ml-finance-python

lab_108.ipynb

(35618B)

 {
  "cells": [
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "# The Efficient Frontier - Part II\n",
     "\n",
     "Let's start by loading the returns and generating the expected returns vector and the covariance matrix"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 1,
    "metadata": {},
    "outputs": [],
    "source": [
     "%load_ext autoreload\n",
     "%autoreload 2\n",
     "%matplotlib inline\n",
     "import edhec_risk_kit as erk\n",
     "\n",
     "ind = erk.get_ind_returns()\n",
     "er = erk.annualize_rets(ind[\"1996\":\"2000\"], 12)\n",
     "cov = ind[\"1996\":\"2000\"].cov()"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "As a first exercise, let's assume we have some weights, and let's try and compute the returns and volatility of a portfolio, given a set of weights, returns, and a covariance matrix.\n",
     "\n",
     "The returns are easy, so let's add this to our toolkit\n",
     "\n",
     "```python\n",
     "\n",
     "def portfolio_return(weights, returns):\n",
     "    \"\"\"\n",
     "    Computes the return on a portfolio from constituent returns and weights\n",
     "    weights are a numpy array or Nx1 matrix and returns are a numpy array or Nx1 matrix\n",
     "    \"\"\"\n",
     "    return weights.T @ returns\n",
     "\n",
     "```\n",
     "\n",
     "The volatility is just as easy in matrix form:\n",
     "\n",
     "```python\n",
     "def portfolio_vol(weights, covmat):\n",
     "    \"\"\"\n",
     "    Computes the vol of a portfolio from a covariance matrix and constituent weights\n",
     "    weights are a numpy array or N x 1 maxtrix and covmat is an N x N matrix\n",
     "    \"\"\"\n",
     "    return (weights.T @ covmat @ weights)**0.5\n",
     "```\n",
     "\n"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 2,
    "metadata": {},
    "outputs": [],
    "source": [
     "l = [\"Food\", \"Beer\", \"Smoke\", \"Coal\"]"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 3,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "Food     0.116799\n",
        "Beer     0.141126\n",
        "Smoke    0.107830\n",
        "Coal     0.414689\n",
        "dtype: float64"
       ]
      },
      "execution_count": 3,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "er[l]"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 4,
    "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>Coal</th>\n",
        "    </tr>\n",
        "  </thead>\n",
        "  <tbody>\n",
        "    <tr>\n",
        "      <th>Food</th>\n",
        "      <td>0.002609</td>\n",
        "      <td>0.002379</td>\n",
        "      <td>0.002061</td>\n",
        "      <td>0.000027</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Beer</th>\n",
        "      <td>0.002379</td>\n",
        "      <td>0.005264</td>\n",
        "      <td>0.001359</td>\n",
        "      <td>0.001728</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Smoke</th>\n",
        "      <td>0.002061</td>\n",
        "      <td>0.001359</td>\n",
        "      <td>0.008349</td>\n",
        "      <td>-0.000733</td>\n",
        "    </tr>\n",
        "    <tr>\n",
        "      <th>Coal</th>\n",
        "      <td>0.000027</td>\n",
        "      <td>0.001728</td>\n",
        "      <td>-0.000733</td>\n",
        "      <td>0.018641</td>\n",
        "    </tr>\n",
        "  </tbody>\n",
        "</table>\n",
        "</div>"
       ],
       "text/plain": [
        "           Food      Beer     Smoke      Coal\n",
        "Food   0.002609  0.002379  0.002061  0.000027\n",
        "Beer   0.002379  0.005264  0.001359  0.001728\n",
        "Smoke  0.002061  0.001359  0.008349 -0.000733\n",
        "Coal   0.000027  0.001728 -0.000733  0.018641"
       ]
      },
      "execution_count": 4,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "cov.loc[l,l]"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 5,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "0.19511097196038385"
       ]
      },
      "execution_count": 5,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "import pandas as pd\n",
     "import numpy as np\n",
     "ew = np.repeat(0.25, 4)\n",
     "erk.portfolio_return(ew, er[l])"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 6,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "0.055059195776437045"
       ]
      },
