Download Comparative study on Calculations of Cohesive Value at Risk Abstract:

Comparative study on Calculations of Cohesive Value at Risk
*
CHEN Yaonian LAN Qiujun
(School of Business Management, Hunan Univ. Changsha, 410082, China)
Abstract: Value at Risk, as a measurement for risks, has some weaknesses such as the measured
relative risk value being not in accordance with the actual feeling of the investors and the lack of
subadditive, and therefore the concept of Coherent Value at Risk has been raised. This thesis, based on
this concept, proposes several calculation methods of Cohesive Value at Risk and makes a comparison
on their effectiveness.
Keywords Cohesive Value at Risk Hyperbolic Distribution Extreme Value Theory
：
The collapse of the Bretton Woods system at the beginning of the 1970s brought the prevalence of
the floating exchange rate in countries all over the world. Prices of financial products such as exchange
rates and interest rates are fluctuating more frequently and even harder to predict. Since the beginning of
the 1980s, along with the rapid development of the information technology, the trend of financial
innovation and liberalization is growing unprecedented and the punctuation of the financial and stocks
market is worsened. At the same time, the derivatives have been brought to use and developed very fast
due to the need of distracting risks. Meanwhile, when they are used more and more for speculation
rather than keeping away from evaluation, the derivatives, as are developed due to the need of avoiding
risks, holds great adventure in themselves. Recently, the closedown of Balin Bank in Britain, the loss in
the scads trade of the Owa Bank in Japan, and things alike, can all be contributed to the derivatives.
Therefore, the adventure of the financial and stocks market, especially the derivatives are becoming
more and more evident and causing increasing attention. And how to test and control those market risks
efficiently has become a question needing to be solved by financial and stocks market, investors and
related supervisors.
It was under such circumstances that the method called VaR was given birth to, a widely used
technology that can be directly applied to testing the overall risk of banks, trust and security institutions
and the portfolios. It stands for the greatest value loss possible for the whole asset combination in a
certain period in the future, due to the fluctuation of the market. If we only consider the fluctuation in
one day, VaR can also be called everyday risk proficiency. In other words, the method of VaR is a
possible or potential loss in testing given investment tool or combination in future fluctuation of asset
prices. The authorities saying put by Philippe Jorion is “the worst predicted loss during a holding period
in given confidence region under normal market circumstances.” In mathematics, it can be expressed by
the α-quartile of the P & L distribution of investment tool or combinations in the following way:
Prob ( p△t≤ VaR)= α
p△t stands for the market value change of fabricate p during the holding period of t under
confidence interval 1-α . The equivalence indicates that the probability of loss value being or over VaR
is α, or we may say that under the probability α, the loss value is over VaR. In the latter we actually take
VaR as the function of α. If we use F( p△t) to indicate the probability distribution function of the
profits of portfolios, then it can be followed that:
VaR=F-1 (α)
The debate over the applicability of VaR, the risk measuring method, has never ceased ever since the
raising-up and prevalence of it. Opponents represented by Nassim Taleb[1] thinks that there are some
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△
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Project supported by Hunan Provincial Natural Science Foundation of China 05JJ30215
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irremediable weaknesses of VaR: (1) It is a method looking backwards. The prediction towards the loss
of the future is based on historical data and the relations of the variables in the past are supposed to be
stable. Obviously, this is not in accordance with the practical situation in many cases. (2)VaR is
performed under certain hypothetical conditions, such as the normal distribution of data, but sometimes
such hypothesis may not be in accordance with the actual situation. (3)The calculation of VaR may be
