Casualty Actuarial Science & Excess Pricing
Pareto Excess Layer Severity & ILF Pricing Formula
Price heavy-tailed casualty layers using Single-Parameter Pareto Limited Expected Value LEV(x), Excess Layer Severity LEV(D + L) - LEV(D), and Increased Limits Factor ILF(L_high, L_base).
Price umbrella and excess layers when empirical loss data above $500k is sparse!
Historical burning cost fails on high umbrella and casualty layers when zero or one mega-claim occurred in the past five years! Casualty actuaries model losses above threshold theta > 0 using the heavy-tailed Single-Parameter Pareto survival curve S(x) = (theta / x)^alpha (for x >= theta, tail index alpha > 0)! Integrating S(t) from 0 to policy limit x >= theta yields Limited Expected Value LEV(x) = theta + LN(x / theta) * theta (if alpha = 1) or theta + (theta / (alpha - 1)) * (1 - (theta / x)^(alpha - 1)) (if alpha != 1)! From LEV(x), two universal pricing formulas emerge immediately: (1) Increased Limits Factor ILF(x_high, x_base) = LEV(x_high) / LEV(x_base), and (2) Expected Severity in Excess Layer L xs D (covering [D, D + L]) = LEV(D + L) - LEV(D)!
Verify on Pareto_Tail_Alpha = 2.0 (B2), Ground_Threshold theta = $100k (B3), Basic_Policy_Limit D = $200k (B4), and Excess_Layer_Width L = $300k (B5 -> Top Limit D + L = $500k): LEV($200k) = 100 + (100 / 1) * (1 - (100 / 200)^1) = $150.00k; LEV($500k) = 100 + 100 * (1 - (100 / 500)^1) = $180.00k; Excess Layer ($300k xs $200k) Expected Loss per claim >= $100k is 180.00 - 150.00 = $30.00k; and Increased Limits Factor ILF($500k vs $200k) is 180.00 / 150.00 = 1.2000 (a +20.0% rate surcharge to raise policy limit 2.5x from $200k to $500k)!
| Pareto Excess Pricing Metric (alpha=2.0, theta=$100k, D=$200k, L=$300k) | Formula | Verified Output |
|---|---|---|
| Basic Limit Expected Severity LEV(D) | theta + (theta/(alpha-1))*(1 - (theta/D)^(alpha-1)) | $150.00k |
| Top Limit Expected Severity LEV(D + L) | theta + (theta/(alpha-1))*(1 - (theta/(D+L))^(alpha-1)) | $180.00k |
| Excess Layer ($300k xs $200k) Severity | LEV(D + L) - LEV(D) | $30.00k / claim |
| Increased Limits Factor ILF(500k, 200k) | LEV(D + L) / LEV(D) | 1.2000x |
Universal Pareto Limited Expected Value LEV(x) Formula (Including alpha = 1)
Handle extreme heavy tail alpha = 1 (infinite unconditional mean, finite capped layer!) smoothly via theta * (1 + LN(x / theta))!
=LET(al_p, B2, th_m, B3, d_bs, B4, IF(OR(al_p <= 0, th_m <= 0, d_bs < th_m, B5 <= 0), "Invalid Pareto Layer", ROUND(IF(al_p = 1, th_m * (1 + LN(d_bs / th_m)), th_m + (th_m / (al_p - 1)) * (1 - ((th_m / d_bs) ^ (al_p - 1)))), 4)))Excess Layer Severity & Increased Limits Factor (ILF) Formula
Test heavier tail alpha = 1.0 in cell B2: LEV($200k) rises to 100*(1 + LN(2)) = 169.3147, LEV($500k) rises to 100*(1 + LN(5)) = 260.9438, and ILF jumps from 1.2000x to 1.5412x!
=IF(ISTEXT(E2), E2, ROUND(E3 / E2, 4))Try the example
Use the original values or change them to test your understanding.