![](\"_static/loss_credit_rating_var.png\")
"
],
"text/plain": [
"')"
]
},
{
"cell_type": "markdown",
"id": "quick-gardening",
"metadata": {},
"source": [
"**Other Considerations**\n",
"\n",
"+ Correlation of Defaults:\n",
" + exogeneous factors may result in borrowers defaulting at the same time and more than they would independently. This increases losses at those times.\n",
"+ Correlation of PD and LGD:\n",
" + simplest (and prevailing) assumption is PD and LGD are uncorrelated\n",
" + more realistic assumption is PD and LGD are correlated\n",
"+ 3rd party providers\n",
" + data and modelling tools can be purchased from 3rd parties\n",
" + Examples: \n",
" + https://theanalyticsboutique.com/demos/\n",
"\n",
"**Process to Determine Loss Distribution**\n",
"\n",
"+ Determine **PD**, **LGD**, LGD distribution, PD correlations, PD/LGD correlations (if any).\n",
"+ Find **EL** and **UL**.\n",
"+ Run Monte Carlo simulations. Determine the **Loss Distribution** of the portfolio.\n",
"+ Find best fit for beta distribution\n",
"+ Find **Economic Capital** through **VaR** (for whatever credit rating desired).\n",
"\n",
"+ Fico needs equity equal to *at least* the **Economic Capital**.\n",
"+ Extreme Value Theory can be used to protect against even greater tail risks.\n",
"\n",
""
]
},
{
"cell_type": "markdown",
"id": "distinguished-sequence",
"metadata": {},
"source": [
"## Probability of Default ##\n",
"\n", "\n", "+ determined through analysis of company financial statements, business/industry risks, and exogenous factors like economic activity.\n", "+ bond/credit ratings are categorized based on PD. The S&P ratings and their analogous PDs are shown below:" ] }, { "cell_type": "code", "execution_count": 41, "id": "elementary-quarter", "metadata": { "tags": [ "hide_input" ] }, "outputs": [ { "data": { "text/html": [ "
![](\"_static/loss_pds.png\")
"
],
"text/plain": [
"'))"
]
},
{
"cell_type": "markdown",
"id": "central-blues",
"metadata": {},
"source": [
"+ credit ratings BBB+ and greater are typically considered investment grade.\n",
"+ rating BBB to CCC+/CCC are consider \"junk\" or speculative.\n",
"+ CCC+/CCC or worse is typically considered a default"
]
},
{
"cell_type": "markdown",
"id": "generous-bidder",
"metadata": {},
"source": [
"+ The distribution of ratings within a portfolio are often unique to the portfolio: what borrowers are targeted? what is risk management quality? etc.\n",
"\n",
"+ For now, it is useful to assume a portfolio with normally distributed PDs with the following parameters:\n",
"\n",
" + $\\mu = .44$%\n",
" + $\\sigma = .2$%\n",
"\n",
"The portfolio has a PD distribution as per below. Note the x-axis increases exponentially, so the normal distribution is skewed."
]
},
{
"cell_type": "code",
"execution_count": 42,
"id": "norman-pontiac",
"metadata": {},
"outputs": [
{
"data": {
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hTgdVrNuAICnzKqZPgJ0rWT/l0cQJrkRMuo9KXs+sarZ3I8HdBCqLezaV3GmAWt7pIFzekuDUaFXv2RPU8O4BcdufVc2246cZwEkpbu/mhPlDUtj+EuDAJNvch213yUic+qXyPqcaZ0KsbwGXVVHv06s4VnciGKoi2XqbgHOqi5cgcU22/pC4Mv3i5n/r80iQ8D1fzfv+BtAsybqdk73PVb1ftTwG9yO4UrKy+LYAV1SxfsrfCykcDzV+DQl1UN1UTnBLuV1rG6umwpnUwiaRcfcKd7+MoF/QUIIWjM0EpzDHApcDB7h7nS9zd/d/EIyiPpagU3hFHbd3C8FI7v8iSM42EnQ0n0Awrlgvr1sLV2X7XU7QR+0iYCTBqbLNBP2kXgROdPdzPb13D6gg6Fs0hyCp+AvB/Tn3cPdhVaxXlRsI6uPfBHezWEzwj3gl8B7wW6CHu7+fuKK7TyL4p/giweuut1YWd/87wWsfTpBQbiRIXP8I9K7sWPVgoNqDgTsI7re5ieA1DwUOcvf/pLD74whOn01j+wsZahL/Bnc/laCP5SsErVmbw9fyOvBD4CivwentdPNgvMYeBD+ePia4EnQjwXh6jwL7ufvfooqvDjYBSwmOl6EEd9/o5u6nuFrWJAXm7lHHICIiIiJVUAubiIiISJZTwiYiIiKS5ZSwiYiIiGQ5JWwiIiIiWU4Jm4iIiEiWU8ImIiIikuWUsImIiIhkOSVsIiIiIllOCZuIiIhIllPCJiIiIpLllLCJiIiIZDklbCIiIiJZTgmbiIiISJZTwiYiIiKS5ZSwiYiIiGQ5JWwiIiIiWU4Jm4iIiEiWU8ImIiIikuWUsImIiIhkOSVsIiIiIllOCZuIiIhIllPCJiIiIpLliqMOoDZat27tnTt3jjoMERERkWp9+OGHS929TV22kZMJW+fOnZkwYULUYYiIiIhUy8xm13UbOiUqIiIikuWUsImIiIhkOSVsIiIiIllOCZuIiIhIllPCJiIiIpLllLCJiIiIZLlqEzYzO93M/mtms81svZlNN7NbzKxZQrkWZvawmS01s7Vm9oaZ7Z1kezuY2V/MbEG4vXFmdmg6X5SIiIhIPkmlhe3nQDnwS+AY4H7gp8DrZtYAwMwMeClcfgVwGlACjDKzXRO29wjwE+A3wAnAAuBVM9uvri9GREREJB+lMnDuie6+JO75aDNbDvwT6AeMBE4CDgb6u/soADMbB8wErgeuDOftC5wD/NjdHwvnjQamAr8PtyMiIiIicaptYUtI1mI+CB87hI8nAV/HkrVwvW8IWt1OjlvvJGAz8HRcuS3AU8AAM2tUo+hFRERECkBtLzo4LHz8LHzsBUxJUm4q0MnMmsaVm+nu65KUawh0r2U8IiIiInmrxgmbmXUgOH35hrvHbujZEliRpPjy8LFFiuVa1jQeERERkXxXo4QtbCl7EdgCXJCRiCrf92Azm2BmE5YsSXaWVkSiNmrUKE4++WQ6derEaaedxnvvvRd1SCIieSHlhM3MdiTok9YVGODu8+IWr2BbK1q8lnHLUym3PMkyANz9QXcvc/eyNm3apBq2iNST+++/nyOPPJJhw4Yxd+5cnnvuOfr168fLL78cdWgiIjkvpYTNzEqAoUAZcJy7T04oMpWgf1qinsAcd18TV66LmTVOUm4T8EWqgYtI9hg+fDiXXnopFRUV3HDDDXzyySdcdNFFbNiwgYEDB/LVV19FHaKISE5LZeDcBsATQH/gFHcfn6TYMKCDmR0Wt15z4MRwWcxLBOOznRFXrhg4E3jN3TfW5kWISHQWLlzIoEGDAPjTn/7Erbfeyr777suDDz7IWWedxfr16xk8eDDuHm2gIiI5LJUWtr8TJFh3AGvNrE/cFBsUdxgwDnjczM4yswHhPANui23I3T8mGNLjbjO7yMyOIBjSowvw27S9KhGpNzfddBNLlizhyCOP5IYbbtg638y45557aNWqFW+++SYjR46MMEoRkdyWSsJ2bPj4K4KkLH66CMDdKwjuWvA6cB/wPMHdEQ5397kJ27sAeAz4A/AK0BE4xt0/qtMrEZF6N23aNB577DGKioq47777aNBg+6+UNm3acPXVVwNw++23RxGiiEhesFw8TVFWVuYTJkyovqCIZNSZZ57JM888w8UXX8w//vGPpGWWLVtGp06dWLduHVOmTKFXr2TdXUVE8peZfejuZXXZRm0HzhWRAjdnzhyGDh1KcXExv/71ryst16pVK8477zwAHn/88foKT0QkryhhE5Faue+++6ioqOCMM86gQ4cOVZY999xzAXjyySd18YGISC0oYRORGlu/fj0PPfQQAFdeeWW15fv27cuuu+7K7NmzGTduXKbDExHJO0rYRKTGXnjhBZYvX853v/tdvve971VbvkGDBpxxRjCaz4svvpjp8ERE8o4SNhGpsX//+98AXHDBBZhZSuscf/zxAIwYMSJjcYmI5CslbCJSI4sWLeK1116juLiYs846K+X1+vbtS5MmTZg0aRLz58/PYIQiIvlHCZuI1MiTTz5JeXk5xx13HK1bt055vUaNGtG/f39ArWwiIjWlhE1EauSZZ54Btl35WRNHH300AKNGjUprTCIi+U4Jm4ik7Ouvv2bcuHHssMMOHHfccTVe/9BDDwVg7Nix6Q5NRCSvKWETkZS98MILAAwYMICmTZvWeP299tqL0tJS5syZw5w5c9IcnYhI/lLCJiIp++9//wvAaaedVqv1GzRowMEHHwyolU1EpCaUsIlISpYvX87o0aMpLi7mhBNOqPV2DjnkEEAJm4hITShhE5GUvPrqq5SXl3PooYfSokWLWm/noIMOAuCDDz5IV2giInlPCZuIpOSVV14BqNXFBvH2339/zIxJkyaxYcOGdIQmIpL3lLCJSLXKy8u3jp0Wu2NBbTVr1ow999yTLVu2MHHixHSEJyKS95SwiUi13n//fZYtW0aXLl3YY4896ry9Aw44ANBpURGRVClhE5FqDR8+HAha11K9d2hVysrKACVsIiKpUsImItV64403gGD8tXTo3bs3gE6JioikSAmbiFRp1apVfPDBBxQVFW29U0Fd7bXXXgB8+umnbNq0KS3bFBHJZ0rYRKRKo0ePpry8nAMPPJDmzZunZZvNmjWjW7dubN68mWnTpqVlmyIi+UwJm4hU6c033wTgiCOOSOt29913XwAmTZqU1u2KiOQjJWwiUqVMJ2zqxyYiUj0lbCJSqUWLFjFlyhR23HHHrXcoSJd99tkHUAubiEgqlLCJSKVGjhwJQN++fWnUqFFat92rVy8A9WETEUmBEjYRqVSmTocCdOnShZKSEubMmcPatWvTvn0RkXySUsJmZrua2d/MbJyZrTMzN7POCWVuDucnmzYklJ1VSblT0vfSRKSuYi1smUjYiouL2X333QGYMWNG2rcvIpJPUm1h6w4MBFYAYysp8zBwUMJ0JLAFGJak/KtJyo9ONXARyayZM2cyc+ZMSktL2X///TOyjx49egA6LSoiUp3iFMuNcfd2AGZ2EXB0YgF3nwfMi59nZj8M9/HPJNtc6u7jaxauiNSX2OnQww8/nKKioozsI3ZfUiVsIiJVS6mFzd0rarn9HwGLCFrTRCSHxE6H9u/fP2P7UAubiEhqMnbRgZl1BA4HnnD3LUmKnBj2h9toZuPVf00ke7g7Y8aMAaBfv34Z248SNhGR1GTyKtHzwu0nOx36EnAFMAA4F9gAPG9m51W2MTMbbGYTzGzCkiVLMhGviITmzJnD/PnzKS0tpWfPnhnbT+yU6IwZMygvL8/YfkREcl0mE7bzgY/d/VujYrr7Fe7+L3cf6+5DgSOACcAtlW3M3R909zJ3L2vTpk3mohYR3nnnHQAOOuggGjTI3NfETjvtxM4778yGDRuYM2dOxvYjIpLrMvJNbGYHAj1I3rr2Le5eDjwL7GpmO2ciJhFJXSxhO/jggzO+L50WFRGpXqZ+Ov8I2Az8pxbreppjEZEaiiVsffv2zfi+Ygnb9OnTM74vEZFclfaEzcwaAmcB/3P3lDqbmVkxcCYwx90XpjsmEUndqlWrmDx5MsXFxRxwwAEZ359a2EREqpfqOGyY2enhn73Dx2PNbAmwxN3jB7w9AWhJJadDzexs4GRgODAXaAdcBnwXOLtG0YtI2o0fP56KigrKyspo3LhxxvenhE1EpHopJ2wEfczi3Rc+jgb6xc3/EbAceLmS7cwE2gJ/IUjs1hJccHCMu2u8NpGI1Wf/NYDu3bsD8OWXX9bL/kREclHKCZu7W4rlTq5m+XggcyNxikid1HfC1rFjR4qKipg/fz4bN26kUaNG9bJfEZFckslhPUQkx2zZsoXx44M7xtVXwlZSUkKnTp1wd2bPnl0v+xQRyTVK2ERkq0mTJrF27Vq6du1K+/bt622/Xbt2BeCrr76qt32KiOQSJWwislV9nw6NUcImIlI1JWwislVUCVuXLl0AJWwiIpVRwiYiW6mFTUQkOylhExEguOH7vHnzMn7D92SUsImIVE0Jm4gA9XfD92TiEzZ33Z1ORCSREjYRAaI7HQrQsmVLmjdvzurVq1m2bFm9719EJNspYRMRINqEzcx0WlREpApK2ESE1atXM2nSJIqLiznwwAMjiUEJm4hI5ZSwicjWG77vv//+9XLD92RiCdvMmTMj2b+ISDZTwiYikZ4OjVELm4hI5ZSwiUhWJGwaPFdEpHJK2EQKXBQ3fE9mt912A4Lx4EREZHtK2EQK3OTJk1mzZg1dunRh5513jiyOjh07AjB37lwqKioii0NEJBspYRMpcNlwOhSgadOmtGzZko0bN7JkyZJIYxERyTZK2EQKXLYkbACdOnUCdFpURCSREjaRAqeETUQk+ylhEylgc+fOZe7cuey000706tUr6nCUsImIVEIJm0gBi/KG78koYRMRSS76b2gRiUw2nQ4FJWwiIpVRwiZSwJSwiYjkBiVsIgVq9erVTJw4MdIbvidSwiYikpwSNpECFX/D9yZNmkQdDgDt27enuLiYxYsXs379+qjDERHJGiklbGa2q5n9zczGmdk6M3Mz65yknFcy7ZdQroGZ3Whms8xsg5lNNLPT0vOSRCQV2XY6FKCoqIhdd90VgHnz5kUcjYhI9ki1ha07MBBYAYytpuwQ4KCEaUZCmf8DbgbuBY4FxgPPmtlxKcYjInWUjQkb6LSoiEgyxSmWG+Pu7QDM7CLg6CrKznf38ZUtNLO2wM+BW9399nD2KDPrDtwKDE8xJhGppWy54XsySthERL4tpRY2d0/nnZgHAA2BxxPmPw7sbWZd0rgvEUkiW274nowSNhGRb8vERQc/NbONYV+3kWZ2SMLyXsBG4IuE+VPDx54ZiElE4mTr6VBQwiYikky6E7bHgUuBI4HBQCtgpJn1iyvTEljp7p6w7vK45SKSQdmcsMUuOpg/f37EkYiIZI9U+7ClxN1/GPd0rJm9CEwB/gD0rcu2zWwwQRK49Re4iNRONidsHTp0AJSwiYjEy+g4bO6+GngFOCBu9gqg1MwsoXisZW05Sbj7g+5e5u5lbdq0SX+wIgUi2274nmiXXXYBlLCJiMSrr4Fz409/TgUaAd0SysT6rn1aLxGJFKhsu+F7otatW1NSUsKKFSs0eK6ISCij39Zm1hw4AXg/bvYIYDNwbkLx84Ap7j4zkzGJFLpsPh0K0KBBA7WyiYgkSLkPm5mdHv7ZO3w81syWAEvcfbSZ/RzYAxgFfA3sRjDeWnvikjN3X2xmdwI3mtlq4CPgTKA/cFIdX4+IVCPbEzYI+rHNnj2b+fPn071796jDERGJXE0uOng24fl94eNooB8wHTg1nHYCVgHvABe6+/sJ6/4KWANcRZDQTQcGuvvLNQleRGomdsP3oqKirLnhezKxCw++/vrriCMREckOKSds7p54kUDi8peAl1LcVjnBlaN/SHX/IlJ37733HhUVFZSVlWXNDd+T0ZWiIiLby74exyKSMblwOhR0paiISCIlbCIFJFcSNrWwiYhsTwmbSIEoLy/P2hu+J1LCJiKyPSVsIgVi8uTJrF69ms6dO2895ZitdNGBiMj2lLCJFIhcOR0K2yds377tsIhI4VHCJlIgcilh23HHHWnRogWbNm1i6dKlUYcjIhI5JWwiBSKXEjbQlaIiIvGUsIkUgHnz5jFnzhyaN2+elTd8T0YXHoiIbKOETaQAxN/wvaioKOJoUqMLD0REtlHCJlIAxo4dC0Dfvn0jjiR1amETEdlGCZtIAYglbIccckjEkaROCZuIyDZK2ETy3MqVK5k8eTIlJSVZfcP3RErYRES2UcImkufeeecd3J0DDjiAHXfcMepwUqarREVEtlHCJpLncvF0KKiFTUQknhI2kTyXqwlb27ZtKS4uZtmyZWzcuDHqcEREIqWETSSPrV+/ng8++AAzy5kBc2MaNGjAzjvvDGhoDxERJWwieez9999n8+bN7L333pSWlkYdTo3ptKiISEAJm0gey9XToTFK2EREAkrYRPJYridssVOiCxYsiDgSEZFoKWETyVNbtmzh3XffBXI/YVu4cGHEkYiIREsJm0iemjhxImvWrKFr165bxzTLNWphExEJKGETyVO5fjoUoH379oASNhERJWwieSqWsB166KERR1J7OiUqIhJQwiaSh9xdLWwiInkkpYTNzHY1s7+Z2TgzW2dmbmadE8qUmdmDZjYtLDPHzJ4wsy5Jtjcr3EbidEp6XpZIYZsxYwZLliyhXbt2dO/ePepwaq1NmzY0aNCApUuXsnnz5qjDERGJTKotbN2BgcAKYGwlZc4CegH3AMcCvwC+C0wws45Jyr8KHJQwjU45chGpVHzrmplFHE3tFRUV0bZtWwAWLVoUcTQiItEpTrHcGHdvB2BmFwFHJynzZ3dfEj/DzN4BZgI/AX6TUH6pu4+vYbwikoJ8OB0as/POO7Nw4UIWLFjArrvuGnU4IiKRSKmFzd0rUiizJMm82cASoEPNQxOR2nB3Ro0aBcBhhx0WcTR1pwsPREQyfNGBme0JtAU+S7L4xLCv20YzG6/+ayLp8eWXXzJ37lxatWrF3nvvHXU4daYLD0REMpiwmVkx8A+CFrZHEha/BFwBDADOBTYAz5vZeZmKR6RQxFrX+vXrR4MGuX8huFrYRERS78NWG/cC3weOd/cV8Qvc/Yr452b2PDAeuAV4PNnGzGwwMBigU6dOmYhXJC+MHDkSgP79+0ccSXqohU1EJEMtbGZ2K0Fy9WN3f6268u5eDjwL7GpmO1dS5kF3L3P3sjZt2qQ3YJE8Ed9/7fDDD484mvTQ7alERDLQwmZmvwJuAK5w93/XYhOe5pBECsa0adNYtGgR7du3p0ePHlGHkxaxFjadEhWRQpbWFjYzuxL4A/Ard7+3BusVA2cCc9xd38oitRQ7HXr44Yfn9Phr8dTCJiJSgxY2Mzs9/LN3+HismS0Blrj7aDM7C7gbGAGMNLM+cauvcvdPw+2cDZwMDAfmAu2AywgG2T27Dq9FpODl2+lQ2L6Fzd3zJhEVEamJmpwSfTbh+X3h42igH3AMYOHjMQllY2UgGEi3LfAXoCWwFpgAHOPur9YgHhGJU1FRsTVhy5cLDgAaN25M8+bNWbVqFStWrKBly5ZRhyQiUu9STtjcvcqfte4+CBiUwnbGA/nz30QkS0yePJnly5fTsWNHunbtGnU4abXzzjuzatUqFixYoIRNRApS7g/SJCLA9sN55NtpQ114ICKFTgmbSJ7Ix/5rMbrwQEQKnRI2kTywZcsWRo8eDeRnwqYWNhEpdErYRPLAxx9/zKpVq+jWrVte3glELWwiUuiUsInkgXw+HQpK2ERElLCJ5IF8u39oIp0SFZFCp4RNJMdt2rSJt99+G4B+/fpFG0yGqIVNRAqdEjaRHPfBBx+wdu1aevTosTWxyTdqYRORQqeETSTHvfbaawAcddRREUeSOS1btqSkpISVK1eyfv36qMMREal3SthEclwsYTv66KMjjiRzGjRoQLt27QC1solIYVLCJpLDVqxYwfvvv09JSUne9l+LiZ3uVcImIoVICZtIDhs5ciQVFRV8//vfp2nTplGHk1G68EBECpkSNpEcVginQ2N04YGIFDIlbCI5yt159dVXgcJI2NTCJiKFTAmbSI764osvmD17Nq1atWL//fePOpyMi7WwKWETkUKkhE0kR8VOhx555JEUFRVFHE3m6aIDESlkSthEclTsdGg+j78WT6dERaSQKWETyUEbN27kzTffBOCYY46JOJr6oYsORKSQKWETyUFjxoxh3bp17LvvvnTo0CHqcOpFbODcRYsWUV5eHnE0IiL1SwmbSA565ZVXADjuuOMijqT+NGrUiJYtW1JeXs6yZcuiDkdEpF4pYRPJQcOHDwcKK2EDnRYVkcKlhE0kx3z++ed8/vnnlJaW0qdPn6jDqVe68EBECpUSNpEc87///Q+AAQMGUFxcHHE09UstbCJSqJSwieSY2OnQ448/PuJI6p8SNhEpVCklbGa2q5n9zczGmdk6M3Mz65yk3A5m9hczW2Bm68PyhyYp18DMbjSzWWa2wcwmmtlpaXg9Inlt7dq1vPXWW5gZAwYMiDqceqdToiJSqFJtYesODARWAGOrKPcI8BPgN8AJwALgVTPbL6Hc/wE3A/cCxwLjgWfNrLB6UIvU0JtvvsnGjRspKyujbdu2UYdT79TCJiKFKtUOMGPcvR2AmV0EfOtO02a2L3AO8GN3fyycNxqYCvweOCmc1xb4OXCru98erj7KzLoDtwLDa/9yRPLbiy++CMBJJ50UcSTRUAubiBSqlFrY3L0ihWInAZuBp+PW2wI8BQwws0bh7AFAQ+DxhPUfB/Y2sy6pxCRSaMrLy3nppZcAOPnkkyOOJhpqYRORQpXOiw56ATPdfV3C/KkECVr3uHIbgS+SlAPomcaYRPLGe++9x5IlS+jcuTN77bVX1OFEQgmbiBSqdCZsLQn6uCVaHrc89rjS3b2aciISJ3Y69OSTT8bMIo4mGi1atKBhw4asWrWKdesSfxuKiOSvnBnWw8wGm9kEM5uwZMmSqMMRqXfxCVuhMjO1solIQUpnwrYCaJFkfqzFbHlcuVL7dhNBYrntuPuD7l7m7mVt2rSpc7AiuWT69OlMnz6dFi1acMghh0QdTqRiFx4oYRORQpLOhG0q0MXMGifM7wlsYluftalAI6BbknIAn6YxJpG8EGtdO/744wvu7gaJYi1sulJURApJOhO2l4AS4IzYDDMrBs4EXnP3jeHsEQRXk56bsP55wBR3n5nGmETywrBhw4DCHc4jnk6JikghSvmnupmdHv7ZO3w81syWAEvcfbS7f2xmTwN3m1kJMBP4KdCFuOTM3Reb2Z3AjWa2GviIIKnrTzhWm4hss2DBAt59910aNWrEMcccE3U4kdNYbCJSiGpybuXZhOf3hY+jgX7h3xcAfwT+AJQCE4Fj3P2jhHV/BawBrgLaA9OBge7+cg3iESkIzz//PO7OgAEDaNasWdThRE4tbCJSiFJO2Ny92nEE3H09cE04VVWunCCp+0Oq+xcpVEOHDgXg9NNPr6ZkYdBFByJSiHJmWA+RQrR48WJGjx5NSUkJJ554YtThZAVddCAihUgJm0gWe/7556moqODoo4+mtLQ06nCygk6JikghUsImksV0OvTb2rVrB8CiRYsoLy+POBoRkfqhhE0kSy1dupRRo0ZRXFxc0Hc3SNSoUSNatmxJeXk5y5YtizocEZF6oYRNJEu98MILlJeXc+SRR9KiRbKbiBQuXXggIoVGCZtIlnr22WAkHZ0O/TZdeCAihUYJm0gWWrRoEW+88QYlJSWccsopUYeTdXThgYgUGiVsIlno6aefpqKigmOPPZZWrVpFHU7W0SlRESk0SthEstATTzwBwDnnnBNxJNlJp0RFpNAoYRPJMp9//jnvv/8+TZs21WC5lVALm4gUGiVsIlnmP//5DwA/+MEPaNy4ccTRZCe1sIlIoVHCJpJF3H3r6dBzzz034miyly46EJFCo4RNJItMmDCBzz//nHbt2tG/f/+ow8laOiUqIoVGCZtIFom1rp111lkUFxdHHE32Ki0tpVGjRqxatYp169ZFHY6ISMYpYRPJEps2bdLp0BSZmU6LikhBUcImkiVefvllli5dyl577UVZWVnU4WQ9XXggIoVECZtIlnjkkUcAuPDCCzGziKPJfmphE5FCooRNJAvMnz+fESNGUFJSwnnnnRd1ODlBFx6ISCFRwiaSBYYMGUJFRQUnn3wyrVu3jjqcnKBToiJSSJSwiUSsoqKCRx99FAhOh0pq1MImIoVECZtIxMaMGcNXX31Fx44dOeqoo6IOJ2eoD5uIFBIlbCIRi11sMGjQIIqKiiKOJnfolKiIFBIlbCIRWrZsGc8++yxmxqBBg6IOJ6folKiIFBIlbCIReuSRR9i4cSPHHHMMXbt2jTqcnNKuXTsAFi1aRHl5ecTRiIhkVloTNjN7y8y8kmlEWKZzFWVK0xmPSDYrLy/nvvvuA+Dyyy+POJrc07BhQ1q1akV5eTnLli2LOhwRkYxK980KLwWaJ8w7CLgTGJYw/5Yk81anOR6RrPXKK68we/ZsunbtyjHHHBN1ODmpffv2LFu2jIULF9K2bduowxERyZi0Jmzu/mniPDP7CbAJeCph0VfuPj6d+xfJJffeey8Al112GQ0aqHdCbbRv356pU6eyYMEC9tlnn6jDERHJmIz+lzCzxsAZwEvuvjyT+xLJJdOnT+f1119nxx135IILLog6nJylCw9EpFBk+mf9qUAz4J9Jlt1iZlvM7BszG2Zme2c4FpGsEeu7dt5559GiRYuIo8ldGtpDRApFuvuwJTofWAz8L27eRuAB4DVgCdAD+CXwrpkd6O6fZTgmkUitWbOGIUOGAMHpUKk9tbCJSKHIWMJmZrsARwJ/dfctsfnuvgC4JK7o2PAK0qnAr4Ckd742s8HAYIBOnTplKmyRjHv44YdZtWoVhxxyCPvuu2/U4eQ03e1ARApFJk+JnhduP9np0O24+1zgbeCAKso86O5l7l7Wpk2b9EUpUo82b97MnXfeCcDPf/7ziKPJfTolKiKFIpMJ24+Aie4+sQbreKaCEckGTz75JHPnzqVnz56ccMIJUYeT83RKVEQKRUYSNjMrA3qSQutaWL4T0Bd4PxPxiGSDiooKbrvtNgCuu+46DeWRBjolKiKFIlN92M4HtgBPJC4wszsIEsVxBBcd7AHcCFQAf8xQPCKRGz58OFOnTqVDhw6cc845UYeTF0pLS2nUqBGrVq1i3bp1NG7cOOqQREQyIu0/8c2sBDgbGOHui5MUmUrQmha7UvRm4B3ge+4+Pd3xiGSLP//5zwBcffXVNGzYMOJo8oOZqZVNRApC2lvY3H0zUOlVAe7+KPBouvcrks3effdd3n77bUpLSxk8eHDU4eSV9u3bM3v2bBYsWEDXrl2jDkdEJCPUiUakHtx6660AXHrppTRr1iziaPJL7MIDXSkqIvlMCZtIhn3wwQe89NJLNG7cmCuvvDLqcPKOEjYRKQRK2EQy7Ne//jUAV1xxBe3atYs4mvzToUMHAObPnx9xJCIimaOETSSDxo4dy6uvvkqzZs247rrrog4nL+2yyy6AEjYRyW9K2EQyxN256aabALjmmmto1apVxBHlJ7WwiUghUMImkiFvvvkmY8aMoUWLFlx99dVRh5O3Ygnb119/HXEkIiKZo4RNJAPiW9euv/56dtppp4gjyl9qYRORQqCETSQDXnnlFd577z3atm3LFVdcEXU4eW2nnXaicePGrFmzhlWrVkUdjohIRihhE0mzzZs3b73A4MYbb6RJkyYRR5TfzEwXHohI3lPCJpJm999/P9OmTWP33Xfn0ksvjTqcgqDToiKS75SwiaTR0qVL+e1vfwvAHXfcoXuG1hNdeCAi+U4Jm0ga/eY3v2HlypUcffTRnHDCCVGHUzDUwiYi+U4Jm0iaTJo0iQceeICioiLuuusuzCzqkAqGEjYRyXdK2ETSwN352c9+RkVFBZdddhk9e/aMOqSCoosORCTfKWETSYPnnnuOUaNG0bJly6192KT+qIVNRPKdEjaROvrmm2+46qqrAPj9739Py5YtI46o8ChhE5F8p4RNpI6uu+465s+fz/e+9z0uueSSqMMpSDvvvDMACxcupLy8POJoRETSTwmbSB2MHDmShx56iJKSEh555BGKioqiDqkgNWzYkLZt21JRUcGiRYuiDkdEJO2UsInU0tq1a7nooosA+PWvf02vXr0ijqiw6cIDEclnSthEaummm25i5syZ7LPPPvziF7+IOpyCp35sIpLPlLCJ1MK4ceP461//SlFREY8++iglJSVRh1TwlLCJSD5TwiZSQ+vXr+fCCy/E3bnuuuvo3bt31CEJSthEJL8pYROpoauuuorPPvuMHj16aMy1LLLrrrsCMG/evIgjERFJPyVsIjXw5JNP8tBDD9GoUSOeeuopdthhh6hDklDHjh0BmDt3bsSRiIikX1oTNjPrZ2aeZFqZUK6FmT1sZkvNbK2ZvWFme6czFpF0mzFjBoMHDwbg7rvvZt999404IonXqVMnAObMmRNxJCIi6Vecoe1eCXwQ93xL7A8L7oj9EtAZuAJYAdwIjDKz/dxd5zMk62zYsIGBAweyZs0aBg4cyMUXXxx1SJIgvoWtoqKCBg10AkFE8kemErbP3H18JctOAg4G+rv7KAAzGwfMBK4nSPZEssq1117LxIkT6datGw899BDB7w7JJo0bN6Z169YsXbqUxYsX0759+6hDEhFJmyh+gp4EfB1L1gDc/RuCVreTI4hHpEpDhw7lvvvuo2HDhjz99NM0b9486pCkEjotKiL5KlMJ2xNmVm5my8zsP2bWKW5ZL2BKknWmAp3MrGmGYhKpsU8++YRBgwYBcPvtt2sIjywXOy2qhE1E8k26T4l+A9wBjAZWAfsDvwTGmdn+7r4YaAnMSrLu8vCxBbAmzXGJ1Ni8efM4/vjjWbt2Leeccw6XX3551CFJNdTCJiL5Kq0Jm7t/DHwcN2u0mY0B3ifom3ZTbbdtZoOBwbDtS1kkU1avXs2JJ57I119/Td++fXn00UfVby0HKGETkXyV8T5s7v4RMAM4IJy1gqAVLVHLuOXJtvOgu5e5e1mbNm3SH6hIaMuWLZx11ll88skn7L777rzwwgs0atQo6rAkBbGETWOxiUi+qc+LDjx8nErQjy1RT2COu+t0qETG3fnZz37G8OHDadWq1dZHyQ1qYRORfJXxhM3MyoA9CE6LAgwDOpjZYXFlmgMnhstEInPPPffw97//nYYNG/LCCy/QvXv3qEOSGtBFByKSr9Lah83MniAYT+0jYCXBRQc3AvOBe8Jiw4BxwONmdh3bBs414LZ0xiNSE48++ihXX301AEOGDKFv374RRyQ11b59e4qLi1m8eDHr169nxx13jDokEZG0SHcL2xSCcdYeA14FfgY8B3zP3ZcCuHsFcALwOnAf8DxQDhzu7up4IpF49NFHueiii3B3brvtNs4+++yoQ5JaKCoq0k3gRSQvpTVhc/db3H0fd9/J3UvcvaO7D3b3BQnllrv7j929pbs3dvcj3H1iOmMRSdUjjzzChRdeiLvz5z//meuuuy7qkKQOdOGBiOQj3WxPCtrDDz/MRRddBMBtt93G9ddfH3FEUle68EBE8lGm7iUqkvUeeughBg8eDAR3Mbj22msjjkjSQRceiEg+UgubFKQHHnhga7J2xx13KFnLI2phE5F8pIRNCkp5eTnXX389l1xyCQB33nkn11xzTcRRSTrFErbZs2dHHImISProlKgUjDVr1nDuuecybNgwiouLue+++/jJT34SdViSZl27dgVg5syZEUciIpI+StikIMyZM4eTTjqJiRMn0qJFC4YOHUr//v2jDksyoHPnzkDQwrZlyxaKi/U1JyK5T6dEJe+99957HHjggUycOJHvfOc7jB8/XslaHtthhx3o0KEDW7Zs0dAeIpI3lLBJXnvqqac47LDDWLRoEf3792f8+PF85zvfiTosybDYadGvvvoq4khERNJDCZvkpXXr1nHllVdy9tlns3HjRi6++GJGjBhBixYtog5N6oESNhHJN+rcIXnnww8/5LzzzmPatGkUFxdz++23c+WVV2JmUYcm9aRbt26AEjYRyR9qYZO8sWXLFv7whz/Qp08fpk2bxp577sl7773HVVddpWStwMRa2L788suIIxERSQ+1sEle+OKLLzj//PMZN24cAFdddRW33HILO+64Y8SRSRR0SlRE8o0SNslp7s7DDz/M1Vdfzdq1a+nQoQNDhgzhyCOPjDo0iZASNhHJNzolKjlr8uTJHH300QwePJi1a9dy1llnMXnyZCVrQtu2bWnSpAkrVqxgxYoVUYcjIlJnStgk5yxatIiLL76Y/fbbjzfeeIPS0lL+85//8OSTT+oqUAHAzNTKJiJ5RQmb5IwNGzZw6623svvuu/Pggw9iZlxxxRV8/vnnnH322VGHJ1lGCZuI5BP1YZOs5+4888wz3HDDDVtv6H388cdz++2306NHj4ijk2ylhE1E8okSNslq7733Htdccw3vvvsuAHvvvTd33HEHRx11VMSRSbZTwiYi+USnRCXrVFRU8Morr3DEEUfQp08f3n33Xdq2bcsDDzzAxx9/rGRNUhIbPFdjsYlIPlALm2SN9evX8+9//5u77rqLadOmAdC0aVMuv/xybrzxRpo3bx5xhJJLunfvDsCMGTMijkREpO6UsEnkFi5cyH333cf999/P0qVLAdh111256qqruOiiiygtLY02QMlJXbp0oaSkhLlz57J27VqaNGkSdUgiIrWmhE0iM3nyZO666y6eeOIJNm3aBEBZWRnXXnstp512GiUlJRFHKLmsuLiY7t2789lnnzFjxgz233//qEMSEak1JWxSr7755hteeOEFHn/8cd544w0gGDPr1FNP5ZprruHggw/WfT8lbXr06MFnn33G9OnTlbCJSE5TwiYZt3r1aoYNG8YzzzzDiBEjtramNWnShB//+MdcddVVWzuIi6TTHnvsAbC1T6SISK5SwiYZsXbtWl555RWefvpphg8fzoYNG4CgNe3www9n4MCBnHnmmbozgWRULGGbPn16xJGIiNRNWhM2MzsdOBsoA9oCc4DngD+5++qwTGdgZiWbaOHuK9MZk9Sf9evX87///Y+nn36al19+mXXr1m1d1rdvX84880xOP/102rdvH2GUUkhiAysrYRORXJfuFrafEyRpvwTmAfsDNwOHm9n33b0iruwtwLCE9VenOR7JsNmzZzNmzBhGjBjBsGHDWLNmzdZlffr04cwzz+SMM86gQ4cOEUYphSq+ha2iooIGDTT0pIjkpnQnbCe6+5K456PNbDnwT6AfMDJu2VfuPj7N+5cMcne++OILxowZw+jRoxkzZszWW0XFlJWVbU3Sdtttt4giFQm0aNGCtm3bsnjxYubOnatjUkRyVloTtoRkLeaD8FFNLDmmoqKCzz77bGtyNmbMGBYsWLBdmdLSUg455BAOO+wwTjnlFF08IFlnr732YuTIkUyZMkUJm4jkrPq46OCw8PGzhPm3mNk/gLXAaOBX7j65HuKRSpSXlzNp0qTtErRly5ZtV6ZNmzYceuihHHbYYRx66KHstddeFBUVRRSxSPXiE7bjjz8+6nBERGolowmbmXUAfg+84e4TwtkbgQeA14AlQA+CPm/vmtmB7p6Y2MW2NRgYDNCpU6dMhl0QvvnmG6ZNm8Znn33GZ599xuTJk3nnnXdYtWrVduU6dOiwNTk79NBD6dGjh8ZJk5yy9957A8FAzSIiuSpjCZuZNQVeBLYAF8Tmu/sC4JK4omPNbAQwFfgVcF6y7bn7g8CDAGVlZZ6hsPOKu7No0aKtSVn89PXXXyddp0uXLlsTtMMOO4wuXbooQZOcpoRNRPJBRhI2M9sReAnoChzm7vOqKu/uc83sbeCATMST78rLy5k1a9a3krJp06axcuXKpOvssMMO7LHHHuy5555bpz59+tCxY8f6DV4kw3r27AkEg+du3rxZtzwTkZyU9oTNzEqAoQRjsR1Vw35pajmrhLuzatUq5syZ863EbMaMGVsHpk1UWlq6XVIWm3bbbTf1PZOC0KxZM7p06cLMmTP5/PPPtyZwIiK5JN0D5zYAngD6AyekOmyHmXUC+gIvpDOebLdx40aWLFnC4sWLk06LFi3a7nnslk7JdOjQgR49enwrMWvXrp1OaUrB22uvvZg5cyaTJ09WwiYiOSndLWx/B84A/gisNbM+ccvmufs8M7sDaACMI7joYA/gRqAiXC9nVVRUsGLFikoTsMRE7JtvvqnR9ps2bcouu+zyrcSsR48eNG/ePEOvSiT37bfffrz00kt8/PHHnHnmmVGHIyJSY+lO2I4NH38VTvF+R3DXg6nAT4FBQFNgGcGAur9z96y7f8y6deuqTcBi05IlS9iyZUvK2y4qKqJt27Zbp3bt2m33PH5q06YNjRs3zuArFclfvXv3BmDChAnVlBQRyU7mnnvdxho3buzdu3fP6D7Wrl3L4sWLt7vVUipKS0srTboSE7LS0lLdKkekHsybN4+OHTvSokULli1bpm4CIlKvzOxDdy+r0zZyMWEzs3oLumHDhlW2fMUnY61bt6ZRo0b1FZqIpMjdad++PYsXL+bLL7+ka9euUYckIgUkHQlbfdzpIO169uzJk08+mdF97LDDDrRr147mzZvr17hIjjMzysrKGD58OBMmTFDCJiI5JycTth133JF99tkn6jBEJIf07t2b4cOH8+GHHzJw4MCowxERqRF1oBKRgnDAAcG43OPHpzTakIhIVlHCJiIF4aCDDgLg/fffr3JMQxGRbKSETUQKQuvWrenRowcbNmzgo48+ijocEZEaUcImIgWjb9++ALzzzjsRRyIiUjNK2ESkYBx88MEAvP322xFHIiJSM0rYRKRgxFrYxo4dS0VFRcTRiIikTgmbiBSMbt260alTJ5YtW8bHH38cdTgiIilTwiYiBcPMGDBgAACvvfZaxNGIiKROCZuIFJSjjz4aUMImIrlFCZuIFJQjjjiCBg0a8M4777B69eqowxERSYkSNhEpKC1atOD73/8+mzdvZvjw4VGHIyKSEiVsIlJwTj/9dACGDh0acSQiIqlRwiYiBecHP/gBAMOHD2ft2rURRyMiUj0lbCJScDp27EifPn1Yt24dw4YNizocEZFqKWETkYJ0/vnnA/Dwww9HHImISPWUsIlIQTrnnHNo3LgxI0eO5Isvvog6HBGRKilhE5GCtNNOOzFw4EAA7rnnnoijERGpmhI2ESlY11xzDQAPPfQQixYtijgaEZHKKWETkYK19957c8opp7BhwwZuueWWqMMREamUEjYRKWi//e1vMTPuvfdepkyZEnU4IiJJKWETkYK23377cckll1BeXs6FF