02 Introduction to Portfolio Analysis - Portfolio Optimization.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Portfolio Optimization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import edhc_risk_kit as erk\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "%load_ext autoreload\n",
    "%autoreload 2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The data used in this notebook is available at http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html#Research."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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>Food</th>\n",
       "      <th>Beer</th>\n",
       "      <th>Smoke</th>\n",
       "      <th>Games</th>\n",
       "      <th>Books</th>\n",
       "      <th>Hshld</th>\n",
       "      <th>Clths</th>\n",
       "      <th>Hlth</th>\n",
       "      <th>Chems</th>\n",
       "      <th>Txtls</th>\n",
       "      <th>...</th>\n",
       "      <th>Telcm</th>\n",
       "      <th>Servs</th>\n",
       "      <th>BusEq</th>\n",
       "      <th>Paper</th>\n",
       "      <th>Trans</th>\n",
       "      <th>Whlsl</th>\n",
       "      <th>Rtail</th>\n",
       "      <th>Meals</th>\n",
       "      <th>Fin</th>\n",
       "      <th>Other</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>1926-07</th>\n",
       "      <td>0.0056</td>\n",
       "      <td>-0.0519</td>\n",
       "      <td>0.0129</td>\n",
       "      <td>0.0293</td>\n",
       "      <td>0.1097</td>\n",
       "      <td>-0.0048</td>\n",
       "      <td>0.0808</td>\n",
       "      <td>0.0177</td>\n",
       "      <td>0.0814</td>\n",
       "      <td>0.0039</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0083</td>\n",
       "      <td>0.0922</td>\n",
       "      <td>0.0206</td>\n",
       "      <td>0.0770</td>\n",
       "      <td>0.0193</td>\n",
       "      <td>-0.2379</td>\n",
       "      <td>0.0007</td>\n",
       "      <td>0.0187</td>\n",
       "      <td>0.0037</td>\n",
       "      <td>0.0520</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-08</th>\n",
       "      <td>0.0259</td>\n",
       "      <td>0.2703</td>\n",
       "      <td>0.0650</td>\n",
       "      <td>0.0055</td>\n",
       "      <td>0.1001</td>\n",
       "      <td>-0.0358</td>\n",
       "      <td>-0.0251</td>\n",
       "      <td>0.0425</td>\n",
       "      <td>0.0550</td>\n",
       "      <td>0.0814</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0217</td>\n",
       "      <td>0.0202</td>\n",
       "      <td>0.0439</td>\n",
       "      <td>-0.0238</td>\n",
       "      <td>0.0488</td>\n",
       "      <td>0.0539</td>\n",
       "      <td>-0.0075</td>\n",
       "      <td>-0.0013</td>\n",
       "      <td>0.0446</td>\n",
       "      <td>0.0676</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-09</th>\n",
       "      <td>0.0116</td>\n",
       "      <td>0.0402</td>\n",
       "      <td>0.0126</td>\n",
       "      <td>0.0658</td>\n",
       "      <td>-0.0099</td>\n",
       "      <td>0.0073</td>\n",
       "      <td>-0.0051</td>\n",
       "      <td>0.0069</td>\n",
       "      <td>0.0533</td>\n",
       "      <td>0.0231</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0241</td>\n",
       "      <td>0.0225</td>\n",
       "      <td>0.0019</td>\n",
       "      <td>-0.0554</td>\n",
       "      <td>0.0005</td>\n",
       "      <td>-0.0787</td>\n",
       "      <td>0.0025</td>\n",
       "      <td>-0.0056</td>\n",
       "      <td>-0.0123</td>\n",
       "      <td>-0.0386</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-10</th>\n",
       "      <td>-0.0306</td>\n",
       "      <td>-0.0331</td>\n",
       "      <td>0.0106</td>\n",
       "      <td>-0.0476</td>\n",
       "      <td>0.0947</td>\n",
       "      <td>-0.0468</td>\n",
       "      <td>0.0012</td>\n",
       "      <td>-0.0057</td>\n",
       "      <td>-0.0476</td>\n",
       "      <td>0.0100</td>\n",
       "      <td>...</td>\n",
       "      <td>-0.0011</td>\n",
       "      <td>-0.0200</td>\n",
       "      <td>-0.0109</td>\n",
       "      <td>-0.0508</td>\n",
       "      <td>-0.0264</td>\n",
       "      <td>-0.1538</td>\n",
       "      <td>-0.0220</td>\n",
       "      <td>-0.0411</td>\n",
       "      <td>-0.0516</td>\n",
