FINANCIAL TIME SERIES AND THEIR FEATURES

Acta oeconomica pragensia 9: (4), str. 7-20, VŠE Praha, 2001. ISSN 0572-3043. FINANCIAL TIME SERIES AND THEIR FEATURES Josef ARLT, Markéta ARLTOVÁ* U...
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Acta oeconomica pragensia 9: (4), str. 7-20, VŠE Praha, 2001. ISSN 0572-3043.

FINANCIAL TIME SERIES AND THEIR FEATURES Josef ARLT, Markéta ARLTOVÁ* University of Economics Prague, Czech Republic

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. 3200 2800 2400 2000 1600 1200 800 0

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This paper was written with the support of grant No. 402/00/0459 of Grant Agency of the Czech Republic.

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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)

E[Pt+1- PtPt, Pt-1, ...] = 0.

(2)

or as

The martingale assumes that non-overlaping price changes in the all leads and lags are non-correlated i.e. linear independent. It does not mean that these price changes could not be dependent. The non-linear dependency can be in such form that price changes are not constant in time. In the past it was assumed that the martingale is the necessary condition of the efficient market. Now, in connection with the hypothesis of the efficient market there is not only the non-ability to predict important but also the relationship of the expected price change and risk (the variance of price changes). From the point of view of investor the positive expected price change with relatively high risk can be interesting also. From the point of view of the market the future price change is still unpredictable which means that the market is efficient. As the martingale puts the restriction only on the expected price 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, σ2). 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 Rt(k) + 1 = (Rt + 1) . (Rt-1 + 1) . ... . (Rt-k+1 + 1) =

Pt Pt −1 Pt − 2 Pt − k +1 P . . ... = t . (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 and rt(k) is normally distributed.

(5)

Pictures 5-10 show the shape of some daily financial time series and their simple gross returns in log transformation rt (further only log returns): Picture 5 - the index of the stock market in Amsterdam (EOE) from 1 June 1986 until 31 December 1997, Picture 6 - the index of the stock market in Tokyo (Nikkei) from 1 June 1986 until 31 December 1997, Picture 7 - the index of the stock market in Prague (PX50) from 7 September 1993 until 13 February 2001, Picture 8 - the exchange rate ATS/USD from 4 January 1971 until 31 December 1998, Picture 9 - the exchange rate GBP/USD from 4 January 1971 until 9 February 2001, Picture 10 - the exchange rate CZK/DEM from 1 January 1991 until 14 February 2001. 1200 1000 800 600 400 200 0 0

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Picture 10: CZK/USD In Tables 1 and 2 there are the basic sample characteristics of daily, weekly and two weekly log returns of the above mentioned time series and of the following time series: the indexes of the stock markets in Frankfurt (DAX), London (FTSE100), Hong Kong (Hang Seng), Singapore (Singapore All Shares), New York (S&P500) from 1 June 1986 until 31 December 1997, the exchange rates FRF/USD from 4 January 1971 until 31 December 1998, CHF/USD from 4 January 1971 until 9 February 2001, JPY/USD from 4 January 1971 until 9 February 2001. These tables will be used for description of the characteristic properties of the financial time series.

Table 1 n

r

~ r

3127 3127 2734 3127 3127 3127 3127 3127 1700 626 626 547 626 626 626 626 626 388 313 313 274 313 313 313 313 313 194

0,000383 0,000349 0,000258 0,000410 0,000571 0,000050 0,000192 0,000489 0,000212 0,001916 0,001742 0,001278 0,002059 0,002837 0,000257 0,000934 0,002416 0,000931 0,003831 0,003483 0,002552 0,004118 0,005674 0,000514 0,001868 0,004833 0,001862

0,000293 0,000258 0,000000 0,000274 0,000222 0,000000 0,000000 0,000384 -0,000206 0,003510 0,003810 0,001433 0,002979 0,005308 0,000622 0,001146 0,003884 0,000108 0,005965 0,008112 0,006552 0,004875 0,014598 -0,000144 0,002801 0,004985 -0,003211

Two-Weekly

Weekly

Daily

Stock Market Index Amsterodam (EOE) Frankfurt (DAX) Paříž (CAC40) Londýn (FTSE100) Hong Kong (HANG SENG) Tokyo (NIKKEI) Singapore (SINGALLS) New York (S&P500) Praha (PX50) Amsterodam (EOE) Frankfurt (DAX) Paříž (CAC40) Londýn (FTSE100) Hong Kong (HANG SENG) Tokyo (NIKKEI) Singapore (SINGALLS) New York (S&P500) Praha (PX50) Amsterodam (EOE) Frankfurt (DAX) Paříž (CAC40) Londýn (FTSE100) Hong Kong (HANG SENG) Tokyo (NIKKEI) Singapore (SINGALLS) New York (S&P500) Praha (PX50)

