Risk management: an analysis of the low-tail behavior of high frequency data for computing value at risk.

ABSTRACT

This paper deals with the analysis of the low-tail behaviour of a sample. This issue is very important in the financial field, where the fat-tailedness characterizes many series. In particular, the tail behaviour affects the estimates of expected losses. The aim of this paper is to analyze the performance of the bootstrap methodology to implement goodness-of-fit tests for a certain proportion of a sample based on the modified Cramer-von Mises statistics, assuming the presence of unknown parameters. An empirical application is carried out using the SP500 index for the period from January 1994 to December 2002.

1. INTRODUCTION

One common assumption in the financial literature is the hypothesis of normality. However, a feature which stands out most prominently is that the kurtosis of financial series is much larger than the normal value, especially for the daily series. This reflects the fact that the tails of the distributions of these series are fatter than the tails of the normal distribution. Put differently, large observations occur much more often than one might expect for a normally distributed variable (see, for example, de Vries, 1994, for foreign exchange markets and Campbell et al., 1997, for stock market).

Given the fat-tailedness observed for many financial time-series, the use of the normal distribution is questionable (Praetz, 1972; Kon, 1994; Peiro, 1994, between others). The description of the distribution of the daily stock-returns is a subject of continuous analysis in the financial literature. However, less has been said about the analysis of the behaviour of a certain proportion of the sample. In particular, the analysis of the low-tail behaviour of the sample is very important for risk management. Some sources of risk affecting the performance of financial policies include interest rate risk, exchange rate risk and credit risk. Many different risk measures have been proposed and have been used for investment decisions, supervisory decisions, risk capital allocation, external regulation and efficient bank operations. In the context of asset management, the most popular downside-risk measure is the Value-at-Risk (VaR). The tail behaviour of the distribution of asset returns is relevant for computing VaR due to that the calculation of VaR deals with the estimation of the lower quantiles of a distribution.

The aim of this paper is to analyze the performance of the bootstrap methodology to implement goodness-of-fit tests of a certain proportion of a sample with a Cramer-von Mises type statistic, when the distribution function is known completely except by a vector of unknown parameters, and the test statistic is constructed with the standarized residuals of a linear regression model. The paper is organized as following. Section 2 describes the modified Cramer-von Mises test statistic for censored data. Section 3 presents a simulation experiment to observe the performance of the designed bootstrap procedures to carry out the goodness-of-fit tests with the modified Cramer-von Mises test statistic. In Section 4, empirical evidence is provided using the SP500 index from January 1994 to December 2002. Section 5 concludes.

2. GOODNESS-OF-FIT TESTS FOR CENSORED DATA

It is well-known the fact that, if it is wished to test the null hypothesis [H.sub.0]: F(x)=[F.sub.o](x), where [F.sub.o](x) is a completely specified distribution function, the traditional Cramer-von Mises test statistic (hereafter, [W.sup.2]) can be used. The asymptotic distribution of [W.sup.2sub.n] is distribution free. Pettitt and Stephens (1976) modified the [W.sup.2.sub.n] so that it could be used to test the goodness of fit of a censored data; that is, to test the goodness of fit of a certain proportion of the random sample when the distribution function is known completely, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Also, they obtained the asymptotic distribution of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] from a straightforward extension of those of Anderson and Darling (1952). However, to assume that the null distribution function is known completely is a very restrictive assumption. For this reason, numerous articles have appeared in the literature related with the asymptotic properties of the Cramer-von Mises type statistics when the null distribution function is F(x,[theta]), a known distribution function but [theta] is a vector of unknown parameters,

[[??]W.sup.2.sub.n]. A comprehensive analysis of the basic theory about the goodness of fit tests when parameters are estimated has been given by Durbin (1973). His main result is that the limit distribution function is no longer free but it depends on the postulated null distribution and, in general on [theta]. It should be also noted that the tabulated asymptotic critical values (Shorack and Wellner, 1986) are inappropriate when parameters of the hypothesized distribution are estimated from the data used for the test. Pettitt (1976) applied Durbin's (1973) results when [theta] is estimated by maximum likelihood procedure from censored data and he derived the modified version of [[??].sup.2.sub.n], denoted [.sub.q][[??].sup.2.sub.n], and its asymptotic properties.

