\( \newcommand{\Frac}[2]{\displaystyle\frac{#1}{#2}} \newcommand{\eins}{{\mathbf 1}} \newcommand{\var}{\text{var}} \newcommand{\cov}{\text{cov}} \newcommand{\w}{{\mathbf w}} \newcommand{\p}{{\mathbf p}} \newcommand{\r}{{\mathbf r}} \newcommand{\bmu}{{\boldsymbol \mu}} \newcommand{\x}{{\mathbf x}} \newcommand{\R}{\mathbb R} \)

The Mean-Variance Portfolio Problem

These pages introduce the geometric intuition for the mean-variance efficient portfolio for a given target return.

The Mean-Variance Efficient Portfolio for a Given Target Return

  • A mean-variance efficient portfolio \(\w^{\text{mv}}\) is the portfolio that shows minimum-variance for some given target return.

Expected Returns

The expected value of the return \(\r_i\) of asset \(i\) is denoted by \(\mu_i=E(r_i)\), the vector of expected returns is denoted by

\( \bmu = E(\r) = \left(\begin{array}{c} \mu_1\\ \mu_2\\ \vdots\\ \mu_n \end{array}\right) \).

The expected return \(\r_{\w}\) of portfolio \(\w\) is

\( E(r_{\w})=\w'\bmu \)

  • Requiring a target return means that we restrict portfolio choice to portfolios \(\w\) with a given expected return \(\bar\mu\),

    \( \w'\bmu\,=\,\bar\mu. \)

  • Mean-variance efficiency analyzes the trade-off between expected return and return variance.

Mean-variance Efficient Portfolio

The mean-variance efficient portfolio \(\w^{\text{mv}}(\bar\mu)\) with respect to a target return \(\bar\mu\) is the portfolio with the lowest possible return variance among all portfolios with a given expected return \(\bar\mu\). It is the solution of the following problem

\(\begin{eqnarray*} \frac{1}{2}\,\w'\Sigma \w &\to& \min_w\\[1em] \text{s.t.}\quad \w'\eins &=& 1,\\[0.2em] \w'\bmu &=& \bar\mu. \end{eqnarray*} \)

  • The weights of the mean-variance efficient portfolio \(\w^{\text{mv}}\) depend on expected returns of the securities in the investment universe as well as on the covariance structure of their returns. In contrast, the weights of the global minimum-variance portfolio \(\w^{\text{gmv}}\) are independent of return expectations \(\bmu\).

    In real-world application, components of \(\bmu\) and \(\Sigma\) must be estimated from data. Especially \(\bmu\) is hard to estimate with accuracy. Thus, generally \(\w^{\text{gmv}}\) shows better robustness against estimation errors than \(\w^{\text{mv}}\).

  • The space of possible weight combinations \(\w\) is now restricted by two linear constraints.

    \( \w'\eins\,=\,1, \)

    weights must sum to one, i.e., the return combinations must be portfolios. This restricts the space of allowed choices to an \((n-1)\)-dim (affine) subspace of the \(\R^n\). The stated restriction is the normal-vector form of a (hyper) plane, the so-called portfolio plane. The vector \(\eins\) is the (Euclidean) normal vector.

    \( \w'\bmu\,=\,\bar\mu, \)

    which means that the choice is further restricted to return combinations with equal expectation \(\bar\mu\). This constraint is again an \((n-1)\)-dim (affine) subspace of the \(\R^n\). The vector \(\bmu\) is the (Euclidean) normal vector of this (hyper) plane.
  • All allowed weight combinations lie in the intersection of these two planes.
    • This is an \(n-2\)-dim subspace of the \(\R^n\), if \(\neg\exists k \in\R\, |\,\bmu = k \eins\), i.e., \(\bmu\) and \(\eins\) are not parallel. In other words, if the \(n\) available assets do not have identical return expectations.
    • If all \(n\) assets do have identical return expectations equal to \(\bar\mu\), then the intersection of the two planes is the entire portfolio plane, i.e., the target-return constraint equals the portfolio constraint. In this case, the mean-variance efficient portfolio equals the global minimum-variance portfolio, \(\w^{\text{mv}}=\w^{\text{gmv}}\).
    • If all \(n\) assets do have identical return expectations not equal to \(\bar\mu\), the target return is not implementable and, thus, the above problem has no solution.
Interactive Chart.
  • Since the variance of any portfolio \(\w\) can be decomposed into

    \( \var(\w) = \var(\w^{\text{gmv}}) + \var(\w-\w^{\text{gmv}}), \)

    searching for the mean-variance efficient portfolio among all portfolios with a given return \(\bar{\mu}\) is equivalent to searching for the minimum-variance portfolio transaction \(\Delta_{\w} = \w-\w^{\text{gmv}}\) among all transactions with a given expectation \(\bar m = \bar\mu-\bar\mu(\w^{\text{gmv}})\)

    \(\begin{eqnarray*} \frac{1}{2}\var(\Delta_{\w}) &=& {\Delta}'_{\w}\Sigma\Delta_{\w} \,\to\, \min_w\\[1em] \text{s.t.}\quad {\Delta}'_{\w}\eins &=& 0,\\[0.2em] {\Delta}'_{\w}\bmu &=& \bar m. \end{eqnarray*} \)

Interaktive Graphik.