       "      <td>-0.0849</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1926-11</th>\n",
       "      <td>0.0635</td>\n",
       "      <td>0.0729</td>\n",
       "      <td>0.0455</td>\n",
       "      <td>0.0166</td>\n",
       "      <td>-0.0580</td>\n",
       "      <td>-0.0054</td>\n",
       "      <td>0.0187</td>\n",
       "      <td>0.0542</td>\n",
       "      <td>0.0520</td>\n",
       "      <td>0.0311</td>\n",
       "      <td>...</td>\n",
       "      <td>0.0163</td>\n",
       "      <td>0.0377</td>\n",
       "      <td>0.0364</td>\n",
       "      <td>0.0384</td>\n",
       "      <td>0.0160</td>\n",
       "      <td>0.0467</td>\n",
       "      <td>0.0652</td>\n",
       "      <td>0.0433</td>\n",
       "      <td>0.0224</td>\n",
       "      <td>0.0400</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>5 rows × 30 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "           Food    Beer   Smoke   Games   Books   Hshld   Clths    Hlth  \\\n",
       "1926-07  0.0056 -0.0519  0.0129  0.0293  0.1097 -0.0048  0.0808  0.0177   \n",
       "1926-08  0.0259  0.2703  0.0650  0.0055  0.1001 -0.0358 -0.0251  0.0425   \n",
       "1926-09  0.0116  0.0402  0.0126  0.0658 -0.0099  0.0073 -0.0051  0.0069   \n",
       "1926-10 -0.0306 -0.0331  0.0106 -0.0476  0.0947 -0.0468  0.0012 -0.0057   \n",
       "1926-11  0.0635  0.0729  0.0455  0.0166 -0.0580 -0.0054  0.0187  0.0542   \n",
       "\n",
       "          Chems   Txtls  ...   Telcm   Servs   BusEq   Paper   Trans   Whlsl  \\\n",
       "1926-07  0.0814  0.0039  ...  0.0083  0.0922  0.0206  0.0770  0.0193 -0.2379   \n",
       "1926-08  0.0550  0.0814  ...  0.0217  0.0202  0.0439 -0.0238  0.0488  0.0539   \n",
       "1926-09  0.0533  0.0231  ...  0.0241  0.0225  0.0019 -0.0554  0.0005 -0.0787   \n",
       "1926-10 -0.0476  0.0100  ... -0.0011 -0.0200 -0.0109 -0.0508 -0.0264 -0.1538   \n",
       "1926-11  0.0520  0.0311  ...  0.0163  0.0377  0.0364  0.0384  0.0160  0.0467   \n",
       "\n",
       "          Rtail   Meals     Fin   Other  \n",
       "1926-07  0.0007  0.0187  0.0037  0.0520  \n",
       "1926-08 -0.0075 -0.0013  0.0446  0.0676  \n",
       "1926-09  0.0025 -0.0056 -0.0123 -0.0386  \n",
       "1926-10 -0.0220 -0.0411 -0.0516 -0.0849  \n",
       "1926-11  0.0652  0.0433  0.0224  0.0400  \n",
       "\n",
       "[5 rows x 30 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Monthly returns of thirty different industry portfolios\n",
    "ind30 = erk.get_ind_returns()\n",
    "ind30.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Introduction to Optimization and The Efficient Frontier\n",
    "\n",
    "### Sharpe Ratio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Sharpe ratio was developed by Nobel laureate William F. Sharpe and is used to help investors understand the return of an investment compared to its risk. The ratio is the average return earned in excess of the risk-free rate per unit of volatility or total risk. Subtracting the risk-free rate from the mean return allows an investor to better isolate the profits associated with risk-taking activities. Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Calculate Sharpe ration with a risk free rate of 3% and 12 periods per year\n",
    "erk.sharpe_ratio(ind30['1996':'2000'], 0.03, 12).sort_values().plot.bar(figsize=(12,6))\n",
    "plt.xlabel('Portfolios')\n",
    "plt.ylabel('Sharpe Ratio')\n",
    "plt.title('Industry Sharpe Ratios from 1996 to 2000');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The Efficient Frontier\n",
    "\n",
    "The efficient frontier is the set of optimal portfolios that offer the highest expected return for a defined level of risk or the lowest risk for a given level of expected return. Portfolios that lie below the efficient frontier are sub-optimal because they do not provide enough return for the level of risk. Portfolios that cluster to the right of the efficient frontier are sub-optimal because they have a higher level of risk for the defined rate of return.  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![title](data/efficient_frontier.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Get expected returns\n",
    "er = erk.annualize_rets(ind30['1996':'2000'], 12)\n",
    "er.sort_values().plot.bar(figsize=(12,6))\n",
    "plt.xlabel('Portfolios')\n",
    "plt.ylabel('Expected Returns')\n",