      "execution_count": 6,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "erk.portfolio_vol(ew, cov.loc[l,l])"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "# The 2-Asset Case\n",
     "\n",
     "In the case of 2 assets, the problem is somewhat simplified, since the weight of the second asset is 1-the weight of the first asset.\n",
     "\n",
     "Let's write a function that draws the efficient frontier for a simple 2 asset case.\n",
     "\n",
     "Start by generating a sequence of weights in a list of tuples. Python makes it easy to generate a list by using something called a _list comprehension_ ... which you can think of as an efficient way to generate a list of values instead of writing a for loop.\n"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 7,
    "metadata": {},
    "outputs": [],
    "source": [
     "import numpy as np\n",
     "\n",
     "n_points = 20\n",
     "weights = [np.array([w, 1-w]) for w in np.linspace(0, 1, n_points)]\n"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 8,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "list"
       ]
      },
      "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "type(weights)"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 9,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "20"
       ]
      },
      "execution_count": 9,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "len(weights)"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 10,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "array([0., 1.])"
       ]
      },
      "execution_count": 10,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "weights[0]"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 11,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "array([0.21052632, 0.78947368])"
       ]
      },
      "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "weights[4]"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 12,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "array([1., 0.])"
       ]
      },
      "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
     "weights[19]"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 13,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "<matplotlib.axes._subplots.AxesSubplot at 0x1a1cc43c18>"
       ]
      },
      "execution_count": 13,
      "metadata": {},
      "output_type": "execute_result"
     },
     {
      "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 = [\"Games\", \"Fin\"]\n",
     "rets = [erk.portfolio_return(w, er[l]) for w in weights]\n",
     "vols = [erk.portfolio_vol(w, cov.loc[l,l]) for w in weights]\n",
     "ef = pd.DataFrame({\"R\": rets, \"V\": vols})\n",
     "ef.plot.scatter(x=\"V\", y=\"R\")"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
     "We can create function that plots the frontier:\n",
     "\n",
     "```python\n",
     "def plot_ef2(n_points, er, cov):\n",
     "    \"\"\"\n",
     "    Plots the 2-asset efficient frontier\n",
     "    \"\"\"\n",
     "    if er.shape[0] != 2 or er.shape[0] != 2:\n",
     "        raise ValueError(\"plot_ef2 can only plot 2-asset frontiers\")\n",
     "    weights = [np.array([w, 1-w]) for w in np.linspace(0, 1, n_points)]\n",
     "    rets = [portfolio_return(w, er) for w in weights]\n",
     "    vols = [portfolio_vol(w, cov) for w in weights]\n",
     "    ef = pd.DataFrame({\n",
     "        \"Returns\": rets, \n",
     "        \"Volatility\": vols\n",
     "    })\n",
     "    return ef.plot.line(x=\"Volatility\", y=\"Returns\", style=\".-\")\n",
     "```\n",
     "\n",
     "A useful summary of the visualization features in pandas is [here](https://pandas.pydata.org/pandas-docs/stable/user_guide/visualization.html)\n"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 14,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
        "<matplotlib.axes._subplots.AxesSubplot at 0x1a1cd7bc88>"
       ]
      },
      "execution_count": 14,
      "metadata": {},
      "output_type": "execute_result"
     },
     {
      "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].values, cov.loc[l,l])"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
    "metadata": {},
    "outputs": [],
    "source": []
   }
  ],
  "metadata": {
   "kernelspec": {
    "display_name": "Python 3",
    "language": "python",
    "name": "python3"
   },
   "language_info": {
    "codemirror_mode": {
     "name": "ipython",
     "version": 3
    },
    "file_extension": ".py",
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