This letter uses the Block Maxima Extreme Value approach to quantify catastrophic risk in international equity markets. Risk measures are generated from a set threshold of the distribution of returns that avoids the pitfall of using absolute returns for markets exhibiting diverging levels of risk. From an application to leading markets, the letter finds that the Nikkei is more prone to catastrophic risk than the FTSE and Dow Jones Indexes.

John Cotter

English

Research Papers in Economics

 
 
 
 
 
 
UCD GEARY INSTITUTE  
DISCUSSION PAPER SERIES 
 
Modelling Catastrophic Risk in 
International Equity Markets: An 
Extreme Value Approach 
 
John Cotter 
Centre for Financial Markets, 
Graduate School of Business, 
University College Dublin 
 
Geary WP2005/15 
April 06, 2005  
 
 
 
 
UCD  Geary  Institute  Discussion  Papers  often  represent  preliminary  work  and  are  circulated  to 
encourage discussion. Citation of such a paper should account for its provisional character. A revised 
version may be available directly from the author. 
 
Any opinions expressed here are those of the author(s) and not those of UCD Geary Institute. Research 
published in this series may include views on policy, but the institute itself takes no institutional policy 
positions. 
 
  
Modelling catastrophic risk in international equity markets: An extreme value 
approach 
 
JOHN COTTER 
University College Dublin 
 
 
Abstract: 
This letter uses the Block Maxima Extreme Value approach to quantify catastrophic 
risk in international equity markets.  Risk measures are generated from a set threshold 
of  the  distribution  of  returns  that  avoids  the  pitfall  of  using  absolute  returns  for 
markets exhibiting diverging levels of risk.  From an application to leading markets, 
the letter finds that the Nikkei is more prone to catastrophic risk than the FTSE and 
Dow Jones Indexes. 
 
Address for Correspondence: 
Dr. John Cotter, 
Centre for Financial Markets, 
Graduate School of Business, 
University College Dublin, 
Blackrock, 
Co. Dublin, 
Ireland. 
E-mail. john.cotter@ucd.ie 
 
April 6 2005  
Modelling catastrophic risk in international equity markets: An extreme value 
approach 
 
I Introduction: 
Tail  returns  can  be  catastrophic  for  investors  and  accurate  modelling  of  these  is 
paramount.  This letter uses the Block Maxima Extreme Value approach to quantify 
catastrophic risk for investors with long and short positions in international equity 
markets.
1  Quantitatively, catastrophic risk occurs if market movements exceed some 
extreme threshold value.  We fit the Generalised Extreme Value (GEV) distribution to 
leading equity indexes that only models the tail returns of a probability distribution 
associated with catastrophic risk.
2  Risk measures are generated from a set threshold 
of the distribution of returns (99% level) that avoids the pitfall  of using  absolute 
returns for markets exhibiting diverging risk.   
 
Catastrophic events relate to extraordinary trading periods that cannot be reconciled 
with  previous and subsequent market movements. Thus, these events distinctively 
belong to a separate distribution distinct from ordinary market movements and should 
be modelled separately.  Extreme Value Theory (EVT) is an optimal approach to 
quantifying the extent of rare catastrophic events. First, and most important, EVT 
dominates  alternative  frameworks  in  modelling  tail  events  (Longin,  2000,  Cotter, 
2004a).  Second, event risk is explicitly taken into account by EVT since it explicitly 
focuses on extreme events.  Third EVT reduces model risk since it does not assume a 
                                                
1 In contrast, in a qualitative sense, catastrophic risk requires market participants such as investors and 
bankers agreeing on the occurrence of extreme events.  Kindleberger (2001) notes that these extreme 
events are a result of irrational speculation in the form of manias and panics, accurately describing the 
large decline in international markets during the 1987 crash. 
2 Alternative applications of EVT include modelling margin requirements (Dewachter and Gielens, 
1999) and stability in foreign exchange markets (Cotter, 2005). 
  
particular model for returns.  Finally, it avoids coarseness and bias in the tail estimates 
and produces a useful risk language for promoting high-risk concepts.   
 
The  letter  is  organised  as  follows:  section  II  describes  the  risk  measures  and 
estimation procedure. Using three leading market indexes from different geographical 
regions, the Dow Jones Industrial Average, the Nikkei 225 and the FTSE All Share, 
section III follows with extreme value estimates and discussion. 
 
