Category Archives: Mathematics

CBO projection of $8 billion from Risk Corridors is baffling

The Congressional Budget Office just issued a report that assumes the Affordable Care Act system of individual policies sold in Exchanges without medical underwriting can remain relatively stable. Tightly bound up with that assumption is its prediction about a controversial ACA program known as “Risk Corridors” that requires profitable insurers to pay the federal government up to 80% of profits they make on policies sold on the Exchanges but that also requires the federal government to pay insurers up to 80% of the losses they suffer from policies sold on the Exchanges.  The CBO now believes it has enough information to predict that Risk Corridors will actually make money — $ 8 billion over three years — for the government at the expense of insurers.

This CBO prediction of $8 billion in federal revenue, which has gained much publicity,  pulls the rug out from critics of the ACA such as Senator Marco Rubio who have introduced legislation that would repeal Risk Corridors as an insurance industry “bailout.” Such a blunting of Senator Rubio’s proposed repeal legislation is crucial in the ongoing battle over the ACA because repeal of Risk Corridors could  result in insurers (who just might not believe the CBO’s numbers) exiting the Exchanges for fear of having no government protection against losses resulting from unfavorable experiences in the new market the government has created. On the other hand, if the CBO is just getting its number wrong, Rubio’s case for repeal of Risk Corridors remains as strong (or problematic) as it ever was. The CBO projection is also important because Risk Corridors nets the government money if and only if the ACA works, insurers are able to make some profits, and a death spiral never takes hold. And this, as readers of this blog are aware, is a prediction about which many have serious doubts. 

Here’s the short version of the rest of this post.

I’ve done the math and I don’t see how the CBO is getting this $8 billion number unless it is assuming either very high enrollment in policies covered by Risk Corridors or very high rates of return made by insurers.  Or it made a mistake. I don’t think the CBO’s own numbers support very high enrollment in policies covered by Risk Corridors and I don’t believe either an emerging reality or the CBO’s own rhetoric justify assuming very high rates of return.  So I think the CBO ought to take a second look at its prediction. People should not yet make policy decisions based on the CBO estimate.

Reader, you now have a choice. I’m afraid that the next several paragraphs of this post become very technical. It’s kind of forensic mathematics in which one attempts to use statistics and numerical methods to deduce the circumstances under which something said could be true.  If that sounds dreadful, scary or tedious, I would not protest too loudly were you to skip ahead to the section titled “How could I be wrong?”  Before you leave, however, realize that what I am attempting to accomplish in the part you skip is a form of proof by contradiction. I prove that if what the CBO was saying were true, then insurers would have to be making 8% profit.  But nobody, including the CBO thinks they will make 8% profit, so the $8 billion number can’t be right.

On the other hand, dear reader, if you liked the Numb3rs television show (including my minor contributions thereto) or math or detective work or just care a lot about the Affordable Care Act, the rest of this post is for you.  What I am about to discuss is not only exciting math, but also the soul of the Affordable Care Act — whether the individual Exchanges without medical underwriting can remain relatively stable.

Forensic mathematics in action

Conceptually, here’s the calculation one needs to do.  What we want to figure out is the distribution of insurer profits (measured as a ratio of expenses divided by premium revenues)  upon which the CBO must be relying. I assume the CBO is using a  member from the “Normal” or “Lognormal” family of distributions because those are typical models of financial returns and there is little reason to think that the distributions of insurer profits (expenses minus revenues) will materially depart from those assumptions.  To continue reading this post, you don’t have to know exactly what those distributions are except that they look for our purposes like the “bell curves” you have seen for many years.  I’ve placed a graphic below showing some normal (blue) and lognormal (red) distributions. Although it should not matter all that much, I’m going to use a lognormal distribution from here on in because the ratio of insurer expenses to premiums should never be negative and the lognormal distribution, unlike its normal cousin, never takes on negative values.

Examples of probability density functions for normal and lognormal distributions
Examples of probability density functions for normal and lognormal distributions

The problem is that there are an infinite number of lognormal distributions from which to choose.  How do we know which distribution the CBO is emulating in its computations?  How do we know just how positive the CBO assumes the individual Exchange market is going to be on average or how dispersed insurer profits are going to be? As it turns out, the complexity of the lognormal distribution can be characterized with just two “parameters” often labeled μ (mu, the mean of the distribution) and σ (sigma, the standard deviation of the distribution).  Once we have those two parameters (just two numbers), we can deduce everything we need about the entire distribution.

