A Partial Asymmetric Price Adjustment Model
and
Fama’s Market Efficiency
by
Tatsuyoshi Mi yakoshi *
Yoshihiko Tsukuda**
Junji Shimada***
February 27, 2006
Keywords: individual actions; market efficiency; asymmetric response; ARCH-type
model
JEL Classification Number:
G12; G14
*Graduate School of International Cultural Studies, Tohoku University
**
***
Graduate School of Economics, Tohoku University
School of Management, Aoyama Gakuin University
Correspondence: Tatsuyoshi Miyakoshi, Graduate School of International Cultural Studies, Tohoku
University, Kawauchi Aoba-ku, Sendai 980-8576, JAPAN
Tel: +81-22-217-7613, Fax:+81-22-217-7613, E-mail: miyakoshi@intcul.tohoku.ac.jp
1
Abstract:
The paper incorporates a partial asymmetric price adjustment model for individual
actions into an ARCH-type model and clarifies the relationship between the price
adjustment speed and the Fama's random walk form of market efficiency. The stock
price instantaneously and fully adjusts to the intrinsic value if and only if the market is
efficient in the Fama's sense. Thereby, the paper provides a hypothesized individual
action with the Fama’s market efficiency. As an operational example, the Tokyo stock
market is found to be inefficient during 1980-2005. The speed of price adjustment is
asymmetric in the 80s but symmetric in the 90s and 2000s.
2
1. Introduction.
The definition of market efficiency by Fama(1976) is based on informational
concept. It is characterized as one in which security prices fully reflect all available
information. In the framework of empirical researches, it is defined as the expectation of
the returns conditional on the previous information is constant, i.e. E{Rt | I t -1 } 
constant, which is a so-called Fama’s random walk form of the market efficiency. The
concept of market efficiency is silent, however, about the market processes that might
deliver the hypothesis on individual actions.
There are, so far, some hypotheses on individual actions, which are not related
to the definition of market efficiency by Fama (1976). The first idea was formalized by
Grossman and Stiglitz (1980). In their theoretical model, the market price does not fully
incorporate all knowable information because informed individuals make returns by
exploiting deviations of prices from security values. Later, Busse and Green (2002)
found that the small profits available to very short horizon traders (i.e., informed
individuals) are consistent with compensation for continuously monitoring information
sources, supporting the theory by Grossman and Stiglitz (1980). Second, without
characterizing individual actions, Busse and Green (2002) empirically find that news
reports about individual stocks on the financial television network CNBC are
incorporated into stock prices within few minutes. They shed light on the degree of
efficiency and conclude that the market is efficient enough that a trader cannot generate
profits based on widely disseminated news unless he acts almost immediately. Third,
Amihud and Mendelson (1987) and Koutmos(1998, 1999) developed and linked the
partial adjustment price model for individual actions to an ARCH-type model. The
stock price is adjusted to intrinsic values (security values) by portfolio mangers and the
adjustment speed is different depending on whether the stock price is over or under the
intrinsic value. Koutmos (1998, 1999) also offered an empirical new finding that the
adjustment speed is faster when the stock price is over the intrinsic value. Motivated by
his new finding, Pagan and Soydemir (2001), Bang and Shin (2003), and Nam et al.
(2003, 2005) have got the same finding as Koutmos (1998, 1999).
The purpose of this paper is twofold. First, we define the market efficiency in
the framework of a partial price adjustment model for individual actions, show the
equivalence between our definition and Fama’s random walk form of the market
efficiency, and thereby, provides a hypothesized individual action with the Fama’s
market efficiency and inefficiency. Second, as an operational example of the model, we