II Risk measures and estimation procedure 
Using techniques from EVT this study fits a GEV to the data using the Block Maxima 
approach that models the maxima, M, for the upper tail for some block of time, for 
example, a year.
3  Assuming a random variable X, for finite samples the following 
three parameter version of the GEV is as follows:   
] [
] [
] [ ￿
￿
￿
￿
￿
> ¥
= ¥ ¥ -
< ¥
= Î ￿
￿
￿
￿
￿
￿ -
=
0 ,
0 ,
0
, , ,
x x s m
x
x x s m
s
m
x m s x
-
- , -
D D x
x
H H     (1) 
where  m  is  the  location  parameter,  s  is  the  scale  parameter  and  x  is  the  shape 
parameter of the extreme value distribution.  The shape parameter is the key to using 
EVT as it separates three types of extreme value distributions according to different 
shapes, the Gumbel (x = 0), Weibull (x < 0) and Fréchet (x > 0) distributions.  The 
latter extreme value distribution is supported in the finance literature as it exhibits a 
fat-tails property, also found for market returns (Cotter and McKillop, 2000).   
 
                                                
3 EVT is commonly applied in the financial economics literature, and for a comprehensive discussion 
of the theoretical framework see Embrechts et al (1997).  Following convention we will focus on the 
maxima but alternatively we could detail the minima for the lower tail of a distribution.  Alternative 
non-pararmetric approaches have also been applied in modelling tail behavour (see Cotter, 2004b)   
The maximum likelihood estimation procedure yields parameter estimates for x, m 
and s  by maximizing the following log-likelihood function with respect to the three 
unknown parameters 
( ) ( ) ( ) Õ Î =
i
i i M x x h x L , log ; , , s m x    
where 
( ) ￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿
￿ -
+ - ￿
￿
￿
￿
￿
￿ -
+ =
- - - x x
s
m
x
s
m
x
s
s m x
1 1 1
1 exp 1
1
; , ,
x x
x h     (2) 
is  the  probability  distribution  function  for  x  ¹ 0  and  0 1 > +
-
s
m x
x .    Confidence 
intervals for these parameters x ˆ,  m ˆ  and s ˆ  can easily be obtained via the profile log 
likelihood function.  
 
The GEV parameters are used to generate the catastrophic risk measures.  Taking an 
extreme  threshold  or  qth  quantile  of  a  continuous  distribution  with  distribution 
function F is 
    ( ) q F xq
1 - =                
where F
-1 
 is the inverse of the distribution function.  The GEV catastrophic risk level, 
xn,k, using the maximum likelihood parameters is: 
( )
( ) ( ) ( )
( ) ( ) ￿ ￿
￿
￿
￿
= - -
¹ - - - -
= - =
-
-
0 1 1 log log ˆ ˆ
0 1 1 log 1 ˆ
ˆ
ˆ
1 1
ˆ
1
, , .
x s m
x
x
s
m
x
s m x
k
k
k H x k n      (3) 
Again  asymmetric  confidence  intervals  for  the  catastrophic  risk  level  can  be 
calculated using the profile log likelihood function. 
 
To illustrate, consider a model using daily returns and a block size corresponding to 
annual maxima (»261 days).  The k-year catastrophic risk level x261,k is defined as 
  { } 1 , 1 , 261 261 > = > k k x M P k             (4) 
  
This is the return level that we expect to exceed only in one year out of every k years, 
on average and has a probability  k 1 .  Now we turn to our estimation and inferences.  
 
III Results and discussion 
The analysis is completed on daily logarithmic returns series from liquid US, Asian 
and  European  markets  between  January  1,  1985  and  December  31.    The  indexes 
chosen are the well-known Dow Jones Industrial Average, the Nikkei 225 and the 
FTSE All Share.  Findings from a representative selection of indexes are given.   
 
In Table 1 summary statistics detailing the first four moments, min and max values 
and the Jarque-Bera normality test are given for the full distribution of returns and for 
a subset of values incorporating 10 percent of the full sample. The latter analysis is to 
investigate the tail behaviour of financial returns, as it this part of the distribution that 
gives rise to catastrophic risk and our application of GEV.  Overall, we find the mean 
of  upper  and  lower  tail  returns  deviate  substantially  from  the  approximately  zero 
mean  of  the  full  distribution  of  returns,  with  the  Nikkei  exhibiting  the  largest 
deviations. Moreover, daily risk is approximately 1% although some very large single 
day returns occur.   
INSERT TABLE 1 HERE 
 
Standard financial time series properties are recorded for the tail and full distributions 
namely,  a  lack  of  normality,  due  to  excess  skewness,  and  excess  kurtosis.    To 
investigate this latter property in more detail, Figure 1 presents QQ plots of quantiles 
of the observed distribution set against the normal distribution for both the full set of 
Dow returns and for the subsets of tail values.  Two obvious points are clear.  First, all  
distributions  exhibit  fat-tails.    Second,  the  fat-tail  characteristic  becomes  more 
pronounced for the tail returns.  These plots drive our application of the GEV and in 
particular, the Fréchet GEV.   
INSERT FIGURE 1 HERE 
 
