Portfolio Optimization

Introduction

The portfolio optimization example as described in Boyd, Vandenberghe, 2009, Convex Optimization can be formulated as a Conic SOCP program as follows.

\[\begin{aligned} \text{maximize}\qquad & \mathbb{E}(p)^Tx \\ \text{subject to}\qquad & \mathbb{E}(p)^Tx + \Phi^{-1}(\beta)\left \lVert \Sigma^{1/2}x \right \rVert _2 \ge \alpha \\ & x \ge 0 \\ & \textbf{1}^Tx = 1 \end{aligned}\]

How to run the example

Ensure that you have the necessary dependencies installed. These can be installed by executing the following commands:

julia> ]
pkg> activate .
pkg> add CSV
pkg> add DataFrames
pkg> add Distributions

pkg> add ConicSolve
julia> exit()

Run the example from the command line

julia example.jl

Explanation

The portfolio optimization example is an example for maximizing expected return given a set of stocks subject to some risk tolerance. A table of stock ticks can be found in portfolio.csv, i.e. the daily close price of seven random stocks.

Data Acquisition

CSV.jl is used to load the file portfolio.csv into a DataFrame object (using DataFrames.jl). The data frame is then converted to a matrix for the solver.

Solve the problem

The problem parameters are calculated using Distributions.jl and Statistics.jl to get the expected return and price change variance. The loss risk constraint contains two tuning parameters $\beta \le 0.5$ (based on the inv. cdf of the std. normal distribution) to control the level of risk and $\alpha$, the loss amount for the level of risk.

Get the solution

For this problem setting the parameter $\eta$ to nothing (i.e. $\sigma$) may give better results.

The solution is x which can be obtained as below: $x \in [0, 1]$ is the optimal weight of each stock in the portfolio.

x = get_solution(solver)