Finance Time Series

1. Application of time series analysis in
Finance

2. 0 500 2000 2500 3000
1. Introduction
The financial market as a part of a market system is created by supply and demand of
money and capital. There are four parts of the financial market: bond market, stock market,
commodity market and exchange market. The financial market is business with credits,
loans, bonds, shares, commodities and currency. The basic information from financial
markets is the price: price of share, commodity price, currency price, bond price etc. The
prices are monitored in certain time frequency and create time series. These time series as
well as time series based on prices or time series which describe prices and their dynamism
are called financial time series.
In comparison with other economic time series, the financial time series have some
characteristic properties and shapes given by the microstructure of the financial market. The
basic feature of the financial time series is a high frequency of individual values. This leads
to the intensification of the influence of nonsystematic factors to the dynamism of these time
series, the result is relatively high volatility which usually changes through time. The
systematic factors create a trend and cycle part of time series, the seasonal part does not play
usually any significant role.
The following pictures illustrate a common shape of financial time series which are
monitored in daily frequency. Picture 1 contains the index of the stock market in Paris
(CAC40) from 9 July 1987 until 31 December 1997, Picture 2 describes development of the
exchange rate CZK/USD from 1 January 1991 until 14 February 2001, Picture 3 describes
development of the exchange rate DEM/USD from 4 January 1971 to 31 December 1998.
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Picture 1: CAC40

3. 1000 1500 2000 2500 3000
Picture 2: CZK/USD
0 1000 2000 3000 4000
Picture 3: DEM/USD
5000 6000 7000
4,1
3,7
3,3
2,9
2,5
2,1
1,7
1,3
2. The classical assumptions and characteristic features of the financial
time series
The basic and primary hypothesis about a behavior of the market is the efficient market
hypothesis. Its first formulations appeared at the beginning and in the first half of 20th
century
[Bachelier (1900), Cowles (1933)]. In later papers from the second half of 20th
century
[Fama (1970), Malkiel (1992)] this hypothesis was specified more deep. It can be shortly
formulated by this way: If the prices "fully reflect" expectations and information of the all
participants of the market, their changes must be unpredictable. The concept of efficiency is
in other definitions specified in accordance with the notion of information.
The hypothesis of efficient market is in a very close connection with the idea of the
martingale model. Its origin is in the theory of probability of 16th
century. This model can be
described in the following way. If Pt is the asset price in time t then the expected asset price
in time t+1 is the asset price in time t under the condition of knowledge of the all prices of
this asset in the history, i.e. in times t - 1, t - 2, ... . From the forecasting point of view the
martingale implies that the best ("minimal mean square error") forecast of the tomorrow's
price is the today's price. Under the consideration that the time series is the realization of
stochastic process {Pt}, i.e. the sequence of random variables ordered in time, than the
martingale can be expressed as
E[Pt+1Pt, Pt-1, ...] = Pt, (1)

4. change and not on the risk, it is not necessary condition of the efficient market.
Martingale (1) and (2) can be expressed as
Pt = Pt-1 + at, (3)
where at is called martingale difference. This form of expression looks like the random walk
model. In the comparison with the martingale there it is assumed that {at} is the white noise
process where the random variables are not only non-correlated but also identically
distributed with zero mean and constant variance. Frequently it is also assumed that {at} is
the strict white noise process where the random variables are independent and identically
distributed with zero mean and constant variance.
Sometimes it is assumed that the distribution of these random variables is normal i.e. at
 N(0, 2
). This idea is clear and attractive from the statistical point of view but it has two
basic defects.
The asset price can not be smaller than zero, the minimal asset net return is therefore Rt
= (Pt - Pt-1)/Pt-1 = -1. As the normally distributed random variable can generate any real
number and it follows that net return Rt is normally distributed, the lower border is not
guaranteed.
The asset gross return for k periods from time t - k to time t can be expressed as the
product of k individual periods gross returns, i.e. as the product of k simple gross returns in
the following way
Pt Pt
.
Pt1
.
Pt2
...
Ptk 1

Rt(k) + 1 = (Rt + 1) . (Rt-1 + 1) . ... . (Rt-k+1 +1) = . (4)
Pt1 Pt2 Pt3 Ptk Ptk
The problem is that the simple asset gross returns are normally distributed but their
product, i.e. the k-periods asset gross return is not normally distributed. The sum of the
simple gross returns is normally distributed but its interpretation is not possible.
These problems can be overcome by consideration that the simple gross returns should
have some distribution of nonnegative random variable. In this connection it is possible to
apply lognormal distribution. The logarithmic transformation of random variable with
lognormal distribution is normally distributed. Therefore, if simple gross return Rt + 1 =
Pt/Pt-1 is lognormally distributed than its logarithm, i.e. rt = ln(Rt + 1) = lnPt - lnPt-1 = pt - pt-1
is normally distributed. The gross return for k periods is the sum of k simple gross returns in
the log transformations, i.e.
rt(k) = rt + rt-1 + rt-2 + ... + rt-k+1 (5)
and rt(k) is normally distributed.

