Advances in Risk Management

OLHA BODNAR 251

The MEWMA chart gives an alarm when

Qt=(Zt−μη)′#−Zt^1 (Zt−μη)>h 4 (13.14)

where#Ztis the covariance matrix ofZt, which is calculated by

Zt=

∑t

j= 1

Var(R(I−R)t−jXj)=

∑t

j= 1

R(I−R)t−j#η(I−R)t−jR (13.15)

In the caser 1 =r 2 =...=rq+q(q+1)/ 2 =rthe formula (13.15) simplifies to

Zt=#η

r(1−(1−r)^2 t)

2 −r

In equation (13.14) the control limith 4 is defined such that the in-controlARL
is equal to a fixed quantityξ. In practice this has to be done by simulations.
It is possible to monitor changes in the covariance matrix of the asset
returns based on the asymptotic MEWMA control chart, namely, MEW-
MAas. In this case the Mahalanobis distance in the equation (13.14) is taken
due to the asymptotic covariance matrixZt;asymp=#ηr/(2−r) instead of
the exact oneZt.

13.4 SIMULTANEOUS STATISTICAL SURVEILLANCE

In this section we use the same notation as in section 13.3. However, instead
of calculating the vectorvˆ(i)for the whole vector of portfolio weightswˆ(i)

we calculate the sequence of{ˆv
(j)
(i)},j∈1,...,p, for each componentwˆ

(j)
(i)cor-
respondingly. Then the two-dimensional control procedures for detecting

shifts in the mean and variance ofˆv((ji))are constructed simultaneously. We
consider the sequence of

η

(j)

(i)=(vˆ

(j)

(i),vˆ

(j)

(i)vˆ

(j)

(i))

with the in-control mean and covariance matrix given by

μη(j)=E(η

(j)

(i))=

(

0,

n ̃−p

n ̃−p− 2

)

, j∈{1,...,p} (13.16)

and

#η(j)=









n ̃−p

n ̃−p− 2

0

0

3(n ̃−p)^2

(n ̃−p−2)(n ̃−p−4)

−

(

n ̃−p

n ̃−p− 2

) 2







 (13.17)