Koul (1991) presented results about the goodness-of-fit tests when parameters are estimated and the observations are the errors of a linear regression model. Owing to [theta] is a vector of unknown parameters and the errors of the linear regression model are not observable random variables the test statistic must be constructed substituting [theta] by its estimate [theta] and using the standardized residuals of the model, denoted by e. In this case, the Cramer-von Mises test statistic is given by:

(1) [[bar.W].sup.2.sub.n] = [n.summation over (i=1)] [{[[??].sub.n] ([e.sub.i]) - F([e.sub.i] [??])}.sup.2]

where [[??].sub.n] is the empirical distribution function of the standardized residuals, [theta] is the maximum likelihood estimate of [theta] and F(x ,[theta]) is the estimated null distribution. Following Pettitt (1976), [[bar.W].sup.2.sub.n] is straightforwardly modified as:

(2) [sub.q][[bar.W].sup.2.sub.n] = [[n.sub.q].summation over (i=1]) [{[[??].sub.n]([e.sub.i]) - F([e.sub.i], [??])}.sup.2]

where [n.sub.q] denotes the number of data of the fixed proportion q of the sample.

3. BOOTSTRAP IMPLEMENTATION OF THE GOODNESS-OF-FIT TESTS FOR CENSORED DATA

We denote the univariate time series of interest as [y.sub.i], where [y.sub.i] could be a return on a financial asset and [y.sub.i] is observed for i = 1, ..., n. We consider [y.sub.i] = [mu] + [[epsilon].sub.i], = [mu] + [sigma][u.sub.i], where [u.sub.1], ..., [u.sub.n] are unobservable independent and identically distributed random variables sampled from a specific distribution function, with zero mean and unitary variance. We wish to test:

[H.sub.0]: The distribution of [u.sub.i] is [F.sub.q] (x; [mu], [[sigma].sup.2], [theta]);

[H.sub.1]: The distribution [u.sub.i], is not of this type,

x [member of] R, i = 1 , ..., [q.sub.n], q is a certain proportion of a sample, n is the sample size and [theta] [member of] [R.sup.n]. Given that [u.sub.i] = [[epsilon].sub.i]/[sigma], this is equivalent to test:

[H.sub.0]: The distribution of [u.sub.i] is [F.sub.q,u] (x,[theta]) ; [H.sub.1]: The distribution of [u.sub.i] is not of this type.

In financial literature, [[epsilon].sub.i] is a measure of news at time i. A positive [[epsilon].sub.i] suggests the arrival of good news, while a negative [[epsilon].sub.i] suggests the arrival of bad news.

In our analysis, we distinguish two cases: (1) Case 1: [[epsilon].sub.i] is sampled from a specific distribution function, with zero mean and a certain constant variance [[sigma].sup.2.sub.0] ; (2) Case 2: [[epsilon].sub.i] is sampled from a specific distribution function, with zero mean and a time-varying variance [[sigma].sup.2.sub.i]. That is, in Case 2 we consider the presence of conditional heteroskedasticity in the errors of the linear regression model. In particular, Case 2 is relevant for financial analyses due to the importance of the analysis of the conditional distribution instead of the unconditional variance to predict the return of the assets when they present dynamic (Baixauli and Alvarez, 2004). Many approaches have been proposed to modelate the dynamic structure of the variance. The most popular one is the class of autoregressive conditional heteroskedasticity (ARCH) models, introduced by Engle (1982). Bollerslev (1986) generalized the ARCH models to the GARCH models, which are capable of describing the feature of volatility clustering and other characteristics of financial time series. GARCH(1,1) model often appears adequate in practice (see Bollerslev et al., 1992).

3.1 Bootstrap Procedure Assuming Constant Variance

In Case 1, we distinguish the following stages in the bootstrap procedure:

Stage 1: Consider a univariate time series of interest [y.sub.i], i = 1, ..., n, following the equation [y.sub.i] = [mu] + [[sigma.sub.0] [[mu].sub.i], where [u.sub.1] , ..., [u.sub.n], are unobservable independent and identically distributed random variables sampled from a postulated null distribution function [F.sub.u], (x,[theta]), with zero mean and unitary variance. Estimate [mu], [[sigma].sub.0],[theta] by quasi-maximum likelihood procedure, using the n observations of the original sample. These estimates are denoted by [??], [??], [??].

