technical-knockout.com : Python for Finance: Part II: 4 Markowitz Portfolio Optimization

Python for Finance: Part II: 4 Markowitz Portfolio Optimization

Portfolio Theory

1952, Harry Markowitz published a paper.
– investors should understand the relationship between securities in their portfolio.

Implementation

import numpy as np
import pandas as pd
from pandas_datareader import data as wb
import matplotlib.pyplot as plt
%matplotlib inline

assets = ['DIS', 'AMZN']
pf_data = pd.DataFrame()

for a in assets:
    pf_data[a] = wb.DataReader(a, data_source = 'google', start = '2010-1-1')['Close']

pf_data.tail()

(pf_data / pf_data.iloc[0] * 100).plot(figsize=(10,5))

import numpy as np

import pandas as pd

from pandas_datareader import data as wb

import matplotlib.pyplot as plt

%matplotlib inline

assets = ['DIS', 'AMZN']

pf_data = pd.DataFrame()

for a in assets:

pf_data[a] = wb.DataReader(a, data_source = 'google', start = '2010-1-1')['Close']

pf_data.tail()

(pf_data / pf_data.iloc[0] * 100).plot(figsize=(10,5))

log_returns = np.log(pf_data / pf_data.shift(1))

log_returns.mean() * 250
// DIS     0.160962
// AMZN    0.268312
// dtype: float64

log_returns.cov() * 250

log_returns.corr()

num_assets = len(assets)

num_assets
// 2

arr = np.random.random(2)
arr
// array([ 0.52831474,  0.47945205])

arr[0] + arr[1]
// 1.0077667888401995

weights = np.random.random(num_assets)
weights /= np.sum(weights)
weights
// array([ 0.47348175,  0.52651825])

weights[0] + weights[1]
// 1.0

log_returns = np.log(pf_data / pf_data.shift(1))

log_returns.mean() * 250

// DIS 0.160962

// AMZN 0.268312

// dtype: float64

log_returns.cov() * 250

log_returns.corr()

num_assets = len(assets)

num_assets

// 2

arr = np.random.random(2)

arr

// array([ 0.52831474, 0.47945205])

arr[0] + arr[1]

// 1.0077667888401995

weights = np.random.random(num_assets)

weights /= np.sum(weights)

weights

// array([ 0.47348175, 0.52651825])

weights[0] + weights[1]

// 1.0

// Expected Portfolio Return:
np.sum(weights * log_returns.mean()) * 250
// 0.2496822910761077

// Expected Portfolio Variance:
np.dot(weights.T, np.dot(log_returns.cov() * 250, weights))
// 0.074759086518318432

// Expected Portfolio Volatility:
np.sqrt(np.dot(weights.T,np.dot(log_returns.cov() * 250, weights)))
// 0.27342107914043212

pfolio_returns = []
pfolio_volatilities = []

for x in range (1000):
    weights = np.random.random(num_assets)
    weights /= np.sum(weights)
    pfolio_returns.append(np.sum(weights * log_returns.mean()) * 250)
    pfolio_volatilities.append(np.sqrt(np.dot(weights.T,np.dot(log_returns.cov() * 250, weights))))
    
pfolio_returns, pfolio_volatilities

pfolio_returns = []
pfolio_volatilities = []

for x in range (1000):
    weights = np.random.random(num_assets)
    weights /= np.sum(weights)
    pfolio_returns.append(np.sum(weights * log_returns.mean()) * 250)
    pfolio_volatilities.append(np.sqrt(np.dot(weights.T,np.dot(log_returns.cov() * 250, weights))))
    
pfolio_returns = np.array(pfolio_returns)
pfolio_volatilities = np.array(pfolio_volatilities)

pfolio_returns, pfolio_volatilities

portfolios = pd.DataFrame({'Return': pfolio_returns, 'Volatility': pfolio_volatilities})

portfolios.head()

portfolios.tail()

portfolios.plot(x='Volatility', y='Return', kind='scatter', figsize=(10,6));
plt.xlabel('Expected Volatility')
plt.ylabel('Expected Return')

// Expected Portfolio Return:

np.sum(weights * log_returns.mean()) * 250

// 0.2496822910761077

// Expected Portfolio Variance:

np.dot(weights.T, np.dot(log_returns.cov() * 250, weights))

// 0.074759086518318432

// Expected Portfolio Volatility:

np.sqrt(np.dot(weights.T,np.dot(log_returns.cov() * 250, weights)))

// 0.27342107914043212

pfolio_returns = []

pfolio_volatilities = []

for x in range (1000):

weights = np.random.random(num_assets)

weights /= np.sum(weights)

pfolio_returns.append(np.sum(weights * log_returns.mean()) * 250)

pfolio_volatilities.append(np.sqrt(np.dot(weights.T,np.dot(log_returns.cov() * 250, weights))))

pfolio_returns, pfolio_volatilities

pfolio_returns = []

pfolio_volatilities = []

for x in range (1000):

weights = np.random.random(num_assets)

weights /= np.sum(weights)

pfolio_returns.append(np.sum(weights * log_returns.mean()) * 250)

pfolio_volatilities.append(np.sqrt(np.dot(weights.T,np.dot(log_returns.cov() * 250, weights))))

pfolio_returns = np.array(pfolio_returns)

pfolio_volatilities = np.array(pfolio_volatilities)

pfolio_returns, pfolio_volatilities

portfolios = pd.DataFrame({'Return': pfolio_returns, 'Volatility': pfolio_volatilities})

portfolios.head()

portfolios.tail()

portfolios.plot(x='Volatility', y='Return', kind='scatter', figsize=(10,6));

plt.xlabel('Expected Volatility')

plt.ylabel('Expected Return')