├── README.md
└── VaR.ipynb


/README.md:
--------------------------------------------------------------------------------
 1 | # Value_at_Risk_Python
 2 | 
 3 | The Value at Risk (VaR) calculation, Python version
 4 | 
 5 | # Model description
 6 | Volatility model: GARCH(1,1)
 7 | 
 8 | VaR model: historical simulation with volatility updating
 9 | 
10 | Backtesting: binomial test
11 | 


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/VaR.ipynb:
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  1 | {
  2 |   "nbformat": 4,
  3 |   "nbformat_minor": 0,
  4 |   "metadata": {
  5 |     "colab": {
  6 |       "name": "VaR.ipynb",
  7 |       "provenance": [],
  8 |       "include_colab_link": true
  9 |     },
 10 |     "kernelspec": {
 11 |       "name": "python3",
 12 |       "display_name": "Python 3"
 13 |     }
 14 |   },
 15 |   "cells": [
 16 |     {
 17 |       "cell_type": "markdown",
 18 |       "metadata": {
 19 |         "id": "view-in-github",
 20 |         "colab_type": "text"
 21 |       },
 22 |       "source": [
 23 |         "<a href=\"https://colab.research.google.com/github/jhihan/Value_at_Risk_Python/blob/master/VaR.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
 24 |       ]
 25 |     },
 26 |     {
 27 |       "cell_type": "code",
 28 |       "metadata": {
 29 |         "id": "av5Mm9F_0egA",
 30 |         "colab_type": "code",
 31 |         "outputId": "5b89f591-decd-42a7-8f8e-cfa99a2c630e",
 32 |         "colab": {
 33 |           "base_uri": "https://localhost:8080/",
 34 |           "height": 309
 35 |         }
 36 |       },
 37 |       "source": [
 38 |         "pip install arch"
 39 |       ],
 40 |       "execution_count": 1,
 41 |       "outputs": [
 42 |         {
 43 |           "output_type": "stream",
 44 |           "text": [
 45 |             "Collecting arch\n",
 46 |             "\u001b[?25l  Downloading https://files.pythonhosted.org/packages/b7/05/241c4238e9ee8cfb78b2053c059fb5ce6d1bc32a1266eb99a39e96a00069/arch-4.10.0-cp36-cp36m-manylinux1_x86_64.whl (710kB)\n",
 47 |             "\u001b[K     |████████████████████████████████| 716kB 2.8MB/s \n",
 48 |             "\u001b[?25hRequirement already satisfied: statsmodels>=0.9 in /usr/local/lib/python3.6/dist-packages (from arch) (0.10.1)\n",
 49 |             "Requirement already satisfied: cython>=0.29 in /usr/local/lib/python3.6/dist-packages (from arch) (0.29.13)\n",
 50 |             "Collecting property-cached>=1.6.2\n",
 51 |             "  Downloading https://files.pythonhosted.org/packages/c0/11/6e91ff5fe0476492f023cebad434a1a34fc513cfa98ddb1f3e5c856d2d99/property_cached-1.6.3-py2.py3-none-any.whl\n",
 52 |             "Requirement already satisfied: pandas>=0.22 in /usr/local/lib/python3.6/dist-packages (from arch) (0.25.2)\n",
 53 |             "Requirement already satisfied: scipy>=1.0 in /usr/local/lib/python3.6/dist-packages (from arch) (1.3.1)\n",
 54 |             "Requirement already satisfied: numpy>=1.14 in /usr/local/lib/python3.6/dist-packages (from arch) (1.17.3)\n",
 55 |             "Requirement already satisfied: patsy>=0.4.0 in /usr/local/lib/python3.6/dist-packages (from statsmodels>=0.9->arch) (0.5.1)\n",
 56 |             "Requirement already satisfied: pytz>=2017.2 in /usr/local/lib/python3.6/dist-packages (from pandas>=0.22->arch) (2018.9)\n",
 57 |             "Requirement already satisfied: python-dateutil>=2.6.1 in /usr/local/lib/python3.6/dist-packages (from pandas>=0.22->arch) (2.6.1)\n",
 58 |             "Requirement already satisfied: six in /usr/local/lib/python3.6/dist-packages (from patsy>=0.4.0->statsmodels>=0.9->arch) (1.12.0)\n",
