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

algo-risk-var