"How much could we lose?" is the question every risk manager is asked. Value at Risk (VaR) gives a compact answer: the loss that will not be exceeded with a given confidence over a given period, under normal market conditions. For example, a one-day 99% VaR of 279,000 means that on 99 days out of 100, losses should not exceed 279,000. Expected shortfall (ES) goes further and asks how large losses are, on average, on the days when that threshold is exceeded. Both are central to market risk management and bank regulation.
What you will learn
- Define VaR and expected shortfall and interpret them correctly
- Calculate parametric VaR and ES for a portfolio
- Estimate VaR by historical simulation
- Understand time scaling, backtesting and the limitations of VaR
Definitions
- VaR(α, T): the loss level that will not be exceeded with probability α over horizon T. Three inputs define it: the confidence level (often 95% or 99%), the horizon (one day, ten days) and the portfolio.
- Expected shortfall (ES): the average loss in the worst (1 − α) of outcomes — the expected loss given that VaR is exceeded. Also called conditional VaR.
VaR says nothing about how bad losses can be beyond the threshold; ES does. This is why the Basel Committee’s revised market risk framework moved from 99% VaR to 97.5% expected shortfall for internal models.
Method 1: Parametric (variance–covariance) VaR
Assume returns are normally distributed with a known standard deviation:
Parametric VaR and ES (normal distribution, zero mean)
VaR = z × σ × Portfolio value
ES = σ × φ(z) ÷ (1 − α) × Portfolio value
where z is the standard normal value for the confidence level (1.645 for 95%, 1.960 for 97.5%, 2.326 for 99%) and φ is the standard normal density.
Worked example: parametric VaR and ES
A portfolio worth 10,000,000 has a daily return standard deviation of 1.2% (illustrative figures).
| Measure | Calculation | Result |
|---|---|---|
| 1-day 95% VaR | 1.645 × 1.2% × 10,000,000 | 197,400 |
| 1-day 99% VaR | 2.326 × 1.2% × 10,000,000 | 279,120 |
| 1-day 97.5% ES | 1.2% × φ(1.960) ÷ 2.5% × 10,000,000 | 280,517 |
| 1-day 99% ES | 1.2% × φ(2.326) ÷ 1% × 10,000,000 | 320,085 |
| 10-day 99% VaR | 279,120 × √10 | 882,655 |
Under the normal distribution, 97.5% ES (280,517) is very close to 99% VaR (279,120). The two diverge when returns have fat tails — which is exactly when ES gives more warning.
For portfolios of several assets, σ is calculated from asset volatilities and correlations, as in modern portfolio theory. The method is fast but relies on normality and stable correlations, and handles options poorly.
Method 2: Historical simulation
Apply actual past changes in market factors to today’s portfolio and read the loss distribution directly — no assumption about its shape.
Worked example: historical simulation
A risk manager revalues today’s 10,000,000 portfolio using the market moves of each of the last 100 trading days. The six worst simulated returns are −3.1%, −2.6%, −2.4%, −2.2%, −1.9% and −1.8% (illustrative figures).
- 95% VaR: with 100 scenarios, the worst 5% are the 5 worst outcomes. Taking the 5th worst loss as the VaR (one common convention): 1.9% × 10,000,000 = 190,000.
- ES beyond VaR: the average of the 4 losses worse than this: (3.1 + 2.6 + 2.4 + 2.2) ÷ 4 = 2.575% → 257,500.
Conventions for choosing the cut-off differ, so firms document their method. Results depend heavily on the historical window chosen — a calm period produces low VaR.
Method 3: Monte Carlo simulation
Generate thousands of random scenarios for risk factors from chosen distributions, revalue the portfolio in each, and read VaR and ES from the simulated losses. It handles complex instruments and non-normal distributions but needs more computation and careful model choices — see Monte Carlo simulation in finance.
Time scaling
The square-root-of-time rule scales one-day VaR to longer horizons: VaR(T days) ≈ VaR(1 day) × √T. It assumes independent, identically distributed returns and an unchanged portfolio; with trending markets or changing positions, it can understate risk.
Backtesting
Backtesting compares VaR forecasts with actual outcomes. With 99% one-day VaR over 250 trading days, about 2.5 exceptions (days when the loss exceeds VaR) are expected. The Basel framework’s traffic-light approach treats 0–4 exceptions as green, 5–9 as yellow and 10 or more as red, with higher capital multipliers for poorer models. Too many exceptions mean the model underestimates risk; far too few may mean it is too conservative.
Limitations
- Not the worst case: VaR says nothing about the size of losses beyond the threshold.
- Model and data risk: results depend on distribution assumptions and the data window.
- Fat tails and correlation breakdowns: extreme moves are more common than the normal distribution implies, and correlations rise in crises.
- Liquidity: VaR assumes positions can be held or closed at market prices.
- Not always subadditive: VaR of a combined portfolio can exceed the sum of parts’ VaRs; ES does not have this problem.
- False comfort: a single number can hide concentrations.
For these reasons, VaR and ES are always combined with stress testing, sensitivity limits and judgement.
Uses of VaR
- Setting and monitoring trading limits.
- Regulatory and economic capital for market risk.
- Comparing risk across desks and portfolios.
- Risk-adjusted performance, relating profits to risk taken.
- Communicating risk to boards and investors.
Common mistakes
- Describing VaR as the "maximum possible loss".
- Comparing VaR figures with different confidence levels or horizons.
- Relying on a short, calm historical window.
- Ignoring exceptions found in backtesting.
- Using VaR without stress tests for extreme events.
Key terms
- Value at Risk (VaR) — the loss not exceeded with a given confidence over a given horizon.
- Expected shortfall (ES) — the average loss when losses exceed the VaR threshold.
- Historical simulation — estimating VaR from past market moves applied to today’s portfolio.
- Backtesting — comparing VaR forecasts with actual losses.
- Exception — a day when the actual loss exceeds VaR.
Frequently asked questions
Is a higher confidence level better?
A higher confidence level gives a larger, more conservative figure, but it is estimated from fewer extreme observations and is less precise.
Why did regulators move to expected shortfall?
Because ES captures the size of losses in the tail and better reflects fat-tailed risks, which VaR ignores beyond its threshold.
Can VaR be used for a personal investment portfolio?
Yes, as a rough guide to possible short-term losses, but long-term investors should focus more on goals, time horizon and diversification.
Key takeaways
- VaR is a loss threshold at a confidence level and horizon; ES is the average loss beyond it.
- Parametric VaR = z × σ × value; historical and Monte Carlo methods avoid the normality assumption.
- Scale with √T cautiously and backtest forecasts against actual losses.
- VaR is not the worst case: combine it with ES, stress tests and limits.
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- Value at Risk (VaR): Historical, parametric, Monte Carlo VaR.
- Conditional VaR (CVaR): Expected loss beyond VaR.