What is the difference between VaR and CVaR?

VaR is the threshold a loss should stay inside on a given share of days. CVaR is the average size of the losses that exceed it. CVaR is always the larger number, and it is the one that describes the tail.

Which mode makes fewer assumptions?

Historical. It sorts real returns and reads the tail directly. Parametric needs only one volatility figure, but it assumes a normal distribution that real returns do not have.

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Sizing & Risk

Value at Risk (VaR) calculator

“Over the next day — or ten — how bad is a bad day for this portfolio, and how bad are the days worse than that?”

Value at Risk states a loss that a given share of days should not exceed. Conditional VaR states the average of the days that do. This calculator runs both, either parametrically from a volatility figure or historically from a sample of returns pasted in.

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Free to use, nothing to sign up for, and nothing you type here is sent anywhere — the arithmetic runs in your browser.

What it computes

z            = normal score at the confidence level (95% -> 1.645, 99% -> 2.326)
sigma over t = daily vol x SQRT(t)
VaR fraction = z x sigma_t - drift x t
CVaR         = sigma_t x phi(z)/(1 - c) - drift x t

What you type in

Field	What it means	Example
Portfolio value	What is at risk.	$100,000
Daily volatility %	The standard deviation of daily returns.	1.5
Daily mean %	Expected daily drift. Usually left at zero.	0
Confidence %	95 or 99 — the share of days the loss should not exceed.	95
Horizon (days)	How far ahead. Volatility scales with the square root of time.	1

Reading the result

VaR — in percent and in currency, at the confidence and horizon chosen.
CVaR (expected shortfall) — always larger than VaR, because it is the average of the losses that break through it.
Historical mode instead sorts the returns pasted in and reads the loss off the tail, assuming no distribution at all.

Worked example

$100,000 portfolio, 1.5% daily volatility, 95% confidence, one day.

z = 1.645, horizon volatility 1.5%
VaR = 2.47% = $2,467 — on about 19 days in 20, a day's loss stays inside this
CVaR = 3.09% = $3,094 — the average loss on the one day in 20 that does not
At 99% over 10 days: volatility scales to 4.74%, VaR = $11,035 and CVaR = $12,642

Ten days is not ten times one day — it is the square root of ten.

What trips people up

VaR is a frequency statement, not a worst case. A 95% one-day VaR is exceeded about once a month by construction; CVaR is the figure that describes those exceedances.
Parametric mode assumes normally distributed returns, and real returns have fatter tails. Historical mode is the honest one when the data is available.
The square-root-of-time scaling assumes each day is independent of the last.

Questions

What is the difference between VaR and CVaR?: VaR is the threshold a loss should stay inside on a given share of days. CVaR is the average size of the losses that exceed it. CVaR is always the larger number, and it is the one that describes the tail.
Which mode makes fewer assumptions?: Historical. It sorts real returns and reads the tail directly. Parametric needs only one volatility figure, but it assumes a normal distribution that real returns do not have.

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