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  #1  
Old 07-11-2017, 05:57 PM
koudai8 koudai8 is offline
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Default (1+q)^(1/12) -1 vs. 1-(1-q)^(1/12)

Hello! I've been looking around and seem to not be able to find an answer. Can anyone tell me if they have ever seen [(1+q)^(1/12)]-1 used in place of 1-[(1-q)^(1/12)]? The q can be death, exiting a policy (in terms of disability, etc). I am not looking for i or d in this case, even though they are similar.
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Old 07-11-2017, 06:58 PM
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Originally Posted by koudai8 View Post
Hello! I've been looking around and seem to not be able to find an answer. Can anyone tell me if they have ever seen [(1+q)^(1/12)]-1 used in place of 1-[(1-q)^(1/12)]? The q can be death, exiting a policy (in terms of disability, etc). I am not looking for i or d in this case, even though they are similar.
What are you doing with the value you are calculating? That will probably tell you if it makes sense or not.
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Old 07-11-2017, 07:20 PM
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"[(1+q)^(1/12)]"

Where did you see this? And what sort of probability theory do you think would support a plus sign??
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Old 07-11-2017, 09:23 PM
DiscreteAndDiscreet DiscreteAndDiscreet is offline
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"[(1+q)^(1/12)]"

Where did you see this? And what sort of probability theory do you think would support a plus sign??
The familiar math works the same way if you set 0Px equal to 1 and define tPx as greater than 1 for negative t and conventional tPx<1 for positive t. The fraction represents projected past/fewer population sizes relative to an observed population at t=0.

I'd expect to see something like -1Px=1/(1-q), where q is a rate of observed decrements with the prior year as the exposure base and where the -1Px gives you the relative size of an inferred past population relative to a known one, but someone, somewhere might have invented a reason to study the q in the backward direction. That's the only rationale I can come up with.
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Old 07-11-2017, 09:36 PM
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OK, I guess. Somehow I doubt that's what the OP had in mind. But if it was, then the formula to use simply depends on the convention you use for q. Right?
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Old 07-12-2017, 02:02 AM
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I saw something like this for UL COI charges years ago, but darned if I remember the explanation for it; I also seem to recall seeing this discussed elsewhere on the AO.
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Old 07-12-2017, 11:21 AM
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I saw something like this for UL COI charges years ago, but darned if I remember the explanation for it; I also seem to recall seeing this discussed elsewhere on the AO.
I don't have the formula at the top of my head, but the thing with coi charges is the need to adjust for amount at risk and what old-timers call benefit of survivorship. I don't think it exactly matches what appears in the OP in any case.

I do hope the person who posed this problem will come back and explain how the formula is intended to be used.

ETA - I wonder how similar the two formulas would be if you took just the first few terms of a Taylor expansion. I leave this as an exercise for the student.
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Old 07-12-2017, 11:58 AM
koudai8 koudai8 is offline
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Thank you much for replying. TBH Im an intern (my first internship) and my manager just asked me to do research and see if I've seen this formula used, but not using i or d. So just seeing it anywhere and using q as either exit rate, mortality rate, etc. Basically, leaving a plan. I'm assuming it relating to pricing or modeling a life or disability plan.

*This isn't my account. One of the other inters let me borrow it. So I actually only have 2 exams passed.
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Old 07-13-2017, 03:42 PM
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Assuming the q is a decrement (which they usually are), the reason I don't think the "plus" version could ever work is because, it doesn't get you to the right place after 12 months.

Take a decrement of 1% as an example. Turning it into a monthly decrement using the "minus" version gives you .000837. If you start with a population of 100,000 and apply that decrement each month you end up with 99,000 left after 12 months. As you would expect you lost exactly 1% over the course of the year.

Using the "plus" version gives you are monthly value of .00083. Again starting with a population of 100,000 and applying the "plus" decrement each month for 12 months, you end up with 99,009.

I think this is as simply as the "plus" version should be used for accumulating (like interest) and the "minus" version for decrementing.
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Old 07-13-2017, 04:51 PM
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Assuming the q is a decrement (which they usually are), the reason I don't think the "plus" version could ever work is because, it doesn't get you to the right place after 12 months.

Take a decrement of 1% as an example. Turning it into a monthly decrement using the "minus" version gives you .000837. If you start with a population of 100,000 and apply that decrement each month you end up with 99,000 left after 12 months. As you would expect you lost exactly 1% over the course of the year.

Using the "plus" version gives you are monthly value of .00083. Again starting with a population of 100,000 and applying the "plus" decrement each month for 12 months, you end up with 99,009.

I think this is as simply as the "plus" version should be used for accumulating (like interest) and the "minus" version for decrementing.
I don't disagree with your conclusion, but you can't argue it this way - you are trying to disprove a method that uses a simplifying assumption by showing that if you make a different assumption, you don't get the same answer.

Using a 1/12 per month uniform or a ^(1/12) multiplicative factor are both reasonable and have their place, but you could "disprove" either of them with this kind of logic.

Using 1+q is an unusual choice, as the force increases through the year, but I could imagine using it in the scope of a larger problem if it simplified the math.
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