Using piecewise Pareto distribution could help create better outcomes in reinsurance treaty pricing, says Saliya Jinadasa
Typically, pricing of non-proportional treaty layers in property and casualty reinsurance is performed with well-known frequency and severity-based approaches, using historical experience. For severity, it is a common practice to fit a single statistical distribution to historical claims. However, this can be challenging when data is sparse or has outliers. A piecewise Pareto distribution helps overcome some of the difficulties in distribution fitting, and can often lead to better outcomes for reinsurers.
The traditional method
Underwriters and pricing actuaries typically fit a single statistical distribution to historical individual claims (adjusted for inflation and structural changes). A best fit is chosen using one or a selection of statistical criteria such as least squares error, Kolmogorov or Anderson value.
If claims are not spread out reasonably well – for example, claims with very few outliers – then distribution fitting becomes difficult. The difficulty stems from trying to fit a single distribution over a wider range, often leading to overestimating or underestimating, specifically at the tail end of the distribution. In some cases, we only get claims closer to the selected deductible, leaving a wider range without any claims.
Pareto distribution and its popularity
The Pareto distribution is one of the most commonly used statistical distributions for fitting large claims. It has two parameters, scale (or threshold) and shape (often denoted as alpha, α). Its popularity and widespread use in the actuarial field, especially within pricing, can be attributed to several reasons:
- Parameter invariance is arguably the most important feature, which implies that as long as we are in the tail, the same alpha parameter applies – regardless of the threshold
- The ease of deriving parameters using empirical data via techniques such as method of moments and maximum likelihood estimation
- It can be used to represent empirical data fairly well over a wide range of values Availability of benchmarks for alpha values to help underwriters and actuaries choose for pricing of different classes of business
- It makes it easier to explain parameter selections and results.
However, despite all the advantages, using one Pareto distribution to fit a range of losses leads to the issues mentioned previously.
Piecewise Pareto distribution
The basic idea of piecewise Pareto distribution is to divide the layer that is being priced into equal chunks called ‘priorities’ (other scales such as log basis can be used) and fit Pareto distributions to layer segments between priorities. We can determine the number of claims exceeding each priority for each historical year, and thus arrive at an average exposure-adjusted exceeding frequency for each priority. There is a link between two subsequent priorities and their exceeding frequencies, providing an alpha parameter value to represent losses between the two priorities. The following formula shows the link and we can use it to derive Alpha parameter values.
Typically, at the lower end of a layer, we find credible enough historical experience to determine reasonable alpha parameter values, whereas at the higher end of a layer this may not be the case. In this case, we can extrapolate exceeding frequencies, together with appropriate alpha parameter values from lower priorities, to determine exceeding frequencies at higher priorities, still using the formula above. We tend to use the same alpha parameter value probably given by a benchmark for higher priorities by taking advantage of the parameter invariance feature of Pareto distribution.
Figure 2 shows one example in which, for a layer, two weight assignments have been set up for historical treaty years in order to derive an overall average exceeding frequency for each priority and a user selection of exceeding frequencies. (In this example, weight assignment W2 discards the oldest treaty year and assigns equal weights to the rest of historical treaty years. The weight assignment W1 assigns equal weights to all historical treaty years.) The straight-lined segment from mid priorities to the end of the layer shows that the same alpha parameter value has been selected. The user selection of exceeding frequencies is more pessimistic for mid priorities, whereas it is more optimistic for the last two priorities.
Once exceedance frequencies are selected and corresponding alpha parameter values are determined, the expected loss (EL) or severity between two subsequent priorities n and n+1 can be determined with the following formula.
Finally, the pure premium or risk premium between priorities n and n+1 can easily be worked out with the general frequency and severity based formula: Risk premium = exceeding frequency at priorityn * ELn
The above steps should be repeated for all priorities, and the risk premium for an excess layer is then given by summing up the risk premium for each priority within the layer.
We should compare the risk premium using the traditional method of fitting a single distribution to gauge how far the risk premium deviate from that of the piecewise Pareto distribution-based method. Such a comparison can be useful for a reinsurer, particularly to stay competitive in a quoting market.
“If claims are not spread out reasonably well – for example, claims with a very few outliers –then distribution fitting becomes difficult”
Advantages of the approach
There are several advantages to treaty pricing with piecewise Pareto distribution. These include:
- It provides great flexibility in pricing, especially for higher layers where experience is sparse
- You can take advantage of features such as parameter invariability of Pareto distribution for extrapolation
- It is easier to calculate expected values and variances
- It can easily be programmed into a function, or you make it formula-based in Excel
- You can take advantage of benchmark alphas for pricing of higher layers.
However, there are certain disadvantages:
- The threshold should be above zero. It is not possible to use Pareto distribution for some proportional, stop loss and multiline treaties, for which losses starting from zero is important
- There are real-life loss distributions in which medium-sized losses are more probable than small or large losses. Distribution functions of such distributions have a turning or inflection point. Pareto distribution does not contain an inflection point.
The author would like to thank Mudit Gupta and Frederic Boulliung for their invaluable feedback to improve the content of this article.
Saliya Jinadasa heads analytics services at Asia Reinsurance Brokers in Singapore
