CONDITIONAL AND UNCONDITIONAL RISK MANAGEMENTESTIMATES FOR EUROPEAN STOCK INDEX FUTURES

JOHN COTTER*University College Dublin

Address for Correspondence:John Cotter,Department of Banking and Finance,Graduate School of Business,University College Dublin,Blackrock,Co. Dublin,Ireland.

E-mail. john.cotter@ucd.iePh. +353 1 706 8900Fax. +353 1 283 5482

The author is a College Lecturer in the Graduate School of Business at UniversityCollege Dublin.

* Acknowledgements: The author would like to thank Donal McKillop and Ronan O’Connor for their helpful comments on this paper.

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CONDITIONAL AND UNCONDITIONAL RISK MANAGEMENTESTIMATES FOR EUROPEAN STOCK INDEX FUTURES

Abstract

Accurate forecasting of risk is the key to successful risk management techniques.

Correct modelling of a variable’s extreme values located at the distributional tails

accounting for the fat-tail phenomena is paramount, and this paper presents an

overview of a theoretically and statistically robust approach to this problem.

Underpinned by Extreme Value Theory that explicitly allows for fat-tailed densities,

this paper presents four precise measures of downside risk. Two Value at Risk and

two Excess Loss Probability estimators evolve from the conditional and unconditional

distributions. Fitting an AR(1)-GARCH(1, 1) filter provides the description of the

current volatility climate. Results are presented for a panorama of twelve stock index

futures traded on European bourses.

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CONDITIONAL AND UNCONDITIONAL RISK MANAGEMENTESTIMATES FOR EUROPEAN STOCK INDEX FUTURES

I INTRODUCTION

Given the fat-tailed characteristic of futures price changes, the assumption of

modelling these returns with the thin-tailed Gaussian distribution is inappropriate.1

Risk measures are underestimated under these conditions leading to invalid risk

management practices. Alternative approaches to measuring risk include historical

simulation, Monte Carlo simulation, and extreme value methods. This paper

primarily relies on the latter to provide four downside risk management techniques.

The validity of this theoretical framework is enhanced by the asymptotic behaviour of

the three distributions that come under the heading of the extreme value distributions.

One of these limiting distributions, the Fréchet distribution, has the fat-tailed

characteristic in line with futures returns. Whilst, futures returns may not exactly fit

the Frechét distribution, this problem is overcome by measuring the tail values with

the semi-parametric Hill index.

Previous applications of extreme value theory in risk management include Longin,

1999a; Danielsson and de Vries, 1997; and McNeil and Frey, 1998; where they

outline extreme value theory assuming that asset price changes are identical and

independently distributed (iid). However, the empirical evidence in the finance

literature (Taylor, 1986; Ding and Granger, 1993; and Ding and Granger, 1996)

provides contrary findings on this issue, but, does support the less restrictive

hypothesis that returns are stationary over time. This paper not only demonstrates

extreme value theory in its traditional form for the iid variable, but also, it presents the

conditions that are required for its correct application in the case of stationarity. This

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extension of the theory allows a stationary variable to be considered an associated iid

variable through the lack of long term dependence in futures returns, and the

imposition of an extremal index that measures the volatility clusters inherent in these

assets.

Whilst, the popular approach in risk management is to present a single risk measure,

namely the highly used Value at Risk (VaR), which outlines the extent of an

investment loss for a holding period and a specific confidence level. This paper

presents two distinct VaR measures, one dealing with the unconditional distribution

and one with the conditional distribution. The former provides risk managers with

information on different worst case scenarios dealing with market risk occurring over

long periods, for example ten years. In contrast, the latter measure details the present

risk facing investment managers conditional on the present risk and return

environment of a futures contract. In addition, two extra measures, based on the

conditional and unconditional distribution of returns, estimating the probability of

different loss thresholds being exceeded are given. These Excess Loss Probability

estimators (ELP) describe the extent of downside risk beyond VaR estimates.

Through a combination of historical simulation and an AR(1)-GARCH (1, 1) process,

the conditional distribution is generated for application with extreme value theory.

Illustrations of these techniques are presented for European stock index futures.

These assets are chosen as they are relatively risky vis-à-vis other more secure

investment assets; and also, give a panorama of risk levels for European bourses.

1 It is assumed that the investor takes a long position. The theoretical underpinnings of the paper canbe equally applied to a short position.

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The paper proceeds in the following section with an outline of extreme value theory in

the case of iid and stationary random variables. Section III details the methods

applied to generate the conditional and unconditional risk management estimates and

in particular, a description of the specific GARCH specification, modelling the

conditional risk and return environment is detailed. The results are presented for an

array of stock index futures from European bourses in section IV. Here, the

dependence structure of futures returns, as well as two conditional and unconditional

risk measures are presented. Finally, concluding comments are documented in

section V.

II EXTREME VALUE THEORY

Extreme Value Theory is detailed for the iid case as well as the less restrictive

stationary case.

Extreme Value Theory for the IID Case

The theoretical concepts underpinning the risk measures in this paper are from

Extreme Value Theory. This theory relies on order statistics where a set of

logarithmic futures returns {R1, R2,..., Rn} associated with days 1, 2.. n, are assumed

to be independent and identically distributed (iid), and belonging to the true unknown

distribution F. For a comprehensive discussion of extreme value results under a wide

range of distributional situations see Gumbell (1958) and Leadbetter et al (1983). As

we are dealing with large negative returns, we can examine the maxima (Mn) of a

sequence of n random variables rearranged in descending order where

Mn = Max{ R1, R2,..., Rn} (1)

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The corresponding density function of Mn is got from the cumulative probability

relationship:

P{Mn ≤ r} = P{R1 ≤ r, … , Rn ≤ r} = Fn(r) -∞ < r < ∞ (2)

The random variables of interest in this analysis are located at the tail of the

distribution, Fn(r). For example, the VaR measures the amount of possible loss

exposure upto the extreme return, r. Losses in excess of this extreme return can be

generated, and related measures of the probability of exceeding the extreme return, r,

are given by Excess Loss Probability (ELP) estimators for a range of possible losses.

