The article proposes a new algorithm for adjusting correlation matrices and for comparison with Finger's algorithm, which is used to compute Value-at-Risk in RiskMetrics for stress test scenarios. The solution proposed by the new methodology is always better than Finger's approach in the sense that it alters as little as possible those correlations that we do not wish to alter but they change in order to obtain a consistent Finger correlation matrix.Stochastic, Volatility, Skewness, Kurtosis, Pricing.

Begoña Subiza

Josep Enric Peris Ferrando

José Angel Silva

Ángel León

English

Research Papers in Economics

ADJUSTING CORRELATION MATRICES
Ángel León, Josep E. Peris, José A. Silva and Begoña Subiza
WP-AD 2000-24
Correspondence to Ángel León: University of Alicante. Departamento Economía
Financiera, Contabilidad y Marketing. E-03071 Alicante (Spain). Tel.: 34 96 590 36
1 1/F a x :3 49 65 9 03 62 1/E - m a i l :a l e o n @ u a . e s .
Editor: Instituto Valenciano de Investigaciones Económicas, s.a.
First Edition November 2000.
Depósito Legal: V-4599-2000
IVIE working papers o¤er in advance the results of economic research under way
in order to encourage a discussion process before sending them to scienti…c journals
for their …nal publication.
* The authors would like to acknowledge the …nancial support provided by DGICYT
grants PB97-0120, PB97-0621 and PB96-0339.
** University of Alicante.
1ADJUSTING CORRELATION MATRICES
Ángel León, Josep E. Peris, José A. Silva and Begoña Subiza
ABSTRACT
This article proposes a new algorithm for adjusting correlation matrices and
for comparison with Finger’s algorithm, which is used to compute Value-at-
Risk in RiskMetrics for stress test scenarios. The solution proposed by the new
methodology is always better than Finger’s approach in the sense that it alters
as little as possible those correlations that we do not wish to alter but they
change in order to obtain a consistent Finger correlation matrix .
Keywords: correlation matrix, Kuhn-Tucker conditions, eigenvalue, Value-
at-Risk.
21I N T R O D U C T I O N
An important problem, arising when using RiskMetrics (RM) for Value-at-Risk
(VaR), is that sometimes it is desirable to alter the correlation matrix in order
to re‡ect a view of markets that di¤ers from the traditional one. An arbitrary
alteration in the correlation matrix however can breakdown the required consis-
tency of the methodology since the new correlation matrix may be inde…nite.
Finger (1997) introduces a methodology in RM to alter some correlations
from a correlation matrix such that the new matrix is still consistent. A prob-
lem that arises in using this algorithm is that the new matrix indicates more
correlations to be altered than the desired ones.
This paper introduces a new algorithm to adjust the correlation matrix and
compare it with Finger’s. In particular, this new methodology uses the Finger’s
correlation matrix and then modi…es, as little as possible, those correlations
that we do not wish to alter by minimizing the distance to the original ones,
subject to the restriction of the correlation matrix being consistent.
The remainder of the paper is organized as follows. Finger’s algorithm is
r e v i e w e di nS e c t i o n2 .I nS e c t i o n3 ,an e wa l g o r i t h mi sp r o p o s e dw i t ht h ep r o o f
given in the Appendix. Finally, in Section 4, both methodologies are applied to
the hypothetical currency correlation matrix example taken from Finger (1997).
