CAS Exam 8 cheat sheet

Advanced Ratemaking

• Experience modification: M = (Ap + Ee·w)/(Ep + Ee·w), where Ap = actual limited losses, Ep = expected limited losses, Ee = expected excess losses, and w = credibility weight on excess (w = Ee/(Ee + Ep·k)).

• Credibility-weighted excess loss provision: excess ratio r = (Ap_x + k·Ee_x)/(Ep_x + k·Ee_x), where k = (1−Z)/Z and Z is the credibility of actual excess experience; applied when excess experience is thin.

• Limited expected value LEV(d) = E[min(X,d)] = ∫₀^d S(x)dx; the excess ratio uses E[(X−d)₊] = E[X] − LEV(d), so limiting losses at d removes the need to model the tail explicitly.

• Increased limits factor ILF(d) = LEV(d)/LEV(b) for a base limit b; ILFs are not additive across layers, so a layer (a,b] factor equals [LEV(b) − LEV(a)]/LEV(base limit).

• Loss elimination ratio LER(d) = LEV(d)/E[X]; LER is the complement of the excess ratio and is used to price deductibles as a credit of 1 − LER(d) on the ground-up expected loss.

• Credibility: Z = √(N/N_full) under square-root partial credibility for frequency; for aggregate loss use Z = √(P/P_full) when P is exposure (premium or counts), capped at 1.

• Bühlmann credibility: Z = n/(n+k) with k = EPV/VHM, where EPV = expected process variance and VHM = variance of hypothetical means; larger between-class heterogeneity raises Z.

• Deductible pricing: with a per-loss deductible d, insured expected loss = E[X] − LEV(d) plus any loss-constant loading; franchise deductibles pay the full loss once it exceeds d, so expected cost = E[(X)·1{X>d}].