A recent post on the PirateGrunt blog on claims reserving inspired me to look into the paper Regression models based on log-incremental payments by Stavros Christofides [1], published as part of the Claims Reserving Manual (Version 2) of the Institute of Actuaries.

The paper is available together with a spread sheet model, illustrating the calculations. It is very much based on ideas by Barnett and Zehnwirth, see [2] for a reference. However, doing statistical analysis in a spread sheet programme is often cumbersome. I will go through the first 15 pages of Christofides' paper today and illustrate how the model can be implemented in R.

Let's start with the example data of an incremental claims triangle:

## Page D5.4
tri <- t(matrix(
  c(11073,  6427, 1839, 766,
    14799,  9357, 2344,  NA,
    15636, 10523,   NA,  NA,
    16913,    NA,   NA,  NA), 
  nc=4, dimnames=list(origin=0:3, dev=0:3)))

The above triangle shows incremental claims payments for four origin (accident) years over time (development years). It is the aim to predict the bottom right triangle of future claims payments, assuming no further claims after four development years.

Christofides model assumes the following structure for the incremental paid claims \(P_{ij}\):
\begin{align}
\ln(P_{ij}) & = Y_{ij} = a_i + b_j + \epsilon_{ij}
\end{align}where i and j go from 0 to 3, \(b_0=0\) and \(\epsilon_{ij} \sim N(0, \sigma^2)\). Unlike the basic chain-ladder method, this is a stochastic model that allows me to test my assumptions and calculate various statistics, e.g. standards errors of my predictions.
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