Measuring the Size of Market Inefficiency:
Trading Systems and Price Bubbles*by
Tatsuyoshi Miyakoshia
Yoshihiko Tsukudab
Junji Shimada c

Keywords: size; market inefficiency; trading system; price bubble
JEL classification numbers: G11; G12; G14; C10.
ab

Faculty of Science and Engineering, Hosei University

Graduate School of Economics, Tohoku University
School of Management, Aoyama Gakuin University
Correspondence: Tatsuyoshi Miyakoshi, 3-7-2, Kajino-cho, Koganei, Tokyo, 184-8584,
Japan. tel: 042-387-6352, email: miyakoshi@hosei.ac.jpc* Earlier versions of this paper were presented at the Humbolt-Copenhagen Conference 2009
(Berlin), Hitotsubashi University Conference 2009 (Tokyo), and Japanese Economic Association
Conference 2011 (Kumamoto). We thank Torben G. Andersen, Tim Bollerslev, Masanobu
Taniguchi, Toshiaki Watanabe, Masamitsu Ohnishi, Isao Ishida, Yasushi Tanizaki, Willian Dunsmirand participants for useful suggestions. The first and second authors received supports from,respectively, Grant-in-Aid 16530204 and Grant-in-Aid 18530153 from the Ministry of Education,
Culture, Sport, Science and Technology of Japan.

Abstract:
The performance of trading systems in the stock markets has been studied by most of theprevious literatures in terms of the costs or equivalently speeds for adjusting the stock
prices. This paper proposes an alternative measure for accessing the trading systems called
the size of market inefficiency which simultaneously takes care of the intrinsic value ofstock and adjustment speed of price, and applies this measure to the stock market indices ofnine countries during the tow sub-periods from1980 to 2010. We find that the trading
system of the Indian market is the worst in the first period but Korea’s is the worst in thesecond sub- period. The trading system of Japan, Canada, and India improve in the second
period, while the trading systems of the USA, France, Italy, Korea, Singapore, andMalaysia become worse in the second period. This paper also shows that the net priceovervaluation (intrinsic value minus fundamental value) is the sum of nominal priceovervaluation (stock price minus fundamental value) and size of market inefficiency. Thisdistinction of net and nominal overvaluations clarifies the role of our measure of market
inefficiency. An application reveals that contribution of the size of market inefficiency tothe net overvaluation is fairly large for the markets of some countries.

11. Introduction
Huge amounts of researches on the efficiency of security markets have been accumulated
since Fama’s (1970) pioneering work. Fama (1970) defined that the market is efficient if
the prices fully and instantaneously reflect all information available at present time.
His concept of market efficiency was extended to explicitly incorporate market
microstructures (or investor’s behavior). Amihud and Mendelson (1987) and Damodaran(1993) propose a partial price adjustment model. They define the market is efficient wheninvestors and specialists can instantaneously and fully adjust the market price of a security
to its intrinsic value. In contrast, the market is inefficient when the price of its security does
not adjust to its intrinsic value. Koutmos (1998, 1999) extend the above mentioned modelto an asymmetric partial price adjustment model, where the adjustment speeds are different
depending on whether the price increases or decreases. Empirical studies of Soydemir(2001), Bahng and Shin (2003), and Nam et al. (2003, 2005) support the asymmetricadjustment model. Most of the previous authors study the performance of trading systems
in the stock markets in terms of the adjustment speeds of stock price but do not pay
attention to intrinsic value.
This paper proposes an alternative measure for accessing the trading systems called
the size of market inefficiency which simultaneously takes care of the intrinsic value ofstock and adjustment speed of price. The size of market inefficiency is defined as thedifference between market price and its intrinsic value. We investigate the performance oftrading systems in terms of the size of market inefficiency. Our approach is more usefulthan the previous approaches for accessing the trading system in the following two senses.
First, the previous approach accesses the trading system only in terms ofadjustment speed and ignores the fluctuations in intrinsic values: the trading system isbetter (worse) if the adjusting speed is higher (lower) regardless how vigorously the
intrinsic value fluctuates. For example, Amihud and Mendelson (1989, 1991), Amihud etal. (1997), Lauterbach (2001), and Chang et al. (2008) assess whether the trading system ofcall market method or that of continuous auction method is better in terms adjustment
speed. However, a higher adjustment speed will produce a large amount of discrepancy
between intrinsic value and market price when the intrinsic value vigorously fluctuates.
We feel that the difference between intrinsic value and market price is a more suitablemeasure for accessing the performance of trading system than the adjusting speed.
2Second, a number of empirical papers including Campbell and Shiller (1987),
Cheung and Lee (1998), Lee (1998), and Black et al. (2003) have estimated fundamental
values by the dividend discount model, and used them to measure the magnitudes ofovervaluation of the market price. The concept of overvaluation is used to evaluate theinvestor’s speculative behaviors and bubble prices. The overvaluation is traditionallydefined by the difference between market prices and fundamental value. A related concept
is the difference between intrinsic value and fundamental value. We call the former as thenominal overvaluation and the latter as net overvaluation. Then, the net overvaluation is
the sum of nominal overvaluation and size of market inefficiency. The concept of net
overvaluation is preferable to the nominal overvaluation for evaluating the overvaluationof the market in practice.
We apply this measure to the stock market indices of nine countries during the tow
sub-periods from1980 to 2010. The result reveals that the trading system of the Indian
market is the worst in the first period but Korea’s is the worst in the second sub- period.
The trading system of Japan, Canada, and India improve in the second period, while thetrading systems of the USA, France, Italy, Korea, Singapore, and Malaysia become worsein the second period. And the contribution of the size of market inefficiency to the net
overvaluation is fairly large for the markets of some countries.
The paper is organized as follows. Section 2 outlines the model and give a precise
definition of the size of market inefficiency. Section3 provides an estimation method for
unknown parameters and unobserved intrinsic value using the observed market price.
Section 4 reports the results of an application of our approach to the stock market indicesfor the nine countries of G7 and Asia. Section 5 provides some concluding remarks. All
proofs of propositions of this paper are given in Appendices.

