{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Approximating real interest rates\n", "Randall Romero Aguilar\n", "\n", "August 2016\n", "\n", "------------------------------------\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The **real interest rate** $r$ is defined in terms of the **nominal interest rate** $i$ and the **inflation rate** $\\pi$ by\n", "\$$1+r = \\frac{1+i}{1+\\pi} \$$\n", "which is usually approximated by \n", "\$$r \\approx i - \\pi \$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Why it works" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* **Taylor approximation**\n", "\n", "This approximation is based on the first-order Taylor approximation of $\\ln(1+x) \\approx x$, which is fairly accurate for **small** values of $x$. Thus, using Taylor approximation we see that the real interest rate is\n", "\\begin{align}\n", "1+r &= \\frac{1+i}{1+\\pi} \\\\\\\\\n", "\\ln(1+r) &= \\ln(1+i) - \\ln(1+\\pi) \\\\\\\\\n", "r &\\approx i - \\pi\n", "\\end{align}\n", "\n", "* **An easier way**\n", "\n", "From the definition of the real interest rate, \n", "\\begin{align}\n", "(1+r)(1+\\pi) &= 1+i \\\\\\\\\n", "1+r+\\pi+r\\pi &= 1+i \\\\\\\\\n", "r &= i - \\pi - r\\pi \\\\\\\\\n", "r &\\approx i - \\pi\n", "\\end{align}\n", "where the last step follows because $r\\pi$ is a small quantity as long as $r$ and $\\pi$ are small.\n", "\n", "\n", "This numerical example illustrates the magnitude of approximation errors that we induce by taking this approximation." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Set up\n", "\n", "* Import necessary libraries" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import seaborn\n", "%matplotlib inline\n", "plt.rc('text', usetex=True)\n", "plt.rcParams['figure.figsize'] = 10, 6" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Generate data" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "a, b = -0.1, 0.3\n", "x = np.linspace(a, b, 120)\n", "y = np.log(1 + x)\n", "absolute_error = np.abs(x - y)\n", "relative_error = np.abs(x / y - 1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* This utility function we'll make it easier to work with the plots below" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def common_options(title, xlab, ylab):\n", " plt.xlim([a, b])\n", " plt.xlabel(xlab)\n", " plt.ylabel(ylab)\n", " plt.title(title)\n", " return" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Plots\n", "\n", "* The function and its