Capitation Problem

[Plant Food]

Here's a problem I made up. Post your answer.

Claim Size Interval/Avg Claim size in interval/probability of claim in
interval
0/0/.90
1-10,000/5,000/.05
10,000-20,000/15,000/.03
20,000-30,000/25,000/.01
30,000-50,000/40,000/.005
50,000-100,000/75,000/.003
100,000-500,000/250,000/.002

Deductible is $20,000
Managed Care decreases cost by 25%
Maximum benefit amount is $200,000
Insurer's portion of the risk is 90%
Fixed costs=5.00
Costs that vary with net claim costs are 4% of net claim costs
costs which vary with premium are 17% of premium

Compute the gross premium per member per month

[PAC]

Is the $200K max benefit amount per occurence or annual aggregate per member?

Is the $20K deductible per member (doubtful) or per group - and if per group, what is the assumed group size?

[Utanapishtim]

Are your given claim probabilities on a monthly or annual basis?

[Plant Food]

Are your given claim probabilities on a monthly or annual basis?

Table 2 on page 125 is annual so is this.

Is the $200K max benefit amount per occurence or annual aggregate per member?


Maximum benefit amount is per covered member per year (see page 131).

Is the $20K deductible per member (doubtful) or per group - and if per group, what is the assumed group size?

It's per member. The example in the paper has a deductible of $50,000 per member.

[Rice]

This is my solution. I make no guarantee that it is correct.


Avg in Cumul Cumul
Range Range Prob Prob Cost Cost
1k-10k 5k .05 .1 250 1875
10k-20k 15k .03 .05 450 1625
20k-30k 25k .01 .02 250 1075
30k-50k 40k .005 .01 200 825
50k-100k 75k .003 .005 225 625
100k-500k 200k* .002 .002 400 400

*Capped to reflect the max benefit


Avg cost = (.75*1075)–(20k*.02) = 406.25

Per month = 406.25/12 = 33.85

After coinsurance = 33.85*.9 = 30.47

Gross premium = (5+(30.47*1.04))/.83 = 44.20

[PAC]

Assuming the deductible and maximum benefit are applicable to the insurer reimbursement amounts, i.e. grossed-up by 4/3 and 10/9...

Cost table deductible = 30K
Max benefit = 300K therefore assume no offset, no capping in cost table ...

Net of coins & deductible managed care cost PMPM = (.75*925-200) * 0.9 / 12 = $37

Gross premium PMPM = $52

[Ugh]

My solution (based on All10 appendix summary):

Deductible-----Yearly Cost---Monthly Cost (pmpm)
20,000----------775.00--------64.58
26,667----------641.66--------53.47
242,222----------15.56--------1.30

Since we're assuming managed care cuts the cost by 25%, the percent of managed care claims in excess of 20,000 is equal to the percent of claims in the above distribution which are excess of 26,667 = 20,000/(1-.25)

Also, the cost of managed care claims greater than 20,000 is 75% of the cost of claims in the above distribution which are greater than 26,667.

So the Net pmpm Claim Cost = .75 * 53.47 = 40.1

Max adjustment:
If the max benefit is 200,000 and the insurers portion is 90% and the deductible is 20,000, then the dollar amount to look up in the table above is 242,222.22 = 200,000/.9 + 20,000

So we subtract 1.30 from the table above from 40.1 and get 38.8

Coinsurance Adj:
38.8 * .9 = 34.9

Gross Prem:
(5 + 34.9*1.04)/(1-.17) = 49.8

[Rice]

How did you calculate the Yearly Costs for the three deductibles?

[Ugh]

You could use the same method as in your solution above (without capping the average cost in the 100-500K range).

However, it seems to me that I get the same answer without going through the details of the Bourdon algorithm and just using the logic from Brown and Smitz's deductible paper.

For example, the yearly claim cost for the 26,667 deductible is equal to the expected ground up losses above 26,667 - 26,667 * probability of loss greater than 26,667.

The given distribution is not exact since it only gives probabilities and average costs for broad ranges. But it seems that Bourden is approximating using the probabilities greater than the lower limit of the range that the deductible falls in.

