# 2019 test 2 q1

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Hi

Can someone please explain how the answer for (d) is obtained? How would you treat a question like this where not all the forces of transition are constant across integer ages? Why is the force of lapsing ignored in the calculations of the forces of mortality and critical illness, and how is the 0.99777777 obtained?

Thanks  by (1.2k points)
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Hi,

Apologies for the late response.

I am not exactly sure how the working would change if the forces were not constant, although I know you could not use the methods in the answers.

So this question is nice because it essentially broadens your decrement understanding (although the hard way of having to think about it a bit).

Think of the decrements as monsters in a room with you. There are 3 monsters, Lapse, Death and CI diagnosis. By referring to my timeline in the PDF attached, I have shown how these monsters interact. Lapses occur only at the beginning of the year and before the other monsters. After Lapses chooses what do do with you, the other monsters come in and have to fight with each other over what Lapses has left behind. In this way, Lapses can be treated as independent, while Death and CI diagnosis are dependent on each other (they occur at the same time).

As a result, we split the timeline into 3 components, 1 - before any decrements occur, 2 - after Lapses occur, 3 - after all decrements occur.

With question d, q50(l) = (aq)50(l) because this monster works alone, so only q50(m) and q50(c) are ones in which we need to use our multiple decrement formula for (see PDF). We need to use point 2 on our timeline as these decrements start after our Lapses, so technically, at the start of when these decrements occur, we need to have incorporated Lapses already. So lives outstanding will be 10 000 - 1 000. The rest of the math follows in the PDF.

Hope that helps

by (860 points)
Thank you Keenan, this makes a lot of sense. Follow-up question: If lapses only occurred at the end of the year instead of throughout the year then how would the calculation change? I'm assuming you won't have to subtract 1000 from 10000 in t(ap)x in that case because the lives will still be exposed to risk of lapse during the year?
by (1.2k points)
Hi,

Just quickly, in this question lapses occur at the beginning (not throughout the year). Yeah, so if lapses occur at the end of the year then there is no need to subtract the 1 000 from the 10 000, but that's just the general idea, there might be question specifics or more information needed in the question to deal with lapses being at the end of the year.
by (860 points)
Yeah if it was throughout the year then would you just have used the normal method to calculate it (same as the other 2 decrements)?
by (1.2k points)
Yes, because now all the decrements are occurring at the same time.
by (860 points)
Thank you for the assistance.
ago by (990 points)
Hi, just on the first comment asking what would happen if lapses occurred at the end of the year: how would we deal with this? Would we first work out the (aq) with the formula including only CI and deaths and then multiply the surviving probability by the rate of lapses?

Because I think the formula only work for decrements occurring constantly over the year right?
by (990 points)
Hi,

(I'm not a tutor but here is my thinking):

Answering your question regarding why the lapsing is not included in the calculations of the forces of decrement during the year:

If a decrement occurs continuously (like death and critical illness) then there will be a force at which that decrement occurs continuously over time. Where a decrement can only occur at discrete time points (as lapsing here) then there is just the probability that the decrement occurs at that exact time point. So that is why there is no force of lapsing during the year, but a probability that lapsing occurs at the beginning of the year. In effect, the force of lapsing during the year is zero at all time points throughout the year, so it has no effect on the dependent probabilities of leaving by other causes during the year i.e. its basically not happening.
by (860 points)

Thanks, do you know how the 0.99777777 figure is obtained?

by (990 points)
I'm going to leave that to a tutor to explain as I can't seem to find a plausible reason but this is my thinking:

The answer of how -LN(0.997777) is obtained:

The formula to work out the force of transition for a particular decrement (say s)  is:

= (aq)^s_x/(aq)_x * (-ln(ap)_x)

so the 0.99777 is the (ap)_x. From the reasoning in the previous answer (given above), we only deal with death and critical illness.

(ap)_x = 1-(aq)_x = 1-(ad)_x/(al)_x = 1 - (8+12)/(10000-1000) = 0.997777

But I'm not sure why we can minus the 1000 from the 10000.