2. The Model and market inefficiency
2.1 The model
We essentially follow the model of Koutmos (1998, 1999), which is itself an extension ofthat of Amihud and Mendelson (1987) by incorporating asymmetric adjustment. Thismodel consists of two parts: the intrinsic value process for a security and the market price3

adjustment process. The model distinguishes the unobserved intrinsic value of a stock (V t )from the observed market price (P t ) of the stock, both expressed in natural logarithms. 1
The intrinsic value process follows a random walk process with drift:Vt =a + Vt −1 + ut ,()

ut | I t −1 ~ N 0, σ ut2 , t =1,..., T,(1)

where a is a constant and I t–1 denotes the information set up to time t–1. We assume that the
disturbance term (u t ) has the EGARCH (Exponential Generalized Autoregressive
Conditional Hetroskedastic) process proposed by Nelson (1991):
log σ u2t = a 0 + a1 zt - 1 + a 2 (|z t - 1|- E(|z t - 1|)) + a 3 log σ u2t - 1
(2)

where
ut =σ u t z t : z t ~ N (0, 1) .

The partial asymmetric price adjustment process of P t represents that adjustment speeds(1 − θ + , 1 − θ − ) are asymmetric in the upturn or downturn markets:
Pt − Pt −1 = (1 − θ + )(Vt − Pt −1 ) + + (1 − θ − )(Vt − Pt −1 ) − , − 1 < θ + , θ − < 1,(3)

where (Vt − Pt −1 ) + = max{ Vt − Pt −1 , 0}, and (Vt − Pt −1 ) − = min{ Vt − Pt −1 , 0}. We note that+θ − (≡ θ ) , equation (3) reduces to theθ + , θ − indicate the costs for adjusting prices. If θ=(symmetric) partial price adjustment process proposed by Amihud and Mendelson (1987, p.
536). Koutmos (1998, p. 280; 1999, p. 86) formulated the asymmetric adjustment to
intrinsic value in (3). After the intrinsic value V t is recognized at t, the market price P t–1 ispartially adjusted to P t by (3). 2
The partial price adjustment (θ + ≠ 0 and/or θ − ≠ 0) in (3) is attributed to thefollowing costs that slow down the price adjustment of a security toward its intrinsic value:(i) the cost of acquiring and processing information by investors; (ii) attempts by market
specialists to create orderly markets and ensure price continuity; and (iii) the particularinstitutional market mechanism by which securities are traded 3, 4 . Moreover, the12This distinction is based on an idea by Black (1986; p. 533).
When Vt > Pt −1 in (3), we have Pt =(1 − θ + )Vt + θ + Pt −1 . Then, the condition −1 < θ + < 1 isnecessary for the price adjustment process to converge. Similarly, the condition −1 < θ − < 1 isalso necessary.
3
See Koutmos (1998, pp. 278, 285).

4

possibility of asymmetric adjustment ( θ + ≠ θ − ) reflects that investors may severely
averse the downside risk, so they react more quickly to bad news. The use of stop-loss
orders is an example of such aversion. Furthermore, traders (i.e., investors) may feel they
are penalized more if they underperform in a falling market than in a rising market. This
fact indicates that θ + > θ − . Similarly, for the market makers (i.e., market specialists), thecost of not adjusting for the downward prices is higher than that of not adjusting for the
upward prices. Market specialists, who are required to maintain price continuity, may findit easier and less costly to do so in the rising market than in the falling market. As the latter
involves the inventory building up with overpriced securities, this also leads to θ + > θ − .
However, the negative adjustment cost ( θ + , θ − < 0) in (3) is difficult to interpret.
Nevertheless, we allow the negative cost to generalize the model as done in the previous
studies.

2.2 The size of market inefficiency and performance of trading systems
We define a concept of market inefficiency in terms of adjustment costs as follows.
Definition 1: (i) The market is said to be inefficient if θ + ≠ 0 and / or θ − ≠ 0 in (3) and+−θ=
0 (adjustment cost is zero). (ii) The market priceefficient if and only if θ=adjustment is said to be asymmetric if θ + ≠ θ − and symmetric if θ + = θ − .
If the market is efficient, there is no adjustment costs of (i), (ii), and (iii) stated in+−θ=0 ) and the market prices instantaneously and fully adjust to theSection 2.1 ( θ=intrinsic value as follows:

Pt − Pt −1 =(Vt − Pt −1 ) + + (Vt − Pt −1 ) − =Vt − Pt −1 , and hence Pt =Vt .

(4)

On the other hand, when the market price is neither instantaneously nor fully adjusted
toward the intrinsic value, the inefficient market and less efficient trading system evolve.
The speeds of price adjustment (1 − θ + or 1 − θ − ) in the upturn and downturn
markets could be a measure of accessing how far inefficient the markets are. Amihud and
Mendelson (1989, 1991), Amihud et al. (1997), Lauterbach (2001), and Chang et al. (2008)

4

The continuous auction method is used in most countries. However, Japan and Singapore have no
specialists that deal from their own accounts to set the price, although there are specialists in theother countries considered in this paper. See Inoue (2006).

5

used the adjustment speeds for evaluating the performance of trading systems. However,
all of the previous studies ignore the role of intrinsic values.
Alternatively, we explicitly incorporate the intrinsic value into the consideration foranalyzing the performance of trading systems. We know that the market price is equal tothe intrinsic value from equation (4) if the market is efficient, but it is not equal if themarket is inefficient. Hence, the difference between the intrinsic value and the market price( Vt − Pt ) can be viewed as an indicator for the magnitude of market inefficiency. By
deleting both sides of (3) from the identity Vt = Vt , equation (3) is expressed as follows:
Vt − P
θ + (Vt − Pt −1 ) + + θ − (Vt − Pt −1 ) − , − 1 < θ + , θ − < 1 .
=
t

(5)

The size of market inefficiency is defined by the quantity ( Vt − Pt ) in this paper. The sizeof market inefficiency depends on both the intrinsic value and the adjustment
costs . (θ + or θ − ) .The condition of θ + = θ − = 0 is equivalent to Vt − Pt =0 in (5). The
market price overshoots the intrinsic value in the sense that Pt > Vt when Vt > Pt −1 if thecondition θ + < 0 and θ + < 0 holds.

3. Measuring the size of market inefficiency and overvaluation of the stock price3.1. Estimation of the size of market inefficiency
In order to determine the size of market inefficiency, we have to estimate theunobservable intrinsic value and the unknown parameters θ + and θ − . The next proposition
derives the relations among the observed market price, unobserved intrinsic value, and thesize of market inefficiency 5.
Proposition 1: (i) When the previous market price P t–1 is under (over) the intrinsic value V t
at the present period t, the market price P t at the present period is adjusted to increase(decrease) as follows:
Vt − Pt −1 ≤ 0 ⇔ Rt ≤ 0 and Vt − Pt −1 > 0 ⇔ Rt > 0,(6)where Rt ≡ Pt − Pt −1 is the return on the market price.
(ii) The size of market inefficiency (Vt − Pt ) is computed by using the return ( Rt ) , and theadjustment speeds (1–θ+ , and 1–θ−) as follows:5

All the proofs of this paper are given in Appendix A.

