{ "cells": [ { "cell_type": "markdown", "id": "78cd42e9", "metadata": {}, "source": [ "# Copula Simulations for Credit Portfolios\n", "\n", "Companion notebook to [Module 9 — Copulas & Counterparty Risk](https://rodoslay.com/credit-models/09-copulas-and-counterparty-risk) at **rodoslay.com**.\n", "\n", "The experiment: a 200-name portfolio with **identical marginals** (PD = 2%, LGD = 45%) and\n", "**identical correlation** (ρ = 0.30), coupled by two different copulas — Gaussian and Student-t (ν = 4).\n", "Then we allocate the losses to securitization tranches and watch the senior tranche discover\n", "what tail dependence means. This is the 2008 mechanism, in about thirty lines of `numpy`." ] }, { "cell_type": "code", "execution_count": 1, "id": "e15207f1", "metadata": { "execution": { "iopub.execute_input": "2026-07-14T02:01:18.641019Z", "iopub.status.busy": "2026-07-14T02:01:18.640776Z", "iopub.status.idle": "2026-07-14T02:01:19.885935Z", "shell.execute_reply": "2026-07-14T02:01:19.884290Z" } }, "outputs": [], "source": [ "import numpy as np\n", "from scipy import stats\n", "import matplotlib.pyplot as plt\n", "\n", "rng = np.random.default_rng(9)\n", "plt.rcParams[\"figure.figsize\"] = (10, 5)\n", "plt.rcParams[\"axes.grid\"] = True\n", "plt.rcParams[\"grid.alpha\"] = 0.3\n", "\n", "N_NAMES, N_SIMS = 200, 200_000\n", "PD, LGD, RHO, NU = 0.02, 0.45, 0.30, 4" ] }, { "cell_type": "markdown", "id": "6e8a6284", "metadata": {}, "source": [ "## 1. One-factor simulation, two copulas\n", "\n", "Both copulas share the factor structure $X_i = \\sqrt{\\rho}\\,M + \\sqrt{1-\\rho}\\,\\varepsilon_i$\n", "(Module 8's machine). The t copula divides every name in a scenario by the **same**\n", "$\\sqrt{\\chi^2_\\nu/\\nu}$ draw — one shared \"chaos multiplier\" per year. That single shared divisor\n", "is what creates tail dependence: a small $\\chi^2$ draw fattens *everyone's* tail at once. Marginal\n", "thresholds are set so each name's PD is exactly 2% under either copula — the marginals are\n", "identical by construction; only the coupling differs." ] }, { "cell_type": "code", "execution_count": 2, "id": "3a6dfe15", "metadata": { "execution": { "iopub.execute_input": "2026-07-14T02:01:19.888642Z", "iopub.status.busy": "2026-07-14T02:01:19.888131Z", "iopub.status.idle": "2026-07-14T02:01:29.160015Z", "shell.execute_reply": "2026-07-14T02:01:29.159121Z" } }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "mean loss — Gaussian: 0.8973% t(4): 0.9050% (theory: 0.9000%)\n" ] } ], "source": [ "def simulate_losses(copula, n_sims=N_SIMS):\n", " M = rng.standard_normal(n_sims) # the economy\n", " eps = rng.standard_normal((n_sims, N_NAMES)) # idiosyncratic noise\n", " X = np.sqrt(RHO) * M[:, None] + np.sqrt(1 - RHO) * eps\n", " if copula == \"gauss\":\n", " thresh = stats.norm.ppf(PD)\n", " elif copula == \"t\":\n", " W = rng.chisquare(NU, n_sims) / NU # one shared divisor per scenario\n", " X = X / np.sqrt(W)[:, None]\n", " thresh = stats.t.ppf(PD, NU) # marginal is t(nu) by construction\n", " defaults = X < thresh\n", " return LGD * defaults.mean(axis=1) # portfolio loss rate per scenario\n", "\n", "loss_g = simulate_losses(\"gauss\")\n", "loss_t = simulate_losses(\"t\")\n", "print(f\"mean loss — Gaussian: {loss_g.mean():.4%} t({NU}): {loss_t.mean():.4%} (theory: {PD*LGD:.4%})\")" ] }, { "cell_type": "markdown", "id": "e49df337", "metadata": {}, "source": [ "Expected losses agree (same marginals — the copula cannot change the mean).