18b_backtest_industries_mkt_model.ipynb

{
 "cells": [
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Backtesting the market model covariance strategy (industry portfolios)\n",
    "\n",
    "Based on the estimated inputs each month, we will consider the optimal portfolio for a mean-variance investor with risk aversion of 3 that is short-sale constrained.  We will compare three covariance matrix estimation strategies:\n",
    "- Estimated pair-wise correlation (1176 covariance matrix parameters each month)\n",
    "- Covariance matrix implied by market model regressions (97 covariance matrix parameters each month)\n",
    "- Equal-weighted (48 covariance matrix parameters each month; assume zero correlations and equal standard deviation)\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "from pandas_datareader import DataReader as pdr\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",
    "import statsmodels.api as sm"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Pull data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Read industry data and clean-up missing data (coded -99.99)\n",
    "ff48 = pdr(\"48_Industry_Portfolios\", \"famafrench\", start=1900)[0]\n",
    "\n",
    "# Clean-up missings\n",
    "for c in ff48.columns:\n",
    "    ff48[c] = np.where(ff48[c]==-99.99, np.nan, ff48[c])\n",
    "ff48 = ff48/100\n",
    "\n",
    "# Add risk-free rate\n",
    "ff3 = pdr('F-F_Research_Data_Factors','famafrench', start=1900)[0]/100\n",
    "df = ff48.join(ff3[['Mkt-RF','RF']])\n",
    "\n",
    "# Keep data since Jan 1970\n",
    "df = df.loc['1970-01':].copy() # There is missing data prior to 1970"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create functions for tangency portfolio (with upper and lower position limits) and capital allocation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def tangency(means, cov, rf, min_wgt, max_wgt):\n",
    "    n = len(means)\n",
    "    def f(w):\n",
    "        mn = w @ means\n",
    "        sd = np.sqrt(w @ cov @ w)\n",
    "        return -(mn - rf) / sd\n",
    "    # Initial guess (equal-weighted)\n",
    "    w0 = (1/n)*np.ones(n)\n",
    "    # Constraint: fully-invested portfolio\n",
    "    A = np.ones(n)\n",
    "    b = 1\n",
    "    cons = [{\"type\": \"eq\", \"fun\": lambda x: A @ x - b}]\n",
    "    # Position limits\n",
    "    bnds = [(min_wgt, max_wgt) for i in range(n)] \n",
    "    # Optimization\n",
    "    TOL = 10**(-10)\n",
    "    wgts = minimize(f, w0, bounds=bnds, constraints=cons, options={'ftol':TOL}).x\n",
    "    return wgts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def capital_allocation(mean, sd, rf, risk_aversion):\n",
    "    return (mean - rf) / (risk_aversion * (sd**2))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Estimate inputs on expanding windows"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Index(['Agric', 'Food ', 'Soda ', 'Beer ', 'Smoke', 'Toys ', 'Fun  ', 'Books',\n",
       "       'Hshld', 'Clths', 'Hlth ', 'MedEq', 'Drugs', 'Chems', 'Rubbr', 'Txtls',\n",
       "       'BldMt', 'Cnstr', 'Steel', 'FabPr', 'Mach ', 'ElcEq', 'Autos', 'Aero ',\n",
       "       'Ships', 'Guns ', 'Gold ', 'Mines', 'Coal ', 'Oil  ', 'Util ', 'Telcm',\n",
       "       'PerSv', 'BusSv', 'Comps', 'Chips', 'LabEq', 'Paper', 'Boxes', 'Trans',\n",
       "       'Whlsl', 'Rtail', 'Meals', 'Banks', 'Insur', 'RlEst', 'Fin  ', 'Other',\n",
       "       'Mkt-RF', 'RF'],\n",
       "      dtype='object')"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df.columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Initial estimation window\n",
    "WINDOW = 180\n",
    "\n",
    "# Define the risky asset list\n",
    "RISKY_ASSETS = df.columns[:-2]\n",
    "\n",
    "# Number of risky assets\n",
    "n = len(RISKY_ASSETS)\n",
    "\n",
    "# Length of out-of-sample window\n",
    "T = len(df)-WINDOW\n",
    "\n",
    "# Maximum and minimum position limits\n",
    "MIN_WGT = 0.0\n",
    "MAX_WGT = None\n",
    "\n",
    "# Risk aversion\n",
    "RAVER = 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Standardize column names"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>r0</th>\n",
       "      <th>r1</th>\n",
       "      <th>r2</th>\n",
       "      <th>r3</th>\n",
       "      <th>r4</th>\n",
       "      <th>r5</th>\n",
       "      <th>r6</th>\n",
       "      <th>r7</th>\n",
       "      <th>r8</th>\n",
       "      <th>r9</th>\n",
