Deductibles and Reinsurance

Deductibles

Given a deductible of \(d\), let \(X\) be claim amount, and \(V\) and \(Y\) be the amount paid by policyholder and insurer respectively. \[X=V+Y\] where \[V=X\land d=\begin{cases} X & \text{if }X<d \\ d & \text{if }X\geq d \end{cases} \] and \[Y=(X-d)_+=\begin{cases} 0 & \text{if }X<d \\ X-d & \text{if }X\geq d \end{cases} \]

Excess of Loss Reinsurance

Given a retention level of \(M\) with no deductibles, let \(X\) be claim amount, and \(Y\) and \(Z\) be the amount paid by insurer and reinsurer respectively. \[X=Y+Z\] where \[Y=X\land M=\begin{cases} X & \text{if }X<M \\ M & \text{if }X\geq M \end{cases} \] and \[Z=(X-M)_+=\begin{cases} 0 & \text{if }X<M \\ X-M & \text{if }X\geq M \end{cases} \]

Mixed Distribution

partly discrete and partly continuous \[F_U(x)=\int_{-\infty}^xf_U(x)dx+\sum_{x_i\in S,x_i\leq x}P(U=x_i)\]

\[E[g(U)]=\int_{-\infty}^{\infty}g(x)f_U(x)dx+\sum_{x_i\in S}g(x_i)P(U=x_i)\]