Today, I will sketch out ideas from the Hierarchical Compartmental Models for Loss Reserving paper by Jake Morris, which was published in the summer of 2016 (Morris (2016)). Jake’s model is inspired by PK/PD models (pharmacokinetic/pharmacodynamic models) used in the pharmaceutical industry to describe the time course of effect intensity in response to administration of a drug dose.
The hierarchical compartmental model fits outstanding and paid claims simultaneously, combining ideas of Clark (2003), Quarg and Mack (2004), Miranda, Nielsen, and Verrall (2012), Guszcza (2008) and Zhang, Dukic, and Guszcza (2012). You find the first two models implemented in the ChainLadder package, the third in the DCL package and the fourth one in an earlier blog post of mine and as part of the brms package vignette.
Jake uses OpenBUGS (Lunn et al. (2000)) for the Bayesian examples in his paper. Thanks to Paul-Christian Bürkner’s brms package (Bürkner (2017)) I can run version of Jake’s model in Stan (Stan Development Team (2017)) easily, as I will show in this post.
Note: Paul and Jake will be at the Stan workshop in London, following the Insurance Data Science conference, on 17 July 2018.
Hierarchical Compartmental Models
Similar to a PK/PD model, Jake describes the state-space process of claims developments with a set of ordinary differential equations (ODE), using \(EX=\) Exposure, \(OS=\) Outstanding claims, \(PD=\) Paid claims (these are the compartments):
\[
\begin{aligned}
dEX/dt &amp; = -k_{er} \cdot EX \\
dOS/dt &amp; = k_{er} \cdot RLR \cdot EX - k_p \cdot OS \\
dPD/dt &amp; = k_{p} \cdot RRF \cdot OS
\end{aligned}
\]
with initial conditions \(EX(0) = \Pi\) (ultimate earned premiums), \(OS(0)=0, PD(0)=0\).
The parameters describe:
\(k_{er}\) the rate at which claim events occur and are subsequently reported to the insurer
\(RLR\) the reported loss ratio
\(RRF\) the reserve robustness factor, the proportion of outstanding claims that eventually are paid
\(k_p\) the rate of payment, i.e. the rate at which outstanding claims are paid
Note, the product of \(RLR\) and \(RRF\) describes the ultimate loss ratio \(ULR =\) ultimate loss / premium, which is a key metric in insurance to assess performance and profitability.
Setting the parameters \(k_{er}=1.7\), \(RLR=0.8\), \(k_p=0.5\), \(RRF=0.95\) produces the following output.
The chart illustrates nicely the different compartments (processes) for a policy over time.
The autonomous system of ODEs above can be solved analytically:
\[
\begin{aligned}
EX(t) &amp; = \Pi \cdot \exp(- k_{er} t) \\
OS(t) &amp; = \frac{\Pi \cdot RLR \cdot k_{er}}{k_{er} - k_p} \cdot \left(\exp(-k_p t) - \exp(-k_{er} t)\right)\\
PD (t) &amp; = \frac{\Pi \cdot RLR \cdot RRF}{k_{er} - k_p} \left( k_{er} \cdot (1 - \exp(-k_p t) - k_p \cdot (1 - \exp(-k_{er}t ) \right) \end{aligned}
\]
In this post I will focus on the last two equations, as they describe the observable data of outstanding claims and cumulative paid claims over time. However, one could expand the model, e.g. to allow for different premium earning patterns.
Get example data
In his paper Jake uses data from the CAS Reserving Database, namely company 337 from the worker’s compensation file.
The following function downloads the data and reshapes it into a more useful format, very similar to the function used in an earlier post for Commercial Auto data.
library(data.table)
CASdata