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($\mu, \sigma^2$)	$e^{\mu + \sigma^2/2}$	$e^{2\mu + 2\sigma^2}$	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 $u$: $E[\min(X,u)] = \int_0^u S(x)\,dx$
• With deductible $d$ and limit $u$: $E[\text{payment}] = E[\min(X, d+u)] - E[\min(X, d)]$
• Coinsurance $\alpha$: $E[\text{payment}] = \alpha \cdot E[(X-d)_+]$
• Loss elimination ratio: $LER(d) = E[\min(X,d)] / E[X]$

Aggregate Loss Models

• Compound model: $S = X_1 + X_2 + \ldots + X_N$ ($N$ random)
• $E[S] = E[N] \cdot E[X]$
• $\text{Var}(S) = E[N] \cdot \text{Var}(X) + \text{Var}(N) \cdot (E[X])^2$
• Compound Poisson: $\text{Var}(S) = \lambda \cdot E[X^2]$