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Reflecting Policy Limits in the FFT Algorithm

Reflecting Policy Limits in the FFT Algorithm. James C. Sandor, FCAS, MAAA American Re-Insurance 2003 CAS Seminar on Reinsurance. Agenda. BRIEFLY review FFT algorithm Describe modification to reflect policy limits An example The algorithm in practice Questions. A Brief Review.

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Reflecting Policy Limits in the FFT Algorithm

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  1. Reflecting Policy Limits in the FFT Algorithm James C. Sandor, FCAS, MAAA American Re-Insurance 2003 CAS Seminar on Reinsurance

  2. Agenda • BRIEFLY review FFT algorithm • Describe modification to reflect policy limits • An example • The algorithm in practice • Questions

  3. A Brief Review • Fast Fourier Transform (FFT) Algorithm • Collective Risk Model • Frequency/Severity Assumptions • Uses the Following Relationship • φZ(t) = PN(φX(t)) • φX is the characteristic function of the severity • PNis the PGF of the frequency • If we know these, we can calculate the characteristic function of the aggregate distribution, φZ(t) • FFT produces approximation of φX • IFFT gets us fz(z) from φZ

  4. A Couple Details • The PGFsfor many common frequency distributions can be found in Loss Models • The severity must be discretized before we can use the FFT to get the characteristic function • Insert favorite method here (rounding, mean-preserving, etc.)

  5. Basic Assumptions Underlying Method • Collective Risk Model • One frequency distribution • One severity distribution • Assumes layer is fully exposed • Layer can be ground-up

  6. Modification • Still one severity distribution • Weighted average of severity distributions at different limits • Weighted by underlying expected frequency

  7. Practical Details • Assumptions • Expected Aggregate Loss • aka the 1st moment of the Aggregate Distribution • Severity Distribution • Frequency Moments relationship • Variance to Mean ratio

  8. Practical Details • Expected Total Loss& • Severity Distribution • Expected Frequency(Both Ground-Up and Layer)Policy Limit Weights = Ground-Up Expected Frequency

  9. An Example Severity Curve: ISO Commercial Auto Light/Medium (SG II)

  10. Severity Distribution • Discretize Severity Distribution(s) • One distribution for each Policy Limit • Unconditional Layer Severities • If Retention > 0 then P(X=0)>0 • Distributions reflect Policy Limit and Layer • Final severity distribution = Weighted average of Individual Severities • Weights are Ground-up frequency

  11. An Example Layer is 750,000 xs 250,000

  12. Why is this correct? • Takes into account probability of hitting policy • Assumes Probability of hitting policy  Expected Frequency • Assumes size of loss and policy limit are independent • That assumption can be easily modified • Use different severity for each limit

  13. Finishing Up • Weighted average severity used as input into FFT algorithm • Taking into account policy limits can have a non-trivial impact on the Aggregate distribution

  14. More Examples • Aggregate Modeling tool in Excel/MATLAB • Stochastic Reinsurance Pricing Model • Also in Excel/MATLAB

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