bali_risk management performance of alternative distribution functions_2002

8/3/2019 Bali_risk Management Performance of Alternative Distribution Functions_2002

1/44

Risk Management Performance of Alternative Distribution Functions

January 2002

Turan G. BaliAssistant Professor of Finance

Department of Economics & Finance

Baruch College, Zicklin School of Business

City University of New York

17 Lexington Avenue, Box 10-225New York, New York 10010

Phone: (646) 312-3506

Fax : (646) 312-3451

E-mail: [email protected]

Panayiotis Theodossiou

Professor of FinanceSchool of Business

Rutgers University

227 Penn Street

Camden, New Jersey 08102

Phone: (856) 225-6594

Fax : (856) 225-6632

E-mail: [email protected]

Key words: value at risk, risk management, extreme value distributions, skewed fat-tailed

distributions

JEL classification: G12, C13, C22


2/44

2


ABSTRACT

This paper compares the risk management performance of alternative distribution functions. The

results indicate that the extreme value and skewed fat-tailed distributions, such as the skewed

generalized t (SGT) and inverse hyperbolic sine (IHS) distributions, improve the standard VaR

models that assume normality. This is because the SGT and IHS distributions put more emphasis

on the tail areas of observed frequency distributions, and hence provide good predictions of

catastrophic market risks during extraordinary periods. The empirical findings lead to a firm

rejection of the hypothesis that one is obliged to use the distribution of extremes only (instead of the

distribution of all returns) to obtain precise VaR measures. The results based on the actual andestimated VaR thresholds as well as the likelihood ratio tests point out that the maximum likely loss

of financial institutions can be more accurately estimated using the generalized extreme value and

skewed fat-tailed distributions. They perform surprisingly well in capturing both the rate of

occurrence and the size of extreme observations in financial markets. In addition, this article

proposes a conditional VaR approach that takes into account time-varying volatility, considers the

non-normality of returns, and deals with extreme events. The results indicate that the actual VaR

thresholds are time-varying to a degree not captured by the conditional normal density, but precisely

estimated by the conditional SGT, IHS, and extreme value distributions.


3/44

3


I. Introduction

The importance of sound risk management for financial institutions was emphasized by

several high-profile risk management disasters in the early 1990s.1

During the past decade, there

has been an increased focus on the measurement of risk and the determination of capital

requirements for financial institutions to meet catastrophic market risk. This increased focus has led

to the development of various risk management techniques. The primary technique is Value at Risk

(VaR), which determines the maximum expected loss on a portfolio of assets over a certain holding

period at a given confidence level (probability).2

The use of VaR techniques in risk management has exploded over the past few years. The

two most popular VaR techniques are the variance-covariance analysis and historical simulation.

The variance-covariance analysis relies on the assumption that financial market returns follow a

multivariate normal distribution. This technique is easy to implement because the VaR can be

computed from a simple quadratic formula with the variances and covariances of returns as the only

inputs. Its major drawback is that financial market returns exhibit skewness and significant excess

kurtosis (fat-tails and peakness), and as such they are not normal. Because of this, the size of actual

losses is much higher than that predicted by the normal distribution. As a result, the variance-

covariance analysis produces VaR thresholds that understate the true risk faced by financial

institutions (tail bias). The variance-covariance approach is particularly weak where a VaR model

should be strong in the prediction of large losses for regulatory purposes and risk control.

Historical simulation does not rely on normality and as such it does not suffer from the tail-

bias problem. By applying the empirical distribution of all assets returns in the trading portfolio, the

outcome will reflect the historical frequency of large losses over the specified data window. Unlike


4/44

4

the variance-covariance analysis, the historical approach can be used in a natural way to compute

VaR for non-linear positions, such as derivative positions. The problem with historical simulation

is that it is very sensitive to the particular data window. In particular, the inclusion of extraordinary

periods, such the stock market booms and crashes, affects significantly the computation of VaR

measures. This is because the empirical return distribution is very dense and smooth around the

mean but discrete in the tails because of a few extremely large price movements. As a result, VaR

measures based on historical simulation exhibit high variances. Moreover, at its lower end, the

empirical return distribution drops sharply to zero and remains there, thus the probability of more

severe losses than the past largest one is assigned the value of zero, which might be considered

imprudent.3

In light of the above, an alternative approach that approximates the tails of the distribution of

returns asymptotically is more appropriate than imposing a symmetric thin-tailed functional form

like the normal distribution. Although VaR models based on the normal distribution provide

acceptable estimates of the maximum likely loss under normal market conditions, they fail to

account for extremely volatile periods corresponding to financial crises. Longin (2000), McNeil and

Frey (2000), and Bali (2001a) show that VaR measures based on the distribution of extreme returns

(extreme value distributions), instead of the distribution of all returns, provide good predictions of

catastrophic market risks during extraordinary periods.

An important contribution of this paper is the application and assessment of several flexible

probability distribution functions in computing VaR measures based on the distribution of all

returns. These distributions are the inverse hyperbolic sine (IHS) of Johnson (1949), the exponential

generalized beta of the second kind (EGB2) of McDonald and Xu (1995), the skewed generalized-t

(SGT) of Theodossiou (1998), and the skewed generalized error (SGED) of Theodossiou (2001).


5/44

5

This paper shows that the SGT and IHS (henceforth referred as skewed fat-tailed distributions)

produce precise VaR measures and compare favorably to the extreme value distributions. The latter

is attributed to the fact that the SGT, IHS, and extreme value distributions provide an excellent fit to

the tails of the empirical return distribution.4

Value-at-risk measures are most often expressed as percentiles corresponding to the desired

confidence level. The 99% confidence level is consistent with VaR standards, but this choice is

really arbitrary. Therefore, we expand the analysis beyond the 99% level and consider the entire tail

of the return distribution. We evaluate the empirical performance of alternative distribution

functions in predicting the 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% VaR thresholds. The results

consistently point to the same conclusions and provide evidence that the extreme value and skewed

fat-tailed distributions perform surprisingly well in modeling the asymptotic behavior of stock

market returns.

Moreover, the paper evaluates the relative performance of aforementioned distributions

based on the unconditional and conditional coverage tests introduced by Kupiec (1995) and

Christoffersen (1998), respectively. VaR measures based on the unconditional extreme value and

skewed fat-tailed distributions do not account for systematic time-varying changes in the

distribution of returns. The conditional coverage test results indicate that the actual thresholds are

time-varying to a degree not captured by the unconditional density functions. This paper extends

the unconditional VaR approach by taking into account the dynamic (time-series) behavior of

financial return volatility. The dynamic behavior of returns is modeled using the autoregressive

absolute GARCH process of Taylor (1986) and Schwert (1989).

The paper is organized as follows. Section II contains a short discussion of the extreme value

and flexible distributions. Section III provides the unconditional value-at-risk models based on


6/44

6

alternative distribution functions. Section IV describes the data. Section V presents the estimation

results. Section VI compares the risk management performance of alternative VaR models. Section

VII proposes a conditional VaR approach. Section VIII concludes the paper.

II.A Extreme Value Distributions

We investigate the fluctuations of the sample maxima (minima) of a sequence of i.i.d. non-

degenerate random variables {X1, X2,, Xn} with common cumulative distribution function (cdf)

F(x), where

M1 =X1, M2 = max (X1,X2),, Mn = max (X1,,Xn), n 2. (1)

Corresponding results for the minima can be obtained from those for maxima by using the identity:

min (X1,,Xn) = max (X1,, Xn). (2)

The exact cdf of the maximum Mn is easy to write:

P(Mnx) =P(X1x, ,Xnx) =Fn(x), x, nN. (3)

According to this, extremes happen near the upper end of the support of the distribution,

hence intuitively the asymptotic behavior ofMn must be related to the cdfF(x) in its right tail. In

this case this tail has finite support. We let

xF= sup{x: F(x) < 1} (4)

denote the right endpoint ofF(x). We immediately obtain, for allx


7/44

7

Here, the well-known Fisher-Tippett (1928) theorem has the following content: if there exist

normalizing constants n > 0 and centering constants n such that

1( )d

n n n

x M H = , n, (7)

for some non-degenerate distribution H, then Hbelongs to the type of one of the three so-called

standard extreme value distributions:

>

=

otherwise0

0if)exp()(:

/1

max,

xxxHFrechet

(8)

=

otherwise))(exp(

0if1)(:

/1max,

x

xxHWeibull (9)

Gumbel: Hmax,0(x) = exp[exp(x)] - 0, < 0, and = 0 we obtain the Frechet, Weibull and

Gumbel families, respectively. The Frechet distribution is fat tailed as its tail is slowly decreasing;

the Weibull distribution has no tail after a certain point there are no extremes; the Gumbel

distribution is thin-tailed as its tail is rapidly decreasing. The shape parameter , called the tail

index, reflects the fatness of the distribution, whereas the parameters ofscale, , and oflocation, ,

determine the average and standard deviation of the extremes along with .

