The Evolution of Capital
Asset Pricing Models
Sheng-Syan Chen, National Taiwan University
Cheng-Few Lee, Rutgers University
Yi-Cheng Shih, National Taipei University
Po-Jung Chen, National Taiwan University
Outline
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Introduction
Intertemporal Models
Supply-Side Effect Models
International CAPM
Equilibrium Models with Heterogeneity
Dividend and Taxation Effect Models
Skewness Effect Models
Behavioral Finance
Liquidity-based Models
Existence of Equilibrium
Empirical Tests
Conclusion
Abstract
Based upon Markowitz (1952, 1959) Mean-Variance Portfolio Theory and six critical
assumptions, Sharpe (1964), Lintner (1965), and Mossin (1966) have derived and
developed the static Capital Asset Pricing Model. During the four past decades, the CAPM
has been the benchmark of asset pricing models, and most empirical apply it to calculate
asset returns and the cost of capital. To relax the original six assumptions, many
researchers have tried to generalize the static CAPM by Sharpe, Lintner, and Mossin. In
addition, many researchers have also tried to develop the dynamic Capital Asset Pricing
Models.
In this paper, we survey the important alternative theoretical models of the Capital
Asset Pricing for last four and half decades. We organize these theoretical models, as
follows: (i) Merton’s Intertemporal CAPM, (ii) Consumption-based Intertemporal CAPM, (iii)
Production-based Intertemporal CAPM, (iv) CAPM with Supply-side Effect, (v) International
Equilibrium CAPM with Heterogeneity Beliefs and Investors, (vi) Equilibrium CAPM with
Heterogeneity Investment Horizon, (vii) CAPM with Dividend and Taxation Effect, (viii)
CAPM with Skewness Effect, and (ix) Behavioral Finance, and Liquidity-based CAPM. The
interrelationship among these models is also discussed in some detail.
The results of this paper might be used as a guideline for future theoretical and
empirical research in capital asset pricing. More specifically, we suggest three possible
directions for future research. To the best of our knowledge, this is one of the most
complete reviews of the evolution of theoretical capital asset models.
1.
Introduction
dPi
  i dt   i dzi
Pi
(1)
n
n

dW   wi  i  r f   r f Wdt   wiW i dzi   y  c dt
i 1
 i 1

(2)
2. Intertemporal Models
2.1 Merton Model
2.2 Consumption-based Models
2.3 Production-based Models
2. Intertemporal Models
2.1 Merton Model
 i  r  i  M  r 
(i=1, 2,…,n),
(3)
where i   iM /  M2 ,  iM is the covariance of the return on the ith asset with
the return on the market portfolio and  M is the expected return on the
market portfolio.
i  r 
 i iM  in nM 
 i in  iM nM 


 n  r 


r

M
2
2
 M 1  nM 
 n 1  Mn 
(4)
2. Intertemporal Models
2.2 Consumption-based Models
U (Ck ,t )   Et 1  Ri ,t 1 U (Ck ,t 1 )

U (Ck ,t 1 ) 
1  Et (1  Ri ,t 1 ) 
  Et 1  Ri ,t 1  M k ,t 1 
U (Ck ,t ) 

(5)
(6)
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.
2. Intertemporal Models
2.2 Consumption-based Models
1  Et 1  Ri ,t 1  M t 1 
(7)
Et 1  Ri ,t 1  M t 1 
 Et 1  Ri ,t 1  Et  M t 1   Covt  Ri ,t 1 , M t 1 
(8)
2. Intertemporal Models
2.2 Consumption-based Models
1  Covt Ri ,t 1 , M t 1 
1  Et Ri ,t 1  
Et M t 1 
1  R f ,t 1
1

Et M t 1 
(9)
(10)
2. Intertemporal Models
2.2 Consumption-based Models
1  Et Ri ,t 1   1  R f ,t 1  1  Covt Ri ,t 1 , M t 1 
1
(Ct  X t )
E
1 
t 1

t
Uc (Ct , X t )  (Ct  X t )
(11)
1
(12)


 Ct St

(13)
2. Intertemporal Models
2.2 Consumption-based Models
M t 1
 Ct 1 
U c (Ct 1 , X t 1 )


