPM can be justified in the Cumulative Prospect Theory framework, despite the
fact that under the Cumulative Prospect Theory, the expected utility theory is invalid.
The behavioral models cannot be tested using data on aggregate consumption
or the market portfolio, because rational utility-maximizing investors neither consume
aggregate consumption nor hold the market portfolio.
This makes it hard to test
behavioral models without having detailed information on the investment strategies of
different market participants.
8. Liquidity-based Models
Financial economists have long suspected that less liquid securities might have
to offer an expected return higher than that justified by the covariance of the
security’s return with the market or other factors, and thus the security might have a
lower price level.
Pastor and Stambaugh (2003) find that stocks whose prices decline when the
market gets more illiquid receive compensation in expected returns.
46
Dividing stocks
into 10 portfolios based on liquidity betas (regression coefficients of stock returns on
market liquidity, with other factors as controls), the portfolio of high-beta stocks
earned 9% more than the portfolio of low beta stocks, after accounting for market,
size, and value-growth effects with the Fama-French three factor model.
Almost this
entire premium is accounted for by spread in liquidity betas and a factor risk premium
estimated across 10 portfolios. The standard asset pricing model is E ( R i )   i  .
Thus, a spread in average returns E ( R i ) across portfolios i is explained if they are
linearly related to the betas i .

A test whether  = 0 is often performed to assess
the significance of the model.
Acharya and Pedersen (2005) perform a similar but more general investigation
as follows:
We are interested in how an asset’s expected return,
Dti  Pt i
,
Pt i1
Rti 
(86)
depends on its relative illiquidity cost, Cti represents transactions costs such as
broker fees and bid-ask spread,
cti 
Cti
,
Pt i1
(87)
on the market return, S i means total shares of security i,
RtM 
 i S i ( Dti  Pt i )
,
 i S i Pt i1
and on the relative market illiquidity,
47
(88)
ctM 
 i S i Cti
.
 i S i Pt i1
(89)
They rewrite the single-beta CAPM in net returns in terms of gross returns and
get a liquidity-adjusted CAPM for gross return.
i
t 1
Et ( R
covt ( Rti1  cti1 , RtM1  ctM1 )
 c )  R  t
,
vart ( RtM1  ctM1 )
i
t 1
f
(90)
where t  Et ( RtM1  ctM1  R f ) is the risk premium. Thus, the conditional expected
gross return is
Et ( Rti1 )  R f  Et (cti1 )  t
covt ( Rti1 , RtM1 )
covt (cti1 , ctM1 )


t
vart ( RtM1  ctM1 )
vart ( RtM1  ctM1 )
covt ( Rti1 , ctM1 )
covt (cti1 , RtM1 )
 t
 t
vart ( RtM1  ctM1 )
vart ( RtM1  ctM1 )
They examine all four channels for a liquidity premium.
.
(91)
First, a security
might have to pay a premium simply to compensate for its particular illiquidity or
transactions cost.
Second, a security might have to pay a premium because it
becomes more illiquid in bad times, i.e. when the market goes down.
If you have to
sell it, (and sellers are the marginal investor) this tendency amounts to a larger beta
than would be measured by the midpoint of a bid-asked spread.
Third, the security’s
price (the midpoint) might decline when markets as a whole become less liquid.
If
“market liquidity” is a state variable, an event that drives up the marginal utility of a
marginal investor, then this tendency will also result in a return premium.
mechanism that Pastor and Stambaugh (2003) investigated.
48
This is the
Fourth, the security
could become more illiquid when the market becomes more illiquid.
Then, they
examine whether the four sources of covariation described above explain the variation
in average returns.
Interestingly, their largest premium  in E ( R i )   i  is the
covariance of liquidity with market return — the chance the stock may get more
illiquid if the market goes down.
9. Existence of Equilibrium
Hart (1974) argued that in deriving these properties of equilibrium prices, it
has been assumed that equilibrium does in fact exist.
Surprisingly, no attempt
appears to have been made to establish the existence of equilibrium in the basic
Lintner-Sharpe model or in more general versions of the model.
Yet, the existence
of equilibrium is not implied by any of the standard existence theorems since these
theorems assume that consumption sets are bounded below, whereas the assumption
that investors can hold securities in unlimited negative amounts implies that
consumption sets are unbounded below.
In his paper, he found the conditions for the
existence of equilibrium in a very general version of the Lintner-Sharpe model.
Moverover, Nielsen (1989) presents simple conditions and a simple proof of the
existence of equilibrium in asset markets where short-selling is allowed and satiation
49
is possible. Unlike standard non-satiation assumptions, the one used here is weak
enough to be reasonable in the mean-variance Capital Asset Pricing Model and in
asset market models where investors maximize expected utility and where total
returns to individual assets may be negative.
10. Conclusion
We survey the evolution of CAPM from 1964 to 2009. According to the
assumptions of Sharpe (1964), Lintner (1965), and Mossin (1966), although they
derived the static CAPM, many scholars tried to get more general asset pricing
models by relaxing the assumption for the real world.
A number of important issues
remain as challenges for future researches. There are two directions:
First, we can try to subsume behavioral finance into asset pricing models, for
example investor sentiment. Obviously, many noise traders do affect stock returns but
we still have no theoretical asset pricing model including their behaviors into a
pricing factor.
In the other direction, we can try more efforts on the supply side of asset
pricing models. In the past, there were relatively few literatures on the supply side.
However, it is important, for example Holmstrom and Tirole (2001) suggested new
determinants of asset prices such as the distribution of wealth within the corporate
50
sector and between the corporate sector and the consumers. Also, leverage ratios,
capital adequacy requirements, and the composition of saving affect the corporate
demand for liquid assets and, thereby, interest rates.
51
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