An alternative approach to determine the type of asymptotic distribution for extremes can be

based on the concept of generalized Pareto distribution (GPD).5

Excesses over high thresholds can

be modeled by the generalized Pareto distribution, which can be derived from the generalized


8/44

8

extreme value (GEV) distribution. The generalized Pareto distribution of the standardized maxima

denoted by Gmax(x) isgiven by Gmax(x) = 1 + ln [Hmax(x)], whereHmax(x) is the GEV distribution:

/1

max, 11),;(

+=

M

MG . (12)

Notice that the generalized Pareto distribution presented in equation (12) nests the standard Pareto

distribution, the uniform distribution on [-1,0], and the standard exponentialdistribution:6

Pareto G x x: ( )max,/

= 1 1 forx 1, (13)

Uniform G x x: ( ) ( )max,/

= 1 1 forx [-1,0] (14)

Exponential G x x: ( ) exp( )max,0 1= forx 0. (15)

To determine whether the generalized Pareto or the generalized extreme value distribution

yields a more accurate characterization of extreme movements in financial markets, Bali (2001b)

proposes a more general extreme value distribution using theBox-Cox (1964) transformation:

1/

max,

exp 1 1

( ; , , ) 1

M

F M

+ = +

(16)

The Box-Cox-GEV distribution in equation (16) nests the generalized Pareto distribution of

Pickands (1975) and the generalized extreme value distribution of Jenkinson (1955). More

specifically, when equals one the Box-Cox-GEV reduces to the GEV distribution given in

equation (11): when = 1Fmax,(x) Hmax,(x). When equals zero, the Box-Cox-GEV converges

to the generalized Pareto distribution presented in equation (12): when = 0 Fmax,(x) Gmax,(x).7


9/44

9

II.B Skewed Fat-Tailed Distributions

This section presents the probability density functions for the skewed generalized-t (SGT),

the skewed generalized error (SGED), the inverse hyperbolic sine (IHS), and the exponential

generalized beta of the second kind (EGB2). As shown in Hansen, McDonald, and Theodossiou

(2001), all four densities accommodate diverse distributional characteristics when used for fitting

data or as a basis for quasi-maximum likelihood estimation (QMLE) of regression models.

The SGT probability density function is

( )

( )( )( )

( )1

1; , , , , 1

2 1 ( )

n k

k

k k k

CSGT y n k y

n k sign y

+ = + +

+ +

(17)

where

( )( ) ( )( )1

2 2 1 ,k

C k n k B k n k = ,

( )( ) ( ) ( )( ) ( )1 .5.5 1

2 1 , 3 , 2k

k n B k n k B k n k S = ,

( ) 2 2 21 3 4S A = + ,

( ) ( ) ( ).5 .5

2 , ( 1) 1 , 3 ,( 2)A B k n k B k n k B k n k = ,

( )1

2 AS = ,

and is are the mean and standard deviation of the random variabley, n and kare positive kurtosis

parameters, is a skewness parameter obeying the constraint ||


10/44

10

generalized error distribution or power exponential distribution of Subbotin (1923) (used by Box

and Tiao [1962] and Nelson [1991]), for n=, =0, k=1 the Laplace or double exponential

distribution, forn=1, =0, k=2, the Cauchy distribution, forn=, =0, k=2 the normal distribution,

and forn=, =0, k= the uniform distribution.

The SGEDprobability density function is

( )( )( )

1; , , , exp

1

k

k k k

CSGED y k y

sign y

= + + +

(18)

where

( )( ) ( )1

2 1 , 2C k k AS = =

( ) 2 2 21 3 4S A = +

( ) ( ) ( ).5 .5 1

1 3k k S =

( ) ( ) ( ).5 .5

2 1 3A k k k =

, , , k, andsign are as defined previously.

The inverse hyperbolic sine (IHS) probability density function is

( )( )( )22 2 2

; , , ,

2

kIHS y k

y

=+ +

( ) ( )( ) ( )22

22 2exp ln ln2

ky y

+ + + + +

(19)

where 1 ,w = ,w w =2.5

.5( ) ,kw e e e

= 2 2 22 2 .5 .5

.5( 2) ( 1)k k kw e e e

+ += + + , w and

w are the mean and standard deviation ofw=sinh(+z/k), sinh is the hyperbolic sine function,zis a

standardized normal variable, and and are the mean and standard deviation ofy.8 Note that

negative (positive) values of generate negative (positive) skewness, and zero values no skewness.

Smaller values ofkresult in more leptokurtic distributions.


11/44

11

The EGB2 probability density function is

( )( )

( )( )

( )

( )2 ; , , ,

1

p y

p qy

eEGB y p q C

e

+

+ +=

+, (20)

where ( ) ( ) ( ) ( )1 ( , ) , ( ) ( ) , 1 ,C B p q p q p q = = = + p and q are positive scaling

constants, B() is the beta function, ( ) ( )lnz d z dz = and ( ) ( )z d z dz = are the psi

function and its first derivative, and and are the mean and standard deviation ofy. The EGB2 is

symmetric for equal values ofp and q, positively skewed for values ofp > q, and negatively skewed

for values ofp < q. The EGB2 converges to the normal distribution for infinite values ofp and q.

III. Value at Risk Models with Alternative Distribution Functions

The traditional VaR models assume that the probability distribution of log-price changes

(log-returns) is normal, an assumption that is far from perfect. However, the distributions of log-

returns of financial assets are usually skewed to the left, have fat-tails, and are peaked around the

mode. Because of the fat tails, extreme outcomes happen much more frequently than would be

predicted by the normal distribution.9

This section presents alternative VaR models based on the

extreme value as well as the aforementioned flexible distributions.10

VaR calculations are performed in an environment where the stochastic process At depends

on a risk factor such as the interest rate, exchange rate, or equity return. The arbitrage-free price of a

financial asset at time t,At, is assumed to be a known function ofRtand the parameters , i.e.,

At=A(Rt, t; ). (21)

The stochastic variation inAtduring an infinitesimal interval dtcan be given by Itos Lemma:

dAt=ARdRt+Atdt+21

2RR t A dt (22)


12/44

12

whereAR =( , )

t

t

A R t

R

and ARR =2

2

( , )

t

t

A R t

R

are the delta and gamma of the asset, respectively.

To compute VaR, one imposes a model on the stochastic differential dRt. For most risk

factors, researchers choose the stochastic differential equation of the form:

dRt= tdt + tdWt. (23)

Assuming that t denotes the length of time interval, the discrete time approximation of the

stochastic process in (23) is written as:

t t tR t t z = + , (24)

wherezis standard normal with mean zero and variance one.

The critical step in calculating VaR measures is the estimation of the threshold point

defining what variation in returns Rt is considered to be extreme. Let be the probability that Rt

will exceed the threshold . That is,

( )Pr Pr ttt

tR z a

t

> = > = =

(25)

where Pr() is the underlying probability distribution. In the traditional VaR model, a = 2.326, and

Normal= tt+ 2.326 t t (26)

where Pr() is the cumulative normal distribution and is 1%.11 The VaR at time tis obtained from

equation (22) by lettingAt=ARR = 0:

VaR (A, , t) =AR Normal. (27)

The risk manager who has exposure to a risk factorRt, which changes by discrete increments

of Rt, needs to know how much capital to put aside to cover at least the fraction 1 of daily

losses during a year. For this purpose, the risk manager must first determine a threshold so that the

event (Rt) has a probability under Pr(). The standard approach does this by using an explicit


13/44

13

distribution that is in general the normal distribution. The alternative approach is to use a

cumulative probability distributionF() based on one of the extreme value and flexible probability

distributions then solve forto obtain the threshold, i.e.,

( )1 1 .F = (28)

As shown in Bali (2001b), the Box-Cox-GEV distribution yields the following VaR threshold:

+=

11ln1

n

NGEVCoxBox (29)

where n andNare the number of extremes and the number of total data points, respectively. Once

the location (), scale (), shape (), and parameters of the Box-Cox-GEV distribution are

estimated one can find the VaR threshold, Box-Cox-GEV, based on the choice of confidence level ().

With the GPD (= 0) and GEV (= 1) distributions, the VaR threshold in equation (29) reduces to:

+=

1

n

NGPD (30)

+=

11ln

n

NGEV . (31)

As will be discussed in the paper, there is substantial empirical evidence that the distribution

of stock returns is typically skewed to the left and leptokurtic, that is, the unconditional return

distribution shows high peaks, fat tails, and more outliers on the left tail. This implies that extreme

events are much more likely to occur in practice than would be predicted by the thin-tailed normal

distribution. This also suggests that the normality assumption can produce VaR numbers that are

inappropriate measures of the true risk faced by financial institutions. In order to overcome the

drawbacks of the normal distribution, we use the flexible distributions that take into account the


14/44

14

non-normality of returns, and deals with events that are relatively infrequent. The VaR threshold is

computed using

tatflexible += , (32)

where a is the cut-off for the standardized cdf associated with probability 1, i.e.,F(a)=1, and

and are the mean and standard deviation parameters of the corresponding flexible distribution.

IV. Data

The data set consists of daily (percentage) log-returns (log-price changes) for the Dow Jones

Industrial Average (DJIA) and S&P500 composite indices. The time period investigation for the

DJIA is May 26, 1896 to December 29, 2000 (28,758 observations) and for the S&P500 is January

4, 1950 to December 29, 2000 (12,832 observations).12

Table 1 shows that the unconditional mean

of daily log-returns for the DJIA and S&P500 are 0.0194% and 0.0341% with a standard deviation

of 1.09% and 0.87%, respectively. The maximum and minimum values are 14.27% and 27.96% for

the DJIA, and 8.71% and 22.90% for the S&P500. The table also reports the skewness and excess

kurtosis statistics for testing the distributional assumption of normality. The skewness statistics for

daily returns are negative and statistically significant at the 1% level. The excess kurtosis statistics

are considerably high and significant at the 1% level, implying that the distribution of equity returns

has much ticker tails than the normal distribution. The fat-tail property is more dominant than

skewness in the sample.