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       "      <th>r40</th>\n",
       "      <th>r41</th>\n",
       "      <th>r42</th>\n",
       "      <th>r43</th>\n",
       "      <th>r44</th>\n",
       "      <th>r45</th>\n",
       "      <th>r46</th>\n",
       "      <th>r47</th>\n",
       "      <th>mkt_rf</th>\n",
       "      <th>rf</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Date</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
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       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1970-01</th>\n",
       "      <td>0.0083</td>\n",
       "      <td>-0.0281</td>\n",
       "      <td>-0.0276</td>\n",
       "      <td>-0.0135</td>\n",
       "      <td>-0.0699</td>\n",
       "      <td>-0.0795</td>\n",
       "      <td>-0.0574</td>\n",
       "      <td>-0.1139</td>\n",
       "      <td>-0.0690</td>\n",
       "      <td>-0.0509</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.0768</td>\n",
       "      <td>-0.0573</td>\n",
       "      <td>-0.1186</td>\n",
       "      <td>-0.0753</td>\n",
       "      <td>-0.0888</td>\n",
       "      <td>-0.1094</td>\n",
       "      <td>-0.1082</td>\n",
       "      <td>-0.0359</td>\n",
       "      <td>-0.0810</td>\n",
       "      <td>0.0060</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1970-02</th>\n",
       "      <td>0.0948</td>\n",
       "      <td>0.0596</td>\n",
       "      <td>0.0386</td>\n",
       "      <td>0.0687</td>\n",
       "      <td>0.0028</td>\n",
       "      <td>0.0612</td>\n",
       "      <td>0.0822</td>\n",
       "      <td>0.0068</td>\n",
       "      <td>0.0028</td>\n",
       "      <td>0.0322</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0181</td>\n",
       "      <td>0.0579</td>\n",
       "      <td>0.0543</td>\n",
       "      <td>0.1554</td>\n",
       "      <td>0.1035</td>\n",
       "      <td>0.0024</td>\n",
       "      <td>0.0918</td>\n",
       "      <td>-0.0481</td>\n",
       "      <td>0.0513</td>\n",
       "      <td>0.0062</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>2 rows × 50 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "             r0      r1      r2      r3      r4      r5      r6      r7  \\\n",
       "Date                                                                      \n",
       "1970-01  0.0083 -0.0281 -0.0276 -0.0135 -0.0699 -0.0795 -0.0574 -0.1139   \n",
       "1970-02  0.0948  0.0596  0.0386  0.0687  0.0028  0.0612  0.0822  0.0068   \n",
       "\n",
       "             r8      r9  ...     r40     r41     r42     r43     r44     r45  \\\n",
       "Date                     ...                                                   \n",
       "1970-01 -0.0690 -0.0509  ... -0.0768 -0.0573 -0.1186 -0.0753 -0.0888 -0.1094   \n",
       "1970-02  0.0028  0.0322  ...  0.0181  0.0579  0.0543  0.1554  0.1035  0.0024   \n",
       "\n",
       "            r46     r47  mkt_rf      rf  \n",
       "Date                                     \n",
       "1970-01 -0.1082 -0.0359 -0.0810  0.0060  \n",
       "1970-02  0.0918 -0.0481  0.0513  0.0062  \n",
       "\n",
       "[2 rows x 50 columns]"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "asset_list = [str(i) for i in range(n)]\n",
    "ret_list   = ['r' + asset for asset in asset_list]\n",
    "df.columns = ret_list + ['mkt_rf','rf']\n",
    "df.head(2)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Estimate expected returns, standard deviations, & correlations based on historical expanding windows\n",
    "\n",
    "- Note: expanding command includes current year in calculation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "est_means = df[ret_list].expanding(WINDOW).mean()\n",
    "est_sds   = df[ret_list].expanding(WINDOW).std()\n",
    "est_corrs = df[ret_list].expanding(WINDOW).corr()\n",
    "est_mkt_std = df['mkt_rf'].expanding(WINDOW).std()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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       "      <th rowspan=\"5\" valign=\"top\">2023-01</th>\n",
       "      <th>r43</th>\n",
       "      <td>0.511543</td>\n",
       "      <td>0.608471</td>\n",
       "      <td>0.538800</td>\n",
       "      <td>0.559481</td>\n",
       "      <td>0.417302</td>\n",
       "      <td>0.605439</td>\n",
       "      <td>0.658273</td>\n",
       "      <td>0.730977</td>\n",
       "      <td>0.643155</td>\n",
       "      <td>0.678362</td>\n",