approximation" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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FXp7d3dazp4e+fb1cdplOk5ZGAU5ERKQacmbsI2b+7BO73Cak4G3bPqhzeb3w1lv2adLP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\nIvMvNk2SonUaXD8AAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(x, y, 'b-', label='True value')\n", "plt.plot(x, x, 'r-', label='Approximation')\n", "common_options('Approximating an interest rate','$x$', '$\\ln(1 + x)$')\n", "plt.legend(loc='lower right', frameon=False)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Absolute errors\n", "\n", "Notice that the approximation error is less than half a percentage point as long as $x$ is less than 10% in absolute value.Beyond that range, the error grows quickly" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "scrolled": false }, "outputs": [ { "data": { "image/png": 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XKPmSPkbOGDNQ0hZJPa21dzXmuDGmn3v4FGvt2MacBwAA5K5vv/Vo6NCgwmFp8uSQunTJ\nzskNNSW1Rc4Y00NSzFq7SNIWY0z3eo5vNsb0cEPcBe5tPY0x3Rs6DwAAyF3hsDRiRIG+/tqrW2+t\n1MknR1JdpDaT7K7VQXJa0SRpraT+9RxfJ6mftXaRtfYK97au1tpVjTgPAADIUX/8Y77efNOvM86o\n0tVXZ9+iv/VJdpDrIGlT3PXOjT1ujBkjaWQjzwMAAHLQvHl+Pfxwng48MKL778/+yQ01pe1kB3cc\n3ChjTPtUlwUAAKSf99/36rrrCrTLLjE98US5iopSXaK2l+zJDpsldXIvd5C0saHjcePhVsnpRh3R\niPPUUlycg/+bjUC91EadJEa9JEa9JEa91EadJNZa9bJpkzRsmBQKSc8/Lx1/fLtWOW+mSXaQmyWp\nl6TFkkokLZAkY0x7a+13dRw/RdIK9/EdJL3j3n5UzfPUZ/36ra35OrJCcXER9VIDdZIY9ZIY9ZIY\n9VIbdZJYa9VLJCJdcklQ69b5df31FTr++EqtX98KBUyRloTbpHatWmtXSj8sJ7LZbWWTpIX1HJ8s\nqcQYUyqnZW5e9eMSnAcAAOSYcePy9OqrfvXvH9ZNN+XW5IaaPLFYVq6zEuMvodr4C7E26iQx6iUx\n6iUx6qU26iSx1qiX+fP9GjYsqK5do3rlle1qnwUj6YuLi5o9RSNtJzsAAADE+/BDr66+ukCFhTE9\n8UQoK0JcSyV9ZwcAAICW2rxZuvzyoMrKPHr00ZAOPjia6iKlBVrkAABAWnN2bgjqs8+8uv76Cp11\nVjjVRUobBDkAAJDW7rgjX6+95teppzK5oSaCHAAASFvz5vn14IPOzg2TJoXkJbnshOoAAABp6b33\ndt65YdddU12i9MNkBwAAkHbWr/doyJCgysul6dNDOuggJjckQoscAABIK5WV0rBhBfryS6/Gjq3U\ngAGRVBcpbRHkAABAWvn1r/O1dKlf55xTpWuvZXJDfQhyAAAgbTz+eEBPPJGnQw+N6N57y+Vp9p4H\nuYEgBwAA0sLSpT7demu+OnWKavr0kNq1S3WJ0h9BDgAApNwXX3g0dGiBYjFp2rRydemSlXvBtzpm\nrQIAgJTavl0aMiSoDRu8GjeuXCecwOSGxqJFDgAApEw0Kl19dYE++MCnIUMqNXRoVaqLlFEIcgAA\nIGXuuSdPL74Y0HHHhfWnP1UwuaGJCHIAACAlXnzRrzvvzFeXLlFNm1auvLxUlyjzEOQAAECbe/99\nr666qkCFhTE98URIu+3G5IbmYLIDAABoU9Xbb5WVefTYYyEddhjbbzUXLXIAAKDNVFRIv/xlgf77\nX6/Gjq3QmWeGU12kjEaQAwAAbSIWk8aMKdA77/h13nlVuu46tt9qKYIcAABoE/fcI82cGVCPHhH9\n5S9sv9UaCHIAACDpFizwacwYac89o3riiZCCwVSXKDsQ5AAAQFJ9/LFXI0cGlZ8vTZ8e0p57MkO1\ntTBrFQAAJM2GDR4NHhzUtm0ezZwpde/ODNXWRIscAABIiooK6Re/KNDnn3t1440VGjQo1SXKPgQ5\nAADQ6mIx6YYbnBmq555bpTFjmKGaDAQ5AADQ6u6/P0+zZgXUs2dE997LDNVkIcgBAIBW9de/+nX7\n7fnae29mqCYbQQ4AALSaf//bq9GjnT1UZ8wIaY89mKGaTMxaBQAAreLrr50ZqqGQ9Pjj5TriCGao\nJhstcgAAoMW2bZMGDw7qf//z6re/rdDpp7OHalsgyAEAgBaJRKQrryzQe+/5NHhwpa68sirVRcoZ\nBDkAANAit92Wr5deCqhPn7AmTKhghmobSnqQM8YMNMb0M8aMaexxY0yp+zM+7rbx1ceSXWYAANA4\nTz4Z0KRJeTrwwIimTQspEEh1iXJLUoOcMaaHpJi1dpGkLcaY7vUc32yM6WGM6SdpgbX2EUklxpi+\n7t1HGGNWS1qTzDIDAIDGee01n266KV+dOkX11FMhdeiQ6hLlnmS3yA2StMW9vFZS/3qOr3OPd427\n31pJJe7l4dbag6y1i5NXXAAA0Bgff+zV0KFBeb3ODNWuXVlmJBWSvfxIB0mb4q53buB4J2vtXXHX\ne0qa6V4ucVvreta4DwAAaEPffuvRz38e1NatHj38cEjHHhtJdZFyVtpOdnC7XVdYa1dJkrV2otsF\n2zmuuxUAALShsjJpyJCgvvjCq5tvrtD557PMSColu0Vus6RO7uUOkjY24Xg/a+0t0g8THDZaa+e5\n9ymRVG8Xa3FxUctKnqWol9qok8Sol8Sol8Sol9qysU6iUemKK6R//UsaMkQaNy5fHk9+k86RjfWS\nSskOcrMk9ZITukokLZAkY0x7a+139RwvtdZOdC/3k7RMzng5Seom6eGGnnj9+q2t+kKyQXFxEfVS\nA3WSGPWSGPWSGPVSW7bWyR/+kK+5c/N0/PFh/elPIW3Y0LTHZ2u9tFRLwm1Su1attSulH8LY5upu\nUkkL6zruXh5vjPnUGLNRzqzWVZIGGWMGSvo07jwAAKANPPZYQA8+6Cwz8thjIeXlpbpEkCRPLJaV\ns0xiJP7a+EuoNuokMeolMeolMeqltmyrk4ULfRo8OKhOnWL629/KdMABzcsO2VYvraW4uKjZSyin\n7WQHAACQeu+959Xw4UHl5UkzZoSaHeKQHMkeIwcAADLUl196dOmlQYVC0rRp5erVK5rqIqEGghwA\nAKjl+++lSy8N6ptvvPrjH8t11lksM5KO6FoFAAA7qayUfvnLoD76yKehQys1cmRVqouEOhDkAADA\nD2Ix6frrC7RkiV+nnValO+6okKfZQ/GRbAQ5AADwgwkT8jRrVkA9e0b08MPl8vlSXSLUp94xcu6O\nCu1r3OyRFKtxfbO1dmorlw0AALShJ58M6J578rX//lHNmBFSYWGqS4SG1BvkrLWPtFVBAABA6ixe\n7NOYMfnq1CmqZ58tU3Exy4xkgibNWjXG7Cpnb9RN1trvk1MkAADQlt5916uhQ4MKBJy14kpKCHGZ\nosEgZ4xpL+kWOXuhxiStk9TBGNNJ0hpJ4wh1AABkpv/8x6NLLnHWinv00XL17s1acZmkoTFy/SS1\nt9aOrec+A40xm621i1u9dAAAIGk2bvTo4osLtWGDV+PGlevMM1krLtM0NGt1ubV2Xn13sNbOlbSi\n9YoEAACSraxMGjw4qLVrvbr66goNG8ZacZmo3iBnrf2u+rIx5kZjTHf3cte67gcAANJbOCyNGlWg\nFSt8GjiwSr/+dWWqi4Rmaso6cuskrXUvbzLGnJ+E8gAAgCSKxaSxY/P10ksB9ekT1r33lsvLqrIZ\nqymzVkskPWKMWSBpeZLKAwAAkuiee/I0fXqeDjssosceCykvL9UlQks0JcitlTRHUk9JoyR9mpQS\nAQCApHjyyYAmTMhXly5RzZwZ0q67prpEaKmmNqZudCc33Kwd3awAACDNvfSSTzfemK/OnZ0Ff/fY\ng7XiskGjg5wb4Dq7VztJ6paUEgEAgFb1zjtejRgRVEGB9OSTIXXrRojLFk3a2cFauy7u37uSUiIA\nANBqPvnEq8suK1RVlbNrQ69eLPibTZoU5AAAQOb48kuPLrooqM2bPbrvvpD694+kukhoZUw4BgAg\nC23aJA0aFNRXX3n1m99U6OKL2bUhGzU5yLF+HAAA6W37dunnPy/UJ5/4NHJkpa66igV/s1VzWuRK\nWr0UAACgVVRVSaWlQa1Y4dMFF1TpD3+okMeT6lIhWehaBQAgS0Sj0nXXFWjhQr/69mXXhlzQ4GQH\nY0x7SSMkxSR5JPU3xsi9HJM0xVr7fTILCQAA6heLSb/7Xb5mzQqoZ8+Ipk0LKRBIdamQbA0GOWvt\nd4pbasQYI2vtxKSWCgAANMl99+Vp8uQ8/ehHET39dJnatUt1idAWmtPgyiqCAACkkenTA7rjjnzt\nu29Us2aF1KlTqkuEttKcIMeQSQAA0sT8+X6NGeNsvTVrVpn23pv2llzSnCA3p9VLAQAAmuy113wa\nNapAhYXSzJkhHXggIS7XNDnIVW/TBQAAUmf5cq8uvzwoj8fZeuvII9l6KxexRRcAABnmww+9uvTS\nQlVUSNOmlevEE9l6K1cR5AAAyCD/+Y+zf+qWLR7df39IZ5zB1lu5rNlBzhjT3Vq7qhH3Gyhpi6Se\n1tq7GnPcGFPqHu5mrR3bmPMAAJDt/vc/jy64oFDffuvVHXeUa9AgQlyuqzfIuYsBd63j8CBJ9QY5\nY0wPSTFr7SJjTEnN8FfjeFf3eidJC6y1/zHGzDLG9JW0ub7zAACQ7TZvli66KKjPP/dqzJgKlZZW\npbpISAMNtch1kjRV0jLVXnakl6RbGnj8IEmvuJfXSuqvncNf/PF17vHNcsLjVPcxJZIObOA8AABk\nrW3bpIsvLtTHH/s0YkSlbryxMtVFQpqoN8hZa9cZY2621i6qeczt6mxIB0mb4q53buB4pxrdpj0l\nzZR0VAPnAQAgK4VC0uDBQa1c6dMll1Tpj3+skIcVXeFqcPmRRCHOvX1u6xdnB7ebdQVdqACAXFVZ\nKQ0bFtSbb/p19tlVuueecnmbswIssla9bwdjTPfGnKSe+22W0z0rOa1vG5twvJ+19pZG3A8AgKwT\niUhXXlmghQv96ts3rIceKpfPl+pSId00NEbuO3cG6bJELWNuq1kvSQvrePws9/hiOWPdFriPa2+t\n/a6e46XW2onu5X6SnpXTvbrT/epTXFzU0F1yEvVSG3WSGPWSGPWSGPVSW0vqJBqVSkulF16QfvIT\nZxuuwsLsqGPeK63LE4s1vJ2HOx5ukKT4O2+SM7t0XgOPHS5nIkNXa+1U97Zl1treiY67wW2WnFa4\njpIutNYuTnSeesTWr9/a4OvKNcXFRaJedkadJEa9JEa9JEa91NaSOonFpFtvzde0aXnq3j2iuXPL\nVJQl2Yf3SmLFxUXNHvXYqCCXgQhyCfALVBt1khj1khj1khj1Ultz6yQWk26/PU/335+vQw6J6Lnn\nytSpU8OPyxS8VxJrSZBr9pBJY8z5zX0sAACo7Z57nBDXrVtUs2eHsirEITkavbODMWa8nOVAOspZ\nU66rpHq7VQEAQOM89FBAEybkq0uXqObOLdPuu2dljxlaWVO26FpQvV2W9MNEBwAA0EKPPx7Q735X\noL32imrOnDLtvTchDo3TlK7VmDFm17jrdW3dBQAAGunpp/266aYC7bZbVHPmhHTAAYQ4NF5TWuSm\nSNpsjJHoWgUAoMXmzvXruusK1LFjTHPmhHTQQdFUFwkZpilBbmT8Lg90rQIA0Hzz5/t11VUFKiqS\nZs8u06GHEuLQdI3uWk2wVVfHVi4LAAA54aWXfBo5skDBoPTss2X68Y8JcWieelvkjDGfSloTd5NH\nzqLAHjk7MrB5PQAATbB4sU/DhweVlyc9/XRIvXoR4tB8DXWtjkzQEieJrlUAAJrq1Vd9uvzyoLxe\nacaMkI49NpLqIiHD1du1WleIc4+tbP3iAACQnZYs8WnIkKAk6YknQurThxCHlmvKZAcAANAMb77p\n0+DBQUWjToj76U8JcWgdBDkAAJJo6VKfLr00qHBYevTRkPr1I8Sh9TR5r1VjTPdkFAQAgGzzzjte\nXXJJUJWV0iOPlGvAAEIcWleTg5yk/q1eCgAAssw773g1aFChysulyZPLdcYZ4VQXCVmoOUHO0+ql\nAAAgi7z1lnTxxU6ImzKlXGefTYhDcjQnyLEJHAAAdVi+3KsBA6RQyGmJI8QhmZoT5AAAQAIrVjjd\nqWVlTog75xxCHJKLWasAALSCZcucEBcKSU8/Lf30p4Q4JF9zWuS+a/VSAACQwd5+26eLLir8oTv1\nootSXSLkiiYHOWvtI8koCAAAmWjpUp8uvjioigpnYgPdqWhLdK0CANBMb73l22mduDPPJMShbRHk\nAABohn/+06ef/zyoqipnx4bTTmOxX7Q9Zq0CANBEr76687ZbhDikSqODnDHmxurtuYwxByStRAAA\npLGFC3267LKgolFp+vQQ224hpZrSIrdO0lr38mZjzPlJKA8AAGnrb3/z6/LLg/J6pSefDKlvX0Ic\nUqspY+RKJD1ijFkgaXmSygMAQFp64QW/Ro0qUF6e9PTTIR1/PCEOqdeUFrm1ksZK2ixplJxgBwBA\n1ps9268FAxnNAAAgAElEQVQRIwpUUCA9+ywhDumjqZMdNlpr50q6WTu6WQEAyFozZgR01VUFKiqS\nZs8u0zHHEOKQPhod5NwA19m92klSt6SUCACANPHIIwHdcEOBOnWKad68MvXqFU11kYCdNGkdOWvt\nurh/70pKiQAASAP33Zen22/P1+67RzV3bkjGEOKQfuoNcsaYUkntGziHR9Jma+3UVisVAAApEotJ\nEybk6Z578rXPPlHNnVumkpJYqosFJFRvkGNfVQBALonFpN/+Nl+TJ+dp//2jmjevTPvtR4hD+kr6\nzg7GmIHGmH7GmDFNOW6M6VHj+nj339LklRYAkKsiEemGG5wQZ0xE8+cT4pD+khrk3DAWs9YukrSl\nemeIho4bY/pJml3jdCOMMaslrUlmmQEAuaeqSrriigI9+WSefvzjiJ5/PqQ99yTEIf0lu0VukKQt\n7uW1kvo35rgb7GoGtuHW2oOstYuTVFYAQA4qL5d++cugnn8+oKOPDmvevDJ17kyIQ2ZIdpDrIGlT\n3PXOTTwer6S+LloAAJpq2zbpkkuCeuUVv04+Oaxnnw1p111TXSqg8ZI+Rq61WGsnui11