Group size queens

Actuaries and underwriters love large groups.  The bigger the better. Small groups and individuals are almost impossible to accurately price.  Big groups allow statistical approximations to approach population realities while the error bars on a small group are massive.  Massive error bars make underwriters and actuaries cry.

The following will have some math and more importantly a lot of statistical intuition, so please bear with me.

Let us imagine that Mayhew Insurance company has 1,000 employees in a single group.  This is a good size group. The group premiums are precisely enough to cover medical and administrative expenses for the year.    Let us also assume that 10% or 100 employees are expensive to cover.  The other 900 are in reasonably decent health or in good health.  Their premiums cover the expenses  of the expensive 10%.

Now let us imagine that some upper level Randian genius decides to emulate the Sears organizational structure. We’ll see how group size changes premium distribution.  Each work  group has to buy all of their own services in order to promote internal competition (thus destroying the theory of the firm as an institution).  That means 20 managers are now buying insurance for their employees independently as a small group.  It will be the same 1,000 people getting covered with the same 100 people who are expensive but what happens with rate structures?

Two things will happen.  First, the administrative costs will increase.  Some tasks are scale invariant while other tasks scale with membership.  As I’ve mentioned before, my work is basically scale invariant.  It takes me the same amount of time and thus costs if I work on a group of three members or three thousand members.  The first group has a cost attribution of several dollars per member while the second group has a cost attribution of pennies.  The medical loss ratio of 80% for small groups and individual markets and 85% for large groups recognizes this difference in group structure.

The second pricing reality is when statistics become important.  We know the general population risk of someone being expensive is 10%.  We would expect 5 people in each group of 50 to be expensive.  That is a naive and incorrect modeling of the situation.

A 50 person group that has complete statistical independence in a binomial distribution will have precisely 5 high cost people in the group 18% of the time.  43% of the time, that group will have less than 5 high cost people (including a 3% chance that either no or one person is high cost).  Conversely, there is a 12% chance that there are 8 or more high cost people in that group.  There is roughly a 50% chance that between 4 and 6 people in this group are high cost individuals.

Now we can make life a little bit more complicated for the actuaries.  Let us assume that health status is not randomly distributed throughout the company.  The health economics, actuaries and underwriters are reasonably representative of company health status.   Customer service and government compliance offices have quite a few people who have well known tales of woe.  Doctor outreach  is the company version of Pharma girls where they seem to exclusively recruit former Division 1 scholarship soccer players and swimmers.

The twenty groups of fifty people apiece would have seven groups with no more than two expensive people in it.  There are another six groups with at least eight particularly sick people in it.  Then there would be seven groups with four to six people who are high cost individuals.

So what does this mean?

Seven  groups would see roughly the same premiums that they saw when there was a single 1,000 member Mayhew Insurance group.  Seven groups that have very few high cost individuals in it, would see significantly lower premiums than they would under the 1,000 member company wide group.  Six groups would see significantly higher prices as they have more than average number of sick people in these small sub-groups.  One of these groups would see at least a doubling of premiums.

Slicing and dicing Mayhew Insurance from a single group to 100 groups would produce an even wider spread of premiums dependent on health status.  Breaking Mayhew Insurance from one group with a thousand members to a thousand groups with one member creates even more variance. Random noise becomes more important in small group sizes.

A large group smooths out the random noise and makes the cost of covering the sick lower due to both lower administrative costs and lower variance costs.  This point will be important on the next post on small group underwriting changes and the potential for rate shock next fall.

 

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50 replies
  1. 1
    David Fud says:

    Sampling error writ large. Good blog as always.

  2. 2
    Waldo says:

    The following will have some math and more importantly a lot of statistical intuition, so please bear with me.

    Spoiler alert!

  3. 3
    WereBear says:

    This is why BIG POOLS work so well! Great explanation.

    My pet peeve of late is how I’m allergic to those “healthy living” programs. Statistically, I understand that people who go to the gym and try to eat in healthful ways are much more likely to have lower health costs that the couch’n’cheeto crowd.

    But I’m an outlier; I live on the third floor, hike and snowshoe, and can’t handle gluten. Every time I fill out one of these things, I look like a couch’n’cheeto person. Very bad data collection!

