Hierarchical Compartmental Models for Loss Reserving

Jake Morris

July 17, 2018

Extending the Growth Curve Approach

Exponential Growth Function

  • Exponential growth for rate of payment \(k_{p}\) defined as: \[GF_{t} = 1-e^{-k_{p} \cdot t}\]

Exponential Growth as an ODE system

  • Define \(EX_{t}\) as exposure % at development time t

  • Assume \(EX_{0} = 1\) and \(EX_{t} + GF_{t} = 1\)

  • Growth function is a solution to ODEs:

  • \[dEX_{t}/dt = -k_{p} \cdot EX_{t}\] \[dGF_{t}/dt = k_{p} \cdot EX_{t}\]

  • \[EX_{t} = e^{-k_{p} \cdot t}\] \[GF_{t} = 1-e^{-k_{p} \cdot t}\]

Exponential Growth Model as an ODE system

  • Assume \(EX_{0} = \text{Premiums} = P\)

  • Define \(PD_{t}\) as cumulative paid claims at time t

  • Allow a multiple of exposure, \(ULR\), to become paid

  • \[dEX_{t}/dt = -k_{p} \cdot EX_{t}\] \[dPD_{t}/dt = k_{p} \cdot ULR \cdot EX_{t}\]
  • \[EX_{t} = P \cdot (1-GF_{t})\] \[PD_{t} = P \cdot ULR \cdot GF_{t}\]

Compartmental Model Representation

  • Systems of ODEs can be written schematically as compartmental models:
  • Readily extensible framework for modelling the claims process

Compartmental Reserving Models

Base Model Schematic

  • Introduce additional compartment for reported claims
    • Define \(k_{er}\) as rate claims occur and are reported

  • Split \(ULR\) into multiplicative components:
    • Reported Loss Ratio (\(RLR\))
    • Reserve Robustness Factor (\(RRF\))

Base Model ODEs

  • \[ \begin{gathered} \begin{align} \tfrac{dEX_{t}}{dt} &= -k_{er} \cdot EX_{t} \\ \tfrac{dOS_{t}}{dt} &= k_{er} \cdot RLR \cdot EX_{t} - k_{p} \cdot OS_{t} \\ \tfrac{dPD_{t}}{dt} &= k_{p} \cdot RRF \cdot OS_{t} \end{align} \end{gathered} \]

Extending the Model

  • Constant rates of reporting and payment rarely hold
    • Delays in reporting post-incident
    • Payment delays for larger claims

  • Allow \(k_{er}\) and \(k_{p}\) to depend on development time, \(t\)
    • Many potential functions


  • \[k_{er}(t) = \beta_{er} \cdot t\] \[k_{p}(t) = \beta_{1,p} / (\beta_{2,p} + t)\]

Extended Model ODEs

  • \[ \begin{gathered} \begin{align} \tfrac{dEX_{t}}{dt} &= -\beta_{er} \cdot t \cdot EX_{t} \\ \tfrac{dOS_{t}}{dt} &= \beta_{er} \cdot t \cdot RLR \cdot EX_{t} - \tfrac{\beta_{1,p}} {(\beta_{2,p} + t)} \cdot OS_{t} \\ \tfrac{dPD_{t}}{dt} &= \tfrac{\beta_{1,p}} {(\beta_{2,p} + t)} \cdot RRF \cdot OS_{t} \end{align} \end{gathered} \]

Hierarchical Compartmental Reserving Model

Model Specification

  • \[ \begin{aligned} \begin{bmatrix} OS_{ij}, PD_{ij} \end{bmatrix}^{T} = \ y_{ij} &\thicksim \mathcal{N}(\mu_{ij},\ \sigma^{2}_{[\delta]}) \\ \mu_{ij} &= P_{i} \cdot f(\delta_{ij}, t_{j}, k_{er}, RLR_{[i]}, k_{p}, RRF_{[i]}) \\ \\ f(\delta_{ij},t_{j},\dots) & = (1 - \delta_{ij}) \cdot \hat {\mbox{OS}}(t_{j}, k_{er}, RLR_{[i]}, k_{p}) \ + \\ & \ \ \ \ \ \ \delta_{ij} \cdot \hat {\mbox{PD}}( t_{j}, k_{er}, RLR_{[i]}, k_{p}, RRF_{[i]}) \\ \\ \delta_{ij} &= \begin{cases} 0 & \text{if } y_{ij} = OS_{ij} \\ 1 & \text{if } y_{ij} = PD_{ij} \end{cases} \\ \\ \begin{pmatrix} RLR_{[i]} \\ RRF_{[i]}\end{pmatrix} &\thicksim \mathcal{N} \left( \begin{pmatrix} \mu_{RLR} \\ \mu_{RRF} \end{pmatrix}, \begin{pmatrix} \sigma_{_{RLR}}^2 & 0\\ 0 & \sigma_{_{RRF}}^2 \end{pmatrix} \right) \end{aligned} \]

Priors

  • Prior assumptions driven by domain knowledge:
    • Profitability
    • Case reserving strength
    • Reporting / Payment speed
  • Priors

‘brms’ Code

Posterior Predictive Checks: Outstanding

Outstanding PPIs

Posterior Predictive Checks: Paid

Paid PPIs

Exercise Questions

  • Is a Normal observation model appropriate?

  • How sensitive is the model to alternative priors?

  • Are \(RLR_{i}\) and \(RRF_{i}\) likely to be independent?

  • How might we increase model flexibility?

Questions?