Now, to solve for two parameters, we often need two relationships. And, thoughtfully, the CBO has given us just enough information.  It has told us how much money in total it intends to raise from Risk Corridors ($8 billion) and the ratio (2:1) between money it collects from profitable insurers and the money it pays out to unprofitable insurers. These two facts help constrain the set of permissible combinations of Risk Corridor populations (the number of people purchasing policies in plans subject to the Risk Corridor program) and insurer profitability distributions. What I want to show is that it takes an extremely high Risk Corridor population in order to get rates of return that are not way larger than most people — including the CBO — think likely to occur.

I first want to calculate the amount of money insurers would pay to HHS under the Risk Corridors program if the total amount of premiums collected were $1. Some of the payments — those by highly profitable insurers  —  will be positive.  Those by highly unprofitable insurers will be negative. To do this I take the “expectation” of what I will call the “payment function” over a lognormal distribution characterized by having a mean of  μ and a standard deviation of  σ.  By payment function, I mean the relationship shown below and created by section 1342 of the ACA, 42 U.S.C. § 18062. This provision creates a formula for how much insurers pay the Secretary of HHS or the Secretary of HHS pays insurers depending on a proxy measure of the insurer’s profitability. The idea is to calculate a ratio of “allowable costs” (roughly expenses) to a “target amount” (roughly premiums).  If the ratio is significantly less than 1 (and outside a neutral “corridor”), the insurer makes money and pays the government a cut. If the result is significantly greater than 1 (and outside the neutral “corridor”), the insurer loses money and receives a “bailout”/”subsidy” from the government.  The program has been referred to with some justification as a kind of “derivative” of insurer profitability, the ultimate “Synthetic CDO.

The graphic below shows the relationship contained in the Risk Corridors provision of the ACA.  The blue line shows the net insurer payment (which could be negative) to the government as a function of this proxy measure of the insurer’s profitability. Ratios in the green zone represent profits for the insurer; ratios in the red zone represent losses. Results are stated as a fraction of  “the target amount,” which, as mentioned above, is, roughly speaking, premium revenue.

How much the insurer pays (positive) or receives (negative) under Risk Corridors as a function of  measurement of profitability
How much the insurer pays (positive) or receives (negative) under Risk Corridors as a function of a ratio-based measurement of profitability

When we do this computation, we get a ghastly (but closed form!) mathematical expression of which I set out just a part in small print below. (It won’t be on the exam). I’ll call this value the totalPaymentFactor. Just keep that variable in the back of your mind.

Excerpt of the formula for insurer total payout
Excerpt of the formula for insurer total payout

I next want to calculate the amount of payments profitable insurers will make to HHS. To do this, we truncate the lognormal distribution to include only situations where the ratio between premiums and expenses is greater than 1. Again, we get a pretty ghastly mathematical expression, a small excerpt of which is shown below. I will call it the expectedPositivePaymentFactor.

Formula for expected negative insurer payments under risk corridors over a truncated lognormal distribution
Formula for expected negative insurer payments under risk corridors over a truncated lognormal distribution

Finally, I want to calculate the amount of payments unprofitable insurers will receive from HHS. To do this, we truncate the lognormal distribution to include only situations where the ratio between premiums and expenses is less than 1. Again, we get a pretty ghastly mathematical expression, which, for those of you who can not get enough, I excerpt below. I will call it the expectedNegativePaymentFactor.

Formula for expected positive insurer payments under risk corridors over a truncated lognormal distribution
Formula for expected positive insurer payments under risk corridors over a truncated lognormal distribution

The CBO has told us in its recent report that the government will collect twice as much from profitable insurers (expectedPositivePaymentFactor) as it pays out to unprofitable ones (expectednegativePaymentFactor).  We can use numeric methods to find the set of μ, σ combinations for which that relationship exists.  The thick black line in the graphic below shows those combinations.

