Multiple decrement-improved ind. mort. rate IAI Q.8-Oct'05

[deepakraomore]

Multiple decrement table is given with two decrements death(d) and retire(r) say
Question asks ----
Following improvements in the mortality experience, it is decided to construct a new table with the
independent rates of mortality reduced by 40%.
Construct the new multiple decrement table.

Sol - in the solution Used the below formulae to calculate the independent rates
\( q^d_x \sim \left(aq\right)^d_x /\left[1-0.5\left(aq\right)^r_x\right]\)
and
\( q^r_x \sim \left(aq\right)^r_x /\left[1-0.5\left(aq\right)^d_x\right]\)

how these formulae come? and is there any other way to calculate independent probability?

 

[Robert Chadburn]

Hi Deepak

This is an alternative approach to calculating independent-dependent probabilities, that assumes decrements have uniform distributions over each year of age (the current course methodology is based on assuming that forces of decrement are constant over each year of age, which is a different thing). The syllabus changed to the current one in 2015, which means that any questions that were set before then were based on the old syllabus, so these questions will not be appropriate for you.


But, yes, there is certainly a way of doing the above calculation under the current syllabus, and you might need to do this in the exam. It's all in Chapter 10. First, the formulae for getting the independent probability from the force of decrement is shown near the end of Section 3. It is basically qx = 1 - e^(mu).

Then, in order to get the forces, and all you have is a set of multiple decrement (dependent) probabilities, you can get the independent forces of decrement using the formulae used in Section 4.4.

Robert

 

[OmoT]

Hello,

Is the independent rate of mortality the "probability-qx" or the "force of decrement- Ux". I am kind of confused.

 

[Robert Chadburn]

The wording used is definitely confusing! Actuaries often refer to the "mortality experience", in rather loose terms, as being "the mortality rates". This mortality experience can be defined in terms of forces of mortality at every (continuous) age (the Ux values), or the probabilities at every integer age (the qx values). So actuaries in practice (loosely) can mean either of these then they say "mortality rates".

But, statistically, Ux is a transition rate, because it is the annual rate at which a person is dying at a specific age x; whereas qx is a mortality probability (because it is the probability that a life, currently aged x, will die over the next year).

Now, what about "independent", you ask? well, there are even two senses of "independence" to consider here!

First, qx is described as the independent probability of dying, in the sense that it is the probability of dying when no other cause of decrement is occurring. This is as opposed to the dependent probability of dying, which is written (aq)-d, the value of which depends on how many decrements from other causes are happening (hence, its value is said to be "dependent" on these other decrements).
For Ux, because this is a rate operating only at an instant of time, then other decrements that are occurring at the same time cannot influence this rate at all. So, all else being equal, the Ux value in a population in which only deaths can occur will be the same as the Ux value for mortality where any number of other decrements are also occurring.

The second kind of independence is best considered by thinking of the Ux values. In life assurance, the presence of withdrawals causes the average mortality of those who don't withdraw to increase (because it tends to be only healthy people who withdraw). In other words, Ux itself will be affected. So mortality and withdrawals are not independent decrements. Both Ux and qx will be affected by this. This is the "independence" of decrements assumption referred to in the Course Notes.

Hope that helps!