Expected Shortfall Calculator
Value at Risk tells you the loss threshold a portfolio should stay inside most of the time. It says nothing about how bad the bad days get. Expected shortfall, also called conditional VaR or CVaR, fills that gap: it is the average loss in the outcomes worse than VaR. Enter a portfolio value, an expected annual return, annual volatility and a confidence level, and this calculator returns both numbers from the parametric normal-distribution formulas, with the full step-by-step working shown.
Parametric estimates under a normal return distribution. Real return tails are fatter than normal, so treat these as a baseline, not a worst case. Not investment advice.
Quick Answer
For a $1,000,000 portfolio with an 8% expected annual return and 15% annual volatility at 95% confidence, Value at Risk is $166,728.04 and expected shortfall is $229,406.92. In plain terms: losses should stay under $166,728.04 in 95% of outcomes, and in the worst 5% they average $229,406.92, which is $62,678.88 beyond the VaR threshold. Both figures come from the parametric normal formulas, so read them as a model baseline, not a worst case.
Key takeaways
- VaR is a threshold. Expected shortfall is the average loss beyond it. You need both to see the tail.
- This tool is parametric: it builds the tail from your expected return and volatility under a normal distribution. It never touches a return history.
- The 97.5% level is the Basel market-risk standard, chosen to replace 99% VaR. It sits in the confidence menu for that reason.
- Real tails are fatter than the normal curve. Treat the output as a floor for planning, then stress it.
- Quick Answer
- How to use the expected shortfall calculator
- The parametric formulas: VaR first, then the tail average
- Worked example: $1,000,000 at 95% confidence
- VaR vs expected shortfall: what each number actually says
- Why Basel moved to 97.5% expected shortfall
- Reading your result: expected shortfall as a share of the portfolio
- The normal-distribution caveat: fat tails and what this model cannot see
- When VaR or expected shortfall comes out negative
- What competing calculators miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
How to use the expected shortfall calculator
Four inputs, one click. The results stay hidden until you click Calculate.
- Portfolio value: the amount you want a tail figure for. Both outputs scale in direct proportion to it, so a $500,000 portfolio at the same settings gives exactly half the dollar figures of a $1,000,000 one.
- Expected annual return: your central estimate, as a percentage. If you are still forming that estimate, the expected rate of return calculator works through probability-weighted return scenarios that can feed this field.
- Annual volatility: the standard deviation of annual returns, as a percentage. Use the same horizon as the return: annual with annual. Mixing a daily volatility with an annual return is the most common input error with this family of tools.
- Confidence level: 95%, 97.5% (the Basel level), 99% or 99.9%. Higher confidence looks at a smaller, worse slice of outcomes, so both VaR and expected shortfall rise.
Read the two headline figures together, then the one-line summary under them: it restates the expected shortfall as the average loss in the worst slice and shows how far beyond VaR that average sits. The numbered working under the results shows the z-score and every substitution, so you can audit the arithmetic by hand.
The parametric formulas: VaR first, then the tail average
Both figures come from the same normal distribution, defined by your expected return μ and volatility σ. At confidence level α, with z = Φ-1(α) the matching standard normal quantile and φ(z) the standard normal density at that point:
ES = P × (σφ(z) ÷ (1 − α) − μ)
The VaR line finds the loss at the edge of the tail: push z standard deviations against the expected return and scale by the portfolio value P. The expected shortfall line goes one step further. The term φ(z) ÷ (1 − α) is the average height of the normal curve across the whole tail beyond z, so instead of reporting the loss at the tail's edge, it reports the tail's centre of gravity. That averaging is the entire conceptual difference between the two measures, and it is why expected shortfall reacts when the tail gets heavier while VaR stays fixed at its threshold.
Two properties follow directly from the formulas. First, expected shortfall is always at least as large as VaR on the same inputs, because an average of values beyond a threshold cannot sit below the threshold. Second, both figures are linear in volatility and in portfolio value: double either one and both outputs double. Expected return enters with a minus sign, which is why a strongly positive expected return can push VaR, and occasionally expected shortfall itself, below zero. That case has its own section below.
