How do I implement the peaks-over-threshold (POT) method step by step?

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AcadiFi · Accepted Answer

POT has six steps. First, collect loss data (positive losses or negative returns). Second, select a threshold u using diagnostic tools: mean excess function should be approximately linear above u, Hill plot should stabilize, and choose u high enough to satisfy asymptotic theory but low enough to retain sample size (typically 10-15% of data above u). Third, extract exceedances y = x - u for all x > u. Fourth, fit GPD to y via MLE yielding xi and beta estimates. Fifth, compute tail quantiles: VaR(p) = u + (beta/xi)*((n/Nu)*(1-p))^(-xi) - 1) where p is confidence level, n is total sample, Nu is exceedances. Sixth, check goodness of fit via QQ plot against GPD quantiles and Anderson-Darling test.

Northmarket Analytics applies POT to 3,000 daily losses. Threshold selected at u = 2.1% (300 exceedances). MLE gives xi = 0.28, beta = 0.95%. 99.5% VaR = 2.1 + (0.95/0.28)*((3000/300)*0.005)^(-0.28) - 1) = 4.76%. Expected shortfall at 99.5% is VaR/(1-xi) + (beta - xi*u)/(1-xi) = 7.1%.

Common pitfalls: picking threshold too low (violates asymptotic assumption), ignoring cluster of exceedances (declustering required for dependent data), and using MLE when sample is small (use probability-weighted moments instead).

How do I implement the peaks-over-threshold (POT) method step by step?

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