  4. 4
    Jerzy Russian says:

    Customer service and government compliance offices have quiet a few people who have well known tales of woe.

    Nice piece. Too bad it will go over the heads of most of the policy makers in Congress.

    Put “quite” in place of “quiet” in the above.

  5. 5
    Baud says:

    If some people have rate shock because of reduction in variance, shouldn’t others have rate declines?

  6. 6
    Richard Mayhew says:

    @Baud: YEP — and that is Part 3 of this series (Pt. 1 was about underwriting techniques, Pt. 2 is statistics getting squirrelly with small groups and Pt. 3 is impact of underwriting changes).

  7. 7
    Baud says:

    @Richard Mayhew:

    You make statistics sexy, RM. Thanks.

  8. 8
    GHayduke (formerly lojasmo) says:

    @WereBear:

    Same boat.

  9. 9
    Soonergrunt says:

    Wow. As always, another great post. Given what you just told us, why would any organization split itself into smaller groups like that? Wouldn’t that increase the organization’s over-all costs, due to the replication of admin costs and the increased employer input for the higher risk groups, or do those organizations do what I suspect they do, and make a flat-rate contribution to employee health plans and make the employees eat the costs?

  10. 10
    WereBear says:

    @GHayduke (formerly lojasmo): Glad to know I’m not the only one!

    Now that the Internet has given me access to cutting edge science, it’s stunning to see how many “science authorities” don’t pay any friggin’ attention to it.

  11. 11
    OzarkHillbilly says:

    @WereBear:

    Statistically, I understand that people who go to the gym and try to eat in healthful ways

    I can’t wait for them to die of nothing.

  12. 12
    Richard Mayhew says:

    @Soonergrunt: They shouldn’t. I was using a really dumb decision to illustrate a point about risk pooling and how big pools smooth things over so statistical methods are actually useful

  13. 13
    Omnes Omnibus says:

    Richard Mayhew, great minds think alike or does E.J. Dionne filch column ideas from Balloon Juice?

  14. 14
    DecidedFenceSitter says:

    @Soonergrunt: Or to read it another way. Each of those smaller organizations are the individual companies that provide insurance, as well as various other groups; the overarching “organization” would be the United States of America.

    It is an argument for universal healthcare.

  15. 15
    Richard Mayhew says:

    @Omnes Omnibus: That was a bog-standard liberal argument, nothing to suggest filching

  16. 16
    Omnes Omnibus says:

    @Richard Mayhew: I know. I just found it interesting that, within a day of your post, a national columnist said more or less the same thing in the same way.

  17. 17
    Glocksman says:

    OT, but I found that linked article on the Sears organization model to be quite informative.

    That business model is bad enough when used in manufacturing and only works when you’re near or at the top of the heap.
    In retail, it’s a recipe for failure.

    There should be an unwritten law of business about adopting ideas parodied in a Dilbert strip.

  18. 18
    Baud says:

    @Omnes Omnibus:

    I enjoyed Dionne’s column on his missing mustard. And when his car got stuck in that open field…hilarious.

  19. 19
    Omnes Omnibus says:

    @Baud: The cat pics are a bit much though.

  20. 20
    WereBear says:

    @OzarkHillbilly: It’s a “correlation is not causation” kind of thing.

    The studies that show vegetarians have a health edge usually don’t factor in the marked lack of stuff like smoking tobacco, drinking to excess, avoiding caffeine and meth, not staying up all night… vegetarians are health conscious people, so they don’t do these things either. So is their health edge entirely due to avoiding meat?

    Seventh Day Adventists are known for their vegetarian ways and their general good health, but they also don’t smoke or drink, and a recent study showed that the ones who eat fish and nuts are healthier than those who don’t.

    So when I don’t check the little box that says “goes to the gym at least 3 times a week,” it doesn’t mean nearly as much as my insurance company thinks it does!

  21. 21
    slippytoad says:

    @WereBear:

    My pet peeve of late is how I’m allergic to those “healthy living” programs.

    Yea, because I have all kinds of time to spend updating a pretend nurse about my efforts to lose weight and so forth.

  22. 22
    Another Holocene Human says:

    @WereBear: Shouldn’t they ask you how much “walking” you do?

