Value at Risk - The Probabilistic Loss Estimate

By EC Assets Research Team, Risk Analytics · Published 15 June 2026

Value at Risk — Value at Risk (VaR) estimates the loss that a portfolio would not exceed with a stated probability over a stated time horizon. A 1-day 99% VaR of 10 million means there is a 1% chance of losing more than 10 million in a single trading day, under the assumed model.

Definition

Value at Risk is a single number with three buried parameters: the time horizon, the confidence level, and the model used to estimate the loss distribution. Reporting VaR without those three is meaningless. A typical institutional convention is 1-day 99% VaR for daily P&L control and 10-day 99% VaR for regulatory capital under Basel rules.

The three-parameter form makes the definition precise: VaR at confidence c, over horizon h, is the loss threshold L such that the probability of losing more than L over horizon h is less than 1 − c. For 99% confidence, that is the 99th-percentile loss of the modelled distribution.

VaR became the industry standard following its adoption by JPMorgan in the early 1990s and its incorporation into the Basel II and Basel III capital frameworks. It remains the most widely cited risk number in institutional reporting despite well-documented theoretical limitations.

The Three Calculation Methods

Three principal methods produce three different numbers. All three are correct under their assumptions; all three are wrong outside those assumptions.

Institutional risk teams typically run all three in parallel. Divergence between them is itself informative: when historical and parametric VaR diverge sharply, it signals that the recent past does not match the textbook distribution assumed by the parametric model.

Confidence Levels and z-Scores

For parametric VaR under the normal distribution, the confidence level maps directly to a multiplier of the portfolio's standard deviation. The one-sided quantiles are:

Worked Example

A portfolio has a daily return mean of 0.05% and daily standard deviation of 1.20%. Compute the parametric 1-day 99% VaR for a position size of 100 million.

The 99th-percentile loss under normality is the mean minus 2.326 times the standard deviation:

Worst 1% return ≈ 0.05% − 2.326 × 1.20% = −2.74%

In dollar terms, against a 100 million portfolio:

1-day 99% parametric VaR ≈ 2.74 million

Reported regulatorily, this becomes 8.66 million for the 10-day horizon (multiplying by the square root of 10), which under Basel III is then multiplied by a stressed-period multiplier of typically three to determine the market-risk capital charge.

Three Persistent Problems

VaR has been criticised since its introduction. The criticisms are valid and have not been resolved by any subsequent variant.

Conditional VaR (CVaR), or expected shortfall, addresses problems 1 and 2. It is now the regulatory preference under Basel III's Fundamental Review of the Trading Book.

Why VaR Still Matters

Despite the criticisms, VaR persists for three reasons.

First, regulatory necessity. Basel III, Solvency II, and most national bank capital regimes use VaR or its close relative CVaR as a primary input to market-risk capital. Funds and banks compute it because they must.

Second, operational consistency. A single number, however imperfect, can be compared across desks, books, and time. Risk officers can set limits ("no desk shall hold a 1-day 99% VaR exceeding 5 million") and monitor compliance daily. The number is interpretable across a hierarchy of stakeholders from trader to board.

Third, a discipline of stating assumptions. To report a VaR, the firm must specify a horizon, a confidence, a model, and a look-back window. The discipline of writing down those choices forces explicit thinking about what risks are being measured - even when the resulting number is fragile.

The right way to use VaR is to triangulate. Compute VaR, CVaR, maximum drawdown, and stress-test scenarios in parallel. Treat VaR as the daily compliance metric; treat CVaR as the tail-shape diagnostic; treat stress scenarios as the regime-change check. No single number captures portfolio risk; VaR is one credible input among several.

VaR vs Expected Shortfall (CVaR)

Standard VaR has a structural limitation: it tells you the threshold loss but not what happens beyond it. Expected Shortfall (also called Conditional VaR or CVaR) addresses this:

Expected Shortfall is mathematically coherent (passes the subadditivity test that VaR fails) and captures tail behaviour that VaR misses. Basel III banking regulations transitioned from VaR to ES specifically because of this limitation.

VaR Limitations Made Concrete

[!warning] The 1998 LTCM blowup illustrated VaR's limits. Pre-crisis LTCM ran with leverage of 25-30x and "modelled" 95% VaR of approximately 1% daily - interpreted as "we won't lose more than 1% on 95 of 100 days." The Russia crisis pushed losses to 8-12% per day for multiple consecutive days. The model wasn't wrong about typical days (95% threshold), but the catastrophic tail was 5-10x worse than the model assumed. The lesson: VaR captures typical days, not crisis days. Strategy survival depends on tail behaviour, not VaR thresholds.

Modern institutional risk frameworks supplement VaR with:

• Stress testing under specific scenarios (1998 Russia, 2008 GFC, 2020 COVID)
• Expected Shortfall as the primary metric
• Coherent risk measures across multiple time horizons

References

1. Jorion, P. (2006). Value at Risk (3rd ed.). McGraw-Hill.
2. McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management (2nd ed.). Princeton University Press.
3. CFA Institute. Risk Management. CFA Program Curriculum.