       "      <td>...</td>\n",
       "      <td>0.646412</td>\n",
       "      <td>0.722035</td>\n",
       "      <td>0.707620</td>\n",
       "      <td>0.688386</td>\n",
       "      <td>0.673652</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.809802</td>\n",
       "      <td>0.687345</td>\n",
       "      <td>0.815519</td>\n",
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       "    <tr>\n",
       "      <th>r44</th>\n",
       "      <td>0.506630</td>\n",
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       "      <td>0.554842</td>\n",
       "      <td>0.587948</td>\n",
       "      <td>0.426100</td>\n",
       "      <td>0.588486</td>\n",
       "      <td>0.620263</td>\n",
       "      <td>0.712092</td>\n",
       "      <td>0.651866</td>\n",
       "      <td>0.660256</td>\n",
       "      <td>...</td>\n",
       "      <td>0.643576</td>\n",
       "      <td>0.713293</td>\n",
       "      <td>0.731827</td>\n",
       "      <td>0.681151</td>\n",
       "      <td>0.695304</td>\n",
       "      <td>0.809802</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.658934</td>\n",
       "      <td>0.763834</td>\n",
       "      <td>0.658314</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>r45</th>\n",
       "      <td>0.510634</td>\n",
       "      <td>0.511379</td>\n",
       "      <td>0.499962</td>\n",
       "      <td>0.483420</td>\n",
       "      <td>0.343210</td>\n",
       "      <td>0.658921</td>\n",
       "      <td>0.710930</td>\n",
       "      <td>0.765997</td>\n",
       "      <td>0.550886</td>\n",
       "      <td>0.723737</td>\n",
       "      <td>...</td>\n",
       "      <td>0.572538</td>\n",
       "      <td>0.697044</td>\n",
       "      <td>0.756175</td>\n",
       "      <td>0.656248</td>\n",
       "      <td>0.680162</td>\n",
       "      <td>0.687345</td>\n",
       "      <td>0.658934</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.695730</td>\n",
       "      <td>0.642791</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>r46</th>\n",
       "      <td>0.567784</td>\n",
       "      <td>0.538609</td>\n",
       "      <td>0.493713</td>\n",
       "      <td>0.520048</td>\n",
       "      <td>0.361555</td>\n",
       "      <td>0.644197</td>\n",
       "      <td>0.715652</td>\n",
       "      <td>0.751342</td>\n",
       "      <td>0.613964</td>\n",
       "      <td>0.694792</td>\n",
       "      <td>...</td>\n",
       "      <td>0.700962</td>\n",
       "      <td>0.740695</td>\n",
       "      <td>0.759741</td>\n",
       "      <td>0.721399</td>\n",
       "      <td>0.667088</td>\n",
       "      <td>0.815519</td>\n",
       "      <td>0.763834</td>\n",
       "      <td>0.695730</td>\n",
       "      <td>1.000000</td>\n",
       "      <td>0.676198</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>r47</th>\n",
       "      <td>0.503825</td>\n",
       "      <td>0.545162</td>\n",
       "      <td>0.460174</td>\n",
       "      <td>0.543610</td>\n",
       "      <td>0.417111</td>\n",
       "      <td>0.593796</td>\n",
       "      <td>0.631020</td>\n",
       "      <td>0.659415</td>\n",
       "      <td>0.623212</td>\n",
       "      <td>0.640508</td>\n",
       "      <td>...</td>\n",
       "      <td>0.572011</td>\n",
       "      <td>0.703582</td>\n",
       "      <td>0.739262</td>\n",
       "      <td>0.614411</td>\n",
       "      <td>0.648534</td>\n",
       "      <td>0.664554</td>\n",
       "      <td>0.658314</td>\n",
       "      <td>0.642791</td>\n",
       "      <td>0.676198</td>\n",
       "      <td>1.000000</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>30576 rows × 48 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "                   r0        r1        r2        r3        r4        r5  \\\n",
       "Date                                                                      \n",
       "1970-01 r0        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r1        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r2        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r3        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r4        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "...               ...       ...       ...       ...       ...       ...   \n",
       "2023-01 r43  0.511543  0.608471  0.538800  0.559481  0.417302  0.605439   \n",
       "        r44  0.506630  0.681498  0.554842  0.587948  0.426100  0.588486   \n",
