Expected Claim Shortfall Calculator
Expected Shortfall, also known as Tail Value at Risk or Conditional Tail Expectation, measures the average size of the worst losses beyond a chosen confidence level. It is the risk measure favoured in modern solvency frameworks because, unlike Value at Risk, it captures how severe the tail is rather than just where it begins. This calculator assumes claim amounts follow a normal distribution and returns the Value at Risk threshold and the Expected Shortfall at your chosen confidence level. Enter the mean claim, the standard deviation, and the confidence level. Treat the normal assumption as an analytic baseline, since real claim severities are often heavier-tailed.
Expected Shortfall formula (normal distribution)
alpha = confidence % / 100
z = standard normal quantile at alpha
VaR = mu + sigma * z
phi(z) = standard normal density = exp(-z^2 / 2) / sqrt(2*pi)
ES = mu + sigma * phi(z) / (1 - alpha)
The z-score is found by inverting the standard normal cumulative distribution (here via a rational approximation of the probit function). Because phi(z) / (1 - alpha) is always greater than z, Expected Shortfall is always larger than Value at Risk at the same confidence level.
Worked example
With mu = $50,000, sigma = $20,000, and 99 percent confidence, the quantile z is about 2.3263. VaR = 50,000 + 20,000 * 2.3263 = $96,526.95. phi(z) = 0.026652, so ES = 50,000 + 20,000 * 0.026652 / 0.01 = $103,304.69. Expected Shortfall exceeds VaR by about $6,777.74.
Expected Shortfall: frequently asked questions
What is Expected Shortfall?
Expected Shortfall (ES), also called Tail Value at Risk or Conditional Tail Expectation, is the average loss in the worst (1 minus confidence) fraction of outcomes. Where Value at Risk answers 'what is the threshold I will not exceed with this confidence', Expected Shortfall answers 'if I do exceed it, how bad is it on average'. It is a coherent risk measure used widely in insurance and banking capital rules.
What formula does this use?
For a normal distribution with mean mu and standard deviation sigma at confidence level alpha, VaR = mu + sigma times z, where z is the alpha-quantile of the standard normal. Expected Shortfall = mu + sigma times phi(z) divided by (1 minus alpha), where phi is the standard normal density. The phi(z) divided by (1 minus alpha) term is always larger than z, so ES exceeds VaR.
Why assume a normal distribution?
The closed-form ES formula requires a distribution. The normal model is the standard textbook starting point and is exact when claims are normally distributed. Real claim severities are often skewed and heavy-tailed, so treat this as an analytic baseline. Enter your fitted mean and standard deviation for the best approximation.
How is the confidence level chosen?
Insurance solvency frameworks commonly use high confidence such as 99 or 99.5 percent for a one-year horizon. The confidence level is a user-editable input here so you can match your own regulatory or internal standard.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Normal distribution.
- U.S. National Institute of Standards and Technology: nist.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.