Value at risk (VaR) and expected shortfall calculator

52,343One-day VaR at 99%

In the normal model, the loss over one day exceeds 52,343 with a probability of 1%. The average loss in that worst 1% – the expected shortfall – is 59,967.

Working
z at this confidence	2.3263
σ of profit and loss over one day	22,500
Expected profit over the horizon	0
VaR = z × σ − expected profit	52,343
ES factor φ(z) / (1 − c)	2.6652
ES = factor × σ − expected profit	59,967

Limits: real returns often have fatter tails than the normal distribution, volatility changes over time, and positions with options are not linear – the model can then understate the loss.

Key facts

Normal model: VaR = V × (z × σ × √T − μ × T); z is about 2.3263 at 99% and 1.6449 at 95%.
Normal expected shortfall at 99% is about 2.6652 × V × σ × √T minus the expected profit – with a mean of zero, about 15% above VaR.
With 40 returns, 95% historical VaR is the 38th-smallest loss and expected shortfall averages the two largest.
Two positions with a correlation below 1 have a combined VaR below the sum of their separate VaRs.
The square-root-of-time rule scales volatility, not the mean, and assumes independent, identically distributed returns.

How is the result calculated?

Normal model. For a position worth $V$ whose daily return is normally distributed with mean $\mu$ and volatility $\sigma$ , loss is minus value times return. Over $T$ days with independent, identically distributed returns, the mean grows with $T$ and the volatility with $\sqrt{T}$ : (IMF, risk measurement, Appendix II (secondary source))

\operatorname{VaR}_c = V\left(z_c\,\sigma\sqrt{T} - \mu T\right)

$z_c$ is the standard normal quantile at confidence $c$ , about 2.3263 at 99% and 1.6449 at 95%. (RBA, VaR methods, §§2.1–2.7 (secondary source)) The calculator computes it to at least 12 decimal places instead of reading it from a table. Expected shortfall averages the quantiles beyond $c$ . (Acerbi and Tasche, expected shortfall) For the normal distribution that average has a closed form, because $\int_{z_c}^{\infty} x\,\varphi(x)\,dx = \varphi(z_c)$ :

\operatorname{ES}_c = \frac{1}{1-c}\int_c^1 \operatorname{VaR}_u\,du = V\left(\sigma\sqrt{T}\,\frac{\varphi(z_c)}{1-c} - \mu T\right)

$\varphi$ is the standard normal density. At 99% the factor $\varphi(z_c)/(1-c)$ is about 2.6652, at 97.5% about 2.3378.

Two positions. With values $V_A, V_B$ , volatilities $\sigma_A, \sigma_B$ and correlation $\rho$ , the standard deviation of the combined daily profit and loss is (Naranjo, portfolio risk (secondary source))

\sigma_P = \sqrt{(V_A\sigma_A)^2 + (V_B\sigma_B)^2 + 2\rho\,(V_A\sigma_A)(V_B\sigma_B)}

and the formulas above apply with $V\sigma$ replaced by $\sigma_P$ and $V\mu$ by $V_A\mu_A + V_B\mu_B$ . Below a correlation of 1 the combined VaR is smaller than the two separate VaRs added up; simply adding them misses diversification. (IMF, risk measurement, Appendix II (secondary source))

Historical simulation. Each past return $r_i$ becomes a loss $L_i = -V r_i$ , and the losses are ranked. (RBA, VaR methods, §§2.1–2.7 (secondary source)) With $N$ returns, VaR is the loss of rank $\lceil cN \rceil$ counted from the smallest – the convention of our value at risk guide. Expected shortfall averages the worst $(1-c)N$ losses; when that is not a whole number, the next loss counts only in part, so the tail always has the probability $1-c$ . (Acerbi and Tasche, expected shortfall) For a horizon of more than one period, both figures are multiplied by $\sqrt{T}$ , which is an approximation.

Worked example

One position, normal model

A £1,800,000 position has a daily volatility of 1.25% and, for one day, a mean of 0. At 99%:

\operatorname{VaR} = 1{,}800{,}000 \times 2.3263 \times 0.0125 \approx \pounds52{,}343

\operatorname{ES} = 1{,}800{,}000 \times 0.0125 \times 2.6652 \approx \pounds59{,}967

Tables that round $z$ to 2.33 give £52,425 instead. Over ten days the volatility grows to $0.0125\sqrt{10}$ , so VaR is about £165,523 and expected shortfall about £189,633. With a mean of 0.02% a day, ten days’ expected profit of £3,600 comes off: VaR about £161,923.

Two positions with a correlation

Position A is worth £600,000 with 1.2% daily volatility, position B £400,000 with 0.8%; the correlation is 0.3.

\sigma_P = \sqrt{7{,}200^2 + 3{,}200^2 + 2 \times 0.3 \times 7{,}200 \times 3{,}200} \approx \pounds8{,}712

The one-day 99% VaR of the two together is about £20,268. On their own the positions have VaRs of about £16,750 and £7,444, which add up to £24,194; the correlation below 1 takes about £3,926 off.

Historical simulation from 40 returns

The example returns in the calculator (illustrative, not market data) applied to £1,000,000 give 40 losses. The four largest are £72,000, £48,000, £36,000 and £25,000. At 95%, the VaR is the loss of rank $\lceil 0.95 \times 40 \rceil = 38$ from the smallest, £36,000. Expected shortfall averages the worst $0.05 \times 40 = 2$ losses:

\operatorname{ES}_{95\%} = \frac{72{,}000 + 48{,}000}{2} = \pounds60{,}000

At 99% only 0.4 losses lie in the tail, so VaR and expected shortfall are both the largest loss, £72,000 – a sign that 40 returns are too few for that confidence level.

