Actuarial Outpost SOA Sample Question 196
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#1
05-04-2012, 11:50 PM
 Jonnyboy Member CAS SOA Join Date: Jan 2011 Location: New York Studying for FAP College: Senior Year Favorite beer: Newcastle Posts: 208
SOA Sample Question 196

This is not a particularly difficult sample problem, but it is an extremely tedious one. While doing timed practice problems I came across this one, and after spending a few minutes setting up the solution I thought to myself "F**k this, the algebra here is going to kill me!" and I moved on. However, after finishing a few other problems I went back to #196 (after looking at the answer to ensure that there was no special trick I was missing), and decided to time how long it would take me to complete it.

Knowing exactly how to solve the problem, and being confident in that method, it took me 9:42 to come up with the correct solution: 3.089. Worse still, the solution is very close to being in between the ranges of the solutions given, which would cause me to panic in an exam setting (I like not having to worry about rounding errors, etc).

For those of you who have taken the exam, how many questions are there that require over 9 minutes to come up with the answer (assuming you know exactly how to solve it)? Alternatively, should I be able to solve this problem faster than I did?
#2
05-05-2012, 08:33 AM
 swifthat Member SOA AAA Join Date: Feb 2012 Posts: 2,197

This question was dumb. As soon as I saw it I knew it was going to be very tedious. It felt like this was the SOA's way of hinting at us to just "memorize" MLE formulas
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#3
06-01-2013, 05:14 PM
 LifeIsAPoissonProcess Member CAS SOA AAA Join Date: Aug 2010 Studying for 8 Favorite beer: DFH 90Min Posts: 6,230

I'm confused on why the losses with non-zero deductibles are calculated conditioned on those losses exceeding the deductible. What part of the question implies truncation of losses below the deductibles?
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#4
06-01-2013, 06:24 PM
 cmind Member SOA Join Date: Feb 2013 Posts: 1,312

Quote:
 Originally Posted by LifeIsAPoissonProcess What part of the question implies truncation of losses below the deductibles?
Isn't a deductible a truncation point by definition?
#5
06-03-2013, 11:24 AM
 francois Member Non-Actuary Join Date: Oct 2012 Posts: 345

I did this question in about 3-4 minutes

The trick is this...

The 3 losses of 750 have a deductible of 200. With deductibles in MLE's, you have to condition on the loss being above the deductible. (also, you can drop the "+1" from the (x+theta)^(a+1), since (x+theta)^1 is just a constant

So, for the losses of 750 the likelihood function is (a^3)*(10000^3a)*(10200^3a) divided by (10750^3a)*(10000^3a).

So, the (10000^3a) part will cancel out, and you're left with (a^3)*(10200^3a) divided by (10750^3a).

But, you can cancel that further once you see that there are 3 losses of 200, which its likelihood function of (a^3)*(10000)^3a divided by (10200^3a). So, the (10200^3a) will cancel out between the 750 losses and the 300 losses.

So, combined, their likelihood function is (a^6)*(10000)^3a divided by (10750)^(3a)

The same thing will happen with the 4 losses of 400 with a 300 deductible, and the 4 losses of 300.

So, you'll have (a^8)*(10000)^4a divided by (10400)^4a for those 8 losses.

Now, all you're left with is the 6 censored observations, which have a likelihood function of (10000^6a) divided by (20000^6a).

Combined, the likelihood is a^14 * (10000)^13a divided by (20000^6a) * (10400^4a) * (10750^3a)

log and take derivative, you have (14 / a) + 13 (ln 10000) - 6 (ln 20000) - 4 (ln 10400) - 3 (ln 10750). So, you get 3.089.

Last edited by francois; 06-03-2013 at 11:29 AM..
#6
06-03-2013, 12:03 PM
 LifeIsAPoissonProcess Member CAS SOA AAA Join Date: Aug 2010 Studying for 8 Favorite beer: DFH 90Min Posts: 6,230

Quote:
 Originally Posted by francois With deductibles in MLE's, you have to condition on the loss being above the deductible.
This is the part I disagree with.

I understand the math just fine, it's the setup I'm having an issue with. This is the only question I've seen where you are supposed to infer truncation rather than it being a given assumption. In the ASM Chapter 30 practice problems, the only time the formula adjusts for truncation is when there is an explicit assumption to do so.

In this question there is no assumption that would prevent an occurrence of a recorded loss being less than the deductible, because there is nothing indicating that losses below the deductible have been truncated.
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#7
06-03-2013, 12:42 PM
 francois Member Non-Actuary Join Date: Oct 2012 Posts: 345

Quote:
 Originally Posted by LifeIsAPoissonProcess This is the part I disagree with. I understand the math just fine, it's the setup I'm having an issue with. This is the only question I've seen where you are supposed to infer truncation rather than it being a given assumption. In the ASM Chapter 30 practice problems, the only time the formula adjusts for truncation is when there is an explicit assumption to do so. In this question there is no assumption that would prevent an occurrence of a recorded loss being less than the deductible, because there is nothing indicating that losses below the deductible have been truncated.
Most likely, they will specifically state that you should consider the deductible as truncation. However, if they don't, I would say you should consider it truncated unless it explicitly says there is no truncation.

And, specifically in this question, treating it as truncated causes the simplifications I mentioned. Thus, you can infer that the writers intended you to condition on the loss being above the deductible.