Investment portfolio selection for greater good

Choosing a portfolio is a daunting task for beginning, amateur investors. It’s difficult to translate rudimentary knowledge and well-meaning, but general advice, into concrete numbers representing security shares in your account. In this post, I’d like to broach this subject and present a method of finding a portfolio that maximizes expected return for given variance level, which we’ll associate with risk, based on historical data.

I’ll provide a very light introduction to the quadratic optimization method, which we’ll use to select the portfolio. I’ll also give a short overview the script that implements described methods in practice.

The accompanying Python code is available at my Github repo. The README.md file explains how to run the analysis.

Please be aware that I am not giving financial advice. Using methods presented in this post may give poor results in practice.

Quadratic optimization

Maximize the gain

The very first task in analytic thinking is formal problem definition; definition of what we want. In the problem, we have a collection of securities - \(S_1\), \(S_2\), …, \(S_n\). We’d like to find a combination of securities that maximizes the return after, for example, a few years. We’ll use vector \(a = [a_1, \ldots, a_n], \sum_{k=1}^N a_k\) to denote a solution.

We don’t know the future, so we, like in statistics, have to make some assumptions. For example, prices tend to follow their historical behavior. This assumption seems to make sense and is also easy to translate into results as price history is usually available on the Internet.

Then, a straightforward answer to our question is to choose a portfolio that would have maximized the return in a given time period in the past. Given historical returns \(r = [r_1, \ldots, r_n]\), find \(a = [a_1, \ldots, a_n], \sum a_i = 1\) that maximizes \(a \cdot r = \sum_{i=1}^n a_i r_i\).

It is a simple method. So simple that it only chooses an elementary vector \(e_j\) with \(j\) being the index of the largest \(r_j\). This drawback of this method stem from not taking risk, i.e. price volatility, into account. We’ll have to introduce that concept into our model.

Risk

We’ll use variance as our measure of volatility. Given a history of monthly price changes, \(h_j = [h_1, \ldots, h_m]\) of security \(S_j\) over the last \(m\) months, we calculate its variance as \(\mathbb{V}\left(h_j\right) = \mathbb{E} \left(h_k - \mathbb{E}(h_k)\right)^2\). The bigger this value, the more volatile the price history.

I’d like to introduce one more tool before we formulate how we want to include risk into our model - covariance. Covariance can be used to measure how two variables, like security prices, depend on each other. The formula is

\[\mathtt{Covar}\,(h_j, h_k) = \mathbb{E}(h_j^2) \mathbb{E}\left(h_k^2\right) -\left(\mathbb{E}\left(h_j\right) \mathbb{E}\left(h_k\right)\right)^2\]

Going back to the model, we’ll elaborate the “maximize expected return” task into “maximize expected return, given that risk is at constant level v”. In more mathematical notation, we’d like to find \(\mathtt{argmax}_a a \cdot r\) such that \(\mathbb{V}(a \cdot h) = v\), where \(v\) is constant. One thing worth knowing is that we can express portfolios’s variance as a linear combination of variances and covariances of individual securities:

\[\mathbb{V}(a \cdot h) = \sum_{j=1}^n a_j^2 \mathbb{V}(h_j) + \sum_{j=1}^n\sum_{k=1}^n a_ja_k \mathtt{Covar}(h_j, h_k)\]

We can express this in even more succint form. Let \(P = [p_{i, j}]\) be a \(n \times n\) matrix such that \(p_{i, j} = \mathtt{Covar}(h_j, h_k)\). Such matrix is called a covariance matrix. Then

\[\mathbb{V}(a \cdot h) = a P a^T\]

Quadratic optimization

We’re done with defining the model of our problem. Now we can turn our heads toward finding a way to solve it. Luckily for us, our problem belongs to a class of problems known as convex optimization problems. They occur quite often in the real world, and efficient solving methods are known and widely available. Formally, the problem of maximizing

\[xR - xPx^{T}\]

with respect to \(x\) is known as quadratic optimization. The code in my repository encodes the data into this problem and uses cvxopt, a Python library, to solve it.

Why is this class of problems called convex optimization problems? It’s because curves defined by that kind of equation are convex. For example, for 2-dimensional \(x\) we get a paraboloid.

Further enhancements

Choosing risk level

Astute readers may notice that there’s no specification of the desired risk level \(v\) in the quadratic optimization equation above. That’s not a mistake, and neither it is not desirable. The straightforward application of an optimization algorithm will some vector \(x\) such that we know that for risk level \(xPx^T\) the expected return, \(xR\) is maximal. But we didn’t choose the risk level.

The situation can be remedied by introducing a weight variable \(\nu\), which will mean how important is the risk factor in the equation. In our program we’ll try out different values of \(\nu\) and solve

\[xR - \nu \cdot xVx^T\]

We still can’t choose the risk-level, but this way we’ll generate various solutions for range of risk-levels from which we can choose the one that we want.

Implementing your own strategy

Suppose that we wanted to provide additional restrictions on \(x\), like - at least 10% of the portfolio is composed of bond funds. That is possible and quite easy to do. Quadratic optimization algorithms allow definition of constraints of the form \(Gx^T \leq h, Ax^T = b\), where \(G,A\) are arbitrary matrices and \(h, b\) are vectors. For example, to encode a constraint that the second security gets at least 20% share we would set: \(G = [0, -1, 0, \ldots, 0]\) and \(h = [-0.2]\).

Code

The accompanying code relies on cvxopt library to solve the quadratic equation. The library’s documentation provides an excellent and succinct source of information on the method itself. My script is split into 3 parts.

First, we need to get raw data, clean it, and transform it into a form that is directly pluggable into the quadratic optimization method. I have used data from bossa website, which contains daily investment fund prices. The functions in the script truncate the data to monthly period, get monthly price changes instead of absolute values etc.

Second, we have get the processed data into an equation and solve it. This is done in calculate_sample_portfolios. The function takes an expected return matrix \(R\), a covariance matrix \(P\), and constraints and outputs optimal portfolios for a wide range of possible risk levels. Functions expected_return, covariance, and generate_constraints are used to generate those arguments.

Third, we have to present the results in a readable fashion. plot_risk_vs_return plots a risk vs return curve of found portfolios. plot_stackplot plots a stackplot that shows the calculated portfolios ordered by their expected return.

The plots weren’t optimized for prettiness, but they are there to provide good overview of the situation.

Credits

The whole idea of using this method for portfolio selection came from convex optimization book.