 

Black line shows combination of mu and sigma that result in the correct ratio of positive and negative insurer payouts under Risk Corridors
Black line shows combination of mu and sigma that result in the correct ratio of positive and negative insurer payouts under Risk Corridors

To determine which point on the black line above, which combination of the parameters μ, σ , is the actual distribution, we need to use our information about the totalPaymentFactor.  The idea is to realize that the totalPaymentFactor must be equal to the quotient of the CBO’s estimated $8 billion and the total premium collected by Risk Corridor plans over the next three years.  But we know that the total premium collected should be equal to the mean premium charged by the Exchanges multiplied by the number of people in Risk Corridor plans. Some math, discussed in the technical notes, suggests that the mean premium under the ACA is about $3,962. And the CBO accounts for 8 million people being in Risk Corridor plans in 2014, 15 million being in Risk Corridor plans in 2015 and 25 million being in Risk Corridor plans in 2016. This means that the total premiums collected by insurers under Risk Corridor plans over the next 3 years should be about $190.2 billion. And this in turn means that the totalPaymentFactor must be 0.042.

Ready?

It turns out that of all the infinite number of lognormal distributions there is only one that satisfies the requirements that (a) the government will collect twice as much from profitable insurers (expectedPositivePaymentFactor) as it pays out to unprofitable ones (expectednegativePaymentFactor) and (b) for which the totalPaymentFactor takes on a value of 0.042. It is a distribution in which the mean value is 0.923 and the standard deviation is 0.113.  I plot the distribution below. A dotted line marks the break even point for insurers.  Points to the left of the break even line correspond with profitable insurers; points to the right correspond with unprofitable insurers.

Lognormal distribution of insurer profitability consistent with CBO data
Lognormal distribution of insurer profitability consistent with CBO data

Here are some factoids about the uncovered distribution.  The  average insurer will have expenses that are 92.3% of premiums and the median insurer will have expenses that are only 91.6% of profits. In other words, they will be making 7.7 cent and  8.4 cents respectively on every dollar of premium they take in.  For reasons discussed below, this is a difficult figure to accept. It is particularly difficult in light of the pessimistic news that is emerging about things such as the age distribution of enrollees , reports from Deutsche Bank that one of the largest insurers in the Exchanges, Humana, expects to receive (not pay!) a lot of money under the Risk Corridors program, the hardly exuberant forecasts of other publicly traded insurers about the ACA, and the recent general downgrading of the insurance sector by Moody’s partly because of the ACA.

Implicit in my finding about the most likely distribution of profitability is an assertion by the CBO that 76% of insurers will be profitable under the ACA while 24% will be unprofitable. About 17% will be sufficiently unprofitable that they will receive subsidies (a/k/a bailouts) from the federal government and 9% will be sufficiently unprofitable that their marginal losses will be covered at 80%. Only 15% of insurers will be “inside” the risk corridor and neither pay nor receive under the program.

How could I be wrong?

I feel  confident that I’ve done the ” gory math part” of this blog post correctly. Mathematica, which is the software I’ve used to do the integral calculus and the numeric components involved just does not make mistakes.  I also feel pretty confident that I understand how the Risk Corridors program works under section 1342 of the ACA.  That’s kind of my day job. And so, readers who skipped down to this part, I do believe that if the CBO were right about the $8 billion, that could only happen if insurers were, on average, earning an implausible 8% in the Exchanges.

If I’m wrong, then, it is because, except for the little issue I will mention at the end, I have made bad assumptions about the total premiums insurers expect to collect over the next three years in policies covered by Risk Corridors. That error could come from two sources. I could have the mean premium per policy wrong or I could have the relevant enrollment wrong. Let’s look at each of these.

Could I be wrong about the mean premium?

I computed the mean premium in the computation above by using data collected by the Kaiser Family Foundation on the ratio of premiums by age under most insurance plans and the typical Silver plan premium for a 21 year old (non-smoker). I then used the original forecast about the age distribution of insureds to compute an expected premium.  I got $3,962.  And this number seems very much in line with earlier HHS estimates, which were that mean premiums would be $3,936. So, I think I have the mean premium correct.

Could I be wrong about the number of people in Risk Corridor plans?

I computed the number of people enrolled in policies covered by Risk Corridors by looking at the CBO’s own figures.  I’m not vouching that the CBO is right in its projections, but this is not the day to argue that point.  The CBO now says (Table B-3, p. 109) that individual enrollment in the Exchanges will be 6 million, 13 million and 22 million respectively over the next three years.   And it says that employment-based coverage purchased through Exchanges (which I assume are SHOP Exchanges) will be 2 million, 2 million and 3 million respectively.  So , by addition, that’s where the figures I used of 8 million,  15 million and 25 million come from.  I’m not aware of anyone else who would purchase a policy subject to Risk Corridors. Again, bottom line, I don’t think I’m doing anything wrong here.

