Applying decrements uniformly

[MindFull]

Hi All,

I feel like I continuously have issues understanding when and how to apply decrements uniformly. For eg. the February 2016 past paper (Paper 2) has a model where children leave and join uniformly. Additionally, weekly fees associated with the average number of children is calculated using an assumption of uniform distribution. My thinking is:
Eg. 24 people join over the course of a year, 50% of those will leave. This means that 2 people join per month... so then 50% *2 = 1 person would be leaving every month. So there should be 6 people by mid year (12 have joined, 6 have left). So if I wanted to calculate the fees for this cohort, it would be 6 people * applicable per child fee * 52 (I think).
The model solution indicates that the 50% should be applied to half of the starting population (24 in my example) = 12*0.5 = 6 and then further if I wanted to calculate the fees associated I would apply 0.5 again to get a result 0f 3. I figure this last bit is because we are trying to calculate the average number of children mid-year, but doesn't my previous answer of 6 already assume that there are 6 children left by mid-year?
Hope this isn't too confusing.
Thanks much.

 

[ntickner]

If 2 people join per month, then on average, there's only one new person there for the full month. And the 50% decrement applies to people that have been there for a full year, so in month 1 it only applies to 1/12, leading to a 1/24 of a person leaving. The second month has another 1/24 for the new people, plus 2/12*50% = 1 for the two that joined in the first month. So each month t, you have (t-1)*2/12*50% + 1/12*50%, and adding that all up over the year, you get to 6. Hope that isn't too confusing!

However, it's easy to overthink this sort of thing. Especially in the exam, where you're expecting tricks from the examiners. Remember the following, though:
- The exam isn't trying to test your knowledge of how to apply decrements, or complicated actuarial concepts.
- It's trying to test your ability to Model them, and Communicate your modelling.
- So, the key is to be clear in what you've assumed, and what you've done, and make sure what you've done in the model is consistent with what you're saying you've done.
- So keep it simple, and make it clear.

If you find yourself getting stuck in details like that, put in a simplifying assumption to make it easier. In this case, you could assume that everybody joins in the middle, and people only leave at the end. So you have 24 joining in July, and they all stay there until December, when 12 leave. You might lose a mark or two for that, but you'll have a consistent model & audit trail, you'll have results to explain, and you'll probably pick up more marks as a result than you lose. And if you have time at the end, you could always go back and try again.

 

[MindFull]

Thanks for the reply but to be honest it is a bit confusing lol. I agree that in the exam I should just assume something and move on but I really do think I have a lack of understanding about UDD and I'd like to get over it since it is so widely used. Just to add to my question, based on my understanding, I get that 6 people have left on average but I think the examiners would say that only 3 people have left on average. So I'm trying to figure out my error.
Thanks so much.

 

[MindFull]

If 2 people join per month, then on average, there's only one new person there for the full month. And the 50% decrement applies to people that have been there for a full year, so in month 1 it only applies to 1/12, leading to a 1/24 of a person leaving. The second month has another 1/24 for the new people, plus 2/12*50% = 1 for the two that joined in the first month. So each month t, you have (t-1)*2/12*50% + 1/12*50%, and adding that all up over the year, you get to 6. Hope that isn't too confusing!

Hey can you explain this part? Specifically the part for the new people in the second and subsequent months.
Thanks.