Advances in Risk Management

254 MONITORING COVARIANCES OF ASSET RETURNS

by the daily returns (Schwert, 1989). It follows from Theorem 1 that for each
i, the random vectorsvˆ(i)have a multivariate standardt-distribution with 16
degrees of freedom independently distributed as their construction is based
on the non-overlapping samples.
In order to obtain the performance of the proposed sequential procedures
the out-of-control situation has to be determined. In our simulation study
the changes are generated by the following model:

Xt∼Np( 0 ,#), t≤ 0

Xt∼Np(0,#), t≥ 1

where

=











1 +a 1 a 2 a 2 a 2

a 2 1 +a 1 a 2 a 2

a 2 a 2 1 +a 1 a 2

a 2 a 2 a 2 1











As a measure of the performance of a control chart the average run length
(ARL) is applied. All multivariate and simultaneous control schemes are
calibrated to have the same in-control ARLs, namely 60. Because no explicit
formula for the in-control and the out-of-controlARLs are available, a Monte
Carlo study is used to estimate these quantities. We estimate the in-control
ARLs based on 10^5 simulated independent realizations of the process. The
control limits of all charts are determined by applying the Regula falsi to the
estimated ARLs. In Table 13.1 the control limits of the multivariate charts
are given for various values of the reference valuekand the smoothing
parameterr, while Table 13.2 contains the control limits for the simulta-
neous schemes. Because the vectorsvˆ(i)are independent and identically
standardt-distributed the control limits of the charts do not depend on
the covariance matrix of asset returns. In the out-of-control state they are
again independent but no longer identically distributed (see Theorem 2).
Consequently they are not directionally invariant. For the MEWMA charts
the control limits increase as the parameterrincreases. Conversely, for the
CUSUM schemes the control limits decrease askincreases. The control limit
for the multivariateT^2 control scheme is 356.8, while for the simultaneous
T^2 it is equal to 37.42. Finally, almost in all cases the control limits of the
simultaneous control charts are much smaller than the corresponding limits
of the multivariate schemes.
In order to study the out-of-control behavior of the proposed control
charts we take various reference valuesk into account. For the mul-
tivariate CUSUM chartsk is chosen as an element of the set {1.4, 1.5,
1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3}, while for the simultaneous schemes from
{1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0}. For the MEWMA and asymptotic
MEWMA charts the smoothing matrix is taken as a diagonal matrix with
equal diagonal elementsr∈{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9}.