








































DYNAMIC ECONOMETRIC MODELS


© 2012 Nicolaus Copernicus University Press. All rights reserved.  
http://www.dem.umk.pl/dem 

DYNAMIC ECONOMETRIC MODELS 
Vol. 12 (2012) 89−103 

Submitted October 23, 2011  ISSN 
Accepted October 29, 2012 1234-3862 

Dominik Krężołek* 

Non-Classical Measures of Investment Risk  
on the Market of Precious Non-Ferrous Metals Using  

the Methodology of Stable Distributions† 

A b s t r a c t. The aim of this article is to present some non-classical risk measures which are 
commonly used in financial investments, including investments in assets from the market of 
precious non-ferrous metals. The time series of log-returns of gold, silver, platinum and palladium 
prices are considered. To properly asses the investment risk the measures based on Value-at-Risk 
methodology have been used (the VaR estimation approach based on values from the tail of the 
distribution). Additionally, the measure comparing expected profits to expected losses from the 
opposite tails distribution has been shown – the Rachev ratio.   It was assumed that the log-returns 
of presented assets belong to the family of stable distributions. The results confirm the validity of 
the use of stable distributions to asses the risk on the precious non-ferrous metals market. 

K e y w o r d s: stable distributions, Value-at-Risk, Expected Shortfall, Median Shortfall, Rachev 
ratio, precious metals. 

J E L Classification: G11, C46. 

Introduction 
 Contemporary financial markets represent very complex area both in terms 
of functional and investigating aspects. There are many ways to increase the 
value of invested capital, i.e. investments in securities, real estate, works of art, 
precious metals, etc. Nevertheless, every possibility in investing money is relat-
ed to uncertainty and risk. Since 2007 the world economy is facing a financial 
crisis resulting mainly from the situation on the U.S. mortgage market. This 
situation has rapidly spread to other markets. The potential impact of the crisis 

                                                 
* Correspondence to: Dominik Krężołek, Department of Demography and Economic Statis-

tics, University of Economics in Katowice, ul. Bogucicka 14, Building „D”, 40-226 Katowice, 
Poland, e-mail: dominik.krezolek@ue.katowice.pl 

† Research supported by the grant number KBN: N N111 299838 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

90 

and the hedging methods against its extension have become the topic of fervent 
discussions both among politicians, practitioners and scientists. 

From the scientific point of view the sudden and unpredictable changes in 
the research area make inadequate that the current mathematical models used 
for describing analyzed reality. This raises the necessity of shifting the re-
searcher’s attention from a classical to non-classical approach. Similar necessity 
also applies to the analysis of investment and risk. The investment uncertainty 
can be considered as a derivative of decisions that have been made by a deci-
sion-maker and understood as the risky ones, with consequences in the future. 
This implies that the probability of occurrence of some particular event may be 
impossible to identify or be identified with some probability. Thus, the signifi-
cant difference between risk and uncertainty becomes clear: a risk can be clearly 
measured whereas an uncertainty is some kind of unmeasurable risk that refers 
to the investment. 

The purpose of this paper is to present some non-classical risk measures, 
widely used in risk analysis of financial markets, and presented therein with 
reference to the risk observed in the market of precious non-ferrous metals. The 
main hypothesis runs that the values of risk measures obtained for stable distri-
butions are closer to the empirical ones if compared to the normal approach. 
The subject of the study are time series represented by the log-returns of the 
prices of gold, silver, platinum and palladium. The main reason for choosing 
this particular market is an increasing interest in investing in precious metals. 
Deepening global crisis made investors look for alternatives to the traditional 
financial assets (stocks, bonds, etc.). The most popular precious metal, from the 
investing point of view, is gold. What investors gain are high availability and 
the possibility to hedge themselves in situations, where the financial system 
shows higher level of uncertainty. Purchasing of gold is more secure than typi-
cal and popular investments, e.g. in exchange rates market. However, the in-
vestment in gold doesn’t mean investment only in gold bars. There are some 
alternative forms, such as golden coins, investment certificates or investing in 
units of funds connected to the companies actively acting on the gold market. 
As a result of still worsening situation on the biggest stock markets, majority of 
investors decide to allocate their money investing in assets that show the up-
ward trend during crisis (e.g. as a hedge against increasing inflation). 

