Directory UMM :Data Elmu:jurnal:J-a:Journal Of Banking And Finance:Vol24.Issue4.2000:

Journal of Banking & Finance 24 (2000) 555±576
www.elsevier.com/locate/econbase

Regulation of the Warsaw Stock Exchange: The
portfolio allocation problem
Wojciech W. Charemza
a

a,*

, Ewa Majerowska

b,1

Department of Economics, University of Leicester, University Road, Leicester LE1 7RH, UK
b
Department of Economics, University of Gdansk, 80-824 Sopot, Poland

Abstract
The paper analyses the risk reduction e€ect of limits which are imposed on stock
exchange price movements. As a result of the maximisation of tradersÕ utility functions

subject to expected price constraints, a model similar to the capital asset pricing model
(CAPM) is developed, where the observed returns are corrected for the appearance of
constraints. An analysis of returns from six securities traded on the Warsaw Stock
Exchange has been carried out. The models have been estimated by the two-limit Tobit
model and compared with the results for the corrected returns. The results show that the
trade barriers increase the portfolio risk. Ó 2000 Elsevier Science B.V. All rights reserved.
JEL classi®cation: G12
Keywords: CAPM; Price constraints; Regulation; Emerging markets

1. Introduction
The problem of the optimal portfolio allocation, especially that of ®nding a
relationship between the rate of return of a risky asset and the level of risk of
this asset is well known in literature. The seminal model is the capital asset
*

Corresponding author. Tel.: +44-2116-252-2899; fax: +44-2116-252-9081.
E-mail addresses: wch@le.ac.uk (W.W. Charemza), em@panda.bg-univ.gda.pl (E. Majerowska).
1
Tel./fax: +48-58-5502549.
0378-4266/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 8 - 4 2 6 6 ( 9 9 ) 0 0 0 8 0 - 1

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W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

pricing model (CAPM) developed by Sharpe (1964) and Lintner (1965)
based on the contributions of Markowitz (1952, 1958). Further applications
and developments of the CAPM model are too numerous to be listed here (for
some important further developments see Black (1972), Cuthberston (1996),
Merton (1973), Thomas and Wickens (1992), and Slade and Thille (1994)). In
all these, and in similar models, it is assumed that there are no restrictions and
limitations concerning price movements. Such models, where the price is assumed to settle at a freely negotiated level, determined by supply and demand,
are herein called unconstrained. Although this is the case for many contemporary markets, nevertheless, in some stock markets, the price of an asset is
regulated in such a way that it cannot move by more than a ®xed percentage
above or below that of the previous session price. Such regulations have been
applied in numerous emerging stock markets (e.g., in China, Lithuania, Poland,
Turkey) and also in some mature ones (France). Evidently if the price is not
allowed to settle at its equilibrium level because of the presence of institutional
constraints, demand may not match supply (or vice versa) and disequilibrium

occurs.
This paper proposes a simple model of the optimal portfolio allocation in
the case where some prices in the market are regulated (disequilibrium) prices
developed from the Sharpe±Lintner version of CAPM. The regulation takes
the form of an imposition of price barriers. An estimated model can be used
for assessing the impact of such price regulations on the relative risk of the
market portfolio. A simple empirical analysis of a stock market in which
disequilibrium prices appear, namely, that of the Warsaw Stock Exchange, is
carried out.

2. Principal assumptions of the model
It is assumed that the market in which the portfolio allocation decisions
are made might be inecient, but the only form of ineciency which might
possibly exist is that caused by price regulation (appearance of price constraints). Charemza et al. (1997) have shown that appearance of such constraints, even if they are binding, does not necessarily imply ineciency.
Following Elton and Gruber (1991), it is assumed that if the price restrictions
are not binding (or more strictly, they are not expected to be binding), the
general assumptions of the ecient market hypothesis hold and, in particular,
that:
1. there are no taxes;
2. there are no transaction costs and other imperfections;

3. investors can borrow and lend at the risk-free rate an unlimited amount at
the same time;
4. the market is not dominated by any individual investor;

