Formula sheets

Mack assumptions $\;E[C_{i,k+1}\mid C_{i,1..k}]=f_kC_{i,k}$, $\operatorname{Var}(C_{i,k+1}\mid\cdot)=\sigma_k^2C_{i,k}$, accident years independent.

Mack variance estimate $\hat\sigma_k^2=\frac{1}{I-k-1}\sum_{i=1}^{I-k}C_{i,k}\left(\frac{C_{i,k+1}}{C_{i,k}}-\hat f_k\right)^2$ $\widehat{mse}(\hat C_{i,I})=\hat C_{i,I}^2\sum_{k=I+1-i}^{I-1}\frac{\hat\sigma_k^2}{\hat f_k^2}\left(\frac{1}{\hat C_{i,k}}+\frac{1}{\sum_{j\le I-k}C_{j,k}}\right)$

Brosius least squares $\;\hat y=a+bx$; credibility form $\;Z\cdot\dfrac{x}{d}+(1-Z)E[y]$.

Clark Weibull growth $\;G(x)=1-\exp\!\left[-(x/\theta)^\omega\right]$; loglogistic $\;G(x)=\dfrac{x^\omega}{x^\omega+\theta^\omega}$.

ODP bootstrap scale $\;\hat\phi=\dfrac{\sum r_{ij}^2}{n-p}$, $r_{ij}=\dfrac{q_{ij}-\hat m_{ij}}{\sqrt{\hat m_{ij}}}$.

Cost-of-capital risk margin $\;RM=\sum_t CoC\cdot C_t\,v^{t+1}$.