This article attempts to clarify certain perceptions
that stock prices do not follow a random walk. This is accomplished by
examining the day of the week effect (Tuesday effect in particular),
month effect (Ramazan effect and December effect). The results of the
study validate the random walk theory and conclude that there is no
effect due to day of week, month as per ANOVA technique used in this
article. Other techniques such as Dickey Fuller Test, Box Pierce
pStatistic, Variance Ratio Test, and Unit Root Test are beyond the
scope of this article and might provide contradictory results.
Our objective in this article is to validate the
random walk hypothesis for the Pakistani Stock Market. We do this by
examining the week of day effect and month effect using a simple
statistical tool of comparing means.
The study assumes that securities at KSE follow
random walk phenomenon. The basis of this study is a discussion held
with various market participants who tend to believe that there exists a
Tuesday Effect and December Effect.
HYPOTHESIS BUILD FOR ANALYSIS
1. Mean
returns are equal on all days.
Ho: Return Monday = Return Tuesday = Return Wednesday
= Return Thursday = Return Friday
2. H1: Return
January = Return February = Return March = Return April = Return May =
Return June = Return July = Return August = Return September = Return
October = Return November = Return December
RATIONALE OF THE STUDY
The random walk theory states that the price changes
in stocks follow a random pattern. That is, the price changes in stocks
are unpredictable; the intense competition amongst the experts ensures
that the stock prices already reflect all the relevant information
available. Therefore, any price change would occur in response to new
information only. This is exactly similar to predicting whether head or
tail would occur on each toss of a coin.
The random walk theory states that all information is
reflected in the current stock prices, therefore, it can be safely
averred that any new information would also take little time to be fully
incorporated in the prices, and market participants, thus, would have
little time to utilize this new information to realize above normal
profits. Because of widespread informational homogeneity every one would
be driven to same actions and the abovenormal profit potential of new
information would quickly be discounted to mere normal level average
return offered by the market i.e. the return offered by an average
stock; any stock/portfolio that displays significantly varied price
change to new information from that of an average stock is offering
abnormal return.
To validate the random walk theory a feasible method
is to test the variances between the returns offered by individual
stocks/portfolios and the market. Any significant variance would mean
that there exist inefficiencies and investors can make abnormal profits
and the random walk theory does not hold, whereas an insignificant
variance between the two values would mean the opposite and would thus
validate the random walk theory.
Another way to test the random walk theory would be
to see if the current stock prices reflect all available information.
This would be accomplished by measuring the speed at which information
is incorporated into stock prices and the amount of noise in the price
process, which is another domain of research not attempted in this
article.
DATA USED FOR THE ANALYSIS
The data consists firstly of the daily closing price
data on the scrip traded at Karachi Stock Exchange (KSE) for the period
19982003. The data was collected on daily basis from the web site of
the KSE. Prices of ten of the top traded stocks and the KSE 100 Index
are selected for the study.
The returns are calculated as the logarithmic
returns. A 'total returns' index is used in the analysis, that is, daily
logarithmic returns dividends reinvested, in order to eliminate any
intraweek effect which might be caused by systematic tendencies for
shares to go down exdividend or pay out dividends on particular days of
the week. The price data adjusted for cash and stock dividends and the
right issues is used for this analysis.
The daily stock return is defined as the return from
the opening (adjusted for stock and cash dividend) of the current
trading day to the close of the current trading day.
TIME PERIOD OF STUDY
The period taken for study is June 01, 1998 to May
30, 2003.
ASSUMPTIONS MADE FOR STUDY
The study is made using the following assumptions:
• The population is normally distributed.
• The variance is same for all the populations
• The assumption of homoskedasticity.
STATISTICAL TOOLS FOR ANALYSIS
To test the hypotheses, One Way — Analysis of
Variance (ANOVA), was used at a significance level of 5%. The acceptance
of the hypotheses would show that the mean returns on all the weekdays
and months are not significantly different from each other and the
rejection would mean that mean returns on at least one day of the week
and in at least one month are significantly different from each other.
RESULTS AND CONCLUSIONS
Hypothesis tested: the mean returns on all days of a
week are equal.
Ho: Return Monday = Return Tuesday = Return Wednesday
= Return Thursday = Return Friday
As can be observed from the Appendix A (except for
MPLC) the null hypothesis is accepted, which implies that there is no
dayoftheweek effect (for example Tuesday Effect) on the Karachi Stock
Exchange.
Hypothesis tested: the mean returns in all months of
a year are equal.
Ho: Return January = Return February = Return March =
Return April = Return May = Return June = Return July = Return August =
Return September = Return October = Return November = Return December.
As can be observed from the Appendix B the null
hypothesis is accepted, which shows that there is no montheffect and
thus, no Ramazan and/or December effect on the Karachi Stock Exchange.
FURTHER SCOPE OF STUDY
Asset pricing as a whole needs further research
especially in context of Pakistani Markets. This study can be further
validated using CoIntegration Techniques such as Vector Auto regression
or Unit Root Test. Dickey Fuller Test, Box Pierce QStatistic and
Variance Ratio test are some concepts, still need to be tested on our
markets which might provide contradictory results.
APPENDIX A
ANOVA
FOR MEASURING EFFECT OF DAY 
SYMBOL 
SOURCE 
SUM OF SQUARES 
DF 
MEAN SQUARE 
F 
SIG. 
KSE 
Between Groups 
0.002 
4 
 