very complicated sometimes. (4)VaR is only an efficient measurement of the market risks when the
market is under normal change. Therefore it cannot be applied to the situation when the financial market
is under extreme price fluctuation, such as the collapse of the stork market. Theoretically, the cause of
those weaknesses of VaR is not itself, but the statistical method it applies. Actually, as a risk
measurement, VaR’s greatest weakness is none of the above ones, but that the relative risk value
measured by it is not in accordance with the real feelings of the investors, and there is no minor add
ability of it to be used to optimize portfolios [2]. In this case, a brand-new measuring method is raised,
which is called CVaR Cohesive Value at Risk.
：
．The Definition of CVaR
1
Artzner and some others proposed the truism system of constant risk measurement in Mathematical
Finance in 1999 [3]. Any risk measurement that is Translation-equivariant, positively homogenous,
subadditive and monotony can be called constant risk measurement. It can make sure that there are
different values for different risks, and the measured value of portfolio with high risks is bigger than that
with low risks, while equal for the same risks. It follows that CVaR indicates the expected value by
investors towards 1 α of the tail of the profit distribution during a certain time T with α as the
confidence interval. The mathematical equivalence of it is [2]:
－
CVaRα (Y ) = E[Y | Y ≥ VaRα (Y )]
．Several calculating methods of CVaR
2
From the definition of CVaR we can see that what CVaR measures is the downward risk, i.e., losses,
and therefore the correct prediction of the tail of the return rate distribution has great importance on the
calculation of CVaR. Now we are going to discuss the ways of calculation for different models of return
rate distribution.
2 1 CVaR Calculation for normal distributed return rate
Suppose that the return rate X of investment tools is in accordance of normal distribution, δ X is the
．
standard different value of the return rate X, W0 is the initial value of the investment tool, and Z α is
the relative value of normal distribution, during a certain period of time, when the confidence interval is
α , CVaR can be expressed as:
CVaRα ( X ) = E[− X ≥ VaRα ( X )] =
=
δX
e
(1 − α ) 2π
E[− X *1{X ≤ −VaRα ( X )} ]
1− α
−VaRα ( X ) 2
2δ X2
As it is under normal distribution,
VaRα ( X ) = W0 * Z α * δ X
then
W
−
δX
CVaRα ( X ) =
e
(1 − α ) 2π
0
2
*Zα 2
2
Therefore, when the return rate X is in accordance with normal distribution, the CvaR value of the
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investment tool is
e
−W0 2 *Zα 2
2
α 2π
times of the standard different value. As long as the standard different
．
value can be got, the CVaR value can be gained as well.
2 2 CVaR Calculation when the tail of the return rate distribution obeys the hyperbolic
distribution
In 1977, Barndorff-Nielsen (1977) introduced the family of generalized hyperbolic functions. The
d-dimensional density function Hd has six parameters: δ and ∆ for multivariate scales, µ for
location, β for skewenss, and α and λ mainly change the tails. We have
Hd( α, β, µ, ∆, λ ) ( x )
K λ −(1 / 2 ) d (α [δ 2 + ( x − µ ) T ]1 / 2 )
k λα (1 / 2) d − λ
β ( x − µ )T ,
=
e
(2π ) (1 / 2) d δ λ K λ (δk ) [δ 2 + ( x − µ )∆−1 ( x − µ ) T ]1 / 2((1 / 2) d −λ )
with k =
α 2 − β∆β T
, ∆ a symmetric, positive definite d × d -matrix with determinant 1 and
K v the modified Bessel function of the third kind.9 It is a well known theorem of Blæsild, that the
family of generalized hyperbolic distributions is closed under regular affine transformations and with
respect to formation of marginal and conditional distributions. Blæsild explicitly describes the
parameters of marginal, conditional or affine transformed hyperbolic distributions. A direct result of this
theorem is the following lemma 1 and 2:
Lemma 1: The parameter β is 0, if the corresponding parameter βˆ of any one dimensional
marginal distribution is 0, i.e. the generalized hyperbolic distribution is elliptic, if all one dimensional
marginal distributions are symmetric. The one dimensional marginal distributions are normal hyperbolic
distributions, if the generalizing parameter λ is 1.A generalized hyperbolic distribution with
parameters β = 0 and λ = 1 is therefore called a symmetric hyperbolic distribution. We denote the
density E and replace the parameter
E d (ξ , δ , µ , ∆) =
α
with
ξ def αδ :

