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DYNAMIC ECONOMETRIC MODELS 
Vol. 9 – Nicolaus Copernicus University – Toruń – 2009 

Marcin Fałdziński 
Nicolaus Copernicus University in Toruń 

Application of Modified POT Method with Volatility Model 
for Estimation of Risk Measures 

A b s t r a c t. The main aim of this paper is the presentation and empirical analysis of the new 
approach which combines volatility models with Peaks over Threshold method that comes from 
extreme value theory. The new approach is applied for estimation of risk measures (VaR and ES) 
in financial time series. For the empirical analysis the financial risk model evaluation was con-
ducted. In this paper the POT method was compared with standard volatility models (GARCH 
and SV) in case of the conditional modeling. 

K e y w o r d s: Extreme Value Theory, Peaks over Threshold, Value-at-Risk, Expected Shortfall.  

1. Introduction 
 Current literature in the area of risk management is very extensive (Szegö, 
2004) and results are frequently ambiguous. The existing approaches for esti-
mating the profit/loss distribution of a portfolio of financial instruments can be 
schematically divided into three groups: nonparametric historical simulation 
methods, parametric methods based on volatility models (GARCH type models) 
and methods based on the Extreme Value Theory. This work focuses on meth-
ods based on volatility models and Peaks over Threshold (POT) method. Previ-
ous results from papers which applied the POT method in risk management 
were the main motivation to deal with a problem of extremes in financial time 
series. The main aim of this paper is to propose and analyse more complex ap-
proach taking into account extremes and non-extremes in risk management of 
financial time series. Most of the attention is focused on the new approach pro-
posed in this paper. 

2. The POT Method and Volatility Models 
 In the Peaks over Threshold method we are interested in excesses over 
a high threshold value u . Detailed description of POT method can be found in: 



Marcin Fałdziński 120 

Embrechts, Klüppelberg, Mikosch  (2003) or  Osińska, Fałdziński (2008). Here 
will be presented McNeil and Frey's approach, which joint volatility models and 
POT method. We assume that tX  is a time series representing daily observa-
tions of log return on a financial asset price. We assume that dynamics of X
process is given as: 

t t t tX Zμ σ= + ,  (1) 

where innovations tZ are the white noise process with zero mean and unit va-
riance. We assume that tμ  is the expected return and tσ  is the volatility of the 
return, where both are measurable to information set 1t −F  at time 1t − . To im-
plement an estimation procedure for the process (1), we need to choose 
a dynamic conditional mean as well as a conditional variance model. Many 
volatility models were proposed in econometric literature, from ARCH/GARCH 
models, and their different modifications and generalization, to SV models. 
McNeil and Frey defined simple risk measures forms for one day horizon with 
relation to process (1) as:  
 1 1 ( )t

q t t qVaR VaR Zμ σ+ += + , (2) 

 1 1 ( )t
q t t qES ES Zμ σ+ += + , (3) 

where ( )t
qVaR Z  is the Value-at-Risk of tZ  process, and ( )t

qES Z is the corres-
ponding Expected Shortfall. The method proposed by them demands minimal 
assumptions for innovations distribution and focus on modeling distribution 
tails using Extreme Value Theory. Generally we can say, that we use two stage 
approach, which can be presented in the following steps: 

1. Fit a GARCH-type model (generally volatility model) to return series. Es-
timate 1tμ +  and 1tσ +  using fitted model and calculate standardized resi-
duals. It means, that fitted model is used to estimate one-day ahead predic-
tions of 1tμ +  and 1tσ + . 