Table 2

Two-Weekly

Weekly

Daily

Exchange Rate ATS/USD FRF/USD DEM/USD JPY/USD CHF/USD GBP/USD CZK/USD CZK/DEM ATS/USD FRF/USD DEM/USD JPY/USD CHF/USD GBP/USD CZK/USD CZK/DEM ATS/USD FRF/USD DEM/USD JPY/USD CHF/USD GBP/USD CZK/USD CZK/DEM

n

r

~ r

7006 7020 7020 7545 7552 7552 2561 2561 1461 1461 1461 1571 1571 1571 528 528 731 731 731 786 786 786 264 264

-0,000105 0,000002 -0,000111 -0,000141 -0,000127 0,000066 0,000115 -0,000014 -0,000542 0,000008 -0,000535 -0,000708 -0,000609 0,000320 0,000556 -0,000069 -0,001083 0,000016 -0,001070 -0,001416 -0,001217 0,000639 0,001111 -0,000139

0,000000 0,000000 0,000000 0,000000 0,000000 0,000000 -0,000057 -0,000054 -0,000236 -0,000236 -0,000387 0,000137 -0,000370 0,000000 -0,000674 -0,000405 -0,000753 -0,000073 -0,000703 0,000000 -0,001109 -0,000221 -0,000341 -0,000561

sr2 0,000128 0,000152 0,000144 0,000084 0,000287 0,000184 0,000102 0,000099 0,000231 0,000667 0,000784 0,000685 0,000497 0,001602 0,000842 0,000704 0,000503 0,001607 0,001504 0,001670 0,001531 0,001160 0,003923 0,001674 0,001697 0,000891 0,003403

sr2 0,000054 0,000040 0,000042 0,000042 0,000054 0,000035 0,000040 0,000018 0,000245 0,000200 0,000207 0,000216 0,000267 0,000186 0,000191 0,000084 0,000494 0,000415 0,000430 0,000476 0,000535 0,000381 0,000394 0,000167

rmin -0,127876 -0,137099 -0,101376 -0,130286 -0,405422 -0,161354 -0,094030 -0,228330 -0,075664 -0,235184 -0,177817 -0,119715 -0,265825 -0,338968 -0,186441 -0,221259 -0,283705 -0,154722 -0,285302 -0,258932 -0,180989 -0,321933 -0,472122 -0,151510 -0,338701 -0,299194 -0,190649

rmin -0,139503 -0,060490 -0,061985 -0,095048 -0,044083 -0,045885 -0,025597 -0,026942 -0,147945 -0,064256 -0,064399 -0,117270 -0,074645 -0,066426 -0,045851 -0,035424 -0,154106 -0,078278 -0,079419 -0,122393 -0,083346 -0,086408 -0,053064 -0,036020

rmax 0,111785 0,072875 0,082254 0,075970 0,172471 0,124303 0,143130 0,087089 0,153905 0,166433 0,117060 0,099044 0,087222 0,173908 0,135191 0,120032 0,072882 0,232827 0,162647 0,132930 0,105412 0,100945 0,143702 0,116835 0,115639 0,085946 0,246683

rmax 0,135003 0,058746 0,058678 0,062556 0,058269 0,038427 0,082083 0,075719 0,136155 0,074154 0,083370 0,067829 0,095310 0,091300 0,079297 0,077601 0,131931 0,087378 0,075318 0,089195 0,084341 0,082989 0,079297 0,080209

SKr -0,692960 -0,946464 -0,529315 -1,589870 -5,003890 -0,212871 -0,247148 -4,299805 1,741261 -1,621082 -1,099043 -0,306842 -2,849465 -2,352888 -0,451233 -1,321831 -3,507218 0,573693 -2,157977 -1,583310 -0,680883 -2,696413 -3,136225 -0,295654 -2,284417 -3,358235 1,038067

SKr 0,050822 0,017741 -0,052521 -0,803652 -0,017725 0,140817 0,957220 2,569938 -0,121309 0,146678 0,047985 -0,835678 -0,109459 0,369001 0,575058 1,251247 -0,077479 0,256978 0,058541 -0,754587 -0,151464 0,122023 0,668604 1,094593

Kr 19,801817 15,070425 10,563364 27,416576 119,27899 14,802470 28,155480 99,711844 19,177069 16,151282 5,665650 1,147525 33,319316 17,746855 4,272873 11,573267 41,822356 4,626460 15,108132 8,170425 1,609288 26,252886 19,873742 0,936578 16,168181 33,456870 4,050073

