VaR calculator

Calculator

How it works

This is parametric (variance-covariance) VaR. It assumes daily returns are normal with a mean of zero, so the loss at a given confidence level is the position times volatility times the matching z-score.

1-day VaR = V × σ × z
h-day VaR = 1-day VaR × √h
1-day ES = V × σ × φ(z) ÷ (1 − c)

The square-root-of-time rule needs returns that are independent and identically distributed from day to day. Volatility clustering, autocorrelation and fat tails all break it, which is why regulators and risk teams treat a scaled ten-day figure as an approximation rather than a measurement.

Expected shortfall is the average loss in the tail beyond VaR. Under a normal distribution it is always larger than VaR at the same confidence level; at 97.5% it comes out close to 99% VaR, one reason Basel's market-risk framework could move to 97.5% ES without a jump in capital for normally distributed portfolios.

Worked example

A $10 million portfolio with 1.2% daily volatility at 99% confidence: z = 2.3263, so one-day VaR is $279,162. Over ten days the square-root rule gives $882,787, about 3.16 times the one-day figure, not ten times. The same numbers are walked through in the square-root-of-time guide.

Related

• Scaling VaR with the square root of timeWhen the rule holds and when it breaks.
• Expected shortfall vs VaRNormal and fat-tailed comparison, and the Basel switch.
• Value at RiskGlossary definition with the exam angle.
• Expected shortfallGlossary definition with the exam angle.
• FRM Part I topic weightsWhere VaR sits in the syllabus.
• All calculatorsDuration and convexity, swap valuation.