How to use the value at risk calculator
- Paste the past returns in percent, one per period, separated by lines, commas or spaces.
- Choose a confidence level: 90, 95, 97.5 or 99 percent.
- Enter the value of the portfolio to see the amounts in dollars.
- Read the historical and normal value at risk, the average loss beyond it and the supporting statistics.
Formula
historical VaR = - percentile of returns at position (n - 1) x (1 - confidence), with linear interpolation
normal VaR = z x standard deviation - mean return
average loss beyond VaR = - mean of the returns at or below the percentile
The historical method reads the loss straight from the sorted data. The normal method assumes the returns follow a bell curve with the same mean and standard deviation. The standard deviation uses n - 1.
Twenty daily returns at 95 percent
The 20 example returns are sorted from -2.6 percent to 1.5 percent. At 95 percent confidence the percentile sits at position 19 x 0.05 = 0.95, between the worst return of -2.6 and the second worst of -2.1. Interpolating gives -2.6 + 0.95 x 0.5 = -2.125, so historical VaR is 2.125 percent, or $2,125.00 on $100,000.
The mean is -0.065 percent and the standard deviation is 1.1659 percent. With z = 1.6449, normal VaR is 1.6449 x 1.1659 + 0.065 = 1.983 percent, or $1,982.67. The average loss in the tail is 2.6 percent.
How to read the number
A one period 95 percent VaR of 2.125 percent says that, if the future resembles the past list, about 5 percent of periods would lose at least that much. It does not say how large the loss in the worst periods could be, which is why the average loss beyond VaR is shown too.
Short lists make the percentile unreliable. With 20 returns the 95th percentile is mostly set by the two worst days.
Limits of both methods
The historical method can only show losses that appear in the data. The normal method can understate risk because real returns have fatter tails than a bell curve. Different tools also interpolate the percentile differently, so results can differ slightly.
This page is an educational estimate. It is not a risk limit and not investment advice.
Assumptions and limits
- Each return is for one period of the same length, in percent.
- The percentile uses linear interpolation between the sorted values.
- The normal method uses the sample standard deviation with n - 1 and the exact normal quantile.
- The portfolio is assumed to be a plain long position. Leverage, fees and taxes are not modeled.
- Between 10 and 5,000 returns are read.
Frequently asked questions
What does 95 percent confidence mean here?
It means the loss shown is the one that was exceeded in about 5 percent of the periods in your list. It is a statement about the data, not a promise about the future.
Why do the two methods differ?
The historical method follows the data exactly, while the normal method smooths it into a bell curve. When the data has fat tails or is lopsided, they can differ noticeably.
Can I scale the result to a longer period?
This page gives a one period figure for the period your returns cover. Scaling to more periods needs extra assumptions that are not made here.
Why is the average loss beyond VaR larger?
VaR marks the start of the worst tail. The average of the returns at or below that point is always at least as bad as VaR, and it shows how deep the tail goes.