A value at risk calculator answers one narrow question: over a stated horizon and at a stated confidence level, what loss threshold is not expected to be exceeded? The answer is a threshold, not a worst case. VaR is silent about the size of the loss once that threshold is breached — its known and much-documented failure, and the reason conditional VaR appears beside every figure on this page.
Arb Digital publishes this alongside the max drawdown calculator and the Sortino ratio calculator. The three are different views of downside: drawdown measures the deepest realised peak-to-trough fall in a path that already happened, Sortino measures return per unit of downside deviation, and VaR states a forward-looking threshold under an assumed distribution. None is a substitute for the others.
What This Value at Risk Calculator Does
It computes VaR three ways from the same inputs, because they disagree in instructive ways.
Parametric VaR, also called the variance-covariance or delta-normal method, assumes returns are normally distributed. It takes the quantile of the standard normal at your confidence level, multiplies by volatility scaled to the horizon, and applies it to the portfolio value. It is fast, transparent and wrong in a specific direction: it understates real market tails.
Historical simulation makes no distributional assumption. It sorts your actual return series and reads off the empirical quantile. It is honest about the shape of the data you gave it and completely blind to anything that data does not contain.
Monte Carlo draws a large sample from a normal distribution with your volatility and drift and reads the quantile off the simulation. With normal draws it converges on the parametric answer, which is exactly the point: Monte Carlo inherits whatever distribution you feed it, and calling it a more sophisticated method does not make its assumptions any weaker.
Alongside each figure the calculator computes conditional VaR — the expected shortfall — which is the average loss conditional on the threshold being breached. It is always larger than VaR and it is the number the threshold conceals.
How to Use It
- Supply your own volatility. This page hardcodes no market data, because volatility changes daily and a baked-in figure would be wrong by tomorrow. Estimate it from your own return history — the standard deviation calculator does that arithmetic.
- Match the volatility to the period. A daily volatility with a horizon of 1 gives a one-day VaR. Entering an annual volatility with a horizon of 1 gives an annual VaR, not a daily one. Mixing the two is the single most common error here.
- Give the historical method enough data. At 99% confidence, a sixty-observation series puts fewer than one observation in the tail, so the quantile is an extrapolation from almost nothing. The fourth grid item counts them for you.
- Read the conditional VaR every time. It is the average of the outcomes beyond the threshold. If it is far above the VaR figure, the tail is where your risk actually lives.
- Compare the three bars. When historical VaR is much larger than parametric, your data has fatter tails than a normal distribution — which is the usual finding, not the exception.
The Formula / How It's Calculated
For the parametric method, with portfolio value V, per-period volatility σ, per-period drift μ, horizon h and confidence c:
VaR = V × ( zc × σ × √h − μ × h )
where zc is the standard normal quantile at c — 1.644854 at 95% and 2.326348 at 99%. The √h term is the square-root-of-time rule, and it assumes returns are independent across periods.
Conditional VaR under the same normal assumption has a closed form using the standard normal density φ:
CVaR = V × ( σ × √h × φ(zc) ÷ (1 − c) − μ × h )
For historical simulation, sort the return series ascending and take the empirical quantile at (1 − c), interpolating between the two neighbouring observations. VaR is that quantile applied to V; CVaR is the mean of every observation at or below it.
For Monte Carlo, draw a large sample from a normal distribution with mean μh and standard deviation σ√h, then take the same empirical quantile and tail mean. This page uses a fixed seed so the figure is reproducible rather than changing on every click.
Worked example, matching the values the page loads with. A $1,000,000 portfolio, 1.5% per-period volatility, zero drift, one period, 99% confidence. The normal quantile is 2.326348, so VaR = 1,000,000 × 2.326348 × 0.015 = $34,895.22, which is 3.4895% of the portfolio. The normal density at that quantile is 0.02665214, so the shortfall multiplier is 0.02665214 ÷ 0.01 = 2.665214 and CVaR = $39,978.21 — some 14.6% larger than the threshold.
The historical series supplied contains sixty observations. The 1% empirical quantile is −3.4385%, giving a historical VaR of $34,385, with exactly one observation in the tail, at −3.94%, so historical CVaR is $39,400. At 95% confidence instead, the quantile is −2.6760% and three observations sit in the tail, averaging −3.2733%.
Stretching the horizon to ten periods multiplies the parametric figure by √10 = 3.16228, giving $110,348.37.
VaR Tells You Nothing About the Tail
This matters most, and it is a property of the definition rather than a shortcoming of any implementation.
VaR is a quantile. It marks the boundary of the worst (1 − c) of outcomes and says nothing whatsoever about the distribution of losses beyond that boundary. Two portfolios can have identical 99% VaR while one loses a little more than the threshold on bad days and the other occasionally loses ten times it. The statistic cannot distinguish them, and a risk report showing only VaR would present them as equivalent.
Conditional VaR — expected shortfall — is the standard response. It averages the losses in the tail rather than reading off its edge, so it responds to tail shape. Under a normal distribution it exceeds VaR by a predictable amount, about 15% at 99% confidence. Under a fat-tailed distribution the gap is much larger, which is the diagnostic.
VaR also fails subadditivity, a property risk measures are usually expected to have: the VaR of a combined portfolio can exceed the sum of the VaRs of its parts. Expected shortfall does not have that defect. The Basel Committee's fundamental review of the trading book sets out why supervisors moved market-risk capital from VaR to expected shortfall.
The Normal Assumption Understates Real Market Tails
Parametric VaR is only as good as the distribution it assumes, and the normal distribution is a poor description of financial returns at the extremes.