  
U c (Ct , X t )
 Ct 

 St 1 


 St 

(14)
M t 1  at  (b0  b1 zt )  ct 1
(15)
Wt 1  Rm,t 1 (Wt  Ct )
(16)
2. Intertemporal Models
2.2 Consumption-based Models
Wt 1
Ct
 Rm,t 1 (1  )
Wt
Wt
wt 1
 1
 rm,t 1  a  1  ct  wt 
 
k
ct  wt    (rm,t i  ct i ) 
1 
i 1
(17)
(18)

i
(19)
2. Intertemporal Models
2.2 Consumption-based Models

ct  wt  Et   (rm,t i
i
i 1
k
 ct i ) 
1 
1

1

U t  1   Ct   EtU t 1







1
(20)

1
1
1 /  







C
1





1  Et   t 1   
1  Ri ,t 1 

  Ct   

1  Rm,t 1 





(21)
(22)
2. Intertemporal Models
2.2 Consumption-based Models

1 / 



  Ct 1 
 

 Rm,t 1 
1  Et  
  Ct 


 


0   log  Et ct 1    1Et rm,t 1  Et ri ,t 1 
(23)

2

1   
2
2
2
(  1)(Vcm )  Vci  2(  1)Vim 
  Vcc  (  1) Vmm  Vii 
2   



(24)
Et ri ,t 1  rf ,t 1
Vii
Vic
  
 (1   )Vim
2

(25)
2. Intertemporal Models
2.2 Consumption-based Models
 k  m 
ct  wt  1    Et   (rm,t i ) 
(26)
1 
i 1

i

ct 1  Et ct 1  rm,t 1  Et rm,t 1  1   Et 1  Et   i rm,t 1i
i 1
(27)
Covri ,t 1 , ct 1   Vic  Vim  1   Vih
(28)
2. Intertemporal Models
2.2 Consumption-based Models



i
Vih  Cov ri ,t 1  Et ri ,t 1 , ( Et 1  Et )  rm,t 1i 
i 1


(29)
Et ri ,t 1  rf ,t 1
Vii
   Vim  (  1)Vih
2
(30)
2. Intertemporal Models
2.3 Production-based Models

E0
1 
t
Ri  dt



t 0 i 0
(31)
dt  yt  it  yt  kt 1

yt  AB t kt
t
(32)
(33)
2. Intertemporal Models
2.3 Production-based Models


Et Rt 1    yt 1 / kt 1   1
1

E0
  uc 
t
t 0
(34)
(35)
t
ct  pt  yt st 1   pt  yt   dt  yt st
(36)
2. Intertemporal Models
2.3 Production-based Models
pt  yt uct   Et  pt 1  yt 1   dt 1  yt 1 uct 1 
(37)
Rt 1  yt 1 , yt    pt 1  yt 1   dt 1  yt 1 / pt  yt 
(38)
2. Intertemporal Models
2.3 Production-based Model

pt  Et   i uct i  / uct d t i
(39)
uct   a ln ct 
(40)
i 1

pt  Et   d t   / 1   d t
i
i 1
(41)
2. Intertemporal Models
2.3 Production-based Model
Rt 1  1 /  dt 1 / dt 
Rt 1  1/   yt 1 / yt 

yt 1  t 1 yt
t
M t 1  b0  zt b1   f t 1
(42)
(43)
(44)
(45)
2. Intertemporal Models
2.3 Production-based Models
M t 1  b0 f t 1  b1  f t 1  zt 1 
(46)
V kt , t   Max u ct , n  nt    Et V kt 1 , t 1  (47)
nt ,kt 1
t 1  H t ,  t 1 
(48)
ct  f t , nt , kt   kt 1
(49)
2. Intertemporal Models
2.3 Production-based Models
M t 1 

t 1
f k  t 1 , nt 1 , kt 1 
(50)

Et r V kt 1 ,t 1 / Fk t 1 , nt 1 , kt 1   0
iE
t 1 k
for all i,
(51)
3. Supply-Side Effect Models
3.1 Demand function of capital assets
3.2 Supply function of securities
3.3 Multiperiod Equilibrium Models
3. Supply-Side Effect Models
3.1 Demand function of capital assets
{bWt 1}
U  a  he
(52)
X j ,t 1  Pj ,t 1  Pj ,t  D j ,t 1 j = 1,…,N, (53)
x j ,t 1  Et X j ,t 1  Et Pj ,t 1  Pj ,t  Et D j ,t 1
j = 1,…,N,
(54)
3. Supply-Side Effect Models
3.1 Demand function of capital assets
wt 1  EtWt 1  Wt  r  Wt  qt1 Pt   qt1 xt 1
(55)

