portfolio_optimization.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Portfolio Optimization - SPDR Sector ETFs (2 year Adj Close Data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "from pandas_datareader import data\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from bokeh.plotting import figure, show\n",
    "from bokeh.models import Range1d, HoverTool\n",
    "from bokeh.io import output_notebook\n",
    "output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Get Data & Calculate Basic Stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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       "      <th>Symbols</th>\n",
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       "      <th>XLE</th>\n",
       "      <th>XLF</th>\n",
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       "      <th>XLP</th>\n",
       "      <th>XLU</th>\n",
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       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date</th>\n",
       "      <th></th>\n",
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       "      <th></th>\n",
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       "      <th></th>\n",
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       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2017-03-06</th>\n",
       "      <td>50.380669</td>\n",
       "      <td>67.775238</td>\n",
       "      <td>23.933243</td>\n",
       "      <td>63.677711</td>\n",
       "      <td>51.244270</td>\n",
       "      <td>51.907421</td>\n",
       "      <td>48.107944</td>\n",
       "      <td>73.214569</td>\n",
       "      <td>84.543297</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-07</th>\n",
       "      <td>50.101414</td>\n",
       "      <td>67.174034</td>\n",
       "      <td>23.865637</td>\n",
       "      <td>63.494835</td>\n",
       "      <td>51.244270</td>\n",
       "      <td>51.841240</td>\n",
       "      <td>48.079876</td>\n",
       "      <td>72.700676</td>\n",
       "      <td>84.270340</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-08</th>\n",
       "      <td>50.101414</td>\n",
       "      <td>65.426811</td>\n",
       "      <td>23.875292</td>\n",
       "      <td>63.273472</td>\n",
       "      <td>51.302525</td>\n",
       "      <td>51.727779</td>\n",
       "      <td>47.387543</td>\n",
       "      <td>72.991562</td>\n",
       "      <td>84.621300</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-09</th>\n",
       "      <td>49.947346</td>\n",
       "      <td>65.858910</td>\n",
       "      <td>23.962215</td>\n",
       "      <td>62.984734</td>\n",
       "      <td>51.292820</td>\n",
       "      <td>51.812870</td>\n",
       "      <td>47.284630</td>\n",
       "      <td>73.389099</td>\n",
       "      <td>84.543297</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-10</th>\n",
       "      <td>50.178452</td>\n",
       "      <td>65.764984</td>\n",
       "      <td>23.942904</td>\n",
       "      <td>63.369705</td>\n",
       "      <td>51.564674</td>\n",
       "      <td>52.049248</td>\n",
       "      <td>47.677574</td>\n",
       "      <td>73.670273</td>\n",
       "      <td>84.670036</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "Symbols           XLB        XLE        XLF        XLI        XLK        XLP  \\\n",
       "Date                                                                           \n",
       "2017-03-06  50.380669  67.775238  23.933243  63.677711  51.244270  51.907421   \n",
       "2017-03-07  50.101414  67.174034  23.865637  63.494835  51.244270  51.841240   \n",
       "2017-03-08  50.101414  65.426811  23.875292  63.273472  51.302525  51.727779   \n",
       "2017-03-09  49.947346  65.858910  23.962215  62.984734  51.292820  51.812870   \n",
       "2017-03-10  50.178452  65.764984  23.942904  63.369705  51.564674  52.049248   \n",
       "\n",
       "Symbols           XLU        XLV        XLY  \n",
       "Date                                         \n",
       "2017-03-06  48.107944  73.214569  84.543297  \n",
       "2017-03-07  48.079876  72.700676  84.270340  \n",
       "2017-03-08  47.387543  72.991562  84.621300  \n",
       "2017-03-09  47.284630  73.389099  84.543297  \n",
       "2017-03-10  47.677574  73.670273  84.670036  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th>Symbols</th>\n",
       "      <th>XLB</th>\n",
       "      <th>XLE</th>\n",
       "      <th>XLF</th>\n",
       "      <th>XLI</th>\n",
       "      <th>XLK</th>\n",
       "      <th>XLP</th>\n",
       "      <th>XLU</th>\n",
       "      <th>XLV</th>\n",
       "      <th>XLY</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",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>2017-03-07</th>\n",
       "      <td>-0.005543</td>\n",
       "      <td>-0.008871</td>\n",
       "      <td>-0.002825</td>\n",
       "      <td>-0.002872</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.001275</td>\n",
       "      <td>-0.000583</td>\n",
       "      <td>-0.007019</td>\n",
       "      <td>-0.003229</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-08</th>\n",
       "      <td>0.000000</td>\n",
       "      <td>-0.026010</td>\n",
       "      <td>0.000405</td>\n",
       "      <td>-0.003486</td>\n",
       "      <td>0.001137</td>\n",
       "      <td>-0.002189</td>\n",
       "      <td>-0.014400</td>\n",