6

Vt − Pt =R t (

1

ξt

− 1)

(7)

where ξt ≡ (1 − θ − ) + (θ − − θ + ) D t , =D t 1 for Vt - Pt-1 ≥ 0, and =D t 0 otherwise .
After the intrinsic value V t is found at the beginning of period t and compared withthe previous market price P t–1 (i.e., V t – P t–1 ), the market price P t at period t is adjusted toincrease when V t – P t–1 > 0 and then the positive return follows (R t > 0), as shown in (3)and (6). Although the intrinsic value V t is directly unobservable, equation (7) indicates thatit can be computed from the observed market price P t , the return R t , and the estimatedparameters θ+ and θ– 6.
The model in this paper induces the process of return to a nonlinear autoregressive
process as stated in the next proposition.
Proposition 2: The return process consisting of (1) and (3) has the following form:
Rt

θ+
θ−
+
=
ξt (a +
R t-1 +
R −t-1 ) + ε t ,
+
−
1−θ
1−θ

(8)

where ε t =ξt ut , Rt+−1 ≡ Max{Rt −1 , 0}, Rt−−1 ≡ Min{Rt −1 , 0} , and ξt is defined inproposition 1.
The conditional expectation of ε t in (8) is not zero and the process of { ε t } is seriallydependent except for the case of θ + = θ − 7. The conditional variance of ε t does not follow
an EGARCH process, unlike that of u t . Considering (6) in Proposition 1, R t ≥ 0 in (8)corresponds uniquely to Vt − Pt-1 ≥ 0 in (7), which decides uniquely ξt . Then, the joint
density function of the returns is expressed from equations (8) and (2) as follows.
Proposition 3: The joint density function of {R 1 , . . . , R T } is given by:

6

Even though Amihud and Mendelson (1987) and Koutmos (1998, 1999) introduced the idea of
intrinsic value Vt as a theoretical concept, they did not compute it.

7

See Lemma 1 in the Appendix 1. If θ + = θ − (= θ ) and ξt = 1 − θ , equation (9) reduces to

Rt = a(1 - θ ) + θ R t -1 + (1 - θ )u t .

7

pdf ( R1 ,.., RT | ω ) ( (1 − θ + )σ ut ) −1f ( ( Rt − (1 − θ + ) µt ) / (1 − θ + )σ ut ) =ΠΦ ( µt / σ ut )t∈T + 
(9)

 ( (1 − θ − )σ ut ) −1f ( ( Rt − (1 − θ − ) µt ) / (1 − θ − )σ ut ) ×Π Φ ( − µt / σ ut )t∈T − 


where

θ+
θ+
µt =
a+
R t-1 +
R -t-1 , T + = {t | Rt ≥ 0, t ∈ N }, T − = {t | Rt ≤ 0, t ∈ N } ,
+
1 -θ
1 θ
N = {1, . . . , T}, and ω = {a, θ + , θ − , a 0 , a1 , a 2 , a 3 } is a vector of unknown parameters, andΦ and φ respectively denote the distribution and density functions of the standard normal
distribution.
We can estimate the unknown parameter vector of ω including ( θ + , θ − ) by the
maximum likelihood method using the joint density of (9) 8. Once we have the estimates ofθ + and θ − , the sizes of market inefficiency is estimated by Vˆt − Pt , where the estimate of
intrinsic value is given as
∧

(

) (

)

−
−
+
Vˆt = Pt − R t (1 − (ξt ) −1 ) , and ξˆt =1 − θˆ + θˆ − θˆ D t .

(10)Since Vt and Pt are expressed in terms of logarithms, the size of market inefficiency is thepercentage discrepancy between the market price and the intrinsic value in levels 9.

3.3. Overvaluation of the stock price and market inefficiency
The concept of overvaluation is generally used to evaluate the speculative market price,where market prices should be compared with their fundamental value. This subsection
analyzes the role of market inefficiency in overvaluation of the stock price bubbles. We
define the over- (under-) valuation of the market price.

8

The conditional density function is different from that of the standard GARCH-typemodel. We are not able to prove an asymptotic normality of the maximum likelihood
estimator for this model. Therefore, for simplicity, we assume that the estimated parameter
will be consistent and asymptotically normal.
9
Noting=the fact Vt log=
Vt *and Pt log Pt * , where Vt ∗ and Pt ∗ are the corresponding
variables in level values, we have Vt − P=log(1 + (Vt ∗ − Pt ∗ ) / Pt ∗ ) ≅ (Vt ∗ − Pt ∗ ) / Pt ∗ ) .
t
8

∗

∗

Definition 2: Let market price and fundamental value in levels be Pt and S t . The∗∗∗∗
market price is said to be overvalued if Pt > St , undervalued if Pt < S t , and normal if

Pt∗ = St∗ .
This definition was used by Campbell and Shiller (1987), Cheung and Lee (1998), Lee(1998), and Black et al. (2003) for comparing market prices and fundamental values. As
shown in Section 3, however, the intrinsic value is generally different from the market
value because of market inefficiency. Hence, we distinguish the size of net overvaluation(Vt ∗ − St∗ ) from the size of nominal overvaluation ( Pt ∗ − St∗ ). The following relation interms of a percentage of the market price clarifies the role of market inefficiency:

Vt ∗ − St∗=Pt ∗Pt ∗ − St∗ Vt ∗ − Pt ∗
+
.
Pt ∗
Pt ∗

(11)

The size of net overvaluation is a sum of the size of nominal overvaluation and that ofmarket inefficiency.
It is necessary to estimate the fundamental value for analyzing how much are theactual markets over- (under-) valued. In this paper, we define the fundamental value of themarket as the discounted sum of future dividends per share:∞

St* = ∑
j =1

Q t, j(1 + ρ t ) j

(12)

,

where Qt , j is dividend per share for the j-periods ahead from time t, ρ t is a discount rate attime t, which is assumed to be constant over the future periods. Assuming that the investors
expect the dividend growth rate (λ t ) at time t will stay fixed over the entire future periods,
we have Q=
Qt (1 + λ t ) j , where Qt denotes the dividend per share at time t. Thet, jfundamental value is expressed as:∞

Qt (1 + λ t ) j

j =1(1 + ρ t ) j

S ≡∑
*
t

1+ λt
=
(
)Q ,
ρ t − λt t

(13)

9

under the condition ρ t > λ t . Finally imposing a simplifying condition that the dividend
growth rate (λ t ) is equal to the GDP growth rate ( g t ) : (i.e. λt = g t ) , the fundamental∗

value is approximated in terms of the GDP growth rate and the initial value ( S0 ) as 10:t

St* = St*−1 (1 + g t ) = S0∗ Π (1 + g j ) .