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

557

5. there is a ®xed number of assets on the market and they are available for all
investors;
6. there is freedom of entry and exit for both buyers and sellers.
The above assumptions imply normality of distribution of the rates of return. Assuming that the portfolio is fully described by the mean and variance
of returns, the distribution is described as symmetric (lack of skewness) and has
a kurtosis value equal to 3. Regarding the investors, the following assumptions
are made:
1. Investors behave rationally, they are risk-averse and maximise expected utility of wealth.
2. They are interested only in two features of security: expected returns and
risk, the latter expressed by the variance of returns.
3. All investors have identical perceptions of each security and homogeneous
expectations.

4. Each investor de®nes a period of time as an investing horizon and these periods are not identical.
5. They have full access to information about the market.
6. Investors are price takers; this means that they assume that their own buying
and selling activity will not a€ect asset prices.
7. All investors can lend and borrow without any limitations at a risk-free rate
r at the same time.
8. The separation principle holds, so that the investors take homogeneous decisions regarding the composition of the risk portfolio at the ecient frontier, and then decide, individually, according to the degree of the particular
investorÕs risk aversion, on the composition of the risk portfolio and the
riskless asset.
Evidently, no empirical markets, let alone an emerging market, is fully
consistent with the above assumptions. It is generally agreed that, in the
absence of price limits, emerging market anomalies causing a lack of eciency
(or more precisely, predictability of returns) are not more severe than those of
mature markets (see, e.g., Claessens et al., 1995; Richards, 1996). In particular, Buckberg (1995) and Harvey (1995a) found emerging markets behaviour
to be consistent with the CAPM model. It is also argued that market ineciencies tend to evolve (diminish) over time and it is possible to capture the
convergence towards market eciencies (see Harvey, 1995b; Emerson et al.,
1997). 2
If the price limits are not reached, the organisation of trade at the Warsaw
Stock Exchange is that of a batch system order-driven market. Orders are
submitted prior to trading and then a price is determined which maximises the

2
It is also argued that the CAPM approach is inconsistent with the theory of ecient markets
(see Reingaum, 1992). Discussion of this is, however, outside the scope of this paper.

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W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

session turnover. As shown by Madhavan (1992) such a system is relatively
transparent, with a minimal degree of market asymmetry and, in particular, in
equilibrium the market can be regarded as semi-strong ecient. This con®rms
the rationale of using CAPM as the equilibrium foundation for analysis of the
Warsaw Stock Exchange.

3. Derivation of the model: an outline
Let us introduce the following basic notation:
the number of investors, i ˆ 1; 2; . . . ; m,
the number of assets, j ˆ 1; 2; . . . ; n,
expected price of asset j at the end of period, de®ned as the

expected value of the price of the jth asset conditional on
information available at the beginning of the period
initial price of asset j,
covariance between prices ofÿassets j
and k at the end of
period, that is, cov…pj;k † ˆ E pj ÿ p~j …pk ÿ p~k † .

m
n
p~j
pj0
cov…pj;k †

The Sharpe±Lintner CAPM model could be used to describe equilibrium in
terms of either returns or prices. Let the expected value of the portfolio at the
end of period for the ith investor be li and the variance r2i . The utility function
of the ith investor is Vi …li ; r2i †, with the usual assumptions:
Vi1 ˆ

oVi
> 0;

oli

Vi2 ˆ

oVi
< 0:
or2i

If the investor is not constrained on any of the prices (or more precisely, if the
quantities traded are not constrained due to the fact that the price for any asset
reaches its upper or lower limit) the problem of his/her decision-making is the
maximisation of the investorÕs utility function with respect to wij in proportion
to the wealth held in asset j by the ith investor:
ÿ


max Vi li ; r2i :
wij

Let the expected value of the portfolio i be described as (see Brennan, 1992)
li ˆ

n
X
jˆ1

wij p~j ÿ R

n 
X
jˆ1


 ij pj0 ;
wij ÿ w

 ij is an endowed fraction of asset j in
where R ˆ r ‡ 1 (r a risk free rate) and w
the ith portfolio. The variance of the portfolio of ith investor is de®ned as