       "      <td>0.004001</td>\n",
       "      <td>0.004165</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-09</th>\n",
       "      <td>-0.003075</td>\n",
       "      <td>0.006604</td>\n",
       "      <td>0.003641</td>\n",
       "      <td>-0.004563</td>\n",
       "      <td>-0.000189</td>\n",
       "      <td>0.001645</td>\n",
       "      <td>-0.002172</td>\n",
       "      <td>0.005446</td>\n",
       "      <td>-0.000922</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-10</th>\n",
       "      <td>0.004627</td>\n",
       "      <td>-0.001426</td>\n",
       "      <td>-0.000806</td>\n",
       "      <td>0.006112</td>\n",
       "      <td>0.005300</td>\n",
       "      <td>0.004562</td>\n",
       "      <td>0.008310</td>\n",
       "      <td>0.003831</td>\n",
       "      <td>0.001499</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2017-03-13</th>\n",
       "      <td>0.002878</td>\n",
       "      <td>0.000143</td>\n",
       "      <td>0.002420</td>\n",
       "      <td>-0.000607</td>\n",
       "      <td>0.000941</td>\n",
       "      <td>0.000727</td>\n",
       "      <td>0.002944</td>\n",
       "      <td>-0.000790</td>\n",
       "      <td>0.002648</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "Symbols          XLB       XLE       XLF       XLI       XLK       XLP  \\\n",
       "Date                                                                     \n",
       "2017-03-07 -0.005543 -0.008871 -0.002825 -0.002872  0.000000 -0.001275   \n",
       "2017-03-08  0.000000 -0.026010  0.000405 -0.003486  0.001137 -0.002189   \n",
       "2017-03-09 -0.003075  0.006604  0.003641 -0.004563 -0.000189  0.001645   \n",
       "2017-03-10  0.004627 -0.001426 -0.000806  0.006112  0.005300  0.004562   \n",
       "2017-03-13  0.002878  0.000143  0.002420 -0.000607  0.000941  0.000727   \n",
       "\n",
       "Symbols          XLU       XLV       XLY  \n",
       "Date                                      \n",
       "2017-03-07 -0.000583 -0.007019 -0.003229  \n",
       "2017-03-08 -0.014400  0.004001  0.004165  \n",
       "2017-03-09 -0.002172  0.005446 -0.000922  \n",
       "2017-03-10  0.008310  0.003831  0.001499  \n",
       "2017-03-13  0.002944 -0.000790  0.002648  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "start_date = '2017-03-05'\n",
    "end_date = '2019-03-05'\n",
    "syms = ['XLE','XLF','XLU','XLI','XLK','XLV','XLY','XLP','XLB']\n",
    "\n",
    "prices = data.DataReader(syms, 'yahoo', start_date, end_date)['Adj Close']\n",
    "N = prices.shape[1]\n",
    "display(prices.head())\n",
    "\n",
    "returns = prices.pct_change()[1:]\n",
    "display(returns.head())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "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>Mean_return</th>\n",
       "      <th>Volatility</th>\n",
       "      <th>Total_return</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Symbols</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>XLB</th>\n",
       "      <td>0.000225</td>\n",
       "      <td>0.010101</td>\n",
       "      <td>0.091093</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLE</th>\n",
       "      <td>0.000027</td>\n",
       "      <td>0.011786</td>\n",
       "      <td>-0.021029</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLF</th>\n",
       "      <td>0.000252</td>\n",
       "      <td>0.010482</td>\n",
       "      <td>0.103904</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLI</th>\n",
       "      <td>0.000390</td>\n",
       "      <td>0.009835</td>\n",
       "      <td>0.186915</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLK</th>\n",
       "      <td>0.000718</td>\n",
       "      <td>0.011997</td>\n",
       "      <td>0.382789</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLP</th>\n",
       "      <td>0.000111</td>\n",
       "      <td>0.007460</td>\n",
       "      <td>0.042433</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLU</th>\n",
       "      <td>0.000376</td>\n",
       "      <td>0.008092</td>\n",
       "      <td>0.188161</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLV</th>\n",
       "      <td>0.000488</td>\n",
       "      <td>0.008992</td>\n",
       "      <td>0.251663</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLY</th>\n",
       "      <td>0.000590</td>\n",
       "      <td>0.009705</td>\n",
       "      <td>0.313292</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         Mean_return  Volatility  Total_return\n",
       "Symbols                                       \n",
       "XLB         0.000225    0.010101      0.091093\n",
       "XLE         0.000027    0.011786     -0.021029\n",
       "XLF         0.000252    0.010482      0.103904\n",
       "XLI         0.000390    0.009835      0.186915\n",
       "XLK         0.000718    0.011997      0.382789\n",
       "XLP         0.000111    0.007460      0.042433\n",
       "XLU         0.000376    0.008092      0.188161\n",
       "XLV         0.000488    0.008992      0.251663\n",
       "XLY         0.000590    0.009705      0.313292"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "stats = pd.concat((returns.mean(),returns.std(),(returns+1).prod()-1),axis=1)\n",
    "stats.columns = ['Mean_return', 'Volatility', 'Total_return']\n",
    "stats"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
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       "\n",
