Module XV·Article I·~3 min read
VaR Concept
Portfolio Risk Management
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Value at Risk: Measuring Potential Losses
Value at Risk (VaR) is a statistical measure that answers the question: "How much can we lose under normal market conditions?" VaR has become the risk management standard in banks and investment funds.
Three Components of VaR
| Component | Description | Typical Values |
|---|---|---|
| Loss Amount | Maximum expected loss | $1M, 5% of portfolio |
| Horizon | Time period | 1 day, 10 days, 1 month |
| Confidence Level | Probability not to exceed | 95%, 99%, 99.5% |
Interpretation of VaR
Example: VaR (1-day, 95%) = $1M
- "On 95% of trading days, the loss will not exceed $1M"
- "On 5% of days (~1 time per month), the loss will be more than $1M"
- "VaR DOES NOT say how much more than $1M the loss will be on bad days"
Methods for Calculating VaR
| Method | Approach | Pros | Cons |
|---|---|---|---|
| Parametric (Variance-Covariance) | Assumption of normal distribution | Fast, simple | Underestimates tail risk |
| Historical Simulation | Using historical data | Requires no assumptions | Past ≠ future |
| Monte Carlo | Simulation of thousands of scenarios | Flexible, accounts for nonlinearity | Computationally intensive |
Parametric VaR: Formula
$ \text{VaR} = \text{Portfolio Value} \times \sigma \times z \times \sqrt{T} $
| Variable | Description |
|---|---|
| $\sigma$ | Portfolio volatility (annualized) |
| $z$ | Z-score for confidence level (1.65 for 95%, 2.33 for 99%) |
| $T$ | Horizon in years (1 day = 1/252) |
Example Calculation
Portfolio = $100M, $\sigma$ = 15%, 1-day 95% VaR:
$ \text{VaR} = $100\text{M} \times 0.15 \times 1.65 \times \sqrt{\frac{1}{252}} = $1.56\text{M} $
Scaling VaR Over Time
$ \text{VaR}(T~\text{days}) = \text{VaR}(1~\text{day}) \times \sqrt{T} $
| Horizon | Multiplier | VaR (if 1-day = $1M) |
|---|---|---|
| 1 day | 1.00 | $1.00M |
| 5 days | 2.24 | $2.24M |
| 10 days | 3.16 | $3.16M |
| 21 days (1 mo) | 4.58 | $4.58M |
| 252 days (1 yr) | 15.87 | $15.87M |
VaR Limitations
| Limitation | Problem | Solution |
|---|---|---|
| Does not show tail | VaR = boundary, not average tail loss | Expected Shortfall |
| Normality assumption | Fat tails in reality | Historical, Monte Carlo |
| Backward-looking | Uses past data | Stress testing |
| Not subadditive | Portfolio VaR can be > sum of parts | ES is subadditive |
| Model risk | Different methods = different results | Multiple approaches |
VaR Limits in Risk Management
| Level | VaR Limit | Action Upon Breach |
|---|---|---|
| Green | Normal operations | |
| Amber | 80–100% of limit | Enhanced monitoring |
| Red | >100% of limit | Mandatory risk reduction |
CIO Recommendations
- VaR as one of the tools — not the only metric
- Supplement with ES and stress tests — VaR cannot see tails
- Calibrate to reality — check via backtesting
- Multiple horizons — 1-day, 10-day, monthly
- Report consistently — standardized reporting
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