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📐 '''Link function''' is a mathematical component within a generalized linear model (GLM) that connects the expected value of a response variable to the linear combination of predictor variables — and in insurance, it serves as a foundational element in [[Definition:Predictive modeling | predictive modeling]] for [[Definition:Pricing | pricing]], [[Definition:Reserving | reserving]], and [[Definition:Risk classification | risk classification]]. Because insurance outcomes such as [[Definition:Claim frequency | claim frequency]] and [[Definition:Claim severity | claim severity]] rarely follow a simple normal distribution, actuaries cannot rely on ordinary linear regression. The link function transforms the expected outcome onto a scale where a linear relationship with rating factors becomes appropriate. For example, when modeling claim counts — which are typically Poisson-distributed — the natural logarithm serves as the link function, ensuring that predicted frequencies remain positive regardless of the combination of risk factors applied. Selecting the right link function is not merely a statistical formality; it shapes how each [[Definition:Rating factor | rating factor]] interacts multiplicatively or additively with the predicted outcome, directly influencing the [[Definition:Premium | premium]] each policyholder is charged.

⚙️ In practice, the link function operates as part of the broader [[Definition:Generalized linear model (GLM) | GLM]] framework that has become the standard actuarial tool for [[Definition:Ratemaking | ratemaking]] across virtually every major insurance market. When an [[Definition:Actuary | actuary]] fits a GLM to historical [[Definition:Loss experience | loss experience]], the model estimates coefficients for variables such as driver age, vehicle type, or geographic zone. The link function determines how these coefficients translate back into predicted values on the original scale. A log link, for instance, means that each coefficient acts as a multiplicative relativity — a structure that aligns naturally with how insurers build [[Definition:Rating algorithm | rating algorithms]] by layering base rates with adjustment factors. Other link functions exist: the logit link is standard for binary outcomes like [[Definition:Propensity model | propensity to lapse]] or [[Definition:Fraud detection | fraud]] indicators, while the inverse link sometimes appears in gamma-distributed severity models. Regulatory environments such as the European Union's [[Definition:Solvency II | Solvency II]] regime and the [[Definition:National Association of Insurance Commissioners (NAIC) | NAIC]]-guided frameworks in the United States increasingly expect insurers to demonstrate that their pricing models are statistically sound, and the choice and justification of link function is a key element of that demonstration during [[Definition:Rate filing | rate filing]] reviews and internal model approvals.

💡 Getting the link function right has tangible consequences for an insurer's competitiveness and financial health. A misspecified link function can distort relativities across risk segments — undercharging high-risk policyholders while overcharging low-risk ones — leading to [[Definition:Adverse selection | adverse selection]] that erodes the [[Definition:Loss ratio | loss ratio]] over time. As the insurance industry moves toward more granular, data-driven pricing — particularly in personal lines markets across North America, Europe, and parts of Asia — the technical rigor behind model choices like the link function increasingly differentiates sophisticated carriers and [[Definition:Managing general agent (MGA) | MGAs]] from those relying on cruder approaches. In the [[Definition:Insurtech | insurtech]] space, where startups often build pricing engines from scratch, understanding link functions is essential for translating machine-learning insights back into interpretable, regulatorily compliant rating structures. Far from being an abstract statistical detail, the link function sits at the intersection of actuarial science, regulatory compliance, and market strategy.

'''Related concepts:'''
{{Div col|colwidth=20em}}
* [[Definition:Generalized linear model (GLM)]]
* [[Definition:Predictive modeling]]
* [[Definition:Rating factor]]
* [[Definition:Ratemaking]]
* [[Definition:Risk classification]]
* [[Definition:Claim frequency]]
{{Div col end}}

Definition:Link function - Revision history

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