V Wt 1   E Wt 1  wt 1 Wt 1  wt 1   qt1S q ,t 1
b
Max wt 1  V(Wt 1 )
2
(56)
(57)
3. Supply-Side Effect Models
3.1 Demand function of capital assets


b

Max 1  r Wt  qt1 xt 1  r  Pt    qt1S qt 1
2
*
1
qt 1  b S
1
xt1  r  Pt 
(58)
(59)
m
Qt 1   qtk1  cS 1Et Pt 1  (1  r*)Pt  Et Dt 1 
k 1
(60)
3. Supply-Side Effect Models
3.2 Supply function of securities
1
Min Et Di ,t 1Qi ,t 1    Qi,t 1 Ai Qi ,t 1  (61)
2
Qi,t 1  A i Pi,t  Et Di,t 1 
1
i
(62)
3. Supply-Side Effect Models
3.2 Supply function of securities
1
i
Qt 1  A
BPt  Et Dt 1 
(63)
where
 A11

1
A
2
A1  




1I


2 I
,B  



1 
AN 


 Q1 

Q 
 ,and Q   2 

 

 
N I 
QN 
3. Supply-Side Effect Models
3.3 Multiperiod Equilibrium Models
Qt 1  cS
1
E P
t t 1
1
i
Qt 1  A


 1  r Pt  Et Dt 1
*
BPt  Et Dt 1 
 (64)
(65)
3. Supply-Side Effect Models
3.3 Multiperiod Equilibrium Models
cS 1  Et Pt 1  Et 1Pt  1  r *   Pt  Pt 1   Et Dt 1  Et 1Dt 
A
 BPt  Et Dt 1   Vt


1
(66)

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 BEt 1 Pt  Et 1 Dt 1 .
1  r cS
*
1

(66´)
 A1 B Pt  Et 1 Pt  


(67)
cS 1 Et Pt 1  Et 1 Pt 1   cS 1  A1 Et Dt 1  Et 1 Dt 1   Vt

4. International CAPM
Without a model showing how assets are priced in a world in which asset
markets are fully integrated, it is impossible to determine whether asset
markets are segmented internationally or not. Stulz (1981a) provide an
intertemporal model of international asset pricing, which admits differences
in consumption opportunity sets across countries. The model shows that the
real expected excess return on a risky asset is proportional to the
covariance of the return of that asset with changes in the world real
consumption rate. It has no barriers to international investment, but it is
compatible with empirical facts, which contradict the predictions of earlier
models and which seem to imply that asset markets are internationally
segmented. Besides, Stulz (1981b) also presents a simple model in which it
is costly for domestic investors to hold foreign assets. The implications of
the model for the composition of optimal portfolios at home and abroad are
*
derived. It is shown that all foreign assets with a beta larger than some beta
plot on either one of two security market lines. Some foreign assets with a
beta smaller than  * are not held by domestic investors even if their
expected return is increased slightly.
4. International CAPM
After the above two papers, Stulz (1982) examines the conditions under
which a risk premium is incorporated in the forward exchange rate. A new
condition for the existence of a risk premium is proposed. He shows that
earlier models of the risk premium, which emphasize either the role of net
foreign investment or of the relative supplies of “outside” assets, are not
suited for assessing the effects of changes in macroeconomic policy. Finally,
Stulz (1984) summarizes that how differences across countries of 1) inflation
rate, 2) consumption baskets of investors, and 3) investment opportunity
sets of investors matter when one applies capital asset pricing models in an
international setting. In particular, the fact that countries differ is shown to
affect the portfolio held by investors, the equilibrium expected returns of
risky assets, and the financial policies of firms. In empirical studies, Chang
and Hung (2000) employ a two-factor international equilibrium asset pricing
model to examine pricing relationships among the world's five largest equity
markets. Their paper suggests that the intertemporal asset pricing model
proposed by Campbell (1993) can be used to explain the returns on the five
largest stock market indices.
5. Equilibrium Models with Heterogeneity
5.1 Heterogeneous Beliefs and Investors
5.2 Heterogeneous investment horizon
5. Equilibrium Models with Heterogeneity
5.1 Heterogeneous Beliefs and Investors



