What is Value at Risk (VaR)? Definition, Formula, and Example
Value at Risk (VaR) is a quantitative risk metric that estimates the maximum expected loss of a portfolio over a defined horizon at a specified confidence level under normal market conditions.
Value at Risk (VaR) is a single-number risk metric that estimates the maximum expected loss of a portfolio over a defined horizon at a specified confidence level, under normal market conditions. A bank reporting a one-day 99% VaR of $50 million is stating: on 99 days out of 100, daily losses will not exceed $50 million. VaR became the dominant institutional risk gauge after JPMorgan's RiskMetrics framework was published in 1994 and is embedded in Basel II/III capital rules.
How Value at Risk Is Calculated
Three accepted methodologies exist, each with tradeoffs:
1. Historical Simulation. Sort the portfolio's past N daily returns from worst to best and pick the percentile matching your confidence level. For a 1-day 95% VaR over 500 trading days, the loss at rank 25 (5% × 500) is the VaR. No distributional assumption — but assumes the past sample contains the realistic worst case.
2. Parametric / Variance-Covariance. Assume returns are normally distributed:
VaR = Portfolio Value × z × σ × √t
where σ is daily volatility, z is the standard-normal quantile (1.645 for 95%, 2.326 for 99%), and t is the horizon in days. Fast and analytical; fails badly when returns have fat tails.
3. Monte Carlo Simulation. Generate thousands of synthetic return paths from a chosen distribution, revalue the portfolio in each scenario, and pick the loss percentile. Most flexible — handles non-linear instruments like options — but computationally expensive.
Worked Example
A long-only equity portfolio holds $100 million of SPY. SPY's realized annualized volatility over the trailing 12 months is 16%. Daily volatility = 16% / √252 ≈ 1.0%. Using parametric VaR at 95% confidence:
1-day 95% VaR = $100M × 1.645 × 1.0% = $1.645M
The 99% VaR with z = 2.326 equals roughly $2.33M. Over a 10-day horizon, scale by √10: 10-day 99% VaR ≈ $7.36M. Regulators require the bank to hold capital against this figure plus a stressed-VaR add-on.
When Risk Managers Use VaR
VaR sits at the center of institutional risk infrastructure:
- Regulatory capital — Basel rules tie market-risk capital charges to a 10-day 99% VaR multiplied by 3 to 4
- Limit setting — desks are assigned daily VaR limits and must reduce exposure when breached
- Performance attribution — risk-adjusted return measures like return-on-VaR
- Portfolio construction — VaR contributions identify which positions drive overall risk
Retail platforms now surface VaR-style metrics in margin models — portfolio margin at brokers like Interactive Brokers uses a stress-test framework conceptually overlapping with VaR.
Limitations and Common Misconceptions
The most-cited flaw is tail blindness: VaR specifies the loss threshold at the confidence boundary, not the magnitude of losses beyond it. A 99% VaR of $50M is consistent with a worst-case loss of $51M or $5 billion. Conditional VaR (CVaR), also called Expected Shortfall, fixes this by averaging losses in the tail and is now the Basel III standard. VaR also assumes the historical correlation structure holds — 2008 demonstrated that correlations converge to 1 in crisis, making diversification benefits vanish exactly when needed.
A second misconception: VaR is not "maximum loss." It is the threshold of a confidence interval. Long-Term Capital Management's 1998 collapse and the 2007–2009 mortgage crisis both involved losses that VaR models had assigned near-zero probability. Treat any VaR figure as a normal-regime estimate that breaks down precisely in the regimes that matter.