2.1 Introduction

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.


2.1.1 Analysis of Distribution of the Extremum

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:

$\displaystyle \mathop {\lim }\limits_{n\to \infty } P\left( {\frac{X_{n:n} -b_n }{a_n }\le x} \right)=G(x),$ (2.1)

where $ X_{n:n} =\max (X_1 ,X_2 ,...,X_n )$. It can be proved that this limiting distribution belongs to the family of the so-called Generalized Extreme Value distributions (GEV), whose distribution function is given as:
    $\displaystyle G(x)=\exp \left\{ {-\left[ {1+\xi \left( {\frac{x-\mu }{\sigma
}} \right)} \right]^{-1/\xi }} \right\},$  
      (2.2)
    $\displaystyle 1+\xi
\sigma ^{-1}(x-\mu )>0.$  

The GEV distribution has three parameters Reiss and Thomas (2000): the location parameter $ \mu$, the scale parameter $ \sigma$, and the shape parameter $ \xi $, which reflects the fatness of tails of the distribution (the higher value of this parameter, the fatter tails).

The family of GEV distributions contains three subclasses: the Fréchet distribution, $ \xi > 0$, the Weibull distribution, $ \xi < 0$, and the Gumbel distribution, $ \xi \to 0$. 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.


2.1.2 Analysis of Conditional Excess Distribution

To analyze rare events, another approach can be used. Consider the so-called conditional excess distribution:

$\displaystyle F_u (y)=P(X-u\le y\left\vert {X>u} \right.)=\frac{F(u+y)-F(u)}{1-F(u)},$ (2.3)

where $ 0\le y<x_0 -u$; and $ x_0 =\mathop {\sup } (x:F(x)<1)$. This distribution (also called the conditional tail distribution) is simply the distribution conditional on the underlying random variable taking value from the tail. Of course, this distribution depends on the choice of threshold $ u$.

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):

$\displaystyle F_u (y)=1-(1+\xi y/\beta )^{-1/\xi },$ (2.4)

where $ \beta =\sigma +\xi (u-\mu )$. The shape parameter $ \xi $ has the same role as in GEV distributions. The generalized parameter $ \beta$ depends on all three parameters of the GEV distribution, as well as on the threshold $ u$.

The family of GPD contains three types of distributions, the Pareto distribution - $ \xi > 0$, the Pareto type II distribution - $ \xi < 0$, and the exponential distribution - $ \xi \to 0$.

The mean of the conditional excess distribution can be characterized by a linear function of the threshold and of the parameters of GPD:

$\displaystyle \mathop{\textrm{E}}(X-u\left\vert {X>u} \right.)=\frac{\beta u}{1-\xi }+\frac{\xi }{1-\xi }u$ (2.5)

for $ \xi < 1$. One of the most common methods of estimating the parameters of GPD is maximum likelihood. However, the GPD depends on the choice of the threshold $ u$. The higher the threshold, the better the approximation of the tail by GPD - this is a desired property. Then one has fewer observations to perform maximum likelihood estimation, which weakens the quality of estimation.

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:

$\displaystyle \mathop e\limits^\wedge (u)=\frac{\sum\limits_{i=1}^n {\max\{(x_i -u), 0\}} }{\sum\limits_{i=1}^n {I(x_i >u)} }.$ (2.6)

We know that for the observations higher than the threshold this relation should be a linear function. Therefore a graphical procedure can be applied. In this procedure the value of $ \mathop
e\limits^\wedge (u)$ is calculated for different values of the threshold $ u$. Then such a value is selected, that for the values above this value the linear relation can be observed.


2.1.3 Examples

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.

Figure 2.1: Histograms of the logarithmic rates of return for WSE indices
\includegraphics[width=0.7\defpicwidth]{STFeva03.ps}\includegraphics[width=0.7\defpicwidth]{STFeva04.ps} \includegraphics[width=0.7\defpicwidth]{STFeva05.ps}\includegraphics[width=0.7\defpicwidth]{STFeva06.ps}

Figure 2.2: Histograms of the logarithmic rates of return for world indices
\includegraphics[width=0.7\defpicwidth]{STFeva07.ps}\includegraphics[width=0.7\defpicwidth]{STFeva08.ps} \includegraphics[width=0.7\defpicwidth]{STFeva09.ps}\includegraphics[width=0.7\defpicwidth]{STFeva10.ps}

Figure 2.3: Histograms of the logarithmic rates of return for exchange rates
\includegraphics[width=0.7\defpicwidth]{STFeva11.ps}\includegraphics[width=0.7\defpicwidth]{STFeva12.ps} \includegraphics[width=0.7\defpicwidth]{STFeva13.ps}

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 - $ ES$ (also known as conditional Value at Risk, expected tail loss). It is defined as:

$\displaystyle ES=\mathop{\textrm{E}}(X-u\left\vert {X>u} \right.).$ (2.7)

So $ ES$ is the expected value of the conditional excess distribution. Therefore the GPD could be used to determine $ ES$.

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).

Table 2.1: The estimates of the parameters of GEV distributions, for the stock indices.
Data $ \xi $ $ \mu$ .$ \sigma$ . 
    .    .    . 
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

4318 STFeva02.xpl

The analysis of the results for stock indices leads to the following conclusions. In most cases we obtained the Fréchet distribution (estimate of the shape parameter is positive), which suggests that underlying observations are characterized by a fat-tailed distribution. For FTSE-100 and DAX indices the estimate of $ \xi $ is negative but close to zero, which may suggest either a Weibull distribution or a Gumbel distribution. In the majority of cases, the WSE indices exhibit fatter tails than the other indices. They also have larger estimates of location (related to mean return) and larger estimates of the scale parameter (related to volatility).

Table 2.2: The estimates of the parameters of GEV distributions, for the exchange rates.
Data $ \xi $ $ \mu$ .$ \sigma$ . 
    .    .    . 
USD/PLN 0 .046 0 .014 0 .005
EUR/PLN 0 .384 0 .015 0 .005
EUR/USD -0 .213 0 .014 0 .004

4324 STFeva03.xpl

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.