Graduation is the process of smoothing the crude

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Unformatted text preview: likely to ˆ ˆ progress smoothly as age varies. Graduation is the process of “smoothing” the crude estimated rates. In this module, we will address three questions: Why do we want smoothed estimates? How to carry out the graduation? how tho decide that a given attempt to graduate the crude estimates is satisfactory? 5/72 Actuarial Statistics – Module 8: Method of Graduation Introduction Reasons for graduation It is intuitively sensible to think that the quantities qx or µx should be a smooth functions of age x . There is some evidence from large investigations to support this. A crude estimate at any age x also carries information about the values at adjacent ages. We need to incorporate the information from adjacent ages into the estimate (reduces the sampling errors). A practical reason: it is very desirable that financial quantities (eg premiums) computed from the life table progress smoothly with age, since irregularities (jumps or other anomalies) are hard to justify in practice. The aims of graduation: 6/72 to produce a smooth set of rates to remove random sampling errors to use the information available from adjacent ages Actuarial Statistics – Module 8: Method of Graduation Introduction Desirable features of a Graduation o o Smoothed or graduated rates (q x or µx + 1 ) need to satisfy: 2 smoothness (sensible, required for premiums and reserves) goodness of fit (adherence to data) suitability for the purpose in hand (see examples next) Smoothness and adherence are usually conflicting If rates are “too smooth (overgraduated), then the rates usually show little adherence to the data If rates follow the observed data too closely then the curve of rates has inadequate smoothness (undergraduated) Statistical tests are available to test smoothness and adherence to data The “art” of graduation lies in finding a satisfactory compromise. 7/72 Actuarial Statistics Module 8: Method of Graduation Introduction Examples 1 Term life insurance policies: the insurance company will pay...
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This document was uploaded on 04/03/2014.

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