The little issue at the end: Could ACA definitions be responsible for the incongruity?

The only other conceivable explanation of the divergence between the CBO figures and my analysis is that I am failing to take a subtlety of Risk Corridors into account.  Remember, careful readers, that sentence earlier up that started out: “The idea is to calculate a ratio of “allowable costs” (roughly expenses) to a “target amount” (roughly premiums).” I stuck in the “roughlies” because the “allowable costs” are not exactly expenses and the “target amount” is not exactly premiums. When you look at the statute and the regulations, you can see that both of these terms are tweaked: basically you subtract administrative costs from both values.  And you subtract reinsurance payments from expenses — but that makes sense because the insurer reduced premiums in anticipation of those reinsurance payments.

So, in the end, I don’t see why these subtleties should affect my analysis in any significant way. But I am not infallible. And I do pledge that if someone points out an error to me, I will dutifully assess it and report it.

Sensitivity Analysis

Out of an abundance of caution, however, I have rerun the numbers on the assumption that premium revenue from policies subject to Risk Corridors is 50% greater than my original estimate either because of an underestimate of per policy costs or a failure to understand that there is some additional group within Risk Corridors protection.  When I do that, though, I find that the ratio of expenses to premiums is 0.943, meaning that insurers are still earning a pretty substantial 5.6%.  Although that is more believable than the earlier figure of 7.7%, it is still pretty high. 

Conclusion

To be honest, it makes me very nervous to say that the CBO did its math wrong or, worse, to accuse it of bad faith.  These are intelligent, educated professionals and they have access to a lot more data and a lot more personnel than I do.  Here at acadeathspiral  it’s just me and my little computer along with some very powerful software.  On the other hand, it’s not as if the CBO hasn’t been wrong before. It assumed earlier that the government would reduce its deficit $70 billion over 10 years as a result of Title VIII of the ACA (the so-called CLASS Act on long term care insurance) when many independent sources believed — rightly as it turned out — that the now-repealed CLASS Act was obviously structured in a way that could never fly.  The CBO assumed in July 2012 that 9 million people would enroll in the Exchanges in 2014, a number that is now down to 6 million. And, while there are explanations for each of these changes, the bottom line is that CBO is fallible too.

So, if I might, I would strongly urge the CBO to double check its numbers and provide more information on the data it relied upon and the methodology it employed in getting to its results.  I’d ask Congress, which has ongoing oversight of the ACA, to insist that the Congressional Budget Office, which is exempt from Freedom of Information Act requests from ordinary citizens, provide further detail.  American healthcare is indeed too important to have policy decisions made on the basis of what could be some sort of mathematical error.

Really Technical Notes

  1. I’m using a reparameterized version of the lognormal distribution that permits direct inspection of its mean and standard deviation rather than the conventional one, which in my opinion is less informative.   The explanation for doing so and the formula for reparameterization is here.
  2. To compute the average premium, I took the premium ratios used by the Kaiser Family Foundation, calibrated it so that a 21 year old was paying the national average payment for a silver plan purchased by a 21 year old. I then computed the expected premium over the distribution of purchase ages originally assumed by those modeling the ACA.
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Could the “low” ACA premiums just be “the winner’s curse” in action?

Those optimistic about the success of the Affordable Care Act have been noting over the past several months that the premiums offered by insurers have been lower than those earlier forecast.  But if one looks carefully at the original rhetoric, the comparison tends to be between some of the lowest premiums offered within a jurisdiction and those originally forecast.  And this metric, according to ACA proponents, is appropriate because they expect consumers to focus purchases on the lowest cost policies.

But what if the lowest premiums are lower than expected not because the mix of purchasers is thought to be fine or because of cost cutting measures enabled by the ACA, but simply because all this metric exposes is the work of the insurers who priced their policies below actual risk?  The “winner’s curse” is the term economists and game theorists give to situations in which, in an atmosphere of uncertainty, people bid on an item in an auction environment. What will often happen is that the “winning” bidder will tend to be one that loses money.

