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🔢 '''Fast Fourier transform''' is a computational algorithm that dramatically accelerates the calculation of convolutions and probability distributions, making it an essential tool in [[Definition:Actuarial science | actuarial science]] and [[Definition:Catastrophe modeling | catastrophe modeling]] where insurers must aggregate large numbers of individual [[Definition:Loss distribution | loss distributions]] into a single portfolio-level view of risk. While the Fourier transform itself is a mathematical operation that converts functions from the time or monetary-loss domain into the frequency domain, the "fast" variant — developed in its modern form by Cooley and Tukey in 1965 — reduces the computational complexity from a level that would be impractical for real-world insurance portfolios to one that modern hardware can handle in seconds.

⚙️ Actuaries apply the fast Fourier transform most commonly when computing the [[Definition:Aggregate loss distribution | aggregate loss distribution]] of an insurance portfolio. The classical collective risk model represents total claims as the sum of a random number of individual losses, each drawn from a [[Definition:Severity distribution | severity distribution]]. Calculating the resulting aggregate distribution directly — through repeated convolution — becomes computationally prohibitive when the expected claim count is large, as is typical in [[Definition:Motor insurance | motor]], [[Definition:Health insurance | health]], or [[Definition:Workers compensation insurance | workers' compensation]] books. The fast Fourier transform sidesteps this by converting the severity distribution to the frequency domain, raising it to the appropriate power using the [[Definition:Probability generating function | probability generating function]] of the claim count distribution, and then inverting back to obtain the aggregate distribution. This approach underpins many [[Definition:Internal model | internal capital models]] used under [[Definition:Solvency II | Solvency II]] and similar regimes, as well as [[Definition:Reinsurance | reinsurance]] pricing engines that need to evaluate the impact of excess-of-loss or stop-loss structures on a ceding company's retained risk.

💡 The practical significance for insurers lies in speed and precision. When a [[Definition:Reinsurer | reinsurer]] prices a treaty or a primary carrier evaluates its [[Definition:Capital adequacy | capital adequacy]], the ability to compute tail probabilities of aggregate losses accurately — and to do so thousands of times within a stochastic simulation — directly affects pricing quality and [[Definition:Reserve | reserve]] adequacy. Without fast Fourier transform techniques, many of the real-time pricing platforms used by modern [[Definition:Insurtech | insurtech]] firms and [[Definition:Managing general agent (MGA) | MGAs]] could not deliver the responsiveness that brokers and policyholders expect. The algorithm also supports [[Definition:Enterprise risk management (ERM) | enterprise risk management]] functions by enabling rapid sensitivity analysis: actuaries can adjust assumptions about frequency or severity and instantly observe the effect on the tail of the aggregate distribution, supporting more informed decision-making at both the underwriting and board levels.

'''Related concepts:'''
{{Div col|colwidth=20em}}
* [[Definition:Aggregate loss distribution]]
* [[Definition:Actuarial science]]
* [[Definition:Catastrophe modeling]]
* [[Definition:Loss distribution]]
* [[Definition:Internal model]]
* [[Definition:Monte Carlo simulation]]
{{Div col end}}

Definition:Fast Fourier transform - Revision history

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