 59 |             "Installing collected packages: property-cached, arch\n",
 60 |             "Successfully installed arch-4.10.0 property-cached-1.6.3\n"
 61 |           ],
 62 |           "name": "stdout"
 63 |         }
 64 |       ]
 65 |     },
 66 |     {
 67 |       "cell_type": "code",
 68 |       "metadata": {
 69 |         "id": "XC7Ce1A_zoF_",
 70 |         "colab_type": "code",
 71 |         "colab": {}
 72 |       },
 73 |       "source": [
 74 |         "# example of ARCH model\n",
 75 |         "import numpy as np\n",
 76 |         "from random import gauss\n",
 77 |         "from random import seed\n",
 78 |         "from matplotlib import pyplot\n",
 79 |         "from arch import arch_model\n",
 80 |         "import pandas as pd\n",
 81 |         "from scipy.stats import binom"
 82 |       ],
 83 |       "execution_count": 0,
 84 |       "outputs": []
 85 |     },
 86 |     {
 87 |       "cell_type": "markdown",
 88 |       "metadata": {
 89 |         "id": "TlKwBVLjQqUY",
 90 |         "colab_type": "text"
 91 |       },
 92 |       "source": [
 93 |         "Read and preprocess the data such as calculating the continously compounded return."
 94 |       ]
 95 |     },
 96 |     {
 97 |       "cell_type": "code",
 98 |       "metadata": {
 99 |         "id": "e_-w3EY12eHo",
100 |         "colab_type": "code",
101 |         "colab": {}
102 |       },
103 |       "source": [
104 |         "# here we use the USD YEN exchange rate as an example\n",
105 |         "# Data is downloaded from https://www.macrotrends.net/2550/dollar-yen-exchange-rate-historical-chart\n",
106 |         "data = pd.read_csv(\"dollar-yen-exchange-rate-historical-chart.csv\", header = 8) "
107 |       ],
108 |       "execution_count": 0,
109 |       "outputs": []
110 |     },
111 |     {
112 |       "cell_type": "code",
113 |       "metadata": {
114 |         "id": "EjaJZtf7Kt7A",
115 |         "colab_type": "code",
116 |         "colab": {
117 |           "base_uri": "https://localhost:8080/",
118 |           "height": 204
119 |         },
120 |         "outputId": "4c3abf6a-2a64-4af9-8756-b9403bb906f2"
121 |       },
122 |       "source": [
123 |         "data.tail()"
124 |       ],
125 |       "execution_count": 4,
126 |       "outputs": [
127 |         {
128 |           "output_type": "execute_result",
129 |           "data": {
130 |             "text/html": [
131 |               "<div>\n",
132 |               "<style scoped>\n",
133 |               "    .dataframe tbody tr th:only-of-type {\n",
134 |               "        vertical-align: middle;\n",
135 |               "    }\n",
136 |               "\n",
137 |               "    .dataframe tbody tr th {\n",
138 |               "        vertical-align: top;\n",
139 |               "    }\n",
140 |               "\n",
141 |               "    .dataframe thead th {\n",
142 |               "        text-align: right;\n",
143 |               "    }\n",
144 |               "</style>\n",
145 |               "<table border=\"1\" class=\"dataframe\">\n",
146 |               "  <thead>\n",
147 |               "    <tr style=\"text-align: right;\">\n",
148 |               "      <th></th>\n",
149 |               "      <th>date</th>\n",
150 |               "      <th>value</th>\n",
151 |               "    </tr>\n",
152 |               "  </thead>\n",
153 |               "  <tbody>\n",
154 |               "    <tr>\n",
155 |               "      <th>12385</th>\n",
156 |               "      <td>2019-10-26</td>\n",
157 |               "      <td>108.6650</td>\n",
158 |               "    </tr>\n",
159 |               "    <tr>\n",
160 |               "      <th>12386</th>\n",
161 |               "      <td>2019-10-28</td>\n",
162 |               "      <td>108.9695</td>\n",
163 |               "    </tr>\n",
164 |               "    <tr>\n",
165 |               "      <th>12387</th>\n",
166 |               "      <td>2019-10-29</td>\n",
167 |               "      <td>108.8665</td>\n",
168 |               "    </tr>\n",
169 |               "    <tr>\n",
170 |               "      <th>12388</th>\n",
171 |               "      <td>2019-10-30</td>\n",
172 |               "      <td>108.8185</td>\n",
173 |               "    </tr>\n",
174 |               "    <tr>\n",
175 |               "      <th>12389</th>\n",
176 |               "      <td>2019-10-31</td>\n",
177 |               "      <td>108.1675</td>\n",
178 |               "    </tr>\n",
179 |               "  </tbody>\n",
180 |               "</table>\n",
181 |               "</div>"
182 |             ],
183 |             "text/plain": [
184 |               "             date     value\n",
185 |               "12385  2019-10-26  108.6650\n",
186 |               "12386  2019-10-28  108.9695\n",
187 |               "12387  2019-10-29  108.8665\n",
188 |               "12388  2019-10-30  108.8185\n",
189 |               "12389  2019-10-31  108.1675"
190 |             ]
191 |           },
192 |           "metadata": {
193 |             "tags": []
194 |           },
195 |           "execution_count": 4
196 |         }
197 |       ]
198 |     },
199 |     {
200 |       "cell_type": "code",
201 |       "metadata": {
202 |         "id": "MpLgfW9CI6fZ",
203 |         "colab_type": "code",
204 |         "colab": {
205 |           "base_uri": "https://localhost:8080/",
206 |           "height": 224
207 |         },
208 |         "outputId": "4526912e-964c-4165-8634-c3031afd8555"
209 |       },
210 |       "source": [
211 |         "# preprocessing the data\n",
212 |         "df=data[(data['date'] >= '2009-10-31') & (data['date'] <= '2019-10-31')]\n",
213 |         "df.rename(columns={' value':'value'}, inplace=True)\n",
214 |         "df['return']=df.value.pct_change()\n",
215 |         "# define a new list which ignore the first (index 0) NAN value\n",
216 |         "return_list=df['return'].tolist()[1:]"
217 |       ],
218 |       "execution_count": 5,
219 |       "outputs": [
220 |         {
221 |           "output_type": "stream",
222 |           "text": [
223 |             "/usr/local/lib/python3.6/dist-packages/pandas/core/frame.py:4238: SettingWithCopyWarning: \n",
224 |             "A value is trying to be set on a copy of a slice from a DataFrame\n",
225 |             "\n",
226 |             "See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n",
227 |             "  return super().rename(**kwargs)\n",
228 |             "/usr/local/lib/python3.6/dist-packages/ipykernel_launcher.py:3: SettingWithCopyWarning: \n",
229 |             "A value is trying to be set on a copy of a slice from a DataFrame.\n",
230 |             "Try using .loc[row_indexer,col_indexer] = value instead\n",
231 |             "\n",
232 |             "See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/user_guide/indexing.html#returning-a-view-versus-a-copy\n",
233 |             "  This is separate from the ipykernel package so we can avoid doing imports until\n"
234 |           ],
235 |           "name": "stderr"
236 |         }
237 |       ]
238 |     },
239 |     {
240 |       "cell_type": "markdown",
241 |       "metadata": {
242 |         "id": "41E2WxK-P9RO",
243 |         "colab_type": "text"
244 |       },
245 |       "source": [
246 |         "GARCH(1,1) model is used to determined the volatility."
247 |       ]
248 |     },
249 |     {
250 |       "cell_type": "code",
251 |       "metadata": {
252 |         "id": "E-RHDDEcN7bf",
253 |         "colab_type": "code",
254 |         "outputId": "3d9c580f-56c8-404f-a281-73d3214e68f1",
255 |         "colab": {
256 |           "base_uri": "https://localhost:8080/",
257 |           "height": 768
258 |         }
259 |       },
260 |       "source": [
261 |         "# Using GARCH(1,1) to determine the volatility\n",
262 |         "model = arch_model(return_list, mean='Zero', vol='GARCH', p=1, q=1)\n",
263 |         "model_fit = model.fit()\n",
264 |         "volatility = model_fit.conditional_volatility\n",
265 |         "print(model_fit.summary())"
266 |       ],
267 |       "execution_count": 6,
268 |       "outputs": [
269 |         {
270 |           "output_type": "stream",
271 |           "text": [
272 |             "Iteration:      1,   Func. Count:      5,   Neg. LLF: -10085.986032212\n",
273 |             "Positive directional derivative for linesearch    (Exit mode 8)\n",
274 |             "            Current function value: -10085.986035013902\n",
275 |             "            Iterations: 5\n",
276 |             "            Function evaluations: 5\n",
277 |             "            Gradient evaluations: 1\n",
278 |             "                       Zero Mean - GARCH Model Results                        \n",
279 |             "==============================================================================\n",
280 |             "Dep. Variable:                      y   R-squared:                       0.000\n",
281 |             "Mean Model:                 Zero Mean   Adj. R-squared:                  0.000\n",
282 |             "Vol Model:                      GARCH   Log-Likelihood:                10086.0\n",
283 |             "Distribution:                  Normal   AIC:                          -20166.0\n",
284 |             "Method:            Maximum Likelihood   BIC:                          -20148.3\n",
285 |             "                                        No. Observations:                 2646\n",
286 |             "Date:                Fri, Nov 01 2019   Df Residuals:                     2643\n",
287 |             "Time:                        08:55:04   Df Model:                            3\n",
288 |             "                              Volatility Model                              \n",
289 |             "============================================================================\n",
290 |             "                 coef    std err          t      P>|t|      95.0% Conf. Int.\n",
291 |             "----------------------------------------------------------------------------\n",
292 |             "omega      6.5006e-07  3.058e-11  2.126e+04      0.000 [6.500e-07,6.501e-07]\n",
293 |             "alpha[1]       0.0500  7.916e-03      6.317  2.674e-10 [3.449e-02,6.551e-02]\n",
294 |             "beta[1]        0.9300  7.033e-03    132.234      0.000     [  0.916,  0.944]\n",
295 |             "============================================================================\n",
296 |             "\n",
297 |             "Covariance estimator: robust\n",
298 |             "\n",
299 |             "WARNING: The optimizer did not indicate successful convergence. The message was\n",
300 |             "Positive directional derivative for linesearch. See convergence_flag.\n"
301 |           ],
302 |           "name": "stdout"
303 |         },
304 |         {
305 |           "output_type": "stream",
306 |           "text": [
307 |             "/usr/local/lib/python3.6/dist-packages/arch/univariate/base.py:260: DataScaleWarning: y is poorly scaled, which may affect convergence of the optimizer when\n",
308 |             "estimating the model parameters. The scale of y is 3.25e-05. Parameter\n",
309 |             "estimation work better when this value is between 1 and 1000. The recommended\n",
310 |             "rescaling is 1000 * y.\n",
311 |             "\n",
312 |             "This warning can be disabled by either rescaling y before initializing the\n",
313 |             "model or by setting rescale=False.\n",
314 |             "\n",
315 |             "  DataScaleWarning)\n",
316 |             "/usr/local/lib/python3.6/dist-packages/arch/univariate/base.py:605: ConvergenceWarning: The optimizer returned code 8. The message is:\n",
317 |             "Positive directional derivative for linesearch\n",
318 |             "See scipy.optimize.fmin_slsqp for code meaning.\n",
319 |             "\n",
320 |             "  ConvergenceWarning)\n"
321 |           ],
322 |           "name": "stderr"
323 |         }
324 |       ]
325 |     },
326 |     {
327 |       "cell_type": "markdown",
328 |       "metadata": {
329 |         "id": "gXxF-vwnUFfU",
330 |         "colab_type": "text"
331 |       },
332 |       "source": [
333 |         "Estimate the Volatility in the next date with the GARCH(1,1) model"
334 |       ]
335 |     },
336 |     {
337 |       "cell_type": "code",
338 |       "metadata": {
339 |         "id": "e3EuHIKqUFGf",
340 |         "colab_type": "code",
341 |         "colab": {
342 |           "base_uri": "https://localhost:8080/",
343 |           "height": 34
344 |         },
345 |         "outputId": "2df7562d-c1c0-4d8f-8ed7-35878213f98b"
346 |       },
347 |       "source": [
348 |         "# Estimate the volatility in the next date\n",
349 |         "omega = model_fit.params['omega'] \n",
350 |         "alpha = model_fit.params['alpha[1]']\n",
351 |         "beta = model_fit.params['beta[1]']\n",
352 |         "vola_estimate = omega + alpha * np.power(return_list[-1] , 2)  + beta * np.power( volatility[-1] , 2 )\n",
353 |         "vola_estimate = np.sqrt(vola_estimate)\n",
354 |         "print('The volatility in the next date=', vola_estimate )\n",
355 |         "#return_scienarios = []\n",
356 |         "#for i in range(0,return_list)"
357 |       ],
358 |       "execution_count": 7,
359 |       "outputs": [
360 |         {
361 |           "output_type": "stream",
362 |           "text": [
363 |             "vola= 0.0038338334199308896\n"
364 |           ],
365 |           "name": "stdout"
366 |         }
367 |       ]
368 |     },
369 |     {
370 |       "cell_type": "markdown",
371 |       "metadata": {
372 |         "id": "4KJXGsPbLYov",
373 |         "colab_type": "text"
374 |       },
375 |       "source": [
376 |         "Value at Risk Calculation: historical simulation with volatility updating"
377 |       ]
378 |     },
379 |     {
380 |       "cell_type": "code",
381 |       "metadata": {
382 |         "id": "V_5xkz4Wl1Pz",
383 |         "colab_type": "code",
384 |         "colab": {}
385 |       },
386 |       "source": [
387 |         "# calculate the predicted value in the next date under different scenario and inporporate volatility updating\n",
388 |         "Market_scienarios = []\n",
389 |         "N_history = len(return_list)\n",
390 |         "Value_current = df.loc[df.index[-1], 'value']\n",
391 |         "df.loc[df.index[-1], 'value']\n",
392 |         "for i in range(0, N_history ):\n",
393 |         "  Market_value = Value_current *  ( 1 + return_list[i] * vola_estimate/volatility[i] )\n",
394 |         "  Market_scienarios.append(Market_value)"
395 |       ],
396 |       "execution_count": 0,
397 |       "outputs": []
398 |     },
399 |     {
400 |       "cell_type": "code",
401 |       "metadata": {
402 |         "id": "jg2kWLUWl1CM",
403 |         "colab_type": "code",
404 |         "colab": {
405 |           "base_uri": "https://localhost:8080/",
406 |           "height": 34
407 |         },
408 |         "outputId": "49688cf9-4221-40db-e352-4e120e34d6d3"
409 |       },
410 |       "source": [
411 |         "# Value at Risk (VaR) determination\n",
412 |         "# CL_Var: condifence level for VaR\n",
413 |         "CL_VaR = 95\n",
414 |         "Value_current = df.loc[df.index[-1], 'value']\n",
415 |         "Value_at_Risk = np.percentile(Market_scienarios,100-CL_VaR) -  Value_current\n",
416 |         "print(\"The 1 day\", CL_VaR,\"% VaR:\", -Value_at_Risk)"
417 |       ],
418 |       "execution_count": 11,
419 |       "outputs": [
420 |         {
421 |           "output_type": "stream",
422 |           "text": [
423 |             "The 1 day 95 % VaR: 0.64802029956887\n"
424 |           ],
425 |           "name": "stdout"
426 |         }
427 |       ]
428 |     },
429 |     {
430 |       "cell_type": "code",
431 |       "metadata": {
432 |         "id": "ZLxs5LYPl04T",
433 |         "colab_type": "code",
434 |         "colab": {
435 |           "base_uri": "https://localhost:8080/",
436 |           "height": 295
437 |         },
438 |         "outputId": "706029cf-b809-49e8-8fa0-d62d6d632966"
439 |       },
440 |       "source": [
441 |         "# Make the histogram of loss for the scenaros considered between current date and the next date\n",
442 |         "import matplotlib.pyplot as plt\n",
443 |         "mss = np.asarray(Value_current - Market_scienarios )\n",
444 |         "\n",
445 |         "plt.xlim([min(mss), max(mss)])\n",
446 |         "\n",
447 |         "plt.hist(mss, bins='auto', alpha=0.5)\n",
448 |         "plt.title('Plot of loss for the scenaros')\n",
449 |         "plt.xlabel('Loss')\n",
450 |         "plt.ylabel('Frequency')\n",
451 |         "plt.axvline( -Value_at_Risk , color='k', linestyle='dashed', linewidth=1)\n",
452 |         "min_ylim, max_ylim = plt.ylim()\n",
453 |         "plt.text(-Value_at_Risk*1.1, max_ylim*0.9, '{}% VaR: {:.2f}'.format(CL_VaR ,-Value_at_Risk))\n",
454 |         "plt.show()"
455 |       ],
456 |       "execution_count": 24,
457 |       "outputs": [
458 |         {
459 |           "output_type": "display_data",
460 |           "data": {
461 |             "image/png": 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DJD0ODIuIVcm0Z4GjIuKv21tvdXV1lLtYZNYWtMW+j6x9kFQbEWUvqlTk7qOI2AA8CBwD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\n",
462 |             "text/plain": [
463 |               "<Figure size 432x288 with 1 Axes>"
464 |             ]
465 |           },
466 |           "metadata": {
467 |             "tags": []
468 |           }
469 |         }
470 |       ]
471 |     },
472 |     {
473 |       "cell_type": "markdown",
474 |       "metadata": {
475 |         "id": "GdR_yJpCELCA",
476 |         "colab_type": "text"
477 |       },
478 |       "source": [
479 |         "Value-at-Risk backtesting:\n",
480 |         "here the binomial one-tailed test is used"
481 |       ]
482 |     },
483 |     {
484 |       "cell_type": "code",
485 |       "metadata": {
486 |         "id": "XCj_nmDdEKAJ",
487 |         "colab_type": "code",
488 |         "colab": {
489 |           "base_uri": "https://localhost:8080/",
490 |           "height": 51
491 |         },
492 |         "outputId": "cec7515f-fb18-45c7-a580-e206caef50f7"
493 |       },
494 |       "source": [
495 |         "# Confidence level for backtesting\n",
496 |         "CL_backtesting = 95\n",
497 |         "# The probability of the VaR beeing exceeded on given day\n",
498 |         "p = 1 - CL_VaR/100\n",
499 |         "N_confidence = binom.ppf( CL_backtesting*0.01, N_history, p)\n",
500 |         "N_exceeded = sum( v < Value_at_Risk + Value_current for v in Market_scienarios)\n",
501 |         "\n",
502 |         "print(\"We look at total\", N_history,\"days. VaR level is reached at\", N_exceeded , \"days.\")\n",
503 |         "if N_exceeded <= N_confidence:\n",
504 |         "  print (\"This VaR model is accepted at\", CL_backtesting,\"% confidence level.\")\n",
505 |         "else:\n",
506 |         "  print (\"This VaR model is rejected at\", CL_backtesting,\"% confidence level.\")"
507 |       ],
508 |       "execution_count": 26,
509 |       "outputs": [
510 |         {
511 |           "output_type": "stream",
512 |           "text": [
513 |             "We look at total 2646 days. VaR level is reached at 133 days.\n",
514 |             "This VaR model is accepted at 95 % confidence level.\n"
515 |           ],
516 |           "name": "stdout"
517 |         }
518 |       ]
519 |     }
520 |   ]
521 | }


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