财务风险管理Ch11

14
One-Factor Model continued

If Ui have standard normal distributions we can set
U i ai F 1 ai2 Z i

where the common factor F and the idiosyncratic component Zi have independent standard normal distributions Correlation between Ui and Uj is ai aj
V2 Mapping to U2
V2 0.2 0.4 0.6 0.8
Percentile 8 32 68 92
U2 −1.41 −0.47 0.47 1.41
20
Example of Calculation of Joint Cumulative Distribution


Probability that V1 and V2 are both less than 0.2 is the probability that U1 < −0.84 and U2 < −1.41 When copula correlation is 0.5 this is M( −0.84, −1.41, 0.5) = 0.043 where M is the cumulative distribution function for the bivariate normal distribution
1 w 1 1
.
10
V1 and V2 Bivariate Normal

Conditional on the value of V1, V2 is normal with mean
V1 m1 m 2 rs 2 s1
and standard deviation s 2 1 r 2 where m1,, m2, s1, and s2 are the unconditional means and SDs of V1 and V2 and r is the coefficient of correlation between V1 and V2
21
Other Copulas


Instead of a bivariate normal distribution for U1 and U2 we can assume any other joint distribution One possibility is the bivariate Student t distribution
m n

1 m
m

i 1
xni y ni 1 m
7


i 1
i
xni y ni , i
Monitoring Correlation continued
EWMA (exponential weighted moving average):
i 1 i
cov
n
cov
0
2
4
6
-6
-4
-2
0
2
4
6
U1 Correlation Assumption
U2
17
Example (page 241)
V1
V2
18
V1 Mapping to U1
V1 0.2 0.4 0.6 0.8
Percentile 20 55 80 95
U1 -0.84 0.13 0.84 1.64
19
f (V2 V1 x) f (V2 )
where f(.) denotes the probability density function
3
Independence is Not the Same as Zero Correlation

Suppose V1 = –1, 0, or +1 (equally likely) If V1 = -1 or V1 = +1 then V2 = 1 If V1 = 0 then V2 = 0 V2 is clearly dependent on V1 (and vice versa) but the coefficient of correlation is zero
12
Generating Random Samples for Monte Carlo Simulation (pages 207-208)


=NORMSINV(RAND()) gives a random sample from a normal distribution in Excel or R 6, R ~ Uniform(0,1) For a multivariate normal distribution a method known as Cholesky’s decomposition can be used to generate random samples
16
The Correlation Structure Between the V’s is Defined by that Between the U’s
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
V1
V2
One-to-one mappings
-6
-4
-2
22
5000 Random Samples from the Bivariate Normal
5 4 3 2 1 0 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 0 1 2 3 4 5
23
5000 Random Samples from the Bivariate Student t
n 1
(1 Bollerslev (1986)
cov
n
x n 1 y n 1 cov
L
n 1
Cov
Cov
L


1
: long - term average
covariance
Correlations and Copulas
Chapter 11
1
Correlation and Covariance

The coefficient of correlation between two variables V1 and V2 is defined as
E (V1V2 ) E (V1 ) E (V2 ) SD (V1 ) SD (V2 )
11
Multivariate Normal Distribution


Fairly easy to handle A variance-covariance matrix defines the variances of and correlations between variables To be internally consistent a variancecovariance matrix must be positive semidefinite

The covariance is E(V1V2)−E(V1 )E(V2)
2
Independence

V1 and V2 are independent if the knowledge of one does not affect the probability distribution for the other
8
Positive SemiDefinite Condition
A variance-covariance matrix, W, is internally consistent if the positive semidefinite condition
w Ww 0
T
holds for all vectors w
w W w : the variance
T
rate of w 1 z 1 w 2 z 2 w N z N
9
Example
The variance covariance matrix
1 0 0.9 0 1 0.9 0.9 0.9 1
is not internally consistent. For example,
cov var
x ,n n
var
y ,n
6
Covariance


The covariance on day n is E(xnyn)−E(xn)E(yn) It is usually approximated as E(xnyn) by assuming E(xn)=0, E(yn)=0
cov
10 5
0 -1 0 -5 0 5 10
-5
-1 0
24
Multivariate Gaussian Copula



We can similarly define a correlation structure between V1, V2,…Vn We transform each variable Vi to a new variable Ui that has a standard normal distribution on a “percentile-to-percentile” basis. The U’s are assumed to have a multivariate normal distribution