(14)

j =0

We have to determine the initial value S0∗ in application as will be explained in the next
section.

4. Applications to the stock markets
4.1 Characteristics of the data
We investigate the stock markets of the nine countries of Japan, USA, France, Italy, Canada,
Korea, Singapore, India, and Malaysia to illustrate how our model works. The monthly
stock price indices are taken from International Financial Statistics (IFS) (line 62.ZF.). The
nine series of stock prices from the G7 and Asian countries are chosen for the reason ofavailability of data. 11 The time span of observation is roughly from January 1969 to June
2009, but different among the countries depending on the availability of data.
Figure 1 illustrates the market prices (in a semi-logarithm scale) adjusted for 100 in
January 2000 and their returns (in percentage) for each country. Figure 1(1) shows thatJapan’s stock prices exhibit an uptrend until the end of the 1980s, but a downtrend in the
1990s. The returns move mildly in the former period in comparison with the latter one.
Based on these visual observations, the sample period is divided into the two sub-periods.
These movements suggest a structural change in the EGARCH model, which isrepresented by equations (1) to (3). Similarly, from Figure 1(1)–(9) we would observe a
structural change for the stock price series of other countries than Japanese series, eventhough the point of change differs from country to country, reflecting the collapse of thestock price bubble or economic crisis. For example, the Japanese stock price collapsed
10

See Appendix B for deriving (4).
German prices are distorted by German integration after October 1990; therefore, we cannot
evaluate the trading systems independent of integration. UK (1994M1~) and Thai (~1996M12)
prices are not available from IFS. The Philippines’ prices in 2007 are 100 times higher than 2003
prices, a trend that has continued: 50 times at 2008M5 price, 100 times at 2008M12 price. Its
movement is too abnormal and thus excluded.
11

10

because of the burst of the stock price bubble. The stock prices of Singapore and Malaysia
collapsed because of the Asian currency crisis in1997, the USA stock price collapsed
because of the dot com bubble burst. The stock prices of other countries may have theirown reason of structural change though we do not mention here. To circumvent thisproblem, in this paper we separate the sample period into the two sub-periods. We analyze
the market prices for the same 10 years of observations over all nine countries in order tocorrectly compare the results across countries.
Table 1 reports the two sub-periods of asmaples and the summary statistics such asmean of returns, standard deviation, skewness, excess kurtosis and Ljung–Box
Q2(12)-statistics for no serial correlation in the monthly squared returns for each of thecountries. At the 5% significance level, both the null hypothesis of no excess kurtosis andno serial correlation for squared returns are rejected for the most countries. The monthly
time series still have the typical features of stock returns, displaying a fat-tailed distribution,a spiked peak, and variance persistence. 12 Therefore, the GARCH-type model expressedby (8) with error terms of (2) seems to be preferable to the AR model for analyzing theseseries.

[INSERT Figure 1, Table 1]

4.2 Adjustment speed, size of market inefficiency and trading systems
This sub-section determines the size of market inefficiency and then evaluates trading
systems in terms of this size after stating the estimation results of the parameters in themodel for each of the nine countries.
Table 2 indicates the estimates of parameters of our model. The estimates of the
GARCH effect (a 3 ) are very close to unity for all countries and for the periods I and II,
which strongly imply persistency of variance. 13 Table 2 also shows that the estimates ofθ + and θ − are statistically significant for most of the countries we examined and for both12

All computations in the paper were performed using the computer package WinRATS-32
Version 4.30.
13
Few of the values of α 3 exceed 1 as in previous studies. For France, θ − < −1 despite (3). Fewof the values of α 2 have significant negative signs, in contrast to previous studies. These results
require careful consideration.

11

periods I and II with a few exceptions. We formally test the null hypothesis of market
efficiency in terms of Definition 2 (H 0 : θ + = 0 and θ − = 0). As shown inTable3, thishypothesis is clearly rejected for all the countries and for the both subsamples except forthe period I of the USA. This fact implies the stock markets are inefficient in the sense of
Definition 2. The results of the t-tests for θ + and θ − shown in Table 2 indicate that one of
the two coefficients is either significantly positive or negative and the other is insignificant
for all nine countries, supporting the asymmetry of θ + and θ − . This indicates that ourmodel with possible asymmetric adjustment speeds is appropriate for analyzing market
inefficiency in these nine countries.

[INSERT Table 2, Table 3]

We evaluate the trading system of each country in terms of the size of market
inefficiency. At first, we look at the trading systems based on the adjustment speed along
the line of studies by Amihud and Mendelson (1989, 1991), Amihud et al. (1997),
Lauterbach (2001), and Chang et al. (2008) for the purpose of comparison with our method.
As seen in Period I of Table 2, both the upward adjustment speed and the downturn
adjustment speed vary in large amounts across the countries and between the two
sub-sample periods. The adjustment speed of the upturn market of Japan is slowest among
the nine countries (1 − θ + =1 – 0.440) , while that of the downturn market of Malaysia isovershooting (1 − θ − =1+0.690) and worst. In period II, on the other hand, the adjustment
speed of the upturn market of Italy (1 − θ + =1 – 0.476) , while that of the downturn market
of Korea (1 − θ − =1-0.581) and those of Canada (1 − θ − =1+767) andFrance (1 − θ − =1+1.656) are overshooting. After observing these results, we feel that it is
not easy to determine the best trading system among the nine countries based on theadjustment speeds.
Figure 2 illustrates the estimated values of the sizes of market inefficiency ( Vˆt − Pt )all over the sample periods for each county. These sizes changes over time for all thecountries, and take both positive and negative values for most countries. The average

12values of the size of market inefficiency over each of the sub-sample period are shown in
Table 4.

