Optimal Portfolio Choice

Minimum variance portfolio

The minimum variance portfolio weights are given by the solution to \[\omega_\text{mvp} = \arg\min \omega'\Sigma \omega \text{ s.t. } \iota'\omega= 1,\] where \(\iota\) is an \((N \times 1)\) vector of ones. The Lagrangian reads \[ \mathcal{L}(\omega) = \omega'\Sigma \omega - \lambda(\omega'\iota - 1).\] We can solve the first-order conditions of the Lagrangian equation: \[ \begin{aligned} & \frac{\partial\mathcal{L}(\omega)}{\partial\omega} = 0 \Leftrightarrow 2\Sigma \omega = \lambda\iota \Rightarrow \omega = \frac{\lambda}{2}\Sigma^{-1}\iota \\ \end{aligned} \] Next, the constraint that weights have to sum up to one delivers: \(1 = \iota'\omega = \frac{\lambda}{2}\iota'\Sigma^{-1}\iota \Rightarrow \lambda = \frac{2}{\iota'\Sigma^{-1}\iota}.\) Finally, plug-in the derived value of \(\lambda\) to get \[ \begin{aligned} \omega_\text{mvp} = \frac{\Sigma^{-1}\iota}{\iota'\Sigma^{-1}\iota}. \end{aligned} \]

Efficient portfolio

Consider an investor who aims to achieve minimum variance given a desired expected return \(\bar{\mu}\), that is: \[\omega_\text{eff}\left(\bar{\mu}\right) = \arg\min \omega'\Sigma \omega \text{ s.t. } \iota'\omega = 1 \text{ and } \omega'\mu \geq \bar{\mu}.\] The Lagrangian reads \[ \mathcal{L}(\omega) = \omega'\Sigma \omega - \lambda(\omega'\iota - 1) - \tilde{\lambda}(\omega'\mu - \bar{\mu}). \] We can solve the first-order conditions to get \[ \begin{aligned} 2\Sigma \omega &= \lambda\iota + \tilde\lambda \mu\\ \Rightarrow\omega &= \frac{\lambda}{2}\Sigma^{-1}\iota + \frac{\tilde\lambda}{2}\Sigma^{-1}\mu. \end{aligned} \]

Next, the two constraints (\(w'\iota = 1 \text{ and } \omega'\mu \geq \bar{\mu}\)) imply \[ \begin{aligned} 1 &= \iota'\omega = \frac{\lambda}{2}\underbrace{\iota'\Sigma^{-1}\iota}_{C} + \frac{\tilde\lambda}{2}\underbrace{\iota'\Sigma^{-1}\mu}_D\\ \Rightarrow \lambda&= \frac{2 - \tilde\lambda D}{C}\\ \bar\mu &= \mu'\omega = \frac{\lambda}{2}\underbrace{\mu'\Sigma^{-1}\iota}_{D} + \frac{\tilde\lambda}{2}\underbrace{\mu'\Sigma^{-1}\mu}_E = \frac{1}{2}\left(\frac{2 - \tilde\lambda D}{C}\right)D+\frac{\tilde\lambda}{2}E \\&=\frac{D}{C}+\frac{\tilde\lambda}{2}\left(E - \frac{D^2}{C}\right)\\ \Rightarrow \tilde\lambda &= 2\frac{\bar\mu - D/C}{E-D^2/C}. \end{aligned} \] As a result, the efficient portfolio weight takes the form (for \(\bar{\mu} \geq D/C = \mu'\omega_\text{mvp}\)) \[\omega_\text{eff}\left(\bar\mu\right) = \omega_\text{mvp} + \frac{\tilde\lambda}{2}\left(\Sigma^{-1}\mu -\frac{D}{C}\Sigma^{-1}\iota \right).\] Thus, the efficient portfolio allocates wealth in the minimum variance portfolio \(\omega_\text{mvp}\) and a levered (self-financing) portfolio to increase the expected return.