The maximal and minimal returns are obtained from the original daily data described above.

Following the extreme value theory, we define the extremes as excesses over high thresholds [see

Embrechts et al. (1997, pp. 352-355)]. Specifically, the extreme changes are defined as the 5 percent

of the right and left tails of the empirical distribution. Panel B of Table 1 shows the means, standard


15/44

15

deviations, maximum and minimum values of the extremes. In addition to the 5% tails, the extremes

are obtained from the 2.5% and 10% tails of the empirical distribution. The qualitative results are

found to be robust across different threshold levels. To save space we choose not to present the

empirical findings based on the 2.5% and 10% tails.13

V.A Empirical Results for the Extreme Value Distributions

Table 2 presents the regression method estimates of the Box-Cox-GEV, generalized Pareto,

and generalized extreme value distributions. The empirical results are clear-cut and allow one to

determine unambiguously the type of extreme value distribution: for both the largest falls and rises

of equity returns, the asymptotic distribution belongs to the domain of attraction of the Box-Cox-

GEV distribution. A likelihood ratio (LR) test between the GEV and Box-Cox-GEV distributions

leads to a firm rejection of the GEV distribution for the DJIA and S&P500 stock indices.14

The LR

test between the GPD and Box-Cox-GEV distributions also indicates a firm rejection of the GPD

distribution in all cases except for the DJIA maximal returns.15

In estimating the parameters of the Box-Cox-GEV distribution, a one dimensional grid

search method and a nonlinear least square estimation technique are used.16

Since one of our goals

in this section is to determine whether the asymptotic distribution of extremes belongs to the domain

of attraction of GEV or GPD, the value of is expected to be between zero and one. The parameter

is estimated by scanning this range in increments of 0.1. When a minimum of the sum of squares

is found, greater precision is desired, and the area to the right and left of the current optimum is

searched in increments of 0.01. As shown in Table 2, max is estimated to be 0.02 and 0.51 for the

DJIA and S&P500, respectively, whereas the corresponding figures formin are found to be 0.66 and

0.94 for the minimal returns. The LR test results indicate that both the generalized Pareto


16/44

16

distribution with = 0 and the generalized extreme value distribution with = 1 are strongly

rejected in favor of the Box-Cox-GEV distribution with 0 < < 1.

The tail index for the GPD and GEV distributions is found to be positive and statistically

different from zero. This implies a rejection of the thin-tailed ( = 0) Gumbel and exponential

distributions with rapidly decreasing tails against the fat-tailed ( > 0) Frechet and Pareto

distributions with slowly decreasing tails, and a fortiori a rejection of the short-tailed ( < 0)

Weibull and uniform distributions.17,18

The asymptotic t-statistics of the estimated shape parameters

() clearly indicate the non-normality of extremes. Another notable point in Table 2 is that the

estimated shape parameters for the minimal returns (min) turn out to be greater than those for the

maximal returns (max). More specifically, the estimated max values are in the range of 0.26 to 0.42

for the DJIA and 0.17 to 0.33 for the S&P500, while for the minimal returns the estimates of min

vary from 0.28 to 0.45 for the DJIA and from 0.39 to 0.46 for the S&P500. Since the higher the

fatter the distribution of extremes, the minimal returns have thicker tails than the maximal returns.

A comparison of the estimated scale parameters () of the Box-Cox-GEV, GPD, and GEV

distributions indicates that both max and min are overestimated by the generalized Pareto and

underestimated by the generalized extreme value distribution. For the DJIA (Panel A of Table 2),

the scale parameters (max, min) are found to be (0.656%, 0.625%) for the Box-Cox-GEV, (0.657%,

0.727%) for the GPD, and (0.409%, 0.489%) for the GEV distribution. The qualitative results turn

out to be the same for the S&P500: max and min are 0.440% and 0.303% for the Box-Cox-GEV,

0.503% and 0.383% for the GPD, and 0.316% and 0.283% for the GEV distribution. Since the

volatility of extremes depends on the scale () and shape () parameters of the asymptotic extremal


17/44

17

distributions, there is no clear evidence whether the maximal returns are more volatile than the

minimal returns or vice versa.

V.B Empirical Results for the Flexible Distributions

Table 3 presents the estimated parameters of the skewed generalized t (SGT), skewed

generalized error (SGED), inverse hyperbolic sign (IHS), exponential generalized beta of the second

kind (EGB2), and normal distributions for the DJIA and the S&P500 log-returns. The estimates are

obtained using the maximum likelihood method and the iterative algorithm described in

Theodossiou (1998).

The first two columns of the table present the estimates for the mean and standard deviation

of log-returns for the DJIA and S&P500. As expected, these estimates are quite similar across

distributions and do not differ much from the simple arithmetic means and standard deviations of

log-returns presented in Table 1.

The third and fifth column present the estimates for the kurtosis parameters kand n (SGT

only). In the case of SGT, the values of kand n are, respectively, 1.76 and 3.39 for the DJIA and

1.60 and 5.21 for the S&P500. In both cases, the values are quite different from those of the normal

distribution ofk= 2 and n = . Both pairs of values indicate that the DJIA and S&P500 log-returns

are characterized by excess kurtosis. Note that in the SGT model, the parameter kcontrols mainly

the peakness of the distribution around the mode while the parameter n controls mainly the tails of

the distribution, i.e., adjusting the tails to the extreme values. The parameter n has the degrees of

freedom interpretation as in the Student-t distribution. The fourth column presents the skewness

parameterwhich is negative and statistically significant for both DJIA and S&P500 log-returns.


18/44

18

Column 6 and 7 present the standardized measures for skewness Sk=E(y-)3/3 and kurtosis

Ku= E(y-)4/4 based on the parameter estimates for k, n, and .19 The standardized skewness

values, Sk, for both series are negative indicating that the distributions of log-returns are skewed to

the left. The standardized kurtosis for the DJIA is not defined because the parametern = 3.39 < 4.

Note that the moments of the SGT exist up to the value ofn. In the case of the S&P500, the

standardized kurtosis is 11.045.

The above results provide strong support to the hypothesis that both the DJIA and S&P500

log-returns are not normal. The normality hypothesis is also rejected by the LR statistics for testing

the null hypothesis of normality against that of SGT. Note that the LR statistics, presented in

column 9, are quite large and statistically significant at the 1% percent level. To test the overall fit

of the SGT, we also use the Kolmogorov-Smirnov statistic (KS). The KS statistics, presented in the

last column of Table 3, are small and statistically insignificant at both the 1% and 5% levels

providing support to the null hypothesis that the data are SGT and IHS distributed.

The SGED estimates for the kurtosis parameterkare close to one (k= 0.93937 for the DJIA

and 1.05063 for the S&P500) and they are considerably lower than those of SGT. This is because

the SGEDs kurtosis is controlled by parameterkonly, thus, to account for the excess kurtosis in the

data the parameterkhas to be smaller than that of SGT.20

The skewness parameteris negative and

statistically significant for both the DJIA and the S&P500 data. The standardized skewness and

kurtosis parameters are smaller than those of SGT. SGEDs nesting property allows us to test the

null hypothesis that the data follow the SGED against the alternative hypothesis that they follow the

SGT. The log-likelihood ratios for testing the latter hypothesis are 316.20 for the DJIA and 165.42

for the S&P500. These ratios, which follow chi-square distribution with one degree of freedom, are

large and statistically significant at the 1% level, thus suggesting that the SGT provides a better fit


19/44

19

than the SGED. Moreover, the KS statistic rejects the null hypothesis that the DJIA and S&P500

log-returns follow the SGED. The superiority of the SGT over the SGED can be attributed to the

fact that it provides a better fit to the tails of the distribution because of the parametern.

The IHS and EGB2 distributions are not linked directly to each other or the SGT, but the

EGB2 is linked with the normal distribution. Specifically, as the parameters p and q approach

infinity the EGB2 converges to the normal distribution. The possible values for kurtosis are limited

to the range [3, 9] for EGB2; see Hansen et al. (2001). Like in the case of SGT and SGED, the

results forIHS and EGB2 indicate that the DJIA and S&P500 log-returns exhibit skewness and

significant excess kurtosis. The LR statistic for testing the null hypothesis of normal distribution

against the alternative hypothesis of EGB2 rejects normality. The LR ratio for IHS does not exist

because the normal distribution is not nested within IHS. The KS statistics indicate that the

hypothesis that the data follow IHS cannot be rejected but the EGB2 hypothesis is rejected. The

latter may be attributed to the inability of EGB2 to model kurtosis values outside the range 3 to 9.