       "        r45  0.510634  0.511379  0.499962  0.483420  0.343210  0.658921   \n",
       "        r46  0.567784  0.538609  0.493713  0.520048  0.361555  0.644197   \n",
       "        r47  0.503825  0.545162  0.460174  0.543610  0.417111  0.593796   \n",
       "\n",
       "                   r6        r7        r8        r9  ...       r38       r39  \\\n",
       "Date                                                 ...                       \n",
       "1970-01 r0        NaN       NaN       NaN       NaN  ...       NaN       NaN   \n",
       "        r1        NaN       NaN       NaN       NaN  ...       NaN       NaN   \n",
       "        r2        NaN       NaN       NaN       NaN  ...       NaN       NaN   \n",
       "        r3        NaN       NaN       NaN       NaN  ...       NaN       NaN   \n",
       "        r4        NaN       NaN       NaN       NaN  ...       NaN       NaN   \n",
       "...               ...       ...       ...       ...  ...       ...       ...   \n",
       "2023-01 r43  0.658273  0.730977  0.643155  0.678362  ...  0.646412  0.722035   \n",
       "        r44  0.620263  0.712092  0.651866  0.660256  ...  0.643576  0.713293   \n",
       "        r45  0.710930  0.765997  0.550886  0.723737  ...  0.572538  0.697044   \n",
       "        r46  0.715652  0.751342  0.613964  0.694792  ...  0.700962  0.740695   \n",
       "        r47  0.631020  0.659415  0.623212  0.640508  ...  0.572011  0.703582   \n",
       "\n",
       "                  r40       r41       r42       r43       r44       r45  \\\n",
       "Date                                                                      \n",
       "1970-01 r0        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r1        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r2        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r3        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "        r4        NaN       NaN       NaN       NaN       NaN       NaN   \n",
       "...               ...       ...       ...       ...       ...       ...   \n",
       "2023-01 r43  0.707620  0.688386  0.673652  1.000000  0.809802  0.687345   \n",
       "        r44  0.731827  0.681151  0.695304  0.809802  1.000000  0.658934   \n",
       "        r45  0.756175  0.656248  0.680162  0.687345  0.658934  1.000000   \n",
       "        r46  0.759741  0.721399  0.667088  0.815519  0.763834  0.695730   \n",
       "        r47  0.739262  0.614411  0.648534  0.664554  0.658314  0.642791   \n",
       "\n",
       "                  r46       r47  \n",
       "Date                             \n",
       "1970-01 r0        NaN       NaN  \n",
       "        r1        NaN       NaN  \n",
       "        r2        NaN       NaN  \n",
       "        r3        NaN       NaN  \n",
       "        r4        NaN       NaN  \n",
       "...               ...       ...  \n",
       "2023-01 r43  0.815519  0.664554  \n",
       "        r44  0.763834  0.658314  \n",
       "        r45  0.695730  0.642791  \n",
       "        r46  1.000000  0.676198  \n",
       "        r47  0.676198  1.000000  \n",
       "\n",
       "[30576 rows x 48 columns]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "est_corrs"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Estimate expanding window betas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.85329756, 0.64945719, 0.79449609, 0.71979884, 0.63519958,\n",
       "       1.18047769, 1.3441222 , 1.0875481 , 0.76938733, 1.11587008,\n",
       "       1.12017745, 0.90252279, 0.76493123, 1.05485055, 1.06203416,\n",
       "       1.15044229, 1.19094843, 1.27712315, 1.34159433, 1.10836224,\n",
       "       1.227806  , 1.22688768, 1.24659834, 1.13153831, 1.09926741,\n",
       "       0.78635256, 0.56689086, 1.12608805, 1.15888612, 0.86436981,\n",
       "       0.51325549, 0.79918658, 1.09121814, 1.25655712, 1.19708705,\n",
       "       1.38794651, 1.25774068, 0.95594785, 0.95716347, 1.06448674,\n",
       "       1.03956069, 1.00074589, 1.029491  , 1.07687571, 0.93938061,\n",
       "       1.23518414, 1.22603083, 1.1274833 ])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def estimate_betas(d,varnames):\n",
    "    df_beta = pd.DataFrame(index=varnames, columns=['alpha','beta'],dtype=float)\n",
    "    for varname in varnames:\n",
    "        y = d[varname]-d['rf']\n",
    "        X = sm.add_constant(d['mkt_rf'])\n",
