Authors: Dr. Thomas Starke, David Edwards, Dr. Thomas Wiecki
Notebook released under the Creative Commons Attribution 4.0 License.
Today's blog post is written in collaboration with Dr. Thomas Starke. It is based on a longer whitepaper by Thomas Starke on the relationship between Markowitz portfolio optimization and Kelly optimization. The full whitepaper can be found here.
In this blog post you will learn about the basic idea behind Markowitz portfolio optimization as well as how to do it in Python. We will then show how you can create a simple backtest that rebalances its portfolio in a Markowitz-optimal way. We hope you enjoy it and get a little more enlightened in the process.
We will start by using random data and only later use actual stock data. This will hopefully help you to get a sense of how to use modelling and simulation to improve your understanding of the theoretical concepts. Don‘t forget that the skill of an algo-trader is to put mathematical models into code and this example is great practice.
Let's start with importing a few modules, which we need later and produce a series of normally distributed returns. cvxopt is a convex solver which you can easily download with
sudo pip install cvxopt.
In [1]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
import cvxopt as opt
from cvxopt import blas, solvers
import pandas as pd
np.random.seed(123)
# Turn off progress printing
solvers.options['show_progress'] = False
Assume that we have 4 assets, each with a return series of length 1000. We can use numpy.random.randn to sample returns from a normal distribution.
In [2]:
## NUMBER OF ASSETS
n_assets = 4
## NUMBER OF OBSERVATIONS
n_obs = 1000
return_vec = np.random.randn(n_assets, n_obs)
In [3]:
plt.plot(return_vec.T, alpha=.4);
plt.xlabel('time')
plt.ylabel('returns')
Out[3]:
These return series can be used to create a wide range of portfolios, which all have different returns and risks (standard deviation). We can produce a wide range of random weight vectors and plot those portfolios. As we want all our capital to be invested, this vector will have to sum to one.
In [4]:
def rand_weights(n):
''' Produces n random weights that sum to 1 '''
k = np.random.rand(n)
return k / sum(k)
print rand_weights(n_assets)
print rand_weights(n_assets)
Next, lets evaluate how many of these random portfolios would perform. Towards this goal we are calculating the mean returns as well as the volatility (here we are using standard deviation). You can also see that there is a filter that only allows to plot portfolios with a standard deviation of < 2 for better illustration.
In [5]:
def random_portfolio(returns):
'''
Returns the mean and standard deviation of returns for a random portfolio
'''
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
sigma = np.sqrt(w * C * w.T)
# This recursion reduces outliers to keep plots pretty
if sigma > 2:
return random_portfolio(returns)
return mu, sigma
In the code you will notice the calculation of the return with:
$$ R = p^T w $$where $R$ is the expected return, $p^T$ is the transpose of the vector for the mean
returns for each time series and w is the weight vector of the portfolio. $p$ is a Nx1
column vector, so $p^T$ turns into a 1xN row vector which can be multiplied with the
Nx1 weight (column) vector w to give a scalar result. This is equivalent to the dot
product used in the code. Keep in mind that Python has a reversed definition of
rows and columns and the accurate NumPy version of the previous equation would
be R = w * p.T
Next, we calculate the standard deviation with
$$\sigma = \sqrt{w^T C w}$$where $C$ is the covariance matrix of the returns which is a NxN matrix. Please
note that if we simply calculated the simple standard deviation with the appropriate weighting using std(array(ret_vec).T*w) we would get a slightly different
’bullet’. This is because the simple standard deviation calculation would not take
covariances into account. In the covariance matrix, the values of the diagonal
represent the simple variances of each asset while the off-diagonals are the variances between the assets. By using ordinary std() we effectively only regard the
diagonal and miss the rest. A small but significant difference.
Lets generate the mean returns and volatility for 500 random portfolios:
In [6]:
n_portfolios = 500
means, stds = np.column_stack([
random_portfolio(return_vec)
for _ in xrange(n_portfolios)
])
Upon plotting those you will observe that they form a characteristic parabolic shape called the ‘Markowitz bullet‘ with the boundaries being called the ‘efficient frontier‘, where we have the lowest variance for a given expected.
In [7]:
plt.plot(stds, means, 'o', markersize=5)
plt.xlabel('std')
plt.ylabel('mean')
plt.title('Mean and standard deviation of returns of randomly generated portfolios')
Out[7]:
Once we have a good representation of our portfolios as the blue dots show we can calculate the efficient frontier Markowitz-style. This is done by minimising
$$ w^T C w$$for $w$ on the expected portfolio return $R^T w$ whilst keeping the sum of all the weights equal to 1:
$$ \sum_{i}{w_i} = 1 $$Here we parametrically run through $R^T w = \mu$ and find the minimum variance
for different $\mu$‘s. This can be done with scipy.optimise.minimize but we have
to define quite a complex problem with bounds, constraints and a Lagrange multiplier. Conveniently, the cvxopt package, a convex solver, does all of that for us. We used one of their examples with some modifications as shown below. You will notice that there are some conditioning expressions in the code. They are simply needed to set up the problem. For more information please have a look at the cvxopt example.
