As a phenomenon, risk represents a latent quantity of money or equivalent values needed as a guarantee. We would like to model in some essential way the approach to potential loss caused by various agents. If the interest focuses on security, it is necessary to determine a limit. The aim of this paper is to refer to relevant literature and show how measure theory can be built as a mathematical discipline into economic theory providing thereby risk managers with a tool by means of which they will be able to link mathematical and economic thought.measure, risk measures, moments, random variable, probability,approximation,variance, decision making, risk function

Dominika Crnjac

English

Research Papers in Economics

241
RISK MEASURE MODELLING
Dr. Dominika Crnjac
Faculty of Electrical Engineering, Osijek
ABSTRACT
As a phenomenon, risk represents a latent quantity of money or equivalent values 
needed as a guarantee. We would like to model in some essential way the approach to 
potential loss caused by various agents. If the interest focuses on security, it is necessary 
to determine a limit. 
The aim of this paper is to refer to relevant literature and show how measure theory 
can be built as a mathematical discipline into economic theory providing thereby risk 
managers with a tool by means of which they will be able to link mathematical and 
economic thought. 
Key  words:    measure,  risk  measures,  moments,  random  variable,  probability, 
approximation,variance, decision making, risk function 
 
INTRODUCTION
   From a statistical point of view, a decision making problem can be considered as a 
game played by two players. One player is reality and the other is a statistician, whereby 
the reality condition denoted by q  is unknown to a statistician.
Let us denote the set of all reality conditions (parameter space) by Ω.
A statistician (operator) takes an action (decision) a, if he/she ﬁ  nds out that the reality 
condition is q . 
Let A be a set of all actions or decisions. 
The result of observation is a random variable1 X for which law of probability  ( ) , f x q  
depends on an unknown parameter q .
If the random variable X takes the value  x, the operator makes a decision  ( ) a d x = , 
whereby  ( ) d X  is the decision function. 
1 Sarapa, N., Teorija vjerojatnosti, Školska knjiga, Zagreb, 1988 
RISK MEASURE MODELLING242
     The decision making procedure itself consists of the following: 
a)  We deﬁ  ne the set of all possible values q  might take in the problem under  
       consideration.
b)  We deﬁ  ne the set of all possible actions or decisions which might be made.
c)  We deﬁ  ne the decision function  ( ) 1 2 , ,..., n a d X X X =  of a random sample       
       { } 1 2 , ,..., n X X X  
In this game the operator (statistician) will have either proﬁ  t or loss, depending on the 
decision that will depend on a and q .
For  the  purpose  of  quantitative  measuring  let  us  introduce  a  loss  function 
( ) ( ) ( ) 1 2 , , ,..., , n L a L d X X X q q =   as  a  numeric  function  which  associates  number 
( ) ( ) ( ) 1 2 , , ,..., , n L a L d x x x q q =  representing loss to every decision  ( ) 1 2 , ,..., n a d x x x =  
from A and every parameter value q  from Ω. 
       A correct decision is a decision for which loss is equal to zero.
Clearly, loss function  ( ) , L a q  is a random variable for which the expected value 
( ) ( ) , E L a q  represents a risk obtained by a decision a when the reality condition is q .
On the basis of the aforementioned, let us introduce a risk function  ( ) , R d q  as the 
expected value of the loss function  ( ) ( ) ( ) 1 2 , , ,..., , n R d E L d x x x q q   =    .
Two cases might occur:
1)  If   X  is a continuous random variable, then   
       
( ) ( ) ( ) ( ) ( ) 1 2 1 1 , ... , ,..., , , ... , ... n n n R d L d x x x f x f x dx dx q q q q
∞ ∞
−∞ −∞
= ∫ ∫
2)  If  X  is a discrete random variable, then  
       
