A variance efficient portfolio is one that keeps the risk
(17.2)
minimal under the constraint that the weights sum to , i.e.,
.
For a variance efficient portfolio, we therefore try to find the value of
that minimizes the Lagrangian
A mean-variance efficient portfolio
is defined as one that has minimal variance among all portfolios
with the same mean. More formally, we have to find a vector
of weights such that the variance of the portfolio is minimal
subject to two constraints:
The Lagrangian function for this problem is given by
As a very simple example consider two differently weighted portfolios containing only two assets, IBM and PanAm.
Figure 17.2 displays the monthly returns of the two portfolios. The portfolio in the upper panel consists of approximately 10% PanAm assets and 90% IBM assets. The portfolio in the lower panel contains an equal proportion of each of the assets. The text windows on the right of Figure 17.2 show the exact weights which were used. We can clearly see that the returns of the portfolio with a higher share of the IBM assets (which have a low variance) are much less volatile.For an exact analysis of the optimization problem (17.4) we distinguish between two cases: the existence and nonexistence of a riskless asset. A riskless asset is an asset such as a zero bond, i.e., a financial instrument with a fixed nonrandom return (Franke et al.; 2001).
Assume that the covariance matrix
is invertible (which implies positive definiteness). This is equivalent to
the nonexistence of a portfolio
with variance
.
If all assets are uncorrelated,
is invertible if
all of the asset returns have positive variances.
A riskless asset (uncorrelated with all other assets)
would have zero variance since it has fixed, nonrandom returns.
In this case
would not be positive definite.
The optimal weights can be derived from the first order condition
(17.4) as
For the case of a variance efficient portfolio
there is no restriction on the mean
of the portfolio . The optimal weights are therefore:
This formula is identical to the solution of (17.3).
Indeed, differentiation with respect to gives
If an asset exists with variance equal to zero, then the
covariance matrix is not invertible. The notation
can be adjusted for this case as follows: denote the return
of the riskless asset by
(under the absence of arbitrage this is
the interest rate), and partition the vector and the covariance
matrix of returns such that the last component is the riskless
asset.
Thus, the last equation of the system (17.4) becomes
This equation may be solved for by plugging it into the condition
.
This is the mean-variance efficient weight vector of the
risky assets if a riskless asset exists.
The final solution is:
The variance optimal weighting of the assets in the portfolio depends on the structure of the covariance matrix as the following corollaries show.
PROOF:
Here we obtain
and therefore
PROOF:
can be rewritten as
The inverse is
Let us now consider assets with different variances. We will see that in this case the weights are adjusted to the risk.
PROOF:
From
we have
and therefore the optimal weights are
.
This result can be generalized for covariance matrices with block structures.