Advances in Risk Management

244 MONITORING COVARIANCES OF ASSET RETURNS

The different estimation procedures and their influences on the distribu-
tional properties of the estimator for the optimal portfolio weights has been
discussed (see, for example, Scwert (1989), Ledoit and Wolf (2003, 2004),
Bodnar and Schmid (2004), Kan and Zhou (2004) and references therein).
For our purposes, given the sample of portfolio asset returnsX 1 ,...,Xn, the
most common estimator of#is chosen:

#ˆ =^1

n− 1

∑n

t= 1

(Xt−X)(Xt−X)′=

1

n− 1

X(I−

1

n

11 ′)X′ (13.2)

withX=(X 1 ,...,Xn) andX=X1/n. Then using the standard plug-in portfo-
lio rule, for example, replace#in (13.1) by#ˆ, the estimator for the GMVP
is given by:

wˆM=

#ˆ−^11

1 ′#ˆ−^11

(13.3)

Bodnar and Schmid (2004) showed that a q-dimensional vector of
the estimator for the linear combinations of the GMVP weights, LwˆM,
follows the multivariate t-distribution with the mean vectorLwMand
the covariance matrix n−p^1 + 1 LRL′/ 1 ′#−^11 , whereLdenotes theq×p

dimensional matrix of constants. This assertion we denote byLwˆM∼tq

(n−p+1,LwM, n−^1 p+ 1 LRL′/ 1 ′#−^11 ), whereR=#−^1 −#− 11 ′#−^1 / 1 ′#−^11.

However, it appears that the distribution of the estimator for the GMVP
weights does depend on the unknown parameter#and thus cannot be
directly monitored. To avoid the problem in the paper it is proposed to
make use of the simple transformation of the estimator given by:

vˆ=

√

n−p

√

1 ′#ˆ−^11

(

L#ˆ−^1 L′−

L#ˆ−^111 ′#ˆ−^1 L′

1 ′#ˆ−^11

)− (^12)
L(wˆM−wM)
(13.4)
In Theorem 1 it is shown that its finite sample distribution depends only
on the current GMVP weights and is independent of#. We also preserve
the nice properties of thet-distribution.
Theorem 1 Let the vectors of portfolio asset returnsX 1 ,...,Xnbe inde-
pendent and identical, normally distributed with the mean vectorμand
covariance matrix#. Letn≥p>q≥1 andn>p+2. Assume#to be pos-
itive definite. Then the vectorvˆ has a multivariatet-distribution with
n−pdegrees of freedom, the mean vector 0 and the covariance matrix
(n−p)I/(n−p−2).
Theorem 1 leads to several interesting procedures that are easily imple-
mented in practice. First, it allows us to construct the confidence intervals