1. Methodology 
 Scientists have developed many advanced methods of measurement and 
analysis of risk. Unfortunately, the more commonly used risk measures have the 
drawback of being unclear in terms of  their theoretical underpinnings, their use 
and interpretations. Therefore, the methodological and practical considerations 
play important role for anyone who is responsible for management of that kind 
of risk. The risk level is influenced by many factors, some of which can be 
completely independent. Every theoretical model should accurately reflect the 



Non-Classical Measures of Investment Risk on the Market of Precious… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

91 

reality, and their construction is a very complex process. The methodological 
background for estimation and assessing risk, developed in the last century, 
requires to be modified due to the set of risk factors which are constantly vary-
ing over time. It results from the dynamic changes on the market, new possibili-
ties of assets allocation and unpredictability of some unexpected events. Such 
events are related to the higher risk level and may be impossible to forecast. 
This is a problem both for individual investors and institutions, which are ex-
posed to huge financial loses or even bankruptcy. 

The literature on the subject recognizes a number of risk measures. In this 
paper the distinction between classical and non-classical measures is presented. 
As for classical ones all measures which represent a canon in risk analysis and 
are widely used in practice are considered here. Most of these measures ap-
peared together with the development of certain scientific theories and after-
wards were modified according to the external factors which determine their 
use. Moreover, the set of classical risk measures includes those based on com-
monly used probability distributions (especially the normal distribution). The 
set of non-classical risk measures encompasses all measures not included in the 
set of classical ones. The non-classical measures might be etymologically relat-
ed to the interdisciplinary nature of science, deriving from the different scien-
tific areas (physics, engineering, bio-medicine, etc.). However, because of their 
mathematical properties they are used in measuring risk on financial markets. 

Due to strong assumption about the normality of distribution of stock re-
turns in the set of classical measures, not met for the real data, some alternative 
measures based on different probability distributions have to be taken into ac-
count. The characteristics of empirical time series as high frequency of data, 
heteroscedasticity of variance, autocorrelation or fat tails of distributions are 
essential in risk analysis. The stable distributions, developed by B. Mandelbrot 
in the 60s,  have the wide application in this matter. Distributions which belong 
to the class of stable ones, are described by the shape parameter allowing to 
model the asymmetry and fatness of the tail of distribution. It makes them use-
ful in many scientific areas (from engineering, through physics to applications 
on financial markets (Borak, Härdle, Weron, 2005). The main difficulty in ap-
plication of stable distributions is that their probability distribution function is 
not clearly defined. To describe stable model the characteristic function is used 
(and so-called Inverse Fourier Transform with respect to this function). If the 
random variable X  has the cumulative distribution function ( )xF , then its 
characteristic function has the form (Rachev, Mittnik, 2000): 

( ) ( )[ ] ( ) ( ),expexp ∫
+∞

∞−

== xdFitxitXEtϕ  (1) 

Thus, for stable random variable the characteristic function is presented as: 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

92 

( )
( )

( )
,

1,ln21

1,
2

1
ln











=



 +−

≠













−−

=

α
π

βσµ

ααπβσµ
ϕ

αα

forttsignitti

fortgtsignitti
t  (2) 

where ( )tsign  is defined as ( )








<⇔−
=⇔
>⇔

=
01

00
01

t
t
t

tsign . 