examine whether the Tokyo Stock Market is efficient.
The organization of the paper is as follows. Section 2 sketches a partial
3
adjustment price model by Amihud and Mendelson (1987) and Koutmos (1998, 1999)
and defines the market efficiency on the model. Section 3 shows the equivalence
between our definition and Fama’s random walk form of the market efficiency. Section
4 provides the estimation results of the Tokyo Stock Market during 1980-2005. Section
5 gives concluding remarks. Appendix gives proofs of propositions.
2. The Model
2. 1 Partial Asymmetric Adjustment Model and Market Efficiency
We follow a partial adjustment price model by Amihud and Mendelson (1987)
and Koutmos (1998, 1999) with some modifications. The model distinguishes the
unobserved intrinsic value of stock (Vt ) from the observed stock price (P t ), both are
expressed in natural logarithms. The process of intrinsic value follows a random walk
process with drift:
(1)
Vt  a  Vt 1  u t , u t | I t 1 ~ N 0,  ut2 , t  1,..., T,
where a is constant and I t -1 denotes the information set of the time t-1. We assume that
the disturbance term (u t ) has the EGARCH process proposed by Nelson (1991)):
log  u2t   0   1 z t - 1   2 (| z t - 1 | - E ( z| t - 1 | ) )  3 l o g u2t - 1
(2)
u t /  ut ~ N (0, 1 ) .
The partial asymmetric adjustment price process of Pt represents that adjustment
costs are asymmetric in up and down markets:
Pt  Pt 1  (1    )(Vt  Pt 1 )   (1    )(Vt  Pt 1 )  , - 1    ,    1,
(3)
where (Vt - Pt-1)+ = max{ Vt - Pt-1, 0}, and (Vt - Pt-1)- = min{ Vt - Pt-1, 0}. If  =  (=
), equation (3) reverts to the basic partial adjustment price process proposed by
Amihud and Mendelson (1987). Koutmos (1998; p.280, 1999; p.86) formulated the
asymmetric adjustment to intrinsic value in (3). None of Amihud and Mendelson (1987)
and Koutmos (1998, 1999), however, investigated the theoretical relationship of partial
adjustment price process to the market efficiency. This paper examines the implications
of the partial asymmetric adjustment model for the market efficiency. The market
efficiency based on the partial adjustment price model is defined as follows:
4
Definition 1: The market is said to be efficient if  + =   = 0 in equation (3) and
inefficient otherwise.
This definition of market efficiency indicates that the speeds of price adjustment
of both positive and negative discrepancy for the intrinsic value are equal to unity,
namely the stock price instantaneously and fully adjusts to the intrinsic value:
Pt  Pt 1  (Vt  Pt 1 )   (Vt  Pt 1 )   Vt  Pt 1 , and Pt  Vt .
(4)
If 0 <  +,   <1, the stock price partially adjusts for the intrinsic value. When  +,   <
0, the stock price overshoots the intrinsic value.
2. 2 The Reduced Model
We have an autoregressive process for the returns.
Proposition 1: The return process consisting of (1) and (3) has the following
expression
Rt  a    R *t -1    R *t -1  u t ,
*
(5)
where Rt *   t1 Rt .
The model (5) with (2) is an EGARCH model. Equation (5) is alternatively
expressed as
Rt   t (a    R t -1    R t -1 )   t ,
(6)
where  t  1    (   ) for R t  0, and  t  1    (   ) otherwise;    (  ) 1  ,
 -  (  ) 1  and t =  t ut . If  =  (= ), equation (6) reduces to
Rt  a(1-  )   R t -1  (1   )u t .
(7)
The process of Rt is apparently similar to that of Rt . However, except for the case of 
*
= , the conditional expectation of t is not zero and the process of {t} is serially
dependent as will be shown in Lemma 1 of Appendix.1 The conditional variance of t
Equation (7) in Koutmos (1998, pp. 280) has mistakes and should be corrected as
equation (6) in this paper. Since both autoregressive coefficients and error terms depend
upon the sign of Rt, the optimal one-step ahead forecast in his equations (8a) and (8b) is
incorrect. The correct formula is shown in proposition 2 of this paper. Moreover, since
1
5
does not follow an EGARCH process unlike to that of ut.
3. The Relationship to Fama’s Efficiency and an Illustrative Example
3. 1 The Equivalence of the two definitions of efficiency
