11.8 Bayesian Models (Incremental)

Model applies CY trend and therefore uses incremental data \(I_{wd}\)

  • Correlated Incremental Trend Model

  • Level Incremental Trend Model

11.8.1 Correlated Incremental Trend

\(I_{wd}\) has a mixed ln-n distribution (11.10)

\[\begin{equation} \begin{array}. I_{wd} \sim \begin{cases} \mathcal{N}(Z_{1,d}, \delta) & \text{if }w = 1 \\ \mathcal{N}(Z_{w,d} + \rho \cdot (I_{w-1,d} - Z_{w-1,d}) \cdot e^{\tau}, \delta) & \text{if } w > 1\\ \end{cases} \\ Z_{w,d} \sim \ln \mathcal{N}(\mu_{w,d}, \sigma_d) \\ \end{array} \tag{11.11} \end{equation}\]

With log-normal mean:

\[\begin{equation} \mu_{wd} = \alpha_w + \beta_d + \tau \cdot (w+d-1) \\ \tag{11.12} \end{equation}\]

Remark. \(\tau\)

  • CY trend parameter

  • Note the \(\tau\) is applied additively in the “log” space (11.12), therefore in the autocorrelation step (11.11) it is applied by \(e^{\tau}\)

Remark. \(\sigma_d\)

  • Subject to the following constaints different from LCL and CCL
\[\begin{equation} \sigma_1 < \sigma_2 < \cdots < \sigma_{10} \tag{11.13} \end{equation}\]
  • Since we’re looking at incremental data, smaller less volatile claims should be settled early

Remark. \(\rho\)

  • Include correlation between AYs similar to CCL method

  • \(\rho\) is the coefficient of correlation between \(I_{w-1,d}\) and \(I_{w,d}\)

    • Note \(\rho\) is applied outside of the “log” space here to the incremental loss (not log)

    • Recall for CCL we apply \(\rho\) to the \(\ln(C_{w-1,d})\) but we can’t do that here due to the chance of negative incremental losses

Priors for \(\{\alpha_w\}\), \(\{\sigma_d\}\), \(\{\beta_d\}\), \(\rho\), and \(\tau\)

  • Disperse priors similar to LCL and CCL unless mentioned below

  • Each \(\beta_d \sim \begin{cases} U(0, 10) & \text{if } d \leq 4 \\ U(0, \beta_{d-1}) &\text{if } d > 4 \\ \end{cases}\)

    • This assure \(\beta_d\) decreases for \(d>4\)

  • \(\tau \sim \mathcal{N}(0,3.2\%)\), which correspond to a precision parameter by JAGs of 1000

    • Without restriction it was forecasting very negative trend which is offset by higher \(\alpha\) and \(\beta\)

    • Meyers expected this to be predominantly negative since the other paid method we’ve tried so far were biased high

  • Each \(\sigma_d \sim \begin{cases} U(0,0.5) & \text{if } d = 1 \\ U(\sigma^2_{d-1},\sigma^2_{d-1} +0.1) & \text{else} \\ \end{cases}\)

    • Limit the speed \(\sigma_d\) can increase, very high \(\sigma_d\) can lead to unreasonably high simulate results

  • \(\delta \sim U(0, \text{Avg Premium})\)

Steps for the method:

  1. Uncorrelated log mean of each cell with CY trend
    \(\mu_{wd} = \alpha_w + \beta_d + \tau \cdot(w+d-1)\)

  2. Draw \(Z_{wd} \sim Lognormal(\mu_{wd},\sigma_d)\)

    • \(\sigma_1 > \sigma_2 > \cdots > \sigma_{10}\)

    • Smaller less volatile claims should be settled early

  3. \(\tilde{I}_{wd} \sim Normal(Z_{wd},\delta)\)

  4. Add correlation between AYs for rows after the first
    \(\tilde{I}_{wd} \sim Normal(Z_{wd} + \rho \cdot (\tilde{I}_{w-1,d} - Z_{w-1,d})\cdot e^{\tau},\delta)\)

Test Results

  • Losses not much smaller than CCL while we would like it to be much smaller as CCL was biased high

  • \(\rho\) is lower than from CCL

  • Strong negative correlation between trend \(\tau\) and level parameters \(\alpha_w + \beta_d\)

    • With small data set it is hard for the model to distinguish the AY level + development vs trend

    • Based on scatter plot of \(\tau\) vs \(\alpha_w + \beta_d\) for several \(d\) and \(w=6\)

  • Average \(\tau\) is negative

  • Model showed no improvement over Mack or ODP

11.8.2 Leveled Incremental Trend

Same as CIT but with \(\rho = 0\)

Results similar to CIT with lower standard deviation