17Ipk2bog5JRORbIkvYzKyjmQ01s2/MbJWZPWdmnaKKR0QK1x//+Ec6duzI+++/z7XXXou7Rx2SiMh2IknYzKwxMBLoAfwI+CGwOzDKzJpEEZOIFK4WLVrw9NNPU1JSwr333stNN92kpE1EskpULWw/AboCp7j7C+7+InASsBtwcUQxiUgBO+igg/jPf/5DgwYN+NOf/sSpp57K3Llzow5LRASILmE7CRjv7ltHq3T3mcA7wMkRxSQiBe70009n2LBhNGvWjBdffJEePXpwySWXMHr0aDZs2BB1eCJSwCyKZn8zWwi86O4XJ8y/DzjD3dtUtX5ZWZlPmDAhkyGKSAGbO3cu1157Lc8+++zWeQ0bNqRHjx506tSJXXbZhSZNmtC4cWOaNGlCUVERZlbtJCLZ55hjjmH33XfP6D7M7EN3L6vLNorTFUwNtQRWJJm/HGhRz7GIiGynY8eOPPPMM0ydOpVHH32U119/nSlTpjBp0iQmTZoUdXgikkZPP/10xhO2dIgqYasxMxsMDAbo1EkXk4pI5vXq1Ys77rgDgJUrV/Lll18ye/ZsFi1axLp161i3bh1r166lvLwcd69yEpHs1L1796hDSElUCdsKkrekVdbyhrs/CDwIwSnRzIUmIvJtpaWl9O7dm969e0cdiogUoKguOpgK9EoyvyfwaT3HIiIiIpLVokrYhgF9zKxrbIaZdQYODpeJiIiISCiqhO0hYBbwopmdbGYnAS8Cc4EHIopJREREJCtFkrC5+1qgPzAD+DfwBDAT6O/ua6KISURERCRbRXaVqLvPAU6Lav8iIiIiuSKym7+LiIiISGqUsImIiIhkOSVsIiIiIllOCZuIiIhIllPCJiIiIpLllLCJiIiIZDklbCIiIiJZztxz7z7qZrYEmJ0weyfgmypWq2x5ZfNbA0trFWBmVfc6o9p2bdZNdZ3a1m11y1X36dluttZ9bZap7jO7frrqvboyqvvMbzdf6r6+6n03d29Tpy24e15MwIO1WV7F/AlRv6bavM6otl2bdVNdp7Z1q7qvn+1ma93XZpnqPrPrp6veVffRbzdf6j5b6z3ZlE+nRF+q5fLq1ss2mYy3LtuuzbqprlPbuq1uueo+PdvN1rqv7bJslC91n656r66M6j7z21Xd17OcPCVaH8xsgruXRR2H1D/VfeFS3Rcu1X1hyqV6z6cWtnR7MOoAJDKq+8Klui9cqvvClDP1rhY2ERERkSynFjYRERGRLKeETURERCTL5XTCZmYdzWyomX1jZqvM7Dkz65TiujuY2V/MbIGZrTezcWZ2aJJy15jZS2E5N7ObK9neOWb2pZktN7OHzGzHhOWdzWy1mX2vVi9Wtqpjvf/JzF4zs2VhfQ6qouxPzGyamW00s+lmdkmSMqr3emRmu5rZ38LP67qwDjunuG4DM7vRzGaZ2QYzm2hmp1VSVnUfoVTrOdXv8Sr2U209h+VOMbOPw+NmtpndZGZFCWWONLOp4ffSf82sZcLy5mGcZ6QaX75Ldz3X5DNeSTzV1nNYrq+ZvRvGstDM7kzy+d/fzN4PvwPeMLPdEpYXh/Fdl2p8kY8rUtsJaAx8DkwBTgFOBiYDXwJNUlj/CWAl8BPgCOA5YD2wX0K5z4D3gPsBB25Osq3vAJuBG4GTgHmJ5YAXgQeift9yfUpDva8GxgL/DOtzUCXlfgJUAH8EDgf+ED7/qeo90vrvBywChgOvhnXYOcV1/whsBH4e1ukDYZ0ep7rPrinVek71e7ySfVRbz2G5AUA5Qef0w4FrgA3An+PKtAjj+CtwTPidNCRhO38FRkT93mbTlO56TvUzXkks1dZzWG6fcN8vhLFcBKwAno4rU0zwP+lJ4ChgFPBWwnauJvg/Vpzy+xV1hdWhoq8K39zucfO6AFuAa6pZd9/wwLgg4Q2eDgxLKNsgbnllCdtlwKdxz28A3ot7fgKwBGgZ9fuW61Nd6j2hPrtTScIW1vVi4J8J8x8lGBG7RPUeWf03iPv7osq+4JOs1zb8Iv9dwvw3gUmq++yaUqnnmnyPJ9l+SvUczvsYGJ1Q7jfAJqB9+Px4YBVQFD4/E1iUEOua+O8tTemt51Q/41XEUm09h/OeJ2g0iD9Gzg9j/G74vFf4vG34/HsEiWPj8PkuBHdcOLQm71cunxI9CRjv7l/EZrj7TOAdglaX6tbdDDwdt+4W4ClggJk1iptfkUIsDQky7ph1wA4AYTPpPcB17r48hW1J1epS76nW50FAG+DxhPn/BloBfcPnqvd6lmL9JTOAoL4S6/RxYG8z6xI+V91ngRTrOeXv8SRSqmcz6wjsV0m5EuDY8HlDYJO7l4fP448HIzhDc3v895akvZ5T/Yx/S6r1bGYlBC2oz7j75rhyzxAkdrH/QQ3Dx9h3xDrAgFisdwLPu/uYymJKJpcTtl4EzYmJpgI9U1h3pruvS7JuQ4LWl5p4D9jXzI4ys12AC4Dx4bJfAvMJTsFJ3dWl3muyD5LsZ2r4GNuP6j139CL49Z34DzOxTlX3uaMu3+Op1nPScuGPxHVx5T4CdjKzQWbWCriUbcfDjwlaf26t8tVIZVKt51Q/45XtA6qv524EiXhiuQ0Ep0Bj5aYTnML9mZm1IDgzNMPdV5jZEcDRQOp910LFNV0hi7QkOG+caDlBf4LarhtbnjJ3f9fM/gq8Fs6aCNxsZrsTnKfu42E7qNRZXeq9JvsgyX62Oz5U7zmlJbAySX0kfuZV97mjLt/jKdVzFeVi82LHw2wzux54GCgCZgHHhxce3Ar8MPynLjWXaj2n+hmvbB9Usp8VpHY8LGfb8bAuvIDlMeD3wDLgVDNrCPwd+KW7L6kinqRyuYUtq7j7tQRN7N2B/d19AXAvcL+7TzGzM83sMwuuTnw68QoiyU2q98Klupd47n4XwT/sPQj6qn1KkKy95e4jzKx/eAXiCjMbkXjVoOQXd38aaA3sCezi7mMJLob4BnjQzPYzs7fD4+EdM9unum3mcsK2guQtKpVl46muC9sy8hpx96Xu/qW7e3jpdk/gd2a2J/Av4AqgM9Cc4Iohqbm61HtN9kGS/SQ9PlTvOWEFUBr2J4qXWKeq+9xRl+/xVOu5snKxeYnHwyp3n+Hu5RYM53ImwWmx1gSd1f8KdCC4qjixv5Qkl2o9p/oZr2wfVLKf+HquqlzLxH24+zp3n+bum8IE/RfATwlaYZ8nuDJ2F4KW+ufDPnKVyuWEbSrbzjvH6wl8msK6XcyscZJ1N/Htc+A1YmZNgbuAn7n7GuBIYIq7v+Huq4H7CDouSs3Vpd5rsg+S7CfWPyHpflTvWW0qQYffbgnzE+tUdZ876vI9nmo9Jy0XjhXWmMqPhwYEdf57d59PcJHDFncfEvbF+ivQNzxupGqp1nOqn/HK9gHV1/OXBP3kEsvtAHStZh9/Jbgq+SOgB8EPubvdfT3BRQhdCYYLqlQuJ2zDgD5m1jU2I3xzDw6XVeUlgis/tg5gaGbFBL+GXnP3jXWM7WaCy4j/GzevSdzfTQmuGJGaq0u9p2ocwaX95ybMP4/gF9Q7lax3M6r3bDWC4EqzZHU6JexcDKr7XFKX7/GU6tnd5xD0T0xWbjPwv0q2f2kYW3yrasMwPgiOB9AxkYpU6znVz/i3pFrP7r4p3M/AuLoEOJ0gWUz6P8jMjicY2uOmhEWx74jUjoeajAGSTVP4Qr8gGKDwZIJLfycCXwFN48rtRjBG128S1n+KoHnzIoLB74YSDJL33YRyZWFlDCQYV+WZ8PnphGOqJJTfi2A8nq5x83oRjB32G4Jf2VOB/0T9HubilIZ6Pyysu8vD+rw3Vp8J5S4hGDfnDwSDO/4+fH5ZJXGp3uvvGIh9/mKDWf80fH5YXJktwCMJ690afsavCev0/rBOT1DdZ9+UYj2n+j3+BfBmbeoZOC6c/0BY7upwH3+pJO52BEnfwXHz2hAM2v13goFUxwDvRv0eZ8OU5npO9TP+JvBFbeqZYPiPDQSD9x4BXBjW97OVvL4dCf4/nRM3ryEwB3iWoDX+vwQXqpRU+V5FXVl1rOhO4QtdFX4YXuDbg+51JsmAt+GbeCewMHzz3wP6JdnHkHD9ZFPnJOVHk5AkhPN/SNCcuorg3HWbqN+/XJ3qWO9vVVafSfZzMTCDoAn8c+DSKmJSvddf/Vf2eXwrocyQhPWKCH7hzg7rdBIJibrqPnumFOs51e/xWSSMNF+TegZ+QPDDcCPBP9rfEA6Sm6Tsv4BHk8w/hmA4iDXASKBb1O9xNkxprueUPuPh/4FZta1n4FCCVtoNBHdquJskDThh2d+T8GMhnH8A8D6wNnzsXd17ZeGKIiIiIpKlcrkPm4iIiEhBUMImIiIikuWUsImIiIhkOSVsIiIiIlmuIBI2M3vIzNzM7qqmXEczKzezTeHI1GnZrmQ31WN+C2/I7XFTuZnNN7NnzGyPqOOT2jGzg8I6/Dr8zl5mZq+b2Y/MrCiuXE8ze8zMZpvZRjP7xszGmtmV4YCn8dvsZGb3mtnnZrbBzNaY2Qdm9isz26n+X6WkW5Lvg7VmNsvMnjezgUnulJA18j5hM7MdCcZQAzgnYbC7RD8keE9KgLPTuF3JUqrHgnIGwYjzhwI3AvsDb+ofce4xs58RDG7bEriBYCyrHxMM0XE/cEJY7gzgY4Kx8v4POJrgu/1d4HcEw3rEtnkowTAQRwP3AMcSDPPwEnAZweDIkj9i3wfHAb8mGMrjSeD18P9C9ol6DJZ6GOPlbIIxXV4JH0+ooux0ggFZZwMT0rVdTdk7qR7zfwIGhXXbPWH+keH8Y6tYdxYJY/lpirw+DyUY4PSeSpZ3A/YBdgfWE4yBV5ykXBvCwW0J7g25mGBsrSZJyjYBjqoipptJMq6XpuybKvs+CJedFh5bf4s6zmRT3rewAT8iGCF5EMGH90fJCplZH4L7eP0L+DfQ28yS3bOyRtuVrKd6LFyrwscqb7gsWecGgpHlr0+20N2/dPdJwM+AYoLBcLckKbfE3WO3GruIIIG7wt3XJim71t1fT1P8kqU8uLXci8BPkty7NHJ5nbCZ2S4Ev6KfdvclBCPin2hmLZIU/xHBrWSeIEjaAM5Pw3YlS