x − µ −1 x − µ T
K 1−(1 / 2) d  ξ 1 +
∆

δ
δ

ξ d1
(2π ) (1 / 2 ) d δ d K 1 (ξ ) 
ξ 1 +


x−µ
δ
x − µ T 
∆−1
δ 
(1 / 2 ) d −1



.
The scale matrix of E is δ ∆ .
We can specialize Blæsild’s theorem to the case of symmetric hyperbolic distributions. Lemma3
then states that the subfamily of symmetric hyperbolic distributions is closed with respect to forming
marginal distributions and regular affine transformations.
Lemma 2: Let X be a d-dimensional random vector distributed according to the symmetric
hyperbolic distribution E d (ξ , δ , µ , ∆ )( X ). Given a partition ( X 1 , X 2 ) of X, with r the
2
dimension of X 1 , and analogously defined partitions ( β 1 ,
representation
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β 2 ) and ( µ1 , µ 2 ) .
Let ∆ have block
 ∆11 ∆12 

∆ = 
 ∆ 21 ∆ 22 
so that ∆ 11 is a r×r -matrix. Then the following holds:
1. The distribution of X 1 is a r-dimensional symmetric hyperbolic distribution
ˆ
E r (ξ , δˆ, µˆ , ∆ˆ ) whit parameters ξˆ = ξ , δˆ = δ | ∆ 11 |1 / 2 r , µˆ = µ 1 and ∆ˆ =| ∆ 11 |1 / r ∆ 11 .
2. Let Y=XA+B be a regular affine transformation of X and
A denote the absolute value of the
determinant of A. Then Y is distributed according to the symmetric hyperbolic distribution
E d (ξ , δ , µ , ∆ ) with ξ = ξ , δ = δ A
+ B and ∆ = A
2/ d
1/ d
, µ = µA
AT ∆A. .
We then need the moments of the distribution. The generalized hyperbolic distribution is an
exponential family with respect to β . So its moments can be derived from the partial derivatives of the
logarithm of the densities norming constant. We get
E ( X − µ ) = 0,
(5)
E (( X − E ( X )) 2 ) = Cov( X ) =
E (( X − E ( X )) 3 ) i , j , k = 0
K 2 (ξ ) 2
δ ∆,
ξK 1 (ξ )
(6)
and
(7)
K 3 (ξ ) 4
δ (∆ i ,l ∆ j ,k + ∆ j , k + ∆ i , k + ∆ k ,l ∆ j ,i ).
(8)
ξ 2 K 1 (ξ )
The interpretation of the 4 parameters is: µ is a parameter of location, ∆ defines only the
E (( X − E ( X )) 4 ) i , j ,k ,l =
shape of the covariance matrix and δ varies only its size, while ξ is independent of transformations
of the parameters of scale and location and determines the relation between the tails and the waist of the
distribution. However, a change in ξ will result in a change of the size of the covariance matrix.
There also exists an efficient generator for generalized hyperbolic random variables. Atkinson
(1982) suggested a two or three envelope rejection algorithm. It uses the fact that the generalized
2
hyperbolic distributions are mixed distributions. If σ is an random variable from an generalized
inverse Gaussian distribution with parameters
λ, δ2
and
κ2, Φ
the Cholesky decomposition of
∆ and Y a d-dimensional standard normal random vector, then X = σYΦ T + µ + σ 2 β∆ is
distributed with respect to H d (α , β , δ , µ , ∆, λ ).
So, we can model the return of the security market by use of hyperbolic distribution. According to
density function, we have not any trouble to get xα when the confident level is α . Moreover, we can
get CVaR:
xα
xE (ζ , δ , µ , ∆)( x )dx
−∞
2 3 CVaR Calculation when the tail of the return rate distribution obeys the
extreme value distribution
．
CVaR = ∫
According to the latest achievement from the theory of extreme value distribution, we can gain the
estimating style of the extreme value distribution totally on the basis of the sample data; therefore we do
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not need to make any certain hypothesis on the type of the extreme value distribution (there are three of
them).
2.3.1 Hill 1975 proposed that if the distribution is in accordance with the changing condition of
（ ）
1 − F(tx)
= x −β
t →∞
F(t)
β >0
lim
x>0
then F(x) can be simulated approximately by a one-step expanding equivalence[5]
F(x) = Pr(X > x) = ax − β
β > 0, x → ∞
Hill also gives the estimated one-step dominance value of β
1 M

