258 MONITORING COVARIANCES OF ASSET RETURNSthan other competitors. In second places the asymptotic MEWMAchart. For
moderate and large values of the out-of-control ARLs this scheme shows
a much better performance than the simultaneous MC1 and MCUSUM
approaches that are in third and fourth places, while for small values all three
charts behave similarly. The worst results are gives by the simultaneousT^2
control chart.
The comparison between the multivariate and the simultaneous charts
leads to interesting results. If the ARL of a multivariate scheme is com-
pared with its simultaneous counterpart, then for almost all considered
shifts the simultaneous approach has the smaller out-of-control ARL for
the process under consideration. While the multivariate charts have some
difficulties for some shift constellations, all the changes are detected by the
simultaneous control schemes. In all of our simulations the MEWMA chart
provides very good results. The simultaneous MEWMA scheme shows the
best performance. A possible explanation of this fact is based on the results
of Woodall and Mahmoud (2005) who showed that the MC1 approach can
build up a large amount of inertia. For that reason we recommend applying
the MEWMA control chart for detecting changes in the covariance matrix of
asset returns that could influence the optimal portfolio weights. For simplic-
ity we have taken the smoothing values of the MEWMA chart as all equal.
This chart has much more flexibility, and improvements can be expected
if different values are chosen. The best results for the EWMA charts are
obtained in the case ofr=0.1, for example, the smallest of the considered
smoothing values. This result is in line with the findings of Frisen (2003),
who argued that the best performance of the EWMA scheme is attained for
small values of the parameterr. For further discussion of optimality for the
control procedures we refer the reader to Pollak (1985), Srivastava and Wu
(1997), Yakir (1997) and Moustakides (2004).
13.6 CONCLUSION
One of the most important problems in portfolio management is monitoring
the covariance structure of asset returns. While Ledoit and Wolf (2003, 2004)
have discussed the influence of the estimation error on the estimator for
the covariance matrix, we have focused on the question of monitoring an
optimal portfolio using the distributional properties of the estimator for the
covariance matrix. This problem has been presented widely in the literature
lately, for example Jobson and Korkie (1989), Gibbons, Ross and Shanken
(1989), Britten-Jones (1999), Bodnar and Schmid (2004) and others. However,
none of these proposals deals with sequential procedures.
In this chapter we derive sequential multivariate and simultaneous pro-
cedures for detecting changes in the covariance matrix of asset returns that
have an influence on the weights of the global minimum variance portfolio.