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.
Output Format
{
"var": {"amount": 1250000, "confidence": 0.95, "horizon_days": 1, "currency": "TWD"},
"cvar": {"amount": 1800000},
"backtest": {"exceedances": 13, "expected": 12.5, "days_tested": 250, "pass": true},
"metadata": {"method": "historical_simulation", "portfolio_value": 50000000}
}Examples
Sample I/O
Input: Portfolio value = $1,000,000. Last 20 sorted daily returns (descending loss):
[-0.050, -0.040, -0.035, -0.030, -0.025, -0.020, -0.015, -0.010, -0.005, 0.000,
0.005, 0.010, 0.015, 0.020, 0.025, 0.030, 0.035, 0.040, 0.045, 0.050]Confidence = 95%, horizon = 1 day.
Expected (Historical Simulation):
- 5th percentile index = floor(20 × 0.05) = 1 → return[1] = -0.040
- VaR = $1,000,000 × 0.040 = $40,000
- CVaR (Expected Shortfall) = mean of returns worse than VaR = (-0.050) × $1M = $50,000
Verify: VaR ≤ CVaR always (tail loss ≥ threshold loss). Count of losses > VaR should be ≤ 5% of observations (1 of 20).
Edge Cases
| Input | Expected | Why |
|---|---|---|
| Normal market conditions | VaR looks adequate | But misses tail events |
| 2008-like crisis in history | Higher VaR from historical method | Captures fat tails if crisis is in window |
| Very short history (30 days) | Unreliable VaR | Insufficient data for tail estimation |
Gotchas
- Normality assumption: Parametric VaR assumes normal returns. Financial returns have fat tails — parametric VaR UNDERESTIMATES tail risk.
- Historical window: Historical simulation is only as good as the history. If the past 250 days were calm, VaR will be low even if a crisis is coming.
- Time scaling: VaR scales with √T only under independence and normality. For volatile or trending markets, this approximation is poor.
- Diversification illusion: VaR from correlated assets using normal-times correlations understates risk. Correlations spike during crises (correlation breakdown).
- Gaming VaR: Traders can structure positions that look safe under VaR but have catastrophic tail risk. This is why regulators also require stress testing.
References
- For Expected Shortfall (CVaR) calculation, see
references/expected-shortfall.md - For VaR backtesting methods, see
references/backtesting.md