nY0xfVNd\nHgBAZtu40aPzzy/UP//p1xlnVGnGjJDatUt1qYCmSXaQ2yxn8WDJaX3b2MTjkpwJDsaY892rG8X2\nYACAFvj6a4/OPTeoVat8uvjiKk2dWq78/FSXCmi6ZAe5WdoRukokLZQkY0z7+o67PHGXl8Ud6yZp\neTIKCwDIfmvXenT22YWy1qeRIyv1l7+Uy9+k5fGB9OGJxZI7oNMYM1zSOkldqxcNNsYss9b2ruf4\nQElTJJVaa+e5t5XKGU/X1Vo7sYGnZZQqAKCWd9+VBgyQvvlGuu026de/ljyehh8HJFmz34VJD3Ip\nElu/fmuqy5B2iouLRL3sjDpJjHpJjHpJLFPqZelSnwYPDur77z0aN65cw4ZVJe25MqVO2hr1klhx\ncVGzgxyNyQCArPfyyz6VlgYVDksPPxzS+eeHU10koFVkzKxVAACaY+ZMv37xi6C8XunJJwlxyC4E\nOQBA1po0KaBrrgmqqEiaPbtMfftGUl0koFXRtQoAyDrRqPTHP+Zr0qQ87blnVLNmhXTwwdFUFwto\ndQQ5AEBWqaqSfvWrAs2ZE9BBB0U0c2ZI++2XlRP7AIIcACB7bNsmDRsW1D/+4VevXhE99VSZOnVq\n+HFApmKMHAAgK2zY4NHAgYX6xz/8OuWUsObMIcQh+xHkAAAZb906j848s1ArV/o0aFCVHn+cfVOR\nGwhyAICMtmqVV2eeWah167y65poK3XdfuQKBVJcKaBuMkQMAZKxFi3waNiyoUEgaP75cQ4cmb7cG\nIB3RIgcAyEjPPOPX4MFBRaPSY48R4pCbaJEDAGSUWEyaODFPd92Vr44dY5oxo0xHH80acchNBDkA\nQMaoqpJuuKFAM2cG1KVLVM88E9JBBxHikLsIcgCAjLB1qzR0aFCvveZXjx4RzZgR0u67s9Avchtj\n5AAAae+rrzw6++xCvfaaXwMGhDVvXhkhDhBBDgCQ5t57z6vTTy/Uhx/6NHRoJWvEAXHoWgUApK0F\nC3wqLXWWF/nd78p15ZVV8nhSXSogfdAiBwBIS9OmBXTZZUHFYtK0aeUaPZoQB9REixwAIK1EItLv\nf5+vyZPztNtuUT35ZEg9ezIzFUiEIAcASBvbtklXXlmgl14KyJiInnoqpC5dmNQA1IUgBwBIC19+\n6dHgwUF98IFPP/lJWNOmhdS+fapLBaQ3xsgBAFJu5UqvBgwo1Acf+DRkSKWeeYYQBzQGLXIAgJSa\nP9+v0aMLVFkp3X57uUpLmdQANBZBDgCQErGY9Oc/52n8+Hy1axfTjBkhnXJKJNXFAjIKQQ4A0OZC\nIenaawv03HMB7btvVDNmhHTYYcxMBZqKIAcAaFP/+59HQ4YEtWqVT0cfHdZjj5WruJiZqUBzMNkB\nANBmVq3y6tRTC7VqlU+XXFKluXNDhDigBQhyAIA2MXeuX+ecU6hvvvHo978v11/+Uq78/FSXCshs\ndK0CAJIqEpHuuCNPDzyQr6KimB59NKT+/ZnUALQGghwAIGm++04aNSqoRYv86tYtqunTQzroICY1\nAK2FIAcASIpPP/XosssKtWaNV337hjV5Mov8Aq2NMXIAgFb38ss+DRjQTmvWeHXVVRV66ilCHJAM\ntMgBAFpNNCrdfXee7rorX8FgTA89FNLAgeFUFwvIWkkPcsaYgZK2SOpprb2rsceNMT2stSsbex4A\nQGpt3SqNHl2gl14KaL/9onr88ZCOOILxcEAyJbVr1RjTQ1LMWrtI0hZjTPfGHDfG9JM0u7HnAQCk\n1urVzqb3L70UUJ8+Yb3yShkhDmgDyR4jN0hOK5okrZXUvzHH3cC2pgnnAQCkyNy50qmnFurTT30a\nNapSzz4bUufOLPILtIVkd612kLQp7nrnJhz3NOE8AIA2Fg5Ld9yRrwcflAoLpcmTQzrvPMbDAW2J\nyQ4AgCbbsMGjkSMLtGSJXwcdJE2bVqaDD6YrFWhryQ5ymyV1ci93kLSxCcdjjbwfAKANrVjh1fDh\nQX35pVennValmTMDqqwkxAGpkOwxcrMklbiXSyQtlCRjTPv6jrviu1brux8AoA3EYtLUqQGdc06h\nvv7ao1tuqdDjj5ezPhyQQp5YLLkDUo0xwyWtk9TVWjvVvW2ZtbZ3PccHSpoiqdRaO6+u+9WDUbYA\n0Iq2bZNKS6WZM6XiYumZZ6R+/VJdKiBreBq+Sx0PTHaQS5HY+vVbU12GtFNcXCTqZWfUSWLUS2K5\nWi/WejV0aIFWr/apd++Ipk4Naa+9dnx35Gq91Ic6SYx6Say4uKjZQY4tugAAdZo1y68BAwq1erVP\nI0dW6vnny3YKcQBSi1mrAIBaysqkW2/N19NP56moKKZp00I6+2yWFgHSDUEOALCT1au9Gj68QB99\n5NOPfxzRI4+E1LUrrXBAOqJrFQDwgzlz/DrllEJ99JFPv/xlpV58sYwQB6QxWuQAANq+Xbr11gI9\n80xA7drFNGVKSOeeS1cqkO4IcgCQ4z780KsRIwr0ySdOV+qUKSGVlNAKB2QCulYBIEfFYtL06