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Markowitz portfolio optimization Description In Markowitz portfolio optimization there are two objectives: to maximize reward
while minimizing risk (i.e. variance). This example plots several points on the
optimal frontier using a blended multi-objective approach, and shows that a point
computed using a lexicographic approach also lies on this frontier. Further explanation of this example:
'Xpress Python Reference Manual'
Source Files By clicking on a file name, a preview is opened at the bottom of this page.
example_markowitz.py # Markowitz portfolio optimization # A multi-objective quadratic programming example # In Markowitz portfolio optimization there are two objectives: to maximize reward # while minimizing risk (i.e. variance). This example plots several points on the # optimal frontier using a blended multi-objective approach, and shows that a point # computed using a lexicographic approach also lies on this frontier. # (c) Fair Isaac Corp., 2024 import xpress as xp import numpy as np from matplotlib import pyplot as plt # The historical mean return on investment for five stocks returns = np.array([0.31, 0.87, 0.31, 0.66, 0.24]) # The historical covariances of the five stocks covariance = np.array([ [0.32, 0.70, 0.19, 0.52, 0.16], [0.70, 4.35, -0.48, -0.06, -0.03], [0.19, -0.48, 0.98, 1.10, 0.10], [0.52, -0.6, 1.10, 2.48, 0.37], [0.16, -0.3, 0.10, 0.37, 0.31] ]) p = xp.problem() # Non-negative variables represent percentage of capital to invest in each stock x = p.addVariables(len(returns)) # All objectives must be linear, so define a free transfer variable for the variance variance = p.addVariable(lb=-xp.infinity) ctrs = [ xp.Sum(x) == 1, # Must invest 100% of capital xp.Dot(x, covariance, x) - variance <= 0 # Set up transfer variable for variance ] p.addConstraint(ctrs) p.addObjective(xp.Dot(x, returns)) # Maximize mean return p.addObjective(variance, weight=-1) # Minimize variance p.setOutputEnabled(False) # Vary the objective weights to explore the optimal frontier weights = np.linspace(0.05, 0.95, 20) means = [] variances = [] for w in weights: p.setObjective(objidx=0, weight=w,sense=xp.maximize) p.setObjective(objidx=1, weight=w-1) # Negative weight because we minimize variance p.optimize() means.append(xp.Dot(p.getSolution(x), returns).item()) variances.append(p.getSolution(variance)) # Now we will maximize profit alone, and then minimize variance while not # sacrificing more than 10% of the maximum profit p.setObjective(objidx=0, priority=1, weight=1, reltol=0.1,sense=xp.maximize) p.setObjective(objidx=1, priority=0, weight=-1) p.optimize() m0 = xp.Dot(p.getSolution(x), returns).item() v0 = p.getSolution(variance) # Plot the optimal frontier and mark the final point that we calculated plt.plot(means, variances) plt.plot(m0, v0, c='r', marker='.') plt.title('Return on investment vs variance') plt.xlabel('Expected return') plt.ylabel('Variance') plt.show() | |||||||||||

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