Worked example: $1,000,000 at 95% confidence
A $1,000,000 portfolio, 8% expected annual return, 15% annual volatility, 95% confidence. The calculator is pre-filled with exactly these values.
Same portfolio, four confidence levels. Every row is a real run of this calculator:
| Confidence | VaR | Expected shortfall | ES beyond VaR |
|---|---|---|---|
| 95% | $166,728.04 | $229,406.92 | $62,678.88 |
| 97.5% (Basel) | $213,994.60 | $270,670.42 | $56,675.82 |
| 99% | $268,952.18 | $319,782.13 | $50,829.95 |
| 99.9% | $383,534.85 | $425,063.51 | $41,528.67 |
Two patterns worth noticing. The gap between expected shortfall and VaR narrows as confidence rises, because far out in a normal tail the outcomes bunch closer to the threshold. And the step from 95% to 99.9% more than doubles expected shortfall, from 22.94% of the portfolio to 42.51%, even though nothing about the portfolio changed. The confidence level is a choice about how much of the tail you want priced in, not a property of the asset. A second check with different inputs: a $500,000 portfolio at 6% expected return and 20% volatility gives a 99% VaR of $202,634.79 and expected shortfall of $236,521.42. Higher volatility with a lower return produces a larger tail on half the money.
VaR vs expected shortfall: what each number actually says
VaR says where the bad region starts. Expected shortfall says what the bad region costs on average. They answer different questions, which is why this tool reports both.
| Question | VaR | Expected shortfall |
|---|---|---|
| What it reports | A loss threshold | The average loss beyond the threshold |
| Blind spot | Everything past the threshold: a 1-in-20 loss and a wipeout look identical | The frequency side: it averages the tail without saying how likely any single point in it is |
| Coherent risk measure? | No. VaR can penalise diversification, because two portfolios can each look safe at the threshold while their combination is not | Yes. Expected shortfall satisfies the coherence axioms, including subadditivity, so diversification never raises it |
| Regulatory status | The old Basel market-risk standard at 99% | The current Basel standard at 97.5% under FRTB |
The coherence point is not academic decoration. A risk measure that can rise when you diversify gives risk managers a reason to concentrate positions, which is the opposite of what a capital rule should reward. Acerbi and Tasche's coherence proof for expected shortfall, and the Artzner framework behind it, are the research base the Basel Committee drew on when it made the switch. In expectation terms, expected shortfall is a conditional expected value: the mean of the loss distribution, conditioned on being in the tail.
Why Basel moved to 97.5% expected shortfall
The Basel Committee's Fundamental Review of the Trading Book (FRTB) replaced 99% VaR with 97.5% expected shortfall as the internal-models market-risk measure. Two reasons drove the change, and both are visible in the formulas above.
First, the 2008 crisis exposed VaR's blind spot in live portfolios: desks reported acceptable VaR while carrying concentrated tail exposure that VaR, by construction, could not see. Expected shortfall prices that exposure because every outcome in the tail feeds the average. Second, coherence: a capital measure should never punish diversification, and VaR can. The 97.5% level was calibrated so that, for a normal distribution, expected shortfall at 97.5% lands close to VaR at 99%: on the worked example the 97.5% expected shortfall is $270,670.42 against a 99% VaR of $268,952.18. Regulators kept the overall capital scale roughly intact while changing what the number sees.
Two honest limits on the comparison. Regulatory expected shortfall is not computed from a tidy normal formula: banks use stressed historical data, liquidity horizons from 10 to 120 days, and desk-level models that supervisors approve. This page computes the same measure by the parametric route for a single portfolio over an annual horizon, which is the right tool for comparing portfolios and sizing a first buffer, not a substitute for a regulatory capital engine. The capital side of that framework connects to the bank risk based capital calculator, which works through risk-weighted capital ratios.
