CT5_2018_04 Question 1

[Aditi Vaidya]

Q: Calculate 2.75_q_84.5 using the method of Constant Force of Mortality.......................Basis: AM92
The method used in the examiner's report was as follows:
2.75_q_84.5 = (1 - 2.75_p_84.5)
= 1 – (0.5_p_84.5 * 1_p_85 * 1_p_86 * 0.25_ p_87) = 1 - ( ((p_84)^0.5) * (1_p_85) * (1_p_86) * ((p_87)^0.25) )

However, since the question mentions the constant force of mortality method, I answer the question in the following manner:
2.75_q_84.5 = (1 - 2.75_p_84.5) = 1 – (0.5_p_84.5 * 1_p_85 * 1_p_86 * 0.25_ p_87)
=
1 – (exp(-INT(0,0.5):0.101417.dt) * exp(-INT(0,1):.111691) * exp(-INT(0,1):0.122884.dt) * exp(-INT(0,0.25):0.135066.dt) )

my answer did not match the examiner's report. Will this method be credited marks as I used the force of mortality!

 

[Lucy England]

Your approach isn't quite right because you're picking up the mu_x values from page 81 of the Tables. These values are the force of mortality at exact age x, not the average force of mortality that would apply over the whole year of age if assuming a constant force of mortality.

If you're assuming a constant force of mortality over each year of age, you'll need to work out the forces (I'll call them mu*_x) by using the relationship:

p_x = exp(-mu*_x)
=> (1 - q_x) = exp(-mu*_x)
=> mu*_x = -ln(1 - q_x)

and you can then use the q_x values from page 79 of the Tables to work out mu*_x.

For these non-integer ages and times probabilities, it's much quicker and easier to use the approach in the Examiners' Report!

 

[Aditi Vaidya]

Thank you Lucy! this was helpful