    "        mm = sm.OLS(y, X, missing='drop').fit()\n",
    "        df_beta.loc[varname] = mm.params.values\n",
    "    return df_beta['beta'].values\n",
    "estimate_betas(df,ret_list)     #Example: these are full sample betas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "est_betas = pd.DataFrame(dtype=float,columns = ret_list, index=df.index)\n",
    "est_betas.iloc[WINDOW]\n",
    "for i in np.arange(WINDOW,WINDOW+T):\n",
    "    # print(i)\n",
    "    est_betas.iloc[i-1] = estimate_betas(df.iloc[:i],ret_list)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Subset data with estimates"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "est_means = est_means.dropna()\n",
    "est_sds   = est_sds.dropna()\n",
    "est_corrs = est_corrs.dropna()\n",
    "est_mkt_std = est_mkt_std.dropna()\n",
    "est_betas = est_betas.dropna()\n",
    "df = df.loc[est_means.index].reset_index()  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Save a copy of the data\n",
    "data = df.copy()\n",
    "\n",
    "# Create dataframe to store the results for each strategy\n",
    "results = pd.DataFrame(dtype=float, \n",
    "    columns=['sr','avg_ret','sd_ret'], \n",
    "    index = ['Pairwise', 'Mkt Model','Equal-weighted'])"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pairwise correlation estimation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Load data\n",
    "df = data.copy()\n",
    "\n",
    "# Make list of weights columns\n",
    "wgt_list= ['wgt'+asset for asset in asset_list] \n",
    "\n",
    "# Initialize empty columns for risky asset weights, risky portfolio return, CAL location, CAL portfolio return\n",
    "for wgt in wgt_list:\n",
    "    df[wgt] = np.nan\n",
    "df['risky_ret'] = np.nan\n",
    "df['wgt_risky'] = np.nan \n",
    "df['port_ret']  = np.nan\n",
    "\n",
    "# Choose optimal portfolios each time period\n",
    "for i in df.index[1:]:                              # Start at 2nd date because estimates include that month's obs\n",
    "    # Pull estimation inputs at each point in time\n",
    "    est_pull_date = df.loc[i-1,'Date']\n",
    "    means = est_means.loc[est_pull_date].values\n",
    "    sds   = est_sds.loc[est_pull_date].values\n",
    "    C     = est_corrs.loc[(est_pull_date,slice(None))].to_numpy()\n",
    "    cov = np.diag(sds) @ C @ np.diag(sds)\n",
    "\n",
    "    # Pull current period returns\n",
    "    rf = df.loc[i,'rf']\n",
    "    rets = df.loc[i,ret_list].values\n",
    "\n",
    "    ##### Note: all portfolio weights considered to be beginning of period weights\n",
    "    ##### (so multiply by contemporaneous realized returns)\n",
    "    wgts = tangency(means, cov, rf, MIN_WGT, MAX_WGT)\n",
    "    df.loc[i,wgt_list] = wgts\n",
    "    df.loc[i,'risky_ret'] = wgts @ rets\n",
    "\n",
    "    # Capital allocation and overall portfolio return\n",
    "    risky_expret = wgts @ means\n",
    "    risky_sd     = np.sqrt(wgts @ cov @ wgts)\n",
    "    df.loc[i,'wgt_risky'] = capital_allocation(risky_expret, risky_sd, rf, RAVER)\n",
    "    df.loc[i,'port_ret']  = rf + df.loc[i,'wgt_risky']*(df.loc[i,'risky_ret'] -rf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "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>Date</th>\n",
       "      <th>rf</th>\n",
       "      <th>risky_ret</th>\n",
       "      <th>wgt_risky</th>\n",
       "      <th>port_ret</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1985-01</td>\n",
       "      <td>0.0065</td>\n",
       "      <td>0.031886</td>\n",
       "      <td>1.126979</td>\n",
       "      <td>0.035110</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1985-02</td>\n",
       "      <td>0.0058</td>\n",
       "      <td>0.072667</td>\n",
       "      <td>1.293530</td>\n",
       "      <td>0.092294</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1985-03</td>\n",
       "      <td>0.0062</td>\n",
       "      <td>0.012670</td>\n",
       "      <td>1.247693</td>\n",
       "      <td>0.014273</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1985-04</td>\n",
       "      <td>0.0072</td>\n",
       "      <td>-0.068889</td>\n",
       "      <td>1.116319</td>\n",
       "      <td>-0.077740</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1985-05</td>\n",
       "      <td>0.0066</td>\n",
       "      <td>0.029510</td>\n",
       "      <td>1.152789</td>\n",
       "      <td>0.033010</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      Date      rf  risky_ret  wgt_risky  port_ret\n",
