Graduation and Mortality Projection
This domain covers turning noisy crude mortality rates into a smooth, usable mortality table, and then projecting that table forward in time.
Why and how to graduate
- Crude rates are erratic, especially at old ages where exposure is thin. Graduation removes sampling noise while preserving the underlying pattern so rates can be used for pricing and reserving.
- Parametric (ad-hoc) graduation fits a formula, for example Gompertz mu = B c^x, Makeham mu = A + B c^x, or the GM(r,s) family combining a polynomial in x with an exponential term.
- Standard table graduation scales a reference table, often with two parameters, so smoothness is inherited from the standard.
- Whittaker-Henderson smoothing minimises a weighted sum of squared deviations from the crude rates plus a roughness penalty on third or higher differences, with the smoothing constant trading fidelity against smoothness.
Testing a graduation
- Chi-square measures adherence to the data; the signs test, runs (grouping of signs) test and serial correlation tests detect systematic bias that a chi-square can miss.
- Degrees of freedom equal the number of age groups minus the number of parameters estimated.
Projection
- Lee-Carter: log m(x,t) = a_x + b_x k_t + error, with k_t typically a random walk with drift; a jump-off correction aligns the model with the latest observed rates.
- CBD M5 models log mortality as a linear function of age with a cohort term.
- In practice, projections converge from the current table to a long-term target over a chosen period.
Worked example
A graduation over six age groups estimates two parameters, so the chi-square has 4 degrees of freedom. The statistic is 8.2 against a 5% critical value of 9.488, so there is no evidence of poor adherence. That alone does not validate the graduation: an over-smoothed curve can also pass, so check the signs and runs tests and the plausibility of the resulting curve.
Traps that cost marks
- Over-smoothing that passes the chi-square but distorts the rates.
- Relying on one test; each test detects a different kind of failure.
- Projecting with Lee-Carter without addressing the jump-off bias.