While the exact distribution of the n random variables may not be known, we rely on

the Fisher-Tippett theorem to examine asymptotic behaviour of the distribution. From

this theorem, there are three types of limit laws and these make up the extreme value

distributions. Formally, taking the sequence of n random variables (returns in this

paper) that are assumed to be iid, from the non-degenerate distribution function H,

and by assumption there is convergence at the limit to the following types of the H

distribution:

Type I (Gumbell): Λ(r) = exp (- e -r ) -∞ < r < ∞

Type II (Frechét): Φ α(r) = 0 r ≤ 0 = exp (-r)(-α) = exp(-r)(-1/γ) r > 0

Type III (Weibull): ψ α(r) = exp (- (- r)(α)) = exp (- (- r)(1/γ) r ≤ 0 = 1 r > 0 (3)and for α > 0.

The types of limit distribution are distinguished by the shape parameter α in (3),

detailing the rate at which its tail converges asymptotically. This tail can decline

exponentially or by a power. The Gumbell distribution has a tail declining

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exponentially and all moments of the distribution are finite. Examples of this type of

extreme value distribution include thin tailed densities such as the normal and

exponential distributions. The Weibull distribution has a bounded tail with all

moments being finite, and examples of this distribution include the uniform or beta

distributions. However of paramount importance, the Frechét distribution offers

characteristics that are noted for the tails of financial returns, namely the fat tail

attribute. This type of extreme value distribution exhibits a tail with a power decline

causing a relatively slow decay for convergence towards the limit, vis-à-vis the

exponential decline of the type I process. Also, for this type of max-stable process,

the relatively slow decline in the tails generates moments that are not necessarily

always finite. In the finance literature, common modelling of derivative first

differences has taken place using ARCH related specifications (see Hull and White,

1998 for an example) which has relatively fat tails and belongs to a Frechét

Distribution.

Fortunately, the necessary and sufficient conditions for a distribution to

asymptotically converge on a type of extreme value distributions can be easily met,

and these are:

Type I (Gumbell): lim n [1 - F(αnr + βn)] = e-r

t → ∞

Type II (Frechét): lim 1 - F(tr) = r-α = r-(1/γ)

t → +∞ 1 - F(t)For r > 0, α > 0.

Type III (Weibull): lim 1 - F(tr + u) = rα = r(1/γ) (4)t → ∞ 1 - F(t + u)

For r > 0, α < 0.

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A vast literature on financial returns (Cotter, 1998; Danielsson and DeVries, 1997;

Kearns and Pagan, 1997; Koedijk and Kool, 1992; and Venkataraman, 1997); and on

derivative first differences (Cotter and McKillop, 2000; Duffie and Pan, 1997; Hull

and White, 1998; and Longin, 1999b) has recognised the existence of fat tailed

characteristics. For this reason the rest of the theoretical underpinnings deals with the

Frechét distribution, and the distribution H can only be a type II distribution in (3)

with the corresponding condition for convergence given in (4). This condition allows

for unbounded moments and represents a tail having a regular variation at infinity

property (Feller, 1971). Practically a distribution of random numbers with the

property of regular variation implies that the unconditional moments of r larger than γ

are unbounded.

Extreme Value Theory for the Stationary Case

The theoretical framework presented in the last section deals with the case of iid

distributions. However, the dependence structure of financial price series indicates

that while the data may be stationary after first differencing, there is second order

dependence that manifests itself in the form of volatility clustering. This

characteristic is such an integral part of the time series analysis of asset market prices,

that the introduction of ARCH and related specifications is still seen as an important

breakthrough in the finance literature as it gave empiricists a tool to describe the

persistence in the data. Thus, analysis and results presented in this paper only require

the acceptance of stationarity and it is for this (amongst other) reasons that the futures

data are expressed in return form using the first difference of the natural logarithm of

all closing daily prices.

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The reasoning behind extending extreme value theory in the strict iid case to the more

reasonable assumption of stationarity for financial returns is fully discussed in

Leadbetter et al (1983). By examining the limit behaviour of the maxima of both

series, one can determine whether the series are associated or not. Support for the

stationary series being an associated iid series implies that the behaviour of both

stationary and true iid series' are such that the extremes have the same qualitative

limiting behaviour. The importance of this should not be understated as both types of

series', the stationary and iid cases, exhibit similar limit behaviour if two conditions

apply. The first condition, D(un), is a distributional mixing condition that indicates

only weak long range dependence in the data under analysis, and is given as:

For any integer p, q, and n,

1 ≤ i1 < ip< j1<… < jq ≤ n

such that j1 – ip ≥ l we have

| P(Max Ri ≤ un) – P(Max Ri ≤ un) P(Max Ri ≤ un)| ≤ αn, l (5) i∈ A1∪ A2 i∈ A1 i∈ A2

where A1 = [i1,… .,ip], A2 = [j1,… .,jq], αn, l → 0 as n → ∝ for some sequence l = ln =

o(n)

This condition is generally supported for financial returns given insignificant first

order dependence. In contrast, a second condition, D'(un), is not present in financial

returns. This is an anti-clustering condition and it is contradicted as the extremal

values indicate ARCH effects and they do cluster together. For empirical evidence,

Ding and Granger (1996) give a comprehensive treatment of the characteristics of

dependence in financial returns. They show insignificant first moment dependence,

coupled with the characteristic of second moment dependence, exhibiting clustering.

