# Proofs

## Optimal Portfolio Choice

### The 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}

### The 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 her optimal balance between risk ($$\omega'\Sigma\omega)$$ and return ($$\mu'\omega$$). In the second case, the investor has a target return she wants 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} Plug-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 $$\bar\mu = \frac{D}{C} = \omega_\text{mvp}'\mu$$.