The definition of market efficiency by Fama(1976) is based on informational
concept. Fama(1976, 16th row on pp.144) argues that ”if the market is efficient, there is
no way to use any information available at time t-1 as the basis for a correct assessment
of an expected value of Rt which is different from the assumed constant equilibrium
expected return, E(R). Since part of the information available at t-1 is the time series of
past returns, there is no way to use the past returns as the basis for a correct assessment
of the expected return from t-1 to t which is other than E(R).” In the framework of this
paper, Fama’s definition of efficiency is equal to
E{Rt | I t -1 }  a .
(8)
In order to find the relationship between the two definitions of efficiency, we
consider the conditional expectation of Rt in (6).
Proposiion 2: The expectation of Rt conditional on It-1 is given by
E{Rt | I t -1 }  a    { (  ) 1 R t -1   t   ut}
(9)
   { (  ) 1 R t -1  (1  )  t   ut}
where  t  a    R t -1    R t -1 ,   (  t /  ut ),    (  t /  ut ), and  and 
respectively denote the distribution and density functions of the standard normal
distribution.
Next corollary readily follows from proposition 2. It shows that the definition of
market efficiency in this paper is equivalent to that of Fama (1976).
the conditional expectation of t is not zero and {t}are serially dependent, then t can
not follow the GARCH model. Though Koutmos (1998) finds empirically useful facts
about the stock market movements, the partial asymmetric adjustment model does not
logically induce the Threshhold GARCH model which is used for his empirical studies.
Extending Koutmos (1998), equation (4) of Koutmos (1999, pp. 86) introduces an error
term in the asymmetric price adjustment process. However, the error term in equation
(4) causes discrepancy of stochastic orders between ut and t in his equations (4) and
(5) because t is expressed as difference of ut and ut-1. In other words, if t is an I(0)
process, then ut becomes an I(1) process.
6
Corollary 1: The market is efficient if and only if equation (8) holds. That is,
E{Rt | I t -1 }  a if and only if
 =  = 0.
(10)
If the investor's adjustment to the intrinsic value is symmetric, i.e.  =  (= ),
we have the auto-regression equation (7). In this case, the first order autocorrelation
of Rt is equal to the coefficient of Rt-1 (i.e.  and Fama's definition of market
efficiency is equivalent to  In many empirical studies, the hypothesis of  is
tested against    In the case of   , however, it is not obvious whether the
equivalence between and  =  = 0 holds. Propositions 3 and 4 are intended to
partially answer this question. As has been mentioned, the error terms in equation (6)
are correlated to the explanatory variables, and it is not easy to derive the explicit
formula of the first order autocorrelation coefficient of Rt. Hence, we examine the
autocorrelation of Rt* in equation (5) instead of equation (6) for the sake of simplicity of
exposition.
Proposition 3: The first two unconditional moments of Rt * exist and have the following
relations:
  E{R * t }      
(11)
 0  E{( R * t   ) 2 }   0   0  2   
(12)
where    E{Rt * } ,    E{Rt * } ,  0   E{( Rt *    ) 2 } , and  0   E{( Rt *    ) 2 } .
Proposition 4: Let the first order autocorrelation of Rt * be  If both  and  are non
negative, the following inequalities hold:
min{   ,   }   *  m a x{ ,   } .
(13)
If  =  (= , Proposition 4 indicates that  =  =0 implies
and ,  > 0 implies However, the converse statement is not justified,
i.e. does not mean  =  =0. In other words, non-existence of the
autocorrelation ( of Rt * in equation (5) doesn't necessarily imply the market
efficiency ( =  = 0). Though we are not able to derive an explicit formula of the first
7
order autocorrelation of Rt in equation (6), we conjecture that doesn't imply the
market efficiency.
3. 3 Statistical Inference of the Model
The joint density function based on equation (6) is expressed as follows.
Proposition 5: The joint density function of {R1, . . . , RT} is given by
p d f ( 1R, . . . , R T |  ) 
{
t T