6keC06RmRWbWSMz2xP4E0GrylvRhiWpCvumHU5wD8kN1RQ/CvjA3ReksOmjgAXuPqGuMUrOG05wX9CyqANJlNcJG8GNW4vYloD9k6AizowvZGaxeW+6+9fuPgMYD5wX33m1ptuVrKd6LCzTCG7kvAH4FNiT4BT4KgALFMdP4XoNEuYn+06Q+tGa4DZFs1Mo2zHFcjUti5kVJRwnDcL5yY4fyS1zwsedI40iiXxP2H4EfO7u48LnbwBf8+3TXicR9GH4V9y8fwKxFpjableyW5X1aGaJ/6jz/fOS704luH/fgcApBEnb8LC1DeAwgoQuftqNoENy/Lw36zVqyUZfsv0x8WuCY2W748fMOkcVoNRa7CrRrLtvZ97+AjCzMqAn8GczK41b9BxwuZl9J2xJg+Af9DpgVFzZVwk+dOeHf9dmu5KlUqlH4Jdsn4T/k6Cvm+SmKe7+ReyJmb0GzCXoMH4m8CFBQhdvGPAy8GDcvNWZDVOqsIygr+luKZSdm2K5WNm9ahDHiQSt8TGDCa5MPSmh3Nc12KZkh47hYyqn0utV3iZsbPtHe0M4JTofuMnM2gEDCN6L+UnKnWpmzWOnTVLdbq2jlvqSSj3eDNwbN29phmOSeuTu683sK4IrCnH31cB2fZjMbBPwtfo2ZQd332JmbwFHmVkjd99YRfE3gIvMrL27L6xm02+E2+zt7h+mEMfk+OdmdgKwScdJXjieoNtEtcdBfcvLUzxm1pBguIb3CDqoJk6fAD8MB8g7lyBZ+2mScj8j6C9xRi22K1kq1XoEZrv7hLhpVjQRSyaEV4F1A5ZEHYvUyK1AK+C2ZAvNrIuZ7QPcRXAh2X3J+h2aWWszOzh8+jDBD7J7zaxJkrKNzSxZ9xjJI2Z2GkEr6T/cfV3U8STK1xa24wk+0Ne6+1uJC83sAYLBFfsRtLTMBB7wcCCWuHJjCS4dPx94pIbbHZWuFyNpp3osTPtZcAcTI+hQfDnBwKt/izQqqRF3H2Nm1wB3mllPYAhBR/EWwBEEQ3Sc4+4vmtn5wOPAeDP7B/A5wZhqhxAMmvt74B13Xx7+sx4GfGRmfwOmEAz5ciBwCTCUoCVO8kPs+6Ah0InglPYZwOsEA2tnn6gHgsvERDA8wyqgcSXLdyLos/YCQcfCX1exrT8SDKTXpQbbHRL1e6ApLceH6jEPJrYNlBk/LQZGAgOqWXcWGjg3Kyfg+8CzBH2NNhOMzfYawdXfDeLK9WJbUrcJ+AYYC1wKNErY5m4E3SC+JBj5fg3wAfALoHkVsdyMBs7NiSnJ98F6giuEnydI2CzqGCubLHwBIiIiIpKl8rIPm4iIiEg+UcImIiIikuWUsImIiIhkOSVsIiIiIllOCZuIiIhIllPCJiIiIpLllLCJiIiIZDklbCIiIiJZTgmbiIiISJb7fwNyc30ug5MmAAAAAElFTkSuQmCC",
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"\n", "\n", "A detailed discussion of [LGD is here](lgd.ipynb). For this exercise, we are just looking for a credible way to model it.\n", "\n", "+ often modelled based on the Beta distribution, as per [Moody's LossCalc](http://www.defaultrisk.com/_pdf6j4/losscalc_methodology.pdf)\n", "+ the Beta distribution isn't great for leases, because it does not allow for recovery gains. [More complex models should be considered for leasing](https://www.cairn.info/revue-finance-2005-2-page-35.html).\n", "+ There is limited data available on equipment recoveries, but [one study shows recoveries in various European countries between 1976 and 2002](https://www.cairn.info/revue-finance-2005-2-page-35.html). The average LGD and standard deviation varied quite widely, so judging by the wind we'll go with:\n", "\n", "$$\n", "\\mu_{LGD} = .4\n", "\\\\\\sigma_{LGD} = .4\n", "$$\n", " \n", "+ the Beta distribution is generally parameterized as follows:\n", "\n", "$$\\beta(\\alpha, \\beta)$$\n", "\n", "+ but it can also be built from the mean and standard deviation:\n", "\n", "$$\n", "\\mu = \\frac{\\alpha}{\\alpha+\\beta}\n", "\\\\\\sigma^2 = \\frac{\\alpha\\beta}{(\\alpha + \\beta)^2(\\alpha+\\beta+1)}\n", "$$" ] }, { "cell_type": "markdown", "id": "severe-publicity", "metadata": {}, "source": [ "We have some helper functions written derive $\\alpha$/$\\beta$ from $\\mu$/$\\sigma$." ] }, { "cell_type": "code", "execution_count": 43, "id": "right-wayne", "metadata": {}, "outputs": [], "source": [ "mu = .4\n", "std = .4\n", "\n", "a, b = beta_params_from_descript(mu, std**2)" ] }, { "cell_type": "code", "execution_count": 44, "id": "refined-congo", "metadata": { "tags": [ "hide_input" ] }, "outputs": [ { "data": { "text/markdown": [ "This results in alpha = 0.2 and beta = 0.3. Plug those into Beta(0.2, 0.3) and you get the following LGD distribution:" ], "text/plain": [ "
\n", "\n", "A loss model must:\n", "\n", "+ determine expected losses at loan level\n", "+ determine expected losses at portfolio level, and how to aggregrate based on correlated outcomes.\n", "\n", "\n", "\n", "Loss models have developed increasing complexity in the past two decades but regulatory requirements of banks and other FIs still center around Basel 2, which has a basic assumption that PD and LGD are uncorrelated. This assumption has been questioned more recently (and intuitively seems weak).\n", "\n", "\n", "\n", "We will show examples for both (though the latter is much more complicated and so the example is partial).\n", "\n", "" ] }, { "cell_type": "markdown", "id": "trained-locator", "metadata": {}, "source": [ "### Model 1: PD / LGD Uncorrelated ###\n", "\n", "+ Assumes that the loss given default on the loan is *not* related to the default. Picture an asset whose value does *not* change regardless of economic circumstances.\n", "\n", "+ The model *does* account for correlation of default among obligors. i.e. there is some exogneous factor(s) that cause bororwers to default at the same time. This makes sense intuitively.