2. EVT is used to estimate ( )qVaR Z  and ( )qES Z  with application of POT 
method for mentioned residuals. 

A Value-at-Risk in the EVT for the Peaks over Threshold method is equal:  
ˆ

ˆ
( ) 1

ˆ u

nVaR u
N

γ
σα α
γ

−⎛ ⎞⎛ ⎞
⎜ ⎟= + −⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

, (4) 

where α  is a tolerance level, u  is a threshold, ˆ ˆ,γ σ  are estimated parameters 
from Generalised Pareto Distribution (GPD), n  is the total number of realiza-
tions and uN  is number of extremes. Because of VaR drawbacks the alternative 
risk measure was developed, which is called Expected Shortfall (ES), and was 



Application of Modified POT Method with Volatility Model …  121

proposed by Artzner et al. (1997; 1999). The Expected Shortfall for the Peaks 
over Threshold method is given by: 

ˆˆ( )( )
ˆ ˆ1 1

VaR uES α σ γα
γ γ

−
= +

− −
, (5) 

In the literature we can find comparisons of  models which estimate VaR and 
ES where extreme value theory is applied (i.e. Brooks, Clare, Dalle Molle, Per-
sand, 2005; Harmantzis, Miao, Chien, 2006; Kuester, Mittnik, Paolella, 2006; 
Osińska, Fałdziński, 2008; Fałdziński, 2008). In all of these papers authors find, 
that EVT is good or very good approach to estimate risk measures. Empirical 
results presented in these papers show that volatility models with application of 
Extreme Value Theory more accurately estimate expected future values of asset 
returns, particularly in case of extremely rare events (i.e. extremes). Standard 
volatility models better fit to mean values in financial time series in contrast to 
models with EVT. Therefore the new approach is based on such combination, 
that extremes are estimated using POT method, and non-extremes are estimated 
using standard volatility models. This combination is an attempt to identify 
extremes in financial time series. The new approach can be written in the fol-
lowing: 

1 1

1 1

-       
- -

               
t t t

t t t

GARCH POT u
N GARCH POT

GARCH u
μ σ
μ σ

+ +

+ +

+ ≥⎧
= ⎨ + <⎩

 (6)  

This new hybrid of volatility model and POT method is based on conditional 
volatility forecast 1tσ + , conditional mean 1tμ +  and on threshold tu . If sum of 

1tμ +  and 1tσ +  forecasts are higher than threshold then we apply McNeil and 
Frey's approach, in other case we have standard GARCH model. Threshold tu  
could be constant or time-varying. Switching mechanism is formulated to iden-
tify whether the forecast of time series return is an extreme value, or not. Fore-
casts of asset returns are very important, the more precise forecast of time series 
is, the extremes are better identified, which is connected to proper switching 
mechanism (6). The idea of new approach is completely based on the forecast 
capability from volatility model for time series (in this case it is GARCH model 
but we also used SV model). 

3. Backtesting 
 A key part for risk measures is necessity to check correctness of estimation 
and simultaneously choose the most precise method for their estimates. Risk 
models need to be validated and backtesting is the class of quantitative methods 
used to e.g. rank a group of models against each other (Dowd, 2005; Alexander, 
2008). For backtesting we used three binominal tests: the failure test ucLR , the 
mixed Kupiec-test indLR K  (Haas, 2001) and the test of independence indLR CH  



Marcin Fałdziński 122 

(Christoffersen, 1998). Unfortunately presented tests have weak power, and 
additionally these methods do not give the opportunity to create a ranking of the 
models. Angelidis and Degiannakis (2006) presented modified Lopez's (Lopez, 
1999) loss function: 

1 1|
1, 1

 if  violation occurs

0                    else
t t t

t

y ES+ +
+

⎧ −⎪Ψ = ⎨
⎪⎩

 (7)  

  ( )2

1 1|
2, 1

 if  violation occurs

0                        else
t t t

t
y ES+ +

+

⎧ −⎪Ψ = ⎨
⎪⎩

 (8) 

To judge which model is the best, we compute the mean absolute er-

ror 1,
1

/
T

t
t

MAE T
=

= Ψ∑
%

% , and the mean squared error 2,
1

/
T

t
t

MSE T
=

= Ψ∑
%

% , where 

T%  is the number of the forecasts, and total loss ( )TL  is the sum of these 
errors (Angelidis, Degiannakis, 2006). The loss function approach is 
based on calculating magnitude of violations (or exceedances), i.e. cases 
when the risk measure underestimates a future value of asset. As we can 
see the presented loss functions measure only the underestimation of risk 
measure. A perfect situation is when an estimated risk measure does not 
underestimate and overestimate1 too much a future value of asset. For 
example, if we would have two estimated risk measures and both of them 
have almost equal value of standard loss functions, then the better risk 
measure is that which has the lower overestimation. An overestimation of 
risk measures was proposed to measure (Fałdziński, 2009). The overes-
timation of loss functions are given in the following:  

