Advances in Risk Management

OLHA BODNAR 247

where all elements ofμηat positionsq+1, 2q+1, 3q,4q−2, 5q−3,...,q+
q(q+1)2 are equal to (n ̃−p)/(n ̃−p−2) and the others are zero. Furthermore,
let denote!={1,q+1, 2q,3q−2, 4q−3,...,q(q+1)2}. Then it follows that
the covariance matrix of the vectorη(i),i=1,...,m, in the in-control state is:

#η=





n ̃−p

n ̃−p− 2

I0

0 # ̃η



 (13.8)

withB(.,.) being the Beta function and

# ̃η=

           

a 0 ... 0 b ... b

0 c ... 00 ... 0

..

.

..

.

..

.

00 ... c 0 ... 0

b 0 ... 0 a ... b

..

. ...

..

.

..

.

b 0 ... 0 b ... a

           

(13.9)

The elements are equal toa=3(n ̃−p)^2 /((n ̃−p−2)(n ̃−p−4)) at positions
(j,j), wherej∈!,b=(n ̃−p)^2 /((n ̃−p−2)(n ̃−p−4))−((n ̃−p)/(n ̃−p−2))^2
at positions (j 1 , j 2 ), wherej 1 , j 2 ∈!andj 1 =j 2 , c=(n ̃−p)^2 /((n ̃−p−2)
(n ̃−p−4)) at positions (j,j), wherej∈{1,...,q(q+1)/ 2 }!, and the others
are zero (Fang and Zhang, Lemma 5.6.3, 1990).
To test if there is a change in the covariance matrix#requires check-
ing whether the mean of the vectorη(i), whose estimator is given in (13.6),
departs significantly fromμη. Hence, at timetthe testing problem is
given by:

H0,t: E(η(t)) =μη against H1,t: E(η(t))=μ 1 =μη (13.10)

No change occurs in the covariance matrix of asset returns and, corre-
spondingly, the GMVP is efficient up to timetifH 0 is valid for alls∈1,...,t.
We refer to this as an in-control state. In other case the portfolio is not longer
efficient, starting at the time, when the first shift has occurred. The corre-
sponding state of the system is called out-of-control. It also means that an
investor has to adjust his portfolio. As a measure of the performance of con-
trol schemes the average run length (ARL) is chosen. By this criteria the best
control chart possesses the smallest out-of-controlARLfor a given in-control
one. Following Schipper and Schmid (2001) the in-control ARL is taken 60
that corresponds to three months returns data.