       "1  1985-01  0.0065   0.031886   1.126979  0.035110\n",
       "2  1985-02  0.0058   0.072667   1.293530  0.092294\n",
       "3  1985-03  0.0062   0.012670   1.247693  0.014273\n",
       "4  1985-04  0.0072  -0.068889   1.116319 -0.077740\n",
       "5  1985-05  0.0066   0.029510   1.152789  0.033010"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = df.dropna()\n",
    "df_pairwise = df.copy()\n",
    "df[['Date']+['rf','risky_ret','wgt_risky','port_ret']].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Avg return of Pairwise:\t\t  0.0164\n",
      "SD(return) of Pairwise:\t\t  0.0845\n",
      "Sharpe ratio of Pairwise:\t  0.1645\n"
     ]
    }
   ],
   "source": [
    "avg_ret = df.port_ret.mean()\n",
    "sd_ret  = df.port_ret.std()\n",
    "avg_rf  = df.rf.mean()\n",
    "sr = (avg_ret - avg_rf)/sd_ret\n",
    "print(f'Avg return of Pairwise:\\t\\t {avg_ret: .4f}')\n",
    "print(f'SD(return) of Pairwise:\\t\\t {sd_ret: .4f}')\n",
    "print(f'Sharpe ratio of Pairwise:\\t {sr: .4f}')\n",
    "\n",
    "# Save results\n",
    "results.loc['Pairwise','avg_ret'] = avg_ret\n",
    "results.loc['Pairwise','sd_ret']  = sd_ret\n",
    "results.loc['Pairwise','sr']      = sr"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Market Model Covariances"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Load data\n",
    "df = data.copy()\n",
    "\n",
    "# Make list of weights columns\n",
    "wgt_list= ['wgt'+asset for asset in asset_list] \n",
    "\n",
    "# Initialize empty columns for risky asset weights, risky portfolio return, CAL location, CAL portfolio return\n",
    "for wgt in wgt_list:\n",
    "    df[wgt] = np.nan\n",
    "df['risky_ret'] = np.nan\n",
    "df['wgt_risky'] = np.nan \n",
    "df['port_ret']  = np.nan\n",
    "\n",
    "# Choose optimal portfolios each time period\n",
    "for i in df.index[1:]:                              # Start at 2nd date because estimates include that month's obs\n",
    "    # Pull estimation inputs at each point in time\n",
    "    est_pull_date = df.loc[i-1,'Date']\n",
    "    means = est_means.loc[est_pull_date].values\n",
    "    sds   = est_sds.loc[est_pull_date].values\n",
    "    mkt_std = est_mkt_std.loc[est_pull_date]\n",
    "    betas = est_betas.loc[est_pull_date].values\n",
    "    C  = np.identity(n)\n",
    "    cov = np.diag(sds) @ C @ np.diag(sds)\n",
    "    for j, asset in enumerate(asset_list):\n",
    "        for k in range(j+1,n):\n",
    "            cov[j, k] = cov[k, j] = betas[j] * betas[k] * (mkt_std**2)\n",
    "\n",
    "    # Pull current period returns\n",
    "    rf = df.loc[i,'rf']\n",
    "    rets = df.loc[i,ret_list].values\n",
    "\n",
    "    ##### Note: all portfolio weights considered to be beginning of period weights\n",
    "    ##### (so multiply by contemporaneous realized returns)\n",
    "    wgts = tangency(means, cov, rf, MIN_WGT, MAX_WGT)\n",
    "    df.loc[i,wgt_list] = wgts\n",
    "    df.loc[i,'risky_ret'] = wgts @ rets\n",
    "\n",
    "    # Capital allocation and overall portfolio return\n",
    "    risky_expret = wgts @ means\n",
    "    risky_sd     = np.sqrt(wgts @ cov @ wgts)\n",
    "    df.loc[i,'wgt_risky'] = capital_allocation(risky_expret, risky_sd, rf, RAVER)\n",
    "    df.loc[i,'port_ret']  = rf + df.loc[i,'wgt_risky']*(df.loc[i,'risky_ret'] -rf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Date</th>\n",
       "      <th>rf</th>\n",
       "      <th>risky_ret</th>\n",
       "      <th>wgt_risky</th>\n",
       "      <th>port_ret</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1985-01</td>\n",
       "      <td>0.0065</td>\n",
       "      <td>0.030422</td>\n",
       "      <td>1.092042</td>\n",
       "      <td>0.032624</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1985-02</td>\n",
       "      <td>0.0058</td>\n",
       "      <td>0.073954</td>\n",
       "      <td>1.284388</td>\n",
       "      <td>0.093336</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1985-03</td>\n",
       "      <td>0.0062</td>\n",
       "      <td>0.010399</td>\n",
       "      <td>1.243305</td>\n",
       "      <td>0.011421</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1985-04</td>\n",
       "      <td>0.0072</td>\n",
       "      <td>-0.069171</td>\n",
       "      <td>1.081089</td>\n",
       "      <td>-0.075364</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1985-05</td>\n",
