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