So in this example we have:

25,000 * .01 + 40,000 * .005 + 75,000 * .003 + 250,000 * .002 - 26,667 * (.01 + .005 + .003 + .002) = 641.66

[PAC]

From a P/C perspective this looks to be a an aggregate excess reinsurance with the following features:
self-insured retention = first $20K of covered benefits
Coinsurance layer = next $200K aggregate of covered benefits at 10/90
Excess of next $200K - reverts to self-insurance
...where covered benefits are those subject to the network/managed care par agreement.

I think the max benefit needs to be grossed-up for managed care savings.

The claim-size distribution in the last layer is skew so I wouldn't use the average probability for the layer to adjust the max benefit.

IRL I expect the deductible would be treated as in Ugh's solution, consistent with the P&C description above.

[Plant Food]

Ugh, the 641.66 is wrong. It's (40,000-26,667)*.005+(75,000-26,667)*.003+(250,000-26,667)*.002=658.33. Or, if you prefer the paper's method it's 925-26,667*.01=658.33.

Taking 75% of that gives 493.75. Net pmpm Claim Cost is 493.75/12=41.15. Now subtract the 1.30 from the table to get 41.15-1.30=39.85

Coinsurance adjustment is .9*39.85=35.86

Gross premium=(5+35.86*1.04)/(1-.83)=50.96

[Rice]

The reading gives an example where managed care slices the cost of each claim by 50%. It states:

Net pmpm Claim Cost @ $50,000 Deductible =
(25/50) * Cost of claims excess of the $100,000 deductible interval
- $50,000 * Probability that claims exceed the $100,000 deductible interval

So I think, in our example, we should take .75 * cost of claims excess of 26,667 - 20,000 * prob that claims exceed 26,667.

But since 26,667 falls in the same range as 20,000 we get:

[.75*1175 - 20,000*.01]/12 = 681.25/12 = 56.77

The author goes on to say:

If the maximum benefit amount is $500,000, the insurer's portion of the risk is 90%, and the Specific Deductible is $50,000, the appropriate dollar amount to look up in the table is $500,000/.9 +$50,000 = $605,556.
...
The adjustment for maximum benefit is then subtracted from the expected net claim cost.

Our max benefit is 200,000 and the deductible is 20,000. So we look up 200,000/.9 + 20,000 = 242,222, which is in the 100k-500k range. So we get:

[.75*500 - 242,222*0]/12 = 375/12 = 31.25

Then 56.77 - 31.25 = 25.52

Adjusting for coins = 25.52*.9 = 22.97

Gross prem = (5+22.97*1.04)/.83 = 34.81

[PAC]

Does the paper include an example adjusting for max benefit with coinsurance and managed care?

[Rice]

Yes, that's what this example was.

[Ugh]

Plant Food -

The 641.66 makes use of the values from the 20K-30K interval. Using the paper's method 641.66 = 1,175 - 26,667*.02

I used those figures because the 26,667 falls in the range of 20-30K.

Did you use the next higher range (30-40K) because the 26,667 is greater than the average of 25,000 in the 20-30K range?
Is the paper clear on this? The paper's example uses the values from the 20-30K interval for calculating the 25,000 deductible claim cost, however, the 25,000 is less than the average claim size in the interval of 27,000.

The decapitation paper by Bourdon's sister Lizzy is much more precise...

[PAC]

This example apparently considers the coinsurance but not the managed care savings when computing the max benefit adjustment.

If the max benefit is the maximum amount in the aggregate that the insurer will pay, then the $200K max benefit should be grossed-up by coinsurance and managed care savings and add-in the grossed-up deductible when pulling from the claim cost table.

Look-up value = 200K / ( 0.9 * 0.75 ) + 20K / 0.75 = 323K

The effect of understating the lookup value is to overstate the reduction in premium due to the max benefit limitation - a relatively small error in this example - but the magnitude of the premium understatement will increase as the estimated managed care savings % increases.

[Plant Food]

Construct Table 3 from this data

Average Claim Size in Interval/Accumulated Probability of Claim in Interval/Accumulated Annual Claim Cost
0/1.00/1,875
5,000/0.100/1,875
15,000/0.050/1,625
25,000/0.020/1,175
40,000/0.010/925
75,000/0.005/725
250,000/0.002/500

where 500=250000*.002
725=500+75000*.003
925=725+40000*.005
1,175=925+25000*.01
1,625=1,175+1500*.03
1,875=1,175+5000*.05

Then use net annual claim cost excess of deductible=accumulated annual claim cost for intervals excess of the deductible-deductible*accumulated probability for intervals in excess of the deductible.