       "      <td>0.0066</td>\n",
       "      <td>0.033783</td>\n",
       "      <td>1.140198</td>\n",
       "      <td>0.037594</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      Date      rf  risky_ret  wgt_risky  port_ret\n",
       "1  1985-01  0.0065   0.030422   1.092042  0.032624\n",
       "2  1985-02  0.0058   0.073954   1.284388  0.093336\n",
       "3  1985-03  0.0062   0.010399   1.243305  0.011421\n",
       "4  1985-04  0.0072  -0.069171   1.081089 -0.075364\n",
       "5  1985-05  0.0066   0.033783   1.140198  0.037594"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = df.dropna()\n",
    "df_mktmodel = df.copy()\n",
    "df[['Date']+['rf','risky_ret','wgt_risky','port_ret']].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Avg return of market model:\t  0.0200\n",
      "SD(return) of market model:\t  0.0940\n",
      "Sharpe ratio of market model:\t  0.1861\n"
     ]
    }
   ],
   "source": [
    "avg_ret = df.port_ret.mean()\n",
    "sd_ret  = df.port_ret.std()\n",
    "avg_rf  = df.rf.mean()\n",
    "sr = (avg_ret - avg_rf)/sd_ret\n",
    "print(f'Avg return of market model:\\t {avg_ret: .4f}')\n",
    "print(f'SD(return) of market model:\\t {sd_ret: .4f}')\n",
    "print(f'Sharpe ratio of market model:\\t {sr: .4f}')\n",
    "\n",
    "# Save results\n",
    "results.loc['Mkt Model','avg_ret'] = avg_ret\n",
    "results.loc['Mkt Model','sd_ret']  = sd_ret\n",
    "results.loc['Mkt Model','sr']      = sr"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equal-weighted"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Load data\n",
    "df = data.copy()\n",
    "\n",
    "# Make list of weights columns\n",
    "wgt_list= ['wgt'+asset for asset in asset_list] \n",
    "\n",
    "# Initialize empty columns for risky asset weights, risky portfolio return, CAL location, CAL portfolio return\n",
    "for wgt in wgt_list:\n",
    "    df[wgt] = np.nan\n",
    "df['risky_ret'] = np.nan\n",
    "df['wgt_risky'] = np.nan \n",
    "df['port_ret']  = np.nan\n",
    "\n",
    "# Choose optimal portfolios each time period\n",
    "for i in df.index[1:]:                              # Start at 2nd date because estimates include that month's obs\n",
    "    # Pull estimation inputs at each point in time\n",
    "    est_pull_date = df.loc[i-1,'Date']\n",
    "    means = est_means.loc[est_pull_date].values\n",
    "    sds   = est_sds.loc[est_pull_date].values\n",
    "    C     = est_corrs.loc[(est_pull_date,slice(None))].to_numpy()\n",
    "    cov = np.diag(sds) @ C @ np.diag(sds)\n",
    "\n",
    "    # Pull current period returns\n",
    "    rf = df.loc[i,'rf']\n",
    "    rets = df.loc[i,ret_list].values\n",
    "\n",
    "    ##### Note: all portfolio weights considered to be beginning of period weights\n",
    "    ##### (so multiply by contemporaneous realized returns)\n",
    "    wgts = (1/n)*np.ones(n)\n",
    "    df.loc[i,wgt_list] = wgts\n",
    "    df.loc[i,'risky_ret'] = wgts @ rets\n",
    "\n",
    "    # Capital allocation and overall portfolio return\n",
    "    risky_expret = wgts @ means\n",
    "    risky_sd     = np.sqrt(wgts @ cov @ wgts)\n",
    "    df.loc[i,'wgt_risky'] = max(0,capital_allocation(risky_expret, risky_sd, rf, RAVER))\n",
    "    df.loc[i,'port_ret']  = rf + df.loc[i,'wgt_risky']*(df.loc[i,'risky_ret'] -rf)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>Date</th>\n",
       "      <th>rf</th>\n",
       "      <th>risky_ret</th>\n",
       "      <th>wgt_risky</th>\n",
       "      <th>port_ret</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>1985-01</td>\n",
       "      <td>0.0065</td>\n",
       "      <td>0.097065</td>\n",
       "      <td>0.285082</td>\n",
       "      <td>0.032318</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>1985-02</td>\n",
       "      <td>0.0058</td>\n",
       "      <td>0.017535</td>\n",
       "      <td>0.414214</td>\n",
       "      <td>0.010661</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>1985-03</td>\n",
       "      <td>0.0062</td>\n",
       "      <td>-0.003635</td>\n",
       "      <td>0.376748</td>\n",
       "      <td>0.002495</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1985-04</td>\n",
       "      <td>0.0072</td>\n",
       "      <td>-0.014421</td>\n",
       "      <td>0.258474</td>\n",
       "      <td>0.001612</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5</th>\n",
