Log Returns.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {
    "toc": true
   },
   "source": [
    "<h1>Table of Contents<span class=\"tocSkip\"></span></h1>\n",
    "<div class=\"toc\"><ul class=\"toc-item\"><li><span><a href=\"#-Adjustification-of-Log-Returns-\" data-toc-modified-id=\"-Adjustification-of-Log-Returns--1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span><font face=\"gotham\" color=\"purple\"> Adjustification of Log Returns </font></a></span></li><li><span><a href=\"#-Relations-of-Log-Return-and-Simple-Return\" data-toc-modified-id=\"-Relations-of-Log-Return-and-Simple-Return-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span><font face=\"gotham\" color=\"purple\"> Relations of Log Return and Simple Return</font></a></span></li></ul></div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "import scipy as sp\n",
    "import scipy.stats\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Adjustification of Log Returns </font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The adjustification of using log return/growth is to avoid positive bias of arithmic return. An example will show the difference."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose you are hold a stock share for four periods."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "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>price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <td>0</td>\n",
       "      <td>100</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>1</td>\n",
       "      <td>200</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>2</td>\n",
       "      <td>150</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>3</td>\n",
       "      <td>90</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      ],
      "text/plain": [
       "   price\n",
       "0    100\n",
       "1    200\n",
       "2    150\n",
       "3     90"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data = pd.DataFrame([100, 200, 150, 90], columns = ['price']);data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To calculate the simple arithmetic return:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "price    0.116667\n",
       "dtype: float64"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "data.pct_change().mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We found a ludicrous outcome, that even if you lost money comparing to your initial investment, you have an average of $11.67\\%$ annual return."
   ]
  },
  {
   "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>price</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <td>0</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>1</td>\n",
       "      <td>0.693147</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>2</td>\n",
       "      <td>-0.287682</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <td>3</td>\n",
       "      <td>-0.510826</td>\n",
       "    </tr>\n",
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      ],
      "text/plain": [
       "      price\n",
       "0       NaN\n",
       "1  0.693147\n",
       "2 -0.287682\n",
       "3 -0.510826"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logReturn = np.log(data) - np.log(data.shift(1)); logReturn"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can that annual log return gives us a fairly precise measurement of the performance, annual growth rate of $-3.51\\%$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "price   -0.03512\n",
       "dtype: float64"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "logReturn.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <font face=\"gotham\" color=\"purple\"> Relations of Log Return and Simple Return</font>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The log return is calculated by\n",
    "\n",
    "$$\n",
    "r_t = \\ln \\frac{P_t}{P_{t-1}} = p_t-p_{t-1}\n",
    "$$\n",
    "\n",
    "where $p_t$ is $\\ln P_t$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We take the exponential to recover single return $R_t$\n",
    "\n",
    "$$\n",
    "e^{r_t} = \\frac{P_t}{P_{t-1}} = 1+R_{t} \\quad \\Longrightarrow \\quad R_t = e^{r_t}-1\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [],
   "source": [
    "R = np.linspace(-1,1,100)\n",
    "r = np.log(R+1)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
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\n",
      "text/plain": [
       "<Figure size 720x720 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize = (10, 10))\n",
    "ax.plot(R, r, label = '$r_t = \\ln(R_t+1)$')\n",
    "ax.plot(R, R, label = '$y =x$', ls = '--')\n",
    "ax.set_xlabel('$R_t$', size = 18)\n",
    "ax.set_ylabel('$r_t$', size = 18)\n",
    "ax.scatter(0, 0)\n",
    "ax.set_xlim([-.7, .7])\n",
    "ax.set_ylim([-1.3, .7])\n",
    "ax.legend(fontsize = 14)\n",
    "ax.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see if $R_t$ is small enough, it can be approximated by $r_t$."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.4"
  },
  "toc": {
   "base_numbering": 1,
   "nav_menu": {},
   "number_sections": true,
   "sideBar": true,
   "skip_h1_title": false,
   "title_cell": "Table of Contents",
   "title_sidebar": "Contents",
   "toc_cell": true,
   "toc_position": {},
   "toc_section_display": true,
   "toc_window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}