16c_rebalancing.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rebalancing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import plotly.graph_objects as go\n",
    "from scipy.optimize import minimize\n",
    "from cvxopt import matrix\n",
    "from cvxopt.solvers import qp as Solver, options as SolverOptions\n",
    "from scipy.stats import multivariate_normal as mvn\n",
    "\n",
    "pd.options.display.float_format = '{:,.4f}'.format"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Two asset simulation with no rebalancing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Single time-series of returns\n",
    "def returns(mn1, mn2, sd1, sd2, c, rf, init_wt1, init_wt2, T):\n",
    "    # Organize inputs\n",
    "    mns = [mn1, mn2]\n",
    "    sds = [sd1, sd2]   \n",
    "    cov = np.array([[sds[0] ** 2, sds[0] * sds[1] * c], [sds[0] * sds[1] * c, sds[1] ** 2]]).reshape(2, 2)\n",
    "\n",
    "    # Generate random sample and do calculations\n",
    "    col_names = ['beg_wgt1','beg_wgt2','ret1','ret2', 'end_wgt1','end_wgt2','realized_pret','exp_pret','sd_pret','sharpe ratio']\n",
    "    df = pd.DataFrame(dtype=float, columns=col_names, index=np.arange(T)+1)\n",
    "    df[['ret1','ret2']]  = mvn.rvs(mns, cov, size=T)\n",
    "    df.loc[1,'beg_wgt1'] = init_wt1\n",
    "    df.loc[1,'beg_wgt2'] = init_wt2\n",
    "    for t in df.index:\n",
    "        if t > 1:\n",
    "            df.loc[t,'beg_wgt1'] = df.loc[t-1,'end_wgt1']\n",
    "            df.loc[t,'beg_wgt2'] = df.loc[t-1,'end_wgt2']\n",
    "        # Calculate realized portfolio return\n",
    "        w1 = df.loc[t,'beg_wgt1']\n",
    "        w2 = df.loc[t,'beg_wgt2']\n",
    "        r1 = df.loc[t,'ret1']\n",
    "        r2 = df.loc[t,'ret2']\n",
    "        rp = w1*r1 + w2*r2\n",
    "        df.loc[t,'realized_pret']=rp    \n",
    "        # Update weights\n",
    "        df.loc[t,'end_wgt1']  = w1 * (1+r1)/(1+rp)\n",
    "        df.loc[t,'end_wgt2']  = w2 * (1+r2)/(1+rp)\n",
    "    df['exp_pret'] = df.beg_wgt1*mns[0] + df.beg_wgt2*mns[1]\n",
    "    df['sd_pret']  = np.sqrt(df.beg_wgt1**2 * sds[0]**2  + df.beg_wgt2**2 * sds[1]**2 + 2*df.beg_wgt1*df.beg_wgt2*c*sds[0]*sds[1])        \n",
    "    df['sharpe ratio'] = (df.exp_pret - rf) / df.sd_pret\n",
    "    return df"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at a single realization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        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>beg_wgt1</th>\n",
       "      <th>beg_wgt2</th>\n",
       "      <th>ret1</th>\n",
       "      <th>ret2</th>\n",
       "      <th>end_wgt1</th>\n",
       "      <th>end_wgt2</th>\n",
       "      <th>realized_pret</th>\n",
       "      <th>exp_pret</th>\n",
       "      <th>sd_pret</th>\n",
       "      <th>sharpe ratio</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.6000</td>\n",
       "      <td>0.4000</td>\n",
       "      <td>0.0572</td>\n",
       "      <td>0.0216</td>\n",
       "      <td>0.6082</td>\n",
       "      <td>0.3918</td>\n",
       "      <td>0.0430</td>\n",
       "      <td>0.0500</td>\n",
       "      <td>0.0904</td>\n",
       "      <td>0.1730</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.6082</td>\n",
       "      <td>0.3918</td>\n",
       "      <td>-0.0014</td>\n",
       "      <td>0.0741</td>\n",
       "      <td>0.5907</td>\n",
       "      <td>0.4093</td>\n",
       "      <td>0.0282</td>\n",
       "      <td>0.0502</td>\n",
       "      <td>0.0916</td>\n",
       "      <td>0.1730</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.5907</td>\n",
       "      <td>0.4093</td>\n",
       "      <td>0.0845</td>\n",
       "      <td>0.0757</td>\n",
       "      <td>0.5927</td>\n",
       "      <td>0.4073</td>\n",
       "      <td>0.0809</td>\n",
       "      <td>0.0498</td>\n",
       "      <td>0.0890</td>\n",
       "      <td>0.1730</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.5927</td>\n",
       "      <td>0.4073</td>\n",
       "      <td>0.2056</td>\n",
       "      <td>-0.0136</td>\n",
       "      <td>0.6400</td>\n",
       "      <td>0.3600</td>\n",
       "      <td>0.1163</td>\n",
       "      <td>0.0498</td>\n",
       "      <td>0.0893</td>\n",
       "      <td>0.1730</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.6400</td>\n",
       "      <td>0.3600</td>\n",
       "      <td>-0.0868</td>\n",
       "      <td>-0.0178</td>\n",
       "      <td>0.6231</td>\n",
       "      <td>0.3769</td>\n",
       "      <td>-0.0620</td>\n",
       "      <td>0.0510</td>\n",
       "      <td>0.0962</td>\n",
       "      <td>0.1729</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>6</th>\n",
       "      <td>0.6231</td>\n",
       "      <td>0.3769</td>\n",
       "      <td>0.0277</td>\n",
       "      <td>-0.0110</td>\n",
       "      <td>0.6321</td>\n",
       "      <td>0.3679</td>\n",
       "      <td>0.0131</td>\n",
       "      <td>0.0506</td>\n",
       "      <td>0.0937</td>\n",
       "      <td>0.1730</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>7</th>\n",
       "      <td>0.6321</td>\n",
       "      <td>0.3679</td>\n",
       "      <td>0.0721</td>\n",
       "      <td>0.0094</td>\n",
       "      <td>0.6460</td>\n",
       "      <td>0.3540</td>\n",
       "      <td>0.0490</td>\n",
       "      <td>0.0508</td>\n",
       "      <td>0.0950</td>\n",
       "      <td>0.1730</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>8</th>\n",
       "      <td>0.6460</td>\n",
       "      <td>0.3540</td>\n",
       "      <td>0.4061</td>\n",
       "      <td>0.0170</td>\n",
       "      <td>0.7162</td>\n",
       "      <td>0.2838</td>\n",
       "      <td>0.2684</td>\n",
       "      <td>0.0512</td>\n",
       "      <td>0.0971</td>\n",
       "      <td>0.1729</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>9</th>\n",
       "      <td>0.7162</td>\n",
       "      <td>0.2838</td>\n",
       "      <td>0.1930</td>\n",
       "      <td>0.0861</td>\n",
       "      <td>0.7349</td>\n",
       "      <td>0.2651</td>\n",
       "      <td>0.1627</td>\n",
       "      <td>0.0529</td>\n",
       "      <td>0.1074</td>\n",
       "      <td>0.1726</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>10</th>\n",
       "      <td>0.7349</td>\n",
       "      <td>0.2651</td>\n",
       "      <td>0.1066</td>\n",
       "      <td>0.0327</td>\n",
       "      <td>0.7481</td>\n",
       "      <td>0.2519</td>\n",
       "      <td>0.0870</td>\n",
       "      <td>0.0534</td>\n",
       "      <td>0.1102</td>\n",
       "      <td>0.1726</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>11</th>\n",
       "      <td>0.7481</td>\n",
       "      <td>0.2519</td>\n",
       "      <td>0.0746</td>\n",
       "      <td>0.0587</td>\n",
       "      <td>0.7509</td>\n",
       "      <td>0.2491</td>\n",
       "      <td>0.0706</td>\n",
       "      <td>0.0537</td>\n",
       "      <td>0.1121</td>\n",
       "      <td>0.1725</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>12</th>\n",
       "      <td>0.7509</td>\n",
       "      <td>0.2491</td>\n",
       "      <td>0.2293</td>\n",
       "      <td>0.0261</td>\n",
       "      <td>0.7832</td>\n",
       "      <td>0.2168</td>\n",
       "      <td>0.1787</td>\n",
       "      <td>0.0538</td>\n",
       "      <td>0.1125</td>\n",
       "      <td>0.1725</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>13</th>\n",
       "      <td>0.7832</td>\n",
       "      <td>0.2168</td>\n",
       "      <td>0.2579</td>\n",
       "      <td>0.0943</td>\n",
       "      <td>0.8059</td>\n",
       "      <td>0.1941</td>\n",
       "      <td>0.2224</td>\n",
       "      <td>0.0546</td>\n",
       "      <td>0.1173</td>\n",
       "      <td>0.1723</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>14</th>\n",
       "      <td>0.8059</td>\n",
       "      <td>0.1941</td>\n",
       "      <td>0.1666</td>\n",
       "      <td>0.0033</td>\n",
       "      <td>0.8284</td>\n",
       "      <td>0.1716</td>\n",
       "      <td>0.1349</td>\n",
       "      <td>0.0551</td>\n",
       "      <td>0.1207</td>\n",
       "      <td>0.1721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>15</th>\n",
       "      <td>0.8284</td>\n",
       "      <td>0.1716</td>\n",
       "      <td>0.1252</td>\n",
       "      <td>0.0247</td>\n",
       "      <td>0.8413</td>\n",
       "      <td>0.1587</td>\n",
       "      <td>0.1079</td>\n",
       "      <td>0.0557</td>\n",
       "      <td>0.1241</td>\n",
       "      <td>0.1720</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>16</th>\n",
