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 determine the frontier.

From its definition, the Efficient Frontier is formed by a set of portfolios offering the highest expected portfolio return for a 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, the problem can be expressed as:

Here,  is a vector of weights,  is the covariance matrix,  is a vector of returns, and  is the expected portfolio return.

We iterate the optimization routine used for finding the optimal portfolio weights over a range of expected portfolio returns, and this results in the Efficient Frontier.

In this recipe, we work with the same dataset as in the previous one, in order to show that the results obtained by both approaches are similar.