242 MONITORING COVARIANCES OF ASSET RETURNSreturns with the higher frequency are used for calculating the covariance
matrix. The question of interest is not only estimation, but also monitoring
the break in the covariance structure of asset returns. We want to consider
this problem in the present chapter in more detail.
Statistical control methods have been recently developed for monitoring
the mean vector and covariance matrix of a random vector (see for example
Bodnar, 2005, for a detailed survey). The dimension of the control problem
depends on the number of assets included in the portfolio and could be
extremely large. This leads to a delay in detecting changes in the portfolio
structure and can lead to large losses for the investor. In order to reduce the
dimensionality of the control problem the optimal portfolio weights (parts of
the investor’s wealth allocated into an asset) are considered in this chapter.
We show that the changes in the covariance matrix of asset returns generates
changes in the weights of the global minimum variance portfolio (GMVP).
Sequential procedures for monitoring financial data-sets have already
been discussed in the financial literature. For instance, Theodossiou (1993)
applied multivariate CUSUM control charts for predicting business failures.
Financial decision strategies based on the MEWMA control scheme are dis-
cussed by Schipper and Schmid (2001) and Schmid and Tzotchev (2004).
Andersson, Bock and Frisen (2003, 2005) made used of the statistical surveil-
lance for the detection of turning points in business cycles. However, nobody
up to now has adopted sequential procedures in asset management, with
the exception of Yashchin, Steinand and Philips (1997). It is our aim, based
on the historical values of the asset returns process, to derive sequential
control schemes for monitoring changes in the covariance matrix of asset
returns that could influence the selection of an optimal portfolio. In order to
reduce the dimensionality of the control problem we focus essentially on the
transformation of the vector of the optimal portfolio weights. Agreat advan-
tage of this suggested approach is that structural breaks in the covariance
matrix lead to shifts in both the mean vector and covariance matrix of this
transformed vector. It also possesses several nice distributional properties
and thus can be easily monitored in practice. We develop the corresponding
sequential procedures.
The remainder of the chapter is organized as follows. In the next sec-
tion we shed light on the relationship between the covariance matrix of
asset returns and the weights of the GMVP. In Theorem 1 it is shown that
the simple transformation of the estimator for the GMVP weights is mul-
tivariatet-distributed. Theorem 2 motivates the application of the control
charts to the transformed vector of the portfolio weights. The link between
covariances and portfolio weights is investigated. The multivariate statisti-
cal surveillance is introduced in section 13.2, while section 13.3 deals with
the simultaneous control schemes. The two approaches are compared in
section 13.4. As the measure of the control charts performance the average
run length (ARL) is used. As no explicit formula for the ARL is available