2.1 Gunnar Benktander Method (GB)
Proposition 2.1 (Benktander Reserve) Reserve under the Benktaner Method
\[R_{GB} = q_kU_{BF}\]
\(q_k = (1 - p_k)\) = % unpaid loss @ \(k\)
\(U_{BF} = C_k + q_kU_0\)
- Weight on \(U_0\) is \(q_k^2\)
Remark. Benktander recognizes actual emergence faster \(\Rightarrow\) Less weight on the a-priori
- BF reduces the use of actual loss to the extent of the complement credibility
\(\begin{align} U_{GB} &= (1-q_k)U_{CL} + q_kU_{BF} & \cdots (1)\\ &= (1-q_k^2)U_{CL} + q_k^2 U_0 & \cdots (2)\\ \end{align}\)
Crediblity weight \(U_{BF}\) and \(U_{CL}\)
- Crediblity weight \(U_{CL}\) and a priori
2.1.1 Method Comparison
| Method | Ultimate (\(U\)) | Reserve (\(R\)) |
|---|---|---|
| Chain Ladder | \(\dfrac{C_k}{p_k}\) | \(q_k U_{CL} = q_k \dfrac{C_k}{p_k}\) |
| BF Method | \(C_k + q_kU_0\) | \(q_k U_0\) |
| GB Method | \(C_k + q_kU_{BF}\) | \(q_k U_{BF}\) |
Theorem 2.1 For an arbitrary starting point \(U^{(0)} = U_0\) and the iteration rule \(R^{(m)} = q_k U^{(m)}\) and \(U^{(m+1)} = C_k + R^{(m)}\), \(m = 0, 1, 2, ...\)
gives credibility mixtures
\[U^{(m)} = (1-q^m_k)U_{CL} + q^m_k U_0\]
\[R^{(m)} = (1-q^m_k)R_{CL} + q^m_k R_0\]
between \(BF\) and \(CL\) which start at \(BF\) and lead via \(GB\) to \(CL\) for \(m= \infty\)2.1.2 MSE
MSE of Benktander is almost as small as the MSE of the optimal credibility in most cases
\[MSE(R_{GB}) < MSE(R_{BF})\]
When \(p_k \in [0, 2c^*]\); \(c^*\) is the optimal credibility
- \(R_{GB}\) doesn’t have the lowet \(MSE\) only when \(p_k > 2c^*\)
Doesn’t hold if \(c^*\) is small and \(p_k\) is large
Remark. \[MSE(R_c) = \mathrm{E}[R_c - R]^2\]
\(R_c = c R_{CL} + (1-c)R_{BF}\)
\(R = U - C_k = C_n - C_k\)
Where:
\(c = 0\) for \(BF\)
\(c = p_k\) for \(GB\)
- \(c = c^*\) for optimal credibility where \(c \in [0, 1]\)