Value at Risk and Expected Shortfall
What a value-at-risk figure claims, the two ways it's calculated, how badly a calm-period estimate has failed in a crisis, and the measure that asks the question value at risk leaves out.
The short version
Value at risk compresses a portfolio's downside into one sentence: on all but the worst 1% of days, the loss should be no more than this much. It can be read straight off past returns or calculated from a bell curve. On 55 years of daily data, the two methods agree for ordinary bad days and part ways in the tail, where the bell curve predicts half as many very bad days as there were. A figure estimated in calm years has failed several times over when a crisis came. And by design it says nothing about how large the loss is on the days it's exceeded. Expected shortfall, the average loss on those days, answers that. Neither is a worst case.
By the end of this lesson, you'll be able to
- State precisely what a value-at-risk figure does and doesn't claim
- Calculate value at risk from past returns and from a bell curve, and explain why they differ in the tail
- Explain what expected shortfall adds and why any estimate from calm periods understates crisis losses
Why this matters
Standard deviation describes risk as a spread. Most people want something more direct: how much could I lose? Value at risk, usually written VaR, is one attempt at an answer. It's one of three ways SEC rules let a company describe its exposure to market risk in its annual report, defined there as the potential loss over a selected period with a selected likelihood.1 It's worth understanding exactly what it says, because the sentence it's usually compressed into, "our value at risk is $2 million", sounds like a limit and isn't one.
What the number means
A Value at risk (VaR). A loss that a portfolio is expected to exceed only on a stated share of days or years, such as 1 in 100. It marks where the worst outcomes begin and says nothing about how large they are. Glossary figure has three parts: a period, a probability, and an amount.
A one-day 1% VaR of $3,720 means: on 99 days out of 100, the loss should be less than $3,720.
Turn it round and it says that on about one day in a hundred, two or three times a year, the loss should be more than $3,720. It doesn't say how much more. The figure marks the edge of the bad days. It isn't a description of them.
Two ways to calculate it
From history
Collect past returns, sort them from worst to best, and read off the one at the chosen percentile. With 1,000 days of returns, the 1% VaR is the tenth worst.
From a bell curve
Assume returns are normally distributed and use the average and standard deviation. On a bell curve, 5% of results fall more than 1.645 standard deviations below the average, and 1% fall more than 2.326 below it.
Bell-curve VaR = Average return − Multiplier × Standard deviation
| Probability | From history | From a bell curve | Days worse than the bell-curve figure | Days a bell curve expects |
|---|---|---|---|---|
| 5%, or 1 day in 20 | −2.04% | −2.05% | 698 | 702 |
| 1%, or 1 day in 100 | −3.72% | −2.91% | 293 | 140 |
| 0.1%, or 1 day in 1,000 | −7.06% | −3.89% | 119 | 14 |
Calculated for this lesson from daily closing values, using a standard deviation of 1.27% a day.2
At the 5% level the two methods agree almost exactly. For an ordinary bad day, the bell curve is a fine description. Further out it fails. At the 1% level it says −2.91% where history says −3.72%, and there were more than twice as many days beyond its figure as it predicted. At one day in a thousand it predicts 14 such days and there were 119. That's the fat tails from Distributions, Standard Deviation, and Fat Tails, expressed as money: the further into the tail you look, the more a bell-curve VaR understates the loss.
The rest of this lesson is for members
Still to come in this lesson: Expected shortfall, The estimate depends on the period it came from, Longer periods, Using it sensibly, the key takeaways, and 2 quiz questions with explanations.
Membership opens every lesson in the curriculum, the capstone projects, the interactive tools, and the members-only channels in the Hyperion Entities Discord. The first lesson of every module stays free.
Go deeper
- Modern Portfolio Theory and the Efficient FrontierPortfolio Theory and Risk (depth track)
- Distributions, Standard Deviation, and Fat TailsMath and Statistics for Investors (depth track)
- Sharpe, Sortino, and Maximum DrawdownPortfolio Theory and Risk (depth track)
- What These Models MissPortfolio Theory and Risk (depth track)
Notes and sources
- Legal Information Institute, Cornell Law School. 17 CFR 229.305 (Item 305 of Regulation S-K): Quantitative and qualitative disclosures about market risk. Checked 2026-10-12. Back to text
- Federal Reserve Bank of St. Louis (FRED), from Nasdaq OMX Group. NASDAQ Composite Index (NASDAQCOM), daily close. Checked 2026-10-12. Back to text
- Aswath Damodaran, NYU Stern School of Business. Historical Returns on Stocks, Bonds and Bills, with annual data through 2025 (updated January 2026). Checked 2026-10-12.