Stage 2: Construct the statistic [sub.q][[??].sup.2.sub.n] using the vector of standardized residuals of the model [[eta].sub.1] , ..., [[eta].sub.n],

where [[eta].sub.i] = ([y.subi] - [??])/[??], and substituting [theta] by [theta].

Stage 3: Draw B=200 bootstrap samples of size n from the estimated null distribution [F.sub.u] (x, [??]). Use these bootstrap samples and the estimates /~,~-0,0 to construct B bootstrap samples [y.sup.*.sub.bl] , ..., [y.sup.*.sub.bn], b = 1, ..., B = 1, ..., B. For each of these series, compute the corresponding vector of standardized residuals [[eta].sup.*.sub.bl] , ..., [[eta].sup.*.sub.bn], obtain the new estimate [[??].sup.*] and evaluate [sub.q][[??].sup.2*.sub.n]. In this way, a sample of B independent (conditional to the original sample) observations of [sub.q][W.sup.2.sub.n] is obtained; i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Stage 4: Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the (1 - [alpha])B-th order statistic of the sample [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], given a significance level [alpha]. Reject the null hypothesis at the significance level [alpha]~ if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Stage 5: Compute the bootstrap p-value as [P.sub.B] = card([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

3.2 Simulation Results Assuming Constant Variance

To assess for the validity of the designed bootstrap procedure in Case 1, we have conducted the following simulation experiment:

We draw a sample [y.sub.1] , ..., [y.sub.n] of size n = 1000 from a normal distribution, considering [mu] = 0 and [[sigma].sup.2.sub.0] = 1. For different significance levels, [alpha] = 10% and [alpha] = 5%, we test the null hypothesis of normality with unknown mean and variance versus the alternative of no normality. We repeat the test R=100 times and compute the empirical size of the test. In order to see the power of the test, we implement the test generating the original sample from a Student t distribution function, with zero mean, unitary variance and g=4 degrees of freedom. We make use of the Student t due to this distribution function captures some stylised facts of high-frequency financial time series and it is widely use in financial studies as a fat-tailed alternative for the normal distribution. In Figure 1, we plot the quantiles of the standard normal distribution against the quantiles of the Student's t distribution with zero mean, unitary variance and 4 degrees of freedom to compare both distributions.

As the QQ-plot does not lie on a straight line, the two distributions differ along some dimension. In particular, the QQ-plot falls on a straight line in the middle but curves upward at the left end and curves downward at the right end. This fact indicates that the Student's t distribution is leptokurtic.

The rejection percentages of the null hypothesis of normality assuming constant variance are reported in Table 1.

The rejection percentages are based on the 5% and 10% significance levels for the 25% and 15% of the random sample, a sample size of 1000 observations and constant variance. Under the null hypothesis of normality, a well-specified test statistic should have an empirical level close to the significance level. As Table 1 shows, the empirical levels are rather close to the significance level in all cases. In order to check if the differences between these levels are statistically significant or not, we use the significance test statistic Z, given by

(3) Z = [absolute value of p - [alpha]]/[square root of [alpha](1 - [alpha])/R]

where p is a rejection rate, ct is the significance level and R is the number of repetitions. The test statistic is well-specified if p [member of] [1.41%, 8.58%] and p [member of] [5.06%, 15.88%], for R=100 and [alpha]=5% and [alpha]=10%, respectively. Hence, given our results, it can be stated that the test statistic is well-specified. Moreover, when we generated under the Student's t distribution with zero mean, unitary variance and 4 degrees of freedom, the rejection percentages of the hypothesis of normality is close to 100%; that is, the power of the test is high.

