Exposed to Risk Calculations

Question 1

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

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

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

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

Answer 1

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

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

P_x,t = ¹/₂(P^*_x-1,t + P^*_x,t)

Question 2

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

if P_x,t is the population aged x last birthday on 1^st 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

Answer 2

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

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

P_x,t = ¹/₂(P^*_x,t + P^*_x+1,t)

Question 3

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

The deaths during the period of investigation, θ_x , 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 P_x(t) which corresponds to the deaths data and which may be used to estimate the force of mortality, μ_x+f at age x + f.

Answer 3

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 P_x-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[P_x-1(t) + P_x(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.

Question 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)