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📐 '''Central limit theorem''' is a foundational result in probability theory stating that the sum (or average) of a large number of independent, identically distributed random variables tends toward a normal distribution, regardless of the underlying distribution of the individual variables — a principle that underpins much of the [[Definition:Actuarial science | actuarial]] and statistical methodology used throughout the insurance industry. Actuaries invoke the central limit theorem when modeling aggregate [[Definition:Insurance claim | claims]] costs, estimating [[Definition:Loss reserve | reserves]], and constructing confidence intervals around expected losses, because it provides the mathematical justification for treating portfolio-level outcomes as approximately normally distributed even when individual claim amounts follow highly skewed distributions.

🔬 In practice, the theorem works as follows: when an insurer writes a sufficiently large and homogeneous book of business — say, thousands of [[Definition:Motor insurance | motor]] or [[Definition:Homeowners insurance | homeowners]] policies — the average claim cost per policy converges toward a predictable value, and the distribution of total claims around that value becomes bell-shaped. This allows actuaries to apply normal-distribution techniques to estimate the probability that aggregate losses will fall within specified ranges, to set [[Definition:Premium | premiums]] with defined confidence levels, and to determine the amount of [[Definition:Capital adequacy | capital]] needed to absorb adverse deviation. The theorem does have important limitations in insurance: it assumes independence among risks, which breaks down during [[Definition:Catastrophe | catastrophic events]] where losses are correlated; it requires a large sample size, which may not hold for low-frequency, high-severity lines like [[Definition:Marine insurance | marine hull]] or [[Definition:Directors and officers liability insurance | D&O liability]]; and it presumes finite variance, a condition violated by certain heavy-tailed claim distributions. Actuaries working in these domains rely on alternative distributional models — [[Definition:Extreme value theory | extreme value theory]], for instance — rather than leaning on central limit theorem approximations.

💡 Despite its limitations, the central limit theorem remains one of the most practically consequential theoretical results in insurance mathematics. It is the reason that diversification works: pooling a large number of independent risks reduces relative volatility, making outcomes more predictable and enabling insurers to charge stable [[Definition:Premium rate | premium rates]]. Regulatory [[Definition:Capital model | capital models]], including the standard formulas under [[Definition:Solvency II | Solvency II]] and the [[Definition:Risk-based capital (RBC) | RBC framework]] in the United States, implicitly or explicitly rely on normality assumptions for aggregated risk modules — assumptions grounded in the central limit theorem. For anyone building [[Definition:Predictive analytics | predictive models]] or [[Definition:Internal model | internal capital models]] in the insurance sector, understanding when the theorem holds, when it breaks down, and what alternatives exist is a matter of technical rigor with direct financial consequences.

'''Related concepts:'''
{{Div col|colwidth=20em}}
* [[Definition:Actuarial science]]
* [[Definition:Law of large numbers]]
* [[Definition:Extreme value theory]]
* [[Definition:Loss distribution]]
* [[Definition:Capital model]]
* [[Definition:Risk pooling]]
{{Div col end}}

Definition:Central limit theorem - Revision history

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