5. 0 500 1000 1500 2000 2500 3000 3500
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Picture 5: EOE
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0 500 1000 1500 2000 2500 3000 3500
0,18
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-0,17

6. Weekly
Londýn (FTSE100) 626 0,002059 0,002979 0,000497 -0,265825 0,087222 -2,849465 33,319316
Hong Kong (HANG SENG) 626 0,002837 0,005308 0,001602 -0,338968 0,173908 -2,352888 17,746855
Tokyo (NIKKEI) 626 0,000257 0,000622 0,000842 -0,186441 0,135191 -0,451233 4,272873
Singapore (SINGALLS) 626 0,000934 0,001146 0,000704 -0,221259 0,120032 -1,321831 11,573267
New York (S&P500) 626 0,002416 0,003884 0,000503 -0,283705 0,072882 -3,507218 41,822356
Praha (PX50) 388 0,000931 0,000108 0,001607 -0,154722 0,232827 0,573693 4,626460
Two-Weekly
Amsterodam (EOE) 313 0,003831 0,005965 0,001504 -0,285302 0,162647 -2,157977 15,108132
Frankfurt (DAX) 313 0,003483 0,008112 0,001670 -0,258932 0,132930 -1,583310 8,170425
Paříž (CAC40) 274 0,002552 0,006552 0,001531 -0,180989 0,105412 -0,680883 1,609288
Londýn (FTSE100) 313 0,004118 0,004875 0,001160 -0,321933 0,100945 -2,696413 26,252886
Hong Kong (HANG SENG) 313 0,005674 0,014598 0,003923 -0,472122 0,143702 -3,136225 19,873742
Tokyo (NIKKEI) 313 0,000514 -0,000144 0,001674 -0,151510 0,116835 -0,295654 0,936578
Singapore (SINGALLS) 313 0,001868 0,002801 0,001697 -0,338701 0,115639 -2,284417 16,168181
New York (S&P500) 313 0,004833 0,004985 0,000891 -0,299194 0,085946 -3,358235 33,456870
Praha (PX50) 194 0,001862 -0,003211 0,003403 -0,190649 0,246683 1,038067 4,050073
Table 2
Ex change Rate n r ~r sr
2 rmin rmax SKr Kr
Daily
ATS/USD 7006 -0,000105 0,000000 0,000054 -0,139503 0,135003 0,050822 47,540368
FRF/USD 7020 0,000002 0,000000 0,000040 -0,060490 0,058746 0,017741 10,343803
DEM/USD 7020 -0,000111 0,000000 0,000042 -0,061985 0,058678 -0,052521 8,187142
JPY/USD 7545 -0,000141 0,000000 0,000042 -0,095048 0,062556 -0,803652 15,032246
CHF/USD 7552 -0,000127 0,000000 0,000054 -0,044083 0,058269 -0,017725 6,897995
GBP/USD 7552 0,000066 0,000000 0,000035 -0,045885 0,038427 0,140817 7,376703
CZK/USD 2561 0,000115 -0,000057 0,000040 -0,025597 0,082083 0,957220 15,761794
CZK/DEM 2561 -0,000014 -0,000054 0,000018 -0,026942 0,075719 2,569938 47,350066
Weekly
ATS/USD 1461 -0,000542 -0,000236 0,000245 -0,147945 0,136155 -0,121309 11,113815
FRF/USD 1461 0,000008 -0,000236 0,000200 -0,064256 0,074154 0,146678 3,019751
DEM/USD 1461 -0,000535 -0,000387 0,000207 -0,064399 0,083370 0,047985 2,612050
JPY/USD 1571 -0,000708 0,000137 0,000216 -0,117270 0,067829 -0,835678 5,556047
CHF/USD 1571 -0,000609 -0,000370 0,000267 -0,074645 0,095310 -0,109459 2,308039
GBP/USD 1571 0,000320 0,000000 0,000186 -0,066426 0,091300 0,369001 3,430552
CZK/USD 528 0,000556 -0,000674 0,000191 -0,045851 0,079297 0,575058 2,680507
CZK/DEM 528 -0,000069 -0,000405 0,000084 -0,035424 0,077601 1,251247 10,810454
Two-Weekly
ATS/USD 731 -0,001083 -0,000753 0,000494 -0,154106 0,131931 -0,077479 5,138250
FRF/USD 731 0,000016 -0,000073 0,000415 -0,078278 0,087378 0,256978 1,539859
DEM/USD 731 -0,001070 -0,000703 0,000430 -0,079419 0,075318 0,058541 1,073864
JPY/USD 786 -0,001416 0,000000 0,000476 -0,122393 0,089195 -0,754587 3,061085
CHF/USD 786 -0,001217 -0,001109 0,000535 -0,083346 0,084341 -0,151464 0,979900
GBP/USD 786 0,000639 -0,000221 0,000381 -0,086408 0,082989 0,122023 1,875184
CZK/USD 264 0,001111 -0,000341 0,000394 -0,053064 0,079297 0,668604 1,584380
CZK/DEM 264 -0,000139 -0,000561 0,000167 -0,036020 0,080209 1,094593 6,014193

