Machine learning (ML) systems increasingly make consequential decisions about individuals, including credit, hiring, and insurance. Research on ML-driven insurance pricing has asked how the pricing mechanism itself can be made accountable. Yet it remains unclear how an individual policyholder's algorithmically determined premium should be explained to them. We argue that a useful explanation must be valid, actionable, and reliable. A valid recommendation lowers the premium according to the model. An actionable recommendation changes only characteristics the policyholder can control. A reliable recommendation has an actual outcome that is likely to meet the target, even in sparsely observed regions of the feature space where the model is uncertain. We formalize these three requirements as axioms and develop a method that turns an algorithmic premium into recommendations that meet them. The method meets the third requirement with a Conformal Prediction constraint, which screens out recommendations that the model is too uncertain about and thereby bounds the rate of unreliable recommendations. We apply four Counterfactual Explanation techniques to 57,832 real Florida flood-insurance claims and 98,058 synthetic driver-telematics policies. We adapt these techniques for insurance applications, so that their recommendations change only controllable characteristics and resemble policyholders observed in the data. In our illustrations, the constrained recommendations for high-risk policyholders are both more reliable and associated with larger predicted risk reductions than the unconstrained ones. These results suggest that insurers can give policyholders explanations that are both actionable and trustworthy. The work contributes to research on algorithmic accountability and offers implications for insurers and regulators.

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