It is quite possible that all we are seeing with “low” ACA pricing, as measured by ACA proponents, is “the winner’s curse” in action.  We may well be looking at insurers who (a) got it wrong or (b) thought the government would most greatly subsidize their losses or (c) for strategic reasons, decided to  sell a “loss leader” in the first year or so of the ACA in order to lock consumers into their networks and their doctors with the idea that they could substantially raise premiums in the future. If this hypothesis is correct, individual policies under the Exchange are a lot less stable than many ACA proponents are asserting.

To summarize the results of the computations shown below, if the mean premium charged by insurers selling a type of policy (Silver HMO Plans, for example ) in a given geographic region (Harris County, Texas, for example) reflects the true risk posed by ACA policy purchasers, about 20% of the low bidders — the ones that I suspect will get a disproportionate share of the business — stand to lose at least 20% on their policies before the Risk Corridor program bails them out.

The big story as the ACA unfolds may be that some insurers — the ones who ended up with the business — simply made an error of exuberance in a new market and priced their policies too low. While these insurers will, thanks to a federal subsidization program for losing insurers called Risk Corridors, not entirely lose their shirts in the first year of the program as a result, they do stand to lose a lot of money that they will likely want to make up in any subsequent years of the Affordable Care Act.

New data analysis finds significant dispersion in plan premiums

This post will contribute some new data analysis that suggests the likelihood of the winner’s curse materializing as well as the magnitude of such a curse. Basically, I have sucked into my computer official government data on the 78,000 plans sold on the federal marketplace and done a lot of number crunching.  The data shows a significant dispersion of prices offered by insurers for plans in the same geographic area,  of the same metal tier and offering the same form of coverage (PPO, POS, HMO, or EPO) .  While this dispersion does not prove that the low prices are outliers reflecting either miscalculation by some insurers or only-temporary use of low prices, it does  suggest a significant possibility that such is the case.

Let’s take an example. Here are the prices offered where I live, Harris County, Texas — mostly Houston — for an HMO Silver Policy to a couple with two kids. The couple has an average age of 50 years old.  We’ll call this hypothetical family “The Chandlers,” as a matter of convenience. The graphic shows the dispersion of premiums normalized so that the lowest price for a given policy is given a value of 1.

Dispersion plot for Harris County, Texas, Silver HMO policies sold to Couple, aged, 50 with two children
Dispersion plot for Harris County, Texas, Silver HMO policies sold to Couple, aged, 50 with two children

As one can see, for the Harris County, Texas policies shown here, although there are three policies that have premiums fairly close to the minimum, there are, however, two policies that have premiums  more than 30% more than the minimum. If the mean premium estimated by insurers is “correct,” the insurer selling a Silver HMO policy at the lowest price will lose about 17%. The implication, if the Harris County plan is representative and if the mean premium is closer to the true risk than the low premium, is that the insurers most likely to win business due to low prices are likely to lose a considerable amount of money.

There are several potential rejoinders to the suggested implication of the graphic.  Let me address each of them in turn.

Might Harris County, Texas be unusual?

One response is that the example for Harris County, Texas Chandlers is unrepresentative. Houston, for example, has some very fancy hospitals and some not so fancy hospitals; so maybe premium dispersion for Harris County simply reflects whether one has access to the fancier hospitals (and the doctors who have admitting privileges to them).  I have considered this possibility and find that, actually, the example I provide is pretty representative. Here, for example, are 20 randomly selected examples. For each plan, I show the amount the low bidder would lose if the average premium is “correct,” the dispersion of premiums, and the plan and purchaser randomly chosen. Of the ones in which there are any significant number of policies available, most of the premiums show a dispersion pattern qualitatively similar to that in Harris County for The Chandlers. Indeed, some of the random examples show dispersion considerably greater than that for the Harris County silver HMO policies. Except where there is little competition for plans and the low bidder is thus selling at the average price, the result presented above does not look like a fluke.

Dispersion Plot and potential losses of low bidder for 20 random plans and purchasers
Dispersion Plot and potential losses of low bidder for 20 random plans and purchasers

I can double check this result by computing for 5,000 random combinations of plans and purchasers the losses of the low bidder if the true risk was equal to the mean premium charged for policies and purchasers of that type. The graphic below shows the “survival function” (or “exceedance curve”) for the resulting distribution of these losses.  The value on the y-axis is the probability that the losses will exceed the value on the x-axis. The results shown below confirm that the situation for Harris County Silver HMO plans sold to The Chandlers is not all that unusual.  As one can see, losses of more than 10% take place more than 30% of the time and losses of more than 20% take place about 17% of the time. A rather scary picture.