VI. VaR Calculations with Alternative Distribution Functions

Table 4 presents the estimated thresholds for the extreme value, normal, and flexible

distributions. The DJIA results show that the extreme tails yield threshold points, Box-Cox-GEV, that

are up to 40% higher than the normal thresholds, Normal. Moreover, the Box-Cox-GEV, GPD, and GEV

thresholds for the extreme negative increments are much greater than those for the extreme positive

increments. The multiplication factors (Box-Cox-GEV/Normal) for the extreme tails (=0.5% and 1%)

of the DJIA are in the range of 1.22 to 1.40 for the minimal returns and 1.12 to 1.28 for the maximal

returns. The VaR thresholds for the Box-Cox-GEV, GPD, and GEV distributions indicate that the


20/44

20

two tails are asymmetric. The multiplication factors for the S&P500 extreme tails (=0.5% and 1%)

are in the range of 1.13 to 1.23 for the maxima and 1.09to 1.22 for the minima.21The VaR measures for the flexible distributions are similar to those of the extreme value

distributions. Specifically, the average ratio of VaR thresholds for the maximal (minimal) DJIA

returns are Box-Cox-GEV/SGT = 1.0013 (1.0207), Box-Cox-GEV/SGED = 0.9887 (1.0028), Box-Cox-

GEV/EGB2 = 1.0271 (1.0497), and Box-Cox-GEV/IHS = 0.9984 (1.0096). The corresponding figures for

the S&P500 are Box-Cox-GEV/SGT = 1.0058 (0.9803), Box-Cox-GEV/SGED = 0.9923 (0.9609), Box-Cox-

GEV/EGB2 = 1.0091 (0.9753), and Box-Cox-GEV/IHS = 1.0012 (0.9679). These findings imply that the

tails of the empirical distribution approximated by the flexible distributions are similar to those of

the extreme value distributions.

The above results indicate that the tail areas obtained from the extreme value and flexible

distributions are quite different and potentially more useful than those of the normal distribution

(i.e., standard approach). Table 5 presents the risk management performance statistics for the

extreme value, normal, and flexible distributions. The results show that the normal VaR thresholds

for both the DJIA and S&P500 at the various tails are quite inadequate. Given that the DJIA data

includes 28,758 daily returns, one would expect 144, 288, 431, 575, 719, and 1,438 returns to fall

respectively into the 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% negative and positive tails. The number of

returns for the normal VaR thresholds falling into the negative (positive) 0.5% tail are 377 (295),

1% tail are 480 (390), 1.5% tail are 585 (479), 2% tail are 672 (552), 2.5% tail are 747 (616), and

5% tail are 1,100 (975). Based on these results, the normal distribution underestimates the actual

VaR thresholds at the 0.5%, 1%, and 1.5% tails and overestimates the VaR thresholds for most of

the remaining tails. The results for the S&P500 (see Panel B of Table 5) are quite similar to those of


21/44

21

the DJIA. The normal VaR estimates have a mean absolute percentage error (MA%E) of 42.53%

for the DJIA and 34.53% for the S&P500.

The Box-Cox-GEV distribution has the best overall performance, although the GPD, GEV,

SGT, and IHS appear to be very similar in performance to the Box-Cox-GEV in both the DJIA and

S&P500 samples. All five distributions estimate the actual VaR thresholds very well. Their mean

absolute percentage errors in the DJIA samples are 2.04% for the Box-Cox-GEV distribution,

2.22% for the GPD, 5.14% for the GEV, 4.92% for the SGT, and 3.81% for the IHS. In the

S&P500 sample, these errors are 3.12% for the Box-Cox-GEV distribution, 4.95% for the GPD,

4.29% for the GEV, 4.26% for the SGT, and 6.59% for the IHS. The remaining two flexible

distributions, SGED and EGB2, perform better than the normal distribution, however, their MA%E

are quite larger than those of the other distributions. The results indicate that the extreme value and

flexible distributions are superior to the normal distribution in calculating value at risk.

Given the obvious importance of VaR estimates to financial institutions and their regulators,

evaluating the accuracy of the distribution functions underlying them is a necessary exercise. The

evaluation of VaR estimates is based on hypothesis testing using the binomial distribution. This

latter test is currently embodied in the MRA.22

Under the hypothesis-testing method, the null

hypothesis is that the VaR estimates exhibit the property characteristics of accurate VaR measures.

If the null hypothesis is rejected, the VaR estimates do not exhibit the specified property, and the

underlying distribution function can be said to be inaccurate. Otherwise, the VaR model is

acceptably accurate.

Under the MRA, banks will report their VaR estimates to their regulators, who observe

when actual portfolio losses exceed these estimates. As discussed by Kupiec (1995), assuming that

the VaR measures are accurate, such exceptions can be modeled as independent draws form a


22/44

22

binomial distribution with a probability of occurrence equal to (say) 1 percent. Accurate VaR

estimates should exhibit the property that their unconditional coverage = q/N equals 1 percent,

where q is the number of exceptions in N trading days. Since the probability of observing q

exceptions in a sample of sizeNunder the null hypothesis is

qNq

q

Nq

= 99.001.0)Pr( , (33)

the appropriate likelihood ratio statistic for testing whether= 0.01 is

LRuc = 2 1 0 01 0 99[ln( ( ) ) ln( . . )] q N q q N q . (34)

Note that theLRuc unconditional coverage test is uniformly the most powerful for a given sample

size and has an asymptotic 2(1) distribution.23

Table 6 presents the likelihood ratio test results from testing the null hypothesis that the

reported VaR estimates are acceptably accurate. According to the LR statistics for the binomial

method, the standard approach that assumes normality of asset returns is strongly rejected for the

0.5%, 1%, 1.5%, 2.5%, and 5% VaRs for both the maximal and minimal returns. The extreme value

distributions (Box-Cox-GEV, GPD, GEV) produce acceptably accurate VaR estimates for all tails

and for all data sets considered in the paper. As shown in Panel B of Table 6, the SGT and IHS

distributions cannot be rejected for all VaR tails and for both the maximal and minimal returns on

S&P 500. In general, the flexible distributions produce acceptably accurate VaR measures except

for few cases for the minimal returns on S&P 500: the 2% VaR estimates of EGB2, and the 2%,

2.5%, and 5% VaR estimates of SGED are found to be inaccurate at the 5% level of significance.

The relative performance of the flexible distributions turns out to be slightly lower for the Dow

Jones Industrial Average. Panel A of Table 6 indicates that the VaR estimates of SGED and EGB2

be inaccurate for the 0.5%, 1%, 2.5%, and 5% VaR tails. However, the VaR measures obtained


23/44

23

from the SGT and IHS distributions are found to be very precise for the Dow 30 index, except for

the 0.5% tail of the minimal returns.

As discussed by Christoffersen (1998), VaR estimates can be viewed as interval forecasts of

the lower tail of the return distribution. Interval forecasts can be evaluated conditionally and

unconditionally, that is, with or without reference to the information available at each point in time.

The LRuc given in equation (34) is an unconditional test statistic because it simply counts

exceedences (or violations) over the entire period. However, in the presence of volatility clustering

or volatility persistence, the conditional accuracy of VaR estimates becomes an important issue. The

VaR models that ignore mean-volatility dynamics may have correct unconditional coverage, but at

any given time, they will have incorrect conditional coverage. In such cases, the LRuc test is of

limited use since it will classify inaccurate VaR estimates as acceptably accurate. Moreover, as

indicated by Kupiec (1995), Christoffersen (1998), and Berkowitz (2001), the unconditional

coverage tests have low power against alternative hypotheses if the sample size is small. This

problem does not exist here since our daily data sets cover a long period of time (28,758

observations for DJIA and 12,832 observations for S&P500).

The conditional coverage test developed by Christoffersen (1998) determines whether the

VaR estimates exhibit both correct unconditional coverage and serial independence. In other words,

if a VaR model produces acceptably accurate conditional thresholds then not only the exceedences

implied by the model occurx % (say 1 %) of the time, but they are also independent and identically

distributed over time. Given a set of VaR estimates, the indicator variable is constructed as

=ursedence occif no exce

nce occursif exceedeIt

0

1(35)


24/44

24

and should follow an iid Bernoulli sequence with the targeted exceedence rate (say 1 %). The LRcc

test is a joint test of two properties: correct unconditional coverage and serial independence,

LRcc = LRuc + LRind, (36)

which is asymptotically distributed as the Chi-squared with two degrees of freedom, 2(2). TheLRind

is the likelihood ratio test statistic for the null hypothesis of serial independence against the

alternative of first-order Markov dependence. We calculated the LRind andLRcc statistics for all the

distribution functions considered in the paper.24

We also computed the first-order serial correlation

coefficient corr(It,It-1), a diagnostic suggested by Christoffersen and Diebold (2000), and test its

statistical significance. The LRcc and corr(It,It-1) statistics indicate a strong rejection of the null

hypothesis, implying that given an exceedence (or violation) on one day there is a high probability

of a violation the next day.

VII. A Conditional VaR Approach with Flexible Distributions

These findings in the previous section suggest that the actual thresholds are time-varying to a

degree not captured by the unconditional VaR models. The earlier results are based on a sound

statistical theory, but do not yield VaR measures reflecting the current volatility background. In light

of the fact that conditional heteroskedasticity and serial correlation are present in most financial

time series, the unconditional VaR estimates cannot provide an accurate characterization of the

actual thresholds.25

We extend the unconditional VaR approach by taking into account the dynamic

behavior of financial return volatility in extreme values. In order to improve the existing VaR

methods we use the absolute GARCH (ABS-GARCH) process of Taylor (1986) and Schwert (1989)

that takes into account time-varying volatility characterized by persistence, considers the conditional

non-normality of returns, and deals with extreme events.


