Loss Models Cheat Sheet

Severity Distributions

Distribution	E[X]	E[X^2]	Key Property
Exponential(theta)	theta	2 theta^2	Memoryless
Pareto(alpha, theta)	theta/(alpha-1)	2 theta^2/((alpha-1)(alpha-2))	Heavy tail
Lognormal(mu, sigma^2)	e^(mu + sigma^2/2)	e^(2mu + 2sigma^2)	Right skewed
Weibull(tau, theta)	theta Gamma(1+1/tau)	theta^2 Gamma(1+2/tau)	Flexible hazard

Coverage Modifications

• Ordinary deductible d: E[(X-d)+] = E[X] - E[min(X,d)]
• Policy limit u: E[min(X,u)] = integral from 0 to u of S(x) dx
• With deductible d and limit u: E[payment] = E[min(X, d+u)] - E[min(X, d)]
• Coinsurance alpha: E[payment] = alpha * E[(X-d)+]
• Loss elimination ratio: LER(d) = E[min(X,d)] / E[X]

Aggregate Loss Models

• Compound model: S = X_1 + X_2 + ... + X_N (N random)
• E[S] = E[N] * E[X]
• Var(S) = E[N] * Var(X) + Var(N) * (E[X])^2
• Compound Poisson: Var(S) = lambda * E[X^2]