Value at Risk Calculator
Estimate the loss a position is not expected to exceed at a given confidence, by the parametric and historical methods.
Value at risk answers one narrow question: over a stated holding period, and on all but the worst slice of outcomes, how much could this position lose? At 95% confidence the answer is the loss expected to be exceeded on one period in twenty. This value at risk calculator estimates it two ways — parametric VaR, from a volatility you supply, or historical VaR, from a series of returns you supply.
Formula: parametric VaR = position value x z-score x volatility over the horizon
Your figures
Parametric works from a volatility you supply. Historical reads the tail out of a series of returns you supply.
The position or portfolio the loss is measured against. Every cash figure scales directly with this.
How much of the distribution sits above the threshold. 95 leaves the worst 5% in the tail. Between 50% and 99.99%.
Parametric method only. Always your own measured figure annualised, whatever frequency you pick below — a monthly volatility of 5.77% is entered as 20. The frequency sets the clock for the holding period, not for this field.
Parametric method only. Counted in the same periods as the frequency below, so 10 with 252 periods a year means ten trading days.
Parametric method only. The clock the holding period is counted on, and what the annualised volatility is scaled down to. The volatility itself is always annualised.
Only used when the frequency is set to Custom.
Results
Value at risk
US$2,072.32
2.0723% of the position at 95.00% confidence
- Loss as a share of the positionThe same figure before it is applied to the position value.
- 2.0723%
- z-score at this confidenceHow many standard deviations into the left tail of a normal distribution the threshold sits.
- 1.6449
- Volatility over the holding periodThe quoted volatility multiplied by the square root of holding periods divided by periods in a year.
- 1.2599%
- Holding periodIn the periods set by the frequency chosen. Ten periods is not ten times the one-period loss, because the scaling is by the square root.
- 1.00
- Position value
- US$100,000.00
What this means
At 95.00% confidence the threshold sits 1.6449 standard deviations into the left tail. Scaling the volatility entered down to the holding period gives 1.2599% over that horizon, and multiplying the two gives 2.0723% of the position — US$2,072.32 on US$100,000.00. That figure is the point where the worst 5.00% of a normal distribution begins, and it is entirely a product of the volatility you supplied and the shape that distribution assumes.
- Value at risk is a threshold, not a worst case. It says the loss is smaller than this on the stated share of periods, and on the rest it says only that the loss is larger — by an amount it does not describe. Expected shortfall answers that second question.
- The parametric estimate assumes returns are normally distributed and describes the whole shape with one number. Real return distributions have fatter tails than a normal does, so it understates the size of the worst outcomes — in the direction that matters.
- A portfolio is treated as a single position with a single volatility. Offsetting holdings are not netted and no correlation between them is modelled, so the figure describes the position value you entered rather than the parts inside it.
- Scaling a volatility across time uses the square-root-of-time rule, which holds when returns are independent from one period to the next. Real series trend and mean-revert, so a market that grinds one way understates the figure and one that whips back and forth overstates it.
Value at risk: US$2,072.32
The link reopens this calculator with the same numbers. Nothing you enter is sent to a server.
Continue with these figures
About the Value at Risk calculation
The two routes disagree by design. The parametric one describes an entire distribution with a single number and reads the threshold off a normal curve, which makes it smooth in the confidence level and available for any horizon. The historical one assumes nothing about shape and simply reads the percentile out of the record, which makes it exactly as good as the record and blind to anything that has not happened yet.
Whichever route is taken, the result is a threshold and not a worst case. It marks the point where the tail begins. What happens past that point — whether the remaining 5% of periods lose slightly more or catastrophically more — is a question no VaR calculator answers, and was never built to answer. That is the gap the expected shortfall calculator exists to fill.
Value at Risk formula
parametric:
σ over the horizon = σ annual × √(holding periods ÷ periods per year)
VaR% = z × σ over the horizon, z = Φ⁻¹(c)
historical:
h = (n − 1) × (1 − c)
threshold = the return at position h, interpolated between neighbours
VaR% = −threshold
both:
VaR = position value × VaR% ÷ 100- c
- The confidence level as a fraction — 0.95 for 95%. The tail being measured holds the remaining 1 − c of the distribution.
- z
- Φ⁻¹(c), the inverse standard normal. It is 1.6449 at 95%, 2.3263 at 99%, and exactly zero at 50%, which is why the confidence level is floored there.
- σ over the horizon
- The annualised volatility rescaled down to the holding period by the square-root-of-time rule. The volatility you enter is always the annual figure; the frequency only says how many of those periods make up a year.
- h
- The fractional position of the percentile within the sorted record, counting the worst observation as position 0 and the best as position n − 1.
The parametric form is one multiplication resting on one assumption. Once returns are taken to be normally distributed, the whole left tail is fixed by the volatility and the confidence level simply chooses a point along it. That is all z is: how many standard deviations into the tail the threshold sits. No feature of the actual distribution is read from data at any stage.
That assumption is the method's weakness, and it fails in the direction that matters. Real return distributions have fatter tails than a normal one — extreme days occur far more often than the curve permits — so the parametric estimate understates large losses at exactly the confidence levels where large losses are the entire point of the exercise. Moving from 95% to 99% raises z by 41%, while the true tail of a financial series typically thickens by more than that.
The historical form interpolates rather than picking an observation, because a percentile rarely lands on one. With n returns sorted from worst to best, the worst is treated as the 0th percentile and the best as the 100th, so the wanted point sits h = (n − 1) × (1 − c) along the sequence and is read linearly between its two neighbours. This matches Excel's PERCENTILE.INC and R's default quantile, so any figure here can be checked in a spreadsheet.
