13_AS_8_lec_a

To determine the premiums we a must not underestimate

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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 Chi-square (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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