PortOpt.MinVol.Optimization.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Load Packages and Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import pandas as pd \n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "\n",
    "assets = pd.read_csv('/home/Portfolio_2.csv',header='infer',index_col='Date') "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 109,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>GOOGL</th>\n",
       "      <th>FB</th>\n",
       "      <th>NFLX</th>\n",
       "      <th>AMZN</th>\n",
       "      <th>BBBY</th>\n",
       "      <th>COST</th>\n",
       "      <th>DLTR</th>\n",
       "      <th>FOX</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2015-04-01</th>\n",
       "      <td>549.489990</td>\n",
       "      <td>81.669998</td>\n",
       "      <td>59.017143</td>\n",
       "      <td>370.260010</td>\n",
       "      <td>73.779182</td>\n",
       "      <td>139.591400</td>\n",
       "      <td>81.300003</td>\n",
       "      <td>31.535204</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-02</th>\n",
       "      <td>541.309998</td>\n",
       "      <td>81.559998</td>\n",
       "      <td>59.154285</td>\n",
       "      <td>372.250000</td>\n",
       "      <td>74.233482</td>\n",
       "      <td>140.882874</td>\n",
       "      <td>81.120003</td>\n",
       "      <td>32.055969</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-06</th>\n",
       "      <td>543.950012</td>\n",
       "      <td>82.440002</td>\n",
       "      <td>60.330002</td>\n",
       "      <td>377.040009</td>\n",
       "      <td>74.900414</td>\n",
       "      <td>141.691223</td>\n",
       "      <td>80.980003</td>\n",
       "      <td>32.287418</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-07</th>\n",
       "      <td>544.859985</td>\n",
       "      <td>82.320000</td>\n",
       "      <td>60.494286</td>\n",
       "      <td>374.410004</td>\n",
       "      <td>74.136803</td>\n",
       "      <td>141.022232</td>\n",
       "      <td>80.580002</td>\n",
       "      <td>32.104183</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-08</th>\n",
       "      <td>548.840027</td>\n",
       "      <td>82.279999</td>\n",
       "      <td>63.049999</td>\n",
       "      <td>381.200012</td>\n",
       "      <td>75.084068</td>\n",
       "      <td>141.226639</td>\n",
       "      <td>83.529999</td>\n",
       "      <td>32.046318</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                 GOOGL         FB       NFLX        AMZN       BBBY  \\\n",
       "Date                                                                  \n",
       "2015-04-01  549.489990  81.669998  59.017143  370.260010  73.779182   \n",
       "2015-04-02  541.309998  81.559998  59.154285  372.250000  74.233482   \n",
       "2015-04-06  543.950012  82.440002  60.330002  377.040009  74.900414   \n",
       "2015-04-07  544.859985  82.320000  60.494286  374.410004  74.136803   \n",
       "2015-04-08  548.840027  82.279999  63.049999  381.200012  75.084068   \n",
       "\n",
       "                  COST       DLTR        FOX  \n",
       "Date                                          \n",
       "2015-04-01  139.591400  81.300003  31.535204  \n",
       "2015-04-02  140.882874  81.120003  32.055969  \n",
       "2015-04-06  141.691223  80.980003  32.287418  \n",
       "2015-04-07  141.022232  80.580002  32.104183  \n",
       "2015-04-08  141.226639  83.529999  32.046318  "
      ]
     },
     "execution_count": 109,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "assets.head(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "assets.tail(5) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculate Daily Returns "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 110,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "    }\n",
       "\n",
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       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>GOOGL</th>\n",
       "      <th>FB</th>\n",
       "      <th>NFLX</th>\n",
       "      <th>AMZN</th>\n",
       "      <th>BBBY</th>\n",
       "      <th>COST</th>\n",
       "      <th>DLTR</th>\n",
       "      <th>FOX</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2015-04-01</th>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-02</th>\n",
       "      <td>-0.014887</td>\n",
       "      <td>-0.001347</td>\n",
       "      <td>0.002324</td>\n",
       "      <td>0.005375</td>\n",
       "      <td>0.006158</td>\n",
       "      <td>0.009252</td>\n",
       "      <td>-0.002214</td>\n",
