9.4 Bootstrap Simulation Procedure

Bootstrap simulation procedure, repeat steps 1 - 5 at least 10,000 times

0. Model our losses, determine mean and residual for each cell

   • This can be based on GLM Framework of simplified GLM

1. Create a sampled \(triangle^*\) from the residuals and the means

   1. Sample with replacement on the Pearson residuals (9.8) from our original triangle from Step 0.

      (Since data needs to be \(iid\) for bootstrap)

      (Note the distribution of the residual will be purely empirical)

   2. Simulated loss: \[\begin{equation} q^*(w,d) = m_{wd} + r_p^* \sqrt{m_{wd}^z} \tag{9.14} \end{equation}\]

      (\(r^*_p\) is the realized sample from i.)

   3. Estimate dispersion factor \(\phi\) for Step 3 as this is based on the original triangle

2. Determine parameters from \(triangle^*\):

   • For GLM Framework, calculate the \(\alpha_w\)’s, \(\beta_d\)’s

   • For Simplified GLM calculate the weighted average LDFs and Ultimate Loss

3. Calculate mean and variance² for the future cells (unpaid): \((m^*_{wd}, \phi m^*_{wd})\)

   • Mean: \(m^*_{wd}\)

     • GLM Framework:

       \(m^*_{wd} = \mathrm{exp} \left [\alpha_w + \sum \limits_{i=2}^d \beta_i \right]\)

     • Simplified GLM

       Back out the \(c^*(w,d)\) by \(\dfrac{Ult_w}{CDF_d}\) then get the \(m^*_{wd}\) (Need to back out from ultimate because we need to complete the cells for unpaid too)

   • Variance: \(\phi m_{wd}^{*}\)

     (for ODP \(z=1\) on the \(m^{*z}_{wd}\))

4. Add process variance: draw losses³ from the following Gamma distribution for the future cells (unpaid):

   \[\begin{equation} Gamma(m_{wd}^*,\phi m_{wd}^*) \tag{9.15} \end{equation}\]

   • Simulate loss from the gamma distribution for each future cells

   • Use \(u \sim U[0,1]\) and \(F^{-1}_{gamma}(u)\)

5. Calculate simulated unpaid: sum the bottom half of triangle

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2. Step 3 and 3 were added in England & Verrall (2002), in their 1999 paper it doesn’t have this step (stopped at 2) and instead just suggest you to add process variance by multiply the results by \(f^{RoF}\)↩

3. Poisson distribution can be used to remain more consistent with the underlying theory of GLM framework, but it is considerably slower to simulate, so gamma is a close substitute that performs much faster in simulation although it can be more skewed than the Poisson. Indeed other distributions could be used as well to better approximate the observed “skewness” of the residuals from the diagnostics↩