       "      <td>0.8413</td>\n",
       "      <td>0.1587</td>\n",
       "      <td>0.1917</td>\n",
       "      <td>0.0068</td>\n",
       "      <td>0.8625</td>\n",
       "      <td>0.1375</td>\n",
       "      <td>0.1624</td>\n",
       "      <td>0.0560</td>\n",
       "      <td>0.1260</td>\n",
       "      <td>0.1719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>17</th>\n",
       "      <td>0.8625</td>\n",
       "      <td>0.1375</td>\n",
       "      <td>-0.0353</td>\n",
       "      <td>0.0657</td>\n",
       "      <td>0.8503</td>\n",
       "      <td>0.1497</td>\n",
       "      <td>-0.0214</td>\n",
       "      <td>0.0566</td>\n",
       "      <td>0.1292</td>\n",
       "      <td>0.1718</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>18</th>\n",
       "      <td>0.8503</td>\n",
       "      <td>0.1497</td>\n",
       "      <td>-0.0782</td>\n",
       "      <td>0.0793</td>\n",
       "      <td>0.8291</td>\n",
       "      <td>0.1709</td>\n",
       "      <td>-0.0546</td>\n",
       "      <td>0.0563</td>\n",
       "      <td>0.1274</td>\n",
       "      <td>0.1719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>19</th>\n",
       "      <td>0.8291</td>\n",
       "      <td>0.1709</td>\n",
       "      <td>0.1650</td>\n",
       "      <td>0.0177</td>\n",
       "      <td>0.8474</td>\n",
       "      <td>0.1526</td>\n",
       "      <td>0.1398</td>\n",
       "      <td>0.0557</td>\n",
       "      <td>0.1242</td>\n",
       "      <td>0.1720</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>20</th>\n",
       "      <td>0.8474</td>\n",
       "      <td>0.1526</td>\n",
       "      <td>-0.0393</td>\n",
       "      <td>0.0484</td>\n",
       "      <td>0.8357</td>\n",
       "      <td>0.1643</td>\n",
       "      <td>-0.0259</td>\n",
       "      <td>0.0562</td>\n",
       "      <td>0.1270</td>\n",
       "      <td>0.1719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>21</th>\n",
       "      <td>0.8357</td>\n",
       "      <td>0.1643</td>\n",
       "      <td>0.0428</td>\n",
       "      <td>0.0372</td>\n",
       "      <td>0.8365</td>\n",
       "      <td>0.1635</td>\n",
       "      <td>0.0418</td>\n",
       "      <td>0.0559</td>\n",
       "      <td>0.1252</td>\n",
       "      <td>0.1720</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>22</th>\n",
       "      <td>0.8365</td>\n",
       "      <td>0.1635</td>\n",
       "      <td>-0.1190</td>\n",
       "      <td>0.0136</td>\n",
       "      <td>0.8164</td>\n",
       "      <td>0.1836</td>\n",
       "      <td>-0.0973</td>\n",
       "      <td>0.0559</td>\n",
       "      <td>0.1253</td>\n",
       "      <td>0.1719</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>23</th>\n",
       "      <td>0.8164</td>\n",
       "      <td>0.1836</td>\n",
       "      <td>0.0926</td>\n",
       "      <td>0.0712</td>\n",
       "      <td>0.8193</td>\n",
       "      <td>0.1807</td>\n",
       "      <td>0.0887</td>\n",
       "      <td>0.0554</td>\n",
       "      <td>0.1223</td>\n",
       "      <td>0.1721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>24</th>\n",
       "      <td>0.8193</td>\n",
       "      <td>0.1807</td>\n",
       "      <td>-0.1082</td>\n",
       "      <td>0.0844</td>\n",
       "      <td>0.7885</td>\n",
       "      <td>0.2115</td>\n",
       "      <td>-0.0734</td>\n",
       "      <td>0.0555</td>\n",
       "      <td>0.1227</td>\n",
       "      <td>0.1721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>0.7885</td>\n",
       "      <td>0.2115</td>\n",
       "      <td>0.1093</td>\n",
       "      <td>-0.0005</td>\n",
       "      <td>0.8054</td>\n",
       "      <td>0.1946</td>\n",
       "      <td>0.0861</td>\n",
       "      <td>0.0547</td>\n",
       "      <td>0.1181</td>\n",
       "      <td>0.1722</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>26</th>\n",
       "      <td>0.8054</td>\n",
       "      <td>0.1946</td>\n",
       "      <td>0.1671</td>\n",
       "      <td>0.0229</td>\n",
       "      <td>0.8252</td>\n",
       "      <td>0.1748</td>\n",
       "      <td>0.1390</td>\n",
       "      <td>0.0551</td>\n",
       "      <td>0.1207</td>\n",
       "      <td>0.1721</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>27</th>\n",
       "      <td>0.8252</td>\n",
       "      <td>0.1748</td>\n",
       "      <td>0.0159</td>\n",
       "      <td>-0.0208</td>\n",
       "      <td>0.8305</td>\n",
       "      <td>0.1695</td>\n",
       "      <td>0.0095</td>\n",
       "      <td>0.0556</td>\n",
       "      <td>0.1236</td>\n",
       "      <td>0.1720</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>28</th>\n",
       "      <td>0.8305</td>\n",
       "      <td>0.1695</td>\n",
       "      <td>0.3783</td>\n",
       "      <td>0.0232</td>\n",
       "      <td>0.8684</td>\n",
       "      <td>0.1316</td>\n",
       "      <td>0.3181</td>\n",
       "      <td>0.0558</td>\n",
       "      <td>0.1244</td>\n",
       "      <td>0.1720</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>29</th>\n",
       "      <td>0.8684</td>\n",
       "      <td>0.1316</td>\n",
       "      <td>0.0575</td>\n",
       "      <td>0.0271</td>\n",
       "      <td>0.8717</td>\n",
       "      <td>0.1283</td>\n",
       "      <td>0.0535</td>\n",
       "      <td>0.0567</td>\n",
       "      <td>0.1301</td>\n",
       "      <td>0.1717</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>30</th>\n",
       "      <td>0.8717</td>\n",
       "      <td>0.1283</td>\n",
       "      <td>0.2223</td>\n",
       "      <td>-0.0009</td>\n",
       "      <td>0.8926</td>\n",
       "      <td>0.1074</td>\n",
       "      <td>0.1936</td>\n",
       "      <td>0.0568</td>\n",
       "      <td>0.1306</td>\n",
       "      <td>0.1717</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    beg_wgt1  beg_wgt2    ret1    ret2  end_wgt1  end_wgt2  realized_pret  \\\n",
       "1     0.6000    0.4000  0.0572  0.0216    0.6082    0.3918         0.0430   \n",
       "2     0.6082    0.3918 -0.0014  0.0741    0.5907    0.4093         0.0282   \n",
       "3     0.5907    0.4093  0.0845  0.0757    0.5927    0.4073         0.0809   \n",
       "4     0.5927    0.4073  0.2056 -0.0136    0.6400    0.3600         0.1163   \n",
       "5     0.6400    0.3600 -0.0868 -0.0178    0.6231    0.3769        -0.0620   \n",
       "6     0.6231    0.3769  0.0277 -0.0110    0.6321    0.3679         0.0131   \n",
       "7     0.6321    0.3679  0.0721  0.0094    0.6460    0.3540         0.0490   \n",
       "8     0.6460    0.3540  0.4061  0.0170    0.7162    0.2838         0.2684   \n",
       "9     0.7162    0.2838  0.1930  0.0861    0.7349    0.2651         0.1627   \n",
       "10    0.7349    0.2651  0.1066  0.0327    0.7481    0.2519         0.0870   \n",
       "11    0.7481    0.2519  0.0746  0.0587    0.7509    0.2491         0.0706   \n",
       "12    0.7509    0.2491  0.2293  0.0261    0.7832    0.2168         0.1787   \n",
       "13    0.7832    0.2168  0.2579  0.0943    0.8059    0.1941         0.2224   \n",
       "14    0.8059    0.1941  0.1666  0.0033    0.8284    0.1716         0.1349   \n",
       "15    0.8284    0.1716  0.1252  0.0247    0.8413    0.1587         0.1079   \n",
       "16    0.8413    0.1587  0.1917  0.0068    0.8625    0.1375         0.1624   \n",
       "17    0.8625    0.1375 -0.0353  0.0657    0.8503    0.1497        -0.0214   \n",
       "18    0.8503    0.1497 -0.0782  0.0793    0.8291    0.1709        -0.0546   \n",
       "19    0.8291    0.1709  0.1650  0.0177    0.8474    0.1526         0.1398   \n",
       "20    0.8474    0.1526 -0.0393  0.0484    0.8357    0.1643        -0.0259   \n",
       "21    0.8357    0.1643  0.0428  0.0372    0.8365    0.1635         0.0418   \n",
       "22    0.8365    0.1635 -0.1190  0.0136    0.8164    0.1836        -0.0973   \n",
       "23    0.8164    0.1836  0.0926  0.0712    0.8193    0.1807         0.0887   \n",
       "24    0.8193    0.1807 -0.1082  0.0844    0.7885    0.2115        -0.0734   \n",
       "25    0.7885    0.2115  0.1093 -0.0005    0.8054    0.1946         0.0861   \n",
       "26    0.8054    0.1946  0.1671  0.0229    0.8252    0.1748         0.1390   \n",
       "27    0.8252    0.1748  0.0159 -0.0208    0.8305    0.1695         0.0095   \n",
       "28    0.8305    0.1695  0.3783  0.0232    0.8684    0.1316         0.3181   \n",
       "29    0.8684    0.1316  0.0575  0.0271    0.8717    0.1283         0.0535   \n",
       "30    0.8717    0.1283  0.2223 -0.0009    0.8926    0.1074         0.1936   \n",
       "\n",
       "    exp_pret  sd_pret  sharpe ratio  \n",