Note that the portfolio weights sum up to one as \[\iota'\left(\Sigma^{-1}\mu -\frac{D}{C}\Sigma^{-1}\iota \right) = D - D = 0\text{ so }\iota'\omega_\text{eff} = \iota'\omega_\text{mvp} = 1.\] Finally, the expected return of the efficient portfolio is \[\mu'\omega_\text{eff} = \frac{D}{C} + \bar\mu - \frac{D}{C} = \bar\mu.\]

Equivalence between certainty equivalent maximization and minimum variance optimization

We argue that an investor with a quadratic utility function with certainty equivalent \[\max_\omega CE(\omega) = \omega'\mu - \frac{\gamma}{2} \omega'\Sigma \omega \text{ s.t. } \iota'\omega = 1\] faces an equivalent optimization problem to a framework where portfolio weights are chosen with the aim to minimize volatility given a pre-specified level or expected returns \[\min_\omega \omega'\Sigma \omega \text{ s.t. } \omega'\mu = \bar\mu \text{ and } \iota'\omega = 1.\] Note the difference: In the first case, the investor has a (known) risk aversion \(\gamma\) which determines their optimal balance between risk (\(\omega'\Sigma\omega)\) and return (\(\mu'\omega\)). In the second case, the investor has a target return they want to achieve while minimizing the volatility. Intuitively, both approaches are closely connected if we consider that the risk aversion \(\gamma\) determines the desirable return \(\bar\mu\). More risk-averse investors (higher \(\gamma\)) will chose a lower target return to keep their volatility level down. The efficient frontier then spans all possible portfolios depending on the risk aversion \(\gamma\), starting from the minimum variance portfolio (\(\gamma = \infty\)).

To proof this equivalence, consider first the optimal portfolio weights for a certainty equivalent maximizing investor. The first-order condition reads \[ \begin{aligned} \mu - \lambda \iota &= \gamma \Sigma \omega \\ \Leftrightarrow \omega &= \frac{1}{\gamma}\Sigma^{-1}\left(\mu - \lambda\iota\right) \end{aligned} \] Next, we make use of the constraint \(\iota'\omega = 1\). \[ \begin{aligned} \iota'\omega &= 1 = \frac{1}{\gamma}\left(\iota'\Sigma^{-1}\mu - \lambda\iota'\Sigma^{-1}\iota\right)\\ \Rightarrow \lambda &= \frac{1}{\iota'\Sigma^{-1}\iota}\left(\iota'\Sigma^{-1}\mu - \gamma \right). \end{aligned} \] Plugging in the value of \(\lambda\) reveals the desired portfolio for an investor with risk aversion \(\gamma\). \[ \begin{aligned} \omega &= \frac{1}{\gamma}\Sigma^{-1}\left(\mu - \frac{1}{\iota'\Sigma^{-1}\iota}\left(\iota'\Sigma^{-1}\mu - \gamma \right)\right) \\ \Rightarrow \omega &= \frac{\Sigma^{-1}\iota}{\iota'\Sigma^{-1}\iota} + \frac{1}{\gamma}\left(\Sigma^{-1} - \frac{\Sigma^{-1}\iota}{\iota'\Sigma^{-1}\iota}\iota'\Sigma^{-1}\right)\mu\\ &= \omega_\text{mvp} + \frac{1}{\gamma}\left(\Sigma^{-1}\mu - \frac{\iota'\Sigma^{-1}\mu}{\iota'\Sigma^{-1}\iota}\Sigma^{-1}\iota\right). \end{aligned} \] The resulting weights correspond to the efficient portfolio with desired return \(\bar r\) such that (in the notation of book) \[\frac{1}{\gamma} = \frac{\tilde\lambda}{2} = \frac{\bar\mu - D/C}{E - D^2/C}\] which implies that the desired return is just \[\bar\mu = \frac{D}{C} + \frac{1}{\gamma}\left({E - D^2/C}\right)\] which is \(\frac{D}{C} = \mu'\omega_\text{mvp}\) for \(\gamma\rightarrow \infty\) as expected. For instance, letting \(\gamma \rightarrow \infty\) implies \(\bar\mu = \frac{D}{C} = \omega_\text{mvp}'\mu\).