Scaling across time uses the square root rather than the period count, for the same reason volatility annualises that way: variance adds, standard deviation does not. Ten days of a 20% annual volatility is 20 × √(10 ÷ 252) = 3.98%, not ten times the one-day 1.26%. That sublinearity holds only while returns are independent from day to day, and it is why a multi-day figure is worth calculating rather than multiplying up from a single-day one.
Value at Risk: a worked example
A $100,000 position carrying a 20% annualised volatility, held for one trading day, at 95% confidence, by the parametric method.
- z-score at 95%Φ⁻¹(0.95) = 1.6449
- Volatility over one day20% × √(1 ÷ 252) = 1.2599%
- Loss as a share of the position1.6449 × 1.2599% = 2.0723%
- Value at risk$100,000 × 2.0723% = $2,072.32
- The same position over ten days20% × √(10 ÷ 252) = 3.9841%, giving 6.5533% or $6,553.25
$2,072.32, or 2.0723% of the position. On the normal distribution assumed here, one trading day in twenty is expected to lose more than that — and the figure says nothing whatever about how much more. Stretching the horizon to ten days raises the loss to $6,553.25 rather than to $20,723, because dispersion grows with the square root of time rather than in step with it.
How to use the Value at Risk calculator
- Pick a method first. Parametric needs a volatility and a holding period; historical needs the return series. Fields belonging to the method you are not using can be left blank.
- Enter the position value as the amount actually exposed. Every cash figure on the page scales directly with it, and a portfolio value at risk figure means entering the whole book as one number, which the calculation then treats as one position.
- Set the confidence level. 95 leaves the worst 5% of periods in the tail; 99 leaves the worst 1% and produces a materially larger number from identical inputs.
- For the parametric method, always quote the volatility annualised, whatever frequency you pick. The frequency sets the clock the holding period is counted on, and the calculator scales the annual figure down to it — so a monthly record whose volatility is 5.77% a month is entered as 20, not 5.77. Entering the monthly figure directly understates the answer by the square root of twelve.
- For the historical method, enter your own periodic returns in percent, oldest first, with losing periods as negatives. Read the 'observations at or below the threshold' row before leaning on the answer.
- Take the loss as a share of the position alongside the cash figure. The percentage is the part that transfers to a differently sized position without being recalculated.
Limitations of the Value at Risk calculator
Every calculation makes assumptions. These are the ones that matter here.
- The parametric estimate assumes normally distributed returns. Markets produce far more extreme days than a normal curve allows, so the figure is an understatement precisely where understating is most expensive.
- The historical estimate contains no loss the record does not contain. With six returns at 95% confidence the answer is almost entirely the single worst one, and one new worst period rewrites it completely.
- A 95% figure is designed to be exceeded on one period in twenty. Roughly twelve breaches across a year of daily data is what the model predicts, not evidence that it has failed.
- A portfolio is treated as a single position with a single volatility. Offsetting holdings are not netted and no correlation is modelled — and correlations between holdings tend to converge upward in exactly the stressed conditions a tail measure is asked about.
- The volatility entered is an input, not a forecast. A figure measured over a calm stretch produces a calm answer, and nothing here checks whether that stretch resembles the period ahead.
- Square-root-of-time scaling assumes each period is independent of the last, which rules out the possibility that a loss and its continuation are related. A sustained sell-off is precisely that case.
Frequently asked questions about Value at Risk
What is the practical difference between parametric VaR and historical VaR?
The parametric method imposes a shape and reads the answer off it, so it can quote any confidence level and any horizon from a single volatility figure. The historical method imposes nothing and reads the answer out of your returns, so it captures skew and fat tails that were actually observed but cannot say anything about a loss the record has never seen. They are best read together: a large gap between them is itself information about how normal the record is.
Why is the 99% figure so much larger than the 95% one?
Because z rises from 1.6449 to 2.3263, a 41% increase, and the loss is directly proportional to it. On the historical method the jump is usually larger still and far less stable, because moving from the 5th percentile to the 1st on a short record often means moving onto the single worst observation and staying there.
Does value at risk tell me my worst case?
No, and this is the measure's defining limitation. It marks where the worst slice begins and is silent about everything inside it — a period losing 1% more than the threshold and one losing 60% more look identical to it. Expected shortfall averages that slice instead of marking its edge, which is why the two are normally quoted as a pair.
How many returns does the historical VaR calculator need?
Enough for the percentile to sit on real observations rather than on the very edge of the record. At 95% confidence the threshold falls at h = (n − 1) × 0.05, so twenty returns puts it 0.95 of the way from the worst observation to the second worst — which means the answer is essentially the worst single return. A hundred or more observations is where the percentile starts describing a tail rather than a point.
Why is the ten-day loss not ten times the one-day loss?
Because independent variances add while standard deviations do not. Ten days of variance is ten times one day's, and the standard deviation of the ten-day return is therefore √10 — about 3.16 — times the one-day figure. That sublinearity is the whole reason a multi-day number is worth calculating, and it disappears if daily returns turn out to be correlated with each other.
The calculator says the percentile is still a gain. What does that mean?
It means every return in the record at and below the chosen percentile was positive, so inverting the sign produces a negative loss. The honest reading is that the series you entered contains no loss that deep, not that the position carries negative risk. It happens on short, strongly positive records, and it is flagged rather than hidden because a negative loss reported without comment reads as a bug.
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