17.3 Efficient Portfolios in Practice

We can now demonstrate the usefulness of this technique by applying our method to the monthly market returns computed on the basis of transactions at the New York stock market between January 1978 to December 1987 (Berndt; 1990).

EXAMPLE 17.2   Recall that we had shown the portfolio returns with uniform and optimal weights in Figure 17.2. The covariance matrix of the returns of IBM and PanAm is

\begin{displaymath}\data{S}= \left( \begin{array}{ll} 0.0034&0.0016\\ 0.0016& 0.0172
\end{array} \right).\end{displaymath}

Hence by (17.7) the optimal weighting is

\begin{displaymath}\widehat c=\frac{\data{S}^{-1} 1_{2}}{1^{\top}_{2}\data{S}^{-1} 1_{2} }
= (0.8957, 0.1043)^{\top}. \end{displaymath}

The effect of efficient weighting becomes even clearer when we expand the portfolio to six assets. The covariance matrix for the returns of all six firms introduced in Example 17.1 is

\begin{displaymath}\data{S}= \left( \begin{array}{rrrrrr}
0.0035& 0.0016& 0.001...
... 0.0003& 0.0010& -0.0004& 0.0021& 0.0063
\end{array} \right).\end{displaymath}

Hence the optimal weighting is

\begin{displaymath}\widehat c=\frac{\data{S}^{-1} 1_{6}}{1^{\top}_{6}\data{S}^{-...
... }
= (0.2504, 0.0039, 0.0409, 0.5087, 0.0072, 0.1890)^{\top}. \end{displaymath}

Figure 17.3: Portfolio of all six assets, equal and efficient weights. 49915 MVAportfol.xpl
\includegraphics[width=1\defpicwidth]{portfall.ps}

As we can clearly see, the optimal weights are quite different from the equal weights ($c_j=1/6$). The weights which were used are shown in text windows on the right hand side of Figure 17.3.

This efficient weighting assumes stable covariances between the assets over time. Changing covariance structure over time implies weights that depend on time as well. This is part of a large body of literature on multivariate volatility models. For a review refer to Franke et al. (2001).

Summary
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Efficient portfolio weighting in practice consists of estimating the covariances of the assets in the portfolio and then computing efficient weights from this empirical covariance matrix.
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Note that this efficient weighting assumes stable covariances between the assets over time.