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

r2i ˆ

559

n
n X
X
wij wik cov…pj;k †:
jˆ1 kˆ1

The sum of wealth proportions in a portfolio for each investor is equal to one
so that
win ˆ 1 ÿ

nÿ1
X

wij :

jˆ1

In order to ®nd the optimal value of the portfolio, it is necessary to calculate
the derivative of the utility function with respect to each asset. Let us calculate
it with respect to the ®rst asset:
oVi
oVi oli
oVi or2i
ˆ
‡ 2
:
owi1 oli owi1 ori owi1

…1†

The ®rst order condition is
"
#
n


X
0
0
Vi1 p~1 ÿ p~n ÿ Rp1 ‡ Rpn ‡ Vi2 2 wik cov…p1;k † ˆ 0:
kˆ1

…2†

So far the result is consistent with the Sharpe±Lintner model. Let us now
assume that the price of the nth asset is constrained as


8
0
0
>
~
if
p
ÿ
p
…1
‡
d†p
=pn0 P d;
>
n
n
n
>
>
<


…3†
if ÿ d < p~n ÿ pn0 =pn0 < d;
p~n ˆ pn
>
>


>
>
: …1 ÿ d†p0 if p~ ÿ p0 =p0 6 ÿ d;
n
n
n
n

where pn is an expected price of asset n in the equilibrium situation and d is the
maximal admissible (by the regulator) price movement. In particular, if the
price constraints are not binding, the price dynamics are described by a martingale process pn ˆ pn0 (see Charemza et al., 1997).
Assume now that the price Ôhits a boundaryÕ (lower or upper limit) with
probability x. Then:

…1  d†pn0 with prob: x;
…4†
p~n ˆ
with prob: 1 ÿ x;
pn
which gives
 
E p~n ˆ x…1  d†pn0 ‡ …1 ÿ x†pn :

…5†

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W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

Let rik ˆ wik
pk0 be the amount invested in asset k, for k ˆ 1; 2; . . . ; n,
pj ÿ pj0 †=pj0 is the expected rate of return of asset j, and rjk is the
lj ˆ …~
covariance of the rates of returns between assets j and k. It can be shown
that:


rjk ˆ cov…rj rk †  E …rj ÿ lj †…rk ÿ lk †
"
!
!#
pj ÿ pj0 p~j ÿ pj0
pk ÿ pk0 p~k ÿ pk0
ÿ
ÿ
ˆE
pk0
pj0
pj0
pk0
"
!
!#
pj p~j
pk p~k
1
ÿ
ÿ
 0 0 cov…pj;k †:
ˆE
pj pk
pj0 pj0
pk0 pk0
Substituting hÿ1
i ˆ ÿVi1 =2Vi2 as the measure of the investorÕs risk tolerance (see
Brennan, 1992) for the constrained price we obtain
!ÿ1
n
n
X
X
pn0
ÿ1
:
…6†
hi
l1 ÿ r ‡ …r ÿ d† 0 ˆ hM rik r1k ; where hM ˆ
p1
kˆ1
iˆ1
For the aggregate market we have


pn0
l ÿ rI ‡ …r ÿ d† 0 I ˆ hM …rX†;
p1

…7†

where X is a variance and covariance matrix and I is a vector of units.
Solving Eq. (6) by substituting hM ˆ …lM ÿ r†=r2M , where lM and r2M are,
respectively, the expected rates of returns and the variance of the market
portfolio, we get (details of computations are included in the discussion paper
version, see Charemza and Majerowska (1998), and are available on request)
r1 ˆ r ‡ b1 …rM ÿ r† ÿ …r ÿ d†

pn0
;
p10

where b1 ˆ

r1M
:
r2M

…8†

For an individual asset j, if the price of nth is constrained, we have
rj ˆ rf ‡ bj …rM ÿ rf † ÿ …rf ÿ d†

pn0
;
pj0

which leads to a solution
8
< rf ‡ bj …rM ÿ rf †
p0
rj ˆ
: rf ‡ bj …rM ÿ rf † ÿ …rf ÿ d† n0
pj
rjM
ˆ 2 :
rM

where bj ˆ

rjM
;
r2M

with prob: 1 ÿ x;
with prob: x;