   1
  Ct 1 



*


Et  
exp
Vart 1   i ,t 1   1

Ct 
2







M
*
t 1
M
RA
t 1

 (  1)
2
Var ck ,t 1
*
t 1
(68)
(69)
6. Dividend and Taxation Effect Models
E(Ri )  rf  b i   (di  rf )
*
(70)
E(Ri )  rf  a  bi  c(d i  rf )
(71)
7. Skewness Effect Models
Sharpe (1964), Lintner (1965), and Mossin (1966), following the work of Markowitz (1959),
develope the first formulations of the mean-variance CAPM. However, many researchers
criticize the widely used mean-variance analysis of portfolio selection and argue that assets
pricing models should subsume the effects of the higher moments. Borch (1969) contends
that any system of upward sloping mean-standard deviation indifference curves can be
shown to be inconsistent with the basic axiom of choice under uncertainty. Feldstein (1969)
shows that Tobin (1958, 1965) is incorrect in asserting that the μ-σ indifference curves of a
risk-averter are convex-downwards whenever the possible investment outcomes are
assumed to follow a two-parameter probability distribution. Although Tobin‘s proof is correct
for normal distributions, for a number of economically interesting distributions, the
indifference curves are not convex, showing that when more than one asset has positive
variance, an analysis in terms of only μ and σ is not strictly possible unless utility functions
are quadratic or the possible subjective probability distributions are severely restricted.
Tsiang (1972) argues that although the mean-standard deviation analysis was at first
introduced by Tobin to explain liquidity preference in the sense of an investment demand
for cash, in his defense of it against its critics, he actually finds that it is quite incapable of
doing what Tobin has expected of it. Furthermore, he claims that the importance of
skewness preference for major risk-takers should obviously be taken into consideration in
problems of investment incentives.
7. Skewness Effect Models
Therefore, Jean (1971) begins a general extension of the two-parameter analysis to three
or more parameters; however, Ingersoll (1975) corrects several errors in Jean’s model
(1971) and derives a normative, individual pricing model for risky securities analogous to
the capital market line within the framework of a perfect market. Finally, Schweser (1978)
clarifies and corrects certain parts of Ingersoll’s correction of Jean’s work.
Although many researchers pay more attention to the skewness effect on capital asset
pricing models, Lee (1977) first employs 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 the skewness effect on capital asset pricing. According to
Sears and Wei (1988) 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. Moreover, Harvey and Siddique
(2000) suggest that if asset returns have systematic skewness, expected returns should
include rewards for accepting this risk. They formalized this intuition with an asset pricing
model that incorporates conditional skewness. Their results show 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.
8. Behavioral Finance
if x  0
 w
V ( x)  

  ( w )

x ( P) 
x  ( P) 
P
P
P

(72)

(73)

 (1  P)
P

if x  0
 (1  P )
 1/ 

 1/ 
9. Liquidity-based Models
D  Pt
R 
i
Pt 1
i
t
i
t
i
(74)
i
Tt
t  i
Pt 1
(75)
i
t
i S ( D  P )
R 
i S P
i
M
t
i
i
t
t
i i
t 1
(76)
9. Liquidity-based Models
t
M
t
i
t 1
Et ( R
 i S i Tt i

