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Actuarial Science - General Maths & Stats > Exposed to Risk Calculations > Flashcards

Flashcards in Exposed to Risk Calculations Deck (4):
1

Age x last birthday 1 June > Age x last birthday 30 June

if Px,t is the population aged x last birthday on 1st January,

and P*x,t is the population aged x last birthday on 30th June, 

give an approximation for Px,t in terms of P*x,t

if Px,t is the population aged x last birthday on 1st January,

and P*x,t is the population aged x last birthday on 30th June, 

Px,t = 1/2(P*x-1,t + P*x,t)

2

Age x nearest birthday 1st Jan > Age x last birthday 1st Jan

if Px,t is the population aged x last birthday on 1st January,

and P*x,t is the population aged x nearest birthday on 1st January, 

give an approximation for Px,t in terms of P*x,t

if Px,t is the population aged x last birthday on 1st January,

and P*x,t is the population aged x nearest birthday on 1st January, 

Px,t = 1/2(P*x,t + P*x+1,t)

3

Calendar Year of Death - Calendar Year of Birth > Exposed to Risk for  μx+f 

The deaths during the period of investigation, θx , have been classified by age x at the date of death, where

x = calendar year of death - calendar year of birth.

Derive an expression for the exposed to risk in terms of the Px(t) which corresponds to the deaths data and which may be used to estimate the force of mortality, μx+f at age x + f.

A census of those aged x next birthday on 1 January in each year would correspond to the classification of deaths.

But we have lives classified by age x last birthday.

However, the number alive aged x next birthday on any date is equal to the number alive aged x - 1 last birthday.

The number alive aged x - 1 last birthday on 1 January in year t is given by Px-1(t).

At the end of year t this cohort will be aged x last birthday.

Thus, using the trapezium rule, the correct exposed to risk at age x in year t is given by

1/2[Px-1(t) + Px(t+1)]

Assuming birthdays are uniformly distributed over the calendar year, the average age at the start of the rate interval will be x - ½.

Therefore the average age in the middle of the rate interval is x.

Assuming a constant force of mortality between x - ½ and x + ½, therefore,
f = 0.

4

Example Exam Q References

March 2006 - B4 (p 69)

September 2006 - B2 (p. 97)

April 2007 - Q4 (p. 125)

Sep 2007 - Q8 (p. 157)

Apri 2008 - Q5 (p. 180)

Sept 2008 - Q6 (p. 209)

April 2009 - Q 6 (p. 236)

Sept 2009 - Q 4 (p. 262)

Apr 2010 - Q6 (p. 288)

Sep 2010 - Q 7 (p. 321)

Sep 2011 - Q 9 (p. 382)

Sep 2012 - Q 3 (p. 433)