…9†

where bj

…10†

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

561

The expected value of the jth return is then
E…rj † ˆ rf ‡ bj …rM ÿ rf † ÿ x…rf ÿ d†

pn0
:
pj0

…11†

As a result we have a model of the rate of return of an individual asset as a
function of the rate of return of a risk-free asset (rf ), the level of systematic risk
and the initial prices of assets jth and nth, where the price of the nth asset is
limited, and standard CAPM where this nth price is not limited.

4. Empirical assessment: Warsaw Stock Exchange
Eq. (11) gives rise to a formulation of a simple CAPM-like empirical model
for a stock market with regulated prices. Ignoring, for high-frequency data
(session-to-session returns) the e€ect of a riskless asset and allowing for constant transactional costs, the model can be formulated as
rt ˆ a ‡ brtm ‡ et ;

…12†

where rtm denotes the session-to-session returns from the market portfolio in
time (session) t and rt is the return from an individual security corrected by the
censored prices. Suppose that there are N ‡ 1 securities included in the market
portfolio, and that they are ordered in such a way that the security investigated
in (12) is the last, (N + 1)th one. Hence, a generalisation of (10) gives rt being
de®ned as
rt ˆ rt ÿ cft ;
where the correction factor cft is
cft ˆ d

PN ÿ
iˆ1


i i
‡
ÿ ÿ
x‡
it zit ÿ xit zit pt w
;
ptN ‡1

…13†

and where rt is the observed, possibly censored, return of the N ‡ 1 security in
time t, d is the relative constraint on price movements (fraction of the last
period price which creates the upper or lower limit for returns, see (3)),
ÿ
pti ; i ˆ 1; 2; . . . ; N ‡ 1, is the price of the ith security in time t, x‡
it and xit are
the probabilities of hitting the upper and lower barrier by the ith price, wi is a
ÿ
weight denoting market share of ith security, and z‡
it and zit are the selector
variables:
z‡
it ˆ

8
<
:

1
0

pi ÿ pi
if t i tÿ1 ˆ d;
ptÿ1
otherwise;

and zÿ
it ˆ

8
<
:

1
0

i
pti ÿ ptÿ1
ˆ ÿd;
i
ptÿ1
otherwise:

if

562

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

Fig. 1. Original returns from Tonsil.

As the empirical observation implies (see also Fig. 1), the frequencies of hits
change over time. In particular, after 1995, the frequency of hits becomes
visibly smaller compared to the earlier years. Hence, sensible estimates for x‡
it
and xÿ
it seem to be the empirical frequencies of the lower and upper hits within
the sample computed in a recursive manner, that is taking into account only
information available up to time t. 3 For the ®rst 100 observations the values of
ÿ
x‡
it and xit are held constant and equal the empirical frequencies of hits for this
period. After the 100th observation the frequencies have been updated recursively. Finally, the value of parameter d is given by market regulations and wi
are the weights of the stock market index.
Model (12) has been estimated for the six longest-established securities
traded at the Warsaw Stock Exchange. The Warsaw Stock Exchange was
established on 16 April 1991, with initially two sessions a week in 1991 and
1992, three sessions a week from the beginning of 1993 until the end of 1994
and ®ve (daily) sessions since then. Detailed descriptive and econometric
analyses of Warsaw Stock Exchange can be found in Gordon and Rittenberg
(1995), Boøt and Miøobeßdzki (1994a,b) and Shields (1997a,b). For our purposes it is important to note that, on a trading session, transactions are
made at a single price, established by the regulator at a level which maximises demand and supply. This single price is established in a t^
atonnement
process, where o€ers to sell and buy are lodged with the ÔauctioneerÕ before
the price is established. If, however, the single price evaluated in this way is
greater or lower than the last session price by more than 10%, it is arti®cially

3

We are grateful to a referee for this suggestion.