       "      <td>1985-05</td>\n",
       "      <td>0.0066</td>\n",
       "      <td>0.057908</td>\n",
       "      <td>0.312556</td>\n",
       "      <td>0.022637</td>\n",
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       "      Date      rf  risky_ret  wgt_risky  port_ret\n",
       "1  1985-01  0.0065   0.097065   0.285082  0.032318\n",
       "2  1985-02  0.0058   0.017535   0.414214  0.010661\n",
       "3  1985-03  0.0062  -0.003635   0.376748  0.002495\n",
       "4  1985-04  0.0072  -0.014421   0.258474  0.001612\n",
       "5  1985-05  0.0066   0.057908   0.312556  0.022637"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "df = df.dropna()\n",
    "df_ew = df.copy()\n",
    "df[['Date']+['rf','risky_ret','wgt_risky','port_ret']].head()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Avg return of Equal-weighted:\t  0.0105\n",
      "SD(return) of Equal-weighted:\t  0.0509\n",
      "Sharpe ratio of Equal-weighted:\t  0.1571\n"
     ]
    }
   ],
   "source": [
    "avg_ret = df.port_ret.mean()\n",
    "sd_ret  = df.port_ret.std()\n",
    "avg_rf  = df.rf.mean()\n",
    "sr = (avg_ret - avg_rf)/sd_ret\n",
    "print(f'Avg return of Equal-weighted:\\t {avg_ret: .4f}')\n",
    "print(f'SD(return) of Equal-weighted:\\t {sd_ret: .4f}')\n",
    "print(f'Sharpe ratio of Equal-weighted:\\t {sr: .4f}')\n",
    "\n",
    "# Save results\n",
    "results.loc['Equal-weighted','avg_ret'] = avg_ret\n",
    "results.loc['Equal-weighted','sd_ret']  = sd_ret\n",
    "results.loc['Equal-weighted','sr']      = sr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparing the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
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       "      <th>sr</th>\n",
       "      <th>avg_ret</th>\n",
       "      <th>sd_ret</th>\n",
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       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>Pairwise</th>\n",
       "      <td>0.1645</td>\n",
       "      <td>0.0164</td>\n",
       "      <td>0.0845</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Mkt Model</th>\n",
       "      <td>0.1861</td>\n",
       "      <td>0.0200</td>\n",
       "      <td>0.0940</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Equal-weighted</th>\n",
       "      <td>0.1571</td>\n",
       "      <td>0.0105</td>\n",
       "      <td>0.0509</td>\n",
       "    </tr>\n",
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       "                    sr  avg_ret  sd_ret\n",
       "Pairwise        0.1645   0.0164  0.0845\n",
       "Mkt Model       0.1861   0.0200  0.0940\n",
       "Equal-weighted  0.1571   0.0105  0.0509"
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     "execution_count": 24,
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   ],
   "source": [
    "results.round(4)"
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Plot the data"
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   "cell_type": "code",
   "execution_count": 25,
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          },
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            "ticks": ""
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           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
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          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
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           "zerolinewidth": 2
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          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
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        },
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      }
     },
     "metadata": {},
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   ],
   "source": [
    "results = results.reset_index()\n",
    "\n",
    "fig = go.Figure()\n",
    "string = \"Sharpe ratio: %{y:0.3f}<br>\"\n",
    "string += \"Average return: %{customdata[0]:0.1%}<br>\"\n",
    "string += \"SD(return): %{customdata[1]:0.1%}<br>\"\n",
    "string += \"<extra></extra>\"\n",
    "trace = go.Bar(x=results['index'], y=results['sr'], \n",
    "    customdata=results[['avg_ret','sd_ret']], \n",
    "    hovertemplate=string)\n",
    "fig.add_trace(trace)\n",
    "fig.layout.yaxis[\"title\"] = \"Sharpe ratio\"\n",
    "fig.layout.xaxis[\"title\"] = \"Strategy\"\n",
    "fig.show()"
   ]
  }
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