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       "\n",
       "    .dataframe thead th {\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Minimizer</th>\n",
       "      <th>Minimum</th>\n",
       "      <th>Maximizer</th>\n",
       "      <th>Maximum</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Mean_return</th>\n",
       "      <td>XLE</td>\n",
       "      <td>0.000027</td>\n",
       "      <td>XLK</td>\n",
       "      <td>0.000718</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Volatility</th>\n",
       "      <td>XLP</td>\n",
       "      <td>0.007460</td>\n",
       "      <td>XLK</td>\n",
       "      <td>0.011997</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Total_return</th>\n",
       "      <td>XLE</td>\n",
       "      <td>-0.021029</td>\n",
       "      <td>XLK</td>\n",
       "      <td>0.382789</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "             Minimizer   Minimum Maximizer   Maximum\n",
       "Mean_return        XLE  0.000027       XLK  0.000718\n",
       "Volatility         XLP  0.007460       XLK  0.011997\n",
       "Total_return       XLE -0.021029       XLK  0.382789"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "extremes = pd.concat((stats.idxmin(),stats.min(),stats.idxmax(),stats.max()),axis=1)\n",
    "extremes.columns = ['Minimizer','Minimum','Maximizer','Maximum']\n",
    "extremes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot Normalized Cumulative Returns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
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       "\n",
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       "    <div class=\"bk-plotdiv\" id=\"1297e639-3f88-43f0-824b-7d8da989ebbe\"></div>\n",
       "</div>"
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     },
     "metadata": {},
     "output_type": "display_data"
    },
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     "data": {
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       "  var render_items = [{\"docid\":\"190bef45-7202-4b5d-82c0-209877d91794\",\"elementid\":\"1297e639-3f88-43f0-824b-7d8da989ebbe\",\"modelid\":\"2b7e5987-c9ca-4031-b8f4-8357256fe3d2\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "2b7e5987-c9ca-4031-b8f4-8357256fe3d2"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "growth = (returns+1.0).cumprod() \n",
    "tx = growth.index \n",
    "syms = growth.columns #symbols\n",
    "esyms = pd.concat((extremes['Minimizer'],extremes['Maximizer'])) #extreme symbols\n",
    "\n",
    "fig = figure(x_axis_type='datetime', y_axis_type='linear', y_axis_label='Cumulative Return',\n",
    "             tools=\"pan,box_zoom,reset\",plot_width=900, plot_height=600)\n",
    "\n",
    "# Plot the extreme stocks\n",
    "for (symb,c) in zip(esyms,{'green','red','blue','purple'}):\n",
    "    fig.line(tx, growth[symb], legend=symb, color=c, line_width=2)\n",
    "    fig.text([tx[-1]], [growth[symb][-1]], [symb], text_font_size='10px', text_baseline='middle', alpha=1.0)\n",
    "    \n",
    "# Equal allocation\n",
    "eq_growth = growth.mean(axis=1)\n",
    "fig.line(tx, eq_growth, color='maroon', legend='equal allocation', line_width=2)\n",
    "\n",
    "# All the other stocks, and the text labels\n",
    "for symb in syms:\n",
    "    if symb not in esyms:\n",
    "        fig.line(tx, growth[symb], color='maroon', alpha=0.25, name=symb)\n",
    "        fig.text([tx[-1]], [growth[symb][-1]], [symb], text_font_size='10px', text_baseline='middle', alpha=0.25)\n",
    "\n",
    "# Expand the y-axis\n",
    "fig.y_range = Range1d(growth.values.min()/1.05,1.05*growth.values.max())\n",
    "fig.legend.location='bottom_left'\n",
    "show(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Mean-Variance Scatter Plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"c8e73164-23a5-4024-98a1-68ac9cf3f43c\"></div>\n",
       "</div>"
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     "metadata": {},
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   "source": [
    "fig = figure(title='Mean-Variance Plot', tools=\"pan,box_zoom,reset\",plot_width=700, plot_height=400)\n",
    "fig.left[0].formatter.use_scientific = False\n",
    "fig.circle(stats['Volatility'], stats['Mean_return'], size=5, color='maroon')\n",
    "fig.text(stats['Volatility'], stats['Mean_return'], syms, text_font_size='10px', x_offset=3, y_offset=-2)\n",
    "fig.xaxis.axis_label='Volatility (standard deviation)'\n",
    "fig.yaxis.axis_label='Mean Daily Return'\n",
    "show(fig)"
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   "source": [
    "### Optimize Weight Allocations Using Gurobi\n",
    "\n",
    "$$\\begin{array}{lll}\n",
    "\\text{minimize}   & x^T \\Sigma x \\\\\n",
    "\\text{subject to} & \\sum_i x_i = 1 & \\text{fixed budget} \\\\\n",
    "                  & r^T x = \\gamma & \\text{fixed return} \\\\\n",
    "                  & x \\geq 0\n",
    "\\end{array}$$\n",
    "\n",
    "$x\\in\\mathbb{R}^N$ is a vector representing the fraction of the budget allocated to each asset. The paramters of the model are the mean returns $r$, a *covariance matrix* $\\Sigma$, and the target return $\\gamma$. The optimal risk-return tradeoff is determined by sweeping $\\gamma$ between the worst and best returns seen above, and computing the portfolio that achieves the target return but with as little risk as possible."
