"algo-risk-var"

Value at Risk (VaR)

Overview

VaR estimates the maximum loss a portfolio can suffer over a given time horizon at a specified confidence level. Example: "95% 1-day VaR of $1M" means there's a 5% chance of losing more than $1M in one day. Three methods: parametric (normal), historical simulation, Monte Carlo.

When to Use

Trigger conditions:

• Quantifying portfolio downside risk for risk management
• Setting trading limits and capital reserves
• Regulatory reporting (Basel III requires VaR-based capital)

When NOT to use:

• When you need to know how bad losses CAN get beyond VaR (use CVaR/Expected Shortfall)
• For illiquid assets with no price history (VaR needs return data)

Algorithm

IRON LAW: VaR Does NOT Tell You How Bad It Gets BEYOND the Threshold
VaR says "95% of the time, losses won't exceed $X." It says NOTHING
about the 5% worst case. A portfolio can have low VaR but catastrophic
tail losses. Always supplement with Expected Shortfall (CVaR) which
measures the average loss in the tail.

Phase 1: Input Validation

Collect: portfolio positions, historical returns (min 250 days for 1Y), confidence level (typically 95% or 99%), time horizon (1 day or 10 days). Gate: Sufficient return history, positions valued at current market.

Phase 2: Core Algorithm

Parametric VaR: VaR = -μ + zα × σ (assumes normal returns). For portfolio: use covariance matrix for portfolio σ.

Historical Simulation: 1. Compute daily P&L from historical returns. 2. Sort P&L ascending. 3. VaR = the (1-α) percentile loss.

Monte Carlo: 1. Fit return distribution (or use historical). 2. Simulate 10,000+ portfolio paths. 3. VaR = (1-α) percentile of simulated losses.

Phase 3: Verification

Backtest: count how often actual losses exceed VaR over the past year. At 95% confidence, exceedances should be ~5%. Use Kupiec or Christoffersen test. Gate: Backtest exceedance rate within acceptable bounds.

Phase 4: Output

Return VaR estimate with backtest results.