Five Bad Days

Question

A market-making book's daily P&L takes four values: it makes 1.5M on 80% of days, loses 2M on 16% of days, loses 6M on 3% of days, and loses 10M on 1% of days. Compute the one-day 95% VaR and the 95% expected shortfall (the average loss across the worst 5% of days).

Accepted Answer

VaR: walk in from the tail. In loss units the distribution is 10 (1%), 6 (3%), 2 (16%), -1.5 (80%). The probability of losing more than 2M is 3% + 1% = 4% le 5%, while any threshold below 2M is breached on at least 20% of days (the whole -2M atom). So textVaR_95% = mathbf2M. The 95th percentile lands inside the 16% atom at -2M, which is typical for discrete P&L: the quantile sits on a lump, not between values. ES: average the worst twentieth. Out of 100 representative days, the worst five are one day at -10M, three at -6M, and one of the sixteen -2M days (only one fits in the 5% budget, the atom gets split): textES_95% = frac10 + 3(6) + 25 = frac305 = mathbf6M. The trap: computing mathbbE[L mid L ge textVaR] without splitting the atom. Averaging over all days losing at least 2M gives frac0.01(10) + 0.03(6) + 0.16(2)0.20 = 3M, which is the average over the worst 20% of days, not the worst 5%. With atoms, "tail conditional expectation" and ES differ, and ES (the average of the worst 1-alpha slice) is the coherent one. Why both numbers exist: VaR says "you lose at least 2M one day in twenty" and is blind to everything beyond; ES says those bad days average 6M, three times the threshol

Five Bad Days

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