MarkowitzStuffForQuantopian.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Learning about the Efficient Frontier - Tests on the optimal portfolio in the Markowitz approach\n",
    "\n",
    "I really liked the Blog post on the efficient frontier [by Thomas Wiecki et al.](http://blog.quantopian.com/markowitz-portfolio-optimization-2/) and a lot of the stuff here is strongly inspired by that post.\n",
    "\n",
    "I wanted to understand several things and I think that the script helps a bit. Stuff that I found interesting:\n",
    "\n",
    "- A random portfolio of more than 5 shares is rarely close to the efficient frontier\n",
    "- The mean has tremendous error bars, just due to the finite length of the sample.\n",
    "- The Algorithm gets the variance down, but it is really bad in optimizing the return.\n",
    "- Beating the random portfolio is hard.\n",
    "\n",
    "Let me import the necessary stuff first."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import datetime as dt\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "from pylab import *\n",
    "import pandas as pd\n",
    "\n",
    "from scipy.stats import norm\n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "# stuff for the optimization\n",
    "import cvxopt as opt\n",
    "from cvxopt import blas, solvers "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def rand_weights(n):\n",
    "    ''' Produces n random weights that sum to 1 '''\n",
    "    k = np.random.rand(n)\n",
    "    return k / sum(k) \n",
    "\n",
    "def random_portfolio(returns):\n",
    "    ''' \n",
    "    Returns the mean and standard deviation of returns for a random portfolio\n",
    "    '''\n",
    "    \n",
    "    mi = np.asmatrix(np.mean(returns, axis=1))\n",
    "    w = np.asmatrix(rand_weights(returns.shape[0]))\n",
    "    C = np.asmatrix(np.cov(returns))\n",
    "    \n",
    "    mu = w * mi.T\n",
    "    sigma = np.sqrt(w * C * w.T)\n",
    "    # This recursion reduces outliers to keep plots pretty\n",
    "    if sigma > 2:\n",
    "        return random_portfolio(returns)\n",
    "    return mu, sigma"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function will allow us to calculate the optimal portfolio at some point. Right now it takes the whole series, but I would like to do it directly on the variance $\\sigma_i$, the correlation matrix $\\mathbf{C}$ and the expected returns $m_i$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def get_return_and_cov_with_err(returns):\n",
    "    '''\n",
    "    Get the mean return and the standard deviation of each series by assuming a standard distribution.\n",
    "    This allows me also to get the errors on those estimates. There is no fitting involved and I simply\n",
    "    use known relations.\n",
    "    '''\n",
    "    n = len(returns)\n",
    "    returns = np.asmatrix(returns)\n",
    "    \n",
    "    S = np.cov(returns)\n",
    "    pbar = np.mean(returns, axis=1)\n",
    "    \n",
    "    return S, pbar\n",
    "\n",
    "def optimal_portfolio(S, pbar, n, rf):\n",
    "    '''\n",
    "    Gives back the optimal portfolio given the standard deviation etc. As optimal we see here the tangential portfolio.\n",
    "    '''\n",
    "    \n",
    "    # Convert to cvxopt matrices\n",
    "    S = opt.matrix(S)\n",
    "    pbar = opt.matrix(pbar)\n",
    "    \n",
    "    N = 50\n",
    "    mus = [10**(5.0 * t/N - 1.0) for t in range(N)]\n",
    "\n",
    "    # Create constraint matrices\n",
    "    \n",
    "    # we have the constraint G dot x <= h. So these constraints imply that the weights are between 0 and 1\n",
    "    G = -opt.matrix(np.eye(n))\n",
    "    h = opt.matrix(0.0, (n ,1))\n",
    "    \n",
    "    # we have the constraint A dot x = b. So this implies that the sum of the weights has to be 1\n",
    "    A = opt.matrix(1.0, (1, n))\n",
    "    b = opt.matrix(1.0)\n",
    "    \n",
    "    # Calculate efficient frontier weights using quadratic programming\n",
    "    portfolios = [solvers.qp(mu*S, -pbar, G, h, A, b)['x'] \n",
    "                  for mu in mus]\n",
    "    ## CALCULATE RISKS AND RETURNS FOR FRONTIER\n",
    "    returns = np.asarray([blas.dot(pbar, x) for x in portfolios])\n",
    "    risks = np.asarray([np.sqrt(blas.dot(x, S*x)) for x in portfolios])\n",
    "    # CALCULATE THE OPTIMAL PORTFOLIO\n",
    "    slope = (returns-rf)/risks\n",
    "    opt_s_ind = slope.argmax()\n",