 {
t T 
1
2   ut

1
2   ut
e x p{
e x p{
1
2  ut
2
1
2  2 ut
2
(R t -    t ) 2 }
(14)
(R t -   t ) }

2
2
where T+ = {t | Rt  0, t  N }, T- = {t | Rt < 0, t  N }, N = {1, . . . , T}, and = {a,
} is a vector of unknown parameters.
The maximum likelihood estimate of the parameter vector  is obtained by
maximizing the likelihood of (14) with respect to  We test the null hypothesis of
market efficiency against the inefficiency of market:
H0 : vs H1 :   or    (15)
The symmetric adjustment speed can be tested as
 H0 : +  -vs H1 : +   -
(16)

3. 4 An Illustrative Example
We investigate the efficiency of the Tokyo Stock Market for illustrating how our
model works. The daily closing stock price data of the Tokyo Stock Exchange Price
Index (TOPIX) are purchased from the Data Base of Nomura Research Institute,
JAPAN. Figure 1 indicates the data of stock prices and returns from January 4, 1980 to
December 2, 2005. It shows the up-trend to the end of 80s, but the down-trend in 90s
and 2000s. The returns move mildly in the former period, while they greatly fluctuate in
the latter one. Based on these visual observations, the sample period is divided into the
two sub-periods: the first is from January 4, 1980 to the end of 80s, the second is from
the beginning of 1990 to December 2, 2005.
8
The model is estimated by the maximum likelihood estimation method using the
joint density of (14) for each of the sub-samples. The results are shown in Table 1. The
estimates of the drift term in equation of (1) are positive in the first sub-sample but
negative in the second, supporting the up-trend of stock prices in the 80s and the
down-trend in the 90s and after. The estimates of 1 and3 reveal asymmetric volatility
and variance persistence, supporting the stylized facts for stock price movements.
Main interests of this study are market efficiency and asymmetric adjustment
speeds to intrinsic values. The estimates of + and  - are positive and significant at 5%
level for each sub-sample period. Table 2 shows that the null hypothesis of market
efficiency (H1: += -=0) is rejected in both sub-samples. Thus, the market is inefficient
in the sense adjustment speed as well as in the Fama’s sense due to Corollary 1. The null
hypothesis of symmetric adjustment speeds (H2: + =  -) in equation (3) is rejected in
the first sub-sample but not in the second.
How do we interpret the findings? In our framework, the observed values of
stock price are adjusted towards the intrinsic values (security values). The adjustment
speeds are (1-+) if Vt  Pt -1  0 (or equivalently R t

0) and (1- ) if Vt  Pt -1 < 0
( R t < 0). In particular, in the first sub-sample the speed is (1-0.284) for the case of
under-intrinsic value (positive return), and (1-0.199) for over-intrinsic value (negative
return). The adjustment speeds are significantly different in this period. Koutmos (1998,
pp. 285) finds the asymmetric adjustment speeds in many stock markets and argues as
“One possibility is that investors have a higher aversion to downside risk, so they react
faster to bad news. The use of stop-loss orders is an example of such aversion. Also,
portfolio managers feel they are penalized more if they under-perform in a falling
market than in a rising market.”
[ INSERT Figure 1 and Tables 1, 2]
4. Concluding Remarks
The concept of market efficiency defined by Fama (1976) is silent about the
market processes that might deliver individual actions. This paper incorporates a partial
adjustment price model by Koutmos (1998) and Amihud and Mendelson (1987) into an
ARCH-type model of the returns and clarifies the relationship between the adjustment
speed and the Fama's market efficiency. We find that the stock price instantaneously
and fully adjusts to the intrinsic value if and only if the market is efficient in the sense
of Fama (1976). By showing the equivalence of the two concepts, this paper gives an
9
economic foundation to the Fama’s random walk form of market efficiency hypothesis.
As an illustrative example, we find that the Tokyo stock market is inefficient
during 1980-2005 in the sense that the adjustment speeds to the intrinsic values are less
than 1. In the 80s, the stock price adjustment is faster when the price is above the
intrinsic value than when the price is below it.
There has never been trial to investigate the market efficiency by using the
ARCH-type model with individual actions hypothesis.
10
Appendix : Proofs of Propositions
This appendix gives proofs of propositions stated without proofs in the text, after
providing a lemma which is useful for recognizing the statistical nature of the partial
adjustment model.
Lemma 1: The conditional expectations of  t and t =  t ut are respectively given as
E{ t | I t 1 }        (1  ) .
(A.1)
E{ t | I t 1 }  (     ) ut ,
(A.2)
where   (  t /  ut ),    (  t /  ut ),  t  a    R t -1    R t -1 , and  and 
respectively denote the distribution and density functions of the standard normal
distribution. Moreover, each process of  t and t is serially dependent.
Proof:
Let us define the set A = { Rt | Rt  0}and its complement Ac = { Rt | Rt < 0}.
We have  t    I A    I AC and t =  t ut    ut I A    ut I AC , where IA and I A C are
respectively the indicator functions of the set A and Ac. The conditional expectations of
IA and I A C on It-1 are obtained by
E{I A | I t 1}  P{ut  - t | I t 1}  1  (- t /  ut )   ,
(A.3)
E{I AC | I t 1 }  1  E{I A | I t 1 }  1   .
(A.4)
and
Then, (A.1) follows from (A.3) and (A.4). Next, we calculate the conditional
expectation of t. Noting that
E{I A u t | I t 1 } 