       "      <td>0.016514</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-06</th>\n",
       "      <td>0.004877</td>\n",
       "      <td>0.010790</td>\n",
       "      <td>0.019875</td>\n",
       "      <td>0.012868</td>\n",
       "      <td>0.008984</td>\n",
       "      <td>0.005738</td>\n",
       "      <td>-0.001726</td>\n",
       "      <td>0.007220</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-07</th>\n",
       "      <td>0.001673</td>\n",
       "      <td>-0.001456</td>\n",
       "      <td>0.002723</td>\n",
       "      <td>-0.006975</td>\n",
       "      <td>-0.010195</td>\n",
       "      <td>-0.004721</td>\n",
       "      <td>-0.004940</td>\n",
       "      <td>-0.005675</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2015-04-08</th>\n",
       "      <td>0.007305</td>\n",
       "      <td>-0.000486</td>\n",
       "      <td>0.042247</td>\n",
       "      <td>0.018135</td>\n",
       "      <td>0.012777</td>\n",
       "      <td>0.001449</td>\n",
       "      <td>0.036610</td>\n",
       "      <td>-0.001802</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               GOOGL        FB      NFLX      AMZN      BBBY      COST  \\\n",
       "Date                                                                     \n",
       "2015-04-01       NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "2015-04-02 -0.014887 -0.001347  0.002324  0.005375  0.006158  0.009252   \n",
       "2015-04-06  0.004877  0.010790  0.019875  0.012868  0.008984  0.005738   \n",
       "2015-04-07  0.001673 -0.001456  0.002723 -0.006975 -0.010195 -0.004721   \n",
       "2015-04-08  0.007305 -0.000486  0.042247  0.018135  0.012777  0.001449   \n",
       "\n",
       "                DLTR       FOX  \n",
       "Date                            \n",
       "2015-04-01       NaN       NaN  \n",
       "2015-04-02 -0.002214  0.016514  \n",
       "2015-04-06 -0.001726  0.007220  \n",
       "2015-04-07 -0.004940 -0.005675  \n",
       "2015-04-08  0.036610 -0.001802  "
      ]
     },
     "execution_count": 110,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "returns = (assets/assets.shift(1))-1\n",
    "returns.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Check Distribution of Returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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N7OSy7V6J1wm8GXi0fnEzYxywot1CDEPWsr3BzHasZ8ec6jgvusuLHABvNrNt6tkxBzrO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\ngXccxyko/x/xCXTO2MEhwAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f41545a0710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "returns.hist(bins=100)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Calculate Log Returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 112,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "log_returns = np.log((assets/assets.shift(1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "log_returns.head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "GOOGL    0.212306\n",
       "FB       0.224318\n",
       "NFLX     0.538202\n",
       "AMZN     0.455631\n",
       "BBBY    -0.420122\n",
       "COST     0.100268\n",
       "DLTR     0.051696\n",
       "FOX      0.047673\n",
       "dtype: float64"
      ]
     },
     "execution_count": 113,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log_returns.mean()*252"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Covariance Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 114,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>GOOGL</th>\n",
       "      <th>FB</th>\n",
       "      <th>NFLX</th>\n",
       "      <th>AMZN</th>\n",
       "      <th>BBBY</th>\n",
       "      <th>COST</th>\n",
       "      <th>DLTR</th>\n",
       "      <th>FOX</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>GOOGL</th>\n",
       "      <td>0.051703</td>\n",
       "      <td>0.037923</td>\n",
       "      <td>0.040084</td>\n",
       "      <td>0.037761</td>\n",
       "      <td>0.006759</td>\n",
       "      <td>0.014809</td>\n",
       "      <td>0.017474</td>\n",
       "      <td>0.016526</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>FB</th>\n",
       "      <td>0.037923</td>\n",
       "      <td>0.063865</td>\n",
       "      <td>0.041359</td>\n",
       "      <td>0.041199</td>\n",
       "      <td>0.006268</td>\n",
       "      <td>0.012379</td>\n",
       "      <td>0.015795</td>\n",
       "      <td>0.018267</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>NFLX</th>\n",
       "      <td>0.040084</td>\n",
       "      <td>0.041359</td>\n",
       "      <td>0.174764</td>\n",