[INSERT Figure 2, and Table 4]

We evaluate the trading systems for the two sub-periods for each country by using an
average of the size of market inefficiency. The worst trading system is India (2.5%) in
period I and Korea (6.1%) in period II, as seen in Table 4. In period II (i.e., after
deterioration of market prices), Japan’s, Canada’s and India’s trading systems become
better in terms of size of market inefficiency. However, the trading systems of the USA,France, Italy, Korea, Singapore, and Malaysia become worse. In period II, most countries
have overshooting adjustment speeds in downturn markets: θ − < 0 , as seen in (5).

4.3. Overvaluation of the stock price and market inefficiency
Since the initial value S0∗ is unknown in practice, we determine it by using the following
strategy. Suppose S s∗ = Ps∗ for a particular period and compute S s∗− j for j ( j = 1,…, J ). If
the condition

S s∗− j (1 − 0.05) ≤ Ps∗− j ≤ S s∗− j (1 + 0.05) ,(15)

holds for all j = 1,…,J, we call this period an initial period, and set s = 0 with S0∗ = P0∗ . WhenPs∗− j satisfies the inequality in (15), we virtually regard that the market price is normal inspite of definition 2. Behind this strategy, we have an idea that the market price might beclose to the fundamental value, and the growth rate of the fundamental value would beroughly equal to that of the market price during the periods of typical economic
environments. Hence, it may be reasonable to choose some value of J greater than one.
Moreover, when the next condition holds:
*
**
,
S0**
+ t (1 + 0.05) < P0 + t for all t from T to T

(16)

for some T* and T**, the stock price Pt ∗ is overvalued at the priods from T* to T**. We
call the quantity ( P0∗+t − S0∗+t ) for these periods as the size of bubble.

13

We chose only one country among the nine countries for each of the periods I and II
for analyzing the size of bubble in order to illustrate how our method works in the realmarket. The market of India is worst in terms of the size of market inefficiency for period I.
However, we chose the second worst market of Japan since there is no available monthly
GDP data for the Indian market. Korea is chosen for the period II because it is worst.
We apply the above strategy for the Japanese market to determine the initial date andvalue. We choose January 1980 as an initial date and set S0∗ = P0∗ . Figure 3(a) indicates thatthe market price P0− j satisfies the condition of (15) at least from 1975 to 1985. Figure 3(a)also shows that P0∗+t meets the condition of (16) approximately from the year of 1985 to
1998.That is, the Japanese market in this period is under the bubble development, and thesize of the bubble is measured by the quantity of P0∗+t − S0∗+t . Similarly, Figure 3(b) illustrates
the Korean market. We choose the year of 1985 as the base year for the market of Korea.
The Korean market is overvalued during the periods approximately from 1986 to1997, andafter these periods this market is undervalued.

[INSERT Figure 3]

The concept of overvaluation is generally used to evaluate investors’ speculative
behaviors. We restrict the analytical period for the overvaluation to the periods of market
inefficiency shown in Table 1. For Japan’s market, as seen in Table 4, in period I, themaximum of 8 percentage points out of the net overvaluation of (V0∗+t − S0∗+t ) / P0∗+t
percentage points relative to the market price is a result of market inefficiency(V0∗+t − P0∗+t ) / P0∗+t . The average of 1.9 percentage points is also a result of market

inefficiency. Similarly, in period II, the maximum of 12.8 percentage points out of Korea’s
net overvaluation, the minimum of –32.7 percentage points, and the average of 6.1
percentage points are the results of market inefficiency. 14

14 Previous papers estimated the fundamental value using annual or quarterly data withnonmonetary variables, whereas this paper is interested in monthly data for the sake of practical use
and hence has to use a simpler model than the previous ones. However, both the present and theprevious definitions of the fundamental value are based on the dividend discount model.

14

We show the detailed contribution of Japan’s market inefficiency (V0∗+t − P0∗+t ) / P0∗+t
to the net overvaluation in Figure 4(a). According to (20), the overvaluation for 1987M4(49 percentage points, ( P0∗+t − S0∗+t ) / P0∗+t shown in Figure 4(a’)) plus the market inefficiency(8.4 percentage points, (V0∗+t − P0∗+t ) / P0∗+t ) yields the net overvaluation (57.4 percentage
points, (V0∗+t − P0∗+t ) / P0∗+t ). The market inefficiency prevents the market price fromincreasing to the intrinsic value by 8.4 percentage points. On the other hand, for Korea,
shown in Figure 4(b’), the overvaluation for 1993M1 (29.1 percentage points) plus themarket inefficiency (11.4 percentage points) yields the net overvaluation (40.5 percentage
points). The market inefficiency prevents the market price from increasing to the intrinsic
value by 11.4 percentage points.

[INSERT Figure 4]

5. Concluding remarks
This paper proposed a new measure of size of market inefficiency for accessing the trading
systems, which simultaneously takes care of the intrinsic value of stock and adjustment
speed of price. Application this measure to the stock market indices of nine countries
during the tow sub-periods from1980 to 2010 reveals that the trading system of the Indian
market is the worst in the first period but Korea’s is the worst in the second sub- period.
The trading system of Japan, Canada, and India improve in the second period, while thetrading systems of the USA, France, Italy, Korea, Singapore, and Malaysia become worsein the second period. This paper distinguished the net overvaluation from the nominal
overvaluation. This distinction clarifies the significant role of our measure of market
inefficiency for analyzing the overvaluation during the periods of bubbles in the markets ofJapan and Korea.

15

Appendix A: Proofs of Propositions
We drive all the propositions stated in this paper without proofs and give a lemma.
Proof of Proposition 1: Rewrite the adjustment process (3) as:

(

)

(

)

(

R t = 1 − θ + ( Vt − Pt-1 ) D t + 1 − θ − ( Vt − Pt-1 ) 1 − D t{(= 1−θ −

) + (θ − θ ) D } ( V − P )
−

)

+

t

t

(A.1)

t-1

= ξ t ( Vt − Pt-1 ) .

Then, the relation of (6) in (i) holds. On the other hand, from (A.1) we obtain
Pt-1 + (R t / ξt ) = Vt . By using this relation, we have (7) in (ii):Vt − Pt =Rtξt) Rt (− (Pt − Pt-1 =1ξt− 1) .