1 1 1 1
1, 1

1

     if   0
0                      if     0

t t t t
t

t

ES y y ES
y

+ + + +
+

+

⎧ − < <⎪Φ = ⎨
≤⎪⎩

 (17) 

( )2
1 1 1 1

2, 1
1

   if   0
0                        if   0

t t t t
t

t

ES y y ES
y

+ + + +
+

+

⎧ − < <⎪Φ = ⎨
≤⎪⎩

 (18) 

We also compute the mean absolute error of overestimation 

1,
1

/
d

over t
t

MAE d
=

= Φ∑ , and the mean squared error 2,
1

/
d

over t
t

MSE d
=

= Φ∑ , 

where 
1

T

i
d

=

= ∑1 , 1 1

1

1     if    0
0    if     0

t t

t

y ES
y

+ +

+

< <⎧
= ⎨ ≤⎩

1  is the number of the positive fore-

                                                 
1 Cases when estimated risk measure is higher than value of asset 



Application of Modified POT Method with Volatility Model …  123

casts (i.e. larger then 0, but smaller than the given risk measure). Similarly we 
could construct the total loss of overestimation ( )over overOTL MAE MSE= + . 
The OTL could also be computed for VaR and SRM, but then we have to 
change 1tES +  for another estimated risk measure in formula (17), (18) and d . 

4. Empirical Analysis 
 The subject of the empirical analysis is the comparison of the estimated 
Value-at-Risk and Expected Shortfall measures for the new approach with vola-
tility models. The comparison is based on selection of the best model for total 
loss TL proposed by Angelidis and Degiannakis and total loss of overestimation 
(OTL). In the analysis the SV model with Gaussian distribution and the 
GARCH model with Gaussian and t-Student error distributions were used. We 
have chosen SV and GARCH models because they represent the most standard 
volatility models. The parameters were estimated with the maximum likelihood 
method in case of  GARCH models and the quasi-maximum likelihood method 
in the case of the SV models. The time series used in the analysis comprise 
3000 observations of log returns (daily data: 07.11.1994 – 31.10.2006). We 
used 5 financial time series (WIG, SP500, DAX, FTSE100, NIKKEI225) that 
represent the stock market returns. For each time series a thousand VaRs and 
ESs were estimated for backtesting purposes. To compute the ES for the volatil-
ity models we used Dowd’s approach (Dowd, 2005). Also we used the time-
varying threshold u , as a result of defining the number of extremes on 10% 
level for all observations in time series. This 10% level is a common standard in 
similar analyses. 
 Results for the binominal tests were computed besides their drawbacks. In 
short we can say that findings were very diverse and it was very difficult to 
make general conclusions that is why we decided no to show them. They are 
available upon request. Based on findings contained in Table 1 we can see that 
SV-POT model has the lowest value for total loss TL (next are N-SV-POT and 
SV models). It should not be a surprise because SV models more take into ac-
count the extremes than standard GARCH model. After the class of SV models 
(i.e. SV, SV-POT and N-SV-POT) we have four variant of GARCH-POT mod-
els with relatively higher values of TL. The end of the total loss ranking con-
tains N-GARCH-POT and GARCH models alternately, but we should point out 
that N-GARCH-POT models are relatively better. On the other hand the class of 
SV models has the highest value of the total loss of overestimation (OTL) for 
value-at-risk and this is the consequence of the same property mentioned be-
fore. The lowest value of OTL for VaR is obtained for GARCH TD model. We 
have this kind of result because the standard GARCH model do not take into 
consideration the extremes like the other models in the analysis and that is why 
the overestimation is the lowest. The next in the ranking are GARCH and N-
GARCH-POT models alternately. 



Marcin Fałdziński 124 

Table 1. Backtesting results for WIG20 and SP500  
WIG20 α=0.05 

Model TL Rank TL OTL VaR Rank OTL VaR OTL ES Rank OTL ES 
GARCH 0.0998 11 2.788 3 7.446 7 

GARCH TD 0.2019 15 2.680 1 10.111 14 
AR-GARCH 0.0927 10 2.980 7 7.778 9 

AR-GARCH TD 0.1874 14 2.812 4 10.416 15 
SV 0.0281 3 4.420 13 7.428 6 

GARCH-POT 0.0498 4 3.219 12 5.913 3 
GARCH-POT TD 0.0525 5 3.115 9 5.733 2 
AR-GARCH-POT 0.0734 7 3.118 10 6.817 4 

AR-GARCH-POT TD 0.0530 6 3.101 8 5.634 1 
SV-POT 0.0069 1 7.631 15 9.425 12 

N-GARCH-POT 0.0823 8 2.909 5 6.905 5 
N-GARCH-POT TD 0.1622 13 2.775 2 9.031 11 
N-AR-GARCH-POT 0.0857 9 3.126 11 7.679 8 

N-AR-GARCH-POT TD 0.1609 12 2.922 6 9.957 13 
N-SV-POT 0.0162 2 6.486 14 8.754 10 

SP500 α=0.05 
Model TL Rank LF OTL VaR Rank OTL VaR OTL ES Rank OTL ES 

GARCH 0.0554 11 1.890 3 5.025 8 
GARCH TD 0.1261 15 1.807 1 6.827 13 
AR-GARCH 0.0495 10 2.080 11 5.379 11 

AR-GARCH TD 0.0997 14 2.003 5 7.168 15 
SV 0.0102 3 2.344 13 3.850 5 

GARCH-POT 0.0194 7 2.038 7 3.450 1 
GARCH-POT TD 0.0191 6 2.038 8 3.461 3 
AR-GARCH-POT 0.0191 5 2.050 9 3.453 2 

AR-GARCH-POT TD 0.0175 4 2.051 10 3.489 4 
SV-POT 0.0015 1 4.048 15 5.108 9 

N-GARCH-POT 0.0435 8 1.890 4 4.762 7 
N-GARCH-POT TD 0.0854 12 1.822 2 6.222 12 
N-AR-GARCH-POT 0.0440 9 2.084 12 5.245 10 

N-AR-GARCH-POT TD 0.0880 13 2.006 6 6.873 14 
N-SV-POT 0.0098 2 2.959 14 4.315 6 

Note:  N-AR-GARCH-POT TD - means the new approach proposed in this paper with  AR(1)-GARCH(1,1) 
model and Peak over Threshold method where t-distribution was applied. Respectively other abbreviation are 
constructed. 

It means that the switching mechanism takes into account non-extremes rather 
than extremes. In case of OTL for VaR, GARCH-POT models are better than 



Application of Modified POT Method with Volatility Model …  125

the class of the SV models, but relatively worse than the other models. The 
lowest values of OTL for Expected Shortfall have been obtained for GARCH-
POT models. Generally we can say that N-GARCH-POT models are relatively 
better than GARCH models in case of OTL for ES. 

