View Full Version : Parallelogram method question

Bicycle Repair Man

03-28-2005, 11:21 AM

At the NEAS seminar, Howard Mahler told us that if we use a different mathematical technique than the paper, but both methods produce the same answer, we will get full credit.

The method I use for the parallelogram method is different. I simply calculate the areas, and then multiply that fraction of the total area by the product of all rate changes to the right of that area, and add up the total.

For example, assuming annual policies and two rate changes, one +10.0% effective 7/1/2000 and one +20.0% effective 7/1/2001, the on-level factor for bringing 2001 to the present rates would be

(1/8)*(1.1)*(1.2) + (3/4)*(1.2) + (1/8)

It seems to me that this is mathematically equivalent to the existing parallelogram method given in the paper and by Mahler. However, I've done a few past exam problems using my technique and the answers seem to be off from the answers given in the study guides, although by only a tiny amount, such that it might be rounding, or it might not be.

Klaymen

03-28-2005, 12:36 PM

With annual rate changes, the 2000 rate level is

7/8 x 1.00 + 1/8 x 1.10 = 1.0125

The current rate level is 1.10 x 1.20 = 1.32

The on-level factor is 1.32 / 1.0125 = 1.3037

C'mon this is easy stuff.

Do you mean to ask what the CLF is for 2001? If so, I've solved problems with this method, and I too have been off by just a little bit. I can not figure out why this method doesn't give me the same answer though.

dumples

03-28-2005, 12:57 PM

MNBridge: When the rate change on an annual policy is in the middle of the year, it affects 1/8 of the earned premium for that year. It's an isosceles right triangle in with length 1/2. b=1/2 h=1/2 ==> A = 1/2*b*h = 1/8

BR: I don't think this is right, but I don't really understand what you're trying to do well enough to explain why. First off I think you wanted on level factor for 2001. That factor is the current rate level (1*1.1*1.2 = 1.32) divided by the average rate level for the year ((1/8)*1+(3/4)*1*1.1+(1/8)*1*1.1*1.2). Without any rounding , this is 1.18385... your answer is 1.19. Perhaps mr. mahler was giving an approximation or you made a mistake learning HIS method. Either way, the way that I described will work every time and you are only doing one more calculation (the division).

Bicycle Repair Man

03-28-2005, 01:23 PM

Yeah, I meant the factor for the 2001 year.

Bicycle Repair Man

03-28-2005, 01:30 PM

BR: I don't think this is right, but I don't really understand what you're trying to do well enough to explain why. First off I think you wanted on level factor for 2001. That factor is the current rate level (1*1.1*1.2 = 1.32) divided by the average rate level for the year ((1/8)*1+(3/4)*1*1.1+(1/8)*1*1.1*1.2). Without any rounding , this is 1.18385... your answer is 1.19. Perhaps mr. mahler was giving an approximation or you made a mistake learning HIS method. Either way, the way that I described will work every time and you are only doing one more calculation (the division).The method I proposed is not Mahler's. His method is the same as yours. Mine is more intuitive to me, and requires fewer calculations, (both positives for the exam as far as I'm concerned) but I see now that they're not mathematically equivalent.

What I'm saying is that 1/8 of the premium written in 2001 needs to be brought up to the current rate level by mulitplying by (1.1)*(1.2), 3/4 of it needs to be multiplied by (1.2), and 1/8 of it doesn't need to be on-leveled at all. I think the difference between the two methods is equivalant to the difference between multiplying by .97 or dividing by 1.03. If I used my method, do you think I'd lose any points?

Klaymen

03-28-2005, 02:13 PM

I was noticing that my answer didn't exactly match MN's and this must be the reason but I'm too lazy to look into it any more.

JustASix

03-28-2005, 02:15 PM

BR.

I also use the method you propose. I am surprised that it produces different results. I thought it was algebraically equivalent. Can anyone provide an example as to why they are different?

Colymbosathon ecplecticos

03-28-2005, 02:44 PM

The two methods are not equivalent. In the correct method, each exposure has the same weight. In the incorrect method, the exposures are premium weighted.

In this example, solving for 2001, the correct answer is:

1 * 1.1 * 1.2

--------------------------------------------- = 1.1839

(0.125*1.000) + (0.75 * 1.1 ) + (0.125 * 1.32)

The alternative proposed is:

(0.125*1.32/1.000) + (0.75*1.32/1.10) + (0.125*1.32/1.32) = 1.1900.

Study the two expressions and you'll see that they are not algebraically equivalent. Do not attempt to use the second method as a shortcut.

Bicycle Repair Man

03-28-2005, 03:08 PM

Makes sense to me. Thanks to all who responded.

Howard Mahler

03-28-2005, 10:12 PM

To make it clear, I said that in general if you use a method that is mathematically equivalent to the method used in a syllabus paper, you should not lose any credit for this.

(Should is not the same as will not.)

Mathematically equivalent means always gets the same answer.

This can be useful if you just can not remember the order of steps used by a particular author, but have no trouble remembering a mathematically equivalent calculation. (Each of us can run into this problem somewhere, sometimes for no obvious reason. What is easy for one person to remember, may not be easy for someone else.)

As has been stated the alternate technique discussed here is not mathematically equivalent. Therefore, I would not use it.

This would have been evident sooner if you had tried it with very large rate increases, for example 50% or 100%. In general, when comparing methods it can be useful to try out extreme cases.

Howard Mahler

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