    "    return np.asarray(portfolios[opt_s_ind]), np.asarray(returns), np.asarray(risks), portfolios\n",
    "\n",
    "def optimal_portfolio_in_blog(returns):\n",
    "    '''\n",
    "    Gives back the optimal portfolio given the standard deviation etc. As optimal we see here\n",
    "    the portfolio with the highest return. This comes directly from a Blog post on Quantopian.\n",
    "    '''\n",
    "    n = len(returns)\n",
    "    returns = np.asmatrix(returns)\n",
    "    \n",
    "    N = 100\n",
    "    mus = [10**(5.0 * t/N - 1.0) for t in range(N)]\n",
    "    \n",
    "    # Convert to cvxopt matrices\n",
    "    S = opt.matrix(np.cov(returns))\n",
    "    pbar = opt.matrix(np.mean(returns, axis=1))\n",
    "    \n",
    "    # Create constraint matrices\n",
    "    G = -opt.matrix(np.eye(n))   # negative n x n identity matrix\n",
    "    h = opt.matrix(0.0, (n ,1))\n",
    "    A = opt.matrix(1.0, (1, n))\n",
    "    b = opt.matrix(1.0)\n",
    "    \n",
    "    # Calculate efficient frontier weights using quadratic programming\n",
    "    portfolios = [solvers.qp(mu*S, -pbar, G, h, A, b)['x'] \n",
    "                  for mu in mus]\n",
    "    ## CALCULATE RISKS AND RETURNS FOR FRONTIER\n",
    "    returns = [blas.dot(pbar, x) for x in portfolios]\n",
    "    risks = [np.sqrt(blas.dot(x, S*x)) for x in portfolios]\n",
    "    ## CALCULATE THE 2ND DEGREE POLYNOMIAL OF THE FRONTIER CURVE\n",
    "    m1 = np.polyfit(returns, risks, 2)\n",
    "    x1 = np.sqrt(m1[2] / m1[0])\n",
    "    # CALCULATE THE OPTIMAL PORTFOLIO\n",
    "    wt = solvers.qp(opt.matrix(x1 * S), -pbar, G, h, A, b)['x']\n",
    "    return np.asarray(wt), returns, risks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Run the code now\n",
    "\n",
    "some important parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "n_assets = 30\n",
    "n_obs = 800\n",
    "n_portfolios = int(1E4)# number of random portfolios that we create\n",
    "rf_year = 0.01#risk free return\n",
    "doy = 200 #days per year\n",
    "n_bins = 200# bins for the distribution fitting"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "##Do some random walk testing with a normal distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a number of normally distributed returns and look at their typical datas."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "return_vec = np.random.randn(n_assets, n_obs)#using the normal distribution\n",
    "# get some numbers on the portfolio\n",
    "s_means = return_vec.mean(axis=1)\n",
    "s_stds = return_vec.std(axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, that we know the important things about each stock, we can turn to the creation of a portfolio. To optimize the portofolio we take two approaches:\n",
    "- One is just a random portfolio This teaches us later on that random portfolios are not that good to find the efficient frontier for larger numbers of portfolios.\n",
    "- The second one is an optimization of the volatility at a given risk. For that we use cvxopt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x10ff2a650>]"
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10bc7a790>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "means, stds = np.column_stack([\n",
    "    random_portfolio(return_vec) \n",
    "    for _ in xrange(n_portfolios)\n",
    "])\n",
    "\n",
    "rf_d = rf_year/doy\n",
    "solvers.options['show_progress'] = False\n",
    "\n",
    "S, pbar = get_return_and_cov_with_err(return_vec)\n",
    "weights, returns, risks, ports = optimal_portfolio(S, pbar, n_assets, rf_d)\n",
    "# get intersection between the portfolio and the risk free one\n",
    "opt_slope = max((returns-rf_d)/risks)\n",
    "plot_risks = linspace(0,max(risks),5)\n",
    "\n",
    "# and plot it up\n",
    "#plt.figure(2)\n",
    "plt.clf()\n",
    "plt.plot(stds,means, 'bo')\n",
    "plt.errorbar(s_stds,s_means, yerr = s_stds/sqrt(n_obs), fmt = 'ro')\n",
    "plt.plot(risks, returns, 'y-o')\n",
    "plt.plot(0, rf_d, 'go')\n",
    "plot(plot_risks, opt_slope*plot_risks+rf_d, 'g-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now run the real datas in Zipline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## First do some tests on the datas"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Prepare to read the data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "DJIA\n",
      "MMM\n",
      "AXP\n",
      "AAPL\n",
      "BA\n",
      "CAT\n",
