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📋 '''Hazard rate''' is an actuarial and statistical measure that expresses the instantaneous probability of an event — such as death, [[Definition:Claim | claim]] occurrence, policy [[Definition:Lapse | lapse]], or system failure — happening at a specific point in time, given that the event has not yet occurred. In the insurance industry, it is a foundational concept in [[Definition:Survival analysis | survival analysis]] and underpins the construction of [[Definition:Mortality table | mortality tables]], [[Definition:Morbidity table | morbidity tables]], and various [[Definition:Loss model | loss models]] used by [[Definition:Actuary | actuaries]] to price products, calculate [[Definition:Reserves | reserves]], and assess [[Definition:Risk | risk]]. Unlike a simple probability that measures the chance of an event over a fixed period, the hazard rate captures how risk intensity evolves continuously over time, making it especially valuable for modeling time-to-event phenomena central to life, health, and general insurance.

⚙️ Mathematically, the hazard rate — also known as the hazard function, force of mortality (in life insurance contexts), or failure rate — is defined as the limit of the conditional probability of the event occurring in a small interval, divided by the length of that interval, as the interval approaches zero. Actuaries use it to move fluidly between related functions: the survival function, the cumulative distribution function, and the probability density function, all of which are derivable from one another given the hazard rate. In [[Definition:Life insurance | life insurance]] pricing, the force of mortality at each age drives the calculation of [[Definition:Net premium | net premiums]] and [[Definition:Policy reserve | policy reserves]]. In [[Definition:Non-life insurance | general insurance]], hazard rate models help estimate the timing of [[Definition:Loss | losses]] within a policy period and inform [[Definition:Incurred but not reported (IBNR) | IBNR]] reserve development patterns. Frameworks such as the Cox proportional hazards model allow actuaries and data scientists to incorporate covariates — age, health status, geographic risk factors, vehicle type — to produce granular, risk-differentiated hazard estimates.

💡 With the rise of [[Definition:Predictive analytics | predictive analytics]] and [[Definition:Machine learning | machine learning]] in insurance, hazard rate estimation has grown more sophisticated and more commercially consequential. Modern [[Definition:Insurtech | insurtech]] platforms apply survival modeling techniques to predict customer [[Definition:Churn | churn]], optimize [[Definition:Renewal | renewal]] strategies, and dynamically price coverage based on real-time risk profiles. In [[Definition:Catastrophe modeling | catastrophe modeling]], time-dependent hazard rates help quantify the probability of event occurrence within specific return periods, feeding directly into [[Definition:Reinsurance | reinsurance]] purchasing decisions and [[Definition:Capital adequacy | capital adequacy]] calculations. Across regulatory regimes — whether under [[Definition:Solvency II | Solvency II]] in Europe, the [[Definition:Risk-based capital (RBC) | RBC]] framework in the United States, or [[Definition:C-ROSS | C-ROSS]] in China — actuarial assumptions about hazard rates influence the determination of [[Definition:Technical provisions | technical provisions]] and required capital, making the accuracy of these estimates a matter of direct regulatory and financial significance.

'''Related concepts:'''
{{Div col|colwidth=20em}}
* [[Definition:Mortality table]]
* [[Definition:Survival analysis]]
* [[Definition:Actuary]]
* [[Definition:Force of mortality]]
* [[Definition:Predictive analytics]]
* [[Definition:Loss model]]
{{Div col end}}

Definition:Hazard rate - Revision history

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