Lecture 06 _ Value at Risk (VaR).ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "c469f1e4-4eab-412b-bf83-b99af0e4eabf",
   "metadata": {},
   "source": [
    "# <center><font color=navy>Data Science for Quantitative Finance</font></center>\n",
    "## <center><font color=navy>Value at Risk (VaR): Mathematical Fundation</font> </center>\n",
    "### <center> Ali Habibnia</center>\n",
    "\n",
    "    \n",
    "<center> Assistant Professor, Department of Economics, </center>\n",
    "<center> and Division of Computational Modeling & Data Analytics at Virginia Tech</center>\n",
    "<center> habibnia@vt.edu </center> "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ab450459",
   "metadata": {},
   "source": [
    "---\n",
    "## **1. Introduction to Value at Risk (VaR)**\n",
    "\n",
    "Value at Risk (**VaR**) is a statistical measure used in risk management to estimate the **worst expected loss** of a financial asset or portfolio over a given time horizon at a specific confidence level\n",
    "\n",
    "Mathematically, it is defined as:\n",
    "\n",
    "$$ P(R_t \\leq -VaR) = \\alpha $$\n",
    "\n",
    "where:\n",
    "- $ R_t $ is the return of the asset or portfolio at time $ t $,\n",
    "- $ \\alpha $ is the significance level (e.g., 1% or 5%),\n",
    "- $ VaR $ is the maximum expected loss at the given confidence level.\n",
    "\n",
    " **Key Intuition**\n",
    "\n",
    "- If the **1-day 99% VaR = $1M**, this means that **there is a 1% probability of losing more than $1M in one day**.\n",
    "- The **higher the confidence level**, the larger the VaR.\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0e44a46",
   "metadata": {},
   "source": [
    "## **2. Methods for VaR Calculation**\n",
    "\n",
    "### **2.1 Parametric (Variance-Covariance) Method**\n",
    "Assuming that **returns follow a normal distribution**, VaR can be estimated as:\n",
    "\n",
    "$$ VaR = Z_{\\alpha} \\cdot \\sigma \\cdot \\sqrt{T} $$\n",
    "\n",
    "where:\n",
    "- $ Z_{\\alpha} $ is the standard normal quantile for confidence level $ 1 - \\alpha $,\n",
    "- $ \\sigma $ is the standard deviation of portfolio returns (volatility),\n",
    "- $ T $ is the time horizon.\n",
    "\n",
    "### **2.2 Historical Simulation Method**\n",
    "This **non-parametric** method calculates VaR directly from historical return data without assuming a normal distribution.\n",
    "\n",
    "Mathematically:\n",
    "\n",
    "$$ VaR = Quantile_{\\alpha}(R_t) $$\n",
    "\n",
    "The process:\n",
    "\n",
    "1. Collect historical daily returns.\n",
    "2. Rank them from worst to best.\n",
    "3. Select the quantile corresponding to $1−\\alpha$\n",
    "\n",
    "Example:\n",
    "\n",
    "Assume we have 100 days of returns sorted from worst to best. The 5th worst return (since 99% VaR means 1% probability) represents the VaR threshold.\n",
    "\n",
    "\n",
    "### **2.3 Monte Carlo Simulation**\n",
    "Monte Carlo methods simulate thousands of possible future price movements based on assumed distributions. VaR is then estimated from the simulated distribution.\n",
    "\n",
    "Returns are simulated using:\n",
    "\n",
    "$$ R_t = \\mu + \\sigma Z_t $$\n",
    "\n",
    "where:\n",
    "- $ \\mu $ is the expected return,\n",
    "- $ \\sigma $ is the volatility,\n",
    "- $ Z_t \\sim N(0,1) $ is a standard normal random variable.\n",
    "\n",
    "Steps:\n",
    "\n",
    "1. Assume a return distribution (e.g., normal or GARCH model).\n",
    "2. Generate many random return paths.\n",
    "3. Compute the worst $\\alpha$% quantile.\n",
    "\n",
    "### **2.4 GARCH-Based VaR**\n",
    "GARCH models **estimate time-varying volatility** in financial markets. Unlike the parametric method, which assumes constant volatility, GARCH models estimate time-varying volatility, which improves risk estimation.\n",
    "\n",
    "GARCH(1,1) model:\n",
    "\n",
    "$$ \\sigma_t^2 = \\omega + \\alpha \\epsilon_{t-1}^2 + \\beta \\sigma_{t-1}^2 $$\n",
    "\n",
    "VaR is computed using:\n",
    "\n",
    "$$ VaR_t = Z_{\\alpha} \\cdot \\sigma_t $$\n",
    "\n",
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "83cb15e5-c681-4437-a8cd-09d5f7d87abc",
   "metadata": {},
   "source": [
    "## **3. Conditional Value at Risk (CVaR) - Expected Shortfall**\n",
    "\n",
    "**What is CVaR?**\n",
    "Conditional Value at Risk (**CVaR**), also known as **Expected Shortfall (ES)**, is an extension of Value at Risk (**VaR**) that accounts for **extreme losses beyond the VaR threshold**.\n",
    "\n",
    "While VaR only tells us the loss threshold at a certain confidence level, CVaR measures the average loss in the worst-case scenarios, i.e., in the $ \\alpha $% tail of the loss distribution.\n",
    "\n",
    "\n",
    "For a given confidence level $ 1 - \\alpha $, CVaR is defined as:\n",
    "\n",
    "$$\n",
    "CVaR_{\\alpha} = E[R_t | R_t \\leq -VaR_{\\alpha}]\n",
    "$$\n",
    "\n",
    "where:\n",
    "- $ R_t $ represents the portfolio return,\n",