i i
 i S Pt 1
(77)
covt ( R  t , R  t )
 t )  r f  t
vart ( R  t ) (78)
i
t 1
i
t 1
i
t 1
M
t 1
M
t 1
M
t 1
M
t 1
i
M
i
M
cov
(
R
,
R
)
cov
(
t
,
t
i
i
t
t 1
t 1
t t 1 t 1 )
Et ( Rt 1 )  r f  Et (t t 1 )  t
 t
M
M
vart ( Rt 1  t t 1 )
vart ( RtM1  t tM1 )
covt ( Rti1 , t tM1 )
covt (t ti1 , RtM1 )
 t
 t
M
M
vart ( Rt 1  t t 1 )
vart ( RtM1  t tM1 )
(79)
10. Existence of Equilibrium
Hart (1974) argues that in deriving the 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 because these theorems assume that consumption sets are
bounded below. By contrast the assumption that investors can hold
securities in unlimited negative amounts implies that consumption sets are
unbounded below. In his paper, he finds 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 CAPM
and in asset market models where investors maximize expected utility and
where total returns to individual assets may be negative.
11. Empirical Tests
Black et al. (1972) and Fama and MacBeth (1973) test the implication of
CAPM and find empirical evidence to support the linear relationship between
risk and return and efficient market; therefore, their empirical studies support
the CAPM. Roll (1977), however, criticizes their empirical results by
declaring that (a) no correct and unambiguous test of the theory has
appeared in the literature, and (b) there is practically no possibility that such
a test can be accomplished in the future. Besides, Cheng and Grauer (1980)
also criticize the tests of Black et al. (1972) and Fama and MacBeth (1973)
based only on the assumption of constant β and stationarity of the
distribution of return; therefore, their paper argues that it makes no sense to
attempt a test of the CAPM based on stationarity because the validity of the
CAPM over time implies stationarity cannot hold in any but a very
degenerate sense. Thus, they find the CAPM generally does poorly in their
tests. Finally, Fama and French (1992) conclude that market capitalization
(a measure of size) and the ratio of the book to the market value equity
should replace beta altogether.
12. Conclusion
We have surveyed the evolution of CAPM from 1964 to 2009. We use both
figures and a table to summarize this paper. Figure 1 shows the research
flow chart, and Table 1 provides the literature summary. Sharpe (1964),
Lintner (1965), and Mossin (1966) derive their original static CAPM according
to the six critical assumptions. Many scholars have tried to get more
generalized asset pricing models by relaxing the assumption to meet the real
world situation. Because of the limitation of six critical assumptions and
possible model misspecification, we should carefully use the original static
CAPM to acquire the required return of an asset and calculate its abnormal
return. Fama and French (2004) argue that the CAPM’s empirical problems
may reflect theoretical failings, the result of many simplified assumptions;
however, they may also be caused by difficulties in implementing valid tests
of the model. Fama and French’s empirical research is based only upon the
original static CAPM, but we believe that empirical research should not only
be based upon the original static CAPM
.
12. Conclusion
In this paper, we have carefully reviewed papers which have extended the
original static CAPM. These papers have been classified into (i) Merton’s
Intertemporal CAPM, (ii) Consumption-based Intertemporal CAPM, (iii)
Production-based Intertemporal CAPM, (iv) CAPM with Supply-side Effect,
(v) International Equilibrium CAPM with Heterogeneity Beliefs and Investors,
(vi) Equilibrium CAPM with Heterogeneity Investment Horizon, (vii) CAPM
with Dividend and Taxation Effect, (viii) CAPM with Skewness Effect, and
(ix) Behavioral Finance, and Liquidity-based CAPM. As a result of our
review, we believe that some important issues remain for future researchers.
Now we discuss these potential important research issues as follows:
First, we can try to subsume behavioral finance into asset pricing models,
for example, investor sentiment. Obviously, many noise traders affect stock
returns, but we still have no theoretical asset pricing model that includes
their behaviors into a pricing factor.
12. Conclusion
Second, we can further explore the supply side of asset pricing models. In the past, there was
relatively few literature on the supply side; however, it is important. Holmstrom and Tirole (2001)
suggest, for example, new determinants of asset prices, such as the distribution of wealth within
the corporate 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.
Third, although Fama and French’s (1996) three-factor model has good empirical performance,
they acknowledge that there are important limitations in their model. Their empirical results still do
not cleanly identify the two consumption-investment state variables of special hedging concern to
investors that would provide a neat interpretation of their results in terms of Merton’s (1973) ICAPM
or Ross’ (1976) APT. Merton’s (1973) ICAPM not only has a complete and solid theoretical
framework but also provides better empirical performance than the static CAPM, such as Fama
and French’s (1996) three-factor model if we can find those solid and robust state variables. We
suggest that future researchers should pay more attention to how to identify those solid and robust
state variables. Moreover, it will make bring Merton’s (1973) ICAPM closer to real world, and its
implication will be useful for empirical studies.
Fourth, the relationship between perspective theory and CAPM needs further research in both