The mus vector produces a series of expected return values $\mu$ in a non-linear and more appropriate way. We will see later that we don‘t need to calculate a lot of these as they perfectly fit a parabola, which can safely be extrapolated for higher values.
In [8]:
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)]
# 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
weights, returns, risks = optimal_portfolio(return_vec)
plt.plot(stds, means, 'o')
plt.ylabel('mean')
plt.xlabel('std')
plt.plot(risks, returns, 'y-o')
Out[8]:
In yellow you can see the optimal portfolios for each of the desired returns (i.e. the mus). In addition, we get the one optimal portfolio returned:
In [9]:
print weights
This is all very interesting but not very applied. We next demonstrate how you can create a simple algorithm in zipline -- the open-source backtester that powers Quantopian -- to test this optimization on actual historical stock data.
First, lets load in some historical data using Quantopian's data (if we are running in the Quantopian Research Platform, or the load_bars_from_yahoo() function from zipline.
In [14]:
from zipline.utils.factory import load_bars_from_yahoo
end = pd.Timestamp.utcnow()
start = end - 2500 * pd.tseries.offsets.BDay()
data = load_bars_from_yahoo(stocks=['IBM', 'GLD', 'XOM', 'AAPL',
'MSFT', 'TLT', 'SHY'],
start=start, end=end)
In [11]:
data.loc[:, :, 'price'].plot(figsize=(8,5))
plt.ylabel('price in $')
Out[11]:
Next, we'll create a zipline algorithm by defining two functions -- initialize() which is called once before the simulation starts, and handle_data() which is called for every trading bar. We then instantiate the algorithm object.
If you are confused about the syntax of zipline, check out the tutorial.
In [12]:
import zipline
from zipline.api import (add_history,
history,
set_slippage,
slippage,
set_commission,
commission,
order_target_percent)
from zipline import TradingAlgorithm
def initialize(context):
'''
Called once at the very beginning of a backtest (and live trading).
Use this method to set up any bookkeeping variables.
The context object is passed to all the other methods in your algorithm.
Parameters
context: An initialized and empty Python dictionary that has been
augmented so that properties can be accessed using dot
notation as well as the traditional bracket notation.
Returns None
'''
# Register history container to keep a window of the last 100 prices.
add_history(100, '1d', 'price')
# Turn off the slippage model
set_slippage(slippage.FixedSlippage(spread=0.0))
# Set the commission model (Interactive Brokers Commission)
set_commission(commission.PerShare(cost=0.01, min_trade_cost=1.0))
context.tick = 0
def handle_data(context, data):
'''
Called when a market event occurs for any of the algorithm's
securities.
Parameters
data: A dictionary keyed by security id containing the current
state of the securities in the algo's universe.
context: The same context object from the initialize function.
Stores the up to date portfolio as well as any state
variables defined.
Returns None
'''
# Allow history to accumulate 100 days of prices before trading
# and rebalance every day thereafter.
context.tick += 1
if context.tick < 100:
return
# Get rolling window of past prices and compute returns
prices = history(100, '1d', 'price').dropna()
returns = prices.pct_change().dropna()
try:
# Perform Markowitz-style portfolio optimization
weights, _, _ = optimal_portfolio(returns.T)
# Rebalance portfolio accordingly
for stock, weight in zip(prices.columns, weights):
order_target_percent(stock, weight)
except ValueError as e:
# Sometimes this error is thrown
# ValueError: Rank(A) < p or Rank([P; A; G]) < n
pass
# Instantinate algorithm
algo = TradingAlgorithm(initialize=initialize,
handle_data=handle_data)
# Run algorithm
results = algo.run(data)
results.portfolio_value.plot()
Out[12]:
As you can see, the performance here is pretty good, even through the 2008 financial crisis. This is most likey due to our universe selection and shouldn't always be expected. Increasing the number of stocks in the universe might reduce the volatility as well. Please let us know in the comments section if you had any success with this strategy and how many stocks you used.
In this blog, co-written by Quantopian friend Dr. Thomas Starke, we wanted to provide an intuitive and gentle introduction to Markowitz portfolio optimization which still remains relevant today. By using simulation of various random portfolios we have seen that certain portfolios perform better than others. Convex optimization using cvxopt allowed us to then numerically determine the portfolios that live on the efficient frontier. The zipline backtest serves as an example but also shows compelling performance.
cvxopt to the Quantopian backtester -- stay tuned!