How do you calculate expected shortfall from a loss distribution, and why is it preferred over VaR?

Expected shortfall (ES), also called conditional VaR (CVaR), measures the average loss in the tail beyond VaR. It answers the question: 'When things go bad, how bad on average?' This makes it strictly more informative than VaR, which only identifies the threshold.\n\nContinuous Normal Distribution:\n\nFor a normal distribution N(mu, sigma) at confidence level alpha:\n\nES_alpha = mu + sigma x phi(z_alpha) / (1 - alpha)\n\nwhere phi(z) is the standard normal PDF evaluated at z_alpha.\n\nAt 99% confidence (z = 2.326):\nphi(2.326) = 0.02656\nES multiplier = 0.02656 / 0.01 = 2.656\n\nCompare: VaR multiplier = 2.326. So ES is about 14% higher than VaR for normal distributions at 99%.\n\nDiscrete Distribution Calculation:\n\nSuppose Northfield Capital has 1,000 historical daily P&L observations sorted from worst to best. At 95% confidence, VaR is the 50th worst loss.\n\nES(95%) = average of the 50 worst losses\n\n| Rank | Loss ($M) |\n|---|---|\n| 1 (worst) | -8.20 |\n| 2 | -7.45 |\n| 3 | -6.90 |\n| ... | ... |\n| 48 | -3.15 |\n| 49 | -3.08 |\n| 50 | -2.95 |\n\nVaR(95%) = $2.95M (the 50th observation)\nES(95%) = average of losses 1 through 50 = $4.62M (hypothetical average)\n\nES captures how the worst losses are distributed in the tail, while VaR only identifies the boundary.\n\nWhy ES Is Preferred:\n\n1. Coherence: ES is a coherent risk measure (satisfies subadditivity). VaR can paradoxically show that merging two portfolios increases risk, violating diversification logic.\n\n2. Tail sensitivity: Two portfolios with identical VaR can have vastly different ES if one has a heavier tail.\n\n3. Regulatory adoption: Basel III/IV uses ES (97.5%) for market risk capital under FRTB.\n\nExam Trap:\nFor discrete distributions with n scenarios at confidence alpha, the number of tail observations is n x (1 - alpha). If this is not an integer, interpolation rules apply. The FRM exam frequently tests whether students correctly identify which observations fall in the tail.\n\nPractice ES calculations in our FRM question bank.