       "1     0.0500   0.0904        0.1730  \n",
       "2     0.0502   0.0916        0.1730  \n",
       "3     0.0498   0.0890        0.1730  \n",
       "4     0.0498   0.0893        0.1730  \n",
       "5     0.0510   0.0962        0.1729  \n",
       "6     0.0506   0.0937        0.1730  \n",
       "7     0.0508   0.0950        0.1730  \n",
       "8     0.0512   0.0971        0.1729  \n",
       "9     0.0529   0.1074        0.1726  \n",
       "10    0.0534   0.1102        0.1726  \n",
       "11    0.0537   0.1121        0.1725  \n",
       "12    0.0538   0.1125        0.1725  \n",
       "13    0.0546   0.1173        0.1723  \n",
       "14    0.0551   0.1207        0.1721  \n",
       "15    0.0557   0.1241        0.1720  \n",
       "16    0.0560   0.1260        0.1719  \n",
       "17    0.0566   0.1292        0.1718  \n",
       "18    0.0563   0.1274        0.1719  \n",
       "19    0.0557   0.1242        0.1720  \n",
       "20    0.0562   0.1270        0.1719  \n",
       "21    0.0559   0.1252        0.1720  \n",
       "22    0.0559   0.1253        0.1719  \n",
       "23    0.0554   0.1223        0.1721  \n",
       "24    0.0555   0.1227        0.1721  \n",
       "25    0.0547   0.1181        0.1722  \n",
       "26    0.0551   0.1207        0.1721  \n",
       "27    0.0556   0.1236        0.1720  \n",
       "28    0.0558   0.1244        0.1720  \n",
       "29    0.0567   0.1301        0.1717  \n",
       "30    0.0568   0.1306        0.1717  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Inputs\n",
    "MN_STOCK = 6  / 100\n",
    "MN_BOND  = 3.5/ 100\n",
    "SD_STOCK = 15 / 100\n",
    "SD_BOND  = 3.5/ 100\n",
    "CORR = -5 / 100\n",
    "RF   = 3.4364 / 100\n",
    "INIT_WGT_STOCK = 0.6\n",
    "INIT_WGT_BOND  = 0.4\n",
    "T = 30\n",
    "\n",
    "# A single realization\n",
    "rets = returns(MN_STOCK, MN_BOND, SD_STOCK, SD_BOND, CORR, RF, INIT_WGT_STOCK, INIT_WGT_BOND, T)\n",
    "rets"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's simulate 1000 realizations of 30 years of investing and collect the realized sharpe ratios\n",
    "\n",
    "- We need to update our function to return the realized Sharpe ratio (Note: Sharpe ratio column above is forward-looking SR)\n",
    "- We need to collect the results of the 1000 simulations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.47477997863223304"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Realized SR for a single realization\n",
    "(rets.realized_pret.mean()-RF)/rets.realized_pret.std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Simulate a single realization and do calculation(s)\n",
    "def sim_calc(mn1, mn2, sd1, sd2, c, rf, init_wt1, init_wt2, T):\n",
    "    # Organize inputs\n",
    "    mns = [mn1, mn2]\n",
    "    sds = [sd1, sd2]   \n",
    "    cov = np.array([[sds[0] ** 2, sds[0] * sds[1] * c], [sds[0] * sds[1] * c, sds[1] ** 2]]).reshape(2, 2)\n",
    "\n",
    "    # Generate random sample and do calculations\n",
    "    col_names = ['beg_wgt1','beg_wgt2','ret1','ret2', 'end_wgt1','end_wgt2','realized_pret','exp_pret','sd_pret','sharpe ratio']\n",
    "    df = pd.DataFrame(dtype=float, columns=col_names, index=np.arange(T)+1)\n",
    "    df[['ret1','ret2']]  = mvn.rvs(mns, cov, size=T)\n",
    "    df.loc[1,'beg_wgt1'] = init_wt1\n",
    "    df.loc[1,'beg_wgt2'] = init_wt2\n",
    "    for t in df.index:\n",
    "        if t > 1:\n",
    "            df.loc[t,'beg_wgt1'] = df.loc[t-1,'end_wgt1']\n",
    "            df.loc[t,'beg_wgt2'] = df.loc[t-1,'end_wgt2']\n",
    "        # Calculate realized portfolio return\n",
    "        w1 = df.loc[t,'beg_wgt1']\n",
    "        w2 = df.loc[t,'beg_wgt2']\n",
    "        r1 = df.loc[t,'ret1']\n",
    "        r2 = df.loc[t,'ret2']\n",
    "        rp = w1*r1 + w2*r2\n",
    "        df.loc[t,'realized_pret']=rp    \n",
    "        # Update weights\n",
    "        df.loc[t,'end_wgt1']  = w1 * (1+r1)/(1+rp)\n",
    "        df.loc[t,'end_wgt2']  = w2 * (1+r2)/(1+rp)\n",
    "    df['exp_pret'] = df.beg_wgt1*mns[0] + df.beg_wgt2*mns[1]\n",
    "    df['sd_pret']  = np.sqrt(df.beg_wgt1**2 * sds[0]**2  + df.beg_wgt2**2 * sds[1]**2 + 2*df.beg_wgt1*df.beg_wgt2*c*sds[0]*sds[1])        \n",
    "    df['sharpe ratio'] = (df.exp_pret - rf) / df.sd_pret\n",
    "    return (df.realized_pret.mean() - rf)/df.realized_pret.std()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Collect N_SIMS runs of the simulation function\n",
    "N_SIMS=1000\n",
    "sims = pd.DataFrame(dtype=float,columns=['sharpe'],index=np.arange(N_SIMS))\n",
    "for s in sims.index:\n",
    "    sims.loc[s] = sim_calc(MN_STOCK, MN_BOND, SD_STOCK, SD_BOND, CORR, RF, INIT_WGT_STOCK, INIT_WGT_BOND, T)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
        {
         "histnorm": "percent",
         "hovertemplate": "<br>%{y:.2}% of simulations <br><extra></extra>",
         "marker": {
          "line": {
           "color": "black",
           "width": 1
          }
         },
         "type": "histogram",
         "x": [
          -0.26087309200613984,
          0.1915897015935255,
          -0.06851580359998041,
          -0.052219610755096256,
          -0.10055039253760853,
          0.03658227152536686,
          0.3375243233139404,
          0.6098935367822141,
          0.5406073145118415,
          0.6451587917898416,
          0.24827078318720525,
          0.46276401910376264,
          0.3413264685251911,
          0.22308345483026598,
          0.5415097952240162,
          0.05128404285383257,
          0.2109831716209627,
          0.22131341229660376,
          0.14373729272469335,
          -0.020568149263346307,
          -0.10901243138876895,
          0.19767119993581084,
          -0.10072088951211512,
          0.25298003679591585,
          0.05842620580507579,
          0.18192561002276939,
          0.1670047847599024,
          0.33318614832828025,
          0.11601124765284915,
          0.08243101994632779,
          0.27956720636950433,
          0.23136844569433293,
          0.16357374204079841,
          -0.10665874506072025,
          0.10698422904255787,
          0.35764107084534597,
          0.032344259874088684,
          0.39576108853520303,
          0.3905070220219514,
          0.3550540867255113,
          0.5364009114681972,
          -0.019115651170512928,
          0.05862634977043911,
          0.0723211156801824,
          0.13930322778324747,
          0.122941792253292,
          0.2317981980592564,
          0.1469550635932528,
          0.31894619642220967,
          0.21599099835652893,
          -0.21012742641141283,
          0.48704285854563,
          0.08584477436738155,
          -0.04871598309919086,
          0.2206635641259447,
          -0.11684984787051621,
          0.41241202690899603,
          0.4448783166980525,
          -0.1271775172923754,
          0.3686335096701995,
          -0.2574565645558747,
          0.25936128265727154,
          -0.09022526942887192,
          -0.04628484995024802,
          0.2710209636059556,
          0.1552914846088319,
          0.3831569415000887,
          0.11163719409060426,
          0.008063044455542167,
          0.5448539532042488,
          0.39182942452232705,
          0.19279246302706823,
          0.05452305532186053,
          0.2062479909245253,
          0.4546811747218056,
          -0.004659240583160851,
          0.7596989251247899,
          0.37233771098058305,
          0.014993610906676884,
          0.1946958139426385,
          0.5070855343916865,
          0.445039185758391,
          0.48097052761888526,
          0.3960074727950691,
          -0.07201625810488518,
          0.43930810757922656,
          0.39960785803886933,
          0.22000018646890307,
          0.24808629351345046,
          0.11115749208379706,
          -0.051447578700940244,
          0.06797510179124508,
          0.0617505693454367,
          -0.1013145365474409,
          -0.043308528205676274,
          0.2652120163271062,
          0.06375065078426606,
          -0.11111442916063702,
          0.04846786054620821,
          -0.028040147448282714,
          -0.15894067764739014,
          0.5985284415625334,
          0.510803062851431,
          0.19652718885588524,
          0.4979130302184389,
          0.07630296497787231,
          0.23409141090881236,
          0.21078022779653052,
          -0.32336285244897955,
          0.2651415095988457,
          0.12110226674475633,
          0.2383621034235461,
          0.3308866291341028,
          0.08250896624410768,
          0.029831038561526548,
          0.19744656911772415,
          -0.05122483869663212,
          0.34548721777989205,