\n", "\n", "**Single Loan**\n", "\n", "$$EL = EAD * PD * LGD$$\n", "\n", "often shown for each dollar of EAD as:\n", "\n", "\\begin{equation}\n", "UL = EAD * \\sqrt(PD*\\sigma^2_{LGD} + LGD^2*\\sigma^2_{PD})\n", "\\\\\\sigma^2_{PD} = (PD)(1-PD)\n", "\\\\\\sigma^2_{LGD} = \\frac{\\alpha\\beta}{(\\alpha+\\beta)^2(\\alpha+\\beta+1)}\n", "\\end{equation}" ] }, { "cell_type": "markdown", "id": "flexible-roulette", "metadata": {}, "source": [ "**Multiple Loans**\n", "\n", "\\begin{equation}\n", "EL_p = \\sum^n_{i=1}{EAD_i*PD_i*LGD_i}\n", "\\\\UL_p = \\sqrt{\\sum_i\\sum_j\\rho_{ij}w_iw_jUL_iUL_j}\n", "\\end{equation}\n", "\n", "where:\n", "\n", "\\begin{equation}\n", "p_{i,j} = 1 \\text{ if }i=j\n", "\\\\w_i = EAD_i / EAD_p\n", "\\\\EAD_p = \\sum^n_{i=0}EAD_i\n", "\\\\UL_i = EAD_i * \\sqrt(PD^2_i*\\sigma^2_{LGD, i} + LGD^2_i*\\sigma^2_{PD,i})\n", "\\end{equation}" ] }, { "cell_type": "code", "execution_count": 46, "id": "increased-bridges", "metadata": { "nbsphinx": "hidden" }, "outputs": [], "source": [ "# Confirm Equivalence of Formulations\n", "p12 = .5\n", "\n", "corrmat = np.ones(shape=(2,2))\n", "corrmat[0, 1] = p12\n", "corrmat[1, 0] = p12\n", "\n", "EAD = 1\n", "EAD1 = EAD / 2\n", "EAD2 = EAD / 2\n", "w1 = EAD1 / EAD\n", "w2 = EAD2 / EAD\n", "w = np.array([w1, w2])\n", "UL1 = .25\n", "UL2 = .25\n", "ULs = np.array([UL1, UL2])\n", "\n", "rloop = np.zeros(shape=(2,2))\n", "\n", "for i in range(len(w)):\n", " for j in range(len(w)):\n", " rloop[i][j] += corrmat[i, j]*w[i]*w[j]*ULs[i]*ULs[j]\n", "\n", "rform = (UL1**2 + UL2**2 + 2*p12*UL1*UL2)*EAD1*EAD2/(EAD**2)\n", "\n", "rnumpy = np.multiply(corrmat, np.multiply(np.outer(w,w), np.outer(ULs, ULs,)))\n", "\n", "assert rloop.sum() == rform == rnumpy.sum()\n", "\n", "try:\n", " p = np.random.randint(1,10,size=(3,3))\n", " corrs_to_corrmat(p)\n", "except AssertionError as e:\n", " assert str(e) == '`p` must be an nx1 array'" ] }, { "cell_type": "markdown", "id": "immune-donor", "metadata": {}, "source": [ "**Example**\n", "\n", "Assumptions:\n", "\n", "+ 5 loans\n", "+ \\$500K total outstanding\n", "+ PD is Bernoulli variable, however, generated randomly via Normal distribution as per [above](#Probability-of-Default).\n", "+ LGD is Beta variable as per [above](#Loss-Given-Default)\n", "+ all loans are for the same asset, therefore, have same LGD distribution\n", "+ PD and LGD are uncorrelated" ] }, { "cell_type": "markdown", "id": "bizarre-conversation", "metadata": {}, "source": [ "First, we create the loan exposures." ] }, { "cell_type": "code", "execution_count": 47, "id": "compound-teacher", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Confirming Portfolio Size: 500,000.00\n" ] } ], "source": [ "# Loan Portfolio\n", "n = 5\n", "loans_t = 5*10**5\n", "randloans = np.random.uniform(0.1,1,n)\n", "EAD = (randloans / randloans.sum())*loans_t\n", "w = EAD / EAD.sum()\n", "print (f'Confirming Portfolio Size: {EAD.sum():,.2f}')" ] }, { "cell_type": "markdown", "id": "broadband-cisco", "metadata": {}, "source": [ "Now, we created the PDs for each loan.\n" ] }, { "cell_type": "code", "execution_count": 48, "id": "civic-professional", "metadata": {}, "outputs": [], "source": [ "# PD for each borrower/loan.\n", "pd_mu = 0.0044\n", "pd_std = 0.002\n", "p_of_d = np.random.normal(pd_mu, pd_std, n)\n", "count, bins = np.histogram(p_of_d, bins=PDs)\n", "binned = np.digitize(p_of_d, bins)\n", "rats = np.array(SPs)[binned]\n", "\n", "p_of_d = np.where(p_of_d<0, 0, p_of_d)\n", "\n", "pd_var = bern.var(p_of_d) # PD for each individual loan is bernoulli distributed and has its own variance." ] }, { "cell_type": "markdown", "id": "stuck-trinidad", "metadata": {}, "source": [ "And the LGDs for each loan." ] }, { "cell_type": "code", "execution_count": 49, "id": "opposed-opinion", "metadata": {}, "outputs": [], "source": [ "# LGD\n", "lgd_mu = .4\n", "lgd_std = .4\n", "\n", "a, b = beta_params_from_descript(lgd_mu, lgd_std**2)\n", "lgd_freeze = beta(a, b)\n", "lgd = lgd_freeze.rvs(size=n)\n", "\n", "lgd_var = np.repeat(lgd.var(), n)" ] }, { "cell_type": "markdown", "id": "coastal-spending", "metadata": {}, "source": [ "Then EL and UL can be calculated as:" ] }, { "cell_type": "code", "execution_count": 50, "id": "gentle-syndicate", "metadata": {}, "outputs": [], "source": [ "el = EAD*p_of_d*lgd\n", "ul = loan_ul(EAD, p_of_d, lgd, pd_var, lgd_var)" ] }, { "cell_type": "markdown", "id": "alien-watch", "metadata": {}, "source": [ "We show the results for each individual loan below:" ] }, { "cell_type": "code", "execution_count": 51, "id": "cheap-camcorder", "metadata": { "tags": [ "hide_input" ] }, "outputs": [ { "data": { "text/html": [ "
![](\"_static/loss_loan_table.png\")
"
],
"text/plain": [
"'))"
]
},
{
"cell_type": "markdown",
"id": "bizarre-female",
"metadata": {},
"source": [
"Now, we must combine the above results to find the likely performance of the entire portfolio. Remember when trying to determine the **Loss Distribubion**, as per formulations above:\n",
"\n",
"+ expected losses are *additive* \n",
"+ unexpected losses are *not additive*\n",
"+ probability of default may be correlated, thereby impacting unexpected loss\n",
"\n",
"We will consider three scenarios:\n",
"\n",
"\n",
"**PD entirely uncorrelated**\n",
"\n",
"$\\rho_{ij} = 0 \\text{ for all } i,j$\n",
"\n",
"\n",
"**PD all perfectly correlated**\n",
"\n",
"$\\rho_{ij} = 1 \\text{ for all } i,j$\n",
"\n",
"\n",
"\n",
"**PD correlated**\n",
"\n",
"$.2 < \\rho_{ij} < .8 \\text{ where }i \\neq j$"
]
},
{
"cell_type": "markdown",
"id": "intermediate-uzbekistan",
"metadata": {},
"source": [
"Scenarios 1 and 2 can be developed rather easily."