Reading your result: expected shortfall as a share of the portfolio
The dollar figure means more once it is set against the portfolio itself. The badge under your result restates expected shortfall as a percentage of portfolio value and bands it:
| ES as % of portfolio | Badge reading | What it usually means |
|---|---|---|
| Not positive | No tail loss in this model | Expected return outweighs volatility at this confidence level. A model statement, not a promise. |
| Up to 10% | Moderate tail | A bad slice costs about a tenth of the portfolio on average. |
| 10% to 25% | Elevated tail | The worked example sits here at 95%. A tail year takes a real bite. |
| Over 25% | Heavy tail | High volatility, a deep confidence level, or both. Size buffers and hedges against this figure, not VaR. |
These bands are a reading aid built for this page, not a regulatory scale: no supervisor bands portfolios this way. Their job is to stop a raw dollar figure floating free of the portfolio it belongs to. When you compare two portfolios, compare them at the same confidence level and as percentages, which is also how the portfolio performance calculator frames return against risk on the performance side.
The normal-distribution caveat: fat tails and what this model cannot see
The parametric method assumes returns follow a normal distribution. Real returns do not, quite: crashes and spikes arrive more often than the bell curve predicts, a pattern usually called fat tails or excess kurtosis.
Parametric (this tool)
- Instant: three inputs, closed-form answer
- Comparable across portfolios and confidence levels
- Shows its working, so the arithmetic is auditable
- Understates crash risk, because the normal tail is too thin
Historical simulation (the usual complement)
- Averages your worst actual past outcomes, no distribution assumed
- Captures the crashes your data actually contains
- Blind to anything worse than your sample's worst days
- Needs a long, clean return history for the same portfolio
The practical reading: use this page's figure as the baseline for comparing portfolios, setting a first capital buffer, or checking how sensitive a portfolio is to volatility, then ask what a stressed historical window would say before treating the number as sufficient. A portfolio whose parametric expected shortfall already eats a quarter of its value has a tail problem under any method. A portfolio that looks calm here can still hide one, because the normal assumption is doing real work in the result.
When VaR or expected shortfall comes out negative
A negative result looks like a bug. Here it is the formula reporting, honestly, that the modelled tail sits in profit territory.
Because expected return enters both formulas with a minus sign, a portfolio with a high expected return and low volatility can have its 95th percentile outcome still positive. A real run of this tool: $250,000 at 10% expected return and 5% volatility gives a 95% VaR of $-4,439.33 and an expected shortfall of $783.91. The threshold is negative while the tail average is still a small loss, because the average reaches deeper into the tail than the threshold does. Push further, to $100,000 at 25% expected return and 10% volatility, and both figures go negative: VaR $-8,551.46, expected shortfall $-4,372.87. The tool prints those figures, and its one-line summary still calls the second one an average loss. Read any negative figure as "no loss at this confidence level in this model", and notice how much of that comfort comes from the expected return you typed in. Garbage return estimates produce confident, wrong tails in both directions.
One formatting note so the output does not trip you: amounts print without thousands separators, exactly as the formula produces them, so a billion-dollar portfolio shows figures like $617904362.32 at 99% confidence (20% return assumptions aside, that run used 8% return and 30% volatility). The cents are exact; only the grouping commas are absent.
What competing calculators miss
Expected shortfall has plenty of definitional pages and very few honest tools. The pattern across the ones that do compute:
- BraveCalculator Expected Shortfall: asks you to type in an "expected shortfall percentage" and multiplies it by your portfolio value. That is circular: the percentage it wants is the answer a real ES tool exists to produce. No VaR, no distribution, no confidence mechanics.
- CalculatorsHub Expected Shortfall: uses the loss-distribution form of the formula with a worked example in dollars of loss, which is legitimate, but it stops at expected shortfall alone: no VaR beside it, no portfolio-and-return framing, and no way to see how the answer moves across confidence levels.
- VaR-first tools (Ryan O'Connell Finance VaR Calculator): the strongest of the set, with a time horizon and a risk-level guide, and it does add an expected shortfall line. Its ES formula drops the expected return term, which is defensible for daily horizons where the mean is near zero and misleading at the annual horizon this page works at.
- Guide-led pages (FasterCapital, Number Analytics): solid explanations of CVaR, coherence and the integral definition, with nothing you can run your own portfolio through.
- The gap this page fills: both measures from the same distribution, the Basel 97.5% level in the menu, the z-score and substitutions printed with the result, a cents-exact run at a non-round portfolio value ($123,456.78 at 7.5% return and 12.34% volatility gives VaR $20,599.94 and expected shortfall $26,356.15 at 97.5%), and the negative-result case explained instead of suppressed.