  u (
-
t
ut )
-1
 (  t /  ut )dut
t


  ut v
-
t
(A.5)
 (v)dv   ut
/  ut
-t
E{I AC ut | I t 1 }   ut ( ut ) -1 ( t /  ut )dut   ut

(A.6)
where    (  t /  ut ) , we have (A.2) . Because  in (A.1) is a function of R t -1 and R t -1 ,
and hence a function of  t 1 , it follows E{ t |  t 1 }  E{ t } . Similarly, we can show the
serial dependence of  t .
□
Proof of Proposition 1: Rewrite the adjustment process (3) as
11




 


R t  1    Vt  Pt -1 D t  1    Vt  Pt -1  1  D t

 
 1          D t Vt  Pt -1 
(A.7)
  t Vt  Pt -1 
where D t  1 for Vt  Pt -1  0, and D t  0 otherwise . Then, noting the
1
facts Vt   t Rt  Pt -1 from (A1), 0 < t  1 from (3), and Vt  Vt -1 
 t1 Rt   t-11 Rt 1  Pt -1  Pt -2  a  u t , we have
 t1 Rt  a  (1   t 1 ) 1  t 1 Rt 1  u t .
(A.8)
Since Vt  Pt -1 ⋚ 0  R t ⋚ 0, the return process has the expression in (5).
□
Proof of Proposition 2: Substituting (A.1) and (A.2) into the equation
E{Rt | I t -1 }  E{ t | I t 1} t  E{ t | I t 1} ,
(A.9)
□
the required result in (9) is obtained.
Proof of Proposition 3: From (5), we have


| Rt |  | a |   | R * t -1 |  | u t |  a *   | R * t -1 | u t ,
*
*

where a* = |a| + 2  ,   max{  ,   } , and ut* = |ut| -
(A.10)
2
 having E{ut* | I t 1 } = 0.
Then,


  j
E{| Rt |}  E{ a *   | R * t -1 | u t }  (1   ) 1 a *  E{u * t - j }
*
*
j 0

 (1   ) 1 a*   .
Since E{u * t -i u * t - j | I t  j }  0 for all i < j , we have

E{R * t }  E{ (a *   | R * t -1 | u t ) 2 }
2

*
  j
 E{{(1   ) 1 a *  u * t - j }2 }
j 0
12
(A.11)

  2j
 (1   ) a *   
2
2
E{u
*2
  i j
t- j
j 0
}  2 
i j
2

E{u * t -i u * t - j }
(A.12)
2
 (1   )  2 a *2 (1   ) 1{E{ ut }  }   ,
2

where the fact E{ ut }   is used. See Nelson (1991) for the existence of E{ ut } .
2
2
(A.11) and (A.12) indicates the existence of the first two moments of Rt * . Equation (11)
directly follows from R * t  Rt *  Rt * . Equation (12) is obtained from
E{( R * t   ) 2 }  E{( Rt
*
   ) 2 }  E{( Rt
 2 E{( Rt
and Rt
*
Rt
*
*
   )( Rt
*
*
   )2}
 ) } ,
 0.
(A.13)
□
Proof of Proposition 4: From (6), the first order auto-covariance of Rt * is given by
 1  E{a                (R *t -1    )    (R *t -1
   )  u t }( R * t 1   )}
   E{(R *t -1    )( R * t 1   )}    E{(R *t -1    )( R * t 1   )}
(A.14)
   { 0      }    { 0      } .
If ,   0, we have
min{   ,   } 0   1  m a x{  ,   } 0 .
The required result is obtained from (A.15).
Proof of Proposition 5: From (6), the return process is rewritten as
      u t
Rt    t