The main characteristic of stable distribution is the shape parameter α  al-
lowing to measure the fatness of the tail of distribution. Remaining parameters 
which describe the stable PDF are skewness parameter 1;1−∈β , scale pa-
rameter 0>σ  and location parameter R∈µ . The practical use of stable distri-
bution is related to complex estimation procedure of all parameters. In literature 
only three types of stable distributions are explicitly defined: normal distribu-
tion (where 2=α ), Cauchy distribution (where 0,1 == βα ) and Lévy distri-

bution (where 1,2
1 −== βα ). The shape parameter plays significant role in 

probability distribution analysis. If it is less than 2, then the variance of the dis-
tribution is infinite and the location parameter is equal to the mean of distribu-
tion. In the case where the shape parameter is less than 1, both variance and 
mean are infinite (Samorodnitsky, Taqqu, 1994). 

The application of stable distributions is extremely justified if the fat tailed 
distributions are considered. In that case the probability that random variable 
takes values at the level significantly outlying from the center of the distribution 
is higher than in the normal case. Hence, the risk measures which are used in 
risk analysis have to take into account such values. The most popular and the 
most widely used risk measure in this case is Value-at-Risk (VaR). Several 
methods of estimating Value-at-Risk are used, but depend on many aspects, 
such as statistical assumptions related to the risk factors, dependency between 
these factors or the portfolio structure. In this paper the quantile based method 
for estimating VaR is used. In this method it is not necessary to assume the ana-
lytical form of function which describes the probability distribution of the risk. 
The historical data used in this method allows to estimate parameters which 
describe the real distribution of return. Afterwards, the −α quantile of the dis-
tribution is estimated which allows to determine the VaR. In this method the 
most important problem is to select the proper probability distribution function, 
especially if the financial time series are analyzed. In addition, some tests of the 
fit of distribution have to be used to verify the convergence between empirical 
and theoretical distributions. 



Non-Classical Measures of Investment Risk on the Market of Precious… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

93 

2. Non-Classical Risk Measures 
The tail analysis of empirical distributions requires to define some class of 

measures which allow to asses risk related to the values significantly distant 
from the central part of the distribution. In this matter the quantile risk measures 
are considered. The idea of quantile risk measures is based on the Value-at-Risk 
(VaR) methodology, which is one of the most popular risk measures used in 
practice. Its principal advantage is that the VaR conveys straightforward infor-
mation about potential loss as a result of an investment. However, such infor-
mation does not include cases where some extreme observation occurs. Hence is 
not a good measure. Artzner, Delbean, Eber and Heat (Artzner et al., 1997) 
have proposed the set of axioms which have to be met by a good risk measure. 
These axioms define a coherent risk measure. If ϑ  is a coherent risk measure 
for a set of random variables Y  (defined on some probability space), such as 

+→ RY:ϑ , a measure ϑ  have to be: 
sub-additive, for any ( ) ( ) ( )212121 :, XXXXYXX ϑϑϑ +≤+∈  
positively homogeneous, for any 0≥λ  and ( ) ( )XXYX λϑλϑ =∈ :  
monotonous, for any YXX ∈21 ,  if only ( ) ( )2121 : XXXX ϑϑ ≤≤  
translation invariant, for any ( ) ( ) cXcXRcYX +=+∈∈ ϑϑ:,  

As was mentioned above, VaR is one of the most popular quantile measure 
used by practitioners. It gives an answer to a question of what is the maximum 
loss incurred by an investor (or institution) from an investment in some speci-
fied time horizon. This loss may occur with some probability α  called toler-
ance level. Maintaining previous assumptions and defining Value-at-Risk at the 
confidence level α , i.e. ( )XVaRα , it is possible to calculate coherent risk 
measures determining expected value (in terms of mean and median) of an in-
vestment exceeding potential loss beyond VaR: the Expected Shortfall and Me-
dian Shortfall. These measures are calculated using formula: 

( ) ( ) ( )[ ],XVaRXXVaRXEXES ααα >−=  (3) 

( ) ( ) ( )[ ],XVaRXXVaRXMedianXMS ααα >−=  (4) 

These measures inform about the possible expected loss beyond the level repre-
sented by VaR. In other words, what follows from (3) and (4) is that for any 
given α  there exists one-step-ahead prediction representing expected value (in 
terms of mean and median) of loss beyond the VaR. 