The condition, D'(un), is given as:

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[n/k]

The relation lim lim sup n ∑ P (R1 > un, Rj > un) = 0 (6)k → ∞ n → ∞ j = 2

There is a way of overcoming the fact that a set of random variables may have a

number of different types of dependency structures by using a parameter that links the

range of dependency relationships together. The parameter used is an extremal index,

θ, and allows an extension of the iid theoretical underpinnings to the case of

stationary variables, not with standing the rejection of condition D'(un) for financial

returns. Formally, dealing with the specific case of a Frechét type distribution and

two related expressions of the iid and stationary cases for the maximum set of returns

that exceed a certain threshold (un):

Lim P (Mn ≤ un (r)) = Φ α(r), r ≤ ∝n → ∝

~Lim P (Μ n ≤ un (r)) = Φ θ

α(r), r ≤ ∝ (7)n → ∝

there exists an extremal index θ that describes the clustering tendency of the high

threshold exceedences in dependent data. This extremal index takes on values

between 0 and 1 (0 ≤ θ ≤ 1), for example, in the iid case θ = 1 and the expressions

above are equivalent.

Again, the Fisher-Tippett theorem demonstrates the asymptotic behaviour of the

distributions in (7) for respective normalising and centring constants, cn and dn. This

is outlined as

cn-1 (Mn – dn) d→

H for cn > 0, -∞ < dn < ∞ (8)

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indicating that the maxima at the limit converges in distribution to H after normalising

and centring. Using (8), the probabilities in (7), for example, P(Mn ≤ un (r)), are

related to the expression in (8) cn-1 (Mn – dn) d→ H as follows:

P(Mn ≤ un) for un = un(r) = cnr + dn

Which can be rearranged as

P(Mn ≤ cnr + dn)

→ P(Mn - dn ≤ cnr)

→ Pcn-1(Mn - dn ≤ r)

Thus the extremal index allows us to overcome the deficiency of volatility clustering

by having a measure that links extremal theory for the iid case to other situations that

incorporate levels of clustering. The relationship between the stationary and

associated iid sequences incorporating the extremal index is

~P(Mn ≤ un) ≈ Pθ(Mn ≤ un) ≈ Fθn(un) (9)

_Provided nF(un) → τ where τ > 0 and τ = τ(r) = r-α = r-(1/γ)

or in terms of the limiting behaviour of an extreme value distribution using the Fisher-

Tippett theorem:

~cn

-1(Mn - dn) d→ H ≈ cn-1(Mn - dn) d→ Hθ (10)

Thus for any stationary sequence that is from the non-degenerate distribution function

H, and assuming that the distribution of F converges at the limit, then the limiting

distribution of F is an extreme value distribution with extremal index, θ. The shape

parameter shape, γ = α-1 > 0 is unchanged from the iid case to the stationary case, so

our extreme value estimates, for example quantiles, remain unchanged.

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The extremal index, θ, that measures the relative levels of volatility clustering

existing in any series is defined for any stationary sequence of variables:

_Lim nF(un) = τn → ∝ ~Lim P (Mn ≤ un (r)) = e-θτ, for τ > 0 (11)

n → ∝

The equivalent statements for the iid case are

Lim nF(un) = τn → ∝

Lim P (Mn ≤ un (r)) = e-τ, for τ > 0 (12)n → ∝

Comparing the two sets of expressions in (11) and (12), we can detail the difference

between the iid and stationary cases as the extremal index, θ; which implies that the

two sequences are asymptotically equivalent. Thus the corresponding maximum of n

random variables from both sequences are equivalent, and θ can be interpreted as the

reciprocal of the mean cluster size of the dependent sequence. The only difference in

the above expressions for the iid and stationary cases is that convergence to the

generalised extreme value distribution is slower for the latter sequence due to the

volatility clustering.

III METHODOLOGY

Risk Management Measures

We now focus on the conditional and unconditional risk measures that are generated

using extreme value theory. Using the distribution, Fn(r), given in (3), two related

measures, the VaR, and the ELP can be noted. Both measures can be calculated for

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the conditional and unconditional distributions of the sequence of returns, providing

risk managers with several pieces of information for use in their strategic responses to

different risk scenarios. The unconditional measures answer questions dealing with

long periods of analysis, for example, different large scale losses over a ten year

period. Risk managers can thus identify the probabilities of a spectrum of losses over

time periods encompassing the time period of the sequence analysed, but also, going

outside this time period. An example, of this for the sequences analysed in this paper

would be a risk resulting over one year, and an exposure that may occur every thirty

years at a certain probability. In contrast to this, the volatility that is expected to

occur at time t + 1, dependent on past values is investigated through the conditional

distribution. These measures provide managers with dynamic risk information on

prospective losses occurring over a short time period. A key issue here in the

conditional estimates is the method in which past volatility is to be modelled. The

expected risk measures at time t + 1 are noted as being influenced by the time varying

past volatility (Hull and White, 1998), of which there are a wide range of model

choices (Duffie and Pan, 1997). An AR(1)-GARCH (1, 1) filter is used in this paper

to obtain a measure of the past volatility and the coefficients required to compute

conditional VaR and excess shortfall measures, and this approach is detailed in the

next sub-section.

First dealing with the unconditional VaR measure, this corresponds to a prescribed

quantile, rp, from the lower tail of the marginal distribution, Fn(r), and is given as:

VaR [Rrp] = rM, n (M/np)γ (13)

Where γ is the Hill (1975) semi-parametric tail index.

