Trader P&L is 60% skill and 40% luck in variance terms, with skill persistent across years and luck fresh each year, both mean zero. Last year's top decile averaged +2.0 standard deviations. How many standard deviations do you predict for them this year, and what does that say about a bonus rule keyed to last year's rank?

Trader P&L is 60% skill and 40% luck in variance terms, with skill persistent across years and luck fresh each year, both mean zero. Last year's top decile averaged +2.0 standard deviations. How many standard deviations do you predict for them this year, and what does that say about a bonus rule keyed to last year's rank?

Approach: Decompose observed performance into a persistent and a transient component, then take the conditional expectation of the persistent part given the observed total.

1.2. Write observed performance as y = s + l with Var(s) = 0.6 and Var(l) = 0.4 in units where Var(y) = 1 and the two parts independent. Next year's outcome is s plus fresh noise, so the best linear predictor is E[s | y] = (Cov(s, y)/Var(y))*y, which is 0.6*y, and a group observed at +2.0 is predicted at 0.6*2.0 = 1.2. The shrinkage factor 0.6 is the signal share of total variance, the rest being luck that does not repeat. The 0.8 drop is regression to the mean, a real statistical effect present even when nothing about the traders changed. The same algebra predicts -1.2 for a bottom decile observed at -2.0, so a bonus rule keyed to last year's rank systematically overpays the lucky and underpays the unlucky, and a manager who fires the bottom decile will observe improvement that would have arrived without the firing.

Follow-up: How would you estimate the 0.6 signal share directly from a panel of trader-year P&L without assuming it?

Key concepts: regression to the mean, shrinkage factor, signal-to-noise ratio, best linear predictor.