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Case Study 4: Excess of Loss Reinsurance Pricing
When you are designing an outward reinsurance program, it is important to consider carefully what your historical data has to say about the likely benefits of various options.

In this example, there are two layers: $1M excess of $0 (1M) and $1M (1Mxs1M). If we aggregate the data from the layers, we have a triangle of $2M excess of $0 (2M).

This case study illustrates:
  1. Models for different layers often have very similar structure.


  2. Trends may be significant in some layers and not significant in others.


  3. The coefficient of variation of the total outstanding is just as high, or higher, in losses limited to $1M as in losses limited to $2M.
The best procedure is usually to construct a model for the aggregate triangle, apply that model to the layers and make appropriate adjustments. In this case, the same model fits all the triangles well. However, the alendar trend is not significantly different from zero in the upper layer 1Mxs1M. As with the segments discussed earlier, we can see from the residual plots that the residuals in the two layers are highly correlated.

Next we model all three triangles together in a composite model in MPTF. The correlation between the residuals is a very high 0.96. The calendar trends for the three triangles are significantly different, so there are no parameters in common. However, the correlation between the residuals creates a correlation between corresponding trend parameters of about 0.95.

Comparison of the forecast distribution for the losses limited to $1M with the forecast distribution for the losses limited to $2M can be used by the insurer to decide on the benefits of alternative limit points. In this case, the coefficient of variation of the total outstanding for losses limited to $1M is the same as the coefficient of variation of the losses limited to $2M. This means changing the limit point will not reduce the requirement for capital above the mean, as a proportion of the mean. Using MPTF, we have seen this “no change in CV” effect in many other examples of layered data.

Highly Correlated
Highly Correlated