Neural networks for insurance pricing with frequency and severity data: a benchmark study from data preprocessing to technical tariff

TL;DR

The paper analyzes frequency-severity pricing for non-life insurance using tabular data, benchmarking GLMs, GBMs, FFNNs, and Combined Actuarial Neural Networks (CANNs) on four motor TPL datasets. It introduces autoencoder-based embeddings for categoricals, and a surrogate GLM approach to translate neural insights into interpretable tariffs. Through a rigorous cross-validated framework, it demonstrates that combining GBM with neural corrections (CANN) often yields the best frequency performance, while severity gains are more dataset-dependent; surrogate GLMs can retain predictive power with enhanced interpretability. The study advances practical pricing by linking preprocessing, model evaluation (including DM tests and Murphy diagrams), and actionable tariff construction via global surrogates and risk profiling. These insights support deploying sophisticated DL models in pricing with robust safeguards for calibration, dominance, and interpretability.

Abstract

Insurers usually turn to generalized linear models for modeling claim frequency and severity data. Due to their success in other fields, machine learning techniques are gaining popularity within the actuarial toolbox. Our paper contributes to the literature on frequency-severity insurance pricing with machine learning via deep learning structures. We present a benchmark study on four insurance data sets with frequency and severity targets in the presence of multiple types of input features. We compare in detail the performance of: a generalized linear model on binned input data, a gradient-boosted tree model, a feed-forward neural network (FFNN), and the combined actuarial neural network (CANN). The CANNs combine a baseline prediction established with a GLM and GBM, respectively, with a neural network correction. We explain the data preprocessing steps with specific focus on the multiple types of input features typically present in tabular insurance data sets, such as postal codes, numeric and categorical covariates. Autoencoders are used to embed the categorical variables into the neural network, and we explore their potential advantages in a frequency-severity setting. Model performance is evaluated not only on out-of-sample deviance but also using statistical and calibration performance criteria and managerial tools to get more nuanced insights. Finally, we construct global surrogate models for the neural nets' frequency and severity models. These surrogates enable the translation of the essential insights captured by the FFNNs or CANNs to GLMs. As such, a technical tariff table results that can easily be deployed in practice.

Figures (30)

• Figure 1: Structure of a feed-forward neural network with $p$-dimensional input layer, hidden layers $\boldsymbol{z}^{(1)}, \ldots,\boldsymbol{z}^{(M)}$, with $q_1,\ldots,q_M$ nodes, respectively. The network has a single output node $\hat{y}$.
• Figure 2: Structure of a Combined Actuarial Neural Network (CANN). The initial model prediction $\hat{y}^{\text{IN}}$ is connected via a skip-connection to the output node of the FFNN.
• Figure 3: Our proposed network structure combines the autoencoder embedding technique from delong2021 and the CANN structure from Schelldorfer2019.
• Figure 4: Example of random grid search with two tuning parameters $t_1$ and $t_2$. The search space $\mathcal{S} = \left[t_{1,\text{min}},t_{1,\text{max}}\right]\times\left[t_{2,\text{min}},t_{2,\text{max}}\right]$ is shown in the figure by the dotted square. The random grid $\mathcal{R}$ consists of nine randomly drawn $s_1,\ldots,s_9$ from $\mathcal{S}$. The optimal $s^{*}\in\mathcal{R}$ is then selected via a cross-validation scheme.
• Figure 5: Representation of the $6$ times $5$-fold cross-validation scheme; figure from Henckaerts2021.