3.1 Loss Ratio Claims Reserve Definition
\(m_k\): Expected loss ratio @ each age \(k\)
Based on incremental column paid loss ratios
\(k \in \{1, ..., n \}\) For \(n\) development periods
\(ELR\): Expected loss ratio:
\[ELR = \sum \limits_{k=1}^n m_k\]
a priori ELR for collective loss ratio approach
Use for the entire triangle
\(p_k\): % Losses emerged for exposure period \(k\)
\[p_k = \dfrac{\sum \limits_{j=1}^{n} m_j}{ELR}\]
Based on column loss ratios \(m_k\)
Loss ratio payout factor or loss ratio lag-factor
\(q_k = 1 - p_k\) is the loss ratio reserve factor
3.1.1 Loss Ratio Claims Reserve Summary
\[\begin{equation} R_i^c = Z_i \times R_i^{ind} + (1-Z_i) \times R_i^{coll} \tag{3.1} \end{equation}\]| \(\mathbf{Z_i}\) | Method |
|---|---|
| 1 | Chainladder; Individual LR |
| 0 | BF; Collective LR |
| \(p_k\) | Benktander (GB)3 |
| \(p_k \times ELR\) | Neuhaus (WN)1 |
| \(\dfrac{p_k}{p_k + \sqrt{p_k}}\) | Optimal Credibility2 |
Remark.
Neuhaus gives low credibility to lines with low loss ratios
Since Neuhaus use loss ratio, \(\Delta\) exposure base will \(\Delta\) result
Neuhaus credibility = expected loss ratio to date
Optimal credibility is capped @ 0.5
- There’s additional assumptions made to get to the optimal credibility formula
This is the same as Mack-2000 but just defined with \(p_k\)
- Note we’re weighting the reserve here
- Benktander and Neuhaus reduce the MSE of the reserve estimate nearly to an optimal level outperforming individual and collective
Proposition 3.1 (Individual Loss Ratio Claims Reserve) Analogous to chainladder
\(\begin{align} R_i^{ind} &= \dfrac{C_{ik}}{p_k} \times q_k \\ &= \dfrac{C_{ik}}{p_k} - C_{ik} \\ &= U_i^{ind} - C_{ik} \\ \end{align}\)Proposition 3.2 (Collective Loss Ratio Claims Reserve) Analogous to BF
\(\begin{align} R_i^{Coll} &= q_k(V_i \times ELR) \\ &= q_k(U_i^{BC}) \\ \end{align}\)
BC = Burning Cost
- \(V_i\) = premium for year \(i\)
3.1.2 Optimal Credibility Weights
Optimal credibility weights for loss ratio claims reserve
\(Z^*_i\) is the credibility that minimizes the \(MSE(R_i^c) = \mathrm{E}[(R_i^c - R_i)^2]\)
Theorem 3.1 Optimal credibility factor \(c^*\) that minimizes \(MSE(R_i^c) = \mathrm{E}[(R_i^c - R_i)^2]\) is
\[Z^*_i = \dfrac{p_i}{q_i} \dfrac{Cov(C_i, R_i) + p_i q_i Var(U_i^{BC})}{Var(C_i) + p_i^2 Var(U_i^{BC})}\]| Impact on \(\mathbf{Z_i^*}\) | Comments | |
|---|---|---|
| Losses emerge | Increase | Since \(\dfrac{p_i}{q_i}\) increases as losses emerge |
| \(\mathrm{Cov}(C_i, R_i)\) increase | Increase | Large covariance implies that \(C_i\) is predictive of \(R_i\) \(\Rightarrow\) More weight on \(CL\) method |
| \(\mathrm{Var}(C_i)\) increase | Decrease | If \(C_i\) is volatile, we want to rely less on \(CL\) method |
| \(\mathrm{Var}(U_i^{BC})\) increases | Increase | Trust \(CL\) method more when a-priori is volatile |
Remark.
Assumes \(U_i^{BC} {\perp\!\!\!\!\perp} \: C_i\) and \(R_i\)
- Large \(Var(U_i^{BC})\) \(\Rightarrow\) \(Z \approx \dfrac{p}{q} \times \dfrac{pq}{p^2} = 1\)
Assumes \(\mathrm{E}\left[ \dfrac{C_{ik}}{U_i} \mid U_i \right] = p_k\) and \(\mathrm{Var}\left( \dfrac{C_{ik}}{U_i} \mid U_i \right) = p_k q_k \beta^2(U_i)\)
Theorem 3.2 Under the additional assumptions above, we have
\[Z_i^* = \dfrac{p_k}{p_k + t_k}\]
Where \(t_k = \dfrac{\mathrm{E}[\alpha^2_i(U_i)]}{\mathrm{Var}(U^{BC}_i) + \mathrm{Var}(U_i) - \mathrm{E}[\alpha^2_i(U_i)]}\)Theorem 3.3 If we assume \(\mathrm{Var}(U_i) = \mathrm{Var}(U_i^{BC})\) then
\[Z_i^* = \dfrac{p_k}{p_k + \sqrt{p_k}}\]
Where the above assumption lead to \(t_k \sim \sqrt{p_k}\)Remark.
We would actually expect the \(\mathrm{Var}(U_i) > \mathrm{Var}(U_i^{BC})\), but the above is just an assumption to make things simplier
- The i’s and the k’s are sort of interchangable
3.1.3 MSE
\[MSE(R^{ind}_i) = \mathrm{E}\left[\alpha_i^2(U_i)\right] \cdot \left( \dfrac{Z_i^2}{p_i} + \dfrac{1}{q_i} + \dfrac{(1-Z_i)^2}{t_i} \right) \cdot q^2_i\]
\(MSE(R^{ind}_i)\) when \(Z_i = 1\)
\(MSE(R^{coll}_i)\) when \(Z_i = 0\)