Turning to the extreme value analysis, maximum likelihood parameters of the fitted 
GEV to the upper tails of the indexes are given in table 2.  The dispersion parameter 
values concur with the summary statistics of the tail distributions, indicating that the 
Nikkei index fluctuates more than its counterparts.  And as expected, the location and 
dispersion estimates increase as interval size increases.  As stated, the most important 
parameter for modelling and distinguishing tail behaviour is the shape parameter.  We 
find all point estimates are positive, and generally there is support for the hypothesis 
of returns converging to the fat-tailed Fréchet distribution at a 95% confidence levels.  
Specifically,  the  fattest  tail  shape  recorded  is  for  the  FTSE  index  at  a  quarterly 
interval with  a shape point estimate of  0.361.   Variation in  tail  shape does  occur 
across markets, and interval of estimation, where no systematic pattern occurs.  For 
example, the shape parameter is reasonably constant across the intervals for the Dow 
whereas  it  decreases  for  the  Nikkei.    This  has  implications  for  the  modelling  of 
catastrophic risk where each asset should be modelled separately, and for different 
frequencies. 
INSERT TABLE 2 HERE 
 
Taking the three EV parameters, we now estimate the catastrophic risk levels for a 
99.9% confidence level and these are given in Table 3.  This allows us to obtain 
information on the size and frequency of catastrophic risk levels.  Here we show the  
estimated 20-month, 20-quarter and 20-semester catastrophic risk levels for the upper 
and lower tail of each index.  These have an attractive inference with for example, a 
20-month catastrophic risk level representing a level that we expect to exceed in one 
month  out  of  every  twenty  months  on  average.    So  for  example,  the  20-month 
catastrophic risk level for the upper tail of the Nikkei index is 5.82% implying that 
positive extreme price movements of this magnitude are expected in this market once 
every 20 months on average.   
INSERT TABLE 3 HERE 
 
Some interesting findings are noted.  Catastrophic risk increases as you increase the 
interval  size  where  investors  would  experience  larger  absolute  returns  from  these 
major  markets.    Furthermore,  with  the exception  of  the  Dow for  monthly  blocks, 
lower and upper catastrophic risk is similar and is within the respective confidence 
intervals for each interval block.  However in terms of identifying the riskiest market 
at  this  interval,  we  find  that  the  Nikkei exhibits  the  largest  levels,  and  the  FTSE 
exhibits the smallest levels, of extreme returns.   
 
An overall portfolio return is driven by these extreme catastrophic values as investor 
performance is frequently the end result of a few exceptional trading days as most of 
the other days only contribute marginally to the bottom line.  Hence correct modelling 
of catastrophic risk is vital.  
References: 
Cotter, J. (2005) Tail Behaviour Of The Euro, Applied Economics, 37, 1 –14. 
Cotter,  J.  (2004a)  Extreme  Risk  in  Futures  Contracts,  Applied  Economic  Letters, 
Forthcoming. 
Cotter, J. (2004b) Downside Risk For European Equity Markets, Applied Financial 
Economics, 14, 707-716. 
Cotter, J. and D.G. McKillop (2000) The Distributional Characteristics of a Selection 
of Contracts Traded on the London International Financial Futures Exchange, Journal 
of Business Finance and Accounting, 27, 487-510. 
Dewachter,  H.  and  G.  Gielens  (1999)  Setting  Futures  Margins:  The  Extremes 
Approach, Applied Financial Economics, 9, 173-181.  
Embrechts, P., Klüppelberg, C., and T. Mikosch (1997) Modelling Extremal Events 
for Insurance and Finance, Springer-Verlag, Berlin. 
Kindleberger C. P. (2000) Manias, Panics and Crashes, 4
th Edition, Wiley, New York. 
Longin, F.M (2000) From value at risk to stress testing: The extreme value approach, 
Journal of Banking and Finance, 24, 1097-1130. 
  
Acknowledgements:  University  College  Dublin’s  Faculty  research  funding  is 
gratefully acknowledged. 
  
Full series             
 
 
 
 
 
 
 
Upper tail 
Upper tail 
Lower tail 
 
Figure 3.  Q-Q plots of Dow Jones Industrial Average returns.  
This  figure  plots  the  quantiles  of  the  observed  distribution  against  the  normal 
distribution (straight line) for the full series and upper and lower 10 percent of returns.   
 