25/44

25

Following the introduction of ARCH models by Engle (1982) and their generalization by

Bollerslev (1986), there have been numerous refinements of this approach to modeling conditional

variance. The symmetric and asymmetric GARCH models, that parameterize the current conditional

variance as a function of the last periods squared shocks and the last periods variance, produce

dramatic increases in variance for very large shocks. The exponential GARCH (EGARCH) model

of Nelson (1991), that defines the current conditional variance as an exponential function of the

lagged shocks, lagged absolute shocks, and lagged log-variance, also overestimates the actual

variance during highly volatile periods. Friedman and Laibson (1989) argue that large shocks

constitute extraordinary events and propose to truncate their influence on the conditional variance.

Taylor (1986) and Schwert (1989) suggest a less drastic approach. They propose an ARCH model

that specifies the conditional standard deviation as a moving average of lagged absolute residuals.

Since the GARCH and EGARCH models may overestimate the actual VaR thresholds, we choose to

use the ABS-GARCH process to be able to estimate extreme return volatility more accurately.

The following AR(1) ABS-GARCH(1,1) conditional mean-volatility specification is utilized

with the generalized error distribution to estimate time-varying conditional VaR thresholds:

tttt zRR ++= 110 (37)

1211101| || ++= ttttt z (38)

)(1/2

|/|)2/1exp[()|(

]/)1[(1v

zvzf

vv

v

tttv

= + (39)

where

2/1)/2(

)/3(

)/1(2

=

v

vv,

1|

1|

=tt

ttt

t

Rz

is drawn from the conditional GED densityfv(zt, t-1),

and can be viewed as an unexpected shock to the stock market, and v is the degrees of freedom or


26/44

26

tail-thickness parameter.26

t|t-1= 0 + 1Rt-1 is the conditional mean, | 1t t is the conditional

standard deviation, and t-1 is the information set at time t1.27

The conditional value-at-risk measure with coverage probability, , at time t is defined as

the conditional quintile,t|t-1(), where

Pr(Rtt|t-1()| t-1) = . (40)

The conditionality of the VaR measure is crucial because of the dynamic behavior of asset returns.

When the innovations are assumed to be Gaussian, the conditional VaR threshold is

t|t-1() = t|t-1 +t|t-1 (41)

where t|t-1and t|t-1 are the estimated conditional mean and volatility of daily log-returns Rt, and

is the critical value for the normal distribution, e.g., = 2.326 for = 1%. When estimating the

conditional VaR thresholds for the flexible distributions, is calculated by integrating the area

under the unconditional probability density function of standardized returns, i.e., ( )| 1 | 1.t t t t t R

An implicit assumption in the context of risk management is that the return series

standardized by the conditional mean and conditional standardized deviation volatility are i.i.d. We

should note that in our case the transformation of conditional volatility is based on the ABS-

GARCH process given in equation (38). If (Rtt|t-1)/t|t-1 is not i.i.d for any transformation of

volatility, then the conditional volatility alone is not sufficient for characterization of conditional

VaR threshold. As discussed in Christoffersen, Hahn, and Inoue (2001), if (Rt t|t-1)/t|t-1 is i.i.d

then the conditional quantile is some linear function of volatility, where the relevant coefficients of

such a linear function [e.g., in eq. (41)] is determined by the common distribution of the

standardized returns.


27/44

27

We compare the risk management performance of alternative distribution functions for the

standardized residuals. The estimated and actual counts as well as the likelihood ratio test statistics

are presented in Table 7.28

The relative performance of distribution functions is not affected when

conditional heteroskedasticity and serial correlation are eliminated from the original data. The

extreme value and flexible distributions outperform the normal distribution in estimating value at

risk and provide the best predictions of catastrophic market risks for all quantiles considered in the

paper. The SGT and IHS distributions produce very similar VaR thresholds for the standardized

returns and perform better than the SGED and EGB2 distributions. The results provide strong

evidence that it is not true as stated by Longin (2000) and Bali (2001a) that one is obliged to use

the distribution of extremes only to be able to obtain precise VaR measures. The same accuracy

level can be obtained from the distribution of all returns if the flexible distributions, which take into

account skewness, leptokurtosis and volatility clustering in the financial data, are used in estimating

conditional VaR thresholds.

VIII. Conclusions

This paper provides strong evidence that the skewed fat-tailed distributions perform as well

as the extreme value distributions in modeling the asymptotic behavior of stock market returns.

They yield a more precise and robust approach to risk management and value-at-risk calculations

than the symmetric thin-tailed normal distribution. A notable point is that the SGT and IHS

distributions produce very similar VaR thresholds, and perform better than the SGED and EGB2

distributions in approximating the extreme tails of the return distribution. It is also important to note

that the Box-Cox-GEV distribution gives a more accurate characterization of the actual data than the

GPD and GEV distributions because the truth is somewhere in between.


28/44

28

The statistical results indicate that given the conditional heteroskedasticity and serial

correlation of most financial data, using an unconditional density function is a major drawback of

any kind of VaR-estimator. Therefore, the unconditional VaR approach is extended to take into

account the dynamic behavior of the mean and volatility of equity returns in extreme values. A

conditional VaR approach is proposed based on the standardized residuals modeled with an

autoregressive absolute GARCH process. The conditional coverage test results indicate that the

actual thresholds are time varying to a degree not captured by the conditional normal distribution.

The conditional skewed fat-tailed distributions, however, perform considerably well in modeling the

time-varying conditional VaR thresholds. Based on the likelihood ratio tests, the normal distribution

is strongly rejected in favor of the extreme value and skewed fat-tailed distributions.


29/44

29

References

Bali, T. G., 2001a. An Extreme Value Approach to Estimating Volatility and Value at Risk.

Journal of Business forthcoming.

Bali, T. G., 2001b. The Generalized Extreme Value Distribution: Implications for the Value atRisk. Working Paper, Baruch College, City University of New York.

Basak, S., and A. Shapiro, 2000. Value-at-Risk Based Risk Management: Optimal Policies and

Asset Prices.Review of Financial Studies 14, 371-405.

Berkowitz, J., 2001. Testing Density Forecasts With Applications to Risk Management.Journal

of Business and Economic Statistics 19, 465-474.

Berkowitz, J., and J. OBrian, 2001. How Accurate Are Value-at-Risk Models at Commercial

Banks.Journal of Finance forthcoming.

Bollerslev, T., 1986. Generalized Autoregressive Conditional Heteroscedasticity.Journal of

Econometrics 31, 307-327.

Booth, G. G., J. P. Broussard, T. Martikainen, and V. Puttonen, 1997. Prudent Margin Levels in

the Finnish Stock Index Futures Market. Management Science 43, 1177-1188.

Box, G., and D. Cox, 1964. An Analysis of Transformations.Journal of the Royal Statistical

Society, Series B, 211-264.

Box, G., and G. C. Tiao, 1962. A Further Look at Robustness Via Bayes Theorem,Biometrika,

49, 419-432.

Christoffersen, P. F., 1998. Evaluating Interval Forecasts.International Economic Review 39,

841-862.

Christoffersen, P. F., and F. X., Diebold, 2000. How Relevant Is Volatility Forecasting For

Financial Risk Management.Review of Economics and Statistics 82, 12-22.

Christoffersen, P. F., J. Hahn, and A. Inoue, 2001. Testing and Comparing Value-at-Risk

Measures.Journal of Empirical Finance 8, 325-342.

Diebold, F. X., T. A., Gunther, and A. S. Tay, 1998. Evaluating Density Forecasts.International

Economic Review 39, 863-883.

Dowd, K., 1998.Beyond Value at Risk: The New Science of Risk Management. John Wiley&Sons.

Duffie, D., and J. Pan, 1997. An Overview of Value at Risk.Journal of Derivatives Spring, 7-49.


30/44

30

Embrechts, P., C. Kluppelberg, and T. Mikosch, 1997. Modeling Extremal Events, Springer: Berlin

Heidelberg.

Engle, R. F., 1982. Autoregressive Conditional Heteroscedasticity with Estimates of the Variance

of United Kingdom Inflation.Econometrica 50, 987-1007.

Falk, M., 1985. Asymptotic Normality of the Kernel Quantile Estimator. The Annals of Statistics

13, 428-433.

Fisher, R. A., and L. H. C., Tippett, 1928. Limiting Forms of the Frequency Distribution of the

Largest or Smallest Member of a Sample.Proc. Cambridge Philos. Soc. 24, 180-190.

Friedman, B. M., and D. I. Laibson., 1989. Economic Implications of Extraordinary Movements in

Stock Prices.Brookings Papers on Economic Activity 2, 137-189.

Gnedenko, B.V., 1943. Sur la distribution limite du terme maximum dune serie aleatoire.Annals

of Mathematics 44, 423-453.

Gourieroux, C., and J. Jasiak, 1999. Truncated Local Likelihood and Nonparametric Tail Analysis. DP 99,

CREST.

Gourieroux, C., Laurent, J. P., and O. Scaillet, 2000. Sensitivity Analysis of Values at Risk.

Journal of Empirical Finance 7, 225-245.

Gumbel, E. J., 1958. Statistics of Extremes, Columbia University Press: New York.

Hansen, B. E. 1994, Autoregressive Conditional Density Estimation,International Economic

Review, 35, 3, 705-730.

Hansen, C. B., J. B. McDonald, and P. Theodossiou, 2001. Some Flexible Parametric Models and

Leptokurtic Data, Working Paper, Brigham Young University.

Harrel, F., and C. Davis, 1982. A New Distribution Free Quantile Estimation.Biometrica 69, 635-

640.

Hendricks, D., 1996. Evaluation of Value-at-Risk Models Using Historical Data.Economic Policy

Review, Federal Reserve Bank of New York April, 39-69.

