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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.
2
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
3
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.
4
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)
5
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
6
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.
7
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.
8
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:
9
[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)
10
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.
11
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
12
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.
13
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
14
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
15
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
16
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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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
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