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The estimation of the lower tail values using the semi-parametric approach is done as

the extreme returns may not exactly fit the Frechét distribution. In terms of

robustness, the Hill index is recognised as the most efficient semi-parametric

estimator (Kearns and Pagan, 1997), and also, it operates analogously with extreme

value theory by dealing with order statistics. This downside risk estimator is:

γ = 1/α = (1/m - 1) ∑ [log (-ri) - log (-rm)] for i = 1,...., m - 1. (14)

Properties of the Hill index are that it is asymptotically normal, (γ - E{γ})/(m)1/2 ≈ (0,

γ2) (Hall, 1982).

While (13) gives a measure for the amount of possible downside risk exposure upto

the extreme return, r, it has been criticised for not describing the losses in excess of

this extreme return that an investor can incur (Artzner et al, 1997). A risk averse

investor is similarly ill-disposed to losses in excess of the VaR as they are to the

measure itself. To overcome this shortcoming in VaR estimates, a measure dealing

with downside risk that is in excess of a particular threshold, u, is measured by the

ELP estimator:

ELP(r) = P(R - u ≥ r | R < u), r ≤ 0 (15)

This measure requires the determination of the tail of the distribution, and this is

given by the inverted VaR formula:

Pr = (rM, n /rp)1/γM/n (16)

Second, we focus on the conditional risk estimates. The unconditional VaR estimates

given in (13) are modified according to extrapolated lead conditional values, thereby

generating a set of predictive conditional VaR estimates. For example, the

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conditional VaR estimate for a one day holding period is obtained with the following

changes to the initial marginal risk estimate:

VaR[Rtrp]= µt + 1 + σt + 1VaR [Zrp] (17)

The series Z is related to our original sequence of returns, R, through a time-varying

GARCH filter. The predictive t + n step ahead conditional VaR forecasts can also be

developed using this approach with examples of adjustments to the one period VaR

forecasts using the square root of time scaling factor and a Monte Carlo based

technique discussed in McNeil (1999). Similarly, the conditional ELP requires a

simple adjustment to the unconditional measure:

ELP[Rtrp]= µt + 1 + σt + 1ELP [Zrp] (18)

GARCH Filtering Procedure

In order to obtain the conditional risk estimates, related measures of the mean and

variance parameters, and more importantly, the sequence Z are required. A number of

alternative methods have previously been applied to financial returns. These include

GARCH-M (Cotter, 1998) and ARMA-GARCH (Barone-Adesi, 1999) and AR-

GARCH (McNeil, 1999) specifications. The latter approach is used here as it fulfils

the objectives of obtaining a forecast of the conditional expected value µt + 1 through

the AR component of the filter, the conditional variance σt + 1 and residual sequence Z

through the GARCH component. Dealing with the method of obtaining Z through

the GARCH filter, we see that a sequence of returns, R, can be related to Z by:

Rt = µt + σt Zt (19)

Where µt is the expectation of the return, Rt, σt is its standard deviation and the

measure of volatility, with both variables conditional on the information upto day t.

Whilst, the sequence R does not have iid properties, it is assumed that the variable

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follows a stationary process, a property that is also required from the theoretical

underpinnings of extreme value theory. The sequence, Z, is the shock to the process,

assumed normally distributed, with zero mean and conditional standard deviation

equalling one. This series introduces randomness into the process by being an (iid)

sequence of values.

Amongst the possible sources of futures returns, R, exhibiting a fat tail characteristic

is the set of stochastic volatility models that come under the heading of GARCH

processes. A comprehensive review of these processes and their applications in

finance is found in Pagan (1996). The main assumption of these models is that the

conditional second moment, σt, is time varying, but with a degree of persistence. In

contrast, the unconditional variance is assumed constant. The persistence focuses on

the fact that periods of high (low) volatility tend to be followed by similar periods of

high (low) volatility, a characteristic known as volatility clustering. Our conditional

VaR measure explicitly adapts the risk values for this feature by assuming that

volatility is not constant over time. Formally, the estimation of the conditional

variance relies on Bollerslev's (1986) linear GARCH (p, q), where the stochastic

volatility term is defined by an ARMA process:

σ2t = α0 + α1(Rt - 1 - µt - 1)2 + βσ2

t - 1 (20)

for α0, α1, and β > 0; and α1 + β > 0 to fulfil the stationary requirement.

Conventionally (and the one adopted here), the specification with support for most

speculative returns series' is the GARCH (1, 1) model where p = q = 1. If p = q = 0,

both the unconditional and conditional variances are the same. Persistence in the

volatility measure is given by the β coefficient. The inclusion of an AR term, φRt, in

the GARCH process gives a general expression of

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σ2t = α0 + α1(Rt - 1 - µt - 1)2 + βσ2

t - 1 + φRt (21)

This expression gives the conditional variance estimate as well as the AR parameter

for the conditional mean return estimate, Ut + 1.

The residual series, Z, are obtained from running a GARCH filter on the original

series of returns, R. This series is then processed by the extreme value methods

applying the Hill index, and is used to calculate the conditional risk measures, VaR

[Zrp], by the quantile and excess loss probability estimators. The specific measure of

volatility, σt, required in the conditional VaR and ELP expressions is also obtained

from the stochastic volatility estimation. Taking (20), the forecasted variance in t + 1

can be calculated at t by

σt+1 = [α0 + α1(Rt - µt)2 + βσ2t]1/2

(22)

with the conditional mean and variance estimates plus the related parameters given by

the GARCH model. Finally, the inclusion of an autoregressive term in the GARCH

process gives the required variable for the expected return of the conditional risk

estimates in (17) and (18). This conditional return is found adaptively by the

following relationship:

µt + 1 = φRt (23)

where |φ| < 1.