( ) ( ) ( ) ( ) ( )
1 2
1 2 1 , ... , ,..., , , ... , .
n
n n
x x x
R d L d x x x f x f x q q q q =∑∑ ∑
Furthermore,  two  decision  functions  1 d   and  2 d   may  be  compared  by  means  of 
corresponding risk functions  ( ) 1, R d q  and ( ) 2, R d q . The decision function with the risk 
function taking a less value is a better one.
In order to be able to select a decision function, it is natural to use maximum values of 
the risk function. 
BAYESIAN APPROACH 
Bayesian  approach  is  simple  and  therefore  interesting  from  the  point  of  view  of 
exploitability in the decision making theory.
Dominika Crnjac243
The reality condition is considered to be a random variable q  with the law of probability 
( ) , x p q  which is a result of operator’s (statistician’s) personal conviction referring to the 
condition of reality. 
Since q  is a random variable, risk function  ( ) , R d q  will also be a random variable, 
whereby the associated law of probability reads ( ) ( ) ( ) , f x f x q p q q = .
In this case the expected risk function value with respect to the law of probability of 
the a priori parameter q  deﬁ  nes a new function (Bayes risk)
( ) ( ) , , r d E R d p q   =   .
According to (1) and (2) we have 
I)     ( ) ( ) ( ) ( ) , , r d R d d p q p q q
∞
−∞
= ∫        and 
II)     ( ) ( ) ( )
1
, ,
n
i i
i
r d R d p q p q
=
=∑ , respectively, depending on the random variable 
being continuous or discrete.
In this paper we will implement this approach into risk theory with special attention 
being paid to risk measure, by which security strategies will be optimised.
Naturally, security function depends on many variables and their bounds and the 
problem of risk is located on the ﬁ  nite time interval [ ] 0,t .
An interesting deﬁ  nition of risk was given in 1989 by Castagnoli, not assuming market 
integrity, but deﬁ  ning risk as a future unacceptable value in the interval [ ] 0,t .
Let us assume that D is a set of acceptable situations, X a random variable of the 
observed situations, and i the feedback instrument. 
Risk will be measured as minimum additional capital C, which should be invested into 
the project in order to have the value of the new situation C and +X acceptable.  
Risk measure can be taken as mapping
: , m D R →   ( ) { } inf : m X C Ci X D = + ∈ ,  whereby  we  accept  all  laws  from  the 
mathematical measure theory2, such as nonnegativity,  subadditivity, translation, etc.
Since  we  deal  with  approximation,  variance  can  be  used,  a  mathematical  notion 
2 P.R. Halmos, Measure Theory, Van Nostrand, Princeton, New Jersey, 1963
RISK MEASURE MODELLING244
frequently used in statistics:
( ) ( ) ( ) p m X E X k X s = − +  or  ( ) ( ) ( ) ( ) ( ) p m X E X E X X s = − + − , where E is 
mathematical expectancy. 
Clearly, measures deﬁ  ned in this way are not subadditive3.
Probability  distribution  cannot  be  found  easily,  so  that  we  encounter  a  deﬁ  nition 
problem by means of family p  of possible situations. Let us take risk measure as expected 
loss from the most unfavourable situation
( ) sup : p
X
m X E P
i
p
  −   = ∈    
   , least upper bound4.
Let us mention some examples of measures of the previously mentioned type:
•  average surplus function (P. Embrechts C. Kluppenberg T. Mikosch, 1997)
                  ( ) ( ) : V f E X f X f = − >
•  retarded measure (Wirch Hardy, 1999), which introduces a concave function5 
[ ] [ ] : 0,1 0,1 d →  with properties  ( ) ( ) 0 0,  1 1 d d = =
                                          
[ ] ( )
0
D E X d P X x dx
∞
  = >   ∫
A general risk theory is studied well in (C. Fishburn, 1977) using much of the stochastic 
theory and utility theory.
With respect to the situation  f  risk is deﬁ  ned as the measure 
                             ( ) ( ) ( )
f
m F f x dF x j
−∞
= − ∫ , where  ( ) ( ) 0,  0,  0 0 y y j j ≥ ≥ = , the 
domain is bounded with f, and F is a distribution function6.
Average risk usefulness model exists if and only if there exists a real function H such 
that distribution F is more favourable than G, i.e. if and only if  
                                                ( ) ( ) ( ) ( ) ( ) ( ) , , H E F m F H E G m G >
 . 
3 H.L. Royden, Real Analysis, Macmillan, New York, 1968
4 See Appendix A and Appendix B.
5 M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, University Press,   Warszaw, 
1985
6 N. Sarapa, Teorija vjerojatnosti, Školska knjiga, Zagreb, 1988
Dominika Crnjac245
Dynamic approach to risk measures 
We have seen how risk is measured in one period. However, very often insurance 
contracts, investments, etc. require a longer period of time, so that time must be clearly 
taken into account when deﬁ  ning risk and modelling risk measure.
Let [ ] 0,t  be a ﬁ  nite time interval and C capital with due time t.
In case of a complete market, whereby there is no arbitration, and there is an interested 
party who cannot invest the whole amount immediately, i.e. in time t=0, so that
                                       ( ) ( ) 0
0 :
C
C E
S t
 
=  
   
, which guarantees certain protection.
( ) 0 S T  is the price of a non risky investment at the market, E expectancy of the bounded 
risk, and C is a risk that has to be measured.
A very interesting proposal was given in 1999 by J. Cvitanić and I. Karatzes. 
If the model depends on probability distribution, strategies g  should be selected aiming 
at the decrease of expected probabilities of net losses;
                                        ( )
() ( )
( )
( )
,
0
inf
x g
C g G x
C X t
m X E
S t ⋅ ∈
  −
=  
   