2 FINGER’S ALGORITHM
Let X be a random vector in Rn that represents n asset returns with a mean ¹
and a covariance matrix ­ =[ ¾ij]n£n: Let C =[ cij]n£n denote the correlation
matrix of X; i.e. C =¡ ­¡ where ¡=diag
³
¾
¡1=2
11 ;:::;¾
¡1=2
nn
´
: Then, a 2 £ 1
3partition of X; according to the assets whose correlations we wish to change and
t h eo n e sw ed on o t ,t h et w os u b s e t sb e i n gd e n o t e da sI and J respectively, is
X =
2
6 6
4
XI
XJ
3
7 7
5
where XI 2 Rm and XJ 2 Rn¡m. We can express C as
2
6 6
4
C11 C12
CT
12 C22
3
7 7
5
where C11 is a m£m matrix containing the correlations of I;C22 is a (n ¡ m)£
(n ¡ m) matrix containing the correlations of J and C12 denotes the correlations
between both I and J: Finally, by MT we denote the transpose of a matrix M:
Let e C11 be the matrix containing the new correlations of I:If C11 is replaced
by e C11 in C; an e wm a t r i x e C is obtained, which, at times, can produce unde-
sirable results; i.e. e C may be inde…nite and is therefore not a true correlation
matrix. Finger’s algorithm, denoted by F; is a method for altering correlations
consistently, such that a new correlation matrix CF 6= e C is obtained when e C is
inde…nite, verifying that CF is non-negative de…nite. The algorithm is de…ned
as follows:
Let Z be the random vector in Rn such that Z =¡ ( X¡¹); then Z » (0;C):
Let Z = 1
m
X
i2I Zi: The random variables (rv’s) XF
i ;i= 1;:::;n are now
de…ned as
XF
i =
8
> > <
> > :
(1 ¡ µi)Zi + µiZ if i 2 I
Zi otherwise
4where µi 2 [0;1]: We can express the new rv’s1,i nam a t r i xw a y ,b yXF = AZ
where A is a n £ n matrix de…ned as
2
6 6
4
A11 0
0I
3
7 7
5
such that A11 is a m £ m matrix whose elements aij are
aij =
8
> > <
> > :
1 ¡ µi +
µi
m
if i = j; i 2 I
µi
m
if i 6= j; i;j 2 I
(1)
and I is the identity matrix. Then, E
¡
XFXT
F
¢
= ACAT gives us the covariance
matrix of XF; denoted as ­F; which can be partitioned as
2
6 6
4
­F
11 ­F
12
­F
21 ­F
22
3
7 7
5 =
2
6 6
4
A11C11AT
11 A11C12
CT
12AT
11 C22
3
7 7
5:
Consider
¡F =
2
6 6
4
¡F
11 0
0I
3
7 7
5
where ¡F
11 = diag
³
±
¡1=2
11 ;:::;±
¡1=2
mm
´
and ±ii denotes the (i;i) element of the
matrix ­F
11.L e t ZF be a new random vector in Rn such that ZF =¡ FXF:
Then E
¡
ZFZT
F
¢
=¡ F­F¡F; denoted as CF; is the covariance matrix of ZF;
which is well de…ned2. By splitting matrix CF in the same way as we did with
1We introduce, here, a di¤erent version of Finger’s algorithm, the only di¤erence is about
de…ning XF
i : In Finger’s paper, µi = µ 8i 2 I while this new version allows µi to be di¤erent
to a better adjusting of the new C11 matrix, obtained through F and denoted as CF
11,t ot h e
matrix e C11:
2It is easy to prove that CF is non-negative de…nite.
5the previous matrices, we can write
CF =
2
6 6
4
CF
11 CF
12
CF
21 CF
22
3
7 7
5 =
2
6 6
4
¦C11¦T ¦C12
CT
12¦T C22
3
7 7
5
where ¦=¡ F
11A11: Notice that when computing CF there may be more than
¡
m2 ¡ m
¢
=2 new correlations in CF as was expected at …rst. At most, there can
be
£¡
m2 ¡ m
¢
=2+m(n ¡ m)
¤
new correlations; i.e., those belonging to CF
12:
Finally, note that A11 is a function of the parameter vector µ 2 Rm; i.e.
A11 = A11 (µ); so that the …rst step to computing CF is to solve the following
constrained minimization program:
min
fµigm
i=12[0;1]
° ° °
£
A11 (µ)C11AT
11 (µ)
¤
¡ e C11
° ° °
2
where k¢k denotes the euclidean norm3.