    If they don’t that’s a really bad questionnaire.

  23. 23
    mai naem says:

    @WereBear: It’s even more pronounced with vegans because you can’t do the artery clogging dairy studd and eggs.. But you also have to bring in the $$$ factor with vegans. Fake meat products aren’t particularly cheap. For your protein you would be dependent on soy,legumes and nuts.

  24. 24
    cintibud says:

    Hey Richard, really appreciate your posts. I am wondering if you or any other knowledgeable posters might be able to comment on my health care insurance situation.

    I work for a smaller company with a little over 100 employees, if that. The company offers PPO plans and HSA. As my wife and I are close to 60 the HSA plan makes no sense, so we go with the PPO.

    Here’s the rub – they only offer single and family plans and my wife cannot get insurance through her work. My monthly contribution to the family plan is 1004 dollars a month. It seems that is quite high for an employee sponsored plan, although the family deductible is lower than most – 1000/year (in network).

    Meanwhile the price for a single employee is only 171.00/Month for the same coverage.

    The difference is big enough that we looked at the exchanges for my wife, even though we do not qualify for any subsidies. We found that we could save close to 200 bucks a month for a gold plan for her while I switched to a single plan through my employer, but that was negated by the tax savings of sticking with my family plan – the premiums paid reduce my taxable income by an equal amount.

    I talked to our HR person and asked if an employee plus one plan could be offered. She indicated that that was looked at every year and every year it was decided that would raise everyone else’s costs too much.

    I would appreciate any comments on my situation. Can I suggest any alternatives to my employer? Do I have any other reasonable alternatives? Will this situation not change until we are eligible for Medicare? My take home pay has been decreasing for several years now due to the Health insurance cost increases being larger than my raises.

    Thanks

  25. 25
    Another Holocene Human says:

    @WereBear: Ditto with multi-vitamin chugging. The people who do this have other good health habits that account for the difference. More careful studies have actually shown increased risk of mortality taking certain vitamins.

    This is not to say that sometimes even Americans need vitamin supplements. Certain medications can deplete you of B-vitamins to the point that you need supplementation beyond that which the US government sprays on your bread. Some people get D-vitamin depleted annually (ask your doctor about this one). Some menstruating females get anemic every period. Some people have conditions like pernicious anemia that require B12 supplementation. (Some idiots are vegans, also, too.)

    I guess since this is a health thread let me share something I found out through my reading a few years back that only really fitness nuts are aware of–a lot of males end up with excessive Fe levels. It doesn’t help that a lot of Caucasian males have a gene that makes them load heme iron. What people don’t realize is that this is a risk factor for increased mortality and that the mineral imbalance is related to high cholesterol. The most effective solution for this problem is bleeding–you know, blood donation.

    Where I work a lot of the fitness nuts are big time blood donors but the non-fitness nuts seem unaware of this. Seems kind of dumb to me to start a potentially toxic statin drug or a wacky impalatable diet before checking if there isn’t a way to help somebody and get those cholesterol headline figures down.

  26. 26
    Another Holocene Human says:

    @mai naem: I doubt long term studies have been done on vegans, especially vegans on your average vegan diet, not rich, I-have-my-own-cook-and-nutritionists lifestyle vegans.

    It often “works” as a short-term crash diet, then followed by serious problems once the B12 runs out, the fatigue kicks in, and the victim starts eating vegan junk food 24/7 (oreos, for example–vegan) instead of lots of preparation intensive fruits and vegetables.

    The fake meat is terrible for you and a lot of vegans also don’t realize that you have to balance amino acids in your meals when you eat that way, causing additional health issues.

    And let’s not even get into people with digestive issues, like gluten issues, rheumatic symptoms from solenin (yup, some people can’t eat potato skin, eggplant, etc, even a little bit), and GERD or IBS caused by FODMAPS like, I dunno, BEANS.

    It doesn’t help that veganism is being sold basically like a religion with a lot of misleading crap and ChaBaD or fundegelical youth outreach-like pitches to mushy middlers who think it’s a weight-loss plan.

  27. 27
    RSA says:

    Good information! Understanding probability distributions is really tough for the average person, and maybe this will help, at least by explaining why it’s important in this context. (This kind of thing ought to be taught in schools.)