      "CVX\n",
      "CSCO\n",
      "KO\n",
      "DIS\n",
      "DD\n",
      "XOM\n",
      "GE\n",
      "GS\n",
      "HD\n",
      "IBM\n",
      "INTC\n",
      "JNJ\n",
      "JPM\n",
      "MCD\n",
      "MRK\n",
      "MSFT\n",
      "NKE\n",
      "PFE\n",
      "PG\n",
      "TRV\n",
      "UTX\n",
      "UNH\n",
      "VZ\n",
      "V\n",
      "WMT\n"
     ]
    }
   ],
   "source": [
    "from zipline.utils.factory import load_bars_from_yahoo\n",
    "\n",
    "symbols = ['IBM', 'GLD', 'XOM', 'AAPL', 'MSFT', 'TLT', 'SHY']\n",
    "dow_symbols = ['MMM', 'AXP', 'AAPL', 'BA', 'CAT', 'CVX', 'CSCO', 'KO', 'DIS', 'DD',\n",
    "              'XOM', 'GE', 'GS', 'HD', 'IBM', 'INTC', 'JNJ', 'JPM', 'MCD', 'MRK', \n",
    "               'MSFT', 'NKE', 'PFE', 'PG', 'TRV', 'UTX', 'UNH', 'VZ', 'V','WMT']\n",
    "end = pd.Timestamp.utcnow()\n",
    "start = end - n_obs * pd.tseries.offsets.BDay()\n",
    "\n",
    "dow_jones = load_bars_from_yahoo(stocks=['DJIA'],\n",
    "                            start=start, end=end)\n",
    "data = load_bars_from_yahoo(stocks=dow_symbols,\n",
    "                            start=start, end=end)\n",
    "\n",
    "close = data.loc[:, :, 'price']\n",
    "dow_close = dow_jones.loc[:, :, 'price']\n",
    "close = (close.fillna(method='ffill')).fillna(method='backfill')\n",
    "n_close = close/close.ix[0,:]\n",
    "n_dow_close = dow_close/dow_close.ix[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "c_rets = close.pct_change()\n",
    "c_rets = c_rets.as_matrix()\n",
    "c_rets = c_rets[1:,:]\n",
    "c_rets = c_rets.T\n",
    "\n",
    "dow_rets = dow_close.pct_change()\n",
    "dow_rets = dow_rets.as_matrix()\n",
    "dow_rets = dow_rets[1:,:]\n",
    "dow_rets = dow_rets.T\n",
    "\n",
    "c_means = c_rets.mean(axis=1)\n",
    "c_stds = c_rets.std(axis=1)\n",
    "\n",
    "dow_mean = dow_rets.mean(axis=1)\n",
    "dow_std = dow_rets.std(axis=1)\n",
    "\n",
    "s= shape(c_rets)\n",
    "n_obs = s[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Get the optimal portfolio and a lot random portfolios here. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.00114538405977\n"
     ]
    }
   ],
   "source": [
    "# the random portfolio\n",
    "cr_means, cr_stds = np.column_stack([\n",
    "    random_portfolio(c_rets) \n",
    "    for _ in xrange(n_portfolios)\n",
    "])\n",
    "\n",
    "# the efficient frontier one\n",
    "S_real, pbar_real = get_return_and_cov_with_err(c_rets)\n",
    "c_weights, c_returns, c_risks, c_ports = optimal_portfolio(S_real, pbar_real, n_assets, rf_d)\n",
    "#c_weights, c_returns, c_risks, c_ports = optimal_portfolio(c_rets,rf_d)\n",
    "c_returns = np.asarray(c_returns)\n",
    "c_risks = np.asarray(c_risks)\n",
    "slope = (c_returns-rf_d)/c_risks\n",
    "opt_slope = max(slope)\n",
    "opt_s_ind = slope.argmax()\n",
    "opt_return = c_returns[opt_s_ind]\n",
    "print opt_return\n",
    "plot_risks = linspace(0,max(c_risks),5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Now look at the trading\n",
    "\n",
    "I do not understand this part for the moment, but I am not sure that it is that important."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "[2015-04-12 15:33:22.357941] INFO: Performance: Simulated 770 trading days out of 770.\n",
      "[2015-04-12 15:33:22.358737] INFO: Performance: first open: 2012-03-19 13:31:00+00:00\n",
      "[2015-04-12 15:33:22.359473] INFO: Performance: last close: 2015-04-10 20:00:00+00:00\n"
     ]
    }
   ],
   "source": [
    "import zipline\n",
    "from zipline.api import (add_history, \n",
    "                         history, \n",
    "                         set_slippage, \n",
    "                         slippage,\n",
    "                         set_commission, \n",
    "                         commission, \n",
    "                         order_target_percent)\n",
    "\n",
    "from zipline import TradingAlgorithm\n",
    "\n",
    "\n",
    "def initialize(context):\n",
    "    '''\n",
    "    Called once at the very beginning of a backtest (and live trading). \n",
    "    Use this method to set up any bookkeeping variables.\n",
    "    \n",
    "    The context object is passed to all the other methods in your algorithm.\n",
    "\n",
    "    Parameters\n",
    "\n",
    "    context: An initialized and empty Python dictionary that has been \n",
    "             augmented so that properties can be accessed using dot \n",