          0.11514947271068215,
          -0.1363334527331793,
          0.2716872340250882,
          0.2889290723863209,
          0.2140369631333153,
          -0.056006397821532945,
          0.05808155837653021,
          -0.09881288817017557,
          0.7528900006061173,
          0.2591776803883819,
          0.12610578741165432,
          0.14311835349312735,
          0.05468336636890351,
          0.2860608509658743,
          0.1846032085827641,
          -0.13915192948546182,
          0.30523577330036633,
          0.19117315301726623,
          -0.08820267039195118,
          0.021878999238935062,
          0.009903575906793489,
          0.47208498933614185,
          -0.1824219778026362,
          0.22077007124394774,
          0.4177941140188532,
          -0.00451108343467112,
          -0.13513883719501735,
          0.12960889245046242,
          -0.20336669669229784,
          0.13371267777836857,
          0.13485519134950052,
          0.27596872411543855,
          0.03979096859758433,
          0.16618177801948378,
          0.11506028343754728,
          0.28132647250079873,
          0.3476328356876838,
          -0.10915673718291438,
          0.3953765095996429,
          0.24351932765109113,
          0.4279529569979097,
          0.14057856187928802,
          0.5501614804293486,
          0.5700093570111281,
          0.23812794259960915,
          0.5205638029911771,
          0.2807787995114587,
          0.03957718416628507,
          0.1223980999427443,
          0.12647451567398837,
          0.0910562781414915,
          0.11652291321810768,
          0.2914772850695559,
          0.015543968709199472,
          0.3040491150762667,
          0.30284332939471836,
          0.10855894961151312,
          0.2655276012034715,
          0.046986376567116225,
          0.35906577226724457,
          -0.095283819979924,
          0.27714657178525265,
          0.37426640631576835,
          0.4814677572177467,
          -0.17903295015354,
          0.5190650235656769,
          0.1160342436093003,
          0.23929027878363932,
          -0.09435220974762583,
          0.5115076159016088,
          0.09935158924110497,
          0.16085362770918982,
          0.2609626053888844,
          0.13004986371695726,
          0.07470601979312641,
          -0.1249057750221536,
          0.2322958252374179,
          0.14317034918562285,
          -0.2485139277601497,
          -0.17437148559591403,
          0.27654944924195807,
          0.046831008027099864,
          0.4193222887145683,
          -0.029726035415961753,
          0.09445117792144853,
          0.2842082451581609,
          0.5243163860664274,
          0.4254497097086789,
          0.2369839007306277,
          0.03985205254551088,
          -0.04897520164893319,
          0.1532526347485982,
          0.42550696171234165,
          0.32735248193190164,
          -0.11798525340236089,
          0.40516208053564556,
          0.10474185241637393,
          0.037661540317348016,
          0.2559785852074088,
          0.4630883330070687,
          0.08698201562798424,
          0.4692916973399143,
          -0.16832733403046088,
          0.1970843402719953,
          -0.0927905432751952,
          -0.2162027887725357,
          -0.04169066170176215,
          0.2731704779920348,
          0.24752358380444164,
          0.17606495350951049,
          0.1675747405827579,
          0.29559567191767777,
          0.050641527803911165,
          -0.040753113261895874,
          0.019743928737315654,
          0.040459885464300835,
          0.19658902513955198,
          0.2529001286051592,
          0.072392678335999,
          0.1215089516413827,
          0.1454970352427743,
          -0.04539483032680989,
          0.39732166170731587,
          0.2144606207972514,
          0.3108712018941677,
          0.23401480545435768,
          0.03679990686906217,
          -0.17047103717206505,
          0.15948148488464214,
          0.31363972428859116,
          0.3516644453925032,
          0.2769896839937136,
          -0.018772684213656833,
          -0.06606521859252358,
          0.27351422222260335,
          0.06171899154080519,
          -0.1474054984981771,
          -0.1345064378794799,
          0.47980101841899353,
          0.17249557500959364,
          0.382356079165369,
          0.3440232123612189,
          0.5189887271781458,
          -0.001304174082006378,
          0.10461643290330343,
          0.03716474285670265,
          0.06731670004065725,
          -0.0056186527163323155,
          -0.19977405538753243,
          0.23182448072455752,
          0.34006945334958116,
          0.10847977968487199,
          0.20839410357929594,
          0.1849609492566078,
          -0.1984676974509069,
          0.19861651964600116,
          0.6075718647832167,
          0.07222079535471479,
          0.29092479293864154,
          -0.05970970051891005,
          -0.04372666999989055,
          0.13919884071864486,
          0.2887784165795139,
          -0.008061531429276231,
          -0.024165678393180026,
          -0.048159088233807504,
          0.14287718658879114,
          -0.16215424012068672,
          0.3004640622953333,
          0.04255578156394182,
          0.4084592078159117,
          0.4646568040578594,
          0.11554422252280575,
          0.11615959390907174,
          -0.08258951776532543,
          0.10864395315217386,
          0.05179353201736299,
          0.1257084985479263,
          0.31135681936672976,
          0.12119464368198152,
          0.055820009736874496,
          -0.061269783991974296,
          0.35287151371894715,
          -0.0027080331659150637,
          0.27445100714857856,
          0.3513961469378921,
          0.50632706632949,
          0.25173412327311034,
          0.30690089308216706,
          -0.12612921147835854,
          0.2305071179548673,
          0.1709889410351702,
          0.04846753812367329,
          0.26518021765315997,
          -0.1899634732925917,
          0.3017213702511125,
          0.12897366716568076,
          0.2484820211539715,
          0.08667286737478655,
          0.4048726000200608,
          0.17264175158876313,
          -0.09369585568703748,
          -0.07825516841798258,
          0.15678795242460924,
          0.27249632536569995,
          0.24771343727525214,
          0.2753271428250911,
          0.30616199516971404,
          0.055170919977566645,
          0.4310918692155354,
          -0.12295937467629932,
          0.13929605875554815,
          0.3297132028441642,
          0.2385409363360837,
          0.36445388241392296,
          0.2743464212149959,
          0.14906663324281472,
          -0.32427187542632563,
          0.09763119901678793,
          -0.19055101238296426,
          0.13433932021608386,
          0.11926339288154887,
          0.24010249157860553,
          -0.03734167870473878,
          0.06941170513094506,
          -0.007386554511398559,
          -0.09389609374597123,
          -0.014744032742424324,
          0.28711888367493815,
          0.020138010474357034,
          -0.2687234531174942,
          0.017131976012125287,
          -0.08970854774421848,
          0.3258613447510893,
          -0.17169262325177112,
          -0.038031894358381266,
          0.17899003628654608,
          0.010750466935163185,
          0.1854889478377452,
          0.30012774475482296,
          0.3391078942598015,
          0.373382460617344,
          0.0010148253140124855,
          0.3536663128891981,
          -0.03815050929600872,
          0.4358560123311787,
          -0.04445362225967447,
          0.2508123814313993,
          0.2491953903938211,
          0.045254358438051,
          -0.037908590304178674,
          0.2339973647354553,
          0.2434193138681726,
          0.024970301786311013,
          0.19287260383433433,
          0.31795235163290936,
          0.37744685506709347,
          0.00957671458114402,
          0.5285709535941113,
          0.07342395630156369,
          -0.11057258714018413,
          -0.0428997028668363,
          0.536639746065931,
          -0.00575396186237581,
          0.2394555581937282,
          0.14653785134793393,
          0.10436821865059978,
          0.14904518114979598,
          0.05863285408580501,
          0.27152930783804274,
          0.05885644737882355,
          -0.06154893843852851,
          0.10574406610108014,
          -0.005993503336080898,
          0.23045091964705652,
          0.17685239902342323,
          0.30093410193589354,
          0.00009119078830038661,
          0.21321933361169323,
          0.08429949194522687,
          0.48431191930075146,