       "      <td>0.048329</td>\n",
       "      <td>0.011944</td>\n",
       "      <td>0.016129</td>\n",
       "      <td>0.022053</td>\n",
       "      <td>0.023956</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AMZN</th>\n",
       "      <td>0.037761</td>\n",
       "      <td>0.041199</td>\n",
       "      <td>0.048329</td>\n",
       "      <td>0.077447</td>\n",
       "      <td>0.004387</td>\n",
       "      <td>0.013007</td>\n",
       "      <td>0.015140</td>\n",
       "      <td>0.020239</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>BBBY</th>\n",
       "      <td>0.006759</td>\n",
       "      <td>0.006268</td>\n",
       "      <td>0.011944</td>\n",
       "      <td>0.004387</td>\n",
       "      <td>0.099406</td>\n",
       "      <td>0.016719</td>\n",
       "      <td>0.031556</td>\n",
       "      <td>0.020358</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>COST</th>\n",
       "      <td>0.014809</td>\n",
       "      <td>0.012379</td>\n",
       "      <td>0.016129</td>\n",
       "      <td>0.013007</td>\n",
       "      <td>0.016719</td>\n",
       "      <td>0.034535</td>\n",
       "      <td>0.021681</td>\n",
       "      <td>0.011282</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>DLTR</th>\n",
       "      <td>0.017474</td>\n",
       "      <td>0.015795</td>\n",
       "      <td>0.022053</td>\n",
       "      <td>0.015140</td>\n",
       "      <td>0.031556</td>\n",
       "      <td>0.021681</td>\n",
       "      <td>0.079864</td>\n",
       "      <td>0.018325</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>FOX</th>\n",
       "      <td>0.016526</td>\n",
       "      <td>0.018267</td>\n",
       "      <td>0.023956</td>\n",
       "      <td>0.020239</td>\n",
       "      <td>0.020358</td>\n",
       "      <td>0.011282</td>\n",
       "      <td>0.018325</td>\n",
       "      <td>0.060808</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          GOOGL        FB      NFLX      AMZN      BBBY      COST      DLTR  \\\n",
       "GOOGL  0.051703  0.037923  0.040084  0.037761  0.006759  0.014809  0.017474   \n",
       "FB     0.037923  0.063865  0.041359  0.041199  0.006268  0.012379  0.015795   \n",
       "NFLX   0.040084  0.041359  0.174764  0.048329  0.011944  0.016129  0.022053   \n",
       "AMZN   0.037761  0.041199  0.048329  0.077447  0.004387  0.013007  0.015140   \n",
       "BBBY   0.006759  0.006268  0.011944  0.004387  0.099406  0.016719  0.031556   \n",
       "COST   0.014809  0.012379  0.016129  0.013007  0.016719  0.034535  0.021681   \n",
       "DLTR   0.017474  0.015795  0.022053  0.015140  0.031556  0.021681  0.079864   \n",
       "FOX    0.016526  0.018267  0.023956  0.020239  0.020358  0.011282  0.018325   \n",
       "\n",
       "            FOX  \n",
       "GOOGL  0.016526  \n",
       "FB     0.018267  \n",
       "NFLX   0.023956  \n",
       "AMZN   0.020239  \n",
       "BBBY   0.020358  \n",
       "COST   0.011282  \n",
       "DLTR   0.018325  \n",
       "FOX    0.060808  "
      ]
     },
     "execution_count": 114,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log_returns.cov()*252"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Correlation Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>GOOGL</th>\n",
       "      <th>FB</th>\n",
       "      <th>NFLX</th>\n",
       "      <th>AMZN</th>\n",
       "      <th>BBBY</th>\n",
       "      <th>COST</th>\n",
       "      <th>DLTR</th>\n",
       "      <th>FOX</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>GOOGL</th>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.659948</td>\n",
       "      <td>0.421682</td>\n",
       "      <td>0.596727</td>\n",
       "      <td>0.094283</td>\n",
       "      <td>0.350461</td>\n",
       "      <td>0.271933</td>\n",
       "      <td>0.294732</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>FB</th>\n",
       "      <td>0.659948</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.391481</td>\n",
       "      <td>0.585806</td>\n",
       "      <td>0.078667</td>\n",
       "      <td>0.263593</td>\n",
       "      <td>0.221170</td>\n",
       "      <td>0.293131</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>NFLX</th>\n",
       "      <td>0.421682</td>\n",
       "      <td>0.391481</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.415408</td>\n",
       "      <td>0.090621</td>\n",
       "      <td>0.207611</td>\n",
       "      <td>0.186670</td>\n",
       "      <td>0.232387</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>AMZN</th>\n",
       "      <td>0.596727</td>\n",