theoretically and empirically, and especially the relationship between skewness type of CAPM and
perspective theory needs to be carefully investigated.
Table 1. Literature Summary
Models
Literature
Results and Contributions
Merton (1973, Econometrica)
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.
Breeden (1979, Journal of Financial Economics)
Breeden (1979) utilizes the same continuous-time economic framework as used
by Merton (1973), shows Merton’s multi-beta pricing equation can be collapsed
into a single-beta equation. The expected return on any asset is proportional to its
beta with respect to aggregate consumption alone.
Campbell (1993, American Economic Review)
Campbell (1993) substitutes consumption out of the model to get a discrete-time
version of the intertrmporal CAPM of Merton (1973).
Campbell and Cochrane (1999, Journal of Political Economy)
Campbell and Cochrane (1999) present a habit persistence model to explain the
dynamic pricing phenomena, that is, using lagged consumption as the state
variable to explain the procyclical variation of stock prices, the long-horizon
predictable of excess stock returns, and the countercyclical variation of stock
market volatility.
Jagannathan and Wang (1996, Journal of Finance)
Jagannathan and Wang (1996) argue that the CAPM holds in a conditional sense
that betas and the market premium vary over time. They add the labor income to
explain the cross-section asset returns.
Lettau and Ludvigson (2001a, Journal of Finance)
Lettau and Ludvigson (2001a) investigate the power of fluctuations in the log
consumption-wealth ratio for forecasting asset returns.
Lettau and Ludvigson (2001b, Journal of Political Economy)
Lettau and Ludvigson (2001b) is the first reexamination 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.
Lewellen and Nagel (2006, Journal of Financial Economics)
Lewellen and Nagel (2006) criticize consumption model on the argument that
the low covariance between the risk premium and the betas. The covariance
between consumption betas and the consumption risk premium obtained from a
series of estimates over small time windows is too small to support the
importance of any conditional variable.
Balvers and Huang (2009, Journal of Financial and Quantitative
Analysis)
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.
Intertemporal CAPM
－Merton Model
Intertemporal CAPM
－Consumption-based Models
Table 1. (Continued)
Models
Literature
Results and Contributions
Balvers, Cosimano, and McDonald (1990, Journal of Finance)
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. 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.
Cochrane (1991, Journal of Finance) (1996, Journal of Political Economy)
Cochrane (1991, 1996) extend the production-based CAPM by deriving from producer’s
first order condition for optimal intertemporal investment demand to describe the asset
returns.
Balvers and Huang (2007, Journal of Financial Economics)
Balvers and Huang (2007) derive the productivity shocks in the marginal value of capital
to obtain an explicit production-based CAPM expression for the asset pricing model.
Black (1976, American Economic Review)
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 shortrun 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.
Grinols (1984, Journal of Finance)
Grinols (1984) extended Merton's intertemporal capital asset pricing model with multiple
consumers to include a description of the supply of traded securities.
Lee, Tsai, and Lee (2009, Quarterly Review of Economics and Finance)
Lee, Tsai, and Lee (2009) first theoretically extend the dynamic, simultaneous CAPM
model of Black (1976) to the existence of the supply effect in the asset pricing process.
They use price, dividend per share and earnings per share to test the existence of supply
effect with U.S. domestic stock markets.
Constantinides (1982, Journal of Business)
Constantinides (1982) argue the equilibrium model of a heterogeneous-household, fullinformation economy under the assumption that the households insure against
idiosyncratic income shocks.
Constantinides and Duffie (1996, Journal of Political Economy)
Constantinides and Duffie (1996) construct a discount factor to represent any asset pricing
anomalies under the assumption that investors have the same power utility function.
Brav, Constantinides, and Geczy (2002, Journal of Political Economy)
Brav, Constantinides, and Geczy (2002) test the stochastic discount factor given by the
equally weighted sum of the household’s marginal rates of substitution to be a valid
stochastic discount factor based on the set of Euler equation of household consumption.
Basak (2005, Journal of Banking and Finance)
Basak (2005) provides a continuous-time pure-exchange framework to study asset pricing
implication of the present of heterogeneous beliefs, within a rational Bayesian setting.
Levy, Levy, and Benita (2006, Journal of Business)
Levy, Levy, and Benita (2006) relax the homogeneous beliefs assumption of CAPM. They
employ the mathematical analysis and numerical simulations to study the effect of the
introduction of heterogeneity of beliefs on asset prices.
Intertemporal CAPM
－Production-based Models
Supply-Side Effect Models
Equilibrium Models with
Heterogeneity Beliefs and Investors
Table 1. (Continued)
Models
Equilibrium Models with
Heterogeneity Investment Horizon
Taxation Effect Models
Skewness Effect Models
Literature
Results and Contributions
Lee (1976, The Review of Economics and Statistics)