          0.43190674795730155,
          0.034974392838336366,
          0.08077604085896699,
          0.2210816811241087,
          0.14302718115008597,
          0.015910723379368167,
          0.07541538967910902,
          0.5275252597522471,
          0.28329774639353045,
          -0.11641945760349913,
          0.037904860786852936,
          0.0788358717486856,
          0.18886889109117752,
          0.03841340382489065,
          0.19632729508265798,
          0.30823756656684026,
          0.10173692549947376,
          0.3371299021754441,
          0.25174308264671746,
          -0.1587231375102808,
          0.3074827920245375,
          0.22484107632076505,
          0.15936603433783164,
          -0.13266233122199952,
          0.1065485714867986,
          0.21631819036849925,
          0.2907343897920689,
          0.05833016556150815,
          0.2289198693877374,
          0.47926648406702493,
          -0.03872674489186899,
          0.09592181111694625,
          0.09416915713207824,
          0.1476272064261537,
          -0.09420158090784982,
          0.2172711721959837,
          -0.0212028875627656,
          -0.06133389051633454,
          0.2237933248652417,
          0.13777872829857074,
          0.16036586428089694,
          0.20339955266441506,
          0.014337763146795886,
          0.21877222632356172,
          0.050651686466494476,
          0.1906699115271112,
          0.014176389752779659,
          0.053678014156391736,
          0.10381429749167734,
          0.50723994884573,
          0.3253985708916695,
          0.5476567334902823,
          0.12317383758793385,
          0.1543394936063198,
          0.003474716054650812,
          0.011907570243765873,
          0.05088089341622074,
          0.11474001778852107,
          0.024014896044995668,
          0.1561616183122272,
          0.20359759059635454,
          0.07083311429940263,
          -0.03739022452993703,
          0.33857686556454003,
          0.3159617475298815,
          0.32602356835990026,
          -0.028254312944942316,
          0.3263149366936159,
          0.5732274809507669,
          -0.22752998642315814,
          -0.1588852067135238,
          0.40110164135047943,
          0.21896577170596218,
          0.2826296799366351,
          0.06279702271145898,
          0.02406000631468318,
          0.030245847634999146,
          0.19556185198542367,
          0.5575442995931591,
          0.047717171961616955,
          0.12947393466485083,
          0.2828173358036786,
          -0.24652544155669276,
          0.21524319361219343,
          0.13425033600754832,
          -0.03404171121983838,
          0.019406896839225734,
          0.41849550189834844,
          -0.4039068898026699,
          0.05863948473584405,
          0.23841229086367383,
          0.6383412586726701,
          0.10422122437764803,
          0.28355449715106684,
          0.3877260548195916,
          -0.09504186301528329,
          0.19120364154669134,
          -0.18790018561414873,
          0.37525627691808794,
          0.1338699248981688,
          0.28907650316934375,
          0.008498883414737013,
          0.3046405130744219,
          0.15350494284540578,
          0.11835827362931634,
          0.004076435976949568,
          0.13667201401655535,
          0.17021159228695087,
          0.38585594580521054,
          0.50431060205308,
          -0.029866084404584874,
          0.06653748346894853,
          -0.0431628608540881,
          0.16459599526601443,
          0.3784154674181747,
          0.26689576110994057,
          0.17186816405955488,
          -0.06659844143100556,
          0.18393407493541694,
          0.25033499529314746,
          0.12228325228367153,
          0.11628443797936128,
          -0.1990025931346032,
          0.3488328712823167,
          0.3956932157316519,
          -0.008056899039079654,
          0.09001898557230276,
          0.034134558143127276,
          0.5045750620228003,
          0.04548924821541443,
          -0.015813300836315755,
          0.0530575538661546,
          0.05623135272799972,
          0.09367040085077018,
          0.2625237983454855,
          0.3276173300194629,
          0.04458169077817545,
          0.4049113557436029,
          0.3452896882207092,
          -0.32193301028786164,
          0.12597805596068898,
          0.5538070748370328,
          0.04237094804108545,
          0.26986514215237817,
          0.2307147481971522,
          -0.0659434234511184,
          0.5433318080447954,
          0.3163900340333181,
          0.27419183460468605,
          -0.027322495080821668,
          -0.05551159801708172,
          0.23905424492529237,
          0.4360876975401094,
          0.32355942987440683,
          0.2650341368997337,
          0.3325573777728848,
          0.2758006460239992,
          0.3496284865745424,
          0.17371086648670336,
          0.3432283428614618,
          -0.10187136545987571,
          0.5012993276594705,
          0.1267118531665905,
          -0.1055372719448239,
          0.01009003897433382,
          0.1129881618225078,
          -0.10965696812222057,
          0.22071303648082352,
          0.1149035071148136,
          0.29397650795199887,
          -0.0495294396273975,
          -0.32895177972695927,
          0.5034530060233187,
          0.1359726228302944,
          -0.03615822648029954,
          0.35046722413520687,
          0.026820929787379627,
          0.07945107478164638,
          -0.022682823259925582,
          0.06789954216651232,
          0.10994562410338572,
          0.5610619099232464,
          0.02238365746846447,
          0.43722255090050527,
          0.26009589232509217,
          0.46355246632423686,
          0.05090133187080338,
          0.1758189616522698,
          0.20776520834322934,
          0.4952289995672961,
          0.5435768784093408,
          -0.044354754173279214,
          0.7540943948443347,
          0.699824072260786,
          0.134795456931279,
          0.07352192710983337,
          0.339473751520377,
          0.6034600079663901,
          0.2467615746894835,
          0.35722200478700794,
          -0.07248969768372017,
          -0.035573584655784043,
          0.0938676795565093,
          0.11363362520837335,
          0.28986003808524724,
          0.24024771583702562,
          0.3234431345211512,
          -0.008680205543256372,
          0.02859309489328533,
          0.545693627038518,
          0.08360164889585263,
          -0.005099837415254196,
          0.5532897270711707,
          0.3292069460069254,
          0.12353086842832937,
          0.10117615074207102,
          0.179778802967844,
          0.19277906797975644,
          0.3070396524691177,
          0.2681814618711873,
          0.14638118382902546,
          0.15115669941893536,
          0.2701419172646906,
          -0.11614282721231317,
          0.247339734144286,
          0.2521943816999499,
          0.0798694711969541,
          0.038003379361874,
          0.05911740131756613,
          0.15014404250843377,
          0.40676734658943936,
          0.011983673720102235,
          0.4490481407737868,
          0.27860073290933446,
          0.3631490028665487,
          0.65395014331847,
          0.12480414891315283,
          0.25991791395503644,
          -0.3130914280368364,
          -0.4135935484923353,
          -0.014451191048937091,
          -0.21049555740573522,
          0.18054318727337368,
          0.27779432007622257,
          0.21156095403846417,
          0.3735429595054929,
          0.05699775121879627,
          0.052704275086156605,
          0.262524992066373,
          0.21150640671178816,
          0.23526533558215346,
          -0.0978479725512518,
          0.24560789658839446,
          0.20296280273484735,
          0.17108157767814877,
          0.32225316481478433,
          0.12230086526145344,
          -0.24676567056090462,
          0.04983993279366532,
          0.17706533274460798,
          -0.08416073304101407,
          0.29082972054933315,
          0.11032788366243763,
          -0.3944856994279392,
          0.2997471251896113,
          0.20733719126422723,
          0.20550025702400068,
          0.18519546851259833,
          0.33923174208824036,
          -0.11096102198157656,
          0.042499618631062885,
          0.27462447612730834,
          0.1744133507289738,
          -0.3271840366190107,
          0.15141529899108916,
          0.12426929345439512,
          0.38446226784705156,
          -0.172662347520135,
          0.3034834402465486,
          -0.0665907570619913,
          0.14390770260309071,
          0.2249262342963521,