3N E W A L G O R I T H M
Now, we shall try to obtain a better correlation matrix than the one presented
in the previous section. Note that C12 6= CF
12; so that our goal is to modify as
little as possible the elements from C12 in the new adjusted correlation matrix.
In order to do so, we present an algorithm which is composed of a two-step
procedure. The …rst step consists of computing CF and the second is to obtain
3The euclidean norm of a matrix A is de…ned as:
kAk =
0
@
p X
i=1
q X
j=1
a2
ij
1
A
1=2
;A =[ aij]p£q :
6a new correlation matrix C¤; de…ned as C¤ = CF + B; where
B =
2
6 6
4
0 B12
BT
12 0
3
7 7
5
being B12 =[ bij] ak n o w nm £ (n ¡ m) matrix. We can now write
C¤ =
2
6 6
4
CF
11 C¤
12
(C¤
12)
T C22
3
7 7
5
with C¤
12 = CF
12 + B12: If we denote the elements in CF
12 by [dij]; then C¤
12 =
[dij + bij]: We try to choose bij such that C¤
12 be approximately equal to C12:
In order to guarantee that the new matrix C¤ is a non-negative de…nite matrix,
we apply the following known result4:
Theorem 1 Let ° be an eigenvalue of C¤ = CF + B; then
° 2
[n
k=1 fz 2 R : jz ¡ »kj·rg
where »1;»2;:::;»n are eigenvalues of CF and r = kBk:
Let » =m i n f»1;»2;:::;»ng; then if » ¸ r we know that »k ¸ r; k =
1;:::;n: So, the following condition is su¢cient to ensure that C¤ is a non-
negative de…nite matrix
»
2 ¸ 2
m X
i=1
n¡m X
j=1
b2
ij:
4See, for instance, Lancaster and Tismenetsky (1985), Chapter 11, p. 388-9.
7We are interested in choosing C¤
12 such that it minimizes kC¤
12 ¡ C12k
2 sub-
ject to C¤
12 being non-negative de…nite. We know that
C¤
12 ¡ C12 = CF
12 ¡ C12 + B12:
Let us call E = CF
12 ¡ C12; which is a m £ (n ¡ m) matrix whose elements,
denoted as [eij]; are known. To obtain the new correlation matrix C¤ we must
solve the following constrained minimization program where the parameters are
the elements of B12; i.e. [bij]:
min
fbijg
m P
i=1
n¡m P
j=1
(eij + bij)
2
s:t: 2
m P
i=1
n¡m P
j=1
b2
ij · »
2
¡1 · dij + bij · 1
(2)
where the number of constraints is m(n ¡ m)+1: Note that the second restric-
tion guarantees that the elements of C¤; speci…cally the elements of C¤
12; must
belong to the interval [¡1;1] since they are correlations.
Remark 1 It must be noted that, since a feasible possibility in the above prob-
lem consists of considering B =0 ; that is C¤ = CF; t h es o l u t i o np r o p o s e db y
the new algorithm is always better than the one proposed by Finger’s approach.
Moreover, the feasible set being closed, bounded and non-empty, program (2)
always has a solution, which is unique due to the strict convexity of the objective
function.
8An important feature of this algorithm is that it is possible to …nd conditions
that ensure that matrix C12 does not change under the computations of matrix
C¤: Thus, from the Kuhn-Tucker (K-T) conditions of (2) given in the Appendix
we know that, if
»
2 ¸ 2
m X
i=1
n¡m X
j=1
e2
ij
then, we do not need to apply the second step, since we obtain that the solution
for program (2) is bij = ¡eij and we directly have:
C¤ =
2
6 6
4
CF
11 C12
CT
12 C22
3
7 7
5:
In other cases the solution is, bij = ¡eij=' ,b e i n g'>1 the value obtained from
the K-T conditions (see Appendix) so that C¤
12 ¡C12 = E +B12 =( 1¡1=')E:
Thus, we directly have:
C¤ =
2
6 6
4
CF
11 C12 +( 1 ¡ 1=')E
(C12 +( 1 ¡ 1=')E)T C22
3
7 7
5:
4F I N G E R ’ S E X A M P L E
We now apply the new algorithm (N-Ag) to an example in the hypothetical cur-
rency correlation matrix taken from Finger5 (1997) and compare it to Finger’s
algorithm (F). Let us consider the following currency correlation matrix:
5See Table 1 from page 4, though shifting the currencies so that the submatrix C11 contains
the Asian currencies whose correlations we wish to alter.
