The tail factor: A source of uncertainty and complexity in loss reserving

The tail factor is a key concept in actuarial science for estimating the reserve for long-tailed claims, such as workers’ compensation, asbestos, or environmental claims. These claims have a long settlement period, which means that the ultimate loss may not be known for many years after the occurrence of the event. The tail factor reflects the uncertainty and variability of the future payments for these claims, which may depend on factors such as legal, regulatory, or social changes.

There are different methods for modelling the tail factor within the loss reserving process, each with its own advantages and limitations. The choice of the method depends on the data quality, availability, and reliability, as well as the context and purpose of the reserve estimation. Some of the common methods are:

?? Bondy-type methods: These are algebraic methods that focus on the relationships between paid and incurred loss. They assume that the tail factor can be expressed as a function of the paid-to-incurred ratio at a certain development period.

?? Methods based on use of benchmark data: These are methods that use external data sources, such as industry data or reinsurance data, to estimate the tail factor. They assume that the tail factor for a given line of business or type of claim is similar to the tail factor observed in the benchmark data.

?? Curve fitting methods: These are methods that fit a mathematical function, such as an exponential or a power function, to the observed loss development factors. They assume that the tail factor can be extrapolated from the fitted curve.

?? Methods based on remaining open counts: These are methods that use the number of open claims at a certain development period to estimate the tail factor. They assume that the tail factor is proportional to the remaining open counts.

?? Methods based on peculiarities of the remaining open claims: These are methods that use the characteristics of the remaining open claims, such as the type, size, or age of the claims, to estimate the tail factor. They assume that the tail factor is influenced by the distribution and behaviour of the remaining open claims.

In addition to these traditional methods, there are also some AI-based models that can be used to model the tail factor, such as:

?? Neural network methods: These are methods that use artificial neural networks, which are computational models that mimic the structure and function of biological neurons, to learn the patterns and relationships between the input and output variables. They assume that the tail factor can be approximated by a nonlinear function of the historical loss development data and other relevant factors.

?? Bayesian methods: These are methods that use Bayesian inference, which is a statistical technique that updates the probability of a hypothesis based on new evidence, to estimate the tail factor. They assume that the tail factor has a prior distribution that reflects the prior knowledge and beliefs, and that the posterior distribution of the tail factor can be updated by incorporating the observed loss development data and other relevant information.

?? Machine learning methods: These are methods that use machine learning algorithms, which are computational methods that learn from data and improve their performance, to estimate the tail factor. They assume that the tail factor can be derived from the features and labels of the training data, and that the model can be validated and tested on the unseen data.

These AI-based models have the potential to capture the complexity and uncertainty of the tail factor, and to provide more accurate and robust estimates. However, they also have some challenges and limitations, such as: (i) The need for large and reliable data sets to train and test the models, and to avoid overfitting or underfitting problems; (ii) The difficulty of interpreting and explaining the results and the logic of the models, especially for black-box models such as neural networks; (iii) The sensitivity and variability of the results to the choice and tuning of the parameters and hyperparameters of the models; (iv) The ethical and regulatory issues related to the transparency, accountability, and fairness of the models. Therefore, it is also important to use professional judgment and consider the context and purpose of the reserve estimation when choosing and applying an AI-based model for modelling the tail factor within the loss reserving.

The tail factor is a crucial but elusive concept in actuarial science, as it involves estimating the unknown and uncertain future payments for long-tailed claims. There is no single or simple method for modelling the tail factor, and each method has its own strengths and weaknesses. Therefore, to obtain a reliable and robust estimate of the tail factor, (re)insurers need to use a combination of complementary methods, and rely on the professional judgment of the actuaries, who can consider the context and purpose of the reserve estimation, and the quality and availability of the data.