Hosking, J. R. M., 1984. Testing Whether the Shape Parameter is Zero in the Generalized ExtremeValue Distribution.Biometrika 71, 367-374.

Hull, J, and A. White, 1998. Value at Risk When Daily Changes in Market Variables Are Not

Normally Distributed. Journal of Derivatives Spring, 9-19.


31/44

31

Jenkinson, A. F., 1955. The Frequency Distribution of the Annual Maximum (or Minimum)

Values of Meteorological Elements. Quarterly Journal of the Royal Meteorology Society 87, 145-

158.

Johnson, N. L. 1949. Systems of Frequency Curves Generated by Methods of Translation,

Biometrica, 36, 149-176.

Johnson, N. L., S. Kotz, and N. Balakrishnan, 1994. Continuous Univariate Distributions, Volume

1, Second Edition, New York: John Wiley & Sons.

Jorion, P., 1996. Risk2: Measuring the Risk in Value at Risk.Financial Analysts

November/December, 47-56.

Jorion, P., 2001. Value-at-Risk: The New Benchmark for Controlling Market Risk. McGraw-Hill:

Chicago.

Kupiec, P. H., 1995. Techniques for Verifying the Accuracy of Risk Measurement Models.Journal of Derivatives 3, 73-84.

Longin, F. M., 2000. From Value at Risk to Stress Testing: The Extreme Value Approach.

Journal of Banking and Finance 24, 1097-1130.

Lopez, J. A., 1998. Methods for Evaluating Value-at-Risk Estimates.Economic Policy Review,

Federal Reserve Bank of New York October, 119-124.

McDonald, J. B. and W. K. Newey 1988. Partially Adaptive Estimation of Regression Models Via

the Generalized t Distribution, Econometric Theory, 4, 428-457.

McDonald, J. B. and Y. J. Xu, 1995. A Generalization of the Beta Distribution with Applications,

Journal of Econometrics, 66, 133-152. Errata 69(1995), 427-428.

McNeil, A. J., and R. Frey, 2000. Estimation of Tail-Related Risk Measures for Heteroscedastic

Financial Time Series: An Extreme Value Approach.Journal of Empirical Finance 7, 271-300.

Nelson, D. 1991. Conditional Heteroskedasticity in Asset Returns: A New Approach,

Econometrica, 59, 347-370.

Pickands, J., 1975. Statistical Inference Using Extreme Order Statistics.Annals of Statistics 3,

119-131.

Prescott, P., and A.T. Walden, 1980. Maximum-likelihood estimation of the parameters of the

three-parameter generalized extreme-value distribution.Biometrica 67, 723-724.

Schwert, G. W., 1989. Why Does Stock Market Volatility Change Over Time?Journal of

Finance 44, 1115-1153.


32/44

32

Smith, R. L., 1987. Estimating Tails of Probability Distributions. The Annals of Statistics 15,

1174-1207.

Subbotin, M. T. H., 1923. On the Law of Frequency of Error, Matematicheskii Sbornik, 31, 296-

301.

Taylor, S., 1986. Modeling Financial Time Series. New York, NY: Wiley.

Theodossiou, P., 1998. Financial Data and the Skewed Generalized t Distribution. Management

Science 44, 1650-1661.

Theodossiou, P., 2001. Skewness and Kurtosis in Financial Data and the Pricing of Options,

Working Paper, Rutgers University.

Topaloglou, N., H. Vladimirou, and S. A., Zenios, 2001. CVaR Models with Selective Hedging for

International Asset Allocation. Working Paper 01-23, HERMES Center of Excellence in

Computational Finance and Economics, School of Economics and Management, University ofCyprus.

Venkataraman, S., 1997. Value at Risk for a Mixture of Normal Distributions: The Use of Quasi-

Bayesian Estimation Techniques.Economic Perspectives, Federal Reserve Bank of Chicago

March/April, 2-13.


33/44

33

Table 1Descriptive Statistics

Panel A shows the descriptive statistics of daily percentage returns on the Dow Jones Industrial

Average (DJIA) and S&P 500 stock market indices. To compute stock market returns ( rt) we use

the formula, rt= lnPtlnPt-1, wherePt is the value of the stock market index at time t. The sample

period, number of observations, maximum and minimum values, mean, standard deviation,

skewness, and excess kurtosis statistics are reported for daily returns.*,**

denote significance at the

5% and 1% level, respectively. Panel B displays the number of extremes, the means, standard

deviations, maximum, and minimum values of extremes. Maximal and minimal returns are defined

as excesses over high threshold, which is set as the 5 percent of the right and left tails of the

empirical distribution.

Panel A: Summary Statistics of Daily Returns

DJIA

(5/ 26/ 189612/ 29/ 00)

S&P 500

(1/ 4/ 5012/ 29/ 00)

# of obs. 28,758 12,832

Maximum 14.27 8.7089

Minimum 27.96 22.899

Mean 0.0194 0.0341

Std. Dev. 1.0898 0.8740

Skewness 1.1761**

1.6216**

Kurtosis 39.553** 45.525**

Panel B: Summary Statistics of Extremes

Maxima n Mean Std. Dev. Maximum Minimum

DJIA 1,438 2.4071 1.1892 14.273 1.5291

S&P 500 642 1.9607 0.7039 8.7089 1.3491

Minima n Mean Std. Dev. Maximum Minimum

DJIA 1,438 2.6000 1.5011 1.5742 27.959

S&P 500 642 1.9741 1.1656 1.3150 22.900


34/44

34

Table 2Regression Method Estimates of the Extreme Value Distributions

Panels A and B show the estimates for the location (), scale (), shape (), and Box-Cox-GEV () parametersof the extreme value distributions. The shape parameter, called the tail index, reflects the weight of the tail ofthe distribution. Asymptotic t-statistics are given in parentheses. The maximized log-likelihood values (Log-L)

are reported. The likelihood ratio test results between the Box-Cox-GEV and GPD, and between the Box-Cox-

GEV and GEV distributions are shown in the last column.

Panel A: Estimat ion Results for the DJIA

Maxima max max max max Log-L LR

Box-Cox-GEV0.01527

(1392.98)

0.00656

(415.21)

0.2641

(217.99)0.02 2861.76 ----------

GPD0.01524

(1382.87)

0.00657

(414.99)

0.2635

(217.70)0.00 2860.54 2.44

GEV0.01882

(1109.47)

0.00409

(179.94)

0.4238

(111.39)1.00 2103.05 1517.42

Minima min min min min Log-L LR

Box-Cox-GEV0.01837

(1158.09)

0.00625

(242.63)

0.3214

(133.90)0.66 1674.28 ----------

GPD0.01597

(544.03)

0.00727

(178.86)

0.2817

(98.619)0.00 1535.55 277.46

GEV0.01999

(966.88)

0.00489

(180.70)

0.4525

(105.29)1.00 839.20 1670.16

Panel B: Estimation Results for the S&P 500

Maxima max max max max Log-L LR

Box-Cox-GEV0.01485

(1013.19)

0.00440

(188.51)

0.2083

(70.621)0.51 892.52 ----------

GPD0.01362

(582.82)

0.00503

(152.53)

0.1701

(51.295)0.00 856.16 72.72

GEV

0.01528

(855.58)

0.00316

(126.71)

0.3291

(59.807) 1.00 574.77 635.50

Minima min min min min Log-L LR

Box-Cox-GEV0.01559

(1296.42)

0.00303

(180.99)

0.4527

(118.10)0.94 660.89 ----------

GPD0.01360

(444.77)

0.00383

(90.081)

0.3906

(65.745)0.00 510.03 301.72

GEV0.01479

(1139.78)

0.00283

(167.18)

0.4578

(108.19)1.00 541.24 239.30


35/44

Table 3Maximum Likelihood Estimates of the Normal and Flexible Distribution

This table presents the parameter estimates of the normal and flexible distributions. Asymptotic t-statistics are given

the estimated mean, standard deviation, skewness, and kurtosis statistics. The sample moments for DJIA and S&P 5

Normal is the likelihood ratio test statistic from testing the null hypothesis that the daily returns are distributed

hypothesis that they are distributed as SGT, SGED, and EGB2. LR-Laplace is the LR statistic for testing the

distributed as Laplace against the alternative hypothesis that they are distributed as SGT and SGED.KSis the Kolm

the null hypothesis that the data follow the corresponding distribution function. *,** denote significance at the 5% and

Panel A: Estimation Results for the DJIA

k n Sk Ku Log-L

SGT0.02156

(3.56)**

1.09422

(66.98)**

1.76012

(34.51)**

0.04640

(6.16)**

3.39491

(24.43)**

0.7862 ------- 38721.37 9

SGED0.01955

(3.34)**

1.02933

(177.66)**

0.93937

(145.19)**

0.04981

(9.30)**

------- 0.2303 6.6108 39037.57 8

EGB20.01937

(3.50)

**

1.00145

(200.31)

**

0.14194

(9.90)

**

0.15613

(9.85)

**

------- 0.1929 5.8557 39025.33 8

IHS0.01985

(3.23)**

1.06217

(108.73)**

1.14131

(85.96)**

0.07399

(6.43)**

------- 0.4382 17.247 38727.52

Normal0.01938

(2.96)**

1.08957

(1034.4)** 2 0 ------- 0 3 43272.76

Panel B: Estimation Results for the S&P 500 Composite Index

k n Sk Ku Log-L

SGT0.03559

(4.82)**

0.85012

(86.63)**

1.60084

(26.42)**

0.02455

(2.23)*

5.21132

(12.79)**

0.1336 11.045 15140.31 2

SGED0.03383

(4.64)**

0.84232

(124.57)**

1.05063

(110.61)**

0.02775

(3.23)**

------- 0.1097 5.6125 15223.02 2

EGB20.03412

(4.81)**

0.83361

(125.93)**

0.23774

(10.03)**

0.25270

(9.80)**

------- 0.1166 5.6238 15191.77 2

IHS0.03502

(4.67)**

0.85512

(85.29)**

1.30407

(49.92)**

0.04021

(2.27)*

------- 0.1712 10.041 15146.84

Normal0.03408

(0.03)

0.87397

(0.87)** 2 0 ------- 0 3 16479.24


36/44

Table 4Estimated VaR Thresholds by the Extreme Value, Normal, and Flexible Distr

Panels A and B present the VaR thresholds estimated by the extreme value, normal, and flexible distributions for

indices. Thresholds for the extreme value distributions are calculated based on the regression method estimates of

normal distribution are computed using the maximum likelihood estimates of the mean and standard deviation para

distributions are estimated by calculating the area under the probability density function.

