alpha.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tail estimation with lognormal distribution\n",
    "\n",
    "Suppose\n",
    "\n",
    "$$\n",
    "\\frac{S_{t+1}}{S_{t}} = A\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "- $S_t$ is the firm size at time $t$\n",
    "- $\\ln A \\sim N(\\mu, \\sigma^2)$\n",
    "\n",
    "\n",
    "We seek the $\\alpha$ satisfying\n",
    "\n",
    "$$\n",
    "\\mathbb E (A^{\\alpha} ) = 1 \n",
    "$$\n",
    "\n",
    "By the expression for the mean of a lognormal distribution, with $\\epsilon \\sim N(0,1)$,\n",
    "\n",
    "$$\n",
    "\\mathbb E (A^{\\alpha} ) =  \\mathbb E (\\exp (\\mu + \\sigma \\epsilon)^{\\alpha} )\n",
    "= \\mathbb E (\\exp (\\mu \\alpha + \\sigma \\epsilon \\alpha) ) \n",
    "= \\exp \\left(  \\alpha \\mu + \\frac{\\sigma^2 \\alpha^2}{2} \\right)\n",
    "$$\n",
    "\n",
    "\n",
    "Hence we seek the $\\alpha$ satisfying\n",
    "$$\n",
    "\\exp (  \\alpha \\mu + \\frac{\\sigma^2 \\alpha^2}{2} ) =1 \n",
    "$$\n",
    "\n",
    "Taking logarithm of both sides yields\n",
    "\n",
    "$$\n",
    "\\alpha \\mu + \\frac{\\sigma^2 \\alpha^2}{2} =0\n",
    "$$\n",
    "\n",
    "Rearranging, we get\n",
    "\n",
    "$$\n",
    "\\alpha = - \\frac{2 \\mu}{\\sigma^2} \n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Tail estimation with Laplacian distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we assume that $\\ln A$ has a Laplacian distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a function called ``laplace`` to generate density its density"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def laplace(x, μ, λ):\n",
    "    return 0.5 * λ * np.exp(- λ * np.abs(x - μ))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Example:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "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": [
    "x_grid = np.linspace(0.01, 10, 200)\n",
    "y_vals = laplace(x_grid, 5.0, 2.0)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "plt.plot(x_grid, y_vals)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $G = \\ln A$.\n",
    "\n",
    "We seek the $\\alpha$ such that $\\mathbb E A^\\alpha = \\mathbb E \\exp(\\alpha G) = 1$\n",
    "\n",
    "In other words, we seek the $\\alpha$ where $m(\\alpha) = 1$, where $m$ is the MGF of $G$.\n",
    "\n",
    "We can obtain this as the MGF of the Laplacian distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def m(α, μ=-0.02, λ=10):\n",
    "    \"Moment generating function, Laplace distribution\"\n",
    "    return np.exp(μ * α) / (1 - α**2 / λ**2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Does there exists an $\\alpha > 0$ such that $m(\\alpha) = 1$?"
   ]
  },
  {
   "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": [
    "alpha_grid = np.linspace(0.01, 3, 100)\n",
    "y_vals = m(alpha_grid)\n",
    "\n",
    "fig, ax = plt.subplots()\n",
    "plt.plot(alpha_grid, y_vals)\n",
    "plt.plot(alpha_grid, np.ones_like(alpha_grid), '--')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For these choices of $\\mu$ and $\\lambda$, the value of $\\alpha$ is around 1.9.\n",
    "\n",
    "What about if $\\mu$ and $\\lambda$ are estimated from IID observations of the growth rate?"
   ]
  },
  {
   "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",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.4"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}