Under a normal distribution a five-standard-deviation daily move has a probability of roughly one in 3.5 million — once in about fourteen thousand years of trading days. Real markets produce such moves far more often than that arithmetic allows. Return distributions have fatter tails and a sharper peak than the normal, so the model is wrong in the region you care about most and right in the middle where you care least.
Volatility clustering compounds it. Large moves arrive in groups, so returns are not independent across periods — which breaks the square-root-of-time rule directly. Scaling a one-day VaR to ten days by multiplying by √10 assumes exactly the independence that volatility clustering rules out, and it typically understates the ten-day figure.
Correlations behave badly too. The Federal Reserve discussion paper Pitfalls in Tests for Changes in Correlations shows that apparent correlation breakdown in stressed markets can be an artefact of conditioning on extreme observations, while noting correlations do genuinely change over time. Either way, a portfolio VaR that relies on stable correlations is relying on an assumption worth stating.
What Each Method Actually Assumes
Naming the assumption matters more than picking a favourite.
Parametric assumes normality, constant volatility over the horizon, and independence across periods. It has no free parameters beyond volatility and drift, so it is easy to audit and easy to be precisely wrong with.
Historical simulation assumes that the future resembles the sample you supplied. It captures whatever fat tails, skew and clustering your data contains — and nothing else. A quiet sample produces a comfortable number. It also has a hard floor: the worst loss it can ever produce is the worst loss in your data.
Monte Carlo assumes whatever distribution you simulate from. Using normal draws, as this page does, means it will converge on the parametric answer with sampling noise around it. That is worth seeing directly, because Monte Carlo is often presented as more robust when it is really more flexible — the robustness would have to come from a better-specified distribution, not from the simulation itself.
A fourth point applies to all three: VaR assumes positions can be valued and, implicitly, exited. In illiquid instruments the marked price and the achievable price differ, and that gap widens precisely in the conditions VaR is meant to describe. FINRA's overview of investment risk covers liquidity and the other risk categories no single number captures.
Arb Digital writes technical finance and data content where the assumptions are stated on the page — the standard that makes a page citable rather than merely rankable.
See Content Marketing Services Talk to Arb DigitalCommon Mistakes to Avoid
- Reading VaR as a maximum loss — it is a threshold that is expected to be exceeded some of the time, and it says nothing about how far it is exceeded by.
- Mismatching volatility and horizon — an annual volatility with a horizon of one period gives an annual figure, not a daily one, and the error is a factor of about sixteen.
- Using historical simulation on a short series — at 99% confidence a sixty-observation sample puts under one observation in the tail, so the quantile is essentially an extrapolation.
- Scaling short horizons to long ones with the square-root rule — it assumes independence across periods, which volatility clustering contradicts, and it usually understates the longer figure.
- Adding the VaR of two portfolios together — VaR is not subadditive, so the combined figure can exceed the sum of the parts. Expected shortfall does not have this problem.
Related Free Tools From Arb Digital
Read this next to the max drawdown calculator for realised peak-to-trough loss and the Sortino ratio calculator for downside deviation — three different views of the same downside. Estimate the volatility input with the standard deviation calculator or the variance calculator, check the quantile with the z-score calculator and the normal distribution calculator, and measure benchmark sensitivity with the beta calculator. Everything else sits in the free online tools hub.
Frequently Asked Questions
It is a loss threshold at a stated confidence level over a stated horizon. A one-day 99% VaR of 34,895 means that under the model's assumptions the loss is expected to exceed that figure on about one day in a hundred. It is a quantile of a distribution, not a maximum and not a forecast of any particular day.
No, and this is its defining limitation. VaR marks the boundary of the tail and says nothing at all about the distribution of losses beyond it. Two portfolios with identical VaR can behave completely differently once the threshold is breached. Conditional VaR, which averages the losses in the tail, is the figure that addresses this.
It is the average loss given that the VaR threshold has been breached. Because it averages the tail rather than reading its edge, it responds to tail shape and is always larger than VaR. Under a normal distribution at 99% confidence it exceeds VaR by roughly 15%; under fatter-tailed data the gap is wider.
Because real return distributions have fatter tails than the normal. A five-standard-deviation daily move has a normal probability of about one in 3.5 million, and markets produce moves of that size far more often. The parametric method is therefore wrong in exactly the region it is meant to describe, and it understates rather than overstates.
Enough that the tail contains a meaningful number of observations. At 99% confidence only one observation in a hundred lands in the tail, so a sixty-observation series gives fewer than one and the quantile is an extrapolation. The calculator counts the tail observations so you can see how thin the estimate is.
The square-root-of-time rule multiplies by the square root of the horizon, but it assumes returns are independent across periods. Volatility clustering means large moves arrive in groups, which breaks that assumption, and the scaled figure usually understates the true multi-period risk.
Because this page draws from a normal distribution, which is the same assumption the parametric method makes. Monte Carlo inherits whatever distribution it simulates from; the simulation adds flexibility, not robustness. Any improvement would have to come from specifying a better distribution, not from the technique itself.
Two reasons. It ignores the tail beyond the threshold entirely, and it is not subadditive, meaning the VaR of a combined portfolio can exceed the sum of its parts. Expected shortfall avoids both problems, which is why banking supervisors moved market-risk capital requirements toward it.
This tool performs a statistical calculation on assumptions and data you supply. It is not investment advice, not a recommendation to buy, sell or hold any security, and its output does not bound your possible losses. Model risk is real and every method here rests on assumptions stated above. Decisions about your money should involve a licensed financial adviser regulated in your jurisdiction.
