Benefit Changes: Range Boundaries

Warning: This is very confusing and shouldn't really appear on the exam because it isn't explicitly discussed in the text. My suggestion is to learn the basic method for calculating the direct effect of a benefit change and only come back to this if you've studied everything else and have time left over at the end.

Part (a) of the exam question below asks you to calculate the direct effect of the benefit change. (This is the same problem I used as my example in the main wiki article: Adjustments to Losses: Coverage and Benefit Levels.)

E (2017.Fall #06)

Strictly speaking, the question did not provide enough information to be able to solve the problem. The examiner's report provided 2 sample answers with slightly different numerical answers. One way or another you had to make an assumption.

Sample Answer 1: This answer followed the same method as outlined in the text and in the practice problems in Adjustments to Losses: Coverage and Benefit Levels. It makes an implicit assumption

Sample Answer 2: This answer followed almost the same method as outlined in the text and in the practice problems in Adjustments to Losses: Coverage and Benefit Levels, but it makes an explicit assumption

The first assumption arose in calculating the current benefit for the 50-75% range. For that row, we had column (5) = 1100 which was the average weekly wage for that range. To calculate the current benefit of 80% subject to the minimum and maximum limits, we used the formula below where 750 and 1875 were the minimum and maximum benefits calculated in Step 1 of my solution...

• current benefit   =   min(MAX(0.8 x (Col 5), 750), 1875)   =   min(MAX(0.8 x 1100, 750), 1875)   =   880

So neither the minimum nor the maximum were relevant because the answer of 880 fell within the acceptable range of (750, 1875). Note however that we used the average weekly wage of 1100 in the formula. Instead, what if you applied the formula separately to the lower and upper boundaries of the range, 50% and 75%. Those percentages correspond to weekly wages $750 and $1125 respectively.

• using the lower boundary of $750: current benefit = min(MAX(0.8 x 750, 750), 1875) = 750    ← the minimum benefit kicks in
• using the upper boundary of $1500: current benefit = min(MAX(0.8 x 1125, 750), 1875) = 900

That's a range of 750 to 900. The true average benefit will be somewhere in between but not necessarily equal to the 880 calculated earlier. The issue is that wages in the lower part of the range were subject to the minimum but wages in the higher part of the range were not. The missing information is how many of the 100 workers in that range were subject to the minimum and how many were not. If you think (hard) for a few minutes, you'll see that the dividing line in the 50-75% range is:

• (min current benefit as % of SAWW) / (% of wages compensation rate)   =   50% / 80%   =   62.5%

This corresponds to a weekly wage of 62.5% x SAWW = 62.5% x 1500 = 937.5, and 80% of anything lower than 937.5 is less than 750 and therefore subject to the minimum. Workers with wages above 937.5 will not be subject to the minimum. The true average wage for the 50-75% range depends on how many workers are above and below 937.5. Sample answer 2 in the examiner's report assumes that 50 are subject to the minimum and 50 are not.

The second assumption arose in calculating the proposed benefit for the 75-100% range. For that row, we had column (5) = 1450 which was the average weekly wage for that range. To calculate the proposed benefit of 80% subject to the minimum and maximum limits, we used the formula below where 1125 and 1500 were the minimum and maximum benefits calculated in Step 1 of my solution...

• proposed benefit   =   min(MAX(0.8 x (Col 5), 1125), 1500)   =   min(MAX(0.8 x 1450, 1125), 1500)   =   1160

So as before neither the minimum nor the maximum were relevant because the answer of 1160 fell within the acceptable range of (1125, 150). But we used the average weekly wage of 1450 in the formula. Let's again apply the formula separately to the lower and upper boundaries of the range, 75% and 100%. Those percentages correspond to weekly wages $1125 and $1500 respectively.

• using the lower boundary of $1125: proposed benefit = min(MAX(0.8 x 1125, 1125), 1500) = 1125    ← the minimum benefit kicks in

• using the upper boundary of $1500: proposed benefit = min(MAX(0.8 x 1500, 1125), 1500) = 1200   

The issue is that wages in the lower part of the range were subject to the minimum but wages in the higher part of the range were not. The missing information is how many of the 95 workers in that range were subject to the minimum and how many were not. If you think (hard) for a few minutes, you'll see that the dividing line in the 75-100% range is:

• (min proposed benefit as % of SAWW) / (% of wages compensation rate)   =   75% / 80%   =   93.75%

This corresponds to a weekly wage of 93.75% x SAWW = 93.75% x 1500 = 1406.25, and 80% of anything lower than 1406.25 is less than 1125 and therefore subject to the minimum. Workers with wages above 1406.25 will not be subject to the minimum. The true average wage for the 75-100% range depends on how many workers are above and below 937.5. Sample answer 2 in the examiner's report assumes that half of the 95 workers in this range are subject to the minimum and the other half are not.

Resolution: I constructed the practice problems so the range boundaries coincide with the dividing line between workers subject to the minimums and maximums and those who are not. That means you can solve the problems using the method outlined in the main part of the wiki article without making any extra assumptions. This is how the example in Werner was constructed, and this is how the 2017.Fall problem should have been constructed.