   ]
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  {
   "cell_type": "code",
   "execution_count": 7,
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       "      <th>Symbols</th>\n",
       "      <th>XLB</th>\n",
       "      <th>XLE</th>\n",
       "      <th>XLF</th>\n",
       "      <th>XLI</th>\n",
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       "      <td>0.000102</td>\n",
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       "      <td>0.000090</td>\n",
       "      <td>0.000144</td>\n",
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       "      <td>0.000014</td>\n",
       "      <td>0.000082</td>\n",
       "      <td>0.000099</td>\n",
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       "      <td>0.000038</td>\n",
       "      <td>0.000032</td>\n",
       "      <td>0.000035</td>\n",
       "      <td>0.000040</td>\n",
       "      <td>0.000041</td>\n",
       "      <td>0.000056</td>\n",
       "      <td>0.000030</td>\n",
       "      <td>0.000041</td>\n",
       "      <td>0.000038</td>\n",
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       "      <td>0.000064</td>\n",
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       "      <td>0.000068</td>\n",
       "      <td>0.000071</td>\n",
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       "      <td>0.000099</td>\n",
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       "Symbols       XLB       XLE       XLF       XLI       XLK       XLP       XLU  \\\n",
       "Symbols                                                                         \n",
       "XLB      0.000102  0.000080  0.000075  0.000084  0.000087  0.000038  0.000010   \n",
       "XLE      0.000080  0.000139  0.000071  0.000075  0.000076  0.000032  0.000008   \n",
       "XLF      0.000075  0.000071  0.000110  0.000079  0.000083  0.000035  0.000002   \n",
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       "XLK      0.000087  0.000076  0.000083  0.000090  0.000144  0.000041  0.000014   \n",
       "XLP      0.000038  0.000032  0.000035  0.000040  0.000041  0.000056  0.000030   \n",
       "XLU      0.000010  0.000008  0.000002  0.000013  0.000014  0.000030  0.000065   \n",
       "XLV      0.000062  0.000057  0.000062  0.000066  0.000082  0.000041  0.000020   \n",
       "XLY      0.000073  0.000068  0.000071  0.000076  0.000099  0.000038  0.000011   \n",
       "\n",
       "Symbols       XLV       XLY  \n",
       "Symbols                      \n",
       "XLB      0.000062  0.000073  \n",
       "XLE      0.000057  0.000068  \n",
       "XLF      0.000062  0.000071  \n",
       "XLI      0.000066  0.000076  \n",
       "XLK      0.000082  0.000099  \n",
       "XLP      0.000041  0.000038  \n",
       "XLU      0.000020  0.000011  \n",
       "XLV      0.000081  0.000064  \n",
       "XLY      0.000064  0.000094  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sigma = returns.cov()\n",
    "Sigma"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Academic license - for non-commercial use only\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
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       "    }\n",
       "\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Variables</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Symbols</th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>XLB</th>\n",
       "      <td>&lt;gurobi.Var XLB&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLE</th>\n",
       "      <td>&lt;gurobi.Var XLE&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLF</th>\n",
       "      <td>&lt;gurobi.Var XLF&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLI</th>\n",
       "      <td>&lt;gurobi.Var XLI&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLK</th>\n",
       "      <td>&lt;gurobi.Var XLK&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLP</th>\n",
       "      <td>&lt;gurobi.Var XLP&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLU</th>\n",
       "      <td>&lt;gurobi.Var XLU&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLV</th>\n",
       "      <td>&lt;gurobi.Var XLV&gt;</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLY</th>\n",
       "      <td>&lt;gurobi.Var XLY&gt;</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                Variables\n",
       "Symbols                  \n",
       "XLB      <gurobi.Var XLB>\n",
       "XLE      <gurobi.Var XLE>\n",
       "XLF      <gurobi.Var XLF>\n",
       "XLI      <gurobi.Var XLI>\n",
       "XLK      <gurobi.Var XLK>\n",
       "XLP      <gurobi.Var XLP>\n",
       "XLU      <gurobi.Var XLU>\n",
       "XLV      <gurobi.Var XLV>\n",
       "XLY      <gurobi.Var XLY>"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Import the Gurobi python module\n",