    "- $ VaR_{\\alpha} $ is the **$ \\alpha $%-quantile loss threshold**,\n",
    "- $ CVaR_{\\alpha} $ is the **expected loss given that losses exceed VaR**.\n",
    "\n",
    "**Why is CVaR Important?**\n",
    "- **Addresses VaR Limitations**: VaR only considers a threshold but does not describe how bad losses can be beyond that point.\n",
    "- **More Conservative Risk Measure**: CVaR accounts for the **severity of tail risk**.\n",
    "- **Better for Portfolio Optimization**: Used in **risk-aware investment strategies**.\n",
    "\n",
    "**Interpretation**\n",
    "- If 99% VaR is \\$500,000, this means that on 1% of worst days, losses exceed \\$500,000.\n",
    "- If 99% CVaR is \\$700,000, this means that when losses exceed VaR, the average loss is \\$700,000.\n",
    "\n",
    "Thus, CVaR provides a more comprehensive view of tail risk than VaR.\n",
    "\n",
    "**CVaR in Risk Management**\n",
    "- Used by financial institutions for **stress testing and extreme risk assessment**.\n",
    "- More reliable than VaR for **hedging strategies** and **regulatory compliance (e.g., Basel Accords).**\n",
    "\n",
    "---\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c46fa14a",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import yfinance as yf\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.stats import norm\n",
    "from arch import arch_model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "5b0067f4",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "[*********************100%***********************]  1 of 1 completed\n"
     ]
    }
   ],
   "source": [
    "# Fetch historical data\n",
    "symbol = \"SPY\"\n",
    "data = yf.download(symbol, start=\"2020-01-01\", end=\"2025-01-01\")\n",
    "returns = data[\"Close\"].pct_change().dropna()\n",
    "\n",
    "# Portfolio settings\n",
    "portfolio_value = 10_000_000  # $10M portfolio\n",
    "confidence_level = 0.99\n",
    "Z_alpha = norm.ppf(confidence_level)  # Z-score for 99% confidence\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "f05fb41d",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ticker\n",
       "SPY    307707.346955\n",
       "dtype: float64"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 1. Parametric (Variance-Covariance) VaR\n",
    "sigma = returns.std(axis=0)  # Standard deviation of returns\n",
    "VaR_parametric = Z_alpha * sigma * portfolio_value\n",
    "VaR_parametric"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "73b10ac6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(-361644.66528306395)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 2. Historical Simulation VaR\n",
    "VaR_historical = np.percentile(returns, (1 - confidence_level) * 100) * portfolio_value\n",
    "VaR_historical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "ace04993",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(-297020.62708196393)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 3. Monte Carlo Simulation VaR\n",
    "simulations = 10000\n",
    "mu = returns.mean().iloc[0]\n",
    "sigma_mc = returns.std(axis=0).iloc[0]\n",
    "simulated_returns = np.random.normal(mu, sigma_mc, simulations)\n",
    "VaR_monte_carlo = np.percentile(simulated_returns, (1 - confidence_level) * 100) * portfolio_value\n",
    "VaR_monte_carlo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "d93f9be0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "np.float64(244306.6871402814)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 4. GARCH-Based VaR\n",
    "garch_model = arch_model(returns * 100, vol=\"Garch\", p=1, q=1, dist=\"t\")\n",
    "garch_fit = garch_model.fit(disp=\"off\")\n",
    "garch_forecast = garch_fit.forecast(start=returns.index[-1], horizon=1)\n",
    "sigma_garch = np.sqrt(garch_forecast.variance.values[-1, :][0]) / 100\n",
    "VaR_garch = Z_alpha * sigma_garch * portfolio_value\n",
    "VaR_garch"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "edbf8f97",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ticker\n",
       "SPY   -563042.145521\n",
       "dtype: float64"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 5. Conditional VaR (Expected Shortfall)\n",
    "CVaR_historical = returns[returns <= VaR_historical / portfolio_value].mean() * portfolio_value\n",
    "CVaR_historical"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "1a75a5a4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/plain": [
       "<Figure size 1000x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot Historical Returns and VaR Thresholds\n",
    "plt.figure(figsize=(10,5))\n",
    "plt.hist(returns, bins=50, alpha=0.75, color='blue', label='Return Distribution')\n",
    "plt.axvline(VaR_historical / portfolio_value, color='red', linestyle='dashed', linewidth=2, label='99% Historical VaR')\n",
    "plt.title('Distribution of SPY Returns with 99% VaR Threshold')\n",
    "plt.legend()\n",
    "plt.show()\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "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": "3.13.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}