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

563

reduced and kept at a level which is exactly 10% above (below) the last
session price. Normally, trading takes place at such regulated price leaving a
part of demand or supply in excess. 4 Hence, for the Warsaw Stock Exchange the parameter d introduced to Eq. (3) above is equal to 10%. The six
companies selected for estimation are: Exbud (EXB, construction services),
Kable (KAB, cable factory), Krosno (KRO, glass factory), Pr
ochnik (PRO,
confection produces), Swarzedz (SWA, furniture factory) and Tonsil (TON,
electronics company). These six companies are the longest established on the
market and, at the beginning of the existence of the Warsaw Stock Exchange, represented the majority of the trade. Over time the number of
companies listed on the stock exchange has grown rapidly. Nevertheless,
these six well-established ÔmatureÕ companies are still regarded as being
representative for the entire market. Other major companies have usually
been introduced to the market at a much later date and their inclusion would
shorten the data sample signi®cantly. We start from almost the beginning of
the series, discarding only the ®rst nine observations; our sample contains
966 data points from the 10th session (25 June 1991) until the end of 1996.
We do not use direct quantitative information concerning the identi®cation
of the disequilibrium trading sessions. Instead, we have assumed that if returns were closer than 0.05% to its upper or lower boundary (that is, if the
published price was equal or higher than 1.095 times the previous session
price, or 0.905 or lower than the previous session price), the upper (lower)
boundary was hit. This 0.05% tolerance limit allows us to account for
rounding errors of published prices. The source of the data was from detailed information published in Gazeta Bankowa (daily) and Rzeczpospolita
(weekly). 5 In a few instances missing observations were interpolated. For
sessions in which trading was suspended and the stock market statistics
denoted zero returns, we randomise the returns by inserting a random
number equal to 10% of the standard deviation of returns. Simple autocorrelation analysis of the returns (of which details are not given here but are
available on request) do not reveal any substantial autocorrelation in the
series, with the largest autocorrelation coecients of about 0.24. The lack of
substantial autocorrelation supports the rationale for the adopted method of
interpolation of zero returns. In some cases, where prices were allowed to go
beyond that limit due to its occasional suspension, we censored the data as if
the upper or lower limit was hit.
4
Occasionally, if demand is greater than supply (or the opposite) by more than ®vefold, the
transactions might be suspended altogether. Also, there are special regulations which allow some of
the securities to be traded at a freely negotiated prices (during extra time trading). For simplicity,
these are ignored herein.
5
Data was collected and made available to us by the Macroeconomic and Financial Data Centre
at the University of Gdansk. Its assistance is gratefully acknowledged.

564

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

Tables 1±3 brie¯y summarise the descriptive characteristics of the series. In
Table 1 the frequencies of limit hits (censoring) are given. In Table 2 the descriptive measures for the series of all returns are presented, together with the
Doornik and Hansen (1994) modi®cation of the Bowman and Shenton (1975)
test of normality. Under the null hypothesis of normality the statistic has a
v2 …2† distribution. Statistics signi®cant at the 0.05 level of signi®cance are
denoted by *. Table 3 gives analogous characteristics computed for the ÔequilibriumÕ returns only, that is for the case where the lower or upper barriers were
not hit.
The statistics con®rm the relative homogeneity of the sample. For all the
series the characteristics are of a similar magnitude, and the distributions of the
returns are close to being symmetric. For equilibrium returns the null hy-

Table 1
Frequencies of limit hits (censoring)
Lower
Upper
Total

EXB

PRO

KAB

KRO

SWA

TON

0.069
0.079
0.148

0.106
0.085
0.190

0.091
0.085
0.176

0.084
0.069
0.153

0.086
0.077
0.162

0.112
0.090
0.202

Table 2
Descriptive statistics and normality tests for all returns
No. of obs.
Mean
SD
Skewness
Ex. kurtosis
Normality