The interval excess the deductible of 26,667 is 30,000-50,000 which corresponds to the value of 40,000.

So if the deductible is 26,667 we use 925-.01*26,667=658.33

You're taking a continuous distribution (claim cost) and approximating it by a discrete distribution. I prefer the first way that I did it because it's how to find the expected value of max(X-d,0). All that actuarial mumbo jumbo about back-sums is confusing.

[Ugh]

I agree that based on the wording of "use the accumulated annual claim cost for intervals excess of the deductible", your method seems correct. However, in the paper's example, they use the 20-30K interval for the 25K deductible.

[Plant Food]

They use the 20-30K interval because the value of 27,000 is greater than 25,000. In the example in the paper, if X is the loss amount then

Pr(20,000<X<30,000)=0.0051600
and
E[X|20,000<X<30,000]=27,000.

We replace this information with a discrete distribution with

Pr(X=27,000)=0.0051600.

The intervals aren't used anymore. Don't even look at them. Replace the continuous with the discrete and ask yourself, what's the expected payment for various deductibles?

[Ugh]

I see what you're doing now. It simplifies everything if you just completely ignore the intervals. Too bad all of the deductibles in the reading were less than the average claim size in the interval.
Thanks PF.

[PAC]

But note that the grossed-up deductible (50K -> 100K) in the reading exactly corresponds to an interval endpoint.

[Rice]

Plant Food wrote:
So if the deductible is 26,667 we use 925-.01*26,667=658.33
But the deductible is not 26,667. It is 20,000. The example in the reading has a deductible of 50,000 and 50% managed care reduction. The corresponding 2nd half of the formula uses 50,000*probability that claims exceed the 100,000 deductible interval, where 50,000 is the deductible as given and 100,000 = 50,000/.5.

In our example, the given deductible is 20,000 and 26,667 = 20,000/.75.

So we replace 50,000 with 20,000 and we replace 100,000 with 26,667 and we get 925-20,000*.01 = 725.

Since the reading does not consider both the managed care savings and max benefit at the same time, I doubt they would ask that question as we are not told how to do it. Or perhaps they would say "state any assumptions you make," in which case it doesn't matter how we do it as long as we explain our methods.

[Mel-o-rama]

To further confuse the issue, the first part of that equation in the paper's example says to multiply the cost of claims by 25/50. I don't understand why, since it seems to be double-counting to me. But to apply that to Plant Food's example, wouldn't this become?:

(3/4 * 925) - (20,000 * .01) = 493.75

About the maximum benefit: Since there is no interval in excess of 242,222, does that mean that subtract zero to account for this? Easy enough if we're not missing something.

Finally, [5 + 493.75 * 1.04] / (1-.17) = 624.70 annual = $52.06 monthly, which is very close to PAC's answer.

I also tried getting the answer by doing something like Brown/Schmitz method and got $51.32.

Am I missing anything?

[PAC]

It's not double counting because you are starting with gross, i.e. before managed care savings, expected losses of $925 which are reduced to a managed care cost level by the factor (1-.25).

Another way to view the first part of the solution - articulating gross of managed care savings figures - is:

(1-.25)*(925 - (26,667*.01) ) = 493.75

The point is that the claim cost table is indexed to claim sizes on a gross of managed care basis while the PMPM's are net of managed care savings.

[Mel-o-rama]

PAC: That makes sense. Thanks. Now I hope they put one of these on the exam.

[Izzy Mandelbaum]

I think Mel-o-rama's answer is the closest to being right so far, except for two points:

1) the net claim cost should be adjusted for the insurer's share of the cost, so multimply the $493.75 by 90% to reflect coinsurance.

2) although it's ambiguous, I think the $5 fixed cost is on a monthly basis, not annual.

Making these adjustments, the answer turns out to be $52.42 (no rounding until the final answer).

Also, keep in mind that the maximum benefit to be looked up in the table is not $242,222, but $322,963, since it must be grossed up for the managed care savings. Since this latter number is so far above the $250,000 average claim size in the largest interval, it's probably safe to say that there are no claims in excess of this amount.