          0.16900059567861914,
          0.3323815884835215,
          0.029174445563285085,
          0.05871280381771259,
          0.12476894237336386,
          -0.13607411642547682,
          0.019736534633806282,
          0.050666762911130814,
          -0.08322575496829394,
          -0.023784223985887254,
          0.1166855157904538,
          -0.16405531423297617,
          0.1666561496144519,
          0.13115355672177867,
          -0.30314254750547,
          0.07078136887470646,
          0.345827219669248,
          0.00543645518813514,
          0.08634756675958326,
          0.1035941012756861,
          -0.07871901168540851,
          -0.025517658384950882,
          0.06427935190108423,
          0.1855381167334397,
          0.1296489740098607,
          0.13025223259738808,
          0.11479420848780159,
          0.405035137924713,
          0.14073987073444197,
          0.3819648808555405,
          0.07604027234061017,
          0.22026909211921247,
          0.2840112494274263,
          0.1275240106201812,
          0.05719939762299134,
          0.5128216808707947,
          -0.10746637278388718,
          0.07364643844180614,
          0.08501746831779773,
          0.6028556071742153,
          0.586963841321364,
          0.3521290143812611,
          0.003962515588037181,
          0.027513539560302037,
          0.08233437705978953,
          0.7206124091810618,
          0.23490193570876736,
          0.23343912422433685,
          0.276895529050774,
          0.014485304508785088,
          -0.05536177078365063,
          0.4815782478717715,
          -0.12528313060061727,
          0.2864872780627862,
          -0.02828741955012698,
          0.45018950831818433,
          -0.048047071363496764,
          0.5431595180187287,
          0.12106135146580427,
          0.03319611547819147,
          -0.3435842630427997,
          0.35411208443016734,
          0.2469033395935989,
          -0.05344605680036748,
          0.47433923890869606,
          0.377369682537109,
          -0.2356994396117962,
          0.2534633534739589,
          -0.2240974378211745,
          -0.47813102648520583,
          0.01839273704323397,
          0.11483212622225542,
          0.17405468628211596,
          0.2857998705820404,
          -0.04783924941668811,
          0.11567312146521815,
          0.23377821166292456,
          0.27590972633206406,
          0.02764627118114243,
          0.04886861072472417,
          0.28972809655497034,
          0.4672213753275654,
          -0.03155506049929679,
          -0.08328238035632271,
          0.36755689000187947,
          0.015703546679358395,
          -0.005120990978596362,
          0.062410879090911975,
          0.29027277540870156,
          -0.053458327831182616,
          0.32972467212885304,
          0.4072723627786441,
          -0.09791721013793214,
          -0.0444404268448141,
          0.4314789151302248,
          0.16331827724961093,
          0.15879568614221476,
          0.2134573326837062,
          -0.13274118783149502,
          0.14151455420688566,
          0.4267249152692666,
          0.17400699264689745,
          0.12587181133768802,
          0.20036198236306382,
          0.14883598036898596,
          0.23942805790625307,
          0.11682450491326146,
          -0.15451842655854786,
          0.19556288996175075,
          0.341350312768404,
          0.1108570348467199,
          0.29741882841389106,
          0.2520665591210462,
          -0.08147440764911133,
          -0.0745675730436441,
          0.025041707690637138,
          0.5469599583003797,
          0.32721226648002094,
          0.1749164331340692,
          0.21079706600841824,
          0.12374365952401177,
          0.24216798342408766,
          0.3034573495431363,
          0.16171313592778713,
          0.10600387669432629,
          -0.04687974593893764,
          -0.07263451521606169,
          0.4523506462080833,
          0.6712488726656649,
          0.28259387227383487,
          0.45699566691885923,
          0.07122717519195293,
          -0.1305009475185933,
          0.32093713629137166,
          0.23522716213858386,
          0.31682028112507743,
          0.40313906264833127,
          0.2257990909360141,
          -0.014184150259625257,
          0.17790127670112799,
          0.5031911486904047,
          0.22661221221593272,
          0.16269233991101711,
          0.31279541022343776,
          0.28122859718512855,
          0.04190556209400241,
          0.12764594911582386,
          0.18786603905083873,
          0.30181021778741046,
          0.10367658924825614,
          0.06590822906469986,
          0.12134134849378155,
          0.2884593914757018,
          0.17649192397261387,
          0.21843208497880817,
          -0.15508602236837726,
          0.04198636273060667,
          0.30550383623680094,
          0.1254002122872674,
          0.12396809776376715,
          0.3216819351442072,
          -0.24409247342268972,
          0.5578456163341655,
          0.22211620935574355,
          0.14052805368429927,
          0.05371881817778582,
          0.12210065686644485,
          0.03783424039003506,
          -0.4807090307772839,
          0.3266070195820538,
          -0.1595880952531443,
          0.03948641798179429,
          0.09775351821616658,
          0.370621644594194,
          0.41053103614203934,
          0.1295431967888197,
          -0.10366234237208576,
          0.04133343237121116,
          -0.16336386515720344,
          0.09369778440267926,
          -0.10500192840713635,
          0.48698214697223857,
          0.5295392346439384,
          0.38028201872083633,
          0.6893915859044303,
          0.2262480079909559,
          0.4250844516343553,
          0.4345029599267342,
          -0.0037389216426236844,
          0.44933931849342074,
          -0.05854662694224228,
          -0.07326988343167505,
          0.38749385836929595,
          -0.08762888433842891,
          0.08738057634049738,
          0.20145599093127395,
          0.2558272761705283,
          0.13496949476058195,
          0.2510586278018191,
          0.28266752485332297,
          0.07138547838513466,
          0.37190881943982873,
          -0.0822834309930183,
          -0.3612711627221914,
          0.3729439548944332,
          0.16330528120773358,
          0.04596665768979264,
          0.5010871715981328,
          0.23063764251540866,
          0.34981475568160136,
          0.33978796203504785,
          0.020939060481930234,
          0.2582310997890838,
          0.389545567274234,
          0.22921123779556737,
          0.30626630080270734,
          -0.2649359326313515,
          -0.03592577864557904,
          -0.07036536463207549,
          0.21541968441838114,
          0.42036478802493416,
          0.2813750164369265,
          0.051628453435645864,
          0.12834068232621787,
          -0.4508528514023116,
          0.362550407977315,
          0.13960048698002014,
          0.24122053220229708,
          0.23215667355952302,
          -0.11606833367082697,
          -0.08483252977462133,
          -0.12944813795966553,
          0.3028227412763489,
          0.16112571818311291,
          0.14468729441114114,
          0.09938643213853633,
          0.07491103092592205,
          0.012681918752999453,
          0.3390959974098456,
          0.1601678435360948,
          0.22236251195754994,
          0.09050789889002143,
          0.22576101817218722,
          -0.20755344892670913,
          -0.05368892822988551,
          0.035704795730892656,
          -0.27811667755533154,
          0.07818062952491114,
          -0.25881386281125895,
          0.463435178314736,
          0.21285767336506792,
          0.11200034253519496,
          0.26845299586547716,
          0.2471729117240727,
          0.05733242835366971,
          0.17864291963931847,
          0.2013461999496642,
          -0.17559243694145724,
          0.33291734842141346,
          0.4739830357722556,
          0.10038424508572164,
          0.33869653601819866,
          -0.0551449745997751,
          0.38150179865955525,
          0.11124075284479673,
          0.3157410819913935,
          0.29207990880296736,
          0.348151321130242,
          -0.1884523286237528,
          0.25965354670215873,
          0.2703885243357596,
          0.3188439520805125,
          0.24020533283292309,
          0.25340014903202,
          0.2901566848633265,
          -0.3141816107844595,
          0.18745688151025341,
          0.3857009451793109,
          0.3151033477468456,
          0.02299428431717082,
          0.16459848928639048,
          -0.09323677467794719,
          0.18726839759079525,