\n", "Everything that differs from here on is *pure dependence structure*." ] }, { "cell_type": "code", "execution_count": 3, "id": "26dd6a8e", "metadata": { "execution": { "iopub.execute_input": "2026-07-14T02:01:29.162421Z", "iopub.status.busy": "2026-07-14T02:01:29.162130Z", "iopub.status.idle": "2026-07-14T02:01:30.314199Z", "shell.execute_reply": "2026-07-14T02:01:30.312937Z" } }, "outputs": [ { "data": { 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 4.5))\n", "bins = np.linspace(0, 0.12, 80)\n", "ax1.hist(loss_g, bins=bins, alpha=0.6, label=\"Gaussian\", density=True, color=\"#1f4d3a\")\n", "ax1.hist(loss_t, bins=bins, alpha=0.5, label=f\"Student-t (nu={NU})\", density=True, color=\"#b3541e\")\n", "ax1.set_xlabel(\"portfolio loss rate\"); ax1.set_title(\"Loss distributions\"); ax1.legend()\n", "\n", "# tail zoom on log scale\n", "ax2.hist(loss_g, bins=bins, alpha=0.6, label=\"Gaussian\", density=True, color=\"#1f4d3a\")\n", "ax2.hist(loss_t, bins=bins, alpha=0.5, label=f\"Student-t (nu={NU})\", density=True, color=\"#b3541e\")\n", "ax2.set_yscale(\"log\"); ax2.set_xlabel(\"portfolio loss rate\"); ax2.set_title(\"Same, log scale — look right\")\n", "ax2.legend()\n", "plt.tight_layout(); plt.show()" ] }, { "cell_type": "code", "execution_count": 4, "id": "57dbb085", "metadata": { "execution": { "iopub.execute_input": "2026-07-14T02:01:30.317027Z", "iopub.status.busy": "2026-07-14T02:01:30.316732Z", "iopub.status.idle": "2026-07-14T02:01:30.523429Z", "shell.execute_reply": "2026-07-14T02:01:30.522259Z" } }, "outputs": [ { "data": { "text/html": [ "
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copulaELVaR 99%VaR 99.9%ES 99%VaR99.9 multiple of EL
0Gaussian0.897%8.100%15.300%11.061%17.1x
1t(4)0.905%14.625%28.350%20.464%31.3x
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" ], "text/plain": [ " copula EL VaR 99% VaR 99.9% ES 99% VaR99.9 multiple of EL\n", "0 Gaussian 0.897% 8.100% 15.300% 11.061% 17.1x\n", "1 t(4) 0.905% 14.625% 28.350% 20.464% 31.3x" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def risk_stats(losses, label):\n", " var99 = np.quantile(losses, 0.99)\n", " var999 = np.quantile(losses, 0.999)\n", " es99 = losses[losses >= var99].mean()\n", " return [label, f\"{losses.mean():.3%}\", f\"{var99:.3%}\", f\"{var999:.3%}\", f\"{es99:.3%}\"]\n", "\n", "import pandas as pd\n", "table = pd.DataFrame([risk_stats(loss_g, \"Gaussian\"), risk_stats(loss_t, f\"t({NU})\")],\n", " columns=[\"copula\", \"EL\", \"VaR 99%\", \"VaR 99.9%\", \"ES 99%\"])\n", "table[\"VaR99.9 multiple of EL\"] = [f\"{np.quantile(l, 0.999)/l.mean():.1f}x\" for l in (loss_g, loss_t)]\n", "table" ] }, { "cell_type": "markdown", "id": "72ae18d0", "metadata": {}, "source": [ "## 2. The tranche experiment — where the copula choice becomes money\n", "\n", "A toy securitization of the same portfolio: **equity** takes the first 5% of losses,\n", "**mezzanine** the next 10% (5–15%), **senior** everything above 15%. The senior tranche\n", "is a pure bet that *many names never default together* — i.e., a bet on the copula." ] }, { "cell_type": "code", "execution_count": 5, "id": "703dfb00", "metadata": { "execution": { "iopub.execute_input": "2026-07-14T02:01:30.526647Z", "iopub.status.busy": "2026-07-14T02:01:30.526377Z", "iopub.status.idle": "2026-07-14T02:01:30.544709Z", "shell.execute_reply": "2026-07-14T02:01:30.543823Z" } }, "outputs": [ { "data": { "text/html": [ "
