Download Study on the Dependence Conditional Risk in Financial Portfolio

Study on the Dependence Conditional Risk in Financial Portfolio
YI Wende1, HUANG Aihua2
1. Dept. of Math & Computer Science, Chongqing University of Arts and Sciences;
2. Dept. of planning finance, Chongqing University of Arts and Sciences Yongchuan Chongqing 402160,
China
[email protected]
Abstract: It’s very interesting to investigate the dependence structure and measures of conditional
dependence risk of time series. In this paper, some dependence probabilities and conditional dependence
risk density functions are proposed to study the conditional dependence risk of time series.
Keywords: Copula; Conditional dependence; Conditional density; Conditional probability
1 Introduction
Dependence risk measurement in portfolio’s risk management is one of the most important issues.
It is well known that linear correlation has some deficiencies that one is that it is not invariant under
non-linear strictly increasing transformation, and the other is that it can not measure the dependence
structure of two time series, such as non-linear correlation, kurtosis, fat-tail and tail-dependence. While
the dependence measures derived from copulas can overcome this shortcoming. From the time when
Embrechts et al. (2001)[1] proposed the application of copula functions in order to model the multivariate
distribution of returns, the copula functions have had broader applications in finance and the theory of
copulas has experienced rapid development[2,3,4] with the development of computer software and
information technology.
This paper presents some conditional dependence risk probabilities and conditional dependence
risk density functions of time series using copula functions to investigate the conditional dependence
risk of financial portfolio assets. Our research focuses on the condition such as X = x or Y = y and
X ≤ x or Y ≤ y and X > x or Y > y or ( X > x, Y > y ) .
The structure of this paper is as follows. Section 2 introduces the notation of copula and some tail
dependence measures. Section 3 presents several conditional dependence risk probability functions and
conditional dependence risk density functions for studying the conditional dependence structure and the
conditional dependence degree of portfolio assets. Section 4 concludes the paper.
2 Copulas and Tail Dependence Measures
2.1 Copulas
The most interesting fact in the copula framework is the Sklar’s (1959)[2] theorem. A copula
function is defined as a multivariate distribution with standard uniform marginal distributions or as a
function that maps values from the unit hypercube to value in the unit interval. If the joint distribution of
X and Y is H ( ⋅, ⋅ ) with continuous marginal distributions F ( ⋅ ) and G ( ⋅ ) , respectively, then there
exists a unique copula C ( ⋅, ⋅ ) such that the joint distribution can be written as
H ( x , y ) = C ( F ( x ) , G ( y ) ) , ( x , y ) ∈ ℜ2 .
From this expression, we can see that any copula C ( ⋅, ⋅ ) pertaining to function H ( ⋅, ⋅ ) can also be
expressed as
C ( u , v ) = H ( F −1 ( u ) , G −1 ( v ) ) ,
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[ ]
where u = F ( x ) , v = G ( y ) , u , v ∈ 0,1 .
2.2 Tail Dependence Measures
The dependence structure of X and Y is characterized by the copula, moreover, the measure of
dependence is invariant under strictly increasing transformations of random variables. For example,
Kendall’s tau (τ ) and Spearman’s rho ( ρ ) are the most widely used scale invariant measure of
concordance.
(1) Tail dependence
Another important concept related to copulas is the concepts of upper tail dependence and lower
tail dependence. The upper tail dependence is defined as
(
)
lim+ λU (α ) = lim+ P X > F −1 (1 − α ) Y > G −1 (1 − α ) ,
α →0
α →0
if this limit exists. The two variables X and Y are said to be asymptotically dependent in the upper tail
if
λU ∈ ( 0,1] ,
and asymptotically independent if
λU = 0 .
In finance the concept of upper tail
dependence can be interpreted by
λU = lim P ( X > VaRα ( X ) Y > VaRα (Y ) ) ,
α → 0+
where VaRα ( X ) = F
−1
(1 − α ) is the value at risk.
In a similar way, the lower tail dependence is defined as
(
)
lim λL (α ) = lim+ P X < F −1 (α ) Y < G −1 (α ) .
α → 0+
In terms of copulas,
α →0
λU and λL can be expressed using the pertaining copula.
1 − 2u + C ( u , u )
,
u →1
1− u
C ( u, u )
.