Table 2. Ranking according to total loss TL for indices 

Model WIG WIG20 SP500 DAX FTSE100 NIKKEI225 
GARCH 10 11 11 9 11 10 

GARCH TD 15 15 15 15 15 15 
AR-GARCH 11 10 10 8 8 11 

AR-GARCH TD 14 14 14 12 14 14 
SV 3 3 3 3 3 3 

GARCH-POT 6 4 7 5 4 7 
GARCH-POT TD 7 5 6 4 5 4 
AR-GARCH-POT 4 7 5 7 6 5 

AR-GARCH-POT TD 5 6 4 6 7 6 
SV-POT 1 1 1 1 1 1 

N-GARCH-POT 9 8 8 11 10 9 
N-GARCH-POT TD 13 13 12 14 12 13 
N-AR-GARCH-POT 8 9 9 10 9 8 

N-AR-GARCH-POT TD 12 12 13 13 13 12 
N-SV-POT 2 2 2 2 2 2 

Table 3. Ranking according to total loss of overestimation  OTL for VaR  

Model WIG WIG20 SP500 DAX FTSE100 NIKKEI225 
GARCH 3 3 3 4 8 11 

GARCH TD 1 1 1 2 2 8 
AR-GARCH 6 7 11 12 12 12 

AR-GARCH TD 4 4 5 11 10 7 
SV 13 13 13 13 13 13 

GARCH-POT 10 12 7 7 4 4 
GARCH-POT TD 8 9 8 8 3 1 
AR-GARCH-POT 12 10 9 5 5 3 

AR-GARCH-POT TD 9 8 10 6 6 2 
SV-POT 15 15 15 15 15 15 

N-GARCH-POT 5 5 4 3 7 9 
N-GARCH-POT TD 2 2 2 1 1 5 
N-AR-GARCH-POT 11 11 12 10 11 10 

N-AR-GARCH-POT TD 7 6 6 9 9 6 
N-SV-POT 14 14 14 14 14 14 



Marcin Fałdziński 126 

 If we compare models at the same class based on total loss TL (Table 2) we 
can state that new approach is placed between McNeil and Frey's method and 
standard volatility models. Similar conclusion can be deduced from the analysis 
of total loss of overestimation OTL for expected shortfall (Table 4). It means 
that the new approach is as good as the specific volatility model. To be accu-
rate, if the forecast of conditional mean and conditional volatility for volatility 
model is more precise, then the new approach is better than two other methods. 
In case of total loss of overestimation OTL for Value-at-Risk (Table 3), we can 
see that volatility models are the best. It should not be a surprise, because VaR 
better fits to small and mean values of financial time series in comparison with 
ES. The new approach which connects volatility models and POT method, like 
before is placed between two other methods, but sometimes this method is  
the best. 

Table 4. Ranking according to total loss of overestimation  OTL for ES 

Model WIG WIG20 SP500 DAX FTSE100 NIKKEI225 
GARCH 8 7 8 8 10 8 

GARCH TD 13 14 13 11 14 14 
AR-GARCH 9 9 11 9 11 9 

AR-GARCH TD 15 15 15 13 15 15 
SV 5 6 5 12 5 5 

GARCH-POT 3 3 1 4 2 4 
GARCH-POT TD 1 2 3 3 1 2 
AR-GARCH-POT 4 4 2 2 3 3 

AR-GARCH-POT TD 2 1 4 1 4 1 
SV-POT 10 12 9 15 7 11 

N-GARCH-POT 7 5 7 5 8 6 
N-GARCH-POT TD 12 11 12 7 12 13 
N-AR-GARCH-POT 11 8 10 6 9 7 

N-AR-GARCH-POT TD 14 13 14 10 13 12 
N-SV-POT 6 10 6 14 6 10 

 
 Generally we can say, that the new approach, which is a hybrid of standard 
volatility models and McNeil and Frey's method, is as precise as the specific 
volatility model and is able to forecast the financial time series. 

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Zastosowanie zmodyfikowanej metody POT z modelami zmienności  
do szacowania miar ryzyka 

Z a r y s  t r e ś c i. Celem artykułu jest prezentacja nowego podejścia mającego na celu połącze-
nie modeli zmienności z metodą Peaks over Threshold (POT), wywodzącą się z teorii wartości 
ekstremalnych. Podejście to opiera się na możliwości szacowania ekstremów na podstawie meto-
dy POT, natomiast wartości średnich na podstawie modeli zmienności. Nowe podejście jest za-
stosowane do estymacji miar ryzyka (VaR i ES) dla finansowych szeregów czasowych. Do oceny 
nowego podejścia wykorzystano procedury testowania wstecznego. W pracy zastosowano metodę 
POT dla stóp zwrotu indeksów rynków finansowych przefiltrowanych za pomocą modeli 
GARCH oraz SV, które porównano z wynikami otrzymanymi tylko za pomocą modeli GARCH 
i SV. 

S ł o w a  k l u c z o w e: teoria wartości ekstremalnych, Peaks over Threshold, miary ryzyka



 


