Finding the efficient frontier using optimization with SciPy

In the previous recipe, Finding the efficient frontier using Monte Carlo simulations, we used a brute force approach based on Monte Carlo simulations to visualize the efficient frontier. In this recipe, we use a more refined method to find the frontier.

From its definition, the efficient frontier is formed by a set of portfolios offering the highest expected portfolio return for certain volatility, or offering the lowest risk (volatility) for a certain level of expected returns. We can leverage this fact, and use it in numerical optimization.

The goal of optimization is to find the best (optimal) value of the objective function by adjusting the target variables and taking into account some boundaries and constraints (which have an impact on the target variables). In this case, the objective function is a function returning portfolio volatility, and the target variables are portfolio weights.

Mathematically,...