]
},
{
"cell_type": "code",
"execution_count": 52,
"id": "toxic-mention",
"metadata": {},
"outputs": [],
"source": [
"p = np.zeros(n)\n",
"\n",
"port_ul1 = port_ul_w_corr(w, ul, p)\n",
"\n",
"p = np.ones(n)\n",
"port_ul2 = port_ul_w_corr(w, ul, p)"
]
},
{
"cell_type": "markdown",
"id": "greater-switch",
"metadata": {},
"source": [
"For Scenario 3, we must prepare a correlation matrix. First we randomly generate the correlation factors."
]
},
{
"cell_type": "code",
"execution_count": 53,
"id": "united-traveler",
"metadata": {},
"outputs": [],
"source": [
"p = np.random.uniform(.2, .8, n)"
]
},
{
"cell_type": "markdown",
"id": "automated-context",
"metadata": {},
"source": [
"Then build a 2-D correlation matrix using our hepler function `corrs_to_corrmat`."
]
},
{
"cell_type": "code",
"execution_count": 54,
"id": "young-component",
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[[1. 0.38 0.38 0.41 0.78]\n",
" [0.38 1. 0.41 0.78 0.78]\n",
" [0.38 0.41 1. 0.78 0.78]\n",
" [0.41 0.78 0.78 1. 0.78]\n",
" [0.78 0.78 0.78 0.78 1. ]]\n"
]
}
],
"source": [
"print (np.around(corrs_to_corrmat(p), 2))"
]
},
{
"cell_type": "markdown",
"id": "incredible-underwear",
"metadata": {},
"source": [
"Then, multiply the correlation matrix by the weights and individual expected losses. This is all done inside the helper function `port_ul_w_corr`."
]
},
{
"cell_type": "code",
"execution_count": 55,
"id": "indie-decade",
"metadata": {},
"outputs": [],
"source": [
"port_ul3 = port_ul_w_corr(w, ul, p)\n",
"port_uls = [port_ul1, port_ul2, port_ul3]"
]
},
{
"cell_type": "markdown",
"id": "synthetic-poultry",
"metadata": {},
"source": [
"This results in the below:"
]
},
{
"cell_type": "code",
"execution_count": 56,
"id": "external-chester",
"metadata": {
"tags": [
"hide_input"
]
},
"outputs": [
{
"data": {
"text/html": [
"![](\"_static/loss_port_table.png\")
"
],
"text/plain": [
"'))"
]
},
{
"cell_type": "code",
"execution_count": 57,
"id": "described-heritage",
"metadata": {
"tags": [
"hide_input"
]
},
"outputs": [
{
"data": {
"text/html": [
"So, in our hypothetical $500k loan portfolio, we should expect loan losses of 0.25% per year on average."
],
"text/plain": [
"![\"Image](\"_static/dec2020_lease_charge_offs.png\")
"
]
},
{
"cell_type": "markdown",
"id": "limiting-personal",
"metadata": {},
"source": [
"\n",
"But we should also note that the Unexpected Loss could be 10x-20x more than the Expected Loss in a given year. Should we hold enough capital to absorb the Unexpected Loss? No, we should hold even more than that.\n",
"\n",
""
]
},
{
"cell_type": "markdown",
"id": "banner-legend",
"metadata": {},
"source": [
"### Model 2: PD / LGD Correlated ###\n",
"+ To be fleshed in greater detail.\n",
"+ [PD and LGD are likely correlated](https://www.moodysanalytics.com/-/media/whitepaper/before-2011/10-05-02-implications-of-PD-lgd-correlation-in-a-portfolio-setting.PDf)\n",
"+ current accepted approaches to risk modelling to not account for this (e.g. Basel 2)\n",
"+ usually considered when the value of the asset is dependent on the economic cycle\n",
"+ models of PD/LGD correlation are generally very complicated. we will use simple assumptions to show the effect on expected loss.\n",
"+ to account for this, EL can incorporate a covariance term for PD / LGD\n",
"\n",
"+ a borrower may default because of exogeneous circumstances that also impact the market value of the asset\n",
"+ a borrower that defaults may be more likely to depreciate an asset through improper use (i.e. poor maintenance, etc).\n",
"\n",
"\n",
"Assumes:\n",
"\n",
"+ Linear relationship between PD and LGD\n",
"+ $R^2$ = .5 [per historical data detailed in Figure 1 here](https://www.researchgate.net/publication/24103510_The_Link_Between_Default_and_Recovery_Rates_Theory_Empirical_Evidence_and_Implications)\n",
"+ all defaults and all assets have the same correlation of PD and LGD\n",
"\n",
"The main adjustment to the loss model is the addition of a covariance term to the Expected Loss.\n",
"\n",
"$$EL = PD*LGD + cov(PD, LGD)$$\n",
"\n",
"where:\n",
"\n",
"$$COV(PD,LGD) = \\rho_{PD,LGD}*\\sigma_{PD}*\\sigma_{LGD}$$\n",
"\n",
"and\n",
"\n",
"$$\\rho_{PD,LGD} = R = \\sqrt{.5}$$\n",
"\n",
"where:\n",
"\n",
"$$cov(PD, LGD) = \\rho_{PD,LGD}*\\sigma_{PD}*\\sigma_{LGD}$$\n",
"\n",
"$$UL = EAD * \\sqrt(PD^2*\\sigma^2_{LGD} + LGD^2*\\sigma^2_{PD})$$\n",
"\n",
"\n",
"For multiple loans:\n",
"\n",
"$$EL_p = \\sum^n_{i=1}({EAD_i*PD_i*LGD_i + cov(PD_i, LGD_i)})$$\n",
"\n",
"$$\\sum^n_{i=1}cov(PD_i, LGD_i) = n*cov(PD, LGD)$$\n",
"\n",
"$$EL_p = (\\sum^n_{i=1}EAD_i*PD_i*LGD_i) + n*cov(PD, LGD)$$\n",
"\n",
"The UL is very complicated and should be derived via simulation.\n",
"\n",
""
]
},
{
"cell_type": "markdown",
"id": "approved-bubble",
"metadata": {},
"source": [
"## Simulating Losses ##\n",
"\n", "\n", "To map the **Loss Distribution**, we must simulate portfolio outcomes based on **Loss Model** we have constructed.\n", "\n", "\n", "**Goal**\n", "\n", "1. Create a sample portfolio from our Loss Model\n", "2. Generate 50,000 simulations of the portfolio performance in a single year.\n", "3. The amount of loss in each simulation will be recorded.\n", "4. A Beta distribution will be fit to the loss record.\n", "\n", "**Process**\n", "\n", "1. Generate `n=20` loans totaling \\$2MM aggregate\n", "2. Generate PD for each borrower/loan from normal distribution as per [here](#Probability-of-Default).\n", "3. Simulate default events\n", " + generate `n` random standard normals, denoted `e`\n", " + generate correlation matrix of defaults across all borrowers for all simulations([values informed by Chapter 7, Appendix B of Ong](https://www.amazon.ca/Internal-Credit-Risk-Models-Performance/dp/1899332030)).\n", " + find transformation matrix via Cholesky method\n", " + correlated defaults found as `e_prime`\n", " + defaults occur when `e_prime` is less than the inverse normal of PD \n", "4. For each default, generate an LGD based on the distribution used earlier\n", " + where the borrower has not defaulted, `LGD = 0%`." ] }, { "cell_type": "code", "execution_count": 58, "id": "broke-solid", "metadata": {}, "outputs": [], "source": [ "from scipy.stats import invgauss\n", "from scipy.linalg import eigh, cholesky, svd\n", "\n", "# Loan amounts\n", "n = 20\n", "n_samples = 50000\n", "\n", "loans_t = 2*10**6\n", "randloans = np.random.uniform(0.1,1,n)\n", "EAD = (randloans / randloans.sum())*loans_t\n", "w = EAD / EAD.sum()\n", "\n", "# PD for each borrower/loan.\n", "pd_mu = 0.0044\n", "pd_std = 0.002\n", "p_of_d = np.random.normal(pd_mu, pd_std, n)\n", "p_of_d = np.where(p_of_d<0, 0, p_of_d)\n", "pd_var = bern.var(p_of_d)\n", "\n", "# Correlation matrix\n", "e = norm.rvs(0, 1, size=(n, n_samples)) # this creates defaults for ALL borrowers in ALL simulations\n", "p = np.random.uniform(.01, .13, n)\n", "corrmat = corrs_to_corrmat(p)\n", "\n", "c = cholesky(corrmat, lower=True)\n", "\n", "e_prime = np.dot(c, e).T\n", "pd_inv = np.repeat(norm.ppf(p_of_d), n_samples).reshape(n_samples, n)" ] }, { "cell_type": "markdown", "id": "intensive-integer", "metadata": {}, "source": [ "If $e\\prime_{i,j} < N^{-1}(PD_{i})$ for any $i,j$ then borrower $i$ defaulted in simulation $j$" ] }, { "cell_type": "code", "execution_count": 59, "id": "confidential-guard", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(50000, 20)\n" ] } ], "source": [ "in_default = e_prime < pd_inv\n", "print (in_default.shape)" ] }, { "cell_type": "markdown", "id": "important-marriage", "metadata": {}, "source": [ "`in_default` is now a boolean array where `True` means default and `False` means no default for every borrower (20) in every simulation (50,000).\n", "\n", "As a check, we can sum all the defaults and divide by the total number of instances of borrowers (n borrowers X n_samples)." ] }, { "cell_type": "code", "execution_count": 60, "id": "considerable-exclusion", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.004452\n" ] } ], "source": [ "default_per = (e_prime < pd_inv).sum() / (n*n_samples)\n", "print (default_per)" ] }, { "cell_type": "code", "execution_count": 61, "id": "proud-excuse", "metadata": { "tags": [ "hide_input" ] }, "outputs": [ { "data": { "text/markdown": [ "The process resulted in 0.45% defaults which aligns closely with the expected value of 0.44% mean of the Normal distribution used." ], "text/plain": [ "