Unformatted text preview: a
beneﬁt in respect of policyholders who die within the speciﬁed
term.
To determine the premiums, we
A. must not underestimate mortality
B. must not overestimate mortality 2 Life annuity: the insurance company agrees to provide a
regular income for the remaining lifetime of the individual who
buys an annuity.
To determine the price of the annuity, we
A. must not underestimate mortality 8/72 B. must not overestimate mortality Actuarial Statistics – Module 8: Method of Graduation
Testing smoothness 1 Introduction
2 Testing smoothness
3 Statistical tests
Preliminaries
Chisquare (χ2 ) test
Standardised Deviations Test
Signs test
Cumulative Deviations Test
Grouping of Signs Test
Serial Correlations Test
4 Methods of graduation
Preliminaries
Graduation by Parametric Formula
Graduation by Reference to a Standard Table
Graphical graduation
Statistical Tests
The Eﬀect of Duplicate policies
9/72 Actuarial Statistics – Module 8: Method of Graduation
Testing smoothness Notation
qx is the true value
qx is the numerical value of the crude (observed) estimate
ˆ
qx is the random variable representing the crude (observed)
˜
estimator
o q x is the graduated value
S
qx is taken from an existing standard table 9/72 Actuarial Statistics – Module 8: Method of Graduation
Testing smoothness Testing Smoothness: a check for undergraduation
Smoothness tested using ﬁnite diﬀerences of the data
First diﬀerences o o o ∆q x = q x +1 − q x
Second diﬀerences
o o o ∆2 q x = ∆q x +1 − ∆q x
Third diﬀerences
o o o ∆3 q x = ∆2 q x +1 − ∆2 q x
Third diﬀerences measure the change in curvature The criterion is
absolute value of third diﬀerences are relatively small compared
o
o
1
to the graduated rates themselves (e.g. q x or µx + 2 )
10/72 Note: Some methods of graduation of rates automatically
ensure smoothness to a speciﬁed level Actuarial Statistics – Module 8: Method of Graduation
Testing smoothness Examples 1 x
40
41
42
43
44 o qx
0.001056
0.0011...
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