λL = lim
u →0
u
λU = lim
(2) Conditional tail risk measures
In risk management, VaR (value-at-risk) usually acts as an important tool of risk measurement.
Briefly speaking, VaR is the maximal potential loss of a position and a portfolio in some investment
horizon under a given confidential level and over a certain period. While VaR is a powerful tool for risk
management, it is not a coherent risk measure and sub-additive. To overcome this problem, a
conditional VaR (denote CVaR or ES)[5]is proposed to modify the risk measure of VaR.
CVaRα = E  X X > VaRα ( X )  ,
(1)
where X is the variable representing the negative return (loss) of asset and FX is its distribution.
Artzner et al. (1998)[6]proposed the concept of the expected shortfall eX
∗
shortfall eX
(α )
(α ) which are respectively defined as
eX (α ) = E  X − VaRα ( X ) X > VaRα ( X )  ,
e∗X (α ) = Median  X − VaRα ( X ) X > VaRα ( X )  .
3 Dependence Conditional Risk Density Functions
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and the median
(2)
(3)
Let random variables X and Y represent the negative returns (loss) value of two financial assets.
If the joint distribution of X and Y is H ( ⋅, ⋅ ) with continuous marginal distributions F ( ⋅ ) and
G ( ⋅ ) , and the unique copula C ( ⋅, ⋅ ) , whose density functions are f ( ⋅) , g ( ⋅) and c ( ⋅, ⋅ ) ,
respectively. Given certain values of X and Y , such as x and y , we consider some dependence risk
density functions under the conditions such as X = x or Y = y and X ≤ x or Y ≤ y and X > x or
Y > y or ( X > x, Y > y ) . These dependence risk density functions give the instantaneous
probabilities for the individual security and the portfolio assets that have attained the values x and y .
P { X ≤ x Y = y} = lim P { X ≤ x y − ∆y < Y ≤ y}
∆y → 0
P { X ≤ x, y − ∆y < Y ≤ y}
∆y → 0
P { y − ∆y < Y ≤ y}
= lim
 x h ( u, t ) du  dt
∫y −∆y  ∫−∞

= lim
y
∆y → 0
∫ g ( t ) dt
y
y −∆y
x
=
∫ h ( u, y ) du
−∞
g ( y)
∫ c ( F ( x ) , G ( y ) ) dF ( x ) .
=
x
−∞
The function f ( x ) =
X Y
g ( y)
c ( F ( x ) , G ( y )) f ( x )
g (y)
(4)
is called as the conditional density function
under the condition Y = y . Similarly, we can obtain the conditional density function gY X ( y ) .
P {X ≤ x Y ≤ y} =
P { X ≤ x , Y ≤ y} H ( x , y ) C ( F ( x ) , G ( y ) )
. (5)
=
=
G (y)
G (y)
G (y)
f X Y( ( x ) =
where C 1 ( u , v ) =
∂C (u , v )
∂u
C1 ( F ( x ) , G ( y ) ) f ( x )
G (y)
(6)
.
P { x < X ≤ x + ∆ x X > x} =
F ( x + ∆x ) − F ( x )
1− F (x)
≈
The function
,
f ( x ) ∆x
.
1− F (x)
(7)
f (x)
is interpreted as a conditional probability density function: it gives the value of
1− F (x)
the conditional probability density function of X at exact value x , given survival to that value. Let’s
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denote it as f X
X
( x ) , which is usually called the hazard rate function. In the follows, we discuss the
dependence scenario of two assets.
P { y < Y ≤ y + ∆y X > x} =
=
gY X ( y ) = lim+
X
1− F (x)
∂H ( x, y )
∂C ( u , v )
, and C 2 ( u , v ) =
.
∂y
∂v
P {x <
,
(9)
X ≤ x + ∆ x , y < Y ≤ y + ∆ y}
1 − F (x )
H ( x + ∆x , y + ∆ y ) − H ( x + ∆ x, y ) − H ( x, y + ∆ y ) + H ( x, y )
1− F (x)
=
hXY
g ( y ) − C 2 ( F ( x ) , G ( y )) g ( y )
< X ≤ x + ∆ x, y < Y ≤ y + ∆ y X > x} =
=
.(8)
∆y
=
{x
1− F (x)
g ( y ) − H Y ( x, y )
1− F (x)
=
P
G ( y + ∆ y ) − G ( y ) −  H ( x , y + ∆ y ) − H ( x , y ) 
P { y < Y ≤ y + ∆y X > x}
∆y → 0
where H Y ( x, y ) =
P { y < Y ≤ y + ∆ y } − P { X ≤ x , y < Y ≤ y + ∆ y}
1− F (x)
(x, y ) =
h ( x , y ) ∆ x ∆y
.
