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Mean - variance optimization. Maximize the quadratic utility for a given risk aversion.

Optimization of some function could be very difficult problem if we are dealing with complex objectives and constraints. But the Convex optimization problem is one of well - known class of problems which is very useful for finance.
A convex problem has the following form:

\begin{split}\begin{equation*} \begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & f(\mathbf{x}) \\ & \text{subject to} & & g_i(\mathbf{x}) \leq 0, i \in \{ 1, \ldots, m \} \\ & & & A\mathbf{x} = b,\\ \end{aligned} \end{equation*}\end{split}

Where $\mathbf{x} \in \mathbb{R}^n$ , and $f(\mathbf{x}), \; g_i(\mathbf{x})$ are convex functions [1].

Maximization of the quadratic utility given risk aversion.

In the portfolio optimization problem we have some amount of money to invest in any of $n$ different assets of some set of assets. We can choose what fraction $w_i,\; i \in \{1, \dots, n\}$, of our money amount to invest in each asset. The portfolio allocation vector is the vector $\mathbb{w} \in \mathbb{R}^n$. We have constraint ${\mathbf 1}^T \mathbb{w} =1$ which means that the components of our vector $\mathbb{w}$ sum up to $1$. We will assume here the case $w_i \ge 0$, i.e. all the weights $w_i$ (the components of the vector $\mathbb{w}$) greater than or equal to $0$. Also, we will assume that we are investing one unit of money.

The portfolio return is the dot product: $$R = \mathbf{r}^T \mathbf{w}$$

The portfolio variance can be calculated as: $$var(R) = \mathbf{w}^T \mathbf{\Sigma} \mathbf{w}$$

Where $\mathbf{r}$ is the vector of stock expected returns, $\mathbf{r} \in \mathbb{R}^n$, $\mathbf{\Sigma}$ is the $n \times n$ covariance matrix calculated for the returns of our asset set, $\mathbf{w}$ is our weight vector.

The variance of portfolio in mean - variance optimization is considered to be the portfolio risk. (Risk is also sometimes given as $std(R) = \sqrt{var(R)}$ ).

The problem of maximizing the quadratic utility for a given risk aversion can be defined as:

\begin{split}\begin{equation*} \begin{aligned} & \underset{\mathbf{w}}{\text{maximize}} & & \mathbf{r}^{T} \mathbf{w} - \lambda \mathbf{w}^T \mathbf{\Sigma} \mathbf{w} \\ & \text{subject to} & & \\ & & & {w}_i \geq 0, i \in \{ 1, \ldots, n \} \\ & & & \mathbf{1}^{T} \mathbf{w} = 1 \\ \end{aligned} \end{equation*}\end{split}

Where $\mathbf{w}$ is our weight vector, $w_i$ is the $i$ - th component of $\mathbf{w}$, $\mathbf{\Sigma}$ is the covariance matrix, $\mathbf{r}$ is the vector of the expected return, $\lambda \gt 0$ is our risk aversion factor value.

Here the optimization objective is the difference of the portfolio expected return and the portfolio variance multiplied by the risk aversion factor, the first constraint means that all the weights in the portfolio are greater than or equal to $0$, the second constraint means that all the weights should sum up to $1$. The covariance matrix should be the symmetric positive definite one.

It is assumed in the mean - variance optimization that the probability distribution of the stock returns to be the multivariate normal one, so the whole distribution of the portfolio returns also will be the normal one.

References:

[1] Boyd, S.; Vandenberghe, L. Convex Optimization (2004).