Exceedance curve of the distribution of losses of low bidders for random plan-purchaser combinations on the assumption that the mean premium represents the true risk
Exceedance curve of the distribution of losses of low bidders for random plan-purchaser combinations on the assumption that the mean premium represents the true risk

In fact, however, the situation may be even worse than depicted in the graphic above. Sometimes the losses computed by this method are low because the low bidder is also the only bidder.  If we consider situations in which there is more than one bidder, here is the resulting survival function (exceedance curve) of the distribution. As one can see in the graphic below, the distribution of risks is shifted slightly to the right.  Now 40% of the low bidders stand to lose at least 10% and about 21% stand to lose at least 20%.

Exceedance curve of the distribution of losses of low bidders for random plan-purchaser combinations where at least two premiums exist on the assumption that the mean premium represents the true risk
Exceedance curve of the distribution of losses of low bidders for random plan-purchaser combinations where at least two premiums exist on the assumption that the mean premium represents the true risk

Maybe the higher priced policies are better?

Another potential explanation for price dispersion is that, even if the policies are priced differently, that does not mean that the cheapest policies are selling for too low a price.  All Silver HMO policies sold in Harris County, Texas to The Chandlers may not be the same.  Some may have different deductibles or different networks.

The first response to this rejoinder is that the actuarial value of the policy — the relationship between expected payments by the insurer and premiums — should be about the same for each metal tier of policies. Silver policies should all have actuarial values, for example, of 70%.  So it should not be the case that one silver policy has cost sharing different than the cost sharing of another silver policy in a way that would affect the premium charged for the policy. Moreover, the calculations underlying this post keep HMOs, PPOs, POS plans and EPOs apart; so it should not be the case that observed premiums differ because, perhaps, the cheaper plans are HMOs whereas the more expensive ones are PPOs.

Of course, cost sharing is not the only way in which policies within a given location, of the same metal tier and sold to the same purchaser could vary.  One policy might offer richer benefits than another.  It could have a richer network with more doctors available or more prestigious and expensive hospitals inside the network. Could that be responsible for a substantial part of the premium dispersion we see?  It’s impossible to tell for sure — the data published by HHS does not attempt to quantify the richness of the network being offered.  I do find it difficult to believe, however, that such differences are responsible for the entirety of differences in excess of 20% between the low bidder and the mean bid, or, for that matter, differences in excess of 40% that sometimes occur between the low bidder and the higher bidders.

Maybe the average premium is meaningless; the low bidder got it right

Of all the potential rejoinders I have considered, the one now forthcoming is the one that is most troubling. There is nothing the data standing by itself can tell us whether most of the insurers have it right and the low insurers are about to lose their shirts or whether the low insurers have been more insightful or have managed to keep costs down such that they will break even (or even make money) selling their policies at low premiums.  And, yet, I am doubtful. One can view the mean or median of the premiums as an “ensemble model” of the true cost of providing care under the Affordable Care Act. And there is research (examples here, here and here) suggesting that ensemble models predict better in many open-textured situations than individual models.  So, while it’s possible, I suppose, that in every jurisdiction the low bidder is predicting more accurately than the group of insurance companies as a whole, such a result would be surprising.  A far simpler explanation is that the low bidder — the one who is likely to win business from price sensitive insureds — is succumbing to “the winner’s curse.”

Maybe the disaggregation of plans is misleading

This is a very technical objection, but consider carefully what I have done.  I have looked at all policies of a given metal tier and a given plan type in a given geographic location sold to a certain family type such as “all silver policies in Harris County, Texas, sold to The Chandlers.”  But, really, plans are sold not to just to The Chandlers but to all family types. So, it could conceivably be that while the plans sold to the family type I am looking at are highly dispersed, the average premiums over all family types (weighted by prevalence of the family type) are far less dispersed.  This strikes me as unlikely — why would an insurer be overcharging one family type relative to another — but you can not rule it out a priori.  Maybe — just maybe — the dispersion we are observing is not real; it is just an artifact of my disaggregation of the data.

