Calendar year rate interval is a rate interval that

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Unformatted text preview: dar year rate intervals Policy year rate intervals 10/35 Actuarial Statistics – Module 7: Exposed to risk Census approximations “Calendar Year” rate interval Calendar Year Rate Intervals A calendar year rate interval is a rate interval that starts on a fixed date in the calendar year. Everyone’s age label changes on that fixed date. Example: age on the birthday in the same calendar year. Investigation over whole year 2008. Life A was born in August 1950 and died on 1 Jan 2008. The death time 1 Jan 2008 and the 58th birthday is in year 2008, so the age label for the death is 58. Life A is included in d58 . Life B was born on 1 Jan 1980 and life C was born 31 Dec 1980. For any time t in 2008, both lives B and C contribute to P (28, t ) if they are alive at time t . This is because t here is in the same calendar year as their 28th birthday. 10/35 Actuarial Statistics – Module 7: Exposed to risk Census approximations “Calendar Year” rate interval q vs µ At the start of the rate interval (1 Jan) for age label x , actual ages range from x − 1 exact (those lives with x th birthdays on 31 December this year) to x exact (those lives with x th birthdays on 1 Jan this year). Hence, if we assume that birthdays are uniformly distributed over the calendar year, the average age at the start of rate interval (1 1 Jan), is x − 2 . This means that in this context: q estimate qx − 1 ˆ 2 µ estimate µ(x − 1 )+ 1 ˆ 2 11/35 2 Actuarial Statistics – Module 7: Exposed to risk Census approximations “Policy Year” rate interval 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 12/35 Actuarial Statistics – Module 7: Exposed to risk Census approximations “Policy Year” rate interval Policy year rate interval I Policy duration - length of time since policy purchased Policy anniversary - integer policy duration (date policy taken out) For example, if an individual bought an endowment policy on 1 August 2006, the third policy anniversary would be 1 August 2009. A policy year rate interval: a rate interval determined by the policy anniversary. Age label changes on a day that is fixed relative to this policy anniversary. 12/35 Actuarial Statistics – Module 7: Exposed to risk Census approximations “Policy Year” rate interval Policy year rate interval II The change in age label need not occur on the policy anniversary. For example, a policy was bought on 1 August 2006. Age definition is age last birthday at the nearest policy anniversary. Age label changes when crossing midway between anniversaries. Example for deaths classified by policy year: dx =no. of deaths, age x last birthday at the policy anniversary preceding (or coincident with) the date of death over the investigation period The classification of deaths by policy year might arise if all we know is age last birthday (or next birthday, etc) at time the policy was purchased the policy duration at death 13/35 Actuarial Statistics – Module 7: Exposed to risk Census approximations “Policy Year” rate interval q vs µ Recall q and µ estimate qy and µy + 1 , respectively, where y is 2 the average age at the start of the rate interval For example, if deaths are “age x last birthday a...
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