Do multi-task pricing models agree with themselves?

In insurance pricing, pure premium is usually expressed as frequency multiplied by severity: the expected number of claims times the expected cost per claim. This identity underpins how rates are built, reviewed and explained to regulators.

Traditional pricing models satisfy the identity by construction, because pure premium is calculated from the frequency and severity models rather than fitted on its own. Many modern machine learning models instead predict frequency, severity and pure premium as separate outputs. Nothing in their training guarantees that these outputs are consistent with each other. When they are not, it is unclear which figure should be used as the price.

In a new paper, we measure this inconsistency on two public motor insurance datasets, from France (freMTPL2) and Belgium (beMTPL97), across nine model families. Our main findings are:

• Classical GLMs and a standard multi-task neural network show small inconsistencies.
• More flexible models, including gradient boosting and several actuarial neural network architectures, show substantially larger ones. Higher predictive accuracy did not imply better consistency.
• Calculating pure premium directly as frequency times severity removes the inconsistency entirely, with no loss of accuracy in our experiments. Where the model structure cannot be changed, adding a consistency penalty during training reduced the gap substantially without a measurable effect on accuracy.

We also document two practical issues. First, a consistency measure from the machine learning literature can give misleading results when applied directly to insurance data, because most policies have no claims. We describe the issue and a correction. Second, in freMTPL2, a widely used benchmark dataset, about a quarter of policies with a reported claim have no recorded claim amount.

Why this matters

Insurers are adopting more flexible models, including tabular foundation models, for pricing. These models can be more accurate, but accuracy measures do not reveal whether a model’s outputs are mutually consistent. An inconsistent model makes prices harder to explain, review and defend. The checks in this paper are simple to run and could be part of standard model validation.

The results are based on two public datasets and should be read with that scope in mind.

Comments from practitioners and researchers are welcome.