Additionally, taking into account certain tails’ characteristics of empirical 
distributions it is necessary to compare if expected profits exceed expected loss-
es. An interesting tool has been proposed by Rachev – the Rachev ratio 

ratioR − (Biglova, Ortobelli, Rachev, Stoyanov, 2004). This measure is calcu-
lated using formula: 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

94 

( )( )
( )( )

|

|

E X X VaR X
R ratio

E X X VaR X
α

β

≥ − −
− =

≤
 (5) 

If the random variable X  represents profit or loss from a given investment, the 
Rachev ratio −R ratio  defines the ratio of expected profit above the VaR value 
(for a given −thα quantile) to expected losses below the VaR value (for a given 

−thβ quantile). If (in a special case) αβ −=1 , then the values of −R ratio  are 
interpreted as follows: 
− 1R ratio− =  – expected profits are equals to expected losses, 
− 1R ratio− >  – expected profits exceed expected losses, 
− 1R ratio− <  – expected profits are lower than expected losses. 

The risk assessment using −R ratio  is based on maximizing its values. 

3. Empirical Analysis 
The practical application of some non-classical risk measures is presented 

here in the relation to the market of precious non-ferrous metals, represented by 
the time series of  2902 daily log-returns of the prices of gold, silver, platinum 
and palladium in period of January 2000 – June 20111. As presented in Table 1, 
there is a significant dispersion in prices of precious non-ferrous metals.  

Table 1.  Descriptive statistics – prices 

  Gold Silver Platinum Palladium 
Mean 622.89    10.84    1 023.26    387.43    

Standard deviation 349.12    7.43    440.08    196.87    
Minimum 255.95    4.07    414.00    148.00    
Maximum 1 552.50    48.70    2 273.00    1 090.00    

Range 1 296.55    44.64    1 859.00    942.00    
 

The highest dispersion level (in terms of range) is observable in prices of 
platinum and gold. Although, if the coefficient of variation is taken into ac-
count, the highest dispersion is related to the prices of silver. The volatility of 
prices of gold and palladium is presented on Figures 1 and 2 respectively. The 
prices of gold shows the behavior similar to the stock indices whereas the prices 
of palladium not.  

As is presented in Figure 1, the price of gold has been increasing reasonably 
till February 2006. Then some significant price changes have been observed. 
The last four years show that the price of gold hadn’t behave in a stable manner. 

 

                                                 
1 Data from London Metal Exchange. 



Non-Classical Measures of Investment Risk on the Market of Precious… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

95 

 

Figure 1.  The volatility of price of gold from January 2000 to June 2011 

The price of palladium has been changing without specifically defined di-
rection. After first two years of growing the price has fallen dramatically return-
ing to upward trend in first half of 2003. The year 2009 initiated the period of 
surge of prices that continues till now. In the next step the log-returns of corre-
sponding prices have been calculated. Figure 3 presents the volatility observed 
in time series of gold log-returns. 

 
Figure 2.   The volatility of price of palladium from January 2000 to June 2011 

Gold

0,00 

600,00 

1 200,00 

1 800,00 

1 501 1001 1501 2001 2501

pr
ice

 [U
SD

]

Palladium

0,00 

400,00 

800,00 

1 200,00 

1 501 1001 1501 2001 2501

pr
ice

 [U
SD

]



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

96 

 
Figure 3.  Volatility of log-return of gold 

As presented in Figure 3, the period starting in the middle of 2005 reveals 
increased level of volatility (clustering and significant jumps in returns). The 
same or even more unstable behavior characterizes returns of palladium. Clus-
tering is observed within entire period, with higher intensity during last year 
(Figure 4). 

