10.2 Stochastic Models for the Chainladder
Stochastic models that have the same estimate of Unpaid losses as the Chainladder
For each model, we can calculate the MSE of prediction and therefore a prediction interval
Remark.
Assume future observations are independent of past observations and the -2 term above goes to 0
S.e. = \(\sqrt{\text{Estimation Variance}}\)
Prediction error includes both the error in estimating our parameters and the process variance (inherent variability in the data being forecast)
- Both bootstrap and MCMC allows us to calculate the prediction error
10.2.1 Mack-1993 (Non-parametric)
Only the first 2 moments of cumulative claims are specified
\[\begin{equation} \mathrm{E}[D_{ij}] = \lambda_j D_{ij-1} \tag{10.1} \end{equation}\] \[\begin{equation} \mathrm{Var}(D_{ij}) = \sigma^2_j D_{ij-1} \tag{10.2} \end{equation}\]Remark.
Mean is the same as the Chainladder
Variance is \(\propto\) claims reported to date \(D_{ij-1}\)
\(\sigma^2_j\) has to be estimated separately from the development factors
The simplicity of this allows the parameter estimate and prediction errors to be obtained from a spreadsheet
- Downside is that without specifying a distribution we don’t get a predictive distribution
10.2.2 Over-dispersed Poisson and Negative Binomial
Separate stream of research that focused on the use of GLM
Incremental claims distribution OD Poisson:
\[\begin{equation} \begin{split} & c_{ij} \mid c, \alpha, \beta, \varphi \sim ODP(m_{ij}, \varphi m_{ij}) \\ & \mathrm{E}[c_{ij}] = m_{ij} \\ & \mathrm{Var}(c_{ij}) = \varphi m_{ij} \\ & \ln(m_{ij}) = c + \alpha_i + \beta_j \: ; \: \alpha_1 = \beta_1 = 0 \end{split} \tag{10.3} \end{equation}\]Remark.
Model result is the same as Chainladder
\(m_{ij}\) mathematically the same as the one defined in Shapland, we can not include the \(c\) term and just use \(\alpha_i\)’s as the individual level parameters instead of having \(\alpha_1 = 0\) and use the \(alpha_i\)’s as adjustments to the level parameter constant \(c\)
- This allows for calculating prediction error
Alternative way of writing ODP
\[\begin{equation} \begin{split} & c_{ij} \mid x,y, \varphi \sim ODP(x_i y_j, \varphi x_i y_j) \\ & \sum_{k=1}^n y_k = 1 \\ & x = \{x_1, x_2,...,x_n\} \:\: \text{row parameters}\\ &x_i = \mathrm{E}[D_{in}] \:\: \text{Expected ultimate cumulative losses up to }n\\ & y = \{y_1, y_2,...,y_n\} \:\: \text{col parameters}\\ & y_i \:\: \text{Proportions of ultimate losses that emerge in each development year} \end{split} \tag{10.4} \end{equation}\]Recall:
\[\begin{equation} \begin{split} &X \sim \text{Poisson}(\mu) \\ &Y = \varphi X \sim ODP(\varphi \mu, \varphi^2 \mu)\\ \end{split} \tag{10.5} \end{equation}\]Where \(\varphi\) is typically > 1 \(\therefore\) over-dispersed
This can be extend to other distribution beyond Poisson
e.g. see over-dispersed negative binomial below
We can get the same predictive distribution (as ODP) with ODNB
- ODNB also make the connection between ODP and Chainladder more apparent
Incremental claims distribution for OD Negative Binomial:
\[\begin{equation} \begin{split} & c_{ij} \mid D_{ij-1}, \lambda_j, \varphi \sim ODNB \\ & \mathrm{E}[c_{ij}] = (\lambda_j - 1)D_{ij-1} \\ & \mathrm{Var}(c_{ij}) = \varphi \lambda_j \mathrm{E}[c_{ij}] \\ \end{split} \tag{10.6} \end{equation}\]Cumulative claims distribution for OD Negative Binomial:
\[\begin{equation} \begin{split} & D_{ij} \mid D_{ij-1}, \lambda_j, \varphi \sim ODNB \\ & \mathrm{E}[c_{ij}] = \lambda_j D_{ij-1} \\ & \mathrm{Var}(c_ij) = \varphi (\lambda_j - 1) \mathrm{E}[c_{ij}] \\ \end{split} \tag{10.7} \end{equation}\]Remark. For both ODP and ODNB
Use a quasi-likelihood approach so that the loss data are not restricted to the positive integers
Reserve estimates are the same as Chainladder
Both are subject to the positivity constraints
- It’s apparent in the ODNB formula that the column sums must be positive or else we’ll have development factor \(\lambda_j < 1\) \(\Rightarrow\) Varaiance to be negative
Advantages (over Kremer):
Does not necessarily break down if there are negative incremental loss values
- In a strict sense, the model requires the incremental losses in a column to be positive otherwise it is more difficult to justify and interpret the inferences (but it doesn’t necessarily break the model)
Gives the same reserve estimate as Chainladder
More stable than the log-normal model of Kremer
Verrall suggest that model can be specified for either incremental or cumulative loss
Advantage of the NB:
- Form of the mean is the same as chainladder
- If we replace the NB and use normal instead we can deal with the problem of negative incremental claims (not discussed in paper)