IV EMPIRICAL FINDINGS

The futures analysed entail the main stock index contract traded on the respective

European exchanges. Basic details on the contracts, their county of origin and the

time period analysed are given in table I. Futures returns are calculated using the first

difference of the natural logarithm of the daily closing prices. The continuous time

17

series of returns (other methods are discussed in Longin, 1999b) is generated for each

contract using prices for the nearest maturing contract, and within this, upto the last

trading day prior to the delivery month before overlapping with the next maturity.

INSERT TABLE I HERE

Futures returns like other speculative assets display a number of common

characteristics including skewness, leptokurtosis and non-normality. The contracts

analysed in this study are no exception and results and related discussion is presented

in (Cotter, 1999). Unit root tests to determine whether stationarity is ensured for the

futures returns are detailed in table II. Tests for the futures returns using the

Augmented Dickey- Fuller and its non-gaussian counterpart, the Phillips Perron tests

with a constant and a time trend included are presented. As all the t-statistics are

above the critical value at the five percent level, -3.41, the null hypothesis of the

presence of a unit root is not accepted.2 Thus, the returns series' are stationary over

time, and the application of extreme value theory is appropriate.

INSERT TABLE II HERE

Percentiles for the contracts analysed are given in table III. The findings in the table

suggest that the risk estimates are similar for all the futures with all the median values

for the full sample of returns being positive. The OMX and MIF30 contracts offer the

most extreme negative returns at the five percentile point with the Swiss future’s

being the least risky at this juncture.

INSERT TABLE III HERE

2 Phillips and Ouliaris (1990) provide the critical values. Stationarity of the returns series analysed isconfirmed by summation of the GARCH coefficients being less than one, that is, α + β < 1.

18

The Frechét characteristic of fat-tails is present in the futures returns as can be seen by

the Quantile-Quantile (Q-Q) plot for the Dutch AEX contract. In figure 1, both lower

and upper percentile values diverge substantially from the corresponding normal

values. The fat-tail characteristic is shown by the lack of convergence of the extreme

values in comparison to an exponential decline.

INSERT FIGURE 1 HERE

Much of extreme value theory discusses sequences of iid variables. However, as we

saw in the theoretical section, the theory applies for stationary returns. In fact,

variations from this restrictive iid case are feasible for other processes including linear

ones with iid or subexponential noise. Before the VaR measures are generated based

on the underpinnings of a Frechét approximation, it is interesting to show the

dependence structure of futures returns. In figures 2 - 4, the sample autocorrelation

function of the Italian MIF30 contract is displayed for 30 lags for the returns series,

squared returns series and absolute returns series. Whilst first order serial correlation

is negligible for the returns themselves, there is significant dependence for the

transformed series'. Second order dependence is demonstrated by the autocorrelation

function of the squared returns corresponding to the volatility clustering

characteristic. The dependence in the absolute returns demonstrates a long memory

property for financial returns. For example, in a comprehensive study of absolute

returns series', Ding and Granger (1996) find that significant dependence for

speculative time series exists in excess of 2700 lags.

INSERT FIGURES 2 - 4 HERE

The related risk management statements are derived from the tail estimates, and

values for the semi-parametric Hill index are given in table IV. Related downside

19

Hill estimates using the GARCH (1, 1) set of residuals are given in table IVa. The

characteristics of these conditional values are similar to the original returns, and the

following discussion focuses on table IV. Initially tail estimates are calculated at

arbitrary thresholds of one and five percent. The Hill estimates are found to be

greater for the one percent threshold than its five percent counterpart. Decreases in

the estimates as you move towards the central location are due to the general finding

that the most extreme returns have different distributional characteristics than the full

sequence of returns. For example, previous studies have shown an estimate in the

range, 1 ≤ γ ≤ 2, (Cotter and McKillop, 2000, Venkateswaran et al, 1993) using all

futures returns for a variety of asset types. Whereas in table IV, some of the contracts

have a Hill value greater than four at the one percent threshold and as a consequence,

there is a suggestion that they display a finite fourth moment coefficient. In contrast,

this does not occur for the extremes at the five percent threshold.

A lack of stability in the Hill estimates would affect the VaR and ELP measures and

their qualitative inference. The extent of the problem is such that a 'Hill horror plot'

has been described by Embrechts et al (1997) showing a wide variability of tail index

estimates for different thresholds. Because of this, this paper takes a pragmatic

approach in its development of risk management measures by combining three

previously supported techniques. First, it calculates Hill estimates for a number of

thresholds that are contained within ten percent of the most extreme sequence of

variables, following DuMouchel (1983). Second, it follows Phillips et al (1996), and

calculates an optimal threshold value for each contract based on a bootstrap procedure

of m = Mn = {λn2/3} where λ is estimated adaptively by λ = γ1/21/2(n/m2(γ1 - γ2)2/3.

20

Finally, it ensures that Hill estimates do not suffer from instability using either of

these two approaches by developing a Hill Plot at different threshold values.

The optimal tail values are given in the last column of table IV. Magnitude wise, the

optimal number of tail values is generally encompassed by the five percentile set of

returns. The optimal Hill estimates remain fairly constant at small intervals around

these values, and are used for the purposes of consistent statistical inference of the

VaR and ELP measures. An illustration of this stability is demonstrated for the IBEX

index in the Hill plot of figure 5. This details the tail estimates over a range of

threshold values and indicates stability in this measure in an interval (50 ≤ m ≤ 100)

around the optimal m (74) cut-off point. Generally, these optimal Hill values range

between two and three (except PS120), verifying previous studies on financial returns

(Loretan and Phillips, 1994).