EXB

PRO

KAB

KRO

SWA

TON

966
0.0028
0.0464
)0.0650
)0.0688
0.8097

966
0.0022
0.0504
)0.0927
)0.3451
6.959

966
0.0028
0.0513
)0.0761
)0.4612
11.016

966
0.0015
0.0482
)0.0742
)0.2619
3.780

966
0.0013
0.0484
)0.0651
)0.2488
3.177

966
0.0016
0.0535
)0.0930
)0.6585
24.711

Table 3
Descriptive statistics and normality tests for equilibrium returns
No. of obs.
Mean
SD
Skewness
Ex. kurtosis
Normality

EXB

PRO

KAB

KRO

SWA

TON

823
0.0023
0.0332
)0.0578
0.4556
7.641

782
0.0050
0.0343
0.1799
0.4621
9.061

796
0.0041
0.0378
0.0458
0.2849
3.404

818
0.0034
0.0352
0.0688
0.2359
2.820

809
0.0026
0.0345
0.0364
0.3490
4.735

771
0.0045
0.0386
)0.0024
0.0117
0.075

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

565

pothesis of normality can only be marginally rejected for two of the series (for
all returns the censored tails clearly makes the distributions non-normal). The
source of some non-normality seems to be excess kurtosis, presumably caused
by a concentration of randomised returns around zero for the days where
trading was suspended and zero returns recorded. Generally, it might be
concluded that the distributions of equilibrium returns are consistent with the
assertion of ecient trading taking place during the sessions where price limits
were not hit.
Corrected returns can vaguely be interpreted as returns which would
have happened if the price limits were not binding. Figs. 1±3 show, respectively, the series of uncorrected returns, corrected returns and the
correction factor computed for one of the companies analysed, Tonsil. Figs.
4 and 5 show distributions of the original and corrected returns. The ®gures
indicate that most of the corrections happen in the early years of the operation of the Warsaw Stock Exchange, where hits of the barriers were
frequent. The distribution of the corrected returns exhibits a much smaller
unconditional variance and greater concentration than that of the original
returns.
Two alternative ways of identi®cation of the market returns variable rtm
were used. Firstly, the published ocial Warsaw Stock Exchange share index,
WIG, computed for all the shares traded on the market was used as the rtm

Fig. 2. Corrected returns from Tonsil.

566

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

Fig. 3. Correction factor.

Fig. 4. Distribution of original returns, Tonsil.

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

567

Fig. 5. Distribution of corrected returns, Tonsil.

variable. Alternatively, it was assumed that the entire market consisted of
only six securities and an arti®cial Laspeyres type index for these six companies only was constructed (WIG6). For each of the six securities the rtm
variable has been constructed by adjusting the WIG6 index by the exclusion
of price and quantity weight information on the modelled security. In other
words, rtm represents the returns from the ®ve remaining shares. The comparison of the original WIG and WIG6 is given in Fig. 6. It shows that until
1994 the development of both indices was almost identical. With the increase
in number of securities traded at the Warsaw Stock Exchange the dynamics
of both indices started to di€er and the prices of new securities, other than
those included in WIG6, were rising faster than the prices of the six securities
analysed herein.
The model (12) has been estimated by the full-information maximum likelihood method in the case where the correction factor was used and, for
comparison, by the two-limit Tobit model (see e.g. Rosett and Nelson, 1975;
Maddala, 1983, pp. 161±162) in case where the returns were left uncorrected
(that is, assuming cf ˆ 0). These methods give, under the normality assumption, the asymptotically unbiased and asymptotically ecient estimators.
Strictly speaking, in both cases the same general likelihood function has been
used, assuming that et  IID N…0; r2e † (see Davidson and MacKinnon, 1993, p.
539):

568

W.W. Charemza, E. Majerowska / Journal of Banking & Finance 24 (2000) 555±576

Fig. 6. WIG and WIG6 (logarithmic scale).

Lˆ


 
 X   l
1
r ÿ a ÿ brtm
r ÿ a ÿ brtm
ln U
/ t
‡
re
re
re
rt ˆrl
rl