Panel A: Estimated VaR Thresholds for the DJIA Index

Maxima Box-Cox-GEV GPD GEV Normal SGT SGED

= 0.5% 3.6046% 3.6047% 3.4817% 2.8260% 3.5320% 3.3033%

= 1% 2.8411% 2.8410% 2.7608% 2.5542% 2.8247% 2.7956%

= 1.5% 2.4550% 2.4548% 2.4199% 2.3839% 2.4571% 2.5005%

= 2% 2.2050% 2.2047% 2.2036% 2.2571% 2.2151% 2.2922%

= 2.5% 2.0237% 2.0235% 2.0455% 2.1550% 2.0374% 2.1313%

= 5% 1.5236% 1.5237% 1.5397% 1.8117% 1.5333% 1.6354%

Minima Box-Cox-GEV GPD GEV Normal SGT SGED

= 0.5% 3.9241% 3.9534% 3.8398% 2.7872% 3.7822% 3.5378%

= 1% 3.0817% 3.0775% 3.0243% 2.5154% 3.0032% 2.9770%

= 1.5% 2.6577% 2.6390% 2.6317% 2.3452% 2.5953% 2.6510%

= 2% 2.3804% 2.3570% 2.3796% 2.2184% 2.3261% 2.4207%

= 2.5% 2.1756% 2.1534% 2.1936% 2.1163% 2.1282% 2.2427%

= 5% 1.5528% 1.5968% 1.5369% 1.7730% 1.5667% 1.6940%


37/44

Panel B: Estimated VaR Thresholds for the S&P 500 Composite Index


= 0.5% 2.7659% 2.7790% 2.7068% 2.2853% 2.7554% 2.6715%

= 1%

2.2934% 2.2926% 2.2537% 2.0673% 2.2865% 2.2891% = 1.5% 2.0411% 2.0336% 2.0246% 1.9307% 2.0306% 2.0642%

= 2% 1.8703% 1.8603% 1.8724% 1.8290% 1.8564% 1.9039%

= 2.5% 1.7413% 1.7317% 1.7572% 1.7471% 1.7251% 1.7792%

= 5% 1.3429% 1.3618% 1.3607% 1.4717% 1.3353% 1.3895%


= 0.5%2.7486% 2.7898% 2.7301% 2.2172% 2.7941% 2.7177%

= 1% 2.2154% 2.2181% 2.2011% 1.9991% 2.3000% 2.3138%

= 1.5% 1.9640% 1.9489% 1.9561% 1.8626% 2.0302% 2.0762%

= 2% 1.8052% 1.7821% 1.8027% 1.7609% 1.8464% 1.9069%

= 2.5% 1.6899% 1.6650% 1.6919% 1.6790% 1.7079% 1.7751%

= 5% 1.2977% 1.3601% 1.3339% 1.4036% 1.2970% 1.3635%


38/44

Table 5Risk Management Performance of the Extreme Value, Normal, and Flexible Dis

Panels A and B compare the risk management performance of the extreme value, normal, and flexible distribu

Average and S&P 500 composite indices. The column actual presents the number of observations that fall in the

tails of the returns distribution. Given that there are 28,758 daily returns on DJIA from May 26, 1896 through De

approximately 144, 288, 431, 575, 719, and 1438 observations to fall into each 0.5%, 1%, 1.5%, 2%, 2.5%, and 5

daily returns on S&P 500 from January 4, 1950 through December 29, 2000 we would expect approximatelyobservations to fall into each 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% tail.

Panel A: Actual and Estimated Counts for the DJIA

Maxima Actual Box-Cox-GEV GPD GEV Normal SGT SGE

= 0.5% 144 142 142 159 295 154 191

= 1% 288 287 287 308 390 291 299

= 1.5% 431 424 425 444 479 424 401 = 2% 575 578 578 579 552 569 518

= 2.5% 719 707 707 686 616 694 617

= 5% 1438 1449 1448 1413 975 1426 122

Minima Actual Box-Cox-GEV GPD GEV Normal SGT SGE

= 0.5% 144 157 154 167 377 170 211

= 1% 288 304 304 316 480 320 328

= 1.5% 431 431 432 436 585 449 431 = 2% 575 571 592 572 672 607 540

= 2.5% 719 717 736 702 747 750 669

= 5% 1438 1475 1401 1503 1100 1450 122

Average MA%E --------- 2.04% 2.22% 5.14% 42.53% 4.92% 14.21


39/44

Panel B: Actual and E stimated Counts for the S&P 500 Composite Index


= 0.5% 64 64 63 69 135 64 74

= 1% 128 125 125 134 197 125 125

= 1.5% 192 195 197 199 251 198 186

= 2% 257 262 264 259 290 266 249

= 2.5% 321 319 329 308 337 330 295

= 5% 642 648 630 630 549 651 594


= 0.5% 64 59 56 62 130 56 65

= 1% 128 137 135 137 178 113 112

= 1.5% 192 198 204 202 223 179 168

= 2% 257 272 280 274 275 247 219

= 2.5% 321 331 339 331 316 321 281

= 5% 642 662 597 619 516 662 592

Average MA%E --------- 3.12% 4.95% 4.29% 34.53% 4.26% 8.45


40/44

Table 6Likelihood Ratio Test Results for the VaR Estimates of the Extreme Value, Normal, and F

This table tests the accuracy of VaR estimates obtained from the extreme value, normal, and flexible distributions

binomial method is implemented in a hypothesis-testing framework and is used to test the null hypothesis that the rep

accurate. Panels A and B show the likelihood ratio test results for the Dow Jones Industrial Average and S&P 500 c

with one degree of freedom at the 5% and 1% level of significance are 2

(1,0.05) = 3.84 and 2

(1,0.01) = 6.63.*

,**

imply ththe corresponding distribution function are not accurate the 5% and 1% level of significance, respectively.

Panel A: LR Test Resu lts for the DJIA


= 0.5% 0.02 0.02 1.56 122.37** 0.71 14.12**

= 1% 0.00 0.00 1.43 33.15** 0.04 0.45

= 1.5% 0.13 0.10 0.37 5.16* 0.13 2.22

= 2% 0.01 0.01 0.03 0.96 0.07 5.99*

= 2.5% 0.20 0.20 1.57 15.88** 0.90 15.56**

= 5% 0.09 0.07 0.46 176.02** 0.10 34.19**


= 0.5% 1.18 0.71 3.57 262.26** 4.54* 27.58**

= 1% 0.93 0.93 2.75 108.26**

3.56 5.49*

= 1.5% 0.00 0.00 0.05 50.01** 0.72 0.00

= 2% 0.03 0.50 0.02 15.79** 1.77 2.24

= 2.5% 0.01 0.41 0.41 1.11 1.36 3.64

= 5% 0.99 1.01 3.06 90.64** 0.11 35.20**


41/44

Panel B: LR Test Results for S&P 500 Composite Index


= 0.5% 0.00 0.02 0.36 59.57** 0.00 1.44

= 1%

0.09 0.09 0.25 31.91

**

0.09 0.09 = 1.5% 0.03 0.11 0.22 16.49** 0.16 0.22

= 2% 0.11 0.21 0.02 4.25* 0.34 0.23

= 2.5% 0.01 0.21 0.53 0.83 0.27 2.19

= 5% 0.07 0.22 0.22 14.76** 0.14 3.81


= 0.5%0.43 1.09 0.07 52.26

**

1.09

0.01 = 1% 0.58 0.35 0.58 17.34** 1.92 2.19

= 1.5% 0.16 0.69 0.47 4.68* 0.98 3.30

= 2% 0.92 2.11 1.17 1.31 0.37 5.92*

= 2.5% 0.33 1.04 0.33 0.07 0.00 5.28*

= 5% 0.68 3.34 0.85 27.66** 0.68 4.14*


42/44

Table 7Risk Management Performance of Alternative Distribution Functions for the Standar

This table compares the risk management performance of alternative distribution functions for the standardized res

that fall in the 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% tails of the return distribution are presented below. Given that the

for the S&P 500 from January 4, 1950 through December 29, 2000 we would expect approximately 64, 128, 192, 25

into each 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% tail. The LR test statistics are reported in parentheses.*,

**denote th

respectively.


