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Unformatted text preview: likely to
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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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