7. 

 4
(rt  ) 4 
Kr  E


(7)
equals to 3. Tables 1 and 2 contain also the point estimates of these parameters for daily,
weekly and two weekly log returns of individual time series. The skewness point estimator
is the following statistics
r
T 
SKr
t1 S3

1 T (rt  r)3
ˆ , (8)
where
T t1
r 
1 T
r
 t r
a S 
T
t
T 
t1
1
(r  r ) 2
, (9)
the kurtosis estimator is the following statistics
r
T 
Kr
t1 S4

1 T (rt  r ) 4
ˆ , (10)
Table 1 shows that in the all time series with exception of the index of Prague stock
market the estimates of skewness are negative and with the growing time aggregation of data
the skewness tends to grow. Regarding the point estimates of the mean are the numbers very
close to zero it is possible to conclude that the distributions are skewed in such way that a
big negative returns are more probable than a big positive returns. Index PX50 has a specific
position as its skewness is positive.
The kurtosis estimates are in all cases the numbers bigger than 3. It means that the real
distributions of daily, weekly and two weekly log returns are more peaked that the normal
distribution, so the low positive and negative returns are more probable that it is expected
under the normality condition. With the growing time aggregation the kurtosis tends to be
lower.
Table 2 contains the basic sample characteristics of daily, weekly and two weekly log
returns of the exchange rates. Even there the distributions are not symmetric, in comparison
with the stock market indexes the skewness is not one-sided, some time series are positively
skewed and some are negatively skewed. The exchange rate CZK/DEM is skewed bigger
than the rest. In contrast with the stock market indexes the time aggregations of the log
returns of the exchange rates do not tend significantly to some change in the skewness. As in
the case of the stock market indexes the kurtosis of the log returns of the exchange rates is

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Picture 11: Hang Seng
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Picture 12: S&P500
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0
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9. Picture 16: S&P500
The above mentioned properties are known a relatively long time (they ware described
by Mandelbrot (1963) a Fama (1965)), the idea to look for the probability distribution which
would catch the properties of the financial time series better than the normal distribution
originated many years ago. It was suggested to apply stable distributions. As a special case
the normal distribution belongs into this class of distributions. This class of distributions was
described by Lévy (1924). The characteristic property of stable random variables is that their
sum has also stable distribution. The non-normal stable distributions catch the high kurtosis,
the non-symmetric shape and the fat tails of the distributions of the financial time series log
returns much better than the normal distribution. This is very closely connected with the
properties of the non-normal stable distributions which have infinite second and higher
moments. The sample variance and kurtosis of the data generated by the non-normal stable
distribution do not converge with the growing sample size. The question of the existence of
variance as well as the question of distribution of returns in the time aggregation split the
financial analytics and researchers into two groups. The opponents of the stable distributions
argument by studies where they try to demonstrate that in the practical examples the variance
converge and that with the growing time period the log returns approach to the normal
distribution.
In this connection some distributions with the final second and higher moments which
catch the properties of the financial time series better than the normal distribution were
introduced. In the last time the mixture of distributions becomes very popular. The log return
might be conditionally normal, conditional on variance parameter which is itself random;
than the unconditional distribution of log returns is a mixture of normal distributions, some
with small conditional variances that concentrate the mass around the mean and others with
large conditional variances that put the mass in the tails of the distribution. The result is a fat-
tailed unconditional distribution with a finite variance and finite higher moments. Since this
property the Central Limit Theorem applies and long- horizon log returns will tend to the
normal distribution.

