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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

‘brms’ Code

fml <-
  loss_train ~ premiums * (
     (1 - delta) * (RLR*ker/(ker - kp) * (exp(-kp*dev) - exp(-ker*dev))) + 
      delta * (RLR*RRF/(ker - kp) * (ker *(1 - exp(-kp*dev)) - kp*(1 - exp(-ker*dev)))) 
   ) 

b1 <- brm(bf(fml,
             ker ~ 1, kp ~ 1,
             RLR ~ 1 + (1 | accident_year),
             RRF ~ 1 + (1 | accident_year),
             sigma ~ 0 + deltaf,
             nl = TRUE),
          data = lossData0[cal <= max(accident_year)], 
          family = brmsfamily("gaussian", link_sigma = "log"),
          prior = c(prior(gamma(4, 5), nlpar = "RLR", lb=0),
                    prior(gamma(4, 5), nlpar = "RRF", lb=0),
                    prior(gamma(3, 2), nlpar = "ker", lb=0),
                    prior(gamma(3, 4), nlpar = "kp", lb=0)),
          control = list(adapt_delta = 0.999, max_treedepth=15),
          seed = 1234, iter = 1000)

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?

1

Hierarchical Compartmental Models for Loss Reserving Jake Morris July 17, 2018