This GitHub repository contains all the work related to my study of efficient frontiers and mean-variance optimisation. My notes are heavily inspired by research and lectures whilst studying at Warwick.
Mean-variance analysis is the problem of finding a portfolio that maximises expected returns for a given level of risk. The variance of an assets return is used as a measure of risk. This theory suggests that an investor can reduce portfolio risk by diversifying their portfolio of assets. The mean-variance framework was first introduced by Markowitz.
We explicitly solve the mean-variance optimisation problems for risk-only portfolios and general portfolios. For general portfolios these can be formulated as follows.
- Given an initial wealth
$x_0>0$ and a minimal desired expected return$\mu_{\text{min}}>0$ , minimise the variance of the return$\sigma_{\bar{\vartheta}}^2$ among all$x_0$ -feasible portfolios$\bar{\vartheta}\in\mathbb{R}^{1+d}$ that satisfy$\mu_{\bar{\vartheta}}\geq\mu_{\text{min}}$ . - Given an initial wealth
$x_0>0$ and a maximal desired variance of the return$\sigma_{\text{max}}^2\geq0$ , maximise the expected return$\mu_{\bar{\vartheta}}$ among all$x_0$ -feasible portfolios$\bar{\vartheta}\in\mathbb{R}^{1+d}$ that satisfy$\sigma_{\bar{\vartheta}}<\sigma_{\text{max}}^2$ .
For risk only-portfolios
- Given an initial wealth
$x_0>0$ and a minimal desired expected return$\mu_{\text{min}}>0$ , minimise the variance of the return$\sigma_{\vartheta}^2$ among all$x_0$ -feasible portfolios$\vartheta\in\mathbb{R}^{d}$ that satisfy$\mu_{\vartheta}\geq\mu_{\text{min}}$ . - Given an initial wealth
$x_0>0$ and a maximal desired variance of the return$\sigma_{\text{max}}^2\geq0$ , maximise the expected return$\mu_{\vartheta}$ among all$x_0$ -feasible portfolios$\vartheta\in\mathbb{R}^{d}$ that satisfy$\sigma_{\vartheta}<\sigma_{\text{max}}^2$ .
An implementation solving the risk-only problems can be seen in the document 23072024_Efficient_Frontier.ipynb. One of the other key problems we look at is the mutual fund theorem. This theorem states that any portfolio on the efficient frontier can be generated by holding a combination of any two given portfolios on the frontier. So even in the absence of a risk-free asset, an investor can achieve any desired efficient portfolio.
We calculate the forward-looking expected returns and covariance of returns using historical data gathered from Yahoo Finance. Given a portfolio, we can then easily compute it's return and variance. We solve the constraint problems by using scipy.optimize.minimize with the method 'SLSQP' (see https://docs.scipy.org/doc/scipy/reference/optimize.minimize-slsqp.html#optimize-minimize-slsqp for more details).
Please view Efficient_Frontier_Curve.png to see the constructed efficient frontier and read Efficient_Frontier_Project.pdf if you are interested in the theory behind efficient frontiers.