       "      <td>0.585806</td>\n",
       "      <td>0.415408</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.050001</td>\n",
       "      <td>0.251497</td>\n",
       "      <td>0.192502</td>\n",
       "      <td>0.294913</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>BBBY</th>\n",
       "      <td>0.094283</td>\n",
       "      <td>0.078667</td>\n",
       "      <td>0.090621</td>\n",
       "      <td>0.050001</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.285347</td>\n",
       "      <td>0.354165</td>\n",
       "      <td>0.261850</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>COST</th>\n",
       "      <td>0.350461</td>\n",
       "      <td>0.263593</td>\n",
       "      <td>0.207611</td>\n",
       "      <td>0.251497</td>\n",
       "      <td>0.285347</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.412833</td>\n",
       "      <td>0.246201</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>DLTR</th>\n",
       "      <td>0.271933</td>\n",
       "      <td>0.221170</td>\n",
       "      <td>0.186670</td>\n",
       "      <td>0.192502</td>\n",
       "      <td>0.354165</td>\n",
       "      <td>0.412833</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.262965</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>FOX</th>\n",
       "      <td>0.294732</td>\n",
       "      <td>0.293131</td>\n",
       "      <td>0.232387</td>\n",
       "      <td>0.294913</td>\n",
       "      <td>0.261850</td>\n",
       "      <td>0.246201</td>\n",
       "      <td>0.262965</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "          GOOGL        FB      NFLX      AMZN      BBBY      COST      DLTR  \\\n",
       "GOOGL  1.000000  0.659948  0.421682  0.596727  0.094283  0.350461  0.271933   \n",
       "FB     0.659948  1.000000  0.391481  0.585806  0.078667  0.263593  0.221170   \n",
       "NFLX   0.421682  0.391481  1.000000  0.415408  0.090621  0.207611  0.186670   \n",
       "AMZN   0.596727  0.585806  0.415408  1.000000  0.050001  0.251497  0.192502   \n",
       "BBBY   0.094283  0.078667  0.090621  0.050001  1.000000  0.285347  0.354165   \n",
       "COST   0.350461  0.263593  0.207611  0.251497  0.285347  1.000000  0.412833   \n",
       "DLTR   0.271933  0.221170  0.186670  0.192502  0.354165  0.412833  1.000000   \n",
       "FOX    0.294732  0.293131  0.232387  0.294913  0.261850  0.246201  0.262965   \n",
       "\n",
       "            FOX  \n",
       "GOOGL  0.294732  \n",
       "FB     0.293131  \n",
       "NFLX   0.232387  \n",
       "AMZN   0.294913  \n",
       "BBBY   0.261850  \n",
       "COST   0.246201  \n",
       "DLTR   0.262965  \n",
       "FOX    1.000000  "
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "log_returns.corr()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Random Weights via Monte Carlo Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def neg_sharpe_ratio(weights, mean_returns, cov_matrix, risk_free_rate):\n",
    "    p_var, p_ret = portfolio_annualised_performance(weights, mean_returns, cov_matrix)\n",
    "    return -(p_ret - risk_free_rate) / p_var\n",
    "\n",
    "\n",
    "def max_sharpe_ratio(mean_returns, cov_matrix, risk_free_rate):\n",
    "    num_assets = len(mean_returns)\n",
    "    args = (mean_returns, cov_matrix, risk_free_rate)\n",
    "    constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})\n",
    "    bound = (0.0,1.0)\n",
    "    bounds = tuple(bound for asset in range(num_assets))\n",
    "    result = sco.minimize(neg_sharpe_ratio, num_assets*[1./num_assets,], args=args,\n",
    "                        method='SLSQP', bounds=bounds, constraints=constraints)\n",
    "    return result    \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Optimization - Minimum Volatility"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def portfolio_volatility(weights, mean_returns, cov_matrix):\n",
    "    return portfolio_annualised_performance(weights, mean_returns, cov_matrix)[0]\n",
    "\n",
    "def min_variance(mean_returns, cov_matrix):\n",
    "    num_assets = len(mean_returns)\n",
    "    args = (mean_returns, cov_matrix)\n",
    "    constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})\n",
    "    bound = (0.0,1.0)\n",
    "    bounds = tuple(bound for asset in range(num_assets))\n",
    "\n",
    "    result = sco.minimize(portfolio_volatility, num_assets*[1./num_assets,], args=args,\n",
    "                        method='SLSQP', bounds=bounds, constraints=constraints)\n",
    "\n",
    "    return result"
   ]
  }
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