    "from gurobipy import *\n",
    "\n",
    "# Instantiate the model\n",
    "m = Model(\"portfolio\")\n",
    "\n",
    "# Create one variable for each asset\n",
    "portvars = [m.addVar(name=symb,lb=0.0) for symb in syms]\n",
    "portvars = pd.Series(portvars, index=syms)\n",
    "portfolio = pd.DataFrame({'Variables':portvars})\n",
    "\n",
    "# Commit the changes to the model\n",
    "m.update()\n",
    "\n",
    "portfolio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Calculate portfolio return & risk"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# The total budget\n",
    "p_total = portvars.sum()\n",
    "\n",
    "# The mean return for the whole portfolio\n",
    "p_return = stats['Mean_return'].dot(portvars)\n",
    "\n",
    "# The (squared) volatility of the portfolio\n",
    "p_risk = Sigma.dot(portvars).dot(portvars) #first line of LP formulation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<gurobi.LinExpr: XLB + XLE + XLF + XLI + XLK + XLP + XLU + XLV + XLY>"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_total"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<gurobi.LinExpr: 0.00022465340564875458 XLB + 2.7122013629363072e-05 XLE + 0.0002519757050215596 XLF + 0.00038989823987543566 XLI + 0.0007178808445970365 XLK + 0.0001106424871688211 XLP + 0.00037627253813230627 XLU + 0.000487707072005415 XLV + 0.0005900809270551753 XLY>"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_return"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<gurobi.QuadExpr: 0.0 + [ 0.00010203412234287192 XLB ^ 2 + 7.969493735245652e-05 XLE * XLB + 7.509984584249656e-05 XLF * XLB + 8.42721697353346e-05 XLI * XLB + 8.66162394166535e-05 XLK * XLB + 3.8096906607490404e-05 XLP * XLB + 1.0021335751199686e-05 XLU * XLB + 6.15967816231019e-05 XLV * XLB + 7.343638575917526e-05 XLY * XLB + 7.969493735245652e-05 XLB * XLE + 0.00013889995028208322 XLE ^ 2 + 7.096116648884983e-05 XLF * XLE + 7.460376616064062e-05 XLI * XLE + 7.647466158829104e-05 XLK * XLE + 3.2135370178993486e-05 XLP * XLE + 7.948108680246474e-06 XLU * XLE + 5.747473956912288e-05 XLV * XLE + 6.817698311028997e-05 XLY * XLE + 7.509984584249656e-05 XLB * XLF + 7.096116648884983e-05 XLE * XLF + 0.00010986376447114481 XLF ^ 2 + 7.911582320112207e-05 XLI * XLF + 8.28690121725639e-05 XLK * XLF + 3.50988009457829e-05 XLP * XLF + 2.3701169886598113e-06 XLU * XLF + 6.191562459060961e-05 XLV * XLF + 7.109852269290956e-05 XLY * XLF + 8.42721697353346e-05 XLB * XLI + 7.460376616064062e-05 XLE * XLI + 7.911582320112207e-05 XLF * XLI + 9.673609003418382e-05 XLI ^ 2 + 9.032071909049454e-05 XLK * XLI + 4.0206564582655964e-05 XLP * XLI + 1.2882017388105205e-05 XLU * XLI + 6.62288865248725e-05 XLV * XLI + 7.568522744094274e-05 XLY * XLI + 8.66162394166535e-05 XLB * XLK + 7.647466158829104e-05 XLE * XLK + 8.28690121725639e-05 XLF * XLK + 9.032071909049454e-05 XLI * XLK + 0.00014393043128908513 XLK ^ 2 + 4.065242838473962e-05 XLP * XLK + 1.4357792935307841e-05 XLU * XLK + 8.191415545890499e-05 XLV * XLK + 9.87481165123039e-05 XLY * XLK + 3.8096906607490404e-05 XLB * XLP + 3.2135370178993486e-05 XLE * XLP + 3.50988009457829e-05 XLF * XLP + 4.0206564582655964e-05 XLI * XLP + 4.065242838473962e-05 XLK * XLP + 5.565431906356036e-05 XLP ^ 2 + 3.0086225760392127e-05 XLU * XLP + 4.135537760990348e-05 XLV * XLP + 3.772258239101758e-05 XLY * XLP + 1.0021335751199686e-05 XLB * XLU + 7.948108680246474e-06 XLE * XLU + 2.3701169886598113e-06 XLF * XLU + 1.2882017388105205e-05 XLI * XLU + 1.4357792935307841e-05 XLK * XLU + 3.0086225760392127e-05 XLP * XLU + 6.548463614173725e-05 XLU ^ 2 + 1.9520346536826984e-05 XLV * XLU + 1.061248696305249e-05 XLY * XLU + 6.15967816231019e-05 XLB * XLV + 5.747473956912288e-05 XLE * XLV + 6.191562459060961e-05 XLF * XLV + 6.62288865248725e-05 XLI * XLV + 8.191415545890499e-05 XLK * XLV + 4.135537760990348e-05 XLP * XLV + 1.9520346536826984e-05 XLU * XLV + 8.085009541896807e-05 XLV ^ 2 + 6.448054843403228e-05 XLY * XLV + 7.343638575917526e-05 XLB * XLY + 6.817698311028997e-05 XLE * XLY + 7.109852269290956e-05 XLF * XLY + 7.568522744094274e-05 XLI * XLY + 9.87481165123039e-05 XLK * XLY + 3.772258239101758e-05 XLP * XLY + 1.061248696305249e-05 XLU * XLY + 6.448054843403228e-05 XLV * XLY + 9.418283992562585e-05 XLY ^ 2 ]>"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_risk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Add objective to minimize risk; Add fixed budget constraint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Changed value of parameter Method to 1\n",
      "   Prev: -1  Min: -1  Max: 5  Default: -1\n"
     ]
    }
   ],
   "source": [
    "# Set the objective: minimize risk\n",
    "m.setObjective(p_risk, GRB.MINIMIZE)\n",
    "\n",
    "# Fix the budget\n",
    "m.addConstr(p_total, GRB.EQUAL, 1)\n",
    "\n",
