The analysis of financial data, usually given in the form of financial time series, has recently received a lot of attention of researchers and finance practitioners, in such areas as valuation of derivative instruments, forecasting of financial prices, risk analysis (particularly market risk analysis).
From the practical point of view, multivariate analysis of financial data may be more appropriate than univariate analysis. Most market participants hold portfolios containing more than one financial instrument. Therefore they should perform analysis for all components of a portfolio. There are more and more financial instruments where payoffs depend on several underlyings (e.g. rainbow options). Therefore, to value them one should use multivariate models of underlying vectors of indices. Risk analysis is strongly based on the issue of correlation, or generally speaking dependence, between the returns (or prices) of the components of a portfolio. Therefore multivariate analysis is an appropriate tool to detect these relations.
One of the most important applications of financial time series models is risk analysis, including risk measurement. A significant tendency, observed in the market, is the occurrence of rare events, which very often lead to exceptionally high losses. This has caused a growing interest in the evaluation of the so-called extreme risk. There are two groups of models applied to financial time series: ``mean-oriented'' models, aiming at modeling the mean (expected value) and the variance of the distribution; and ``extreme value'' models, aiming at modeling tails (including maximum and minimum) of the distribution.
In this chapter we present some methods of time series analysis, both univariate and multivariate time series. The attention is put on two approaches: extreme value analysis and copula analysis. The presented methods are illustrated by examples coming from the Polish financial market.
The analysis of the distribution of the extremum is simply the analysis of the random variable, defined as the maximum (or minimum) of a set of random variables. For simplicity we concentrate only on the distribution of the maximum. The most important result is the Fisher-Tippet theorem (Embrechts, Klüppelberg, and Mikosch; 1997). In this theorem one considers the limiting distribution for the normalized maximum:
The family of GEV distributions contains three subclasses: the
Fréchet distribution, , the Weibull distribution,
, and the Gumbel distribution,
. In financial
problems one usually encounters the Fréchet distribution. In
this case the underlying observations come from a fat-tailed
distribution, such as the Pareto distribution, stable distribution
(including Cauchy), etc.
One of the most common methods to estimate the parameters of GEV distributions is maximum likelihood. The method is applied to block maxima, obtained by dividing the set of observations into subsets, called blocks, and taking maximum for each block.
The main weakness of this approach comes from the fact that the maxima for some blocks may not correspond to rare events. On the other hand, in some blocks there may be more than one observation corresponding to rare events. Therefore this approach can be biased by the selection of the blocks.
To analyze rare events, another approach can be used. Consider the so-called conditional excess distribution:
It can be proved (Embrechts, Klüppelberg, and Mikosch; 1997) that the conditional excess distribution can be approximated by the so-called Generalized Pareto distribution (GPD), which is linked by one parameter to the GEV distribution. The following property is important: the larger the threshold (the further one goes in the direction of the tail), the better the approximation. The distribution function of GPD is given by Franke, Härdle and Hafner (2004) and Reiss and Thomas (2000):
The family of GPD contains three types of distributions, the
Pareto distribution - , the Pareto type II distribution
-
, and the exponential distribution -
.
The mean of the conditional excess distribution can be characterized by a linear function of the threshold and of the parameters of GPD:
To choose the threshold, one can use the procedure, based on the fact that for GPD the mean of the conditional excess distribution is a linear function of the threshold. Therefore, one can use the following function, which is just the arithmetic average of the observations exceeding the threshold:
Consider the logarithmic rate of returns for the following stock market indices:
In addition we studied the logarithmic rates of return for the following exchange rates: USD/PLN, EUR/PLN, EUR/USD.
The financial time series of the logarithmic rates of return come from the period January 2, 1995 - October 3, 2003, except for the case of exchange rates EUR/PLN and EUR/USD, where the period January 1, 1999 - October 3, 2003 was taken into account. Figures 2.1-2.3 show histograms of those time series.
The most common application of the analysis of the extremum is the estimation of the maximum loss of a portfolio. It can be treated as a more conservative measure of risk than the well-known Value at Risk, defined through a quantile of the loss distribution (rather than the distribution of the maximal loss). The limiting distribution of the maximum loss is the GEV distribution. This, of course, requires a rather large sample of observations coming from the same underlying distribution. Since most financial data are in the form of time series, the required procedure would call for at least the check of the hypothesis about stationarity of time series by using unit root test e.g. Dickey-Fuller test, (Dickey and Fuller; 1979). The hypothesis of stationarity states that the process has no unit roots. With the Dickey-Fuller test we test the null hypothesis of a unit root, that is, there is a unit root for the characteristic equation of the AR(1) process. The alternative hypothesis is that the time series is stationary. To verify stationarity hypotheses for each of the considered time series, the augmented Dickey-Fuller test was used. The hypotheses of a unit root were rejected with the level of significance lower than 1%, so all time series in question are stationary.
One of the most important applications of the analysis of
conditional excess distribution is the risk measure called
Expected Shortfall - (also known as
conditional Value at Risk, expected tail loss). It is defined as:
Then for each time series the parameters of GEV distributions were
estimated using maximum likelihood method. The results of the
estimation for GEV are presented in Table 2.1 (for
stock indices) and in Table 2.2 (for exchange
rates).
Data | ![]() |
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. | ![]() |
. | ||||
. | . | . | |||||||
WIG | 0 | . | 374 | 0 | . | 040 | 0 | . | 012 |
WIG20 | 0 | . | 450 | 0 | . | 037 | 0 | . | 022 |
MIDWIG | 0 | . | 604 | 0 | . | 033 | 0 | . | 011 |
TECHWIG | 0 | . | 147 | 0 | . | 066 | 0 | . | 012 |
. | . | . | |||||||
DJIA | 0 | . | 519 | 0 | . | 027 | 0 | . | 006 |
S&P 500 | 0 | . | 244 | 0 | . | 027 | 0 | . | 007 |
FT-SE 100 | -0 | . | 048 | 0 | . | 031 | 0 | . | 006 |
DAX | -0 | . | 084 | 0 | . | 041 | 0 | . | 011 |
Data | ![]() |
![]() |
. | ![]() |
. | ||||
. | . | . | |||||||
USD/PLN | 0 | . | 046 | 0 | . | 014 | 0 | . | 005 |
EUR/PLN | 0 | . | 384 | 0 | . | 015 | 0 | . | 005 |
EUR/USD | -0 | . | 213 | 0 | . | 014 | 0 | . | 004 |
The analysis of the results for the exchange rates leads to the
following conclusions. Three different distributions were
obtained, for USD/PLN - a Gumbel distribution, for EUR/PLN - a
Fréchet distribution, for EUR/USD - a Weibull distribution.
This suggests very different behavior of underlying observations.
The location and scale parameters are almost the same. The scale
parameters are considerably lower for the exchange rates than for
the stock indices.