What is Value at Risk (VaR)?

Value at Risk (VaR) is the maximum expected loss at a specified confidence level over a specified time horizon. A 1-day VaR₉₅ of $2,500 means that over the last N days, only 5% of days had losses of $2,500 or more — you expect similar-magnitude losses to occur roughly 12-13 times per year. VaR is the industry-standard portfolio risk metric used by every major bank and regulator (Basel III, Solvency II).

Calculation type: Historical simulation or parametric Method version: 1.0 Date reviewed: 2026-07-04 Live scanner: /portfolio/risk/

Formula

Historical simulation VaR

VaR_α = −percentile(historical_pnl, 1−α)

For α = 95%: sort N days of portfolio P&L worst-to-best. VaR₉₅ = negative of the P&L at the (1−0.95) × N = 0.05 × N position. Displayed as a positive loss dollar value.

Parametric VaR (assumes normal returns)

VaR_α = Z_α × σ × Portfolio_Value

Where Z₉₅ = 1.645, Z₉₉ = 2.326, and σ = daily volatility. Faster to compute but underestimates fat-tail risk.

Multi-day scaling

VaR_α(T) = VaR_α(1) × √T

The "square-root-of-time" rule. 10-day VaR = 1-day VaR × √10 ≈ 3.16×. Valid under i.i.d. return assumption.

Worked example

Portfolio: 100 shares SPY at $520 = $52,000 mark-to-market. Historical 500-day daily P&L series shows:

Worst 5 days: -$1,890, -$1,720, -$1,650, -$1,580, -$1,510
Days 6-25 (positions 6-25 in worst-to-best sort): losses ranging -$1,480 to -$620
Position 25 in sorted array (5th percentile): -$620

VaR₉₅ 1-day = $620
VaR₉₅ 10-day = $620 × √10 = $1,961

Interpretation: 95% of days, this portfolio loses less than $620 in a single day. But 5% of days (approximately 1 day every 3 weeks) it loses $620 or more. Over a 10-day period, expected worst-case loss at 95% confidence is $1,961.

VaR₉₉

At the 1st percentile (position 5 in the sorted array): loss = $1,510. So VaR₉₉ 1-day = $1,510. Much larger loss but much rarer (roughly 2-3 times per year).

How institutions use VaR

Regulatory capital

Under Basel III, banks compute VaR₉₉ at a 10-day horizon and hold regulatory capital equal to 3× to 4× that VaR value. VaR is central to regulatory risk management for global banks.

Position sizing

A common retail rule: portfolio VaR₉₅ ≤ 2% of capital. If your VaR₉₅ exceeds 2% of capital, reduce position size on high-beta names first. This provides a quantitative check on gut-instinct sizing.

Risk limits

Institutional risk-management teams set daily VaR limits for trading desks. When a desk's VaR exceeds limit, they must reduce positions. This prevents runaway risk accumulation.

Stress testing

VaR is complemented by stress tests: hypothetical scenarios (e.g., "what if COVID-day returns?") that VaR cannot capture. Institutional practice: VaR + stress tests + backtesting.

VaR limitations

Doesn't tell you the tail

VaR is a boundary metric. It tells you the loss at the 5% or 1% threshold but not the average loss inside the tail. If VaR₉₅ = $620 but the average of the worst 5% is $1,200, VaR under-represents the actual downside. This is why regulators increasingly favor Conditional VaR (CVaR).

Not sub-additive

Combining two portfolios can increase or decrease VaR unpredictably. This means VaR at the trader level does not sum to VaR at the desk level. CVaR fixes this: it is mathematically sub-additive.

Assumes stable regime

Historical VaR uses the past to estimate the future. If regime shifts (new market conditions, new correlations), historical VaR under-estimates future risk. Stress tests are the standard complement.

Not intended for options overlay

Basic historical VaR doesn't capture option payoff asymmetries. Delta-adjusted VaR or Greek-based VaR incorporate option greeks. For pure underlying portfolios (like this site's Portfolio CC Optimizer where positions are equity), basic historical VaR is appropriate.

Historical vs Parametric VaR

Historical Simulation

Method: Use actual return history. No distributional assumption.

Pros: Captures fat tails naturally. Simple to explain.

Cons: Bounded by the historical sample. Missing extreme events that happened outside the window.

Best for: Portfolios with adequate return history and stable regime.

Parametric (Variance-Covariance)

Method: Assume normal returns. Compute portfolio volatility, multiply by Z-score.

Pros: Fast. Scales to any horizon. Only need mean and vol per position.

Cons: Underestimates fat-tail risk. Assumes independence across positions (unless full covariance matrix used).

Best for: Real-time risk monitoring where speed matters.

Related terms and tools

Live Portfolio VaR & Stress Test — compute VaR for your specific positions.
CVaR / Expected Shortfall — the "average loss in the tail" complement.
Portfolio Correlation Heatmap — concentration risk view.
Beta-weighted delta — aggregate directional exposure.
Wheel Monte Carlo Simulator — forward-looking distribution analysis.
Monte Carlo Simulation glossary.

Sources: Basel III regulatory framework, Jorion "Value at Risk" (3rd ed.), Hull "Risk Management and Financial Institutions" (Chapter 22). Educational only, not investment advice. Page last reviewed 2026-07-04.