          0.11890234462498715,
          0.42001285529507565,
          0.24992965207602286,
          0.3158617059786659,
          -0.0037545016342583424,
          -0.21180437862759946,
          -0.0654653672947821,
          0.2012307403460645,
          0.3674480617236586,
          0.11380575720490504,
          0.13361695778977029,
          -0.12050942079087135,
          0.09206220584775296,
          0.516356666771374,
          0.030739045343041303,
          0.28395647484194486,
          0.13065592521263686,
          -0.23814397992704908,
          -0.31263468918349246,
          -0.06634936326028411,
          0.18560684323065926,
          -0.06819587238284182,
          0.2746171402230343,
          0.21415796991864408,
          0.28039914363199936,
          0.10862912051114136,
          -0.11049706580731335,
          0.3005736461604867,
          -0.13191292811238947,
          0.07536086116147381,
          -0.10914227843968385,
          0.07735316536639093,
          -0.14115498483783165,
          0.2844081494703737,
          0.046344468969506135,
          0.2540977715367282
         ]
        }
       ],
       "layout": {
        "template": {
         "data": {
          "bar": [
           {
            "error_x": {
             "color": "#2a3f5f"
            },
            "error_y": {
             "color": "#2a3f5f"
            },
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "bar"
           }
          ],
          "barpolar": [
           {
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "barpolar"
           }
          ],
          "carpet": [
           {
            "aaxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "baxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "type": "carpet"
           }
          ],
          "choropleth": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "choropleth"
           }
          ],
          "contour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "contour"
           }
          ],
          "contourcarpet": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "contourcarpet"
           }
          ],
          "heatmap": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmap"
           }
          ],
          "heatmapgl": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmapgl"
           }
          ],
          "histogram": [
           {
            "marker": {
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "histogram"
           }
          ],
          "histogram2d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2d"
           }
          ],
          "histogram2dcontour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2dcontour"
           }
          ],
          "mesh3d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "mesh3d"
           }
          ],
          "parcoords": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "parcoords"
           }
          ],
          "pie": [
           {
            "automargin": true,
            "type": "pie"
           }
          ],
          "scatter": [
           {
            "fillpattern": {
             "fillmode": "overlay",
             "size": 10,
             "solidity": 0.2
            },
            "type": "scatter"
           }
          ],
          "scatter3d": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatter3d"
           }
          ],
          "scattercarpet": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattercarpet"
           }
          ],
          "scattergeo": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergeo"
           }
          ],
          "scattergl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergl"
           }
          ],
          "scattermapbox": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattermapbox"
           }
          ],
          "scatterpolar": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolar"
           }
          ],
          "scatterpolargl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolargl"
           }
          ],
          "scatterternary": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterternary"
           }
          ],
          "surface": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "surface"
           }
          ],
          "table": [
           {
            "cells": {
             "fill": {
              "color": "#EBF0F8"
             },
             "line": {
              "color": "white"
             }
            },
            "header": {
             "fill": {
              "color": "#C8D4E3"
             },
             "line": {
              "color": "white"
             }
            },
            "type": "table"
           }
          ]
         },
         "layout": {
          "annotationdefaults": {
           "arrowcolor": "#2a3f5f",
           "arrowhead": 0,
           "arrowwidth": 1
          },
          "autotypenumbers": "strict",
          "coloraxis": {
           "colorbar": {
            "outlinewidth": 0,
            "ticks": ""
           }
          },
          "colorscale": {
           "diverging": [
            [
             0,
             "#8e0152"
            ],
            [
             0.1,
             "#c51b7d"
            ],
            [
             0.2,
             "#de77ae"
            ],
            [
             0.3,
             "#f1b6da"
            ],
            [
             0.4,
             "#fde0ef"
            ],
            [
             0.5,
             "#f7f7f7"
            ],
            [
             0.6,
             "#e6f5d0"
            ],
            [
             0.7,
             "#b8e186"
            ],
            [
             0.8,
             "#7fbc41"
            ],
            [
             0.9,
             "#4d9221"
            ],
            [
             1,
             "#276419"
            ]
           ],
           "sequential": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ],
           "sequentialminus": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ]
          },
          "colorway": [
           "#636efa",
           "#EF553B",
           "#00cc96",
           "#ab63fa",
           "#FFA15A",
           "#19d3f3",
           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "xaxis": {
         "title": {
          "text": "Realized Sharpe Ratio"
         }
        },
        "yaxis": {
         "title": {
          "text": "Percent of Simulations"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the distribution of output\n",
    "fig = go.Figure()\n",
    "trace= go.Histogram(x=sims.sharpe, histnorm='percent',hovertemplate=\"<br>%{y:.2}% of simulations <br><extra></extra>\")\n",
    "fig.add_trace(trace)\n",
    "fig.update_traces(marker_line_width=1, marker_line_color='black')\n",
    "fig.layout.xaxis[\"title\"] = \"Realized Sharpe Ratio\"\n",
    "fig.layout.yaxis[\"title\"] = \"Percent of Simulations\"\n",
    "fig.show()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### N asset simulation with no rebalancing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def returns(means, sds, corr, rf, init_wgts, T):\n",
    "    # Inputs    \n",
    "    cov = np.diag(sds) @ corr @ np.diag(sds)\n",
    "    n = len(means)\n",
    "    \n",
    "    # Set up dataframe\n",
    "    beg_wgt_names = ['beg_wgt'+str(i) for i in np.arange(n)]\n",
    "    end_wgt_names = ['end_wgt'+str(i) for i in np.arange(n)]\n",
    "    ret_names = ['ret'+str(i) for i in np.arange(n)]\n",
    "    col_names = beg_wgt_names + ret_names + end_wgt_names +['realized_pret','exp_pret','sd_pret','sharpe ratio']\n",
    "    df = pd.DataFrame(dtype=float, columns=col_names, index=np.arange(T)+1)\n",
    "\n",
    "    # Generate random sample and do calculations\n",
    "    df[ret_names] = mvn.rvs(means, cov, size=T)\n",
    "    df.loc[1,beg_wgt_names] = init_wgts\n",
    "    for t in df.index:\n",
    "        if t > 1:\n",
    "            df.loc[t,beg_wgt_names] = df.loc[t-1,end_wgt_names].values\n",
    "        # Calculate realized portfolio return + beginning of period E[r] and SD[r]\n",
    "        wgts = df.loc[t,beg_wgt_names].values\n",
    "        rets = df.loc[t,ret_names].values\n",
    "        rp = wgts @ rets\n",
    "        df.loc[t,'realized_pret']=rp        \n",
    "        df.loc[t,'exp_pret'] = wgts @ means\n",
    "        df.loc[t,'sd_pret']  = np.sqrt(wgts @ cov @ wgts)         \n",
    "        # Update weights\n",
    "        for i in np.arange(n):\n",
    "            w = df.loc[t,'beg_wgt'+str(i)]\n",
    "            r = df.loc[t,'ret'+str(i)]\n",
    "            df.loc[t,'end_wgt'+str(i)]  =  w * (1+r)/(1+rp)\n",
