April 2005 CT5 Q13

[omurice]

Hi,

How are the dependent mortality rates calculated in the exam report? Why are they different from the independent mortality rates?

Thanks a lot in advance

 

[Joe Hook]

Hi,

We have independent probabilities in the question which for age 40 say are: 0.000788 for death and 0.1 for surrender. We need to work out the constant force of mortality and surrender over the year as both are operating together i.e. the life could die or surrender at any time during the year.

Remembering that q_x = 1-exp(-mu_x) we can rearrange to make mu_x the subject:

mu_x = -ln(1 - q_x)

Similarly we could say sigma_x = -ln(1 - s_x)

For age 40 this gives me a force of mortality of 0.000788 and a force of surrender of 0.105361.

We can then use our standard formula for calculating dependent probabilities i.e

(Force of (mortality/surrender) / total force of decrements) * prob of leaving the population

Dependent prob of death = (mu_x)/(mu_x + sigma_x) * (1 - exp(-mu_x-sigma_x)) = 0.000748
Dependent prob of surrender = (sigma_x)/(mu_x + sigma_x) * (1 - exp(-mu_x-sigma_x)) = 0.09996

Note that I get a slightly different number for death using excel for my calculation...presumably because the examiners have rounded somewhere in there answer. However, I wouldn't expect these small differences to be a problem in the exam.

The same approach can then be used for the other ages.

Does this help?
Joe

 

[omurice]

Yes thank you so much!