Mildenhall Aggregate Loss Tools

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Example 13: Loss and ALAE in Excess Reinsurance Layers

Notes

• Example 13 determines the distribution of loss plus allocated in a reinsured layer under the two common treatments of ALAE in reinsurance: pro rata in addition and included in loss. Loss and ALAE is modeled on a per claim basis using a copula relationship, with marginals either input by the user or selected from a built-in distribution.
• The model only includes claims closed with payment (CWP). Claims closed without payment (CWOPs) can be handled separately and combined with the results from this analysis as a mixture, where the weight to a CWOP is the probability a claim closes without payment. Obviously, this only effects the "included" reinsurance structure.
• To input parameters:
  • For a lognormal enter mu, sigma, T
  • For a Pareto enter alpha, beta (alpha ~ 10,000-20,000 and beta ~ 1.5-2.5 is a reasonable range)
  • For a mixture of exponential distributions enter mean, wt, mean, wt, ...
  • For a sample enter Loss₁, Loss₂, ... (for example, a sample of losses)
  • For a density enter x₁, Pr(x₁), ... (for example, RMS losses)
• Inputs:
  
  1. Loss and ALAE distributions. If you select a built-in distribution (PO1, etc.) parameters are not required.
  2. Num L2: number of buckets to use to discretize the severity distributions will be 2^Num L2, so 8 corresponds to 256 buckets.
  3. Unit: the size of each bucket. Leave as zero and the model will compute an appropriate bucket size.
  4. Policy limit: the per occurrence indemnity limit.
  5. ALAE Treatment on the primary policy.
  6. Effective ALAE treatment: if ALAE is outside the primary policy, select an "effective" ALAE limit, reflecting the reality that the insurer can always tender policy limits.
  7. Reinsurance layer and Attachment
  8. Copula relationship between loss and ALAE, together with appropriate tau value.
• Model shows the various severity distributions, and the distribution and statistics for loss plus ALAE under the pro-rata and included assumptions. It also shows the distribution and statistics for a naive approach to included and pro rata. Simple Included simply scales the layer and attachment down by the average ALAE load, computes reinsured losses from the indemnity distribution and then adds the load back. Simple Pro Rata computes the reinsured layer from the indemnity distribution and then loads for ALAE.
• Frees Eg: This example sets up the model with parameters from Frees and Valdez. The model agreement with Frees and Valdez is not particularly reassuring; I am continuing to look into this. They price $750,000 xs 250,000 on a $1M policy limit at $16,762 per ground up claim, with a standard error of $597. This model prices the same layer at $13,750 using ALAE pro-rata in addition with an effective ALAE limit of $5M. This is more than three standard errors off F&V's number which is worrying.
• Add Noise Example: a very important example which illustrates how the sum of two correlated random variables can be approximated by the sum of the independent marginals plus "noise". Noise is a normally distributed random variable with mean zero and the appropriate variance. The variance can be computed from the desired correlation coefficient and the variances of the marginal distributions.

  It is interesting to compare the approximation to the actual sum for different copula relationships between the marginals. The Clayton copula (used by default), is pinched towards the origin: correlation is stronger for smaller values of the marginals (see Example 11). Comparing the actual sum of the marginals with the copula vs. the "add noise" approximation, shows that the Clayton slightly thicker in the left tail and thinner in the right tail, as one would expect.

  The Gumbel copula is almost the exact opposite of the Clayton: it exhibits closest correlation for larger values of the marginals. Again, looking closely, you can see that the actual sum is thicker in the right tails than the approximation, driven by the shape of the copula.

  This example highlights that knowing the marginals and correlation coefficient is not enough to determine the bivariate distribution; it is not even enough to determine the distribution of the sum of the marginals!

  To display the desired behaviour, the example uses marginal distributions which are more like aggregate distributions for a whole book than severity distributions; they are too symmetric for a severity distribution.

References

• Frees E. and E. Valdez, "Understanding Relationships Using Copulas" NAAJ Vol 2, No 1 pages 1-25 (1998)