    "    df['sharpe ratio'] = (df.exp_pret - rf) / df.sd_pret\n",
    "    return df"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Inputs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Risk-free rate\n",
    "RF = 0.01\n",
    "\n",
    "# Expected returns\n",
    "MNS = np.array([0.06, 0.065, 0.08])\n",
    "\n",
    "# Standard deviations\n",
    "SDS = np.array([0.15, 0.165, 0.21])\n",
    "\n",
    "# Correlations\n",
    "C  = np.identity(3)\n",
    "C[0, 1] = C[1, 0] = 0.75\n",
    "C[0, 2] = C[2, 0] = 0.75\n",
    "C[1, 2] = C[2, 1] = 0.75\n",
    "\n",
    "# Sample length\n",
    "T = 30"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A single realization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        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>beg_wgt0</th>\n",
       "      <th>beg_wgt1</th>\n",
       "      <th>beg_wgt2</th>\n",
       "      <th>ret0</th>\n",
       "      <th>ret1</th>\n",
       "      <th>ret2</th>\n",
       "      <th>end_wgt0</th>\n",
       "      <th>end_wgt1</th>\n",
       "      <th>end_wgt2</th>\n",
       "      <th>realized_pret</th>\n",
       "      <th>exp_pret</th>\n",
       "      <th>sd_pret</th>\n",
       "      <th>sharpe ratio</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.3333</td>\n",
       "      <td>0.3333</td>\n",
       "      <td>0.3333</td>\n",
       "      <td>0.0437</td>\n",
       "      <td>0.0106</td>\n",
       "      <td>0.1172</td>\n",
       "      <td>0.3291</td>\n",
       "      <td>0.3186</td>\n",
       "      <td>0.3523</td>\n",
       "      <td>0.0572</td>\n",
       "      <td>0.0683</td>\n",
       "      <td>0.1599</td>\n",
       "      <td>0.3648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.3291</td>\n",
       "      <td>0.3186</td>\n",
       "      <td>0.3523</td>\n",
       "      <td>-0.1156</td>\n",
       "      <td>0.0076</td>\n",
       "      <td>-0.0136</td>\n",
       "      <td>0.3033</td>\n",
       "      <td>0.3346</td>\n",
       "      <td>0.3621</td>\n",
       "      <td>-0.0404</td>\n",
       "      <td>0.0686</td>\n",
       "      <td>0.1609</td>\n",
       "      <td>0.3645</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.3033</td>\n",
       "      <td>0.3346</td>\n",
       "      <td>0.3621</td>\n",
       "      <td>-0.0051</td>\n",
       "      <td>-0.0776</td>\n",
       "      <td>-0.0172</td>\n",
       "      <td>0.3123</td>\n",
       "      <td>0.3194</td>\n",
       "      <td>0.3683</td>\n",
       "      <td>-0.0338</td>\n",
       "      <td>0.0689</td>\n",
       "      <td>0.1617</td>\n",
       "      <td>0.3643</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.3123</td>\n",
       "      <td>0.3194</td>\n",
       "      <td>0.3683</td>\n",
       "      <td>0.3596</td>\n",
       "      <td>0.1598</td>\n",
       "      <td>0.1806</td>\n",
       "      <td>0.3452</td>\n",
       "      <td>0.3012</td>\n",
       "      <td>0.3536</td>\n",
       "      <td>0.2299</td>\n",
       "      <td>0.0690</td>\n",
       "      <td>0.1619</td>\n",
       "      <td>0.3642</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.3452</td>\n",
       "      <td>0.3012</td>\n",
       "      <td>0.3536</td>\n",
       "      <td>-0.1279</td>\n",
       "      <td>0.0049</td>\n",
       "      <td>-0.2347</td>\n",
       "      <td>0.3444</td>\n",
       "      <td>0.3462</td>\n",
       "      <td>0.3095</td>\n",
       "      <td>-0.1256</td>\n",
       "      <td>0.0686</td>\n",
       "      <td>0.1607</td>\n",
       "      <td>0.3645</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   beg_wgt0  beg_wgt1  beg_wgt2    ret0    ret1    ret2  end_wgt0  end_wgt1  \\\n",
       "1    0.3333    0.3333    0.3333  0.0437  0.0106  0.1172    0.3291    0.3186   \n",
       "2    0.3291    0.3186    0.3523 -0.1156  0.0076 -0.0136    0.3033    0.3346   \n",
       "3    0.3033    0.3346    0.3621 -0.0051 -0.0776 -0.0172    0.3123    0.3194   \n",
       "4    0.3123    0.3194    0.3683  0.3596  0.1598  0.1806    0.3452    0.3012   \n",
       "5    0.3452    0.3012    0.3536 -0.1279  0.0049 -0.2347    0.3444    0.3462   \n",
       "\n",
       "   end_wgt2  realized_pret  exp_pret  sd_pret  sharpe ratio  \n",
       "1    0.3523         0.0572    0.0683   0.1599        0.3648  \n",
       "2    0.3621        -0.0404    0.0686   0.1609        0.3645  \n",
       "3    0.3683        -0.0338    0.0689   0.1617        0.3643  \n",
       "4    0.3536         0.2299    0.0690   0.1619        0.3642  \n",
       "5    0.3095        -0.1256    0.0686   0.1607        0.3645  "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = len(MNS)\n",
    "df = returns(MNS, SDS, C, RF, (1/n)*np.ones(n), T)\n",
    "df.head()"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### N asset simulation with rebalancing to constant weights\n",
    "\n",
    "- With rebalancing to constant weights, we do not need to track weights each period"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "def returns_rebalancing(means, sds, corr, rf, wgts, T):\n",
    "    cov = np.diag(sds) @ corr @ np.diag(sds)\n",
    "    n = len(means)\n",
    "    \n",
    "    # Set up dataframe\n",
    "    ret_names = ['ret'+str(i) for i in np.arange(n)]\n",
    "    col_names = ret_names + ['realized_pret','exp_pret','sd_pret','sharpe ratio']\n",
    "    df = pd.DataFrame(dtype=float, columns=col_names, index=np.arange(T)+1)\n",
    "\n",
    "    # Generate random sample and do calculations\n",
    "    df[ret_names] = mvn.rvs(means, cov, size=T)\n",
    "    for t in df.index:\n",
    "        df.loc[t,'exp_pret'] = wgts @ means\n",
    "        df.loc[t,'sd_pret']  = np.sqrt(wgts @ cov @ wgts)  \n",
    "        rets = df.loc[t,ret_names].values\n",
    "        df.loc[t,'realized_pret']=wgts @ rets \n",
    "    df['sharpe ratio'] = (df.exp_pret - rf) / df.sd_pret\n",
    "    return df"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A single realization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        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>ret0</th>\n",
       "      <th>ret1</th>\n",
       "      <th>ret2</th>\n",
       "      <th>realized_pret</th>\n",
       "      <th>exp_pret</th>\n",
       "      <th>sd_pret</th>\n",
       "      <th>sharpe ratio</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.0144</td>\n",
       "      <td>0.1803</td>\n",
       "      <td>0.1215</td>\n",
       "      <td>0.1054</td>\n",
       "      <td>0.0683</td>\n",
       "      <td>0.1599</td>\n",
       "      <td>0.3648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>-0.1968</td>\n",
       "      <td>0.1288</td>\n",
       "      <td>-0.1079</td>\n",
       "      <td>-0.0586</td>\n",
       "      <td>0.0683</td>\n",
       "      <td>0.1599</td>\n",
       "      <td>0.3648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>-0.1262</td>\n",
       "      <td>-0.1384</td>\n",
       "      <td>-0.1558</td>\n",
       "      <td>-0.1402</td>\n",
       "      <td>0.0683</td>\n",
       "      <td>0.1599</td>\n",
       "      <td>0.3648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.3170</td>\n",
       "      <td>0.3527</td>\n",
       "      <td>0.2288</td>\n",
       "      <td>0.2995</td>\n",
       "      <td>0.0683</td>\n",
       "      <td>0.1599</td>\n",
       "      <td>0.3648</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>0.2730</td>\n",
       "      <td>0.1953</td>\n",
       "      <td>0.1805</td>\n",
       "      <td>0.2163</td>\n",
       "      <td>0.0683</td>\n",
       "      <td>0.1599</td>\n",
       "      <td>0.3648</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     ret0    ret1    ret2  realized_pret  exp_pret  sd_pret  sharpe ratio\n",
       "1  0.0144  0.1803  0.1215         0.1054    0.0683   0.1599        0.3648\n",
       "2 -0.1968  0.1288 -0.1079        -0.0586    0.0683   0.1599        0.3648\n",
       "3 -0.1262 -0.1384 -0.1558        -0.1402    0.0683   0.1599        0.3648\n",
       "4  0.3170  0.3527  0.2288         0.2995    0.0683   0.1599        0.3648\n",
       "5  0.2730  0.1953  0.1805         0.2163    0.0683   0.1599        0.3648"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = len(MNS)\n",
    "df = returns_rebalancing(MNS, SDS, C, RF, (1/n)*np.ones(n), T)\n",
    "df.head()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "venv",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.10.6"
  },
  "orig_nbformat": 4,
  "vscode": {
   "interpreter": {
    "hash": "cc788b38d8cb434c82f45ef9d481d0310e17b79321e4c8cbfbfe90f656a3f023"
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}