9HKD MYR PHP THB ARS DEM GBP
1:0000 ¡0:2100 0:1400 ¡0:1500 ¡0:2600 ¡0:1400 0:0600 HKD
1:0000 0:2200 0:1000 0:1900 0:3100 ¡0:0800 MYR
1:0000 0:0700 ¡0:2500 0:1600 0:0400 PHP
1:0000 ¡0:1200 0:0900 0:0400 THB
1:0000 0:1800 ¡0:1300 ARS
1:0000 0:2200 DEM
1:0000 GBP
where the currencies are Argentine Peso (ARS), German Mark (DEM), British
Pound (GBP), Hong Kong Dollar (HKD), Malaysian Ringgit (MYR), Philippine
Peso (PHP) and Thai Baht (THB). Let I denote the Asian currencies in the
matrix, i.e. I ´f HKD, MYR, PHP, THBg: Following Finger (1997), C11 is
changed to e C11; whose correlations are set to 0:85; so that the new correlation
sub-matrix for Asian currency markets properly describes the market behavior.
10By changing only the I0s coe¢cients, the new correlation matrix e C is
HKD MYR PHP THB ARS DEM GBP
1:0000 0:8500 0:8500 0:8500 ¡0:2600 ¡0:1400 0:0600 HKD
1:0000 0:8500 0:8500 0:1900 0:3100 ¡0:0800 MYR
1:0000 0:8500 ¡0:2500 0:1600 0:0400 PHP
1:0000 ¡0:1200 0:0900 0:0400 THB
1:0000 0:1800 ¡0:1300 ARS
1:0000 0:2200 DEM
1:0000 GBP
It is shown that e C is not a true correlation matrix since its minimum eigenvalue
is -0.04. The next step consists of introducing an algorithm, F or N-Ag, to
adjust the above correlation matrix in a consistent way. The solution6 of µ for
our example is
[0:8199;0:7786;0:7026;0:7956]
T :
6In order to obtain the µi ’s vector that corresponds to our example, we have used the
GAUSS library ”Constrained Optimization”. In Finger’s algorithm, a unique parameter µ is
estimated, whose value is 0:7874:
11Then, the correlation matrix CF is
HKD MYR PHP THB ARS DEM GBP
1:0000 0:8357 0:8661 0:8521 ¡0:2625 0:1166 0:0443 HKD
1:0000 0:8520 0:8656 ¡0:0784 0:2705 ¡0:0108 MYR
1:0000 0:8348 ¡0:2487 0:1990 0:0368 PHP
1:0000 ¡0:2058 0:1872 0:0369 THB
1:0000 0:1800 ¡0:1300 ARS
1:0000 0:2200 DEM
1:0000 GBP
N-Ag provides the following matrix C¤ for our example
HKD MYR PHP THB ARS DEM GBP
1:0000 0:8357 0:8661 0:8521 ¡0:2620 0:0668 0:0473 HKD
1:0000 0:8520 0:8656 ¡0:0263 0:2781 ¡0:0243 MYR
1:0000 0:8348 ¡0:2489 0:19150 :0374 PHP
1:0000 ¡0:1891 0:1684 0:0375 THB
1:0000 0:1800 ¡0:1300 ARS
1:0000 0:2200 DEM
1:0000 GBP
12Comparisons between both algorithms are made by using the mean absolute
error (MAE) and the root mean square error (RMSE) as summary statistics,
where the error is de…ned as e C12¡Y12 for Y12 = CF;C¤: The summary statistics
are:
MAE RMSE
F 0:0735 0:1165
N-Ag 0:0592 0:0939
We can observe that N-Ag scores better than F, as expected, under both statis-
tics.
