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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[0])) 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[2] / m1[0]) # 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)

### sgilles33 commented Jan 22, 2021

Thank you very much for this approach of the Markowitz model under Python.
I have a little problem with line 99, Python tells me that the stds variable is not known
and the graph presenting the efficient frontier does not appear.
Do you have a solution?
Thank you

Merci beaucoup pour cette approche du modèle de Markowitz sous Python.
J'ai un petit soucis ligne 99, Python m'indique que la variable stds n'est pas connue
et le graphique présentant la frontière efficiente n'apparait pas.
Auriez-vous une solution ?
Merci

### Percolat commented Feb 17, 2021

Bonjour,

Tente le bout de code suivant à partir de la ligne 93
La fonction plot_portfolio crée une première figure directement dans la fonction.
Alors que la seconde figure est plotée à l'extérieur de la fonction optimal_portfolio

C'est un peu traite mais tout le reste fonctionne parfaitement !

Merci :)

#Main program :
fig = plt.figure()
plot_portfolio(1000)
weights, returns, risks = optimal_portfolio(return_vec)
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.