β =  ∑ lnX (i) − lnX (M +1) 
 M i =1

(2.3.1)
−1
X (i) > X (M +1)
X(i) —— the first statistical value of the sample by decreasing order
M—— the number of the critical sample, the meaning of which is that when X > X (M +1) ,
ax − β equals P(X > x) approximately.
2.3.2 The new way of ascertain
1
β
=
β 、Xp、P by Danielson(1997)
M
Xi
1
ln
∑
M i =1 X M +1
1
 M β
X P = X M +1  
 np 
∧

MX
P =  M +1 
n  XP 
∧
β
From the above we can see that the most important is about how to ascertain X M
、
，this critical
value meaning counting from the statistical number No. M, X1 X2… X M , all of those data ( in
increasing order) being used to estimate β . There are two ways to select the value of M and X M :
one is draft (drawing the picture of β M); The other is called the way of optimization by computers.
Here we use the way of regression to make regression directly on the equivalence of 2.3.1 and take a
reference of the first method of draft that is easy to conduct.
2.3.3 Calculation of CVaR
Here we have to combine the method of the extreme value theory and RiskMetricsTM to calculate
－
[7]
the value of CVaR. The core of it is as follows: when variable X < X M +1 in the tail of the return rate
order, we use the distribution function F(x) =
M  XS 


n  X 
β
, raised in extreme value theory, to describe
approximately P(X > x) ; when X ≥ X M +1 , we use the normal distribution to do it, that is to say, to
make up a combined distribution to calculate CVaR. We only take into account one side of the tail as the
distribution of return rate is symmetrical, and we only take into account the left side for the sake of
convenience. Considering that average value of return rate may not be 0, here we have discharged the
hypnotized condition of µ=0 in RiskMetricsTM. The probability distribution function of this method is:
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 M X M +1 β
 ( X )
F(x) =  n
X −µ
 Φ(
)
δ