1.519 
0.194 

Within Groups 
0.418 
1,326 
 
 
 

Total 
0.420 
1,330 
 
 
 
DGKC 
Between Groups 
0.021 
4 
0.005 
2.771 
0.026 

Within Groups 
2.532 
1,338 
0.002 
 
 

Total 
2.553 
1,342 
 
 
 
ENGRO 
Between Groups 
0.001 
4 
 
0.356 
0.840 

Within Groups 
1.094 
1,341 
0.001 
 
 

Total 
1.095 
1,345 
 
 
 
FFCJ 
Between Groups 
0.007 
4 
0.002 
1.391 
0.235 

Within Groups 
1.705 
1,329 
0.001 
 
 

Total 
1.712 
1,333 
 
 
 
FFCL 
Between Groups 
0.002 
4 
0.001 
0.978 
0.419 

Within Groups 
0.766 
1,341 
0.001 
 
 

Total 
0.768 
1,345 
 
 
 
HUBC 
Between Groups 
0.003 
4 
0.001 
0.521 
0.721 

Within Groups 
1.704 
1,340 
0.001 
 
 

Total 
1.706 
1,344 
 
 
 
ICI 
Between Groups 
0.005 
4 
0.001 
1.340 
0.253 

Within Groups 
1.197 
1,321 
0.001 
 
 

Total 
1.202 
1,325 
 
 
 
MPLC 
Between Groups 
0.065 
4 
0.016 
7.816 
 

Within Groups 
2.756 
1,322 
0.002 
 
 

Total 
2.821 
1,326 
 
 
 
PSOC 
Between Groups 
0.005 
4 
0.001 
1.146 
0.333 

Within Groups 
1.420 
1,341 
0.001 
 
 

Total 
1.425 
1,345 
 
 
 
PTC 
Between Groups 
0.003 
4 
0.001 
0.920 
0.451 

Whhin Groups 
1.130 
1,341 
0.001 
 
 

Total 
1.133 
1,345 
 
 
 
SNGP 
Between Groups 
0.008 
4 
0.002 
1.868 
0.114 

Within Groups 
1.515 
1,341 
0.001 
 
 

Total 
1.523 
1,345 
 
 
 
APPENDIX
B
ANOVA FOR MEASURING EFFECT OF MONTH

KSE100 
Between Groups 
0.002 
11 
0 
0.58 
0.85 

Within Groups 
0.418 
1,319 
0 
 
 

Total 
0.42 
1,330 
 
 
 
DGKC 
Between Groups 
0.015 
11 
0.001 
0.73 
0.71 

Within Groups 
2.537 
1,331 
0.002 
 
 

Total 
2.553 
1,342 
 
 
 
ENGRO 
Between Groups 
0.011 
11 
0.001 
1.29 
0.23 

Within Groups 
1.083 
1,334 
0.001 
 
 

Total 
1.095 
1,345 
 
 
 
FFCJ 
Between Groups 
0.008 
11 
0.001 
0.55 
0.87 

Within Groups 
1.704 
1,322 
0.001 
 
 

Total 
1.712 
1,333 
 
 
 
FFCL 
Between Groups 
0.01 
11 
0.001 
1.58 
0.1 

Within Groups 
0.759 
1,334 
0.001 
 
 

Total 
0.768 
1,345 
 
 
 
HUBC 
Between Groups 
0.013 
11 
0.001 
0.94 
0.5 

Within Groups 
1.693 
1,333 
0.001 
 
 

Total 
1.706 
1,344 
 
 
 
ICI 
Between Groups 
0.006 
11 
0.001 
0.61 
0.82 

Within Groups 
1.196 
1,314 
0.001 
 
 

Total 
1.202 
1,325 
 
 
 
MPLC 
Between Groups 
0.02 
11 
0.002 
0.87 
0.57 

Within Groups 
2.801 
1,315 
0.002 
 
 

Total 
2.821 
1,326 
 
 
 
PSOC 
Between Groups 
0.014 
11 
0.001 
1.22 
0.27 

Within Groups 
1.411 
1,334 
0.001 
 
 

Total 
1.425 
1,345 
 
 
 
PTC 
Between Groups 
0.003 
11 
0 
0 .28 
0.99 

Within Groups 
1.13 
1,334 
0.001 
 
 

Total 
1.133 
1,345 
 
 
 
SNGP 
Between Groups 
0.01 
11 
0.001 
0.8 
0.65 

Within Groups 
1.513 
1,334 
0.001 
 
 

Total 
1.523 
1,345 
 
 
 