    "plt.title('Industry Expected Returns from 1996 to 2000');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to calculate the efficient frontier, we need to have an estimate of the expected returns and the covariance matrix for the set of risky securities which will used to build the optimal portfolio.  These parameters are difficult (impossible) to forecast, and the optimal portfolio calculation is extremely sensitive to these parameters.  \n",
    "\n",
    "The function `portfolio_return` returns the matrix muliplication between the transposed vector of weights and the returns. The function `portfolio_vol` calculates the matrix multiplication between the transposed vector of weights, the covariance matrix of the returns, and the weights."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "cov = ind30['1996': '2000'].cov()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Calculate the return and volatility given a set of weights and returns.**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Return: 0.19511097196038385\n",
      "Volatility: 0.055059195776437045\n"
     ]
    }
   ],
   "source": [
    "weights = np.repeat(1/4, 4) # Vector of weights\n",
    "l = ['Food', 'Beer', 'Smoke', 'Coal'] # Names of the returns\n",
    "print('Return:', erk.portfolio_return(weights, er[l]))\n",
    "print('Volatility:', erk.portfolio_vol(weights, cov.loc[l, l]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Two Asset Frontier"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "l = ['Fin', 'Beer']\n",
    "erk.plot_ef2(25, er[l], cov.loc[l, l]);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### N Asset Frontier\n",
    "\n",
    "The objective is to maximize the expected return, and minimize volatility. The function `minimize_vol` in the edhec_risk_kit module takes a set of target returns, the returns from a portfolio, and the covariance matrix for that portfolio and calculates a set of weights for various asset allocations such that the volatility for a particular combination is minimized and the return is maximized. Subsequently, the efficient frontier is plotted using `plot_ef` in the edhec_risk_kit module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "l = ['Smoke', 'Fin', 'Games', 'Coal']\n",
    "erk.plot_ef(25, er[l], cov.loc[l, l])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementing Markowitz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The capital market line (CML) represents portfolios that optimally combine risk and return. Capital asset pricing model (CAPM) depicts the trade-off between risk and return for efficient portfolios. It is a theoretical concept that represents all the portfolios that optimally combine the risk-free rate of return and the market portfolio of risky assets. Under CAPM, all investors will choose a position on the capital market line, in equilibrium, by borrowing or lending at the risk-free rate, since this maximizes return for a given level of risk. The point where the CML is tangent to the efficient frontier is called the Maximum Sharpe Ratio Portfolio.\n",
    "\n",
    "![title](data/sharpe_portfolio.gif)\n",
    "\n",
    "Note that the efficient frontier changes shape dramatically when a risk-free asset is introduced."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Locating the Max Sharpe Ratio Portfolio"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = erk.plot_ef(20, er, cov)\n",
    "plt.gca().set_xlim(left=0)\n",
    "rf = 0.1\n",
    "w_msr = erk.msr(rf, er, cov)\n",
    "r_msr = erk.portfolio_return(w_msr, er)\n",
    "vol_msr = erk.portfolio_vol(w_msr, cov)\n",
    "cml_x = [0, vol_msr]\n",
    "cml_y = [rf, r_msr]\n",
    "plt.gca().plot(cml_x, cml_y, marker='o');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Along the minimum-variance frontier, the left-most point is a portfolio with minimum variance when compared to all possible portfolios of risky assets (shown in red in the plot below). This is known as the global minimum-variance portfolio. An investor cannot hold a portfolio of risky (note: risk-free assets are excluded at this point) assets with a lower risk than the global minimum-variance (GMV) portfolio.  \n",
    "\n",
    "The portion of the minimum-variance curve that lies above and to the right of the global minimum variance portfolio is known as the Markowitz efficient frontier as it contains all portfolios that rational, risk-averse investors would choose.\n",