  28. 28
    Another Holocene Human says:

    @Omnes Omnibus: Has Dionne ever recanted his assertions that bishops’ delicate fee-fees are more important that women’s bodies, lives, and the security and financial well-being of their children?

    Access to contraception and appropriate gynecological care including abortion is a labor issue and a working family issue and an economic justice issue, unlike you’re a straight male Catholic, then it’s an icky ladyparts issue distracting us from the real issues, as if millions of female heads of household with their children trying to claw their way out of poverty don’t exist.

  29. 29
    Richard Mayhew says:

    What is sounds like is your employer is willing to pay a good amount of the premium for person #1 (the employee) and then offer very little assistance for anyone else. That is not uncommon.

    There is not a significant cost difference between Employee plus 1 adult versus Employee plus 1 Adult plus kids. There is some difference, but not a whole lot, so offering Employee plus 1 Adult might get you in the $900s instead of $1,005. Admin costs might eat that savings up if not many people take that option. Odds are there aren’t many good options. Maybe going Silver instead on the Exchange without subsidy if you need to save money?

  30. 30
    Omnes Omnibus says:

    @Another Holocene Human:

    Has Dionne ever recanted his assertions that bishops’ delicate fee-fees are more important that women’s bodies, lives, and the security and financial well-being of their children?

    Fuck if I know. I am not subscribing the the guy’s entire world view. I was just pointing out the similarity between his column and Richard’s post from yesterday.

  31. 31
    Sister Rail Gun of Warm Humanitarianism says:

    @Another Holocene Human:

    Seems kind of dumb to me to start a potentially toxic statin drug or a wacky impalatable diet before checking if there isn’t a way to help somebody and get those cholesterol headline figures down.

    Kickbacks. My ex-doctor pushed Lipitor so hard that I bluntly asked where Pfizer was sending her on vacation this year.

    I sometimes have an urge to drop in and tell her that my cholesterol numbers are just fine now, since my new doc is treating my thyroid imbalance. But then I remember that it would be pointless since thyroid meds don’t provide any financial incentives for the doctors.

  32. 32
    cintibud says:

    @Richard Mayhew: Thanks Richard. I didn’t know the cost savings of a +1 were so minor. Going silver then might be the best option as my wife is in excellent health and has very low medical expenses, Just get worried that can’t last for someone over 60

  33. 33
    jayackroyd says:

    Actuaries and underwriters love large groups. The bigger the better.

    How about everybody? Is that good for you? No?

    jes’ sayin’…..

  34. 34
    raven says:

    @cintibud: I heard that!

  35. 35
    Richard Mayhew says:

    @jayackroyd: Jay — completely agreed with your implied policy recommendation — how do we get there?

  36. 36
    jayackroyd says:

    @Richard Mayhew: I just think it’s interesting that these very helpful posts ultimately make an extended argument for, well, you know, a universal social insurance solution.

    And that this obviously good public policy is unattainable.

    The answer to your question is to choose better elected officials.

  37. 37
    Mnemosyne says:

    @jayackroyd:

    And that this obviously good public policy is unattainable.

    What timescale are you specifying for “unattainable,” though?

    Unattainable in, say, the next calendar year? Of course. Unattainable in the next decade? I think that’s arguable, especially as people who had no insurance or junk insurance realize what PPACA is doing for them and start clamoring for more.

  38. 38
    Mnemosyne says:

    @WereBear:

    I had the same reaction as AHH — it’s a badly written questionnaire if they’re asking about gym use and not daily exercise. How do people who walk 5 miles a day but never darken the door of a gym answer that question?

  39. 39
    boatboy_srq says:

    @WereBear: @slippytoad: @GHayduke (formerly lojasmo): A worthwhile program of this kind will allow for that with a handful of programs.

    For WereBear, for example, a pedometer linked to the program’s sytem will add all those stairs, hikes and snoshoeing events into your profile, and those steps will be counted toward your participation. My employer counts 40% of program participation in walked/run/climbed/whatever distances. A worthwhile program will also count routine physical exams, inoculations, etc as well as diet information toward participation as well.

