Aim/purpose
–
The primary aim of the paper is to verify the hypothesis on the normal
distributions of 65 stock index returns, while the secondary aims are to examine normal
distributions for specific years (for six indexes) and for bull and bear markets (for DJIA),
demonstrate that the distribution of rates of return for individual indexes can be normal
in short time intervals, and then rank analyzed indexes according to the proximity of the
distribution of their rates of return to the normal distribution.
Design/methodology/approach
–
The research sample consists of the value of 65 stock
indexes from various time intervals. The sample includes both developed markets and
emerging markets. The following rates of return were tested for the normality of the rat
e
of return distribution: close
-close, open
-open, open
-close and overnight, which were
calculated for daily, weekly, monthly, quarterly and yearly data. Statistical tests of di
f-
ferent properties and forces were used: Jarque
–
Bera (JB), Lilliefors (L), Crame
r von
Mises (CVM), Watson (W), Anderson
–
Darling (AD). In the case of six indexes of d
e-
veloped markets (DJIA, SP500, DAX, CAC40, FTSE250 and NIKKEI225), normality
tests of rates distribution were calculated for individual years 2013
-2016 (daily data). In
case of the DJIA index, the normality tests of the distribution of returns for individual
bull and bear markets were analyzed (daily data, rates of return close
-close). In the last
part of the paper the analyzed indexes were ranked due to the convergence of
their return
to normal distribution with the use of the following tests: Jarque
–
Bera, Shapiro
–
Wilk
and D’Agostino
-Pearson.
Findings
–
The distribution of daily and weekly returns of equity indexes is not a normal
distribution for all analyzed rates of ret
urn. For quarterly and annual data compression
the smallest number when there were no reasons to reject the null hypothesis was o
b-
served for overnight returns compared to close
-close, open
-close and open
-open returns.
For the daily, weekly and monthly over
night rates of return, the null hypothesis was rejected for all analyzed indexes. The fo
llowing general conclusion can be formulated:
the higher the data compression (from dail
y to yearly), the fewer rejections of
H
0
hy-
pothesis. The distribution of daily returns can
be normal only in given (rather short) time
intervals, e.g., particular years or up or do
wn waves (bull and bear markets). The posi-
tion of the index in the ranking is not depende
nt on the date of its first publication, and
hence on the number of rates of return possible
to calculate for analyzed index, but only
on the distribution of its rates of return.
Research implications/limitations
– The main limitations of the obtained results are
different time horizons of each of the analyzed indexes (from the first date in a data base
until 30.06.2017). The major part of the retu
rns of the analyzed indexes differs from the
normal distribution, which question the possi
bility of unreflective implementation in
practice of economic such models as CAPM
and its derivatives, Black–Scholes options
valuation, portfolio theory and efficient market hypothesis, especially in long time horizons.
Contribution/value/contribution
– The contribution of this paper is verification of the
statistical hypothesis regarding normal dist
ribution of rates of return: (1) other than
close-close, i.e. open-open, open-close and ove
rnight with the use of various statistical
tests, various data compression (daily, w
eekly, monthly, quarterly, yearly) for 65 in-
dexes, (2) for six stock exchange indexes in each of the years from the period of 2013-
2016 (daily data) and (3) for individual up
and down waves for the DJIA index (daily
data). In addition, other papers focused only
on one or two statistical tests, while five
different tests were implemented in this paper.
This paper is the first to create a ranking
of stock market indexes due to the normal distribution.
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