# Multiple state model question

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edited

Hi, can you please explain why the upper bound for the outer integral with respect to t is 45 and not 44? Surely the benefit is only payable for a maximum of 44 years as the life must be sick for at least 1 year in order to receive the benefit?

Thanks

by (100 points)

I believe that the memo is in fact correct and that your issue about evaluating the integral when t is 44.5 is not an issue in the original expression i wrote

by (100 points)
on second thought you must be correct. The upper bound on integral one must be 44

by (2.9k points)

The policyholder needs to be sick for one year. Because of the way they defined the bounds of that second integral, they are starting the integral directly after the 1 year has passed for the person being sick.

Notice the "+1" in the discount term. This is accounting for that year that the policyholder was sick for and could be taken outside the integral to better see this point.

As the first year has been taken into account, the maximum time that the policyholder can then be paid (after the first year) is 44 years, resulting in the upper bound of the second integral.

by (1.1k points)
edited by

Hi Murray, thanks for the reply!

I'm still a bit confused, what happens if t=44,5, then won't the inner integral run from 0 to -0,5?

In the lecture slides with an identical question (see attached), the 1 is subtracted from the maximum possible term (30) of the benefit payments in the outer integral with respect to t, in order to account for the fact that the life has to be sick for at least 1 year, so why is that not the case here?

I'm under the impression that the extra factor of 1P_20+t^SS x V arises after one simplifies the inner integral with bounds 1 to 45-t, similar to the example in the lectures?

by (2.9k points)

For the inner integral, and please check this and call me out if I have this wrong as my MAM2000W is a little rusty, I believe this isn’t an issue because you need to integrate that integral before the first one (resulting in a function which should be solved without the bounds being a problem). I will try and work on that for you in the next few days.

As for the example, this looks a little different to what the original question is. In the original question you are including the first year of sickness (basically paying from day 1 of being sick provided they are sick for more than a year) where this example seems to only be paying an annuity after the first year has been complete (so after a year of being sick, the person is finally paid their annuity).

This would change the answer to what you see in the example (and the steps that follow) and explains why the two answers are not the same. I fear the phrasing of the example may have been misleading (and might be causing much of the confusion here). But I’m very glad you are asking this as, hopefully, we can give you a clear understanding of this complicated section for the test/exam and beyond so please keep asking more questions!

by (1.1k points)

Thank you so much Murray. I apologise for my ignorance but I'm still a bit confused.

How would we be able to differentiate the two cases? The only difference in wording between the two questions (as far as I can tell) is that in the lecture example it requires the life to be sick for at least 1 year and in the original example the life has to be sick continuously for at least 1 year. Is that the keyword that one should look out for? The example in the lecture slides is the same one in (iii) from the ActEd notes on page 10 of Chapter 24, with the same wording.

Also, is the inner integral not just a continuous annuity function with term 44-t contingent on a life aged 21+t once you take out the V^t and V^1 discount factors? Doesn't this already assume that benefit payments only starts when the life has been sick for at least 1 year since it's contingent on a life aged 21+t and not 20+t? When you replace that inner integral with the above annuity function (like in the first line of the answer in the lecture slide), the term will be negative beyond t=44, is there a rationale for the negative term in the annuity?

No rush though, please do take your time. I completely understand that this is quite a busy and hectic period for tutors, but I would appreciate any further discussion/advice before the final conties exam.

by (1.1k points)

To visualise what I'm confused about, if the original integral was re-written as follows, what happens when t is between 45 and 44 since the term of the annuity that is being summed in continuous time will be negative? If the outer bound was 44 instead of 45 then everything would make sense and the annuity will be 0 after t=44.

by (1.1k points)
Hi, could I please get an update on this question? Thanks.
by (2.9k points)

After looking over my MAM1000W notes again, I found that it is possible for these bounds to occur (especially if you rearrange the terms again as they have simplified the expression). As you can see, my new expression is valid and integrable by the fundamental theorem of calculus (all expressions are continuous over all values of t).

But, again after much thought, that doesn’t mean that the expression in the question is correct. The question specifically states that all benefit payments under this policy cease at age 65 exact.

This leads me to think that, in the final year of the policy, a person cannot claim for the sickness benefit if they become sick in the final year.

So I believe you are correct that the bound should be 44 and not 45. It’s the only thing that makes sense for this question. All my MAM1000W work is here for interest sake because it would be a shame to put it to waste XD

by (1.1k points)
Thanks so much Murray, so in the exam do you reckon that I should just go with my method, i.e. what the lecture slides and Acted Notes say (where the upper bound for t should be n-1 if the person has to be sick for at least 1 year in order to receive any benefits) ?
by (2.9k points)
Yes. Stick with the way that the lecture slides and the ActEd notes say for your exam answers. Just makes sure you read the question carefully and don't just blindly write this method (I don't think you will but it's an easy way to lose marks you should get as your understanding seems solid with this work).
by (1.1k points)
Noted, thanks again for the help!