INSERT FIGURE 5 HERE

A spectrum of unconditional VaR estimates is presented in table V. These estimates

can be easily interpreted, for example, there is a 95% probability that the loss on the

BEL20 contract is less than or equal to 1.42%. Losses in excess of this figure would

occur with a frequency of 1 in 20 days (n = 1/p) on average. As expected a priori, the

estimates of risk exposure increase as you move to more extreme quantiles. For

example, at the lower confidence interval, the MIF30 contract displays the most

inherent risk, but the risk patterns change as you move to higher intervals. Dealing

with a time period of 200 days, the PSI20 contract now entails the highest levels of

risk with forecasted losses in excess of 7% at a frequency of 1 in 200. This analysis

demonstrates a richness that can be gained in the interpretation of downside risk

measures using extreme value methods that generate a spectrum of results.

21

INSERT TABLE V HERE

Again using the marginal distribution of returns, the related ELP estimates taking

deterministic thresholds ranging from –5% to 0.5% are given in table VI. An

illustration of these findings is as follows, the probability on any given day of having

a negative return in excess of 5% for the BEL20 contract is 0.15%. Also, as expected

there is a negative relationship between threshold size and the probability of

exceeding that threshold. A similar conclusion on the relative riskness of the

contracts is made with the VaR analysis as the MIF30 and PSI20 contracts indicate

high probabilities of large losses vis-à-vis the other indexes.

INSERT TABLE VI HERE

Switching our attention to the one day forecasted risk management measure, we

present two sets of results. The conditional VaR and related ELP estimates are given

in tables VII and VIII respectively. As all returns are calculated upto February 28

1999, the procedure described in the methodology section is followed giving a

forecast of the risk management estimates for 1 March 1999.3 The residual series, Z,

and the conditional variance and mean estimates are obtained from the AR(1)-

GARCH (1, 1) specification. Using these, the dynamic risk management estimates

are calculated using (17) and (18). It is clear from both sets of results that the

conditional and unconditional risk values differ considerably. In table VII and VIII,

the OMX contract exhibits the highest VaR (2.57%) and ELP (0.80%) at the initial

quantile and probability levels in contrast to the unconditional values of the MIF30

and PSI20 contracts. These results are no surprise as the conditional measures are

3 The forecasts for the Danish KFX and Swiss contracts are exceptions to this, dealing with 19December 1998 and 1 July 1999 respectively.

22

dealing with the trading environment leading upto March 1, and any considerations

affecting specific contracts at that time would be expected to influence these contracts

only.

V SUMMARY AND CONCLUSION

This paper presents four market risk measures accounting for the fat-tailed

characteristic of futures returns. Extreme value methods based on order statistics

model the tail values of a distribution in an unconditional and conditional setting.

Only two assumptions are imposed on the characteristics of our data sets, namely that

the limiting density of the returns approximates the Frechét distribution, and that they

follow a stationary path. A semi-parametric tail estimate, the Hill index, is used to

deal with the distribution approximation issue.

Unconditional VaR and ELP estimates are obtained using the marginal sequence of

returns. These describe the losses that may occur over long periods of analysis. Also,

current one day VaR and ELP estimates are presented relying on the simulation of the

conditional distribution of returns using an AR(1)-GARCH (1, 1) filter.

The general findings for speculative returns are confirmed from an application to

European stock index futures. Namely, they are fat-tailed, with strong (weak) second

(first) order dependence, and stationary over time. In addition, the paper shows the

range of risk inherent in a group of these contracts, and demonstrates that conditional

and unconditional risk measures differ substantially, thereby providing distinct pieces

of information for risk managers. This is understandable as there is no reason to

expect that the static and dynamic risk environment to be equivalent.

23

The risk management process is aided considerably by the findings in this paper.

First, using the indexes as a proxy for the stocks quoted on the respective exchanges,

investors can infer the risk patterns across all markets. For instance, of the European

markets analysed here, the Portuguese stock exchange appears to incur the largest

unconditional levels of volatility. Investors would use this information in their

trading decision making process by comparing with the risk levels inherent on the

other exchanges.

Second, by concentrating on the conditional measures, investors can update their

decision making process by focusing on current volatility levels. These volatility

levels are time varying and periods of high (low) volatility result in high (low) VaR

and ELP estimates. This paper generally computes conditional risk management

estimates for one particular time period, namely, March 1, 1999. However, this

process can be updated to give appropriate risk measures for today’s date by applying

the relevant data. Again, comparisons across the stock index contracts identify the

exchanges with the highest risk levels, although the analysis now would focus on the

current period. This paper finds that the Swedish stock market environment on March

1, 1999 is comparatively risky as proxied by the OMX index, thereby resulting in the

highest VaR estimates. The investor can use the methodology presented in this paper

to determine at today’s date if the OMX contract remains the stock index future with

the highest risk levels.

