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portfolio.py


#!/usr/bin/python

# Copyright 2019, Gurobi Optimization, LLC

# Portfolio selection: given a sum of money to invest, one must decide how to
# spend it amongst a portfolio of financial securities.  Our approach is due
# to Markowitz (1959) and looks to minimize the risk associated with the
# investment while realizing a target expected return.  By varying the target,
# one can compute an 'efficient frontier', which defines the optimal portfolio
# for a given expected return.
#
# Note that this example reads historical return data from a comma-separated
# file (../data/portfolio.csv).  As a result, it must be run from the Gurobi
# examples/python directory.
#
# This example requires the pandas (>= 0.20.3), NumPy, and Matplotlib
# Python packages, which are part of the SciPy ecosystem for
# mathematics, science, and engineering (http://scipy.org).  These
# packages aren't included in all Python distributions, but are
# included by default with Anaconda Python.

from gurobipy import *
from math import sqrt
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt

# Import (normalized) historical return data using pandas
data = pd.read_csv('../data/portfolio.csv', index_col=0)
stocks = data.columns

# Calculate basic summary statistics for individual stocks
stock_volatility = data.std()
stock_return = data.mean()

# Create an empty model
m = Model('portfolio')

# Add a variable for each stock
vars = pd.Series(m.addVars(stocks), index=stocks)

# Objective is to minimize risk (squared).  This is modeled using the
# covariance matrix, which measures the historical correlation between stocks.
sigma = data.cov()
portfolio_risk = sigma.dot(vars).dot(vars)
m.setObjective(portfolio_risk, GRB.MINIMIZE)

# Fix budget with a constraint
m.addConstr(vars.sum() == 1, 'budget')

# Optimize model to find the minimum risk portfolio
m.setParam('OutputFlag', 0)
m.optimize()

# Create an expression representing the expected return for the portfolio
portfolio_return = stock_return.dot(vars)

# Display minimum risk portfolio
print('Minimum Risk Portfolio:\n')
for v in vars:
    if v.x > 0:
        print('\t%s\t: %g' % (v.varname, v.x))
minrisk_volatility = sqrt(portfolio_risk.getValue())
print('\nVolatility      = %g' % minrisk_volatility)
minrisk_return = portfolio_return.getValue()
print('Expected Return = %g' % minrisk_return)

# Add (redundant) target return constraint
target = m.addConstr(portfolio_return == minrisk_return, 'target')

# Solve for efficient frontier by varying target return
frontier = pd.Series()
for r in np.linspace(stock_return.min(), stock_return.max(), 100):
    target.rhs = r
    m.optimize()
    frontier.loc[sqrt(portfolio_risk.getValue())] = r

# Plot volatility versus expected return for individual stocks
ax = plt.gca()
ax.scatter(x=stock_volatility, y=stock_return,
           color='Blue', label='Individual Stocks')
for i, stock in enumerate(stocks):
    ax.annotate(stock, (stock_volatility[i], stock_return[i]))

# Plot volatility versus expected return for minimum risk portfolio
ax.scatter(x=minrisk_volatility, y=minrisk_return, color='DarkGreen')
ax.annotate('Minimum\nRisk\nPortfolio', (minrisk_volatility, minrisk_return),
            horizontalalignment='right')

# Plot efficient frontier
frontier.plot(color='DarkGreen', label='Efficient Frontier', ax=ax)

# Format and display the final plot
ax.axis([0.005, 0.06, -0.02, 0.025])
ax.set_xlabel('Volatility (standard deviation)')
ax.set_ylabel('Expected Return')
ax.legend()
ax.grid()
plt.show()

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