Don’t Put All Your Eggs in One Basket

The Model

The purpose is to determine what fraction of a portfolio to invest in each of several possible assets with the goal of minimizing the volatility of the portfolio, subject to a target return.

To frame the question mathematically, suppose f is an n -dimensional vector of the fractions that I’ll invest in each of n financial assets. And let C denote the covariance matrix of the daily returns of the assets, an n x n matrix. Let r be the n- dimensional vector of the expected returns of each of the assets. The target return is determined by r*.

The optimization problem to solve is:

Since we want to minimize the function for the volatility of the portfolio (the risk level), the three constraints of the problem are:

• The sum of the fractions should add up to one.
• The portfolio should attain the target return, r* .
• Each stock’s fraction should be less than 100%.

Because the objective function that we want to minimize is a quadratic, this class of problem is called a quadratic program in the operations research and numerical optimization community. It’s important to note that because it’s a convex function , there’s always a unique solution.

The Data

I used a dataset ² of stock prices between February 20 and April 18, 2019. I selected a few stocks in the tech industry — AMZN, GOOGL, MSFT, IBM, FB and NFLX (Amazon, Google, Microsoft, IBM, Facebook and Netflix) — and created a chart to visualize their correlation:

Most of the stocks are strongly correlated, with the exception of Netflix (NFLX), but remember this is only a 2-month period.

The Optimal Portfolio Allocation

To solve the problem, I needed to compute the covariance matrix C of the daily returns and the estimate returns r for each of the stocks in the S&P500 index. I decided to use 9% as my target return.

Once I computed the expected returns and the expected volatilities (and covariances) of the daily returns, I was ready to solve the optimization problem.

Visit this repo to see my Python code, which will run as-is on Watson Studio Desktop. (You will just need to get the dataset into your project.) Here’s a snippet:

The optimization model chose a total of 28 stocks out of the 500 stocks in the S&P500 index, including these as the top 3:

1. COTY — 11.14%
2. SRE — 9.14%
3. CHD — 8.74%

Portfolio Optimal Allocation

To solve the optimization problem, I used the decision optimization tool CPLEX from Python, inspired on a notebook from the CPLEX github repo³.