QGd\ndlqhPvnEpxEjKvXXv5YR4oAMQoscAOSgLVuk668v0IsvBtSxY0xTppTptNMiqS4WgCYiyAFAjlm6\n1KcrrijQl196ddxxYU2aVK599qEVDshEdK0CQI4Ih6Xx4/N07rlB/e9/Ho0dW6F580KEOCCD0SIH\nADngP//xaPTooJYt82m//aJ66KGQjj46mupiAWghWuQAIIvFYtLMmX717dtOy5b5dO65VVq8eDsh\nDsgStMgBQJbavFm68cYCzZ8f0C67xPTAAyFdeGFYnmZvzw0g3RDkACALvf66T1dfXaCvv/bq6KPD\nevDBcu2/P2PhgGxDkAOALFJWJt1+e76mTs2TzxfT2LEVuuaaSvn5tAeyEr/aAJAlVq3yavToAq1e\n7dNBB0X04IPl6t6dsXBANmOyAwBkuHBYuvvuPJ1xRqFWr/aptLRSCxeWEeKAHECLHABksI8/9urq\nqwv07rs+7b13VPfeG9JJJ7FDA5AraJEDgAwUiUgPPBBQ//6Fevddny66qEqvvrqdEAfkGFrkACDD\nrF3r0dVXO4v7FhdHdffdIfZJBXIULXIAkCEiEWnSpIBOPtlZ3Pe886q0ZMl2QhyQw2iRA4AM8Mkn\nXv3qVwVascKn3XaL6oEHynXOOeFUFwtAihHkACCNhcPSpEl5uuuuPFVUeHTeeVX6058q1Lkzi/sC\nIMgBQNp67z2vrr22QO+954yFu+uucp1xBq1wAHYgyAFAmgmFpIkT8zRpUp4iEY8uuqhKt91Wro4d\nU10yAOmGIAcAaeTNN326/voCrV3rVZcuUd11V0g//SmTGQAkRpADgDSwebP0xz/m66mn8uTxxDRy\nZKVuvrlCu+yS6pIBSGcEOQBIoVhMmjvXr9/+Nl8bNnh16KER3X13uXr1YnstAA0jyAFAiqxb59HN\nNxfo1Vf9CgZj+s1vKjRqVKUCgVSXDECmIMgBQBurqJAeeCBP996bp/Jyj/r2DWvChHLtvz9LigBo\nGoIcALSh117zaezYAq1Z49Xuu0d1773lOvfcsDyeVJcMQCYiyAFAG/jmG49++9t8PfdcQF5vTKWl\nzmSGXXdNdckAZDKCHAAkUVWV9MgjAd11V762b/eoV6+I7ryzXEccwWQGAC1HkAOAJHn9dZ9uvTVf\nn3ziU6dOUf3hDxUaPLhKXm+qSwYgWyT948QYM9AY088YM6axx+u4bbz7b2myywwALfHf/3o0fHiB\nLrigUKtXe/WLX1Tqrbe2a8gQQhyA1pXUjxRjTA9JMWvtIklbjDHd6zm+2RjTo57HjDDGrJa0pqHn\nXbiwdV8HADRGWZl05515Ov74dnrhhYCOOiqiBQvKdOedFWyvBSApkv234SBJW9zLayX1r+f4Ovd4\nXY8Zbq09yFq7uKEnPeUUafToAm3a1JKiA0DjxGLSc8/5dcIJ7TRxYr7at4/pgQdCevHFMv34x4yF\nA5A8yQ5yHSTFx6nOjTjevo7HlNTXRRvvqKOk2bMDOvHEdpo/n2GAAJJn5UqvzjknqJEjg1q/3qNr\nr63QW29t10UXhelGBZB0GfMxY62d6Ha3djbG9K3vvm+9Jf3mNxXautWjYcOCGjasQN98wyJNAFrP\nf//r0RVXFGjAgHZ6+22/Tj+9Sm+8sV233lrJ/qgA2kyym6s2S+rkXu4gaWMDxzdIitV8jDvBYaO1\ndp57jhJJdXax+v3O5tM//7k0bJg0f35AS5YENGGCNHy4cvqv5OLiolQXIe1QJ4lRL4kVFBRpwgTp\n7rul8nKpZ0/pnnukk04KSMrdvbV4v9RGnSRGvbSuZAe5WZJ6yQldJZIWSJIxpr219ru6jks6KsFt\na91/u0l6uKEnXr9+qzp1kubOlR5/PKA77sjXyJEeTZsW1t13V8iY3Bu3UlxcpPXrt6a6GGmFOkmM\neqmtqkp6/vki/e53UW3Y4NVee0V1660VuvBCpwt1/fpUlzB1eL/URp0kRr0k1pJwm9S2KWvtSkky\nxvSTtNlau8o9tLCu49X3SXDbIGPMQEmfxp2nQV6vNHRolf75z+0666wqvfOOX337Fmr8+DyFQq33\nWgFkp1hMmj/frz592mn0aCkU8uimmyr05pvbNWgQ4+AApJYnFsvKTZpjdSX+l15y9jn86iuvunSJ\n6k9/Ktepp0bauHipwV9CtVEniVEvjrfe8um22/K1fLlPfn9MI0d6dMUV27T77ln5udlsvF9qo04S\no14SKy4uavZA/pz7W/K00yJ6443tuvLKSn31lUeDBxdqyJACff45kyEAON57z6uLLw7qZz8r1PLl\nPp11VpWWLNmuBx4QIQ5AWsm5ICdJu+wi/f73FVq8uEzHHRfWSy8F1KdPO91zT57Ky1NdOgCpsnat\nRyNGFKhfv3ZavNivE08M6+9/365HHy1Xt24EOADpJyeDXLWDD47q+edDmjQppF12iWn8+HydeGI7\nvfiiX9nZ4wwgkc8/9+hXvyrQCSe00/PPB9S9e0SzZpVp7tyQevXKvYlRADJHTgc5SfJ4pAsuCGvp\nUqe79euvPRo6NKiBA4P68MOcrx4gq331lUdjxuTr2GPb6ZlnAurWLapp00J6+eUynXxyRB5GXABI\ncyQVV1GR0936+uvbdcopYb3xhjO79cYb8/Xtt3yaA9nk6689uvXWfB1zTDs98USeunSJadKkkF57\nrUxnnx0mwAHIGAS5Grp1i+mpp0J65pkydesW1fTpeTrmmHb685/zVFaW6tIBaIkvv/To5pvz1bt3\nO02dmqc99ojp3ntDeuON7brggrB8vlSXEACahiBXh379Inr11TJNmFCuYDCmcePydfzx7TRzpl+R\n3FitBMgaX3zhdKEec0w7PfZYnvbaK6a//CWkt97arksuCcvPlswAMhRBrh6BgPTLX1Zp6dLt+tWv\nKrRxo0fXXBNU376FeuUVHxMigDT3ySdeXX11wQ9dqHvvHdN994X05pvbdemlYQVyd0ctAFmCINcI\nu+4q/frXlXrrre0aNKhKH3/s1eDBhTrnnKDefpu+GCDd/PvfXg0dWqA+fQr17LPOJIYHHnAC3MUX\nE+AAZA+CXBPsu29M999frldfLdNpp1Xp7bf9OvvsQv3850H9+99UJZBKsZj06qs+XXBBUP37t9OL\nLwZ05JFRPf64M4nhoovoQgWQffhYa4ZDDolq+vRyLVtWqdtvz9eCBX4tWODX6adXacyYSh1+OOtO\nAW0lHJZeeMGvBx7I0/vvOy3kffqEddVVlSwhAiDrEeRaoHdvZ0Hh11/3afz4fP397wH9/e8BnX12\nlW64oVKHHkqgA5Ll+++lJ58MaNq0PH3xhVdeb0znnlul0aMrdeSR/O4ByA0EuRbyeKSTToroJz8p\n0z/+4QS6+fMDmj8/oNNPr9L11/OlArSmzz7z6JFH8vTUUwFt3+5RYWFMQ4dWatSoSh1wADOQAOQW\nglwr8Xikvn0j+ulPy7RggU9//vOOFrp+/cK67roKHX00gQ5ojlhMeuMNn6ZODejll/2KRj3ac8+o\nrruuUpddVqmOHVNdQgBIDYJcK/N4pFNPjeiUU8r0+us+/fnPeVq0yK9Fi/w69lhn3E7//hF5mRsB\nNGj7dmnOnICmTQvo44+d8W9HHhnRyJGVOuecsPLyUlxAAEgxglySVHe5nnRSSEuX+nTffXlauNCv\npUv9Miai0aMrdf75fBEBiXzyiVfTpwf07LMBffedR35/TOefX6Vhwyp11FFRJjAAgIsg1waOPTai\nY48N6cMPvZo0KU/z5vl1zTVB3XFHVEOHVmnIkCp17szYHuS2ykrpb3/z64knAvrnP52Ppt13j+qG\nGyr1i19UaY89+B0BgJo8sezcniC2fv3WVJehTl9+6dHDDzuDtbdt86igIKYLL6xSaWmVDj44eePo\niouLlM71kgrUSWJtWS+rV3v11FMBzZrl14YNzpiDPn3C+sUvqnTaaem1eC/vl8Sol9qok8Sol8SK\ni4ua3c/ASK0U2GefmG67rULvvrtNt99erj32iGnGjDz95CftNHBgUC++6Fc4nOpSAsmzfbs0c6Zf\nZ58d1AkntNOkSXmKRDwaObJSb765TXPnhnT22ekV4gAgHdG1mkJFRdKIEVUaNqxKr7zi1yOPBLRk\niV9Llvi1115RDRlSpcGD6VJCdojFpKVLfXr2Wb/+3/9zlg6RpJ/8JKzBg6t0+ulh5eenuJAAkGEI\ncmnA55NOPz2s008Py1qvHn/cGeQ9YUK+7r47TwMGOF90J58ckY+tXZFh1q3zaM6cgGbNCuizz5xO\ngH33jWrkyEpdckmV9t+fP1QAoLkIcmnGmKjGjavQr39doblzA3r88YD++lfnZ599orrkkipdckmV\n9tuPLz+kr/XrPXrhBb/mzAloxQrnr4/CwpguuqhKgwZV6YQTWIIHAFoDQS5N7bKLdPnlzozWd9/1\nasaMgObNC2jixHxNnJivE08M66KLqnTWWWHtskuqSwtI330n/f3vfj3/fECvveZTJOKR1xvTSSeF\nNXAg71UASAaCXJrzeKTu3aPq3r1Cf/hDhV54wa+ZMwN64w2/3njDr7FjYzrzzLAuuKBKffpE5Od/\nFG3o+++ll192xrz94x8+VVU5496OPDKigQOrdN55YcZ4AkAS8bWfQXbZRbr00rAuvTS807ij2bOd\nn912i+qcc8I699ywjj6ariskx/r1Hr30kl9/+5tfr7++I7wddlhE554b1tlnV6mkhPAGAG2BdeQy\nXCwmvf22T88959f8+TvW4dpnn6jOOiusM8/cEepYv6c26iSxmvWyZo1HL7/s18sv+/X22z5Fo054\nO+KIiM48M6xzzqnSgQdm5WfJTni/JEa91EadJEa9JNaSdeQIclkkHJaWLPHpuecC+utf/dq61Xlf\n7L57VGecEdbgwXk65JCtrM0Vhw+VxDp0KNLf/lamV15xwtuaNc4fCB5PTL17O+HtjDPCOTfjlPdL\nYtRLbdRJYtRLYgS52nIyyMWrqHBC3Ysv+vXSS35t2uR8Ee+6a0x9+4Z16qlh9esXVseOKS5oivGh\nssNXX