Lee (1976) first prove the observed function form of CAPM can become
nonlinear and show that either the likelihood ratio method or constant elasticity
of substitution function methods can employed to improve the explanatory
power of CAPM.
Levhari and Levy (1977, The Review of Economics and
Statistics)
Levhari and Levy (1977) investigate the empirical implications of
heterogeneous investment horizons.
Lee, Wu, and Wei (1990, Journal of Financial and Quantitative
Analysis)
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.
Brennan (1970, National Tax Journal)
Brennan (1970) first propose an extended form of the single period CAPM
model that accounted for the differential taxation of dividends over capital gains.
Litzenberger and Ramaswamy (1979, Journal of Financial
Economics)
Litzenberger and Ramaswamy (1979) extend the model of Brennan (1970) to
account for restrictions on investors’ borrowing. The model is the standard twoparameter pricing models adjusted for differential taxation of dividends and
interest income relative to capital gains.
Borch (1969, Review of Economics Studies)
Borch (1969) contended that any system of upward sloping mean-standard
deviation indifference curves can be shown to be inconsistent with the basic
axiom of choice under uncertainty.
Feldstein (1969, Review of Economics Studies)
Feldstein (1969) showed that Tobin (1958, 1965) was incorrect in asserting that
the μ-σ indifference curves of a risk-averter are convex-downwards whenever
the possible investment outcomes are assumed to follow a two-parameter
probability distribution. Although Tobin's proof is correct for normal
distributions, for a number of economically interesting distributions the
indifference curves are not convex shows that when more than one asset has
positive variance, an analysis in terms of only μ and σ is not strictly possible
unless utility functions are quadratic or the possible subjective probability
distributions are severely restricted.
Jean (1971, Journal of Financial and Quantitative Analysis)
Jean (1971) began a general extension of the two-parameter analysis to three or
more parameters.
Tsiang (1972, American Economic Review)
Tsiang (1972) argued that although the mean-standard deviation analysis was at
first introduced by Tobin to explain liquidity preference in the sense of an
investment demand for cash, in his defense of it against its critics, he actually
finds that it is quite incapable of doing what Tobin has expected of it.
Furthermore, he claimed that the importance of skewness preference for major
risk-takers should obviously be taken into consideration in problems of
investment incentives.
Ingersoll (1975, Journal of Financial and Quantitative Analysis)
Ingersoll (1975) corrects several errors in Jean’s model (1971) and derives a
normative, individual pricing model for risky securities analogous to the capital
market line within the framework of a perfect market.
Table 1. (Continued)
Models
Skewness Effect Models
Literature
Results and Contributions
Schweser (1978, Journal of Financial and Quantitative Analysis)
Ingersoll (1975) developed a normative multidimensional security pricing model for
individual investor in which he corrected errors in an earlier attempt by Jean (1971) at
developing such a model. Schweser (1978) clarified and corrected certain parts of
Ingersoll’s correction of Jean’s work.
Sears and Wei (1988, The Financial Review)
Sears and Wei (1988) indicated that 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.
Harvey and Siddique (2000, Journal of Finance)
Harvey and Siddique (2000) suggested that if asset returns have systematic skewness,
expected returns should include rewards for accepting this risk. They formalized 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.
Kahneman and Tversky (1979, Econometrica)
Kahneman and Tversky (1979) developed the prospect theory to describe that people
behave more in accordance with a psychologically based theory rather than seek to
maximize the expected utility.
Tversky and Kahneman (1992, Journal of Risk and Uncertainty)
Tversky and Kahneman (1992) modified the prospect theory by using a cumulative
distribution function for the domain of gains and losses rather than separate decisions
called Cumulative Prospect Theory.
Levy (Forthcoming, European Financial Management)
Levy suggested 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.
Pastor and Stambaugh (2003, Journal of Political Economy)
Pastor and Stambaugh (2003) find that stocks whose prices decline when the market gets
more illiquid receive compensation in expected returns. Dividing stocks into 10 portfolios
based on liquidity betas, the portfolio of high-beta stocks earned more than the portfolio of
low beta stocks, after accounting for market, size, and value-growth effects.
Acharya and Pedersen (2005, Journal of Financial Economics)
Acharya and Pedersen (2005) performed a similar but more general investigation on four
channels for a liquidity premium. Their largest premium is the covariance of liquidity with
market return — the chance the stock may get more illiquid if the market goes down.
Hart (1974, Journal of Economic Theory)
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.
Nielsen (1989, Review of Economic Studies)
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 Pricing Model and in asset market models
where investors maximize expected utility and where total returns to individual assets may
be negative.
Behavioral Finance
Liquidity-based Models
Existence of Equilibrium