    Speaking as one of the semi-healthy folks (gym membership that I don’t use nearly enough right now, take transit to work which adds 2000 steps/day to my count, former smoker whose quit was tracked by the program), while I’m not thrilled with either the programs or the monetary benefits provided, every little bit counts these days and any program worth participating in will provide a benefit for just continuing what you’re doing. My peeve is that at a solid 195# I crack the “overweight” threshold and get counseled on diet – when what I actually need is recognition that muscle mass alters BMI in ways the default calculations can’t accommodate.

  40. 40
    Elizabelle says:

    @raven:

    tell it, bro.

  41. 41
    Richard Mayhew says:

    @jayackroyd: you’re too smart to be that lazy

  42. 42
    Bill Arnold says:

    @Soonergrunt:

    Given what you just told us, why would any organization split itself into smaller groups like that?

    Read the linked story about the Sears re-org along these lines. It didn’t work for them.

  43. 43
    StringOnAStick says:

    @Another Holocene Human: Actually, there has been some longer term studies on vegans; less heart issues, higher cancer rate than the general population.

    And I hear you on the ‘veganism as a personal religion” thing. A good friend of ours died of cancer; his widow used part of his memorial service to attempt to guilt trip everyone into “committing to one vegan meal a week” in his memory, and if you didn’t then you were dishonoring him. The vegan attendees thought it was righteous, and everyone else thought it was obnoxious and something the person we were memorializing would have never, ever done; he simply was not concerned about how other people chose to eat. His wife however, has made it real clear that unless you become vegan, you are a bad person. She once told me that she’d rather that all animals humans use for food be extinct, including honey bees.

  44. 44
    WereBear says:

    @StringOnAStick: His wife however, has made it real clear that unless you become vegan, you are a bad person. She once told me that she’d rather that all animals humans use for food be extinct, including honey bees.

    Ah. I’m familiar with the religious fervor.

    And just how “humanitarian” is animal genocide, hmmmm?

  45. 45
    jayackroyd says:

    @Mnemosyne: “Unattainable” was sarcastic. A social insurance program obviously attainable now, in policy terms. The barriers are graft and corruption, and you’re guess is as good as mine when those barriers can be circumvented in the interests of obviously good public policy.

  46. 46
    Mnemosyne says:

    @StringOnAStick:

    I’m lucky that, so far, my friends who have become vegans did it strictly for health reasons (he has a serious family history of cardiac problems; she has a family history of Type II diabetes even at normal weight) and are more than happy to eat whatever vegetarian dishes are available in restaurants (though they ask the waiter to leave off any obvious things like cheese or eggs).

    But, yes, the evangelistic type is definitely prevalent here in California. If you want me to eat the way you do, you need to do it by convincing me that it tastes good. If something tastes like crap, I’m not going to eat it no matter how “moral” it is.

  47. 47
    Mnemosyne says:

    @jayackroyd:

    The barriers are graft and corruption, and you’re guess is as good as mine when those barriers can be circumvented in the interests of obviously good public policy.

    Uh-huh. “Graft and corruption,” not a political party that is deliberately blocking any progress forward.

  48. 48
    jayackroyd says:

    @Richard Mayhew: Right now I think it’s important to keep pointing out that it doesn’t have to be this way. Tedious as that may be. It’s not unrealistic to try to extend the large populations of the socially insured; the recent CBO news story on people locked into jobs by insurance requirements has the obvious implication that expanding government pools. Lowering Medicare eligibility to 55 and raising the income ceiling level for CHIP coverage for kids 2 and other would expand the socially insured population.

    But these half measures wouldn’t be necessary, or at least the discussion would be more interesting if a public option were actually put to a vote.

  49. 49
    jayackroyd says:

    @Mnemosyne: Yes, graft and corruption. You don’t think that’s true? The Senate’s sudden inability to pass anything, even with a 60 seat Dem majority, reflected deep policy concerns about well designed health care systems?

    As I just said to Richard, Reid could introduce a public option today, could introduce age 55 Medicare eligibility today, could propose legislation to Federalize Medicaid today. And then they could run on those issues.

    When we are told that the PPACA was the best we could get, how can we be talking about anything other than graft and corruption? There were no GOP votes for the bill, remember?

  50. 50

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