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# Nick3523/Markowitz.py

Created October 31, 2020 14:32
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 #!/usr/bin/env python # coding: utf-8 get_ipython().run_line_magic('matplotlib', 'inline') import numpy as np import matplotlib.pyplot as plt import cvxopt as opt from cvxopt import blas, solvers import pandas as pd # ### Param's : np.random.seed(42) ## Number of assets n_assets = 5 ## Number of observations n_obs = 1000 return_vec = np.random.randn(n_assets, n_obs) #Randomly generates returns on each asset, for each observations (like a timeseries) # ### Functions : #Produces n random weights that sum to 1 def rand_weights(n): k = np.random.rand(n) return k / sum(k) #Returns a random portfolio defined by the mean and standard deviation of returns def random_portfolio(returns): p = np.asmatrix(np.mean(returns, axis=1)) w = np.asmatrix(rand_weights(returns.shape)) C = np.asmatrix(np.cov(returns)) mu = w * p.T #mean returns of portfolio sigma = np.sqrt(w * C * w.T) #std returns of portfolio return mu, sigma #Generate random portfolios, defined by their mean and std of returns, using the previous function def generate_portfolios(n_portfolios=500): means, stds = np.column_stack([random_portfolio(return_vec) for _ in range(n_portfolios)]) return means, stds #Plot the randomly generated portfolios def plot_portfolio(n_portfolios=500): means, stds = generate_portfolios(n_portfolios) fig = plt.figure() plt.plot(stds, means, 'o', markersize=5) plt.title("Représentation graphiques des portefeuilles du marché") plt.ylabel('Rendements') plt.xlabel('Risques') plt.savefig("Ptfs-Marchés.png",dpi=144) #Compute the optimal portfolio, based on Markowitz theory - Source : https://plotly.com/python/v3/ipython-notebooks/markowitz-portfolio-optimization/ def optimal_portfolio(returns): n = len(returns) returns = np.asmatrix(returns) N = 100 mus = [10**(5.0 * t/N - 1.0) for t in range(N)] #Used to avoid linear mean # Convert to cvxopt matrices S = opt.matrix(np.cov(returns)) pbar = opt.matrix(np.mean(returns, axis=1)) # Create constraint matrices G = -opt.matrix(np.eye(n)) # negative n x n identity matrix h = opt.matrix(0.0, (n ,1)) A = opt.matrix(1.0, (1, n)) b = opt.matrix(1.0) # Calculate efficient frontier weights using quadratic programming portfolios = [solvers.qp(mu*S, -pbar, G, h, A, b)['x'] for mu in mus] ## CALCULATE RISKS AND RETURNS FOR FRONTIER returns = [blas.dot(pbar, x) for x in portfolios] risks = [np.sqrt(blas.dot(x, S*x)) for x in portfolios] ## CALCULATE THE 2ND DEGREE POLYNOMIAL OF THE FRONTIER CURVE m1 = np.polyfit(returns, risks, 2) x1 = np.sqrt(m1 / m1) # CALCULATE THE OPTIMAL PORTFOLIO wt = solvers.qp(opt.matrix(x1 * S), -pbar, G, h, A, b)['x'] return np.asarray(wt), returns, risks #Main program : plot_portfolio(1000) weights, returns, risks = optimal_portfolio(return_vec) fig = plt.figure() plt.plot(stds, means, 'o') plt.plot(risks, returns, 'y-o') plt.title("Frontière de l'efficience") plt.ylabel('Rendements') plt.xlabel('Risques') plt.savefig("Frotnière-efficience.png",dpi=144) print (weights) print(np.sum(weights)) #100% is invested in n_assets (5 assets in this example)

Parfait merci.