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24

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27

Figure 1

Q-Q Plot of AEX Futures Contract

Observed Value

.08.06.04.020.00-.02-.04-.06-.08

Exp

ecte

d N

orm

al4

3

2

1

0

-1

-2

-3

-4

28

Figure 2

Autocorrelation of MIF30 Daily Returns Between 1/12/94 - 1/3/99

Lag Number

29

27

25

23

21

19

17

15

13

11

9

7

5

3

1

AC

F.1

0.0

-.1

Coefficient

Confidence Limits

29

Figure 3

Autocorrelation of MIF30 Daily Squared Returns Between 1/12/94 - 1/3/99

Lag Number

29

27

25

23

21

19

17

15

13

11

9

7

5

3

1

AC

F.3

.2

.1

0.0

-.1

Coefficient

Confidence Limits

30

Figure 4

Autocorrelation of MIF30 Daily Absolute Returns Between 1/12/94 - 1/3/99

Lag Number

29

27

25

23

21

19

17

15

13

11

9

7

5

3

1

AC

F.3

.2

.1

0.0

-.1

Coefficient

Confidence Limits

31

Figure 5

Hill Plot for Lower Tail Returns of IBEX Contract

32

Table I

Sample of European Stock Index Futures AnalysedContract Country No. Observations Time PeriodBEL20 Belgium 1368 1 Dec. 1992 - 28 Feb. 1999KFX Denmark 1709 1 June 1992 - 18 Dec. 1998CAC40 France 2672 1 Dec. 1988 - 28 Feb. 1999DAX Germany 2151 1 Dec. 1990 - 28 Feb. 1999AEX Holland 2672 1 Dec. 1988 - 28 Feb. 1999MIF30 Italy 1107 1 Dec. 1994 - 28 Feb. 1999OBX Norway 1629 1 Dec. 1992 - 28 Feb. 1999PSI20 Portugal 651 1 Oct. 1996 - 28 Feb. 1999IBEX35 Spain 1760 1 June 1992 - 28 Feb. 1999OMX Sweden 2347 1 Mar. 1990 - 28 Feb. 1999SWISS Switzerland 1716 1 Dec. 1990 – 30 June 1999FTSE100 United Kingdom 3846 1 June 1984 - 28 Feb. 1999

33

Table II

Unit Root Tests for European Stock Index Futures ReturnsContract Dickey Fuller Phillips PerronBEL20 -8.17 -1156.75KFX -21.24 -1648.93CAC40 -15.42 -2466.20AEX -11.28 -2631.20DAX -14.30 -1973.46MIF30 -7.64 -1142.23OBX -22.22 -1550.45PSI20 -9.46 -546.65IBEX35 -11.26 -1634.70OMX -10.67 -2364.04FTSE100 -17.91 -2321.14SWISS -16.74 -1709.49

34

Table III

Percentiles of Stock Index Futures Returns and their tail equivalentsPercentile 5 10 25 50 75 90 95

BEL20 -1.42 -0.95 -0.40 0.00 0.59 1.08 1.39Lower -3.28 -2.78 -1.83 -1.42 -1.19 -1.00 -0.96KFX -1.77 -1.22 -0.49 0.00 0.63 1.30 1.80Lower -4.00 -3.41 -2.28 -1.77 -1.42 -1.28 -1.25CAC40 -1.98 -1.40 -0.63 0.00 0.78 1.49 1.95Lower -4.13 -3.57 -2.63 -1.98 -1.60 -1.47 -1.43DAX -1.97 -1.27 -0.55 0.02 0.69 1.48 1.99Lower -4.29 -3.54 -2.64 -1.96 -1.54 -1.40 -1.32AEX -1.69 -1.09 -0.44 0.05 0.62 1.21 1.69Lower -4.36 -3.54 -2.39 -1.68 -1.28 -1.15 -1.13MIF30 -2.68 -1.84 -0.82 0.00 0.96 2.12 2.83Lower -5.45 -4.46 -3.26 -2.68 -2.15 -1.94 -1.88OBX -1.41 -0.97 -0.23 0.00 0.43 1.11 1.61Lower -4.34 -3.16 -1.96 -1.41 -1.16 -1.08 -1.04PSE20 -2.29 -1.46 -0.49 0.06 0.86 1.79 2.51Lower -7.16 -4.87 -3.46 -2.27 -1.67 -1.49 -1.47IBEX35 -2.24 -1.60 -0.64 0.00 0.90 1.76 2.33Lower -5.81 -4.30 -2.87 -2.23 -1.84 -1.69 -1.64OMX -2.27 -1.61 -0.74 0.00 0.87 1.68 2.31Lower -5.97 -4.37 -2.86 -2.27 -1.87 -1.70 -1.66SWISS -1.36 -0.93 -0.38 0.07 0.57 1.08 1.39Lower -3.02 -2.51 -1.84 -1.36 -1.09 -0.98 -0.96FTSE100 -1.61 -1.13 -0.55 0.02 0.67 1.24 1.65Lower -3.53 -2.77 -2.06 -1.61 -1.31 -1.19 -1.16Notes: a) Lower tail estimates are based on ten percent of the full sample size.

35

Table IV

Downside Tail Estimates for Stock Index FuturesContract 1% γ1% 5% γ5% m γmBEL20 14 4.87 68 2.75 63 2.81

(1.30) (0.33) (0.35)KFX 17 4.05 85 2.61 72 2.65

(0.98) (0.28) (0.31)CAC40 27 4.82 134 2.91 100 2.97

(0.93) (0.25) (0.30)AEX 27 4.01 134 2.30 95 2.72

(0.77) (0.20) (0.28)DAX 22 3.27 108 2.66 83 2.92

(0.70) (0.26) (0.32)MIF30 11 4.14 56 3.37 55 3.31

(1.25) (0.45) (0.45)OBX 16 2.21 82 2.00 71 2.04

(0.55) (0.22) (0.24)PSI20 6 2.29 33 2.00 41 1.91

(0.94) (0.35) (0.30)IBEX35 18 3.03 88 2.62 74 2.62

(0.71) (0.28) (0.30)OMX 24 2.59 117 2.68 88 2.59

(0.53) (0.25) (0.28)FTSE100 39 2.65 193 2.89 126 2.99

(0.43) (0.21) (0.27)SWISS 17 3.73 86 2.79 72 2.81

(0.90) (0.30) (0.33)Notes: a) Hill standard errors are in parenthesis.

