OLHA BODNAR 243we estimate this quantity within an extensive Monte Carlo study. Final
remarks are presented in section 13.5. The proofs of all results are given
in the Appendix.
13.2 COVARIANCE STRUCTURE OF ASSET RETURNS AND
OPTIMAL PORTFOLIO WEIGHTS
We consider a portfolio consisting ofpassets. The weight of thei-th asset in
the portfolio is denoted bywi. Thep-dimensional vector of portfolio weights
w=(w 1 ,...,wp)′specifies the investment policy. It is assumed that the whole
investor’s wealth is shared between the selected assets and the possibility
of short-selling. Mathematically, it means that the sum of the weights is one,
w′ 1 =1, where 1 denotes thep-dimensional vector of ones, and the portfolio
weights are not obviously positive.
Suppose the vector of asset returns possesses the second moment. Its
mean we denote byμand the covariance matrix by#, which is assumed
to be positive definite. Then the expected return of the portfolio is given
byw′μand its variance is equal tow′#w. Following the seminal paper of
Markowitz (1952) and his mean–variance analysis, the optimal portfolio is
selected by minimizing the portfolio variance for the given level of portfolio
return, or by maximizing the expected return for the given risk.
In the present study we make use of the weights of the global minimum
variance portfolio (GMVP) for monitoring the covariance structure of asset
returns. The main advantage of the suggested approach is that we control
only the (p−1) -dimensional vector of the portfolio weights instead of mon-
itoring thep(p+1)/2 vector of the variances and covariances (see Bodnar
(2005) for details). It leads to the significant reduction of the dimensionality
of the control problem and improves considerably its power properties.
The weights of the GMVP are:
wM=#−^11
1 ′#−^11(13.1)obtained by minimizing the portfolio’s variance subject tow′ 1 =1. This port-
folio corresponds to the case of a fully risk averse investor. Clearly more
complex portfolio weights could be used for monitoring the covariance
matrix of asset returns, like the weights of the optimal portfolio in the sense
of maximizing the expected quadratic utility, the weights of the tangency
portfolio, and others. The goal however is to keep this simple.
The situation becomes more difficult with the practical implementation of
the model. The covariance matrix#of asset returns is usually an unknown
parameter, and, thus, the investor cannot determine his portfolio policy.
Instead he has to estimate the quantity by previous observations. This
approach leads to heavy discussion in financial and econometric literature.