3m0eLFfixb5tGRJQN9/79xeWOhMWBgwIKz+/SPaffes/MxoFN4viVEvtVEniVEvibUkyDFG\nLkvl50v9+0fUv39E4XCF3nzTp9deK9T/+38xPf98QM8/H5DP57Su/PSnEfXtG9YRR0QZV5dDtm6V\n3nzTp9dfd8a6WbtjkcKSEumCCyrVv39YJ54YUUFBCgsKAKgTLXI5pLi4SN9+u1UffeT9octs5Urv\nD+OddtstqpNOiuikk5wv7333zcr3xk5y6a/Dbdukd97x6a23fPrnP53/+0jE+b8vLIzp2GMj6tfP\naak95phdcqZemiKX3i9NQb3URp0kRr0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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(x, absolute_error, 'b-')\n", "common_options('Absolute approximation error', '$x$', '$| x - \\ln(1 + x) |$')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "* Relative errors\n", "\n", "Here what we are interested in the size of the approximation error relative to the original interest rate. For example, for a 20% interest rate, interest payments computed with the appoximated formula would be off by 10 percent." ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "image/png": 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1ULtUozaJT+0Sn9olvsa2S2lpdGWDRx7JAWD48BJuvLEEtxu2bGnqKJuXXivx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\nQhsAQANCGwBAA0IbAEADfwE7VIM110Gb9QAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.plot(x, relative_error, 'b-')\n", "common_options('Relative approximation error', '$x$', '$| x / \\ln(1 + x) - 1|$')" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Constant difference between nominal interest and inflation\n", "In this example, we assume that the nominal interest rate is always 10percentage points above the inflation rate. Notice that in such case, for high inflation rates the approximation deteriorates quickly." ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" }, { "data": { "image/png": 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0ULyjExGRxswXiUTqdKExZoK1dkyM49lv27btrtsb8rjs7KZs27Y73mEI8PXX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\nkgjJGwAAQBIheQMAAEgi/wcrfJowEAj+bQAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "a, b = -0.05, 0.50\n", "inflation = np.linspace(a, b, 121)\n", "nominal = inflation + 0.10\n", "r = (1+nominal) / (1+inflation) - 1\n", "\n", "plt.plot(inflation, r, 'b-', label='True value')\n", "plt.plot(inflation, 0.10+np.zeros_like(inflation), 'r-', label='Approximation')\n", "common_options('Approximating the real interest rate,\\nassuming $i=10\\%+\\pi$', 'inflation $\\pi$', 'Real interest rate $r$')\n", "plt.legend(loc='lower left', frameon=False)" ] } ], "metadata": { "anaconda-cloud": {}, "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": "3.6.4" } }, "nbformat": 4, "nbformat_minor": 1 }