Value at Risk (VaR) in Python

Overview

This repository presents three methods, mathematically detailed and implemented in Python, for estimating Value at Risk (VaR), a key metric in financial risk management.

VaR/
│
├── 1.VaR.Montecarlo.ipynb          # Monte Carlo Simulation approach
├── 2.VaR.Parametric.ipynb          # Parametric (Variance-Covariance) approach
├── 3.VaR.Historical.ipynb          # Historical Simulation approach

What is VaR?

Value at Risk (VaR) measures the maximum expected loss of a portfolio over a given time horizon at a specified confidence level. Formally, for confidence level $1 - \alpha$, it is defined as:

$$ VaR_{1-\alpha}(X) := \inf_{t \in \mathbb{R}} \left( t : \mathbb{P}(X \le t) \ge 1 - \alpha \right) $$

where $X$ is the portfolio return distribution, $\alpha$ the significance level (e.g., $\alpha = 0.05$ for 95% confidence), and $t$ the loss threshold exceeded with probability $\alpha$.

In practice, if VaR(95%) = $10,000, there is a 95% chance that losses will not exceed $10,000 and a 5% chance they will.

Methods Comparison

Method	Assumptions	Formula & Key Points
Monte Carlo Simulation	- In this implementation: returns are simulated from a normal distribution. - In general: any return distribution can be modeled. - Model parameters (mean $\mu$, volatility $\sigma$) are constant. - Sufficient number of simulations to approximate distribution.	Simulate $N$ portfolio return paths. Deterministic part: $P_0 \mu T$ Stochastic part: $P_0 \sigma Z \sqrt{T}$ $$\Delta P = P_0(\mu T + \sigma Z \sqrt{T})$$ $$VaR = -\text{Percentile}(\Delta P, (1-\alpha) \times 100)$$
Parametric (Variance-Covariance)	- Portfolio returns are normally distributed. - Constant mean and variance over time. - Portfolio is linear (no significant derivatives).	Analytical solution using $\sigma_p$ and $Z_\alpha$. Time adjustment via $\sqrt{T/252}$. $$VaR = P_0 \sigma_p Z_\alpha \sqrt{\frac{T}{252}}$$
Historical Simulation	- Historical returns are representative of future risks. - No specific distributional assumptions. - Past extreme events must be reflected in historical data.	Empirical quantile approach. VaR based on historical percentiles. $$VaR = -\text{Percentile}(X, 100 \times (1-\alpha)) \times P_0$$

Acknowledgments

This project was inspired by the tutorials of Ryan O'Connell, CFA, FRM. You can check out his videos here:

• 🎥 VaR in Python: Monte Carlo Method
• 🎥 VaR in Python: Parametric Method
• 🎥 VaR in Python: Historical Method