8.1 Introduction

2 Key Parts:

  1. Convert a triangle to one level of trend and layer of losses

    • This is then used to determine LDFs at this base layer

  2. Convert LDFs at a Base layer to LDFs at any other layer

Claim size models (distribution for individual claims)

  • Exponential (in this paper)

  • Different distribution for each column in the triangle

    Requires a distribution of losses at each age, which can be difficult

    (Last section try to address this)

Definition 8.1 Sahasrabuddhe notations

\(B\): Base layer, LDFs are determined at this layer

\(X\): Layer of interest, ultimately we want LDFs for this layer

\(\Phi_{ij}\): Cumulative loss distn in row \(i\) age \(j\)

\(LEV(X; \Phi_{ij})\): Limited Expected Value, average loss with distn \(\Phi_{ij}\) capped at \(X\)

\(F_{ij}^X\): LDF to ultimate for cell \(ij\) with losses capped at \(X\)

\(C^L_{ij}\): Cumulative paid to date for AY \(i\), age \(j\) in layer \(L\)

8.1.1 Claim Size Model

Distribution of the individual losses (this underpins calculations we do below)

Properties of \(\Phi_{ij}\)

  1. Have parameters to adjust for inflation (e.g. Lognormal, Weibull, Exponential, Pareto)

    • Assumes that the only difference between the distribution of cells in the same column is trend (\(T_{ij}\))

      \(\therefore\) We have \(\Phi_{ij} \sim f(\Phi_{nj},T_{ij},T_{nj})\)

      • i.e. Distribution of \(C_{ij}\) only depends on the things above

    • This allows us to choose one distribution for each column

      Specifically the distribution is selected for row n in each column and we use the trend factors to select the distribution for any cell in the same column

  2. Mean (and limited means) can be calculated with reasonable effort

Definition 8.2 (Increased Limits Factors) \[S_{ij}(L_a, L_b) = \dfrac{\text{Expected Losses for Layer }L_a}{\text{Expected Losses for Layer }L_b} = \dfrac{\mathrm{E}\left[ C^{L_a}_{ij} \right]}{\mathrm{E}\left[ C^{L_b}_{ij} \right]} = \dfrac{LEV(L_a; \Phi_{ij})}{LEV(L_b; \Phi_{ij})}\]

  • Both for cell \(C_{ij}\)

  • \(L_a\): Layer \(a\), deductible \(d_a\) and policy limit \(p_a\)

  • \(L_b\): Layer \(b\), deductible \(d_b\) and policy limit \(p_b\)