    "             notation as well as the traditional bracket notation.\n",
    "    \n",
    "    Returns None\n",
    "    '''\n",
    "    # Register history container to keep a window of the last 100 prices.\n",
    "    add_history(100, '1d', 'price')\n",
    "    \n",
    "    # Turn off the slippage model\n",
    "    set_slippage(slippage.FixedSlippage(spread=0.0))\n",
    "    # Set the commission model (Interactive Brokers Commission)\n",
    "    set_commission(commission.PerShare(cost=0.01, min_trade_cost=1.0))\n",
    "    context.tick = 0\n",
    "    \n",
    "def handle_data(context, data):\n",
    "    '''\n",
    "    Called when a market event occurs for any of the algorithm's \n",
    "    securities. \n",
    "\n",
    "    Parameters\n",
    "\n",
    "    data: A dictionary keyed by security id containing the current \n",
    "          state of the securities in the algo's universe.\n",
    "\n",
    "    context: The same context object from the initialize function.\n",
    "             Stores the up to date portfolio as well as any state \n",
    "             variables defined.\n",
    "\n",
    "    Returns None\n",
    "    '''\n",
    "    # Allow history to accumulate 100 days of prices before trading\n",
    "    # and rebalance every day thereafter.\n",
    "    context.tick += 1\n",
    "    if context.tick < 100:\n",
    "        return\n",
    "    # Get rolling window of past prices and compute returns\n",
    "    prices = history(100, '1d', 'price').dropna()\n",
    "    returns = prices.pct_change().dropna()\n",
    "    _, n_secs = returns.shape\n",
    "    try:\n",
    "        # Perform Markowitz-style portfolio optimization\n",
    "        S, pbar = get_return_and_cov_with_err(returns.T)\n",
    "        weights,_,_,_ = optimal_portfolio(S, pbar, n_secs, 0)\n",
    "        #weights, _, _ = optimal_portfolio_in_blog(returns.T)\n",
    "        # Rebalance portfolio accordingly\n",
    "        for stock, weight in zip(prices.columns, weights):\n",
    "            order_target_percent(stock, weight)\n",
    "    except ValueError as e:\n",
    "        # Sometimes this error is thrown\n",
    "        # ValueError: Rank(A) < p or Rank([P; A; G]) < n\n",
    "        pass\n",
    "        \n",
    "# Instantinate algorithm        \n",
    "algo = TradingAlgorithm(initialize=initialize, \n",
    "                        handle_data=handle_data)\n",
    "# Run algorithm\n",
    "results = algo.run(data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot it up. Some comments about the plots:\n",
    "\n",
    "- In the first figure we compare the Dow Jones benchmark (in green) with the portfolio (in blue)\n",
    "- In the second plot we plot the return as a function of the variance. The red dots are the indiviual stocks with statistical errorbars. The yellow parabole is the efficient frontier. The yellow dot is our 'tangential' portfolio. The green dot is the benchmark. The blue dots are random portfolios each containing some portion of each of the 30 stocks. The straight green line is the tangente."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "770\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1145da750>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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       "rkJggg==\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11473cb10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "pjUBaGxQ3CQAAAAASUVORK5CYII=\n"
      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11461f850>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# get some numbers of the portfolio\n",
    "n_port = results.portfolio_value/results.portfolio_value[0]\n",
    "p_rets = results.portfolio_value.pct_change()\n",
    "p_std = p_rets.std()\n",
    "p_mean = p_rets.mean()\n",
    "\n",
    "# and plot it\n",
    "plt.figure(1)\n",
    "plt.clf()\n",
    "n_obs = len(n_port)\n",
    "print n_obs\n",
    "plot(arange(n_obs),n_port.values)\n",
    "plot(arange(n_obs),n_dow_close.values)\n",
    "\n",
    "plt.figure(2)\n",
    "plt.clf()\n",
    "plt.plot(cr_stds, cr_means, 'bo')\n",
    "plt.errorbar(c_stds, c_means, yerr = c_stds/sqrt(n_obs), fmt= 'ro')\n",
    "plt.plot(dow_std, dow_mean, 'go')\n",
    "plt.plot(0, rf_d, 'go')\n",
    "plot(plot_risks, opt_slope*plot_risks+rf_d, 'g-')\n",
    "plot(c_risks,c_returns, 'y-o')\n",
    "plot(p_std,p_mean, 'yo')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}