= 0.5% 64 64 (0.00) 65 (0.01) 67 (0.12) 89 (8.63)** 51 (2.91) 55 (1

= 1% 128 136 (0.46) 138 (0.72) 144 (1.86) 144 (1.87) 111 (2.47) 110 (2

= 1.5% 192 200 (0.29) 203 (0.57) 201 (0.37) 197 (0.11) 176 (1.47) 161 (5

= 2% 257 270 (0.70) 278 (1.77) 265 (0.28) 247 (0.37)

243 (0.75) 220 (5 = 2.5% 321 342 (1.41) 349 (2.47) 331 (0.33) 296 (2.01) 303 (1.03) 278 (6

= 5% 642 630 (0.22) 618 (0.92) 618 (0.92) 541 (17.46)** 630 (0.22) 579 (6


= 0.5% 64 59 (0.28) 55 (1.38) 63 (0.02) 123 (42.71)** 63 (0.02) 64 (0

= 1% 128 137 (0.01) 126 (0.04) 131 (0.06) 180 (18.70)** 120 (0.55) 113 (1

= 1.5% 192 198 (0.69) 207 (1.09) 209 (1.40) 234 (8.52)** 180 (0.84) 163 (4

= 2% 257 272 (0.02) 266 (0.34) 262 (0.11) 291 (4.51)*

243 (0.75) 210 (9

= 2.5% 321 331 (0.55) 348 (2.30) 336 (0.73) 352 (3.03) 324 (0.03) 266 (10

= 5% 642 662 (0.15) 627 (0.35) 623 (0.57) 595 (3.63) 654 (0.25) 576 (7

Average

MA%E--------- 3.63% 6.12% 4.51% 22.21% 6.43% 12.87


43/44

43

Endnotes

1Orange County, Barings Bank, Metallgesellschaft, Showa Shell, Proctor and Gamble, Daiwa Bank, Nat West,

Sumitomo Corporation Some of the worlds largest financial entities have lost billions of dollars in financial

markets. In most cases, senior management poorly monitored the exposure to market risks. To address this

problem, the worlds leading banks and financial firms have turned to value at risk, an easy-to-understand

method for calculating and controlling market risks.2 For example, if the given period of time is one day and the given probability is 1 %, the VaR measure would

be an estimate of the decline in the portfolio value that could occur with a 1 % probability over the next trading

day. In other words, if the VaR measure is accurate, losses greater than the VaR measure should occur less than

1 % of the time.3

In addition to the variance-covariance analysis and historical simulation, fully non-parametric approaches

have been proposed and determine the empirical quantile (the historical VaR) or a smoothed version of it

[Harrel and Davis (1992), Falk (1985), Jorion (1996), and Gourieroux et al. (2000)]. Recently, semi-parametric

approaches have been developed. They are based on either extreme value distributions [Longin (2000), McNeil

and Frey (2000), and Bali (2001a)] or local likelihood methods [Gourieroux and Jasiak (1999a)]. Topaloglou,

Vladimirou, and Zenios (2001) develop a conditional VaR model in the context of scenario analysis. For a

comprehensive survey on value at risk models, see Duffie and Pan (1997), Dowd (1998), and Jorion (2001). For

testing and comparing value-at-risk models, see Hendricks (1996), Lopez (1998), and Christoffersen, Hahn, and

Inoue (2001). The reader may wish to consult Basak and Shapiro (2000), Berkowitz (2001), and Berkowitz and

OBrian (2001)for shortcomings of VaR as a risk management tool.4

Based on the actual and estimated VaR thresholds as well as the likelihood ratio test results, the extreme

value,IHSand SGTdistributions perform much better than the normal distribution in capturing both the rate of

occurrence and the extent of extreme events in financial markets. Thus, they provide a more natural and robust

approach to risk management calculations.5

The significance of GPD in extreme value theory is first observed by Pickands (1975).6

In equations (13)-(15), the shape parameter, , determines the tail behavior of the distributions. For> 0, thedistribution has a polynomially decreasing tail (Pareto). For = 0, the tail decreases exponentially(exponential). For< 0, the distribution is short tailed (uniform).7

The reader may wish to consult Pickands (1975) and Smith (1987) for parameter estimation of the generalized

Pareto distribution (GPD), Prescott and Walden (1980), Hosking (1984), and Booth et al. (1997) for parameterestimation of the generalized extreme value distribution (GEV), and Bali (2001b) for parameter estimation of

the Box-Cox-GEV distribution.8

The IHS parameterization used in Johnson (1949) and Johnson et al. (1994) is w = sinh((-+z)/k). Our parameterization makes the interpretation of the scaling parameters and k easier and comparable acrossmodels.9

Hull and White (1998) and Venkataraman (1997) show that the risk management performance of standard

VaR models that assumes normality increases if one uses a mixture of normal distributions with quasi-Bayesian

estimation techniques.10 Note that unlike the normal and flexible VaR models, the extreme value VaR models are based on the

distribution of extreme returns only.11

The thresholds for the standard approach, Normal, are computed using the estimated mean and volatility

parameters of the normal distribution as well as the critical values: 2.5758, 2.326, 2.1701, 2.0536, 1.960, and1.645 for the 0.5%, 1%, 1.5%, 2%, 2.5%, and 5% VaR tails, respectively.12

In addition to the Dow Jones and S&P 500 stock market indices, at an earlier stage of the study, we use the

Nasdaq, S&P 100, and New York Stock Exchange (NYSE) indices. The risk management performance of

alternative distribution functions turns out to be similar to those obtained for the S&P 500. To preserve space,

we decide not to present the empirical results from the Nasdaq, S&P 100, and NYSE indices. They are available

from the authors upon request.13

It is well known in the extreme value literature that the maxima (and minima) of a random variable over fixed

time interval has a generalized extreme value (GEV) distribution and all exceedences (defined as all


44/44

44

observations above a high threshold) will have a generalized Pareto distribution (GPD). At an earlier stage of

the study, we obtained the extremes overn trading days (where n = 23 days or 1 month) and alternatively using

the mean excess function approach described in Embrechts et al. (1997). Since the qualitative results turn out

to be very similar we choose not to present them. They are available from the authors upon request.14

The likelihood ratio (LR) statistic is calculated asLR = -2 [Log-L*

- Log-L], where Log-L*

is the value of the

log likelihood under the null hypothesis, and Log-L is the log likelihood under the alternative. This statistic is

distributed as 2 with one degree of freedom.15

Note that the parameter of the Box-Cox-GEV distribution is estimated to be max = 0.02, which is very closeto zero, for the maximal returns on DJIA.16

The reader may wish to consult Bali (2001b) for a detailed discussion about the maximum likelihood and

regression method estimation of the Box-Cox-GEV distribution.17

Although not presented in the paper, the maximized log-likelihood values of the Gumbel and Exponential

distributions are found to be much lower than those of the GEV and GPD distributions for both the maximal

and minimal returns on DJIA and S&P 500.18

Although the Frechet and Pareto distributions provide a relatively more precise approximation of the tails

compared to the Gumbel, Exponential, Weibull, and Uniform distributions, the LR statistics given in Table 2

indicate that neither the Frechet nor the Pareto distribution yields an accurate characterization of extreme

movements in financial markets because they are strongly rejected against the Box-Cox-GEV distribution.19 The formulas for computing Sk and Ku are presented in Hansen et al. (2001). These formulas are complex

functions of the parameter estimates for skewness and kurtosis.20

The SGED is derived asymptotically from SGTby setting n = .21

According to a controversial item in the Bank for International Settlements (BIS) guidelines, financial

institutions are required by BIS to calculate a 99 % upper bound for their daily losses due to market risk, and

then are required to multiply this figure by at least 3. This multiplication factor is justified essentially by the

observed non-Gaussian nature of frequency distributions. The new VaR model proposed here is potentially

more precise, and yields a more satisfactory estimate for this factor. Based on the VaR estimates presented in

our tables, the BIS multiplication factor of 3 seems to be excessive since the extreme value approach implies up

to 40 % greater VaR than the traditional approach.22 In August 1996, the U.S. bank regulatory agencies adopted the market risk amendment (MRA) to the 1998

Basle Capital Accord. The MRA, which became effective in January 1998, requires that commercial banks withsignificant trading activities set aside capital to cover the market risk exposure in their trading accounts. The

market risk capital requirements are to be based on the VaR estimates generated by the banks own risk

management models.23

In this paper, we test the risk management performance of alternative distribution functions for 0.5 %, 1

%,, 5 % VaR tails. Therefore, our likelihood ratio tests are not limited to 1 % VaRs.24

We decide not to present theLRind andLRcc statistics since they are found to be significant at the 5% or 1%

level without any exception. They are available upon request.25

Density forecast evaluation techniques described in Diebold, Gunther, and Tay (1998), and Berkowitz (2001)

can be used to identify the weaknesses of the existing unconditional VaR models that use the distribution of all

returns.26

Note that forv = 2, the GED yields the normal distribution, while forv = 1 it yields the Laplace or the double

exponential distribution. Ifv < 2, the density has thicker tails than the normal, whereas for v > 2 it has thinnertails.27

We do not present the maximum likelihood parameter estimates of the GED-AR(1)-ABSGARCH(1,1) model

in order to preserve space. They are available from the authors upon request.28

To save space we choose not to present the estimated VaR thresholds. They are available upon request.

bali_risk management performance of alternative distribution functions_2002

Documents