 
Figure 4.  Volatility of log-return of palladium  

Log-return Gold

-0,1

-0,06

-0,02

0,02

0,06

0,1

log-return Palladium

-0,2

-0,1

0

0,1

0,2



Non-Classical Measures of Investment Risk on the Market of Precious… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

97 

One of the most important matter related to the risk analysis of financial 
markets is that, whether there exists a significant relation between assets. It is of 
great importance especially in portfolio analysis. To see that, Table 2 presents 
the correlations between log-returns of metals analyzed, whereas Table 3 shows 
the descriptive statistics. 

Table 2. Correlations 

  Gold Silver Platinum Palladium 
Gold    1.00000        0.56024        0.44953        0.37513     
Silver    0.56024        1.00000        0.45243        0.42328     

Platinum    0.44953        0.45243        1.00000        0.57408     
Palladium    0.37513        0.42328        0.57408        1.00000     

The highest values of correlation are between pairs of returns of platinum 
and palladium and between gold and silver. The lowest values correspond to the 
relation between gold and palladium. It can be caused by a natural market prop-
erties of these precious metals. Gold and silver are considered as investment 
precious metals while platinum and palladium are more related to industrial 
environment (especially the latter). 

Table 3. Descriptive statistics – returns 

  Gold Silver Platinum Palladium 
Mean 0.00058 0.00065 0.00048 0.00019 

Standard deviation ( Nσ ) 0.01151 0.02107 0.01603 0.02392 
Minimum -0.07972 -0.18693 -0.17277 -0.17859 
Maximum 0.06841 0.18279 0.11129 0.16799 

Range 0.14813 0.36971 0.28406 0.34658 

As presented in Table 3, the highest value of expected return corresponds to 
the variables silver and gold. The comparable level reaches the return of plati-
num, whereas the return of palladium oscillates around zero. The dispersion 
analysis showed that the most risky (in terms of standard deviation) are invest-
ments in palladium and silver (in terms of range as well) while the safest one 
correspond to gold (Figure 5). 

Further analysis related directly to risk measurement requires to match the 
theoretical probability distribution to the real market data. Classical risk analy-
sis is based on the normal distribution, so this case has to be verified. To test the 
normality of log-returns of the selected precious metals the following goodness-
of-fit methods have been used: Jarque-Bera test and Anderson-Darling test. The 
results of comparing empirical and normal distribution show that the assump-
tion of normality has to be rejected (at the significant level of 0.01). Therefore 
the empirical distributions do not belong to the family of normal ones. The re-
jection of normality requires the use of risk measures which are not based on 
this strong assumption and allow to use other statistical distributions. 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

98 

 

 
Figure 5.  Expected return vs. Risk 

To solve this problem, the family of stable distribution is used. To estimate the 
parameters of stable distribution, the Maximum Likelihood Methodology is 
applied and the results are presented in Table 4. 

Table 4. The parameters of stable distributions 

  α β σ μ 
Gold 1.63592 -0.04440 0.00620 0.00063 
Silver 1.60458 -0.03883 0.01074 0.00103 

Platinum 1.58989 -0.09054 0.00790 0.00061 
Palladium 1.54917 -0.01818 0.01192 0.00049 

Note: the parameters are significant at the level of 0.01. 

As was mentioned in section “Methodology”, the parameter α  plays the 
most important role. It is responsible for the thickness of tail of distribution. 
As results from calculations, each of log-returns distributions of analyzed assets 
is fat-tailed. The index of stability takes the highest value for the variable gold, 
and the lowest for the variable palladium. Moreover, each of the distributions 
has a left-sided asymmetry which means that the left-side tail is heavier than the 
right-side one. If α  takes values below 2, the location parameter is equal to the 
mean of distribution. Therefore the more accurate information about the ex-

Expected return vs. Risk

palladium

silver

platinum

gold

0,00000

0,00020

0,00040

0,00060

0,00080

0,00000 0,00500 0,01000 0,01500 0,02000 0,02500 0,03000

Standard dev iation

M
ea

n



Non-Classical Measures of Investment Risk on the Market of Precious… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