36

Table IVa

Downside Tail Estimates for AR(1)-GARCH(1, 1) Filtered Stock Index FuturesContract 1% γ1% 5% γ5% m γmBEL20 14 2.84 68 2.82 63 2.84

(0.76) (0.34) (0.36)KFX 17 3.57 85 3.04 72 3.07

(0.87) (0.33) (0.36)CAC40 27 3.21 134 3.12 100 3.36

(0.62) (0.27) (0.34)AEX 27 4.36 134 2.38 107 2.66

(0.84) (0.21) (0.26)DAX 22 3.37 108 2.67 91 2.85

(0.72) (0.26) (0.30)MIF30 11 5.30 56 3.58 55 3.56

(1.60) (0.48) (0.48)OBX 16 2.15 82 2.98 70 2.90

(0.54) (0.33) (0.35)PSI20 6 4.49 33 2.49 41 2.27

(1.83) (0.43) (0.35)IBEX35 18 3.63 88 3.24 73 3.26

(0.86) (0.35) (0.38)OMX 24 2.61 117 2.68 88 2.60

(0.53) (0.25) (0.28)FTSE100 39 4.16 193 3.06 130 3.28

(0.67) (0.22) (0.29)SWISS 17 3.51 86 2.73 77 2.88

(0.85) (0.29) (0.33)Notes: a) Hill standard errors are in parenthesis.

37

Table V

Unconditional VaR Estimates using Optimal Tail Threshold for European Stock IndexFuturesContract rp95 rp96 rp97 rp98 rp99 rp99.5BEL20 1.42 1.54 1.71 1.97 2.53 3.23KFX 1.75 1.90 2.12 2.47 3.21 4.17CAC40 1.99 2.15 2.37 2.71 3.43 4.33AEX 1.82 1.98 2.20 2.55 3.29 4.25DAX 2.10 2.27 2.50 2.88 3.65 4.62MIF30 2.67 2.86 3.12 3.52 4.34 5.36OBX 1.40 1.56 1.80 2.20 3.09 4.34PSI20 2.22 2.49 2.90 3.59 5.15 7.41IBEX35 2.23 2.43 2.71 3.16 4.12 5.37OMX 2.23 2.43 2.72 3.18 4.16 5.43FTSE100 1.63 1.75 1.93 2.21 2.79 3.52SWISS 1.35 1.46 1.62 1.87 2.39 3.06

38

Table VI

Unconditional ELP Estimates using Optimal Tail Threshold for European Stock IndexFuturesContract Pr5% Pr4% Pr3% Pr2% Pr1% Pr0.5%BEL20 0.15 0.27 0.62 1.93 13.51 94.73KFX 0.31 0.56 1.20 3.50 21.97 137.90CAC40 0.33 0.63 1.49 4.95 38.81 304.08AEX 0.32 0.59 1.29 3.88 25.55 168.37DAX 0.40 0.76 1.77 5.78 43.76 331.16MIF30 0.63 1.31 3.41 13.04 129.29 1282.27OBX 0.37 0.59 1.06 2.42 9.97 40.99PSI20 1.06 1.62 2.81 6.10 22.92 86.13IBEX35 0.60 1.08 2.30 6.65 40.86 251.19OMX 0.62 1.10 2.32 6.64 40.01 240.89FTSE100 0.17 0.34 0.80 2.70 21.45 170.41SWISS 0.13 0.23 0.53 1.65 11.54 80.93

39

Table VII

Conditional VaR Estimates using Optimal Tail Threshold for European Stock IndexFuturesContract rp95 rp96 rp97 rp98 rp99 rp99.5BEL20 0.76 0.81 0.89 1.00 1.24 1.54KFX 0.40 0.43 0.47 0.53 0.65 0.81CAC40 0.79 0.85 0.93 1.06 1.32 1.64AEX 0.99 1.07 1.19 1.39 1.80 2.33DAX 0.90 0.97 1.07 1.23 1.56 1.98MIF30 1.57 1.67 1.81 2.03 2.47 2.99OBX 0.12 0.13 0.14 0.15 0.18 0.22PSI20 0.28 0.30 0.34 0.41 0.55 0.75IBEX35 0.79 0.84 0.92 1.04 1.29 1.59OMX 2.57 2.80 3.12 3.64 4.74 6.18FTSE100 0.44 0.47 0.51 0.57 0.70 0.86SWISS 0.74 0.80 0.88 1.01 1.28 1.62

40

Table VIII

Conditional ELP Estimates using Optimal Tail Threshold for European Stock IndexFuturesContract Pr5% Pr4% Pr3% Pr2% Pr1% Pr0.5%BEL20 0.21 0.27 0.44 1.08 6.85 48.13KFX 0.09 0.13 0.27 0.84 6.78 56.64CAC40 0.13 0.25 0.61 2.29 23.29 238.86AEX 0.17 0.30 0.62 1.79 11.25 71.02DAX 0.18 0.31 0.67 2.04 14.45 103.93MIF30 0.36 0.78 2.16 9.14 107.74 1270.57OBX 0.05 0.06 0.08 0.16 0.89 6.35PSI20 0.12 0.19 0.35 0.86 4.13 19.88IBEX35 0.15 0.31 0.77 2.88 27.51 263.50OMX 0.80 1.39 2.90 8.26 49.87 302.16FTSE100 0.05 0.09 0.21 0.75 7.12 68.98SWISS 0.08 0.15 0.32 0.99 7.20 52.88