Q.E.D.
-1

Proof of Proposition 2: Noting that Vt = ξ t Rt + Pt -1 from (7), we see that
Vt − Vt −1 = ξ t-1 Rt - ξ t--11 Rt -1 + Pt -1 - Pt -2 = a + u t(A.2)

because of (1). We have from (A.2)

ξt−1 Rt =a + (1 − ξt −1 )ξt−−11 ( Rt+−1 + Rt−−1 ) + u t(A.3)

where Pt-1 - Pt-2 = R t-1 = R +t-1 + R -t-1 . Considering (6) in Proposition 1, R +t-1 =R t-1 ≥ 0 in (9)corresponds uniquely to Vt-1 - Pt-2 ≥ 0 in (7), which determines uniquely the value of ξ t–1((≡ 1 - θ) + (θ - θ ) D ). Similarly, R =R ≤ 0 determines the value of ξ

--

+

t-1

t-1

t-1

According to (A.3), the return process follows the expression in (8).

t-1 .

Q.E.D.

Next, we prove the following lemma, which states the expectations of ξt and ε tconditional on the information set I t −1 .
Lemma 1: The conditional expectations of ξ t and ε t = ξ t u t are respectively given asE{ξt | I t −1}= (1 − θ + )Φ + (1 − θ − )(1 − Φ ) ,(A.4)

E{ε t | I t −=(θ − − θ + )σ utϕ ,
1}

(A.5)

where µt = a + (1 - θ + ) -1θ + R +t-1 + (1 - θ ---) 1θ R t-1 , Φ = Φ ( µ t / σ ut ), φ = φ ( µ t / σ ut ),and Φ and φ respectively denote the distribution and density functions of the standard
normal distribution. Moreover, each process of ξ t and ε t is serially dependent.

16

Proof: Let us define the set A = { R t | R t ≥ 0} and its complement Ac = { R t | R t < 0}. We
note that ξ t =1 − θ + ( ≡ ξ + for all t) if Rt =Rt+ , and ξ t =1 − θ − ( ≡ ξ − for all t) if Rt =Rt− .
Then, we can write thatε t ξ=
ξ + ut I A + ξ −ut I AC ,
ξ t = ξ + I A + ξ − I AC and=
t ut

(A.6)

where I A and I A C are respectively the indicator functions of the sets A and Ac.
The conditional expectations of I A and I A C on I t–1 are obtained by:
E{I A | I t −1 } = P{u t ≥ -µ t | I t −1 } = 1 − Φ (-µ t / σ ut ) = Φ ,(A.7)

E{I A C | I t −1 } = 1 − E{I A | I t −1 } = 1 − Φ .

(A.8)

and

Then, (A.4) follows from (A.7) and (A.8).
Next, we calculate the conditional expectation of ε t . Noting that:E{I A ut | I t −1} = ∫

∞

- µt

ut (σ ut ) -1ϕ (µ t / σ ut ) dut
,

(A.9)

∫ u (σ ) ϕ (µ / σ )du = −σ ϕ (µ / σ ) ,(A.10)∞= σ=vϕ (v) dv σ utϕ (µ t / σ ut )ut ∫- µ t /σ ut

E{I AC ut | I t −1} =

- µt

−∞

-1

t

ut

t

ut

t

ut

t

ut

we have (A.2). Because Φ in (A.1) is a function of R +t −1 and R −t −1 , and hence Φ is a functionof ξ t −1 , it follows that E{ξ t | ξ t −1 } ≠ E{ξ t } . Similarly, we can show the serial
dependence of ε t .

Q.E.D.

(

Proof of Proposition 3: From (9) and ξt ≡ 1 − θ−) + (θ − θ ) D , the return process is−+trewritten as: (1 − θ + ) µ t + (1 − θ + )utRt = 
(1 − θ − ) µ + (1 − θ − )utt

for R t ≥ 0(A.11)

for R t < 0.

The conditional density of R t given I t–1 is written as:17 ( (1 − θ + )σ ut ) −1f (( Rt − (1 − θ + ) µt ) / (1 − θ + )σ ut )Φ ( µt / σ ut )
pdf(R t ;ω| I t-1 ) =  ( (1 − θ − )σ ) −1f (( R − (1 − θ − ) µ ) / (1 − θ − )σ )
ut
t
t
ut


Φ (− µt / σ ut )for Rt ≥ 0(A.12)

for Rt ≤ 0

Substituting equation (A.12) into the following relation:

pdf(R 1 ,..., R T |ω ) = Π pdf(R t ;ω|I t-1 ) ,(A.13)t ∈T

the required joint density in (9) is obtained.

Q.E.D.

Appendix B: Derivation of (13)
From equation (12) the growth rate of the fundamental value( η t ) is given byS∗
Qt 1 + λt ρt −1 − λt −1Q=η t ≡ ∗t −1−1 ≅ t −1 ,St −1
Qt −1 1 + λt −1 ρt − λt
Qt −1
where an approximation(A.14)

1 + λt ρt −1 − λt −1≅ 1 is used. Equation (A.14) implies that the
1 + λt −1 ρt − λtgrowth rate of fundamental value equals the growth rate of dividend ( η t = λt ). This leads
to η=
gt by a simplifying assumption. Then, equation (15) follows.
λ=ttAppendix C: Estimation procedure of Koutmos
This appendix gives some comments on the estimation procedure of Koutmos (1998, 1999).
“Equation (8)” in Koutmos (1998, p. 280) imposes strong approximation and then should
be developed as equations (8) and (9) in this paper. (i) As shown by Lemma 1 in Appendix
A in this paper, the conditional expectation of ε t is not zero and { ε t } is serially dependent.
Hence, ε t cannot follow the GARCH-type model. (ii) Extending Koutmos (1998),“equation (4)” of Koutmos (1999, p. 86) introduces an error term in the asymmetric priceadjustment process. However, the error term in “equation (4)” causes a discrepancy in thestochastic orders between ut and ε t in his “equations (4) and (5)” because ε t is expressedas the difference between ut and ut −1 . In other words, if ε t is an I(0) process, then ut
becomes an I(1) process. (iii) “Equation (7)” of Koutmos (1998, p. 280) and “(5)” of
Koutmos (1999, p. 86) seem to be approximated in a crude manner. If they might use the18approximation of θ + / (1 − θ + ) ≅ θ + around θ + =0 , his formula becomes similar to theformulation (8) in this paper. However, his formulation omits ξt in (8).
Although his model with rough approximation would provide a useful model forpractitioners of empirical analysis, the partial asymmetric adjustment model of Koutmos(1998, 1999) does not logically induce the threshold GARCH model that is used for his
empirical analysis.

19

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Quarterly, Nomura Institute of Capital Market Research, 2006, winter, 112–120 (inJapanese).

20

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21

Table 1. Summary Statistics for Monthly Returns in Two periods
country

Period I :Period II :sample period

N of obs.

mean aSt.dev.