 
-8  -6  -4  -2  0  2  4  6  8 
-4 
-3 
-2 
-1 
0 
1 
2 
3 
4 
Observed 
Quantiles 
N 
o 
r 
m
a 
l  
 
-8  -6  -4  -2  0  2 
-3 
-2 
-1 
0 
1 
2 
3 
Observed  
N 
o 
r 
m
a 
l  
 
-2  0  2  4  6  8 
-3 
-2 
-1 
0 
1 
2 
3 
Observed Quantiles 
N 
o 
r 
m
a 
l   
Table 1.  Summary statistics for daily index series 
Index  Mean  Std D  Min  Max  Skew  Kurt  J-B 
DOW  0.0524  1.06  -25.64  9.67  -3.74  92.32  1397123 
Upper  1.76  0.76  1.12  9.67  4.14  33.55  17404 
Lower  -1.80  1.50  -25.64  -0.97  -10.36  154.85  408088 
FTSEALL  0.0387  0.87  -11.91  5.70  -1.31  20.32  53380 
Upper  1.45  0.59  0.98  5.70  3.28  18.83  5104 
Lower  -1.53  0.96  -11.91  -0.91  -6.04  55.85  51071 
NIKKEI  0.0043  1.34  -16.14  12.43  -0.17  13.01  17453 
Upper  2.39  1.17  1.40  12.43  3.23  20.12  5820 
Lower  -2.48  1.15  -16.14  -1.45  -4.93  50.80  41392 
Notes: The summary statistics are presented for each index as well as the upper and 
lower 10 percent of realisations.  Mean, min, max, standard deviation (Std D) values 
are presented in percentages.  Normality is formally examined with the Jarque-Bera 
(J-B) test which a critical value of 3.84.  All the skewness (Skew), kurtosis (Kurt) and 
normality coefficients are significant at the 5 percent level.   
  
Table 2.  Parameter estimates for upper tail of index series 
Index  Block 
Length  x ˆ    s ˆ    m ˆ   
DOW  Month  0.167 [0.075, 0.271]  0.546 [0.495, 0.609]  1.411 [1.339, 1.478] 
  Quarter  0.168 [0.025, 0.354]  0.716 [0.607, 0.869]  1.908 [1.739, 2.057] 
  Semester  0.170 [-0.002, 0.382]  0.825 [0.665, 1.091]  2.241 [1.959, 2.477] 
FTSEALL  Month  0.259 [0.150, 0.373]  0.388 [0.352, 0.432]  1.107 [1.059, 1.151] 
  Quarter  0.361 [0.147, 0.586]  0.456 [0.389, 0.555]  1.434 [1.340, 1.513] 
  Semester  0.214 [-0.007, 0.510]  0.644 [0.514, 0.859]  1.800 [1.585, 1.978] 
NIKKEI  Month  0.294 [0.172, 0.423]  0.872 [0.791, 0.973]  1.688 [1.582, 1.784] 
  Quarter  0.114 [-0.042, 0.337]  1.407 [1.184, 1.715]  2.696 [2.354, 3.003] 
  Semester  0.052 [-0.231, 0.301]  1.765 [1.392, 2.351]  3.434 [2.795, 3.993] 
Notes: Extreme value parameters, the tail index (x), the scale parameter (s), and the 
location  parameter  (m)  are  estimated  via  maximum  likelihood  methods.    95% 
confidence  intervals  are  given  in  [].    Block  lengths  of  192  extremes (month),  64 
extremes (quarter) and 32 extremes (semester) are used.    
Table 3. Estimated catastrophic risk levels of index series 
Index  Tail  Month  Quarter  Semester 
DOW  Lower  4.32 [3.71, 5.29]  6.47 [4.90, 10.09]  9.04 [6.05, 19.40] 
  Upper  3.51 [3.14, 4.07]  4.66 [3.92, 6.25]  5.45 [4.33, 8.40] 
FTSE  Lower  3.21 [2.80, 3.87]  4.07 [3.29, 5.97]  5.02 [3.63, 10.69] 
  Upper  2.84 [2.50, 3.39]  3.89 [3.03, 6.04]  4.47 [3.49, 7.87] 
NIKKEI  Lower  5.46 [4.77, 6.60]  7.09 [5.85, 9.71]  8.81 [6.62, 16.69] 
  Upper  5.82 [4.95, 7.28]  7.67 [6.38, 10.65]  9.10 [7.33, 14.59] 
Notes: The table shows the estimated 20-month, 20-quarter and 20-semester 
catastrophic risk levels for the upper and lower tail of each index.  95% confidence 
intervals are given in [].   

Downside Risk For European Equity Markets,

From value at risk to stress testing: The extreme value approach,

Modelling Extremal Events for Insurance and Finance,

Setting Futures Margins: The Extremes Approach,

Tail Behaviour Of The Euro,

The Distributional Characteristics of a Selection of Contracts Traded on the London International Financial Futures Exchange,

Modelling Catastrophic Risk in International Equity Markets: An Extreme Value Approach

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Modelling Catastrophic Risk in International Equity Markets: An Extreme Value Approach

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