Mean Variance Analysis.

We now finally encounter a classical topic in finance, mean variance analysis. This has had to wait because we needed the tool of a linear algebra class before dealing with this.


The mean variance choice is one of the oldest finance areas, dating back to work of Markowitz. The basic assumption is that risk is measured by variance, and that the decision criterion should be to minimize variance given expected return, or to maximize expected return for a given variance.

Mean variance analysis is very simple when expressed in vector format.


\begin{displaymath}\mathbf{e} = E \left[
r_1 \\

be the expected return for the $n$ assets, and

\begin{displaymath}\mathbf{V} =
\sigma_{11} & \ldots...
...& \\
\sigma_{n1} & \ldots &\sigma_{nn} \\

be the covariance matrix.

\begin{displaymath}\sigma_{ij}= \textrm{cov}(r_i, r_j) \end{displaymath}

A portfolio of assets is expressed as

\begin{displaymath}\mathbf{\omega} = \left[
\omega_1\\ \vdots \\ \omega_n

To find the expected return of a portfolio:

\begin{displaymath}E[r_p] = \mathbf{\omega}^\prime \mathbf{e} \end{displaymath}

and the variance of a portfolio:

\begin{displaymath}\sigma_p = \mathbf{\omega}^\prime \mathbf{V} \mathbf{\omega}

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Mean variance portfolios.

In the case where there are no short sales constraints, the minimum variance portfolio for any given expected return has an analytical solution and is therefore easy to generate.

The portfolio given the expected return $E[r_p]$ is found as

\begin{displaymath}\mathbf{\omega}_p = \mathbf{g} + \mathbf{h} E[r_p] \end{displaymath}

For the mathematics of generating the unconstrained MV frontier, see chapter 3 of Huang and Litzenberger (1988).

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Short sales constraints

In real applications, it is often not possible to sell assets short. In that case we need to add a constraint that all portfolio weights shall be zero or above.

When constraining the short sales, we need to solve a quadratic program.

\begin{displaymath}\min \mathbf{\omega}^\prime \mathbf{V} \mathbf{\omega} \end{displaymath}

subject to

\begin{displaymath}\mathbf{\omega}^\prime\mathbf{1} = 1 \end{displaymath}

\begin{displaymath}\mathbf{\omega}^\prime\mathbf{e} = E[r_p] \end{displaymath}

\begin{displaymath}\omega_i \in [0,1] \ \ \forall \ i \end{displaymath}