Summary A data set from a Belgian telematics product aimed at young drivers is used to identify how car insurance premiums can be designed based on the telematics data collected by a black box installed in the vehicle. In traditional pricing models for car insurance, the premium depends on self‐reported rating variables (e.g. age and postal code) which capture characteristics of the policy(holder) and the insured vehicle and are often only indirectly related to the accident risk. Using telematics technology enables tailor‐made car insurance pricing based on the driving behaviour of the policyholder. We develop a statistical modelling approach using generalized additive models and compositional predictors to quantify and interpret the effect of telematics variables on the expected claim frequency. We find that such variables increase the predictive power and render the use of gender as a rating variable redundant.

We present a fully data driven strategy to incorporate continuous risk factors and geographical information in an insurance tariff. A framework is developed that aligns flexibility with the practical requirements of an insurance company, its policyholders and the regulator. Our strategy is illustrated with an example from property and casualty (P&C) insurance, namely a motor insurance case study. We start by fitting generalized additive models (GAMs) to the number of reported claims and their corresponding severity. These models allow for flexible statistical modeling in the presence of different types of risk factors: categorical, continuous and spatial risk factors. The goal is to bin the continuous and spatial risk factors such that categorical risk factors result which capture the effect of the covariate on the response in an accurate way, while being easy to use in a generalized linear model (GLM). This is in line with the requirement of an insurance company to construct a practical and interpretable tariff that can be explained easily to stakeholders. We propose to bin the spatial risk factor using Fisher's natural breaks algorithm and the continuous risk factors using evolutionary trees. GLMs are fitted to the claims data with the resulting categorical risk factors. We find that the resulting GLMs approximate the original GAMs closely, and lead to a very similar premium structure.

Pricing actuaries typically operate within the framework of generalized linear models (GLMs). With the upswing of data analytics, our study puts focus on machine learning methods to develop full tariff plans built from both the frequency and severity of claims. We adapt the loss functions used in the algorithms such that the specific characteristics of insurance data are carefully incorporated: highly unbalanced count data with excess zeros and varying exposure on the frequency side combined with scarce, but potentially long-tailed data on the severity side. A key requirement is the need for transparent and interpretable pricing models which are easily explainable to all stakeholders. We therefore focus on machine learning with decision trees: starting from simple regression trees, we work towards more advanced ensembles such as random forests and boosted trees. We show how to choose the optimal tuning parameters for these models in an elaborate cross-validation scheme, we present visualization tools to obtain insights from the resulting models and the economic value of these new modeling approaches is evaluated. Boosted trees outperform the classical GLMs, allowing the insurer to form profitable portfolios and to guard against potential adverse risk selection.

We discuss how to fit mixtures of Erlangs to censored and truncated data by iteratively using the EM algorithm. Mixtures of Erlangs form a very versatile, yet analytically tractable, class of distributions making them suitable for loss modeling purposes. The effectiveness of the proposed algorithm is demonstrated on simulated data as well as real data sets. KEYWORDSMixture of Erlang distributions with a common scale parameter, censoring, truncation, expectation-maximization algorithm, maximum likelihood.

In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of losses is essential. Modelling the whole range of the losses using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the loss distribution. A possible solution is to combine two distributions in a splicing model: a light-tailed distribution for the body which covers light and moderate losses, and a heavy-tailed distribution for the tail to capture large losses. We propose a splicing model with a mixed Erlang (ME) distribution for the body and a Pareto distribution for the tail. This combines the flexibility of the ME distribution with the ability of the Pareto distribution to model extreme values. We extend our splicing approach for censored and/or truncated data. Relevant examples of such data can be found in financial risk analysis. We illustrate the flexibility of this splicing model using practical examples from risk measurement.