Kr 47,540368 10,343803 8,187142 15,032246 6,897995 7,376703 15,761794 47,350066 11,113815 3,019751 2,612050 5,556047 2,308039 3,430552 2,680507 10,810454 5,138250 1,539859 1,073864 3,061085 0,979900 1,875184 1,584380 6,014193

2.1 The normality assumption One of the basic assumptions which is adopted by theoretical and empirical studies concerning the analysis of the financial time series is the normality of the log returns with constant mean µ and variance σ2, i.e. rt ∼ N(µ,σ2). This distribution is symmetric so the skewness  (r − µ ) 3  SK r = E  t 3    σ

(6)

 (r − µ ) 4  K r = E t 4    σ

(7)

equals to zero and kurtosis

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 SKˆ r =

1 T

T

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,

(8)

where r=

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a Sr =

1 T

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(9)

t =1

the kurtosis estimator is the following statistics 1 Kˆ r = T

T

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S r4



,

(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 bigger than 3, with the growing time aggregation the kurtosis tends to be lower.

Pictures 11-14 show the shapes of distributions of daily log returns of the indexes of the stock markets in Hong Kong (Hang Seng), New York (S&P500) and the exchange rates FRF/USD, JPY/USD. Pictures 11 - 14: The Shapes of Real Distributions and Normal Distributions 50

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Picture 14: JPY/USD These pictures show that the real distributions of daily, weekly and two weekly returns are skewed and more peaked in comparison with the normal distribution. These distributions have also fat tails which means that the probabilities of extremely high and extremely low returns are bigger than in the case of the normal distribution. This property is seen on Pictures 15 and 16, where the variability of daily, weekly and two weekly log returns of exchange rate FRF/USD and index S&P500 as well as the variability of values generated by normal distributions are showed. It is also seen that with the growing time aggregation the real variability tends to be closer to the normal variability. It follows that in the real life a very low returns (close to zero) and a very high returns (negative or positive) are more probable than the normal distribution suggests.

Picture 15: FRF/USD

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 longhorizon log returns will tend to the normal distribution.

2.2 The linearity assumption The second basic assumption in the classical analysis of the financial time series is that the log returns are non-correlated, identically distributed with zero mean and constant variance or independent, identically distributed with zero mean and constant variance. In the first case it is the white noise process and in the second one the strict white noise process. But the reality is more complex. The condition of zero mean is not usually fulfilled. But more serious problem is the condition of non-correlation in log returns. In many cases also this condition is not fulfilled. The linear dependency can be expressed by linear models of the class ARMA. These models are based on the so called linear processes which are created by linear combinations of non-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 sample correlation coefficients of the squares of log returns rt2 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 Amsterodam (EOE) Frankfurt (DAX) Paříž (CAC40) Londýn (FTSE100) Hong Kong (HANG SENG) Tokyo (NIKKEI) Singapore (SINGAPORE ALL SHARES) New York (S&P500) Praha (PX50)

Correl. Coeff. -0,0495 -0,0946 -0,0423 -0,1989 -0,0806 -0,1299 -0,1070 -0,1077 0,3653

P-value 0,0056 0,0000 0,0271 0,0000 0,0000 0,0000 0,0000 0,0000 0,0000

Exchange Rate ATS/USD FRF/USD DEM/USD JPY/USD CHF/USD GBP/USD CZK/USD CZK/DEM

Correl. Coeff. -0,1224 -0,0014 -0,0108 -0,0691 -0,0079 0,0374 -0,0312 -0,0224

P-value 0,0000 0,9079 0,3635 0,0000 0,4930 0,0012 0,1147 0,2567

The properties of the financial time series follow directly from the economic body of market and from the behaviour of economic agents - investors. The behaviour of these investors is based not only on the levels of the economic variables but also on the risks and on the interaction of these factors. The characteristic features of the financial time series can not be expressed by linear models as these models suppose only one type of dependence, correlation dependence. On the other hand the class of non-linear models consider the other types of dependence. In contrast to linear models whose construction is based on the linear combination of noncorrelated, identically distributed random variables which have not to be necessary independent (but linear combination creates only correlation dependence), non-linear models 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 nonnormality 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.

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Abstract Time series of prices as well as time series based on prices or time series which describe prices and their dynamism are called financial time series. These time series have some typical properties. There are two basic assumptions: normality and linearity of log returns of the financial time series. The distributions of log returns are usually skewed and more peaked that the normal distribution. The characteristic features of these time series can not be expressed by linear models as they suppose only a correlation dependence. The solution of the problems of non-normality and non-linearity in the financial time series can be in a very close connection.

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