 , whereby x is the initial 
capital, and   ( ) G x  a set of acceptable situations (strategies).
This measure is agreeing provided that investment strategy  X  and liability  C  are 
proportional.
This stochastic problem is solved under certain circumstances.
Uncertainty is determined through family ( ) a a A P
∈  strategies.
For the purpose of controlling risk, we determine the interval of possible measures 
with bounds:
  ( )
() ( )
( )
( )
,
0
sup inf
x g
g G x a A
C X t
m X E
S t ⋅ ∈ ∈
    −
=            
, whthe most unfavourable strategy (lower
 bound) and  ( )
() ( )
( )
( )
,
0
inf sup
x g
g G x a A
C X t
M X E
S t ⋅ ∈ ∈
    −
=            
, upper bound of the worst strategy.
           
Being familiar with laws of mathematical analysis enables us to divide interval [ ] 0,t  
into subintervals [ ] , 1 n n+ , where n comes from the set of natural numbers and take into 
account that  1 n X + V is a change of portfolio values in the interval[ ] , 1 n n+ .
Thus,  time  interval  [ ] 0,t   does  not  make  a  unit  any  more,  but  it  is  divided  into 
subintervals, so that every subinterval is considered separately as a partition Π
RISK MEASURE MODELLING246
                                      ( )
1
1 sup :
n
n
n n p n
n
X
m X F E x P
i
+
Π +
      = − ∈Π    
     
V
V V
,
whereby  n i  is a feedback instrument.
If  { } 0, max ,0
n n m m C Π Π > −  ensures alleviation of losses to the investor.
If  { } 0, min ,
n n n m m C Π Π < −  gives a possibility of withdrawal from the account, putting 
up with the expected loss.
     It can be easily seen that by means of real series ( ) , n n a b  the risk measure interval 
is given,
                                         
1
1
inf :
sup :
k k k
k
k k
n
n n F F
n
n
n n F
F n
X
a E x k K
i
X
b E x k K
i
+
∈Π
+
∈Π
      = − ∈    
     
      = − ∈    
     
V
V
V
V
                                          
                                           
{ } { } min ,0 ,max ,0 n n n n a C b C   − −  
for which there holds everything given in the following appendices.
Appendix A
Let ( ) n n N a
∈  be a real series and S a set of all its accumulation points (S may also be 
an empty set). Element  L of the set  R∞ is called the upper limit (limes superior, upper 
accumulation point) of the series ( ) n n N a
∈  , and it is determined in the following way
( )
( )
( )
n +
, if the series   is unbounded above;
  , if the series    is bounded above and the set S is empty (i.e. if  lim );
supS, if the series  is bounded above and the set S is not empt
n n N
n n n N
n n N
a
L a a
a
∈
∈ → ∞
∈
+∞
= ∞ = −∞
y. 

 


 
The lower limit (limes inferior, lower accumulation point) of the series ( ) n n N a
∈  is 
deﬁ  ned analogously.
Dominika Crnjac247
The upper limit of the series ( ) n n N a
∈  is denoted by lim sup n n a
→+∞ , and the lower limit of 
the same series by lim inf n n a
→+∞ .
The upper and the lower limit are unambiguously determined (as elements of the set
R∞) for every real series and each of them is a series accumulation point in case the series 
is ﬁ  nite.
Hence,  if lim sup n n a R
→+∞ ∈ ,  we  have lim sup max n n a S
→+∞ = ,  and  if lim inf n n a R
→+∞ ∈ ,  then
lim inf min n n a S
→+∞ = .
In case a real series ( ) n n N a
∈  is unbounded above i.e. below, the symbol+∞, i.e.  −∞ 
is often referred to as its accumulation point. If this is taken into consideration, then the 
sequence of the set R  naturally expanded to the setR∞, the upper and the lower limit can 
be deﬁ  ned with no bounds in the following way: 
                                     lim sup sup
def
n n a S
→+∞ =  ,            lim inf inf
def
n n a S
→+∞ = ,
but also in this way:
                                      lim sup max
def
n n a S
→+∞ =  ,           lim inf min
def
n n a S
→+∞ = .
Appendix B 
      1.    Let  n a  and  n b  (n N ∈ ) be real series. If for n great enough n a ≤ n b , then
                                         
lim inf lim inf n n n n a b
→+∞ →+∞ ≤
 and  
lim sup lim sup n n n n a b
→+∞ →+∞ ≤
2.  Let ( ) n n N a
∈  and ( ) n n N b
∈  be two real series. Excluding the cases with meaningless 
notions, the following inequalities hold:
               