Frequently asked questions
Expected shortfall is the average loss in the outcomes worse than Value at Risk at a chosen confidence level. At 95% confidence it answers: in the worst 5% of outcomes, how much is lost on average? It is also called conditional VaR or CVaR.
Parametrically, under a normal distribution of returns. The tool takes your portfolio value, expected return and volatility, finds the z-score for your confidence level, computes VaR, then averages the whole tail beyond that z-score using the standard normal density. It does not sort a list of your past returns, so it needs no return history from you.
VaR is a threshold: the loss that is exceeded only in the worst tail of outcomes, 5% of the time at 95% confidence. Expected shortfall is the average of the losses beyond that threshold. Expected shortfall is always at least as large as VaR for the same inputs, and it reacts to how severe the tail is, which VaR cannot see.
The Basel market risk framework (FRTB) replaced 99% VaR with 97.5% expected shortfall for internal-models capital because expected shortfall is a coherent risk measure and captures tail severity instead of stopping at a threshold. The 97.5% level was chosen so capital under the new measure would be broadly comparable to the old 99% VaR in normal conditions. That is why 97.5% (Basel) is one of the four levels in the tool.
It is the average loss in the worst 1% of outcomes under the model. On the worked example portfolio it is $319,782.13, against a 99% VaR of $268,952.18: if the year lands in that worst 1%, the model expects a loss near $319,782.13, not merely at the $268,952.18 threshold.
Yes, and the tool prints the negative figure rather than hiding it. With a high expected return and low volatility, the modelled tail can sit in profit territory: a $250,000 portfolio at 10% expected return and 5% volatility gives a 95% VaR of $-4,439.33 and expected shortfall of $783.91. Read a negative VaR as the model finding no loss at that confidence level, not as a guaranteed gain.
No. It is an average of the worst slice of outcomes, so individual outcomes inside that slice are worse than the expected shortfall figure. It is also a model estimate: this page assumes normal returns, and real market tails are fatter, so a historical-simulation estimate on stressed data would usually be higher.
Neither. It is the parametric (variance-covariance) method: closed-form formulas from your expected return, volatility and confidence level, computed with an inverse-normal approximation in your browser. Historical simulation would need a series of past portfolio returns and averages the worst actual outcomes; Monte Carlo would simulate thousands of paths. The parametric method is instant and comparable across portfolios, at the cost of the normal-distribution assumption.
Add This Calculator to Your Website
Embed this calculator on your own site with the code below. Pick a size to preview it live, then copy the code.
Powered by Calculator Zilo
References
- Basel Committee on Banking Supervision. Minimum capital requirements for market risk (BCBS d457). Bank for International Settlements, 2019. Accessed October 21, 2026.
- Basel Committee on Banking Supervision. Explanatory note on the revised minimum capital requirements for market risk. Bank for International Settlements, 2016. Accessed October 21, 2026.
- Basel Committee on Banking Supervision. Minimum capital requirements for market risk: the published standard. Bank for International Settlements, 2016. Accessed October 21, 2026.
- Acerbi, C. and Tasche, D. On the coherence of expected shortfall, Journal of Banking and Finance 26(7), 2002. Accessed October 21, 2026.
- Rockafellar, R. T. and Uryasev, S. Optimization of conditional value-at-risk, Journal of Risk 2(3), 2000. Accessed October 21, 2026.
- Basel Committee on Banking Supervision. Fundamental review of the trading book (BCBS 219, consultative document). Bank for International Settlements, 2012. Accessed October 21, 2026.
- Board of Governors of the Federal Reserve System. SR 11-7 Attachment: Supervisory Guidance on Model Risk Management. Accessed October 21, 2026.
- Assessing the Finalised Basel III Banking Regulation Regime. Expected shortfall under the finalised Basel III market risk framework. Springer. Accessed October 21, 2026.
- U.S. Securities and Exchange Commission, Investor.gov. Asset Allocation and Diversification. Accessed October 21, 2026.
- OpenStax, Rice University. The Standard Normal Distribution, Statistics. Accessed October 21, 2026.