    "\n",
    "The function `plot_ef` in the `erk` module allows one to plot the GMV portfolio by setting the `show_gmv` parameter to True and the equal weight portfolio (EW, shown in yellow) by setting `show_ew` to True as well."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "erk.plot_ef(20, er, cov, show_cml=True, riskfree_rate=0.1, show_ew=True, show_gmv=True);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that very small differences in the expected returns, which are likely present if we attempt to forecast them, can have a dramatic effect on the asset allocation in a portfolio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Returns: [0.1168, 0.1158]\n",
      "Portfolio: [0.7504, 0.2496]\n"
     ]
    }
   ],
   "source": [
    "# Calculate the asset allocation for returns in-sample (!)\n",
    "l = ['Food', 'Steel']\n",
    "print('Returns:', er[l].round(4).tolist())\n",
    "print('Portfolio:', erk.msr(0.1, er[l], cov.loc[l, l]).round(4).tolist())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Portfolio: [0.5793, 0.4207]\n",
      "Portfolio: [0.0, 1.0]\n",
      "Portfolio: [1.0, 0.0]\n"
     ]
    }
   ],
   "source": [
    "# Attempt to guess the returns\n",
    "print('Portfolio:', erk.msr(0.1, np.array([0.11, 0.12]), cov.loc[l, l]).round(4).tolist())\n",
    "print('Portfolio:', erk.msr(0.1, np.array([0.10, 0.13]), cov.loc[l, l]).round(4).tolist())\n",
    "print('Portfolio:', erk.msr(0.1, np.array([0.13, 0.10]), cov.loc[l, l]).round(4).tolist())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercises\n",
    "\n",
    "Use the EDHEC Hedge Fund Indices data set. Perform the following analysis based on data since 2000 (including all of 2000)\n",
    "\n",
    "**What was the Monthly Parametric Gaussian VaR at the 1% level (as a +ve number) of the Distressed Securities strategy?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3.14"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hfi = erk.get_hfi_returns()\n",
    "(erk.var_gaussian(hfi['2000':]['Distressed Securities'], level=1)*100).round(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**What was the 1% VaR for the same strategy after applying the Cornish-Fisher Adjustment?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.97"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(erk.var_gaussian(hfi['2000':]['Distressed Securities'], level=1, modified=True)*100).round(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**What was the Monthly Historic VaR at the 1% level (as a +ve number) of the Distressed Securities strategy?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.26"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(erk.var_historic(hfi['2000':]['Distressed Securities'], level=1)*100).round(2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using data during the 5 year period 2013-2017 (both inclusive) for the 30 industry return data, estimate the expected returns as well as the covariance matrix for the `Books`, `Steel`, `Oil`, and `Mines` industries. Assume the risk free rate over the 5 year period is 10%."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "l = ['Books', 'Steel', 'Oil', 'Mines']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "cov = ind30['2013':'2017'].cov()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**What is the weight of Steel in the EW Portfolio?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.25\n"
     ]
    }
   ],
   "source": [
    "print(0.25)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**What is the weight of the largest component of the MSR portfolio?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.786011784161686 Books\n"
     ]
    }
   ],
   "source": [
    "msr_weights = erk.msr(0.1, er[l], cov.loc[l, l])\n",
    "print(msr_weights.max(), l[msr_weights.argmax()])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**What is the weight of the largest component of the GMV portfolio?**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.47702487032166246 Books\n"
     ]
    }
   ],
   "source": [
    "gmv_weights = erk.gmv(cov.loc[l, l])\n",
    "print(gmv_weights.max(), l[gmv_weights.argmax()])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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