    "# Select a simplex algorithm (to ensure a vertex solution)\n",
    "m.setParam('Method', 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Solve minimum risk model (no target return specified yet)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimize a model with 1 rows, 9 columns and 9 nonzeros\n",
      "Model has 45 quadratic objective terms\n",
      "Coefficient statistics:\n",
      "  Matrix range     [1e+00, 1e+00]\n",
      "  Objective range  [0e+00, 0e+00]\n",
      "  QObjective range [9e-06, 4e-04]\n",
      "  Bounds range     [0e+00, 0e+00]\n",
      "  RHS range        [1e+00, 1e+00]\n",
      "Presolve time: 0.01s\n",
      "Presolved: 1 rows, 9 columns, 9 nonzeros\n",
      "Presolved model has 45 quadratic objective terms\n",
      "\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0    0.0000000e+00   0.000000e+00   0.000000e+00      0s\n",
      "       8    3.8045026e-05   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 8 iterations and 0.01 seconds\n",
      "Optimal objective  3.804502599e-05\n"
     ]
    },
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Variables</th>\n",
       "      <th>Minimum risk</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Symbols</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>XLB</th>\n",
       "      <td>&lt;gurobi.Var XLB (value 0.016992525171768545)&gt;</td>\n",
       "      <td>0.016993</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLE</th>\n",
       "      <td>&lt;gurobi.Var XLE (value 0.06444679427624088)&gt;</td>\n",
       "      <td>0.064447</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLF</th>\n",
       "      <td>&lt;gurobi.Var XLF (value 0.1525880775584346)&gt;</td>\n",
       "      <td>0.152588</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLI</th>\n",
       "      <td>&lt;gurobi.Var XLI (value 0.0)&gt;</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLK</th>\n",
       "      <td>&lt;gurobi.Var XLK (value 0.0)&gt;</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLP</th>\n",
       "      <td>&lt;gurobi.Var XLP (value 0.24680602143341732)&gt;</td>\n",
       "      <td>0.246806</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLU</th>\n",
       "      <td>&lt;gurobi.Var XLU (value 0.4385785242246754)&gt;</td>\n",
       "      <td>0.438579</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLV</th>\n",
       "      <td>&lt;gurobi.Var XLV (value 0.0)&gt;</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLY</th>\n",
       "      <td>&lt;gurobi.Var XLY (value 0.08058805733546331)&gt;</td>\n",
       "      <td>0.080588</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                             Variables  Minimum risk\n",
       "Symbols                                                             \n",
       "XLB      <gurobi.Var XLB (value 0.016992525171768545)>      0.016993\n",
       "XLE       <gurobi.Var XLE (value 0.06444679427624088)>      0.064447\n",
       "XLF        <gurobi.Var XLF (value 0.1525880775584346)>      0.152588\n",
       "XLI                       <gurobi.Var XLI (value 0.0)>      0.000000\n",
       "XLK                       <gurobi.Var XLK (value 0.0)>      0.000000\n",
       "XLP       <gurobi.Var XLP (value 0.24680602143341732)>      0.246806\n",
       "XLU        <gurobi.Var XLU (value 0.4385785242246754)>      0.438579\n",
       "XLV                       <gurobi.Var XLV (value 0.0)>      0.000000\n",
       "XLY       <gurobi.Var XLY (value 0.08058805733546331)>      0.080588"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m.optimize()\n",
    "\n",
    "portfolio['Minimum risk'] = portvars.apply(lambda x:x.getAttr('x'))\n",
    "portfolio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 50% Max return portfolio (find the weights that result in a return that is 50% of the max possible - 100% allocation in top performer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Optimize a model with 2 rows, 9 columns and 18 nonzeros\n",
      "Model has 45 quadratic objective terms\n",
      "Coefficient statistics:\n",
      "  Matrix range     [3e-05, 1e+00]\n",
      "  Objective range  [0e+00, 0e+00]\n",
      "  QObjective range [9e-06, 4e-04]\n",
      "  Bounds range     [0e+00, 0e+00]\n",
      "  RHS range        [4e-04, 1e+00]\n",
      "Iteration    Objective       Primal Inf.    Dual Inf.      Time\n",
      "       0    3.8045026e-05   0.000000e+00   0.000000e+00      0s\n",
      "       4    3.8791037e-05   0.000000e+00   0.000000e+00      0s\n",
      "\n",
      "Solved in 4 iterations and 0.01 seconds\n",
      "Optimal objective  3.879103733e-05\n"
     ]
    },
    {
     "data": {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Variables</th>\n",
       "      <th>Minimum risk</th>\n",
       "      <th>50% Max</th>\n",
