Advances in Risk Management

OLHA BODNAR 245

fortheGMVPweights. Forexample, takenL=e 1 =(1,0, ..., 0)asthepdimen-
sional vector with the first element being 1 and the rest zeros, we obtain the
two-sided 1−αconfidence interval for the first weight of the GMVP:



wˆM;1−

tn−p;1−α/ 2

√

n−p

√

e′ 1 Reˆ 1

1 ′#ˆ−^11

,wˆM;1+

tn−p;1−α/ 2

√

n−p

√

e′ 1 Reˆ 1

1 ′#ˆ−^11





where tn−p;1−α/ 2 is the 1−α/2 quantile of the standard univariate
t-distribution. Consequently it can be used as a tool for controlling the
weights of the GMVP. It leads to a decision whether the portfolio should
be adjusted or not.
Second, it permits us to apply the sequential control procedures for mon-
itoring the efficiency of the GMVP using the distributional properties of the
random vectorsvˆ.
Third, which is also a main feature for our purposes, the breaks in the
covariance matrix of asset returns lead to the changes in the mean vector and
the covariance matrix ofvˆ. Thus, by monitoring these two parameters we
control both covariance structure of returns and the efficiency of the GMVP.
To show this we need the following result. It is assumed thatXt∼Np(μ,#)
fort≤t 0 , andXt∼Np(μ ̃,# ̃) fort>t0,which leads to the following GMVP
weights:

w=#−^11 / 1 ′#−^11 for t≤t 0 and w ̃=# ̃−^11 / 1 ′# ̃−^11 for t>t 0

The vectorv ̃ˆwe define similar tovˆusing#ˆ ̃−^1 instead of#ˆwith# ̃ˆ−^1 being
the estimator of# ̃(see (13.2). In Theorem 2 the influence of the changes in the
covariance structure of assets returns on the mean vector and the covariance
matrix of the vectorvˆ ̃is presented:

Theorem 2 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#and# ̃to

be positive definite. Then:

(a) The expectation ofvˆ ̃is equal to

E(vˆ ̃)=

√

n−p

√

H ̃(−)

22

b ̃−^12 A(w ̃M;q−wM;q)

whereA=diag(a 11 ,...,aqq) with (13.5)

aii=

B

(

n−p−q−i

2 +1,

n−p− 1

2

)

B

(

n−p−q−i+ 1

2 ,

n−p

2

) ,i=1,q.