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13_AS_7_lec_a - Actuarial Statistics Module 7 Exposed to...

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Actuarial Statistics – Module 7: Exposed to risk Actuarial Statistics Benjamin Avanzi c University of New South Wales (2013) School of Risk and Actuarial Studies [email protected] Module 7: Exposed to risk 1/35
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Actuarial Statistics – Module 7: Exposed to risk Plan 1 Introduction Central vs Initial Exposed to Risk Complete data Incomplete data 2 Census approximations Introduction “Calendar Year” rate interval “Policy Year” rate interval 3 Examples Definition of x Trapezium approximation Estimation Principle of correspondence Calendar year rate intervals Policy year rate intervals 2/35
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Actuarial Statistics – Module 7: Exposed to risk Introduction Central vs Initial Exposed to Risk 1 Introduction Central vs Initial Exposed to Risk Complete data Incomplete data 2 Census approximations Introduction “Calendar Year” rate interval “Policy Year” rate interval 3 Examples Definition of x Trapezium approximation Estimation Principle of correspondence Calendar year rate intervals Policy year rate intervals 3/35
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Actuarial Statistics – Module 7: Exposed to risk Introduction Central vs Initial Exposed to Risk Central versus Initial Exposed to Risk Central exposed to risk (waiting times) are natural and intrinsically observable—just record the time spent under observation by each life Initial exposed to risk is more complicated unless we can use the idealized binomial model in which N lives are observed for a whole year without censoring Central exposed to risk is generally preferable The initial exposed to risk can be reasonably approximated by E x E c x + 1 2 d x . In this module we focus on the estimation of CENTRAL Exposed to Risk. 3/35
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Actuarial Statistics – Module 7: Exposed to risk Introduction Complete data 1 Introduction Central vs Initial Exposed to Risk Complete data Incomplete data 2 Census approximations Introduction “Calendar Year” rate interval “Policy Year” rate interval 3 Examples Definition of x Trapezium approximation Estimation Principle of correspondence Calendar year rate intervals Policy year rate intervals 4/35
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Actuarial Statistics – Module 7: Exposed to risk Introduction Complete data With complete data If we know dates of birth dates of entry into observation dates and reason (e.g. death) of exit form observation then we can calculate E c x exactly (see module 6). Note: this allows for all movements into (increments) and out (decrements) of the study for all causes E c x does not depend on the decrement under study (death, lapse, surrender, sickness, . . . ) exposure is measured in years conventions: either the day of entry or the day of exit counts (and we usually assume 365.25 years in a year) 4/35
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Actuarial Statistics – Module 7: Exposed to risk Introduction Incomplete data 1 Introduction Central vs Initial Exposed to Risk Complete data Incomplete data 2 Census approximations Introduction “Calendar Year” rate interval “Policy Year” rate interval 3 Examples Definition of x Trapezium approximation Estimation Principle of correspondence Calendar year rate intervals Policy year rate intervals 5/35
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