9.10 Using Multiple Models

Use different methods (Paid/Inc’d Dev, BF, etc) and assigning weights by AYs

• Models should be reviewed and finalize individually before blending with weights

Method 1:
In the process variance step of bootstrap, use the same underlying \(u \sim U(0,1)\) to draw from each model then weight the models by a set of deterministic %’s

• Use the same random variable or else we would reduce the variability of the outcomes

Method 2:
Run each model independently for each simulation (i.e. use different \(u \sim U(0,1)\)) then for each AY use the weights to randomly select one of the modeled results

• Results will be a mixture of the various models

Other considerations:

• Should consider both the mean and standard deviation (or CoV) in each model result when selecting weights

• Can also select the weights using Bayesian methods to account for the quality of each model’s forecast

• Perform the same model output review as in the above section for the best estimate

• Also review the IBNR by AYs to look for inconsistencies (e.g. negative IBNR) and compare to deterministic results

9.10.1 Additional Useful Output

Using the best estimate total unpaid mean, s.e., and CoV from above to fit to Normal, LogNormal, and Gamma distribution.

We can use these fitted distribution to:

1. Assess quality of fit

2. Parameterize a DFA model

3. Smooth out extreme values

9.10.2 Estimated Cash Flow Results

Since bootstrap generates simulation for each cell in the bottom half of the triangle we can use this to get cash flow forecasts by CY and their variability as well

We can review the s.e. and CoV similar what we did in the diagnostics section

9.10.3 Estimated Ultimate Loss Ratio Results

We can estimate mean and the variability of ultimate loss ratios by AYs

Compile a similar table as before 9.7 but for loss ratio

Useful for projecting pricing risk in a risk model

9.10.4 Estimated Unpaid Claims Runoff Results

Project unpaid claims out by CY similar to the cash flow projection

Useful for calculating risk margins using the cost of capital method

9.10.5 Distribution Graphs

Plot the distribution of the simulated unpaid in a histogram

• Or smooth the histogram with a Kernel density function (for each point it takes a weighted average of the points around it, giving less weight to points further from it)

• For each point it takes a weighted average of the points around it; giving less weight to points further from it

9.10.6 Correlation

Correlate the loss distribution over several LoB

• Multivariate distribution requires the same underlying distribution which doesn’t work here for ODP

Method 1: Location Mapping

• When sampling the residuals, sample from the same place in the triangle for all the lines we want to correlate

• Disadvantages:

  • Requires all LoB to have the same size triangle with no missing values or outliers

  • Cannot stress the correlations among the LoBs (Can only use the historical correlations)

Method 2: Re-Sorting

• Use Iman-Conover algorithms or Copulas (Not explained in paper)

• Advantages:

  • Can accommodate different shapes and sizes

  • Can make different correlation assumptions

  • Can strengthen the correlation for extreme events (e.g. t Copula vs normal Copula)

• Calculate correlation matrix using Spearman’s Rank Order and re-sorting based on the ranks of unpaid claims by AYs

  • Look at p-value for each correlation parameter to see that they’re significantly different from zero

Additional comments:

• Using residuals to correlate LoBs (both location mapping& re-sorting) are liable to create correlations close to zero

• Reserve Risk: Correlate total unpaid by correlating the incremental paid

  May or may not be a reasonable approximation

  Risk not modeled is contagion risk, where a single event results in claims in multiple lines of business (can change correlation assumptions to address this)

• Pricing Risk: Correlate loss ratios over time

  Not as likely to be close to zero

  Use different correlation assumption than for reserve risk