  t   u t
f o rR t  0
.
f o rR t  0
(A.15)
□
(A.16)
The conditional density of Rt given It-1 is written by
13
1
1


2
 2   exp{  2  2 2 (R t -   t ) } forR t  0
ut
ut

pdf(R t ;  | I t -1 )  
(A.17)

1
1

2
exp{

(
R


)
}
for
R

0
t
t
t

2

2  2 ut
 2   ut
Substituting equation (A16) into the following relation
pdf(R 1 ,..., R T |  )   pdf(R t ;  | I t -1 ) ,
(A.18)
t T
the required joint density in (14) is obtained.
14
□
References
Amihud, Y., and Mendelson, H.(1987), "Trading Mechanisms and Stock Returns:
An Empirical Investigation", The Journal of Finance, 42,533-553.
Bahng, J. S. and S. Shin (2003), "Do stock price indices respond asymmetrically?
Evidence from China, Japan, and South Korea", Journal of Asian Economics, 14,
541–563
Busse, J.A.and Green, T.C.(2002), " Market efficiency in real time",Journal of Financial
Economics, 65, 415–437.
Fama, E.F. (1976), Foundations of Finance, Basic Books, New York.
Grossman, S.and Stiglitz, J.(1980), " On the impossibility of informationally efficient
markets", American Economic Review, 70, 393–408.
Koutmos, G.. (1998), "Asymmetries in the conditional mean and the conditional
variance: evidence from nine stock markets", Journal of Economics and Business,
50, 277-290.
Koutmos(1999) , "Asymmetric Price and Volatility Adjustments in Emerging Asian
Stock Markets", Journal of Business Finance and Accounting, 26,83-101.
Nam, K., C. S. Pyun and S. W. Kim (2003), "Is asymmetric mean-reverting pattern in
stock returns systematic? Evidence from Pacific-basin markets in the
short-horizon", International Financial Markets, Instituitions and Money,
13,481-502.
Nam, K., Washer,K.M. and Chu, Q.C.(2005), "Asymmetric return dynamics and
technical trading strategies",Journal of Banking & Finance, 29, 391-418.
Nelson, D. B. (1991), "Conditional Heteroskedasticity in Asset Returns:A New
Approach", Econometrica, 59, 347-370.
Pagan, J. A. and Soydemir, G. A. (2001), "Response asymmetries in the Latin American
equity markets", International Review of Financial Analysis, 10, 175–185.
15
Figure 1. TOPIX and Its Returns
TOPIX
4000
3000
2000
1000
0
80/1/5
85/1/5
90/1/5
%
95/1/5
00/1/5
05/1/5
00/1/5
05/1/5
Returns
20
10
0
-10
-20
80/1/5
85/1/5
90/1/5
95/1/5
16
Table 1. Estimates of Parameters
Periods
1/4/80 –12/ 28/89
1/4/90 – 12/2/05
a
0.043*
-0.026
(4.77)
(-1.47)

0.284*
0.111*

(19.14)
(6.70)

0.199*
0.116*

(9.90)
(5.97)
1
-0.151*
-0.098*
(-10.71)
(-10.76)
0.933*
0.964*
(92.30)
(176.96)
3
Notes: The numbers in parentheses denote t-statistics. The asterisk "*" is
significant at 1% level.
Table 2. Testing Results
Null Hypothesis
Distibution
1/4/80 –
12/ 28/89
1/4/90 –
12/2/05
H1: =0

380.41*
58.22*
H2 : 

16.72*
0.07
Note : The asterisk "*" is significant at 1% level. The critical values of 2(2)
and 2(1) distributions are respectively 9.21 and 6.34 at 1% level.
17