1− F ( x)
lim +
(10)
P (x < X ≤ x + ∆x, y < Y ≤ y + ∆y X > x )
∆ x∆ y
∆x→ 0
∆y → 0+
=
h ( x, y )
1 − F (x)
=
c ( F ( x ) , G ( y )) f
1− F (x)
(x ) g ( y) .
(11)
Similarly, the corresponding density functions under the condition Y > y can be calculated, omitted
here.
Now we consider the scenario under the condition ( X > x, Y > y ) .
P (x < X ≤ x + ∆x X > x ,Y > y ) =
F ( x + ∆ x ) − F ( x ) −  H ( x + ∆ x , y ) − H
1 − F (x ) − G ( y ) + H (x, y )
=
f X XY ( x ) = lim+
∆x → 0
=
f ( x ) ∆ x − H X ( x, y ) ∆ x .
1 − F ( x ) − G ( y ) + H ( x, y )
( x , y ) 
(12)
P ( x < X ≤ x + ∆x X > x , Y > y )
∆x
f ( x ) − C1 ( F ( x ) , G ( y ) ) f ( x )
1 − F ( x ) − G ( y ) + C ( F ( x ), G ( y ))
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.
(13)
P ( x < X ≤ x + ∆x, y < Y ≤ y + ∆ y X > x , Y > y ) =
=
P {x < X ≤ x + ∆ x, y < Y ≤ y + ∆y}
1 − F ( x ) − G ( y ) + H ( x, y )
h ( x, y ) ∆x ∆y
1 − F ( x ) − G ( y ) + H ( x, y )
=
c ( F ( x ) , G ( y )) f ( x ) g ( y ) ∆ x ∆y
1 − F ( x ) − G ( y ) + C ( F ( x ), G ( y ))
hXY XY ( x, y ) = lim+
(14)
P ( x < X ≤ x + ∆x, y < Y ≤ y + ∆y X > x , Y > y )
∆x∆y
∆x →0
∆y → 0+
=
;
c ( F ( x ) , G ( y )) f
(x) g ( y)
.
1 − F ( x ) − G ( y ) + C ( F ( x ), G ( y ))
(15)
In this section, we achieve some dependence conditional probability functions and its density
functions. These results are very useful in risk management, since the risk analysts need to analyse the
conditional dependence structure and conditional dependence measure of portfolio assets according to
given risk conditions. Especially, the conditional dependence risk probability and density functions have
particular meaning when the threshold values equal some certain values such as x = 0, y = 0 and
x = VaRα ( X ) , y = VaRα ( Y ) .
4 Conclusions
In this paper, we have studied the conditional dependence risk probability and the conditional
dependence risk density functions. We mainly focus on the scenarios under the conditions such as
X = x or Y = y and X ≤ x or Y ≤ y and X > x or Y > y or ( X > x, Y > y ) . A member of
conditional dependence risk relationships is completely characterized by the marginal distribution and
the copulas of random variables. These results can be applied to investigate the conditional dependence
structure and the conditional dependence measure of portfolio assets.
References
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properties and pitfalls[M]. In: Dempster, M., Moffatt, H. (Eds.), Risk Management. Cambridge
University Press, New York, pp. 176-223.
[2]R. B. Nelsen. An Introduction to Copulas[M]. Springer, New York, 1998.
[3]R. B. Nelsen. Dependence and order in families of Archimedean copulas[J]. Journal of Multivariate
Analysis, 1997, 60, 111-122.
[4]C. Genest, L. P. Rivest. On the multivariate probability integral transformation[J]. Statistics and
Probability Letters 53(2001) 391-399.
[5] C. Acerbi, D. Tasche, On the coherence of expected shortfall[J]. Journal of Banking and Finance,
2002,26,1487-1503.
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