3.3 Bootstrap Procedure Assuming Time-Varying Variance

In order to capture the dynamic patterns in conditional volatility in Case 2, we assume a GARCH (1,1) model, widely used in the financial literature. Hence, the time-varying variance is expressed as [[sigma].sup.2.sub.i] = [bar.[omega]] + [beta][[sigma].sup.2.sub.i-1] + [gamma][[epsilon].sup.2.sub.i-1]. In applications of the GARCH (1,1) model to high-frequency financial time series, it is often found that the estimates of [beta] and [gamma] are such that their sum is close to 1. Also, the parameters should satisfied [bar.[omega]] > 0, [beta] [greater than or equal to] 0 and [gamma] [greater than or equal to] 0 to guarantee that [[sigma].sup.2.sub.i] [greater than or equal to] 0. The stages of the designed bootstrap procedure are the following:

Stage 1: Consider a univariate time series of interest [y.sub.i], i = 1, ..., n, following the equation [y.sub.i] = [mu] + [[sigma].subi][u.sub.i], where [u.sub.1], ..., are unobservable independent and identically distributed random variables sampled from a postulated null distribution function [F.sub.u] (x, [theta]), with zero mean and unitary variance. Estimate [mu], [bar.[omega]], [beta], [gamma] and [theta] by quasi-maximum likelihood procedure (with the BHHH procedure of Berndt et al., 1974), using the n observations of the original sample. The estimates are denoted by [??], [??], [??], [??], [??].

Stage 2: Compute the vector of standardized residuals of the model [[eta].sub.1], ..., [[eta].sub.n], where [[eta].sub.i] = ([y.sub.i] - [??]~)/[[??].sub.i], and construct [sub.q][[bar.W].sup.2.sub.n] using [[eta].sub.i] and substituting [theta] by [??].

Stage 3: Draw B=200 bootstrap samples of size n from the estimated null distribution [F.sub.u] (x, [theta]). Use these bootstrap samples and the estimates [??], [??], [??], [??] and [??] to construct B bootstrap samples [y.sup.*.sub.b1], ..., [y.sup.*.sub.bn], b = 1, ..., B. For each of these series, compute the corresponding vector of standardized residuals [[eta].sup.*.sub.b1], ..., [[eta].sup.*.sub.bn], obtain the new estimate [[??].sup.*] and evaluate [sub.q][[bar.W].sup.2*.sub.n]. In this way, a sample of B independent (conditional to the original sample) observations of [sub.q][[bar.W].sup.2.sub.n] is obtained; i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Stage 4: Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the (1 - [alpha])B-th order statistic of the sample [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], given a significance level [alpha]. Reject the null hypothesis at the significance level [alpha] if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Stage 5: Compute the bootstrap p-value as [p.sub.B] = card([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) / B, b = 1, ..., B.

3.4 Simulation Results Assuming Time-Varying Variance

Analogous to the previous case, to examine the performance of the designed bootstrap procedure in Case 2, we have conducted a simulation experiment. In this case, we draw a sample [y.sub.1], ..., [y.sub.n] of size n=1000 from a normal distribution, considering [mu] = 0 and [bar.[omega] = 0.0002, [beta] = 0.82 and [gamma] = 0.09. Figure 2 shows a realization of 1000 observations from this model.

[FIGURE 2 OMITTED]

Figure 2 exhibits one common characteristic of time series, which is known as volatility clustering. It can be observed that large changes tend to follow large changes, and small changes tend to follow small changes.

Table 2 shows the rejection percentages of the hypothesis of normality when we assume time-varying variance. In particular, following the pattern of the series shown in Figure 2, a GARCH (1,1) model is assumed to capture the conditional variance. Moreover, we compute the power of the test, generating the original sample from a Student t distribution function, with zero mean, unitary variance and g=4 degrees of freedom.

Table 2 clearly indicates that the test statistic is well-specified. As it can be observed the empirical size of the test is close to the significance level when [alpha]=5% and [alpha]=10%, fixed the q=25% and q=15% of the random sample. This fact can be confirmed using the significance test statistic Z. As we have computed above, the rejection percentage p must belong to the interval [1.41% , 8.58%] given [alpha]=5% and to the interval [5.06% , 14.93%] given [alpha]=10%, for R=100. Finally, it can be observed that the power is high when the conditional variance is modelated.

4. MEASURING VALUE AT RISK

Value at Risk (VaR) is a popular measure of market risk, which associates the maximum amount that can be lost during a period to a determined statistic likelihood level. The calculus of VaR involves dealing with the confidence level, the time horizon and the true underlying conditional distribution of asset returns. In the financial literature, numerous articles have appeared focused on the treatment of fat tails and VaR estimation (Hull and White, 1998; Rockafellar and Uryasev, 2002, between others).

The aim of our empirical application is to provide empirical evidence of the effects of capturing appropriately the low-tail behaviour in the context of risk measurement.

The data set results from the stock-exchange index SP500 for the period January 1994 December 2002. In particular, the time period is divided into two smaller periods: from January 1994 to May 2001 and from June 2001 to December 2002. The first time period is used to estimate the model, to fit the low tail behaviour of the SP500 distribution and to compare "in-sample" predictive power of VaR. The second time period is used to compare "out-sample" predictive power of VaR. The daily return index is computed as [R.sub.t] = 100. log([S.sub.t]/[S.sub.t-1]), where St is the closing price at day t.

First of all, in Table 3 we report summary statistics, two normality tests and the selected model to capture the presence of dynamic structure in the mean and the conditional behaviour of the variance. In particular, the results for Jarque-Bera normality test and Kolmogorov-Smirnov normality test with Lilliefors correction (Lilliefors, 1968) are presented.

As Table 3 shows, the SP500 distribution exhibits excess kurtosis, which is a common characteristic of stock return distributions. Moreover, the series fails to pass both normality tests. The results for the Box-Ljung statistic for the series, Q(5) and Q(20), and for the squared series, [Q.sup.*](5) and [Q.sup.*](20), constructed with 5 and 20 sample autocorrelations, allow to conclude that the series presents smooth dynamic in the mean while the squared series shows strong evidence of autocorrelation. As consequence, we examine the performance of alternative models for conditional heteroskedastic time series. In order to capture the dynamic structure in the mean, we have fitted various autorregresive moving-average models: AR(1), AR(2), MA(1), MA(2), ARMA(1,1), ARMA(1,2) and ARMA(2,1) models. Analogous, we have used several GARCH specifications to analyze the behaviour of the conditional variance: ARCH(l), ARCH(2), ARCH(3), ARCH(4), GARCH(1,1), GARCH(1,2) and GARCH(2,1) models. All these models have been estimated using the maximum likelihood procedure and they have been compared with the Schwarz Information Criterion (SIC).

Table 4 consists of two panels. In Panel I we compare the accuracy of "in-sample" VaR estimates computed assuming each of the possible specifications: the normal, the Student's t, the Logistic and the Edgeworth-Sargan distribution. All the VaR estimates have been calculated one-step-ahead and using 99% coverage level. In order to carry out the accuracy comparison, we have used several methods, commonly used in the related literature to evaluate VaR estimates: (i) Comparison of the proportion of failures [??], (times that VaR estimates are exceeded by losses) with the confidence level [alpha]. (ii) The L[R.sub.PF] method, proposed by Kupiec (1995). The null hypothesis "the empirical size of the test ([??]) is equal to the nominal size [alpha]" is tested versus the alternative "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]", using the following likelihood ratio test statistic based on the binomial distribution:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where, z denotes the number of times the loss lies under the estimated VaR([alpha], [DELTA]t). Under the null hypothesis, L[R.sub.PF] is asymptotically distributed as [chi square](1). (iii) The L[R.sub.CC] method, proposed by Christoffersen (1998) as a test of the conditional coverage level in contrast to L[R.sub.PF], that ignores the presence of the time-dependence. The test of the null hypothesis "[mathematicalE, = a" versus the alternative "[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]", taking into account the presence of time-dependence, is carried out using the test statistic defined as:

(5) L[R.sub.CC] = L[R.sub.PF] + L[R.sub.ind]

Under the null hypothesis of correct conditional coverage level is asymptotically distributed as [chi square] (2). The L[R.sub.ind] statistic is a likelihood ratio statistic of the null hypothesis of serial independence against the alternative of first-order Markov dependence. (iv) QPS, proposed by Lopez (1999). In this case, the accuracy of VaR estimates is gauged by how well they minimize a loss function that represents the evaluator's concerns. Different loss functions can be considered, as for example, the quadratic probability score (QPS), developed by Brier (1950), which assigns a quadratic numerical score when a VaR estimate is exceeded by its corresponding portfolio loss. In this case and given a sample of size T, QPS is defined as:

(6) QPS = 1/T [T.summation over (t=1)]2[([P.sub.t] - [Q.sub.t+1]).sup.2],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], [??] is the estimated selected distribution function, CV([alpha],[??])= [[??].sup.-1] ([alpha]/100) is the unconditional quantile of interest and, [Q.sub.t+1] is an indicator variable that equals to one if the VaR estimate is exceeded by its corresponding asset or portfolio loss and, zero otherwise. It must be highlight that QPS [member of] [0,2] and smaller values of QPS indicate more accurate forecasts.

Panel I shows that, the differences between the percentage of exceeding [??] and [alpha]=0.01 under the normal assumption are higher than under the other specifications. Moreover, given a 5% significance level, results for L[R.sub.PF] show that the null hypothesis of equality between the proportion of exceedings and nominal size 0.01 is rejected in almost all cases when normality is assuming, while it is accepted when one of the alternative specifications is assumed. Similar results can be observed using L[R.sub.CC] test statistic. Panel I also shows the value of the loss function under each distributional assumption. As it can be observed, the maximum QPS value for each index is the associated to the normality assumption. Hence, when the normal distribution is assumed the VaR estimates are less accurate than under the other specifications. If we look the results of the goodness-of-fit tests reported in Panel II, we can observe that, given a significance level equals to 5%, the normal distribution is rejected while the Student's t, Logistic and Edgeworth-Sargan distribution are accepted to model the low-tail behaviour of the SP500 distribution. In particular, the biggest p-value is the associated to the Edgeworth-Sargan distribution. Hence, results of Panel I and II allow us to conclude that for our series, the fact of choosing a functional form that captures well the low-tail behaviour of the observed distribution helps to obtain accurate VaR estimates.

Our empirical analysis concludes with a comparison of "out-sample" predictive power of VaR, using the data from June 2001 to December 2002. Table 5 reports the results of using the evaluation methods when normality is assumed as well as when the conditional behaviour of this index is captured. Given the goodness-of-fit test p-values, the Logistic is the selected functional form to fit the overall observed distribution of SP500 while the Edgeworth-Sargan distribution is the specification that better fits to the data of the low tail. Moreover, it can be observed that the worst results are those associated to the normality assumption: the biggest percentage of exceedings, rejection of the null hypothesis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] using L[R.sub.PF] and L[R.sub.CC] at 5% significance level and the biggest QPS. Quite good results are obtained when the Logistic is assumed. However, as it was expected, the best results are those obtained when the Edgeworth-Sargan distribution is assumed because fits well to the data of the low tail of the SP500 observed distribution. When the Edgeworth-Sargan distribution is assumed, the percentage of exceeding is very similar to the nominal size 1%, the null hypothesis [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is accepted using L[R.sub.PF] and L[R.sub.CC] at 5% significance level and the value of QPS is the smallest.

Also, Panel 1 and 2 in Table 5 report graphical comparisons of "out-sample" predictive power of VaR estimates when the normal, Student's t, Logistic and Edgeworth-Sargan distributions are used. In Panel 1, VaR estimates under the normal and the Logistic are compared and it can be stated that VaR estimates using the Logistic are slightly better than when normality is considered. However, as Panel 2 reveals, the improvement in VaR estimates is higher when the Edgeworth-Sargan distribution is assumed. This graphical comparison is analytically supported by the evaluation method results summarized in Table 6.

5. CONCLUSIONS

In this paper we have proposed a new approach to implement goodness-of-fit tests for a censored sample based on the bootstrap methodology and using the modified Cramer-von Mises test statistic. Also we have examined the performance of the proposal by several simulation experiments.

We have considered two scenarios. In the first scenario, we have assumed that the variance is constant while in the second we have allowed the variance to change along the time. The latest is more relevant in financial literature since the volatility clustering is a common characteristic of financial series, especially in high frequency data.

From our experiments we have found that the empirical size is rather close to the nominal size. Concerning to the power of the test, the test rejects the null hypothesis of normality more than 95% of the time with data generated from model based on Student's t distribution. It must be highlighted that our proposal could be used to test whatever null distribution depending on the objective of the analysis. In this sense, this proposal can be included in software focus on the creation of a suite of tools to enable investors to make financial decisions.

Finally, we have provided empirical evidence using data from the SP500 index from January 1994 to December 2002. In particular, we have described an application of the analysis of the low-tail behaviour of the SP500 in the context of VaR measurement. We have found that it is possible to gain accuracy in VaR estimates when a good assumption of the underlying true distribution is made. Moreover, we have included both "in-sample" and "out-sample" predictive power comparisons of VaR estimates using the most common evaluation methods.

TABLE 1: REJECTION PERCENTAGES OF THE HYPOTHESIS OF NORMALITY
ASSUMING CONSTANT VARIANCE

                    Data from Normal
                 Distribution (n=1000)

[alpha]    qn [??][??][??]     q[??] [??][??]

[??]             [??]               [??]
[??]             [??]               I[??]

              Data from t-student with g=4
                        (n=1000)

[alpha]   q[??] [??][??][??]   q[??] [??]I[??]

[??]           [??][??]           [??][??]
[??]           [??][??]           [??][??]

TABLE 2: REJECTION PERCENTAGES OF THE HYPOTHESIS OF
NORMALITY ASSUMING TIME-VARYING VARIANCE

                 Data from Normal
              Distribution (n=1000)

[alpha]   qn [??][??][??]   qn [??]I[??]

[??]           [??]             [??]
[??]           [??]             [??]

           Data from t-student with g=4
                     (n=1000)

[alpha]   qn [??][??][??]   qn [??]I[??]

[??]         [??][??]           [??]
[??]         [??][??]           [??]

TABLE 3: DESCRIPTIVE MEASURES, SELECTED MODEL AND DIAGNOSIS
(SP500: 3/1/1994-31/5/2001)

         Skewnes                  Jarque-         KSs
Index       s       Kurtosis       Bera       Lilliefors

SP500    -0.2755     7.7207       1750.6         0.068
                                  (0.000)       (0.000)

Index     Q(5)        Q(20)       Q * (5)      Q * (20)

SP500    11.754      29.862       216.64        436.01
         (0.038)     (0.072)      (0.000)       (0.000)

Index                             Selected model

SP500   [R.sub.t] = [[phi].sub.0] + [h.sup.1/2.sub.t] [[eta].sub.t]
        [h.sub.t] = [omega] + [[alpha].sub.1][[epsilon].sup.2.sub.t-1]
          + [[beta].sub.t][h.sub.t-1]

        [Q.sub.r]   [Q.sub.r]   [Q.sub.r] *   [Q.sub.r] *
Index      (5)        (20)          (5)          (20)

SP500     7.166      27.499        8.972        19.239
         (0.209)     (0.122)      (0.110)       (0.506)

TABLE 4: ACCURACY OF VaR FROM DIFFERENT DISTRIBUTIONS FORMS
(SP500: 3/1/1994-31/5/2001)

Panel I: Distributional forms

Method            Normal           Student t

([??])            0.0172            0.0118
L[R.sub.PF]   8.0371 (0.0045)   0.5967 (0.4398)
L[R.sub.CC]   9.1581 (0.0102)   1.1237 (0.5703)
QPS               0.0397            0.0264

Method           Logistic             E-S

([??])            0.0145            0.0134
L[R.sub.PF]   3.3719 (0.0663)   2.0149 (0.1558)
L[R.sub.CC]   4.1678 (0.1244)   2.6965 (0.2597)
QPS               0.0307            0.0300


Panel II: Results of goodness-of-fit tests

Index             Normal           Student t

SP500              0.000             0.280

Index            Logistic             E-S

SP500              0.310             0.695

TABLE 5: OUT-SAMPLE ACCURACY OF VaR FOR SP500 (1/6/2001-31/12/2002)

Evalutaion
Method            Normal           Logistic             E-S

[??]              0.0226            0.0175             0.0125
L[R.sub.PF]   4.7112 (0.0411)   1.8880 (0.1694)   0.02442 (0.6211)
L[R.sub.CC]   5.1276 (0.0770)   2.1386 (0.3432)   0.3714 (0.8305)
QPS               0.0455            0.0353             0.0256

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Author Profiles:

Dr. Susana Alvarez earned her Ph.D. at the University of Alicante in 2002. Currently she is a lecturer at University of Murcia. Her research interests are in financial modeling and risk management.

Dr. J.Samuel Baixauli earned his Ph.D. at the University of Valencia in 2002. Currently he is associated professor at University of Murcia. His research interests are in risk management and corporate finance.