What do the results mean for you?

VaR is a threshold, not the worst case: a one-day 99% VaR of £52,343 still leaves a 1% chance in the model of losing more on a given day. Expected shortfall says how large those losses are on average. Always quote currency, horizon and confidence level together. (RBA, VaR methods, §§2.1–2.7 (secondary source)) For exam practice, work a question by hand first and then check the figure here; the working table under the result shows each step.

This calculator supports study for the FRM® Exam Part I, run by the Global Association of Risk Professionals (GARP). (GARP, FRM Exam information) The methods, the time scaling and expected shortfall are explained with worked examples in the value at risk guide.

What are the limits of VaR and expected shortfall?

Normal tails are thin. Heavy tails and positions with options can make the normal, linear approximation unreliable. (IMF, VaR approaches, pp. 19–20 (secondary source))
History repeats only what it contains. Historical simulation can show only scenarios inside the observation window you paste. (RBA, VaR methods, §§2.1–2.7 (secondary source))
Time scaling needs strong assumptions. The square-root-of-time rule assumes independent, identically distributed returns and unchanged positions; serial dependence, changing volatility and non-normal tails undermine it. (IMF, risk measurement, Appendix II (secondary source))
Regulatory figures follow their own rules. The Basel internal models approach uses 97.5% expected shortfall calibrated to a period of stress, with ten-day risk-factor changes rather than one-day figures scaled by √10. (BCBS, MAR33)

Which mistakes are common in VaR questions?

Scaling the mean with √T. The mean grows with T; only the volatility grows with √T. Multiplying a mean-adjusted one-day VaR by √10 gives the wrong figure.
Adding up VaRs of positions. That is right only with a correlation of 1; otherwise it overstates the combined VaR.
Counting the boundary loss in expected shortfall. In the 40-return example, averaging the three largest losses instead of two gives £52,000 instead of £60,000.
Mixing up the quantile convention. Some textbooks interpolate between observations or take a different rank, so check which convention a question uses before comparing answers.

What this tool covers

Last updated

30 October 2026

Based on

Textbook formulas for the normal model and historical simulation (see the sources below)
Your own position values, volatilities, correlation, confidence level, horizon or returns

Includes

Value at risk and expected shortfall, with the working
One position or two correlated positions (normal model), or pasted returns (historical simulation)
Square-root-of-time scaling to a longer horizon, and a histogram of pasted returns

Not included

Monte Carlo simulation, options and other non-linear positions
Changing volatility, fat-tailed distributions and backtesting
Regulatory capital calculations

Frequently asked questions

How do you calculate value at risk with the normal model?

Multiply the position value by z × σ × √T and subtract the expected profit μ × T × V. For £1,800,000 with 1.25% daily volatility at 99% over one day that is about £52,343.

What is the difference between VaR and expected shortfall?

VaR is the loss threshold at a confidence level; expected shortfall is the average loss beyond it. In the one-day 99% example, VaR is about £52,343 and expected shortfall about £59,967.

How is historical VaR calculated?

Turn each past return into a loss, rank the losses and take the one of rank ⌈cN⌉ from the smallest. With 40 returns at 95% that is the 38th; expected shortfall averages the two largest losses.

Can I turn one-day VaR into ten-day VaR with √10?

Only with a mean of zero and independent, identically distributed returns. The mean grows with T, so with a mean the volatility part is scaled with √10 and ten days’ expected profit is subtracted.

How does correlation change the VaR of two positions?

Below a correlation of 1, the combined VaR is smaller than the two separate VaRs added up. In the example, £16,750 and £7,444 add up to £24,194, but together the VaR is about £20,268.

Why do textbook answers sometimes differ slightly?

Tables round z, for example to 2.33 at 99%, which gives £52,425 instead of £52,343 in the example. Historical VaR also depends on the rank convention; some books interpolate between observations.

Does the calculator save my returns?

No. The calculation runs in your browser. Your numbers appear only in the page address after the # sign, so you can share a result; lists of more than 300 returns are left out. Your numbers are not sent to us.

Sources

Colleen Cassidy and Marianne Gizycki, Reserve Bank of Australia (methods overview, secondary source): Measuring Traded Market Risk: Value-at-risk and Backtesting Techniques — 2. Value-at-risk (November 1997). Accessed 29 October 2026.
Michael Papaioannou, International Monetary Fund (secondary source): A Primer for Risk Measurement of Bonded Debt from the Perspective of a Sovereign Debt Manager — Appendix II (Working Paper 2006/195) (August 2006). Accessed 29 October 2026.
R. Barry Johnston, Jingqing Chai and Liliana Schumacher, International Monetary Fund (secondary source): Assessing Financial System Vulnerabilities — §III.C, pp. 19–20 (Working Paper 2000/076) (April 2000). Accessed 29 October 2026.
Carlo Acerbi and Dirk Tasche, arXiv: On the coherence of Expected Shortfall — arXiv v5; definitions 2.1, 2.2 and 2.6; proposition 3.2 (2 May 2002). Accessed 29 October 2026.
Lorenzo Naranjo, Washington University in St. Louis (lecture notes, secondary source): Portfolios of Risky Assets — expected return, covariance, portfolio variance and the minimum-variance frontier. Accessed 29 October 2026.
Basel Committee on Banking Supervision, Bank for International Settlements: Basel Framework MAR33 — Internal models approach: capital requirements calculation (§§33.3–33.5, 33.9 and 33.42). Accessed 29 October 2026.
Global Association of Risk Professionals (GARP): FRM Exam information: format and exam topics. Accessed 26 October 2026.

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