99 

pected return of empirical data is provided rather by the value of location pa-
rameter  than by the value of mean (which is more suitable if the assumption  

 

 
Figure 6. Fitting of normal distribution – platinum 

of normality is not rejected). The scale parameter σ  can be interpreted as 
a dispersion parameter (just like the standard deviation in normal case) as the 
relation between location parameter of stable distribution and standard deviation 
of normal distribution is as follows: 2Nσσ = . The fitting of normal and sta-
ble distributions to the empirical data of variable platinum is presented using 
histograms and QQ-plots (Figures 6–9). 

 

 
Figure 7. Fitting of stable distribution – platinum 

-0.15 -0.1 -0.05-1.38778 10-170.05 0.1

10

20

30

-0.15 -0.1 -0.05-1.38778 10-170.05 0.1

10

20

30



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

100 

The Figures 6–7 show that stable distribution is better fitted to the empirical 
data of platinum than in the normal case. QQ-plots confirm that in terms of 
quantile analysis. The more dotted line covers the solid one the better fit to the 
theoretical distribution. This property of QQ-plot is rejected for normal case. 
 

 
Figure 8. QQ-plot of normal distribution – platinum 

 
Figure 9. QQ-plot of stable distribution – platinum 

Further risk analysis is based on the non-classical measures arising from the 
methodology of Value-at-Risk. The concept of VaR is based on the estimation 
of quantile of arbitrary distribution. In the first step the Value-at-Risk for each 
variable is calculated. For calculation one-period forecast is used. The results 
presented in Table 5. 
  

-0.04 -0.02 0.02 0.04

-0.06

-0.04

-0.02

0.02

0.04

-0.04 -0.02 0.02 0.04

-0.04

-0.02

0.02

0.04



Non-Classical Measures of Investment Risk on the Market of Precious… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

101 

Table 5. Value-at-Risk estimates 
Risk 

measure Type of distribution Quantile Gold Silver Platinum Palladium 

Value-
at-Risk 

empirical distribu-
tion 

0.01 -0.03298 -0.05946 -0.04992 -0.07466 
0.05 -0.01727 -0.03093 -0.02266 -0.03747 

normal distribution 0.01 -0.02414 -0.04929 -0.03684 -0.05663 
0.05 -0.01815 -0.03344 -0.0262 -0.04074 

stable distribution 0.01 -0.03638 -0.06713 -0.05238 -0.08346 
0.05 -0.01651 -0.02931 -0.02218 -0.03453 

Table 5 presents Value-at-Risk estimates for quantiles 0.01 and 0.05 for 
three types of distributions. The results obtained for the stable model are similar 
to those obtained for the empirical distribution. However, comparing results of 
normal estimates, the differences are significant. For example, the VaR for gold 
at 0.01 confidence level for normal distribution is -0.02414 while for empirical 
and stable distribution is -0.03289 and -0.03638 respectively. Therefore, taking 
into account the normal case, potential loss seems to be lower than it actually is 
(comparing to the empirical values). Moreover, the results are corroboration of 
the fitting quality of stable models to the empirical data. 

Table 6. Expected Shortfall estimates 
Risk 

measure Type of distribution Quantile Gold Silver Platinum Palladium 

Expected 
Shortfall 

empirical distribu-
tion 

0.01 -0.04349 -0.09007 -0.07171 -0.09979 
0.05 -0.02707 -0.05209 -0.04001 -0.05930 

normal distribution 0.01 -0.02715 -0.05739 -0.04209 -0.06523 
0.05 -0.02207 -0.04352 -0.03286 -0.05051 

stable distribution 0.01 -0.04429 -0.09507 -0.08766 -0.08540 
0.05 -0.03563 -0.06547 -0.06267 -0.06288 

Similar results obtained for the Expected Shortfall (Table 6) and Median 
Shortfall (Table 7). Expected loss exceeding VaR (in terms of mean and medi-
an) is similar for normal and stable distributions while it differs for the normal 
one. 

Table 7. Median Shortfall estimates 
Risk 

measure Type of distribution Quantile r-gold r-silver r-platinum r-palladium 

Median 
Shortfall 

empirical distribu-
tion 

0.01 -0.04017 -0.07918 -0.06704 -0.09038 
0.05 -0.02359 -0.04459 -0.03377 -0.05055 

normal distribution 0.01 -0.02591 -0.05627 -0.03951 -0.06276 
0.05 -0.02147 -0.04145 -0.03137 -0.05051 

stable distribution 0.01 -0.04341 -0.07851 -0.07346 -0.09073 
0.05 -0.02594 -0.03991 -0.03263 -0.04142 

Using values of VaR for a given pairs of quantiles, the results obtained for 
−R ratio are presented in Table 8 and Table 9. 



Dominik Krężołek 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

102 

Table 8. R-ratio estimates (quantiles 0.99 vs. 0.01) 
Risk 

measure Type of distribution Quantile r-gold r-silver r-platinum r-palladium 

R-ratio 

empirical distribu-
tion 0.99 vs.0.01 1.22435 1.12766 0.99684 1.00492 

normal distribution 0.99 vs. 0.01 1.13321 1.19392 0.98224 1.02183 

stable distribution 0.99 vs. 0.01 1.21334 1.14118 0.99733 1.01432 

Taking into account quantiles 0.99 and 0.01 the values of ratio obtained for 
stable distribution are closer to the values obtained from empirical distribution. 
This shows that the approach based on stable distribution is more adequate to 
the real behavior of stock returns. 

Table 9. R-ratio estimates (quantiles 0.95 vs. 0.05) 
Risk 

measure Type of distribution Quantile r-gold r-silver r-platinum r-palladium 

R-ratio 

empirical distribu-
tion 0.95 vs.0.05 1.01766 1.04638 1.00231 1.01233 

normal distribution 0.95 vs. 0.05 1.12235 1.11321 0.99112 1.09121 

stable distribution 0.95 vs. 0.05 1.02152 1.08199 0.99984 1.04523 

The data presented in Table 9 confirmed the results obtained for quantiles 0.99 
and 0.01 – the advantage of using stable approach over the normal one.  

Conclusions 
The application of non-classical risk measures plays very important role in 

financial market analysis. Both scientists and practitioners confirm that the risk 
analysis has to be extended beyond the normal case, and this approach covers 
not only financial market but also other ones, e.g. non-ferrous metals market. 
This paper has presented the analysis of daily log-returns of gold, silver, plati-
num and palladium time series. As confirmed, the assumption of normality of 
distribution for each variables has been rejected and the stable models have 
been applied. The results show that stable distributions are better fitted to the 
empirical data than the normal ones, complying with leptokurtosis, heavy tails 
(the range of parameter α  reflecting thickness of the tail is 1.55–1.64) and 
asymmetry. 

In the case of risk measurement the methodology of Value-at-Risk is the ba-
sis of calculating non-classical risk measures. Both VaR and measures such as 
Expected Shortfall and Median Shortfall prove discrepancy between normal and 
empirical distributions. That is the reason why the other type of distributions 
should be taken into account. Similar results for calculating coherent risk 
measures were obtained by Trzpiot, Krężołek (2009) for analyzing daily log-



Non-Classical Measures of Investment Risk on the Market of Precious… 

DYNAMIC ECONOMETRIC MODELS 12 (2012) 89–103 

103 

returns on Polish financial market using stable models and by Krężołek (2010) 
on precious non-ferrous metals market, but using geo-stable distributions (more 
precisely – Asymmetric Laplace distribution). The results, although straight-
forward in terms which class of theoretical distribution should be chosen, have 
to be interpreted very carefully. Moreover, considering −R ratio the results 
obtained for stable distributions are closer to the values for empirical one com-
paring to the normal ones. 

References 
Artzner, P., Delbaen, F., Eber, J. M., Heath, D. (1997), Thinking Coherently, Risk, 10, 68–71. 
Biglova, A., Ortobelli, S., Rachev, S. T., Stoyanov, S. (2004), Different Approaches to Risk 

Estimation in Portfolio Theory, Journal of Portfolio Management, 31, 1, 106. 
Borak, Sz., Härdle, W., Weron, R. (2005), Stable Distributions, Springer, Berlin.  
Hammoudeh, S., Malik, F., McAleer, M. (2011), Risk Management of Precious Metals, Quarterly 

Review of Economics and Finance, 51, 435–441.  
Krężołek, D. (2010), Kwantylowe oraz koherentne miary ryzyka – analiza empiryczna na rynku 

metali nieżelaznych z wykorzystaniem rodziny statystycznych rozkładów stabilnych 
(Quantiles and Coherent Risk Measures – An Empirical Analysis on the Non-ferrous Met-
als’ Market Based on the Family of Stable Distributions), Finanse, Rynki Finansowe i Ub-
ezpieczenia. Skuteczne Inwestowanie (Finance, Financial Markets and Insurance. Effec-
tive Investment), 29, 433–444.  

Rachev, S., Mittnik, S. (2000), Stable Paretian Models in Finance, Series in Financial Economics 
and Quantitative Analysis, John Wiley & Sons Ltd., England. 

Samorodnitsky, G., Taqqu, M. S. (1994), Stable Non-gaussian Random Processes. Stochastic 
Models with Infinite Variance, Chapman & Hall, New York, 1994. 

Trzpiot, G. (2010), Wielowymiarowe metody statystyczne w analizie ryzyka inwestycyjnego (Mul-
tivariate Statistical Methods in Investment Risk Analysis), Wydawnictwo PWE, Warszawa. 

Trzpiot, G., Krężołek, D. (2009), Quantiles Ratio Risk Measures for Stable Distributions Models 
in Finance (Kwantylowe miary ryzyka z wykorzystaniem rozkładów stabilnych 
w finansach), Studia Ekonomiczne, 53 (Economic Studies, 53), Zeszyty Naukowe Akade-
mii Ekonomicznej w Katowicach, 109–120.  

Nieklasyczne mierniki ryzyka inwestycyjnego na rynku szlachetnych 
metali nieżelaznych z wykorzystaniem metodologii rozkładów stabilnych 

Z a r y s  t r e ś c i. Celem artykułu jest prezentacja wybranych nieklasycznych miar ryzyka, które 
mają szerokie praktyczne zastosowanie w przypadku inwestycji finansowych, w tym inwestycji w 
walory notowane na rynkach surowców (nieżelazne metale szlachetne). Przedmiotem badania są 
szeregi czasowe reprezentowane przez stopy zwrotu cen złota, srebra, platyny oraz palladu. Do 
oceny ryzyka inwestycyjnego wykorzystano miary wyznaczone w oparciu o metodologię Value-
at-Risk stosując estymację opartą na wartościach z ogona rozkładu. Dodatkowo do oceny ryzyka 
wykorzystano wskaźnik Rachev’a. Przyjęto założenie, że stopy zwrotu wykorzystanych zmien-
nych należą do rodziny rozkładów stabilnych. Wyniki badania potwierdzają stosowność wyko-
rzystania rozkładów stabilnych do oceny ryzyka na rynku metali nieżelaznych. 

S ł o w a  k l u c z o w e: rozkłady stabilne, Value-at-Risk, Expected Shortfall, Median Shortfall, 
wskaźnik Rachev’a, metale szlachetne 




	Introduction
	1. Methodology
	2. Non-Classical Risk Measures
	3. Empirical Analysis
	Conclusions
	References