       "      <th>Maximum</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Symbols</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>XLB</th>\n",
       "      <td>&lt;gurobi.Var XLB (value 0.0)&gt;</td>\n",
       "      <td>0.016993</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLE</th>\n",
       "      <td>&lt;gurobi.Var XLE (value 0.020234015473259148)&gt;</td>\n",
       "      <td>0.064447</td>\n",
       "      <td>0.020234</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLF</th>\n",
       "      <td>&lt;gurobi.Var XLF (value 0.13152335096907236)&gt;</td>\n",
       "      <td>0.152588</td>\n",
       "      <td>0.131523</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLI</th>\n",
       "      <td>&lt;gurobi.Var XLI (value 0.0)&gt;</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLK</th>\n",
       "      <td>&lt;gurobi.Var XLK (value 0.0)&gt;</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLP</th>\n",
       "      <td>&lt;gurobi.Var XLP (value 0.1408217202904382)&gt;</td>\n",
       "      <td>0.246806</td>\n",
       "      <td>0.140822</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLU</th>\n",
       "      <td>&lt;gurobi.Var XLU (value 0.4789198346441817)&gt;</td>\n",
       "      <td>0.438579</td>\n",
       "      <td>0.478920</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLV</th>\n",
       "      <td>&lt;gurobi.Var XLV (value 0.05243945790981197)&gt;</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.052439</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>XLY</th>\n",
       "      <td>&lt;gurobi.Var XLY (value 0.17606162071323658)&gt;</td>\n",
       "      <td>0.080588</td>\n",
       "      <td>0.176062</td>\n",
       "      <td>0.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                                             Variables  Minimum risk  \\\n",
       "Symbols                                                                \n",
       "XLB                       <gurobi.Var XLB (value 0.0)>      0.016993   \n",
       "XLE      <gurobi.Var XLE (value 0.020234015473259148)>      0.064447   \n",
       "XLF       <gurobi.Var XLF (value 0.13152335096907236)>      0.152588   \n",
       "XLI                       <gurobi.Var XLI (value 0.0)>      0.000000   \n",
       "XLK                       <gurobi.Var XLK (value 0.0)>      0.000000   \n",
       "XLP        <gurobi.Var XLP (value 0.1408217202904382)>      0.246806   \n",
       "XLU        <gurobi.Var XLU (value 0.4789198346441817)>      0.438579   \n",
       "XLV       <gurobi.Var XLV (value 0.05243945790981197)>      0.000000   \n",
       "XLY       <gurobi.Var XLY (value 0.17606162071323658)>      0.080588   \n",
       "\n",
       "          50% Max  Maximum  \n",
       "Symbols                     \n",
       "XLB      0.000000      0.0  \n",
       "XLE      0.020234      0.0  \n",
       "XLF      0.131523      0.0  \n",
       "XLI      0.000000      0.0  \n",
       "XLK      0.000000      0.0  \n",
       "XLP      0.140822      0.0  \n",
       "XLU      0.478920      1.0  \n",
       "XLV      0.052439      0.0  \n",
       "XLY      0.176062      0.0  "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Add the return target\n",
    "ret50 = 0.5 * extremes.loc['Mean_return','Maximum']\n",
    "fixreturn = m.addConstr(p_return, GRB.EQUAL, ret50)\n",
    "\n",
    "m.optimize()\n",
    "\n",
    "portfolio['50% Max'] = portvars.apply(lambda x:x.getAttr('x'))\n",
    "portfolio['Maximum'] = [1.0 if x == 'XLU' else 0.0 for x in syms]\n",
    "portfolio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot Returns of Different Portfolios"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"a1e888f7-ef80-4809-8631-da13c84e2365\"></div>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "    \n",
       "  var docs_json = 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risk\"},\"renderers\":[{\"id\":\"7507ddfd-ca2c-4760-89c2-81d94d951936\",\"type\":\"GlyphRenderer\"}]},\"id\":\"64feeb4e-677a-4337-8efb-5a61759105e7\",\"type\":\"LegendItem\"},{\"attributes\":{},\"id\":\"9473c283-ff0b-4f22-9625-5874c1b3b170\",\"type\":\"DatetimeTickFormatter\"},{\"attributes\":{\"base\":24,\"mantissas\":[1,2,4,6,8,12],\"max_interval\":43200000.0,\"min_interval\":3600000.0,\"num_minor_ticks\":0},\"id\":\"c1264c29-1a1c-467f-a4d0-fa87f146562c\",\"type\":\"AdaptiveTicker\"},{\"attributes\":{\"mantissas\":[1,2,5],\"max_interval\":500.0,\"num_minor_ticks\":0},\"id\":\"c6106282-7f95-49df-b2b5-b3d3af1e6ab0\",\"type\":\"AdaptiveTicker\"},{\"attributes\":{\"days\":[1,4,7,10,13,16,19,22,25,28]},\"id\":\"6645e649-d4d6-43f6-8a40-8e9522b6b2ee\",\"type\":\"DaysTicker\"},{\"attributes\":{\"days\":[1,8,15,22]},\"id\":\"fe0dcdcc-04a0-49b0-a22f-7bd4fc5fa4fd\",\"type\":\"DaysTicker\"},{\"attributes\":{\"days\":[1,15]},\"id\":\"da61baab-6c99-41a2-acf8-db44c2acd3bf\",\"type\":\"DaysTicker\"},{\"attributes\":{\"months\":[0,1,2,3,4,5,6,7,8,9,10,11]},\"id\":\"9f2ac66f-81ab-4a05-b96f-2af1263ddfcf\",\"type\":\"MonthsTicker\"}],\"root_ids\":[\"3fe9f9dc-57c0-4ec3-8e74-65153f342e2d\"]},\"title\":\"Bokeh Application\",\"version\":\"0.12.16\"}};\n",
