Measuring and Reporting Value at Risk
Measuring and Reporting Value at Risk
Value at Risk (VaR) quantifies the maximum expected loss over a specified time period at a given confidence level. VaR is the industry standard for risk reporting, regulatory capital calculation, and limit setting across derivatives portfolios.
Definition and Key Concepts
VaR Definition
VaR statement: "There is a 95% probability that the portfolio will not lose more than $X over the next day."
Key parameters:
| Parameter | Common Values |
|---|---|
| Confidence level | 95%, 99%, 99.5% |
| Time horizon | 1 day, 10 days, 1 year |
| Holding assumption | Constant portfolio |
VaR Calculation Methods
| Method | Approach | Strengths | Weaknesses |
|---|---|---|---|
| Parametric (variance-covariance) | Assumes normal distribution | Fast, closed-form | Fat tails understated |
| Historical simulation | Uses actual historical returns | Non-parametric, captures fat tails | Past may not repeat |
| Monte Carlo | Simulates random scenarios | Flexible, handles complexity | Computationally intensive |
Related Metrics
| Metric | Definition |
|---|---|
| Expected Shortfall (CVaR) | Average loss beyond VaR |
| Marginal VaR | Contribution of position to total VaR |
| Incremental VaR | Change in VaR from adding position |
| Component VaR | How much position contributes to portfolio VaR |
How It Works in Practice
Parametric VaR
Formula: VaR = Portfolio Value × σ × z × √t
Where:
- σ = portfolio standard deviation
- z = z-score for confidence level (1.65 for 95%, 2.33 for 99%)
- t = time horizon in appropriate units
Example:
- Portfolio: $100 million
- Daily volatility: 1.5%
- Confidence: 95%
VaR = $100,000,000 × 1.5% × 1.65 × 1 = $2,475,000
Historical VaR
Process:
- Collect historical returns (e.g., 500 days)
- Calculate portfolio P/L for each historical day
- Sort P/L from worst to best
- Find the Nth percentile loss
Example (95%, 500 days): 5th worst day = day 25 from bottom If 25th worst P/L was -$2.3M, VaR = $2.3M
Monte Carlo VaR
Process:
- Define return distributions and correlations
- Generate 10,000+ random scenarios
- Calculate portfolio value in each scenario
- Find percentile of resulting distribution
Advantages for derivatives:
- Handles non-linear payoffs (options)
- Incorporates stochastic volatility
- Can model exotic structures
Worked Example
Portfolio composition:
| Position | Notional | Delta | Vega |
|---|---|---|---|
| Long S&P 500 | $50M | 50,000 | — |
| Long SPX puts | 500 contracts | -15,000 | +$50K |
| Short SPX calls | 300 contracts | -12,000 | -$30K |
| Net | 23,000 | +$20K |
Risk parameters:
- S&P 500 daily volatility: 1.2%
- IV daily volatility: 2%
- Correlation (spot, IV): -0.5
Parametric VaR Calculation
Delta VaR (linear): = Net delta × Spot × Spot volatility × z = 23,000 × $1.00 × 1.2% × 1.65 = $455,400
Vega VaR: = Net vega × IV volatility × z = $20,000 × 2% × 1.65 = $660
Combined VaR (with correlation): VaR² = Delta_VaR² + Vega_VaR² + 2 × ρ × Delta_VaR × Vega_VaR VaR² = 455,400² + 660² + 2 × (-0.5) × 455,400 × 660 VaR² = 207,388,960,000 + 435,600 - 300,564,000 VaR² = 207,089,831,600 VaR = $455,000 (approximately)
Negative correlation reduces combined VaR (when stocks drop, IV rises, helping long vega position).
Historical VaR Analysis
Using 500 trading days:
| Percentile | Historical P/L | Implied Daily VaR |
|---|---|---|
| 1% (day 5) | -$1,850,000 | $1,850,000 |
| 5% (day 25) | -$980,000 | $980,000 |
| 10% (day 50) | -$650,000 | $650,000 |
Observation: Historical VaR higher than parametric due to fat tails.
VaR Scaling
1-day to 10-day: 10-day VaR = 1-day VaR × √10
$455,000 × 3.16 = $1,438,000
Regulatory note: Basel requires 10-day, 99% VaR for market risk capital.
Risks, Limitations, and Tradeoffs
VaR Limitations
| Limitation | Description |
|---|---|
| Not a maximum loss | Losses can exceed VaR |
| Confidence level arbitrary | 99% VaR ignores 1% tail |
| Time horizon fixed | Doesn't capture intraday risk |
| Assumes constant positions | Ignores dynamic hedging |
| Model dependent | Parameters affect results |
Backtesting VaR
Process: Compare VaR predictions to actual P/L.
Exceedance test: At 99% confidence, expect 2-3 exceedances per year (250 trading days × 1%).
| Exceedances | Interpretation |
|---|---|
| 0-4 | Model performing well |
| 5-9 | Model may underestimate risk |
| 10+ | Model inadequate |
VaR Volatility
VaR itself is volatile and procyclical:
| Market Condition | Volatility | VaR |
|---|---|---|
| Calm | Low | Low |
| Stressed | High | High (when least helpful) |
Common Pitfalls
| Pitfall | Description | Prevention |
|---|---|---|
| Stale data | Using old volatility estimates | Update parameters daily |
| Correlation breakdown | Correlations spike in crisis | Stress test correlations |
| Non-linearity ignored | Delta-only for options | Include gamma/vega |
| Concentration blind | VaR may hide concentration | Supplement with stress tests |
VaR Reporting
Daily Risk Report
| Metric | Value | Limit | Utilization |
|---|---|---|---|
| 95% 1-day VaR | $455,000 | $1,000,000 | 46% |
| 99% 1-day VaR | $720,000 | $1,500,000 | 48% |
| 99% 10-day VaR | $2,280,000 | $5,000,000 | 46% |
| Expected Shortfall (99%) | $950,000 | N/A | — |
VaR Attribution
| Risk Factor | Component VaR | % of Total |
|---|---|---|
| Equity delta | $400,000 | 88% |
| Volatility | $55,000 | 12% |
| Interest rates | $5,000 | 1% |
| Correlation benefit | -($55,000) | -12% |
| Total | $455,000 | 100% |
Trend Analysis
Track VaR over time:
| Date | 95% VaR | Comment |
|---|---|---|
| Jan 1 | $380,000 | Normal |
| Jan 15 | $420,000 | Position increase |
| Feb 1 | $580,000 | Vol spike |
| Feb 15 | $455,000 | Normalized |
Checklist and Next Steps
VaR implementation checklist:
- Select methodology (parametric, historical, Monte Carlo)
- Determine confidence level and time horizon
- Source historical data or volatility estimates
- Calculate correlation matrix
- Implement calculation engine
- Validate against test portfolios
Reporting checklist:
- Produce daily VaR reports
- Compare to limits
- Track exceedances
- Attribute to risk factors
- Escalate limit breaches
- Archive for regulatory review
Validation checklist:
- Backtest regularly
- Compare methods (if multiple)
- Stress test assumptions
- Review model annually
- Document limitations
Related articles:
- For tail-risk protection, see Tail-Risk Hedging Strategies
- For scenario analysis, see Stress Testing and Scenario Analysis