13Appendix
Since the objective function and the constraints from program (2) are convex,
if
©
b¤
ij
ª
veri…es Kuhn-Tucker (K-T) conditions then
©
b¤
ij
ª
is a global minimum.
Moreover, strict convexity of the objective function implies the unicity of such a
minimum and, since the feasible set is compact and non-empty, and the objective
function continuous, the existence of a solution is always ensured. We can
rewrite (2) as
min
fbijg
m P
i=1
n¡m P
j=1
(eij + bij)
2
s:t: 2
m P
i=1
n¡m P
j=1
b2
ij · »
2
dij + bij ¡ 1 · 0
¡(dij + bij + 1) · 0:
The Lagrangian of this problem is given by
L =
m X
i=1
n¡m X
j=1
(eij + bij)
2 + ¸1
0
@2
m X
i=1
n¡m X
j=1
b2
ij ¡ »
2
1
A
+
m X
i=1
n¡m X
j=1
¸
ij
2 (dij + bij ¡ 1) ¡
m X
i=1
n¡m X
j=1
¸
ij
3 (dij + bij + 1):
K-T conditions are:
[i]2 ( eij + bij)+4 ¸1bij + ¸
ij
2 ¡ ¸
ij
3 =0 ; 8i;j;
[ii:1] ¸1
Ã
2
m P
i=1
n¡m P
j=1
b2
ij ¡ »
2
!
=0 ;
[ii:2] ¸
ij
2 (dij + bij ¡ 1)=0 ; 8i;j;
[ii:3] ¸
ij
3 (dij + bij + 1)=0 ; 8i;j;
14[iii] ¸1 ¸ 0;¸
ij
2 ¸ 0;¸
ij
3 ¸ 0; 8i;j;
[iv:1]2
m P
i=1
n¡m P
j=1
b2
ij ¡ »
2 · 0;
[iv:2] dij + bij ¡ 1 · 0; 8i;j;
[iv:3] ¡ (dij + bij + 1) · 0; 8i;j:
Solution from K-T conditions:
Note that from [ii:2] and [ii:3];¸
ij
2 and ¸
ij
3 cannot be di¤erent from zero at
t h es a m et i m ef o ra n yg i v e ni;j:
We study the following cases according to the possible values of [iii]:
Case 1: If for some i;j we have ¸
ij
2 > 0 (which implies ¸
ij
3 =0 ), then from
[ii:2] we obtain that bij = 1 ¡ dij ¸ 0 and, by substituting in [i]:
2(1 ¡ cij)+4 ¸1bij + ¸
ij
2 =0 ;
which is not possible, since all elements are non-negative and at least one, ¸
ij
2 ,
is strictly positive.
Case 2: If for some i;j we have ¸
ij
3 > 0, by reasoning in a similar way as in
Case 1, we prove that this is not possible.
So, we know that, for all i;j we must necessarily have:
Case 3: for all i;j ¸
ij
2 =0 ;¸
ij
3 =0 :
Case 3.1: If ¸1 =0 ; from [i] we obtain that
b¤
ij = ¡eij; 8i;j
is the solution if and only if 2
m P
i=1
n¡m P
j=1
e2
ij · »
2:
Case 3.2: If ¸1 > 0; this corresponds to the fact that 2
m P
i=1
n¡m P
j=1
e2
ij >»
2
15and then, the solution is
b¤
ij = ¡eij='; 8i;j
where
' =
v u u u t
2
m P
i=1
n¡m P
j=1
e2
ij
»
2 > 1:
16REFERENCES
Finger, C. (1997) A methodology to stress correlations, RiskMetrics Monitor,
Fourth Quarter, 3-11.
Lancaster, P. and Tismenetsky, M. (1985) The Theory of Matrices, Academic
Press, Inc.
17

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