Efficient Frontier Curve Fit

Efficient-Frontier-Curve.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "plt.rcParams['figure.figsize'] = (10.0, 7.0)\n",
    "from scipy.optimize import curve_fit\n",
    "import cvxopt as opt\n",
    "from cvxopt import blas, solvers\n",
    "solvers.options['show_progress'] = False\n",
    "np.random.seed(123)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "## NUMBER OF ASSETS\n",
    "n_assets = 4\n",
    "\n",
    "## NUMBER OF OBSERVATIONS\n",
    "n_obs = 1000\n",
    "\n",
    "return_vec = np.random.randn(n_assets, n_obs) + 0.000001"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "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",
    "    p = 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 * p.T\n",
    "    sigma = np.sqrt(w * C * w.T)\n",
    "    \n",
    "    # This recursion reduces outliers to keep plots pretty\n",
    "    if sigma > 2:\n",
    "        return random_portfolio(returns)\n",
    "    return mu, sigma\n",
    "\n",
    "def optimal_portfolio(returns):\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",
    "    print x1\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\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0555080150868\n"
     ]
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    {
     "data": {
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     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
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      ],
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1102829d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n_portfolios = 500\n",
    "means, stds = np.column_stack([\n",
    "    random_portfolio(return_vec) \n",
    "    for _ in xrange(n_portfolios)\n",
    "])\n",
    "\n",
    "weights, returns, risks = optimal_portfolio(return_vec)\n",
    "\n",
    "# Quadratic Fit\n",
    "quadratic = np.poly1d(np.polyfit(returns, risks, 2))\n",
    "\n",
    "# Try fitting a Hyperbola\n",
    "def hyperbola(x, x0, y0, a, b):\n",
    "    \"\"\"\n",
    "    formula used: \n",
    "        (y - y0)^2 / a^2 - (x - x0)^2 / b^2 = 1\n",
    "    \"\"\"\n",
    "    return y0**2 + np.sqrt(a**2 * ((x - x0)**2 / b**2 + 1))\n",
    "\n",
    "\n",
    "# Initial params guess, these seem pretty sensitive\n",
    "guess = 0.05, 0.05, np.mean(risks) / 2, np.mean(returns) / 2\n",
    "\n",
    "params, _ = curve_fit(hyperbola, returns, risks, guess)\n",
    "\n",
    "plt.plot(means, stds, 'r+')\n",
    "plt.xlabel('mean')\n",
    "plt.ylabel('std')\n",
    "plt.plot(returns, risks,  'y-o', label='Efficient Frontier')\n",
    "\n",
    "plt.plot(returns, quadratic(returns), 'b-+', label='quadratic')\n",
    "plt.plot(returns, hyperbola(returns, *params), 'g-x', label='hyperbolic')\n",
    "plt.legend(loc=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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