Skewness a

Ex-Kurt aQ2(12) b

Japan

1980M1 - 1989M12

120

1.529*

3.323-0.3940.985*

61.54*

USA

1990M2 - 2000M1120

1.228*

3.081-0.094

1.576*

12.81

Canada1990M9 - 2000M8

120

1.067*

4.223-1.345*

7.301

6.34

France1990M9 - 2000M8

120

1.253

5.456

-0.323-0.224

14.62

Italy

1990M4 - 2000M3

120

0.923

6.046

0.037

-0.04922.56*

Korea

1979M4 - 1989M3

120

1.662*

5.390

0.096

1.951*

13.19

Singapore

1986M2 - 1996M1

120

1.184

8.080

-2.479*15.938*

2.39

Malaysia

1987M3 - 1997M2

120

1.133*

6.112-2.071

11.497

3.36

India

1987M7 - 1997M6

120

1.781*

8.344

0.567*

1.434*

24.30*

Japan

1990M1 - 1999M12

120-0.4385.152-0.0470.855

21.68*

USA

2000M2 - 2008M5

100

0.197

3.386

-0.983*2.201*

22.14*

Canada2000M9 - 2007M10

86

0.406

3.935

-1.1301.543

30.35*

France2000M9 - 2008M5

93-0.2425.323-0.862*

1.868*

27.53*

Italy

2000M4 - 2007M10

91

0.029

4.053

-1.640*5.469*

11.44

Korea

1989M4 - 1999M3

120-0.4187.652-0.0030.949*

140.91*

Singapore

1996M2 - 2006M1

120

0.114

7.507

-0.123

1.829*

42.33*

Malaysia

1997M3 - 2007M2

120

0.007

7.300

0.029

2.206*

136.26*

India

1997M7 - 2007M6

120

1.020

6.419

-0.636*0.000

20.03#

Notes: "*" denotes significant at 10% level. Distributed as χ (12) under the null hypothesis of non-serial correlation with lags of up to 12. "*" ,”#” denotes
significant at 5% , 10% levels where the critical values are 21.03 and 18.55.
a

b

2

22

Table 2. Estimates of Parameters (1) Period I :CountryJapan
USA
CanadaFrance
Italy
Korea
Singapore
Malaysia
India

a

θ+ a

θ− a

α0

α1

α2

α3

1.043

0.440**

0.203

0.049

-0.347

0.170

0.974

(0.676)(0.102)(0.169)(0.009)(0.136)(0.078)(0.000)

0.759

0.131

0.115

1.003

0.210

-0.3030.609(1.039)(0.198)(0.383)(0.783)(0.256)(0.227)(0.234)0.604

0.353**

0.317-0.032-0.202-0.0061.012(1.524)(0.100)(0.220)(0.019)(0.092)(0.204)(0.000)-0.605

0.093**

0.126**

0.035-0.327

0.004

0.990

(1.492)(0.039)(0.027)(0.006)(0.119)(0.049)(0.000)0.017-0.0560.190**

0.111-0.313

0.004

0.975

(0.950)(0.175)(0.095)(0.002)(0.128)(0.039)(0.000)1.037

0.082**

0.051

0.186

-0.418

0.129

0.948

(0.273)(0.009)(0.094)(0.000)(0.055)(0.002)(0.000)0.577

0.037**-0.119-0.012-0.398

0.171

0.988

(0.591)(0.015)(0.116)(0.021)(0.018)(0.056)(0.000)3.633

0.115**-0.690**0.117-0.191

0.066

0.989

(0.320)(0.052)(0.140)(0.002)(0.017)(0.029)(0.000)

2.910

0.229

0.360**

0.689

0.131

0.138

0.857

(2.949)(0.219)(0.164)(0.655)(0.196)(0.081)(0.137)

Notes : Robust standard errors are in parentheses. a "*" denotes significant at 10% level and "**" denotes

significant at 5% level.
23

Table 2. Estimates of Parameters (2) Period II :CountryJapan
USA
CanadaFrance
Italy
Korea
Singapore
Malaysia
India

a

θ+ a

θ− a

α0

α1

α2

α3

-0.5450.199*

0.247**

0.202-0.377-0.2170.942(0.397)(0.108)(0.049)(0.014)(0.088)(0.035)(0.000)-0.4930.270*-0.062

0.409

0.087

-0.2290.859(1.064)(0.152)(0.314)(0.321)(0.213)(0.127)(0.105)

1.316

0.164

-0.767**0.129-0.496-0.2440.946(0.270)(0.102)(0.172)(0.044)(0.117)(0.017)(0.000)

1.752

0.135

-1.656**

0.306

0.165

-0.2880.747(0.464)(0.116)(0.708)(0.314)(0.321)(0.182)(0.177)-0.4790.476**

0.255**

0.956-0.929-0.4460.727(0.517)(0.044)(0.117)(0.012)(0.207)(0.212)(0.001)-2.8630.390**

0.581**

0.010-0.217-0.1651.008(0.963)(0.151)(0.051)(0.009)(0.059)(0.152)(0.000)1.344

0.332**-0.052

0.076

0.235

-0.1190.973(1.361)(0.131)(0.284)(0.141)(0.122)(0.081)(0.028)0.980

0.278**

0.093

0.094

0.391

-0.0910.971(1.133)(0.120)(0.273)(0.140)(0.157)(0.102)(0.032)0.037

0.153*-0.0720.007-0.277

0.003

1.001

(1.135)(0.079)(0.152)(0.004)(0.105)(0.009)(0.000)Notes : Robust standard errors are in parentheses. a "*" denotes significant at 10% level and "**" denotes

significant at 5% level.
24

Table 3. Testing Results
Period I

Period II

H1 : θ + =0 & θ− = 0

H1 : θ+ =0 & θ− = 0( χ2(2) )( χ2(2) )Japan

18.93**

35.00**

USA

0.45

4.26*

Canada473.42**25.63**

France33.53**

7.31**

Italy

4.13*

140.42**

Korea

82.11**

167.45**Singapore

11.99**

9.18**

Malaysia

27.90**

5.44#

India

5.85*

5.67#

Notes : "**" ,” * ”, ”#” denotes significant at 1%,5% and 10% level. The critical values of χ (2) distributions are respectively 9.21, 5.99 and 4.61.
2