X < X M +1
X ≥ X M +1
The relationship of function XP —— P is:
1

M β
X
(
)
 M +1
nP
XP = 
 µ + Z Pδ

M +1
n
M +1
P≥
n
P<
Φ
—— the short form of standard normal distribution function expression.
Zp —— the position of p in the standard normal distribution.
Through regression analysis of the historical data we can find out the relationship between X and
XP. From the definition of CVaR:
（）
CVaRα (Y ) = E[Y | Y ≥ VaRα (Y )]
We can get
CVaRα = ∫
Xα
XdF ( X )
． Proofs and Calculation
−∞
3
3
．1 Proof study of market return rate distribution in China
Data used here are mainly taken from CSMAR, co-developed by the Chinese accounting and
financial research center at Hongkong University of Science and Technology and Tai’an Information and
Technology Limit. Corp. in Shenzhen and the research sample for Chinese stock market used here is
daily return rate of plutonic index of the Shenzhen stock market, dating from April 3rd, 1991, to July 31st,
2002, including 2822 initial data and 2821 data of daily return rate. The formula for the return rate is as
follows:
Pt − Pt −1
Pt −1 and t is the time period. The sample features after analysis are
r is daily return rate, P is the closing price,
r=
as follows:
Data
Maximum
Minimum
．2619
－0.2167
0
Average
Value
Standard
Deviation
Slanting
Degree
Peak
Degree
0.00077
0.02588
1.1257
15.5103
In order to show clearly the specific differences between the actual return rate distribution and
normal distribution, we have standardized the sample data, R =
(r − µ )
δ
, among which, R stands for
the standardized daily return rate, µ stands for the average value of the initial data, and δ
for the standard deviation of the initial data. Therefore we can draw the following data:
Maximum
Minimum
Average
Value
0
Square
Deviation
1
stands
Data
10.0923
-8.4045
In
this way, we can tell the
differences between the actual return rate distribution and normal distribution easily through the draft
246
rather than just tell the verification of normal distribution.
1. 0
.9
.8
.7
.6
.5
.4
标标标标标标
标标标标标标
.3
.2
.1
0. 0
-5. 0
-3. 0
-1. 0
1. 0
3. 0
5. 0
Fig1. The comparison between the actual return rate distribution and standard normal
distribution
In Fig1 the horizontal ordinate stands for the square deviation, while the vertical one indicates the
frequency; the full curve indicates the actual return rate distribution, while the dotted curve indicates the
standard normal distribution. Through the comparison, we can see clearly the high peak tail of the actual
return rate distribution.
3 2 Calculation and Comparative Analysis
Calculated by the matlab software, we can gain the following results:
．．
Actual
Distribution
Normal
Distibution
Extreme Value
Distribution
Hyperbolic
Distribution
99.9%
99.5%
99%
98.5%
98%
97%
95%
VaR
0.1675
0.0806
0.0673
0.0561
0.0501
0.0440
0.0328
CVaR
0.1862
0.1134
0.0944
0.0835
0.0758
0.0662
0.0545
VaR
0.0787
0.0653
0.0595
0.0555
0.0522
0.0480
0.0417
CVaR
0.0825
0.0724
0.0676
0.0641
0.0615
0.0577
0.0523
VaR
0.1446
0.0805
0.0650
0.0560
0.0499
0.0430
0.0355
CVaR
0.1761
0.1142
0.0939
0.0823
0.0752
0.0668
0.0577
VaR
0.1536
0.0802
0.0659
0.0553
0.0519
0.0433
0.0345
CVaR
0.1789
0.1137
0.1143
0.0843
0.0757
0.0662
0.0557
S standards for the redundant standard deviation, S =
k
∑
(X
j=1
− Xj)
2
nj
k −1
S
0.0327
0.0054
0.0035
， and there is no standard
deviation in actual distribution. X nj stands for the return rate calculated in different distribution, and
，
X j stands for the return rate in actual distribution, k=52 which means what we calculated is just S
in the tail.
1 When we calculate the CVaR in extreme value distribution, we simulate the return rate
distribution by extreme value distribution on the left of where the accumulated probability is 0.01879,
but normal distribution on the right of this probability. However, both S values of CVaR and VaR
calculated by Hyperbolic Distribution and extreme value distribution are smaller than the S value in
．
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normal distribution, and at the same time, the CVaR and VaR values are even closer to the CVaR and
VaR values in normal distribution, so we can say that Hyperbolic Distribution imitates the left tail of the
return rate distribution better than the normal distribution.
2 The closer the actual return rate distribution towards the tail, the more serious the thick tail
situation becomes, the greater the differences between the normal distribution and the actual distribution
are, and the better the and the Hyperbolic Distribution and extreme rate distribution imitates. We can be
sure that the tail of the return rate distribution obeys approximately the Hyperbolic Distribution and
extreme rate distribution. It does not exist in the distribution. It is not ideal to imitate by normal
distribution the middle part of the return rate distribution with extreme rate distribution, as there is the
bottle circumstance, but as it does not affect much the CVaR and VaR values calculated, we do not make
further research here.
3 No matter in which distribution or under which confidence interval, CVaR is more sensitive than
VaR towards the tail distribution. This is because that VaR is just a decimal point in tail distribution,
while CVaR is a conditional average value, and also the expected value of loss over VaR. We take
positive value of both CVaR value and VaR value, and at the same time, CVaR value is bigger than VaR
value under the same conditions.
4 Compared with the extreme value distribution, the Hyperbolic Distribution imitates better the tail
of the return rate distribution and the CVaR value taken in Hyperbolic Distributions closer to the real
one. Meanwhile, the S value in Hyperbolic Distribution is smaller than that in the extreme value
distribution, which indicates that it is better to use Hyperbolic Distribution the calculate CVaR.
．
．
．
4 Conclusions
As a risk measurement, VaR has some weaknesses such as the measured relative risk value being
not in accordance with the actual feeling of the investors and the lack of minor add ability, and therefore
the concept of constant risk value has been raised. This thesis, based on this concept, proposes several
calculation methods of constant risk value and makes a comparison on their effectiveness. Due to the
thick peak tail of the return rate distribution and the sensitivity of the CVaR value towards the tail
distribution, here we use the extreme rate distribution and the Hyperbolic Distribution to imitate the
return rate distribution, so as to solve the problem of thick tail of the return rate distribution. Through
calculation, we find out that compared with the normal distribution, the CVaR value in the Hyperbolic
Distribution and extreme rate distribution is even closer to the real value, while the CVaR value
calculated in the Hyperbolic Distribution is even better than that in the extreme rate distribution.
1.
2.
3.
4.
5.
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，
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