( )
( )
lim inf lim sup lim sup lim sup lim sup ,
lim inf lim inf lim inf lim inf lim sup .
n n n n n n n n n n n
n n n n n n n n n n n
a b a b a b
a b a b a b
→+∞ →+∞ →+∞ →+∞ →+∞
→+∞ →+∞ →+∞ →+∞ →+∞
+ ≤ + ≤ +
+ ≤ + ≤ +
RISK MEASURE MODELLING248
3.  If  0 n a ≥ ,  0 n b ≥  for n great enough, then, with the same restriction as before, 
       the following inequalities hold:
       
( )
( )
lim inf lim sup lim sup lim sup lim sup ,
lim inf lim inf lim inf lim inf lim sup .
n n n n n n n n n n n
n n n n n n n n n n n
a b a b a b
a b a b a b
→+∞ →+∞ →+∞ →+∞ →+∞
→+∞ →+∞ →+∞ →+∞ →+∞
⋅ ≤ ⋅ ≤ ⋅
⋅ ≤ ⋅ ≤ ⋅
4.  For every real series ( ) n n N a
∈       
                 ( ) ( ) lim inf lim sup ,  lim sup lim inf n n n n n n n n a a a a
→+∞ →+∞ →+∞ →+∞ − = − − = −
,
        
        If ( ) 0  1,2,... n a n > = , then 
    
                 
1 1 1 1
lim inf ,  lim sup
lim sup lim inf n n
n n n n n n a a a a →+∞ →+∞
→+∞ →+∞
= =
,
         where, in addition to the previously adopted convention  1
0 =
+∞
, it is taken  
        that 
1
0
= +∞.
5.  Let ( ) n n N a
∈  and ( ) n n N b
∈  be two real series. If         
         ( )    and   for   great enough, then  n n b n b n → +∞ → +∞ ↑
           
   
       
1 1
1 1
lim inf lim inf lim sup lim sup .
n n n n n n
n n n n
n n n n n n
a a a a a a
b b b b b b
− −
→+∞ →+∞ →+∞ →+∞
− −
− −
≤ ≤ ≤
− −
INSTEAD OF A CONCLUSION
This paper partially gives a mathematical set of instruments to managers who can  
link a mathematical to an economic thought. 
Risk measure assessment (interval) is given explicitly. 
Dominika Crnjac249
LITERATURE   
1.  P. Artzner, F. Delbaen, J.M. Eber, D. Heath, Coherent risk measures, Mathematical 
Finance, 1999, Vol. 9, pp. 203 228
2.  E.  Castagnoli,  Qualche  riﬂ  essione  sull’utilitá  attesa,  Proceed.  13th  Italian 
Conference of the Association for Mathematics Applied to Economic and Social 
Sciences, 1989, pp. 993 995
3.  J. Cvitanik, I. Karatzas, On dynamic measures of risk, Finance and Stochastics, 
1999, Vol. 3, pp. 451 482
4.  C. Fishburn, Mean risk analysis with risk associated with below target returns, 
The American Economic Review, 1977, Vol. 67, pp. 116 126
 
5.  P.R. Halmos, Measure Theory, Van Nostrand, Princeton, New Jersey, 1963
 
6.  S. Karlin, Mathematical Methods and Theory in Games, Pergamon Press, 
         London Paris, 1959                                                                         
7.  M. Kuczma, An Introduction to the Theory of Functional Equations and  
     Inequalities,1985, University Press, Warszawa
8.  J. von Neumann, O. Morgenstern: Theory of Games and Economic Behavior, 
     Princenton University Press, 1953
 9.  H.L. Royden, Real Analysis, Macmillan, New York, 1968
 
10. N. Sarapa, Teorija vjerojatnosti, Školska knjiga, Zagreb, 1988 
      
11. S.D. Stoller, Operations Research: Process and Strategy, University of California  
      Press, Berkley and Los Angeles, 1964
      
12. J.L. Wirch, M.R. Hardy, A synthesis of risk measures for capital adequacy, 
      Insurance: Mathematics and Economics, 1999, Vol. 25, pp. 337 347
      
13. Williams J.D., The compleat strategist, being a primer on the theory of games of
      strategy, Mc Graw Hill Book Co., New York, 1966 
    
 
RISK MEASURE MODELLING

Risk Measure Modelling

http://www.efos.hr/repec/osi/journl/PDF/InterdisciplinaryManagementResearchIII/IMR3a8.pdf

Risk Measure Modelling

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics