lab_4_homework.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "d30576c4",
   "metadata": {},
   "source": [
    "## Laboratory 4 - Models in credit and operational risk\n",
    "**Bartłomiej Jamiołkowski 04.04.2022**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4beed743",
   "metadata": {},
   "source": [
    "## Exercise 1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb1b700a",
   "metadata": {},
   "source": [
    "**What events that may cause losses do we consider as part of operational risk?**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b84e5ef1",
   "metadata": {},
   "source": [
    "As part of operational risk, losses may occur as a result of the following events:\n",
    "\n",
    "- theft of property or information;\n",
    "- robbery;\n",
    "- physical and computer hacking;\n",
    "- discrimination of employees;\n",
    "- occupational health and safety violations resulting in accidents;\n",
    "- salary claim;\n",
    "- disclosure of customers' personal data;\n",
    "- aggressive trading strategy to maximize commissions;\n",
    "- incorrect assessment of the customer's profile;\n",
    "- incorrectly constructed templates, contracts, regulations;\n",
    "- physical destruction of assets as a result of random events;\n",
    "- acts of vandalism or terrorism;\n",
    "- software errors;\n",
    "- equipment malfunction;\n",
    "- telecommunications problems;\n",
    "- errors when entering data into the system;\n",
    "- errors in the execution, settlement and processing of transactions;\n",
    "- negligence in monitoring and reporting;\n",
    "- improper management of customer accounts."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "08a7a7f1",
   "metadata": {},
   "source": [
    "**Explain the difference between the following operational risk assessment approaches: BIA, TSA and AMA.**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d8f9e6f6",
   "metadata": {},
   "source": [
    "The difference between the BIA, TSA and AMA approaches to operational risk is visible in several areas:\n",
    "- the largest banks are obliged to use only the AMA method, which means that these methods are not interchangeable in some institutions;\n",
    "- calculation of required capital. In the case of the base indicator method, it is calculated as 15% of the average of the net positive interest and non-interest results from the previous 3 years. A different approach is presented by the standard method, where the required capital is determined as the sum of the results from 8 business lines weighted by coefficients. The advanced measurement method is different, which takes into account 7 categories of events for up to 8 business lines, creating 56 risk areas used to calculate the operational value at risk;\n",
    "- TSA does not take into account a negative total score."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c043d90c",
   "metadata": {},
   "source": [
    "**What is the LDA method, which is used as part of the advanced measurement method for operational risk?**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4d6a2da1",
   "metadata": {},
   "source": [
    "The LDA (Loss Distribution Approach) method involves modeling the observed distribution of losses in order to predict the range of potential future losses. It consists of the following processes:\n",
    "- estimation of the severity of losses;\n",
    "- estimation of loss frequency;\n",
    "- calculation of capital charge;\n",
    "- calculation of confidence interval;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d23f003",
   "metadata": {},
   "source": [
    "## Exercise 2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "18a9890e",
   "metadata": {},
   "source": [
    "**Libraries used**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "b3dd30c1",
   "metadata": {},
   "outputs": [],
   "source": [
    "options(warn=-1)\n",
    "library(\"readxl\")\n",
    "library(\"nortest\")\n",
    "library(\"ptsuite\")\n",
    "library('lubridate')\n",
    "library('ggplot2')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7f3306ab",
   "metadata": {},
   "source": [
    "**Data import**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "d7eee08e",
   "metadata": {},
   "outputs": [],
   "source": [
    "basel_op <-  read_excel(\"BaselOP.xlsx\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "022cf91d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>ID</th><th scope=col>Date</th><th scope=col>Business Line</th><th scope=col>Risk Category</th><th scope=col>Gross Loss Amount</th><th scope=col>Recovery Amount</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>1         </td><td>03/01/2009</td><td>AG        </td><td>CPBP      </td><td> 8045.29  </td><td> 6926.03  </td></tr>\n",
       "\t<tr><td>2         </td><td>06/01/2009</td><td>CB        </td><td>CPBP      </td><td> 6689.97  </td><td> 5933.89  </td></tr>\n",
       "\t<tr><td>3         </td><td>11/01/2009</td><td>AG        </td><td>CPBP      </td><td>32965.18  </td><td>24815.44  </td></tr>\n",
       "\t<tr><td>4         </td><td>11/01/2009</td><td>AG        </td><td>CPBP      </td><td>17775.68  </td><td>15029.16  </td></tr>\n",
       "\t<tr><td>5         </td><td>12/01/2009</td><td>RBR       </td><td>EDPM      </td><td>  767.43  </td><td>  767.43  </td></tr>\n",
       "\t<tr><td>6         </td><td>14/01/2009</td><td>AG        </td><td>CPBP      </td><td>41426.22  </td><td>34513.52  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllll}\n",
       " ID & Date & Business Line & Risk Category & Gross Loss Amount & Recovery Amount\\\\\n",
       "\\hline\n",
       "\t 1          & 03/01/2009 & AG         & CPBP       &  8045.29   &  6926.03  \\\\\n",
       "\t 2          & 06/01/2009 & CB         & CPBP       &  6689.97   &  5933.89  \\\\\n",
       "\t 3          & 11/01/2009 & AG         & CPBP       & 32965.18   & 24815.44  \\\\\n",
       "\t 4          & 11/01/2009 & AG         & CPBP       & 17775.68   & 15029.16  \\\\\n",
       "\t 5          & 12/01/2009 & RBR        & EDPM       &   767.43   &   767.43  \\\\\n",
       "\t 6          & 14/01/2009 & AG         & CPBP       & 41426.22   & 34513.52  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| ID | Date | Business Line | Risk Category | Gross Loss Amount | Recovery Amount |\n",
       "|---|---|---|---|---|---|\n",
       "| 1          | 03/01/2009 | AG         | CPBP       |  8045.29   |  6926.03   |\n",
       "| 2          | 06/01/2009 | CB         | CPBP       |  6689.97   |  5933.89   |\n",
       "| 3          | 11/01/2009 | AG         | CPBP       | 32965.18   | 24815.44   |\n",
       "| 4          | 11/01/2009 | AG         | CPBP       | 17775.68   | 15029.16   |\n",
       "| 5          | 12/01/2009 | RBR        | EDPM       |   767.43   |   767.43   |\n",
       "| 6          | 14/01/2009 | AG         | CPBP       | 41426.22   | 34513.52   |\n",
       "\n"
      ],
      "text/plain": [
       "  ID Date       Business Line Risk Category Gross Loss Amount Recovery Amount\n",
       "1 1  03/01/2009 AG            CPBP           8045.29           6926.03       \n",
       "2 2  06/01/2009 CB            CPBP           6689.97           5933.89       \n",
       "3 3  11/01/2009 AG            CPBP          32965.18          24815.44       \n",
       "4 4  11/01/2009 AG            CPBP          17775.68          15029.16       \n",
       "5 5  12/01/2009 RBR           EDPM            767.43            767.43       \n",
       "6 6  14/01/2009 AG            CPBP          41426.22          34513.52       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(basel_op)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "87d373b2",
   "metadata": {},
   "source": [
    "**Selection of PS business line**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "c32abd15",
   "metadata": {},
   "outputs": [],
   "source": [
    "ps_basel_op <- basel_op[which(basel_op$'Business Line' == 'PS'),]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "9a833c4e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>ID</th><th scope=col>Date</th><th scope=col>Business Line</th><th scope=col>Risk Category</th><th scope=col>Gross Loss Amount</th><th scope=col>Recovery Amount</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>23        </td><td>08/02/2009</td><td>PS        </td><td>DPA       </td><td>12716.87  </td><td>10148.41  </td></tr>\n",
       "\t<tr><td>28        </td><td>16/02/2009</td><td>PS        </td><td>EPWS      </td><td> 2623.35  </td><td> 2468.11  </td></tr>\n",
       "\t<tr><td>30        </td><td>17/02/2009</td><td>PS        </td><td>IF        </td><td>54797.76  </td><td>36427.16  </td></tr>\n",
       "\t<tr><td>38        </td><td>28/02/2009</td><td>PS        </td><td>DPA       </td><td> 3781.29  </td><td> 3290.28  </td></tr>\n",
       "\t<tr><td>42        </td><td>05/03/2009</td><td>PS        </td><td>DPA       </td><td>28302.43  </td><td>21642.07  </td></tr>\n",
       "\t<tr><td>45        </td><td>07/03/2009</td><td>PS        </td><td>DPA       </td><td> 5574.01  </td><td> 5054.74  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllll}\n",
       " ID & Date & Business Line & Risk Category & Gross Loss Amount & Recovery Amount\\\\\n",
       "\\hline\n",
       "\t 23         & 08/02/2009 & PS         & DPA        & 12716.87   & 10148.41  \\\\\n",
       "\t 28         & 16/02/2009 & PS         & EPWS       &  2623.35   &  2468.11  \\\\\n",
       "\t 30         & 17/02/2009 & PS         & IF         & 54797.76   & 36427.16  \\\\\n",
       "\t 38         & 28/02/2009 & PS         & DPA        &  3781.29   &  3290.28  \\\\\n",
       "\t 42         & 05/03/2009 & PS         & DPA        & 28302.43   & 21642.07  \\\\\n",
       "\t 45         & 07/03/2009 & PS         & DPA        &  5574.01   &  5054.74  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| ID | Date | Business Line | Risk Category | Gross Loss Amount | Recovery Amount |\n",
       "|---|---|---|---|---|---|\n",
       "| 23         | 08/02/2009 | PS         | DPA        | 12716.87   | 10148.41   |\n",
       "| 28         | 16/02/2009 | PS         | EPWS       |  2623.35   |  2468.11   |\n",
       "| 30         | 17/02/2009 | PS         | IF         | 54797.76   | 36427.16   |\n",
       "| 38         | 28/02/2009 | PS         | DPA        |  3781.29   |  3290.28   |\n",
       "| 42         | 05/03/2009 | PS         | DPA        | 28302.43   | 21642.07   |\n",
       "| 45         | 07/03/2009 | PS         | DPA        |  5574.01   |  5054.74   |\n",
       "\n"
      ],
      "text/plain": [
       "  ID Date       Business Line Risk Category Gross Loss Amount Recovery Amount\n",
       "1 23 08/02/2009 PS            DPA           12716.87          10148.41       \n",
       "2 28 16/02/2009 PS            EPWS           2623.35           2468.11       \n",
       "3 30 17/02/2009 PS            IF            54797.76          36427.16       \n",
       "4 38 28/02/2009 PS            DPA            3781.29           3290.28       \n",
       "5 42 05/03/2009 PS            DPA           28302.43          21642.07       \n",
       "6 45 07/03/2009 PS            DPA            5574.01           5054.74       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(ps_basel_op)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "165ed892",
   "metadata": {},
   "source": [
    "**Determination of net start (Net loss)**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "202c8a8a",
   "metadata": {},
   "outputs": [],
   "source": [
    "ps_basel_op[\"Net Loss\"] <- rep(NA, length(ps_basel_op$ID))\n",
    "for(i in 1:length(ps_basel_op$ID)){\n",
    "    ps_basel_op$'Net Loss'[i] = ps_basel_op$'Gross Loss Amount'[i] - ps_basel_op$'Recovery Amount'[i]\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "10f973e4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>ID</th><th scope=col>Date</th><th scope=col>Business Line</th><th scope=col>Risk Category</th><th scope=col>Gross Loss Amount</th><th scope=col>Recovery Amount</th><th scope=col>Net Loss</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>23        </td><td>08/02/2009</td><td>PS        </td><td>DPA       </td><td>12716.87  </td><td>10148.41  </td><td> 2568.46  </td></tr>\n",
       "\t<tr><td>28        </td><td>16/02/2009</td><td>PS        </td><td>EPWS      </td><td> 2623.35  </td><td> 2468.11  </td><td>  155.24  </td></tr>\n",
       "\t<tr><td>30        </td><td>17/02/2009</td><td>PS        </td><td>IF        </td><td>54797.76  </td><td>36427.16  </td><td>18370.60  </td></tr>\n",
       "\t<tr><td>38        </td><td>28/02/2009</td><td>PS        </td><td>DPA       </td><td> 3781.29  </td><td> 3290.28  </td><td>  491.01  </td></tr>\n",
       "\t<tr><td>42        </td><td>05/03/2009</td><td>PS        </td><td>DPA       </td><td>28302.43  </td><td>21642.07  </td><td> 6660.36  </td></tr>\n",
       "\t<tr><td>45        </td><td>07/03/2009</td><td>PS        </td><td>DPA       </td><td> 5574.01  </td><td> 5054.74  </td><td>  519.27  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllllll}\n",
       " ID & Date & Business Line & Risk Category & Gross Loss Amount & Recovery Amount & Net Loss\\\\\n",
       "\\hline\n",
       "\t 23         & 08/02/2009 & PS         & DPA        & 12716.87   & 10148.41   &  2568.46  \\\\\n",
       "\t 28         & 16/02/2009 & PS         & EPWS       &  2623.35   &  2468.11   &   155.24  \\\\\n",
       "\t 30         & 17/02/2009 & PS         & IF         & 54797.76   & 36427.16   & 18370.60  \\\\\n",
       "\t 38         & 28/02/2009 & PS         & DPA        &  3781.29   &  3290.28   &   491.01  \\\\\n",
       "\t 42         & 05/03/2009 & PS         & DPA        & 28302.43   & 21642.07   &  6660.36  \\\\\n",
       "\t 45         & 07/03/2009 & PS         & DPA        &  5574.01   &  5054.74   &   519.27  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| ID | Date | Business Line | Risk Category | Gross Loss Amount | Recovery Amount | Net Loss |\n",
       "|---|---|---|---|---|---|---|\n",
       "| 23         | 08/02/2009 | PS         | DPA        | 12716.87   | 10148.41   |  2568.46   |\n",
       "| 28         | 16/02/2009 | PS         | EPWS       |  2623.35   |  2468.11   |   155.24   |\n",
       "| 30         | 17/02/2009 | PS         | IF         | 54797.76   | 36427.16   | 18370.60   |\n",
       "| 38         | 28/02/2009 | PS         | DPA        |  3781.29   |  3290.28   |   491.01   |\n",
       "| 42         | 05/03/2009 | PS         | DPA        | 28302.43   | 21642.07   |  6660.36   |\n",
       "| 45         | 07/03/2009 | PS         | DPA        |  5574.01   |  5054.74   |   519.27   |\n",
       "\n"
      ],
      "text/plain": [
       "  ID Date       Business Line Risk Category Gross Loss Amount Recovery Amount\n",
       "1 23 08/02/2009 PS            DPA           12716.87          10148.41       \n",
       "2 28 16/02/2009 PS            EPWS           2623.35           2468.11       \n",
       "3 30 17/02/2009 PS            IF            54797.76          36427.16       \n",
       "4 38 28/02/2009 PS            DPA            3781.29           3290.28       \n",
       "5 42 05/03/2009 PS            DPA           28302.43          21642.07       \n",
       "6 45 07/03/2009 PS            DPA            5574.01           5054.74       \n",
       "  Net Loss\n",
       "1  2568.46\n",
       "2   155.24\n",
       "3 18370.60\n",
       "4   491.01\n",
       "5  6660.36\n",
       "6   519.27"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(ps_basel_op)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37d17abd",
   "metadata": {},
   "source": [
    "**Loss turnout**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "4963d0f3",
   "metadata": {},
   "outputs": [],
   "source": [
    "loss_frequency <- data.frame(\n",
    "    half_of_a_year = seq(1, 20),\n",
    "    number_of_loss_events = rep(0, 20)           \n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "36d47124",
   "metadata": {},
   "outputs": [],
   "source": [
    "for(i in 1:length(ps_basel_op$Date)){\n",
    "    int1 = interval(dmy(\"08/02/2009\"), dmy(\"08/08/2009\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int1 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[1] = loss_frequency$number_of_loss_events[1] + 1\n",
    "    }\n",
    "    int2 = interval(dmy(\"09/08/2009\"), dmy(\"08/02/2010\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int2 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[2] = loss_frequency$number_of_loss_events[2] + 1\n",
    "    }\n",
    "    int3 = interval(dmy(\"09/02/2010\"), dmy(\"08/08/2010\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int3 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[3] = loss_frequency$number_of_loss_events[3] + 1\n",
    "    }\n",
    "    int4 = interval(dmy(\"09/08/2010\"), dmy(\"08/02/2011\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int4 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[4] = loss_frequency$number_of_loss_events[4] + 1\n",
    "    }\n",
    "    int5 = interval(dmy(\"09/02/2011\"), dmy(\"08/08/2011\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int5 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[5] = loss_frequency$number_of_loss_events[5] + 1\n",
    "    }\n",
    "    int6 = interval(dmy(\"09/08/2011\"), dmy(\"08/02/2012\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int6 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[6] = loss_frequency$number_of_loss_events[6] + 1\n",
    "    }\n",
    "    int7 = interval(dmy(\"09/02/2012\"), dmy(\"08/08/2012\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int7 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[7] = loss_frequency$number_of_loss_events[7] + 1\n",
    "    }\n",
    "    int8 = interval(dmy(\"09/08/2012\"), dmy(\"08/02/2013\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int8 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[8] = loss_frequency$number_of_loss_events[8] + 1\n",
    "    }\n",
    "    int9 = interval(dmy(\"09/02/2013\"), dmy(\"08/08/2013\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int9 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[9] = loss_frequency$number_of_loss_events[9] + 1\n",
    "    }\n",
    "    int10 = interval(dmy(\"09/08/2013\"), dmy(\"08/02/2014\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int10 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[10] = loss_frequency$number_of_loss_events[10] + 1\n",
    "    }\n",
    "    int11 = interval(dmy(\"09/02/2014\"), dmy(\"08/08/2014\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int11 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[11] = loss_frequency$number_of_loss_events[11] + 1\n",
    "    }\n",
    "    int12 = interval(dmy(\"09/08/2014\"), dmy(\"08/02/2015\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int12 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[12] = loss_frequency$number_of_loss_events[12] + 1\n",
    "    }\n",
    "    int13 = interval(dmy(\"09/02/2015\"), dmy(\"08/08/2015\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int13 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[13] = loss_frequency$number_of_loss_events[13] + 1\n",
    "    }\n",
    "    int14 = interval(dmy(\"09/08/2015\"), dmy(\"08/02/2016\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int14 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[14] = loss_frequency$number_of_loss_events[14] + 1\n",
    "    }\n",
    "    int15 = interval(dmy(\"09/02/2016\"), dmy(\"08/08/2016\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int15 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[15] = loss_frequency$number_of_loss_events[15] + 1\n",
    "    }\n",
    "    int16 = interval(dmy(\"09/08/2016\"), dmy(\"08/02/2017\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int16 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[16] = loss_frequency$number_of_loss_events[16] + 1\n",
    "    }\n",
    "    int17 = interval(dmy(\"09/02/2017\"), dmy(\"08/08/2017\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int17 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[17] = loss_frequency$number_of_loss_events[17] + 1\n",
    "    }\n",
    "    int18 = interval(dmy(\"09/08/2017\"), dmy(\"08/02/2018\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int18 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[18] = loss_frequency$number_of_loss_events[18] + 1\n",
    "    }\n",
    "    int19 = interval(dmy(\"09/02/2018\"), dmy(\"08/08/2018\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int19 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[19] = loss_frequency$number_of_loss_events[19] + 1\n",
    "    }\n",
    "    int20 = interval(dmy(\"09/08/2018\"), dmy(\"30/12/2018\"))\n",
    "    if(dmy(ps_basel_op$Date[i]) %within% int20 == TRUE & ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "       loss_frequency$number_of_loss_events[20] = loss_frequency$number_of_loss_events[20] + 1\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "1494ce40",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>half_of_a_year</th><th scope=col>number_of_loss_events</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 1</td><td>25</td></tr>\n",
       "\t<tr><td> 2</td><td> 9</td></tr>\n",
       "\t<tr><td> 3</td><td>13</td></tr>\n",
       "\t<tr><td> 4</td><td>17</td></tr>\n",
       "\t<tr><td> 5</td><td>35</td></tr>\n",
       "\t<tr><td> 6</td><td>27</td></tr>\n",
       "\t<tr><td> 7</td><td>42</td></tr>\n",
       "\t<tr><td> 8</td><td>41</td></tr>\n",
       "\t<tr><td> 9</td><td> 9</td></tr>\n",
       "\t<tr><td>10</td><td>14</td></tr>\n",
       "\t<tr><td>11</td><td>38</td></tr>\n",
       "\t<tr><td>12</td><td>18</td></tr>\n",
       "\t<tr><td>13</td><td>11</td></tr>\n",
       "\t<tr><td>14</td><td>18</td></tr>\n",
       "\t<tr><td>15</td><td>22</td></tr>\n",
       "\t<tr><td>16</td><td>22</td></tr>\n",
       "\t<tr><td>17</td><td>21</td></tr>\n",
       "\t<tr><td>18</td><td>28</td></tr>\n",
       "\t<tr><td>19</td><td>21</td></tr>\n",
       "\t<tr><td>20</td><td>22</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " half\\_of\\_a\\_year & number\\_of\\_loss\\_events\\\\\n",
       "\\hline\n",
       "\t  1 & 25\\\\\n",
       "\t  2 &  9\\\\\n",
       "\t  3 & 13\\\\\n",
       "\t  4 & 17\\\\\n",
       "\t  5 & 35\\\\\n",
       "\t  6 & 27\\\\\n",
       "\t  7 & 42\\\\\n",
       "\t  8 & 41\\\\\n",
       "\t  9 &  9\\\\\n",
       "\t 10 & 14\\\\\n",
       "\t 11 & 38\\\\\n",
       "\t 12 & 18\\\\\n",
       "\t 13 & 11\\\\\n",
       "\t 14 & 18\\\\\n",
       "\t 15 & 22\\\\\n",
       "\t 16 & 22\\\\\n",
       "\t 17 & 21\\\\\n",
       "\t 18 & 28\\\\\n",
       "\t 19 & 21\\\\\n",
       "\t 20 & 22\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| half_of_a_year | number_of_loss_events |\n",
       "|---|---|\n",
       "|  1 | 25 |\n",
       "|  2 |  9 |\n",
       "|  3 | 13 |\n",
       "|  4 | 17 |\n",
       "|  5 | 35 |\n",
       "|  6 | 27 |\n",
       "|  7 | 42 |\n",
       "|  8 | 41 |\n",
       "|  9 |  9 |\n",
       "| 10 | 14 |\n",
       "| 11 | 38 |\n",
       "| 12 | 18 |\n",
       "| 13 | 11 |\n",
       "| 14 | 18 |\n",
       "| 15 | 22 |\n",
       "| 16 | 22 |\n",
       "| 17 | 21 |\n",
       "| 18 | 28 |\n",
       "| 19 | 21 |\n",
       "| 20 | 22 |\n",
       "\n"
      ],
      "text/plain": [
       "   half_of_a_year number_of_loss_events\n",
       "1   1             25                   \n",
       "2   2              9                   \n",
       "3   3             13                   \n",
       "4   4             17                   \n",
       "5   5             35                   \n",
       "6   6             27                   \n",
       "7   7             42                   \n",
       "8   8             41                   \n",
       "9   9              9                   \n",
       "10 10             14                   \n",
       "11 11             38                   \n",
       "12 12             18                   \n",
       "13 13             11                   \n",
       "14 14             18                   \n",
       "15 15             22                   \n",
       "16 16             22                   \n",
       "17 17             21                   \n",
       "18 18             28                   \n",
       "19 19             21                   \n",
       "20 20             22                   "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "loss_frequency"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "e726f883",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAANlBMVEUAAAAEdLpNTU1oaGh8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\nRK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ggplot(loss_frequency, aes(x = half_of_a_year, y = number_of_loss_events)) +\n",
    "geom_bar(stat = \"identity\", width=0.7, fill=\"#0474BA\") +\n",
    "ggtitle(\"Number of loss events per half of a year\") +\n",
    "ylab('Number of events') + xlab('Half of a year') +\n",
    "theme_minimal()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cc534271",
   "metadata": {},
   "source": [
    "**Fitting the best possible distribution of the number of operational events**\n",
    "\n",
    "a) Pearson's chi-square test\n",
    "\n",
    "H0: the data can be described by the Pearson distribution\n",
    "\n",
    "H1: the data cannot be described by the Pearson distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "70120dde",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tChi-squared test for given probabilities\n",
       "\n",
       "data:  loss_frequency$number_of_loss_events\n",
       "X-squared = 84.351, df = 19, p-value = 3.27e-10\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "chisq.test(loss_frequency$number_of_loss_events) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1c4f34a",
   "metadata": {},
   "source": [
    "The p-value is less than 0.05, which means that H0 is rejected. The observations cannot be described by the Pearson distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "55fd4607",
   "metadata": {},
   "source": [
    "b) Kolmogorov-Smirnov test\n",
    "\n",
    "H0: the data can be described by a normal distribution\n",
    "\n",
    "H1: the data cannot be described by a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "4a3f24b0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tOne-sample Kolmogorov-Smirnov test\n",
       "\n",
       "data:  loss_frequency$number_of_loss_events\n",
       "D = 1, p-value < 2.2e-16\n",
       "alternative hypothesis: two-sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ks.test(loss_frequency$number_of_loss_events, \"pnorm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "399c3ed1",
   "metadata": {},
   "source": [
    "The p-value is less than 0.05, which means that H0 is rejected. The observations cannot be described by a normal distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "41fc0fce",
   "metadata": {},
   "source": [
    "c) Anderson-Darling test\n",
    "\n",
    "H0: the data can be described by a normal distribution\n",
    "\n",
    "H1: the data cannot be described by a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "f1dd48ee",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tAnderson-Darling normality test\n",
       "\n",
       "data:  loss_frequency$number_of_loss_events\n",
       "A = 0.48079, p-value = 0.2069\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ad.test(loss_frequency$number_of_loss_events)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "be52cf59",
   "metadata": {},
   "source": [
    "The p-value is less than 0.05, which means that H0 is rejected. The observations cannot be described by a normal distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9b6cc5dc",
   "metadata": {},
   "source": [
    "The problem assumes an **exponential distribution**."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ef13d2c6",
   "metadata": {},
   "source": [
    "**Severity of losses**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "47c77100",
   "metadata": {},
   "outputs": [],
   "source": [
    "loss_severity <- c()\n",
    "for(i in 1:length(ps_basel_op$Date)){\n",
    "    if(ps_basel_op$'Net Loss'[i] > 0.00){\n",
    "        loss_severity <- append(loss_severity, ps_basel_op$'Net Loss'[i])\n",
    "    }\n",
    "}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "2acd50e9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAAOVBMVEUAAAAyjMQ2kMhNTU1o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\nzJCc62GG5FwPMyTnepghOdfD/g+ys/sM0KYKNwAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "loss_severity <- data.frame(\n",
    "loss = loss_severity\n",
    ")\n",
    "\n",
    "ggplot(loss_severity, aes(x = loss)) +\n",
    "geom_density(fill=\"#0474BA\", alpha=0.8) +\n",
    "ggtitle(\"Loss severity distribution\") +\n",
    "ylab('Density') + xlab('Loss') +\n",
    "theme_minimal()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "40c317d8",
   "metadata": {},
   "source": [
    "**Matching the best possible distribution of operational event severity**\n",
    "\n",
    "a) Pearson's chi-square test\n",
    "\n",
    "H0: the data can be described by the Pearson distribution\n",
    "\n",
    "H1: the data cannot be described by the Pearson distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "afdd307e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tChi-squared test for given probabilities\n",
       "\n",
       "data:  loss_severity$loss\n",
       "X-squared = 1644488224, df = 452, p-value < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "chisq.test(loss_severity$loss) "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4e57822e",
   "metadata": {},
   "source": [
    "The p-value is less than 0.05, which means that H0 is rejected. The observations cannot be described by the Pearson distribution.\n",
    "\n",
    "b) Kolmogorov-Smirnov test\n",
    "\n",
    "H0: the data can be described by a normal distribution\n",
    "\n",
    "H1: the data cannot be described by a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "e7e84646",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tOne-sample Kolmogorov-Smirnov test\n",
       "\n",
       "data:  loss_severity$loss\n",
       "D = 1, p-value < 2.2e-16\n",
       "alternative hypothesis: two-sided\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ks.test(loss_severity$loss, \"pnorm\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3805cc6f",
   "metadata": {},
   "source": [
    "The p-value is less than 0.05, which means that H0 is rejected. The observations cannot be described by a normal distribution.\n",
    "\n",
    "c) Anderson-Darling test\n",
    "\n",
    "H0: the data can be described by a normal distribution\n",
    "\n",
    "H1: the data cannot be described by a normal distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "aabe6e47",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "\tAnderson-Darling normality test\n",
       "\n",
       "data:  loss_severity$loss\n",
       "A = 157.62, p-value < 2.2e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ad.test(loss_severity$loss)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f9f7cf0",
   "metadata": {},
   "source": [
    "The p-value is less than 0.05, which means that H0 is rejected. The observations cannot be described by a normal distribution.\n",
    "\n",
    "**Observing the visualizations in the task, an exponential distribution is assumed.**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e8106bbe",
   "metadata": {},
   "source": [
    "## Exercise 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fca93252",
   "metadata": {},
   "source": [
    "**Determination of 99.9 OpVaR and 99.9 OpES using the Monte Carlo method.**\n",
    "\n",
    "The task uses the established distribution from task 2, i.e. the exponential distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "id": "ff436e8c",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "'OpVaR: 7.114293'"
      ],
      "text/latex": [
       "'OpVaR: 7.114293'"
      ],
      "text/markdown": [
       "'OpVaR: 7.114293'"
      ],
      "text/plain": [
       "[1] \"OpVaR: 7.114293\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "'OpES: 7.814895'"
      ],
      "text/latex": [
       "'OpES: 7.814895'"
      ],
      "text/markdown": [
       "'OpES: 7.814895'"
      ],
      "text/plain": [
       "[1] \"OpES: 7.814895\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "set.seed(123) \n",
    "          \n",
    "losses <- rexp(10000, rate = 1)\n",
    "OpVaR <- quantile(losses, 0.999)\n",
    "es_vector <- c()\n",
    "for(i in 1:10000){\n",
    "    if(losses[i] > OpVaR){\n",
    "        es_vector <- append(es_vector, losses[i])\n",
    "    }\n",
    "}\n",
    "OpES <- mean(es_vector)\n",
    "sprintf(\"OpVaR: %f\", OpVaR)\n",
    "sprintf(\"OpES: %f\", OpES)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 165,
   "id": "edf8495d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA0gAAANICAMAAADKOT/pAAAAPFBMVEUAAAAA/wAEdLpNTU1o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\nste5AAAAAElFTkSuQmCC",
      "text/plain": [
       "Plot with title \"Exponential distribution of losses with OpVaR at 99.9% & OpES at 99.9%\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hist(losses,\n",
    "     breaks = 50,\n",
    "     freq = FALSE,\n",
    "     col = '#0474BA',\n",
    "     main =\"Exponential distribution of losses with OpVaR at 99.9% & OpES at 99.9%\",\n",
    "     xlab =\"Losses\")\n",
    "\n",
    "lines(seq(0, 10, by=0.5),\n",
    "      dexp(xvals, rate = 1),\n",
    "      lwd=3,\n",
    "      col=\"red\")\n",
    "\n",
    "abline(v = OpVaR, lwd=3,lty = 3, col=\"orange\")\n",
    "abline(v = OpES, lwd=3, lty = 3, col=\"green\")\n",
    "\n",
    "legend(\"top\", c(\"Theoretical distribution\", \"OpVaR\", \"OpES\"),\n",
    "       bty = \"n\", lwd = 3, col = c(\"red\", \"orange\", \"green\"))"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
   "version": "3.6.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}