25

Table 4. Vt −Pt : Discrepancy of Vt from Pt
Period I
Max

Min

Average

Japan

0.080

-0.023

0.016

USA

0.015

-0.012

Canada0.061

FrancePeriod II
Average in

Average in

Max

Min

Average

0.019

0.033

-0.055-0.003

0.011

0.002

0.004

0.027

0.000

0.006

0.006

-0.105

0.007

0.017

0.062

0.000

0.009

0.009

0.013

-0.019

0.001

0.005

0.120

0.000

0.016

0.016

Italy

0.000

-0.038-0.006

0.006

0.066

-0.068

0.009

0.019

Korea

0.015

-0.011

0.002

0.003

0.128

-0.327-0.026

0.061

Singapore

0.056

0.000

0.003

0.003

0.123

0.000

0.015

0.015

Malaysia

0.150

0.000

0.010

0.010

0.097

-0.025

0.007

0.013

India

0.103

-0.115-0.001

0.025

0.021

0.000

0.007

0.007

| Vt − Pt |26

| Vt − Pt |

Figure 1. Stock Prices and Their Returns(1) Japan-Price(1) Japan-Return

1000

30
20

100

10
0

10

-10
-20

1
1969M1

1979M1

1989M1

1999M1

2009M1-30
1969M1(2) USA-Price

1979M1

1989M1

1999M1

2009M1(2) USA-Return
30

1000

20
100

10
0
-10

10

-20
1
1969M1

1979M1

1989M1

1999M1

2009M1-30
1969M1(3) Canada-Price

1979M1

1989M1

1999M1

2009M1(3) Canada-Return

1000

30
20

100

10
0

10

-10
-20

1
1969M1

1979M1

1989M1

1999M1

2009M1-30
1969M1(4) France-Price

1979M1

1989M1

1999M1

2009M1(4) France-Return

1000

60
40

100

20
0

10

-20
1
1969M1

1979M1

1989M1

1999M1

2009M1-40
1969M2

1979M2

Notes : Each stock prices are adjusted such that 2000M1 = 100.

27

1989M2

1999M2

2009M2

Figure 1. Stock Prices and Their Returns (continued)(5) Italy-Price(5) Italy-Return
30

1000

20
100

10
0

10

-10
-20

1
1969M1

1979M1

1989M1

1999M1

2009M1-30
1969M1(6) Korea-Price

1979M1

1989M1

1999M1

2009M1(6) Korea-Return
30

1000

20
100

10
0
-10

10

-20
1
1978M1

1988M1

1998M1

2008M1-30
1978M1(7) Singapore-Price

1988M1

1998M1

2008M1(7) Singapore-Return

1000
20
100

0
-20

10
-40
1
1985M1

1995M1

2005M1-60
1985M2(8) Malaysia-Price
1000
100
10
1
1980M1

1990M1

2000M1

1995M2

2005M2(8) Malaysia-Return
30
20
10
0
-10
-20
-30
-40
1980M1

28

1990M1

2000M1

Figure 1. Stock Prices and Their Returns (continued)(9) India-Price
1000
100
10
1
1969M1

1979M1

1989M1

1999M1(9) India-Return
40
30
20
10
0
-10
-20
-30
2009M1
1969M2

1979M2

Notes : Each stock prices are adjusted such that 2000M1 = 100.

29

1989M2

1999M2

2009M2

Figure 2. Sizes of Market Inefficiency(1) Japan : V t -P t

0.1

0.03

(2) USA : V t -P t

0.02

0.05

0.01

0

0

-0.05

-0.01

-0.1
1980M2 1985M2 1990M3 1995M3-0.02
1990M3(3) Canada : V t -P t

0.1
0.05

0.15

1995M3

2000M42005M4(4) France : V t -P t

0.1

0
0.05

-0.05

0

-0.1
-0.15
1990M100.1

1995M10

2000M11

2005M11(5) Italy : V t -P t

-0.05
1990M10 1995M10 2000M11 2005M11

0.2

0.05

0.1
0

0

-0.1

-0.05

-0.2
-0.3

-0.1
1990M5

0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1986M3

1995M5

2000M6

2005M6(7) Singapore : V t -P t

-0.4
1979M5

0.2(6) Korea : V t -P t

1984M5

1989M6

1994M6(8) Malaysia : V t -P t

0.15
0.1
0.05
0
1991M3

1996M4

2001M4-0.05
1987M430

1992M4

1997M5

2002M5

Figure 2. Sizes of Market Inefficiency (continued)
0.15(9) India : V t -P t

0.1
0.05
0
-0.05
-0.1
-0.15
1987M8

1992M8

1997M9

2002M9Notes : The size of market inefficiency Vt − Pt
in level.

31

implies (Vt − Pt ) / Pt∗∗∗of market prices

Figure 3. Market Price vs Fundamental Value(a) Japan
1000
100
10

1980

1985

1975
1
1970M1 1975M1 1980M1 1985M1 1990M1 1995M1 2000M12005M1(b) Korea
1000
100

1979

1981

10

1985
1
1978M1

1983M1

1988M1

1993M1

1998M1

Note: The red circle means the analytical periods.

32

2003M1

2008M1

Figure 4. Contribution of market inefficiency against over-evaluation orunder-evaluation (%)(a) Japan
100(V*0+t - P*0+t)/P*0+t(P*0+t - S*0+t)/P*0+t

50

0

-50

-100
1985M1

1988M1

1991M2

1994M2(a’) Japan
60
50
40
30
(V*0+t - P*0+t)/P*0+t(P*0+t - S*0+t)/P*0+t(V*0+t - S*0+t)/P*0+t

20
10
0
-10
1985M1

1987M1

1989M1

33

1997M2

Figure 4. Contribution of market inefficiency against over-evaluation orunder-evaluation (%) (continued)
(b)

Korea
100

(V*0+t - P*0+t)/P*0+t(P*0+t - S*0+t)/P*0+t

50
0
-50
-100
-150
-200

1986M2

1989M2

1992M2

1995M2

1998M2(b’) Korea
50
40
30
20
10
0
-10
-20
-30
-40
1 9 9 1 M1 2

(V*0+t - P*0+t)/P*0+t(P*0+t - S*0+t)/P*0+t(V*0+t - S*0+t)/P*0+t
1 9 9 3 M1 2

1 9 9 5 M1 2

Note: ( P0∗+t − S0∗+t ) / P0∗+t + (V0∗+t − P0∗+t ) / P0∗+t =(V0∗+t − S0∗+t ) / P0∗+t
The (V0∗+t − S0∗+t ) / P0∗+t is a net over-evaluation if positive and a net under-evaluation if negative.

34