       "  var render_items = [{\"docid\":\"3d783fbf-769e-4475-8452-0a72d14d03d2\",\"elementid\":\"a1e888f7-ef80-4809-8631-da13c84e2365\",\"modelid\":\"3fe9f9dc-57c0-4ec3-8e74-65153f342e2d\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "3fe9f9dc-57c0-4ec3-8e74-65153f342e2d"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = figure(x_axis_type='datetime',y_axis_type='log',tools=\"pan,box_zoom,reset\",plot_width=900, plot_height=600)\n",
    "\n",
    "pgrowth = returns.dot(portfolio.loc[:,'Minimum risk':'Maximum'])\n",
    "pgrowth = (pgrowth + 1).cumprod()\n",
    "\n",
    "for (t,c) in zip(['Minimum risk','50% Max','Maximum'],['red','green','blue']):\n",
    "    fig.line(pgrowth.index, pgrowth[t], legend=t, color=c)\n",
    "\n",
    "# fig.y_range = Range1d(1.0,1.1*pgrowth.max().max())\n",
    "fig.legend.location=\"top_left\"\n",
    "show(fig)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Plot weight allocations for different portfolios"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 936x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.rcParams[\"figure.figsize\"] = (13,8)\n",
    "\n",
    "plt.bar(x=syms, height=portfolio['Minimum risk'])\n",
    "plt.xlabel('ETF')\n",
    "plt.ylabel('Asset Weight Allocation')\n",
    "plt.title('Weight Allocations for Minimum Risk Portfolio');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 936x576 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.bar(x=syms, height=portfolio['50% Max'])\n",
    "plt.xlabel('ETF')\n",
    "plt.ylabel('Asset Weight Allocation')\n",
    "plt.title('Weight Allocations for 50% Max Return Risk Portfolio');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The efficient frontier"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "from math import sqrt\n",
    "# Turn off Gurobi's output\n",
    "m.setParam('OutputFlag',False)\n",
    "\n",
    "minret = extremes.loc['Mean_return','Minimum']\n",
    "maxret = extremes.loc['Mean_return','Maximum']\n",
    "riskret = extremes.loc['Volatility','Minimizer']\n",
    "riskret = stats.loc[riskret,'Mean_return']\n",
    "returns = np.unique(np.hstack((np.linspace(minret,maxret,100),riskret)))\n",
    "\n",
    "# Iterate through all returns\n",
    "risks = returns.copy()\n",
    "for k in range(len(returns)):\n",
    "    fixreturn.rhs = returns[k]\n",
    "    m.optimize()\n",
    "    risks[k] = sqrt(p_risk.getValue())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div class=\"bk-root\">\n",
       "    <div class=\"bk-plotdiv\" id=\"7132db01-8722-4b2f-9e43-51a7a2b50307\"></div>\n",
       "</div>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
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       "  var render_items = [{\"docid\":\"1a997ab8-8fbc-43dc-bac6-05bf906fdd89\",\"elementid\":\"7132db01-8722-4b2f-9e43-51a7a2b50307\",\"modelid\":\"dee7a225-691c-4bef-97ac-09fe9a654cd1\"}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    var attempts = 0;\n",
       "    var timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        embed_document(root);\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "      attempts++;\n",
       "      if (attempts > 100) {\n",
       "        console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\")\n",
       "        clearInterval(timer);\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
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     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = figure(title='Efficient Frontier', tools=\"pan,box_zoom,reset\",plot_width=900, plot_height=600)\n",
    "fig.left[0].formatter.use_scientific = False\n",
    "\n",
    "# Individual stocks\n",
    "fig.circle(stats['Volatility'], stats['Mean_return'], size=5, color='maroon')\n",
    "fig.text(stats['Volatility'], stats['Mean_return'], syms, text_font_size='10px', x_offset=3, y_offset=-2)\n",
    "\n",
    "# Divide the efficient frontier into two sections: those with\n",
    "# a return less than the minimum risk portfolio, those that are greater.\n",
    "tpos_n = returns >= riskret\n",
    "tneg_n = returns <= riskret\n",
    "fig.line(risks[tneg_n], returns[tneg_n], color='red')\n",
    "fig.line(risks[tpos_n], returns[tpos_n], color='blue')\n",
    "\n",
    "fig.xaxis.axis_label='Volatility (standard deviation)'\n",
    "fig.yaxis.axis_label='Mean return'\n",
    "fig.legend.orientation='bottom_left'\n",
    "show(fig)"
   ]
  }
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