10. correlated, identically distributed random variables. Their typical property is that only the
conditional means are time dependent, other characteristics of location and variability are
time invariant.
If we take a look on pictures 5 - 10 we can see that changing variability (volatility) is a
common property of log returns of the all pictured time series. Sometimes the volatility
changes in very short time periods so some log returns look like extreme values, sometimes
the volatility stay on certain level for longer time and than changes, it changes in clusters.
These findings are not new, Mandelbrot (1963) was the first who described them.
From the changing log returns volatility a very interesting disclosures follow. It was for
example revealed that the changing log returns volatility can be in a connection with the log
returns mean and autocorrelation. Another disclosure is that a very high volatility frequently
follows the negative log return. This is illustrated by Table 4 which contains
t
sample correlation coefficients of the squares of log returns r 2
and log returns in the first
lags rt-1 and correlation P-values. In the case of the stock market indexes the P-values
indicate that all correlation coefficients are different from zero, the negative values of the
sample correlation coefficients show the above mentioned property. In the case of the
exchange rates the situation is different as the P-values do not indicate in majority cases that
the correlation coefficients are different from zero.
Table 4
Stock market index Correl. Coeff. P-value
Amsterodam (EOE) -0,0495 0,0056
Frankfurt (DAX) -0,0946 0,0000
Paříž (CAC40) -0,0423 0,0271
Londýn (FTSE100) -0,1989 0,0000
Hong Kong (HANG SENG) -0,0806 0,0000
Tokyo (NIKKEI) -0,1299 0,0000
Singapore (SINGAPORE ALL SHARES) -0,1070 0,0000
New York (S&P500) -0,1077 0,0000
Praha (PX50) 0,3653 0,0000
Exchange Rate Correl. Coeff. P-value
ATS/USD -0,1224 0,0000
FRF/USD -0,0014 0,9079
DEM/USD -0,0108 0,3635
JPY/USD -0,0691 0,0000
CHF/USD -0,0079 0,4930
GBP/USD 0,0374 0,0012
CZK/USD -0,0312 0,1147
CZK/DEM -0,0224 0,2567

11. are based on some non-linear function of series of independent, identically distributed
random variables, so they assume more general form of dependence. The general
representation can be expressed in the following form
rt = f(at, at-1, at-2, ...), (11)
where random variables at have zero mean and unit variance. But models used in the practice
are based on more restrictive representation
rt = g(at-1, at-2, ... ) + ath(at-1, at-2, ...). (12)
The function g(.) represents the mean of rt conditional on past information, since
Et-1(rt) = g(at-1, at-2, ...). The function h(.)2
is the variance of rt conditional on past
information, since Et-1[(rt - Et-1(rt)]2
= h(at-1, at-2, ...)2
. Models with non-linear g(.) are said to
be non-linear in mean, models with non-linear h(.)2
are said to be non-linear in variance.
In this connection it is useful to note that the above mentioned mixture of distributions
is the model non-linear in variance. It is evident that the solutions of the problems of non-
normality and non-linearity in the financial time series can be in a very close connection as
the non-normality can be expressed by suitable non-linear models.
References:
1 Arlt, J. (1999): Moderní metody modelování ekonomických časových řad, Grada
Publishing.
2 Bachelier, L. (1900): Theory of Speculation, in Cootner, P. (ed.), The Random
Character of Stock Market Prices, MIT Press, Cambridge, MA, 1964; Reprint.
3 Campbell, J. Y., Lo, E. W., MacKinlay, A. C. (1997): The Econometrics of Financial
Markets, Princeton University Press.
4 Cowles, A. (1933): Can Stock Market Forecasters Forecast? Econometrica, 1, 309-324.
5 Fama, F. (1965): The Behavior of Stock Market Prices, Journal of Business, 38, 34-
105.
6 Fama, F. (1970): Efficient Capital Markets: A Review of Theory and Empirical Work,
Journal of Finance, 25, 383-417.
7 Franses, H. P., van Dijk, D. (2000): Non-Linear Time Series Models in Empirical
Finance.
8 Lévy, P. (1924): Théorie des Erreurs. La Loi de Gauss et Les Lois Exceptionelles, Bull.
Soc. Math., 52, 49-85.
9 Malkiel, B. (1992): Efficient Market Hypothesis, in Newman, P., M. Milgate, and
J. Eatwell (eds.), New Palgrave Dictionary of Money and Finance, Macmillan, London.
10 Mandelbrot, B. (1963): The Variation of Certain Speculative Prices, Journal of

12. Thank you