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trancheexpected loss (Gaussian)expected loss (t, nu=4)t / Gaussian
0Equity 0-5%16.1804%12.0902%0.7x
1Mezzanine 5-15%0.8453%2.4341%2.9x
2Senior 15-100%0.0044%0.0672%15.4x
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" ], "text/plain": [ " tranche expected loss (Gaussian) expected loss (t, nu=4) \\\n", "0 Equity 0-5% 16.1804% 12.0902% \n", "1 Mezzanine 5-15% 0.8453% 2.4341% \n", "2 Senior 15-100% 0.0044% 0.0672% \n", "\n", " t / Gaussian \n", "0 0.7x \n", "1 2.9x \n", "2 15.4x " ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "def tranche_losses(port_loss, attach, detach):\n", " \"\"\"Loss rate on a tranche with given attachment/detachment points.\"\"\"\n", " return np.clip(port_loss - attach, 0, detach - attach) / (detach - attach)\n", "\n", "tranches = [(\"Equity 0-5%\", 0.00, 0.05), (\"Mezzanine 5-15%\", 0.05, 0.15), (\"Senior 15-100%\", 0.15, 1.00)]\n", "rows = []\n", "for name, a, d in tranches:\n", " eg = tranche_losses(loss_g, a, d).mean()\n", " et = tranche_losses(loss_t, a, d).mean()\n", " rows.append([name, f\"{eg:.4%}\", f\"{et:.4%}\", f\"{et/max(eg,1e-12):.1f}x\"])\n", "pd.DataFrame(rows, columns=[\"tranche\", \"expected loss (Gaussian)\", f\"expected loss (t, nu={NU})\", \"t / Gaussian\"])" ] }, { "cell_type": "markdown", "id": "125f34a9", "metadata": {}, "source": [ "Read the last column. The equity tranche barely notices the copula — it absorbs the *first*\n", "losses, which happen under any dependence structure. The **senior tranche's expected loss moves by\n", "an order of magnitude**, because it only loses in joint-tail scenarios — exactly the scenarios the\n", "Gaussian copula prices as vanishing. Rate a senior tranche AAA under a Gaussian view of the world,\n", "live in a t-copula world, and you have recreated 2008 in a `numpy` array.\n", "\n", "## Exercises\n", "\n", "1. **Clayton lower-tail.** Implement the Clayton copula via the gamma-frailty construction\n", " (draw $W \\sim \\text{Gamma}(1/\\theta)$, set $U_i = (1 + E_i/W)^{-1/\\theta}$ with $E_i$ standard\n", " exponential). Repeat the tranche table. Clayton's dependence is *only* in the loss corner —\n", " does the senior tranche fare better or worse than under the symmetric t?\n", "2. **Correlated LGD.** Make LGD rise with the factor: $\\text{LGD} = 0.45 + 0.15\\,\\Phi(-M)$.\n", " How much does 99.9% VaR move? (This is the \"second correlation people forget\" from the module.)\n", "3. **Sensitivity to ν.** Sweep ν from 3 to 30 and plot senior-tranche EL against it. Where does\n", " the t copula effectively become Gaussian?\n", "4. **Base correlation.** Find the *Gaussian* ρ that reproduces the t-copula's senior-tranche EL.\n", " Notice it differs from the ρ that matches the mezzanine — you have discovered why traders\n", " quote \"correlation skew,\" and why a single-ρ Gaussian model cannot fit all tranches at once." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.15" } }, "nbformat": 4, "nbformat_minor": 5 }