Formula sheets

Stop-loss premium $\;E[(S-d)_+]=\sum_{s>d}(s-d)P(S=s)$; recursion $\;E[(S-d-1)_+]=E[(S-d)_+]-(1-F_S(d))$

Excess-of-loss reinsurance on $X\sim\text{Exp}(\theta)$ with retention $r$: $\;E[\text{reinsurer}]=\theta e^{-r/\theta}$

Bühlmann–Straub $\;Z_i=\dfrac{m_i}{m_i+k}$, $\hat\mu$ = credibility-weighted mean

Generalized Pareto tail $\;\bar F_u(y)=\left(1+\xi y/\beta\right)^{-1/\xi}$; Hill estimator $\;\hat\xi=\dfrac1k\sum_{i=1}^k\ln\dfrac{X_{(n-i+1)}}{X_{(n-k)}}$

TVaR $\;TVaR_\alpha=VaR_\alpha+\dfrac{E[(X-VaR_\alpha)_+]}{1-\alpha}$

Mack process/parameter variance — see CAS Exam 7 guide.