I would, of course, love to aggregate the data and see if the high degree of dispersion persists. The difficulty with this cure comes with the problem of weighting the data.  We don’t know the distribution of policies sold among family types.  We don’t know, for example, whether The Chandlers constitute 2% of policies sold or 5% of policies sold. So, I can’t  perform a perfect aggregation of the data. One way to get a feel for the objection, however, is to simply take an unweighted average of the premiums for all the family types identified in the database and aggregate it that way.  This is far from perfect, and we could spend a lot of time refining it, but it should provide a clue as to whether the disaggregation of plans is significantly responsible for the high degree of observed dispersion.

The graphic below shows the exceedance curve for losses of the low bidder assuming the mean premium is the true risk based on an unweighted average of family types purchasing the policies.  One can see that 20% of the low bidders will lose at least 20% if it turns out that the mean premium charged for similar policies reflected the true risk. Upwards of 35% will lose more than 10%. A quick comparison of this curve with those above shows that it is essentially the same.  There is nothing that I can see suggesting that the fundamental result shown in this blog entry — high dispersion of premiums among what should be similar policies and the potential for significant losses by low bidders — is an artifact of the methodology I have employed.

Exceedance curve for losses of low bidder assuming mean premium is true risk for aggregated purchaser types
Exceedance curve for losses of low bidder assuming mean premium is true risk for aggregated purchaser types

Conclusion

In the end, even the extensive data that the government is put out is insufficient to determine definitively whether the lower priced insurers in the individual Exchanges are about to lose money. There are more optimistic interpretations of the observed premium dispersions: maybe it is the low bidders who are “getting it right” or maybe the low bidders have just found ways to keep costs down through better negotiating or cheaper care networks. But if these optimistic explanations prove insufficient, what this post shows is that while some insurers will likely do just fine there are a substantial minority of insurers who are about to get bitten by the “winner’s curse” and get a large volume of purchasers for whom the premiums charged will be insufficient to defray the expenses incurred.

 Technical Notes

The data used here was taken directly from the United States Department of Health and Human Services. It was analyzed using Mathematica software, which was also used to produce the graphics shown here.

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Insurer losses in Exchanges of 10% not unlikely

Experts who have taken a look at the Affordable Care Act have separately considered the effects of three possible sources of unexpected losses by insurers selling policies in the individual Exchanges: purchasers being older than originally projected, more purchasers being women than originally projected, and purchasers having poorer health than originally projected.  And, at least with respect to the potential for age-based problems, the prestigious Kaiser Family Foundation has given supporters of the ACA considerable comfort by saying, worst case, older purchasers might result in only a 2.5% increase in insurer costs.  But no one to my knowledge — until now — has carefully considered the combined effects of these three sources of potential cost increases and, most likely, pressure for future premium increases.

I have now made an effort to consider the effects of these three sources of insurer losses acting together. Based on that effort, which represents the culmination of work over the past month, I believe it quite possible that insurer losses could amount to 10%, approximately 4% due to purchasers being older than expected, 1% due to greater purchases by women, particularly those in their 20s and 30s, and another 5% due to purchasers having poorer health than expected.

There are four major caveats that should be emphasized up front.  (1) These figures are estimates with large error bars; and anyone pretending to great exactitude in this field, particularly as much of the best data is not yet available, is, I suspect, likely pursuing more of a political agenda than a scholarly one. Losses could be close to zero; losses could be in the 15% range. Still, as I am going to show, significant losses are a serious possibility. (2) These losses are computed without consideration of “risk corridors” under section 1342 of the Affordable Care Act. That provision basically calls on taxpayers to pay insurers losing money on the Exchanges a significant subsidy. After consideration of Risk Corridors, average net insurer losses could range anywhere from close to zero to around 6%-7%. (3) These are national figures.  There are states such as West Virginia in which the age distribution is considerably worse right now than it is nationally.  One should not expect any of the rates of insurer losses (or profits) to be uniform across states or, indeed, across insurers. The figures developed here are an attempt at a  rough average. (4) The figures are based on the last full release of data by HHS on enrollment in the Exchanges; if matters change and, for example, the proportion of younger enrollees grows or the proportion of men grows, the loss rates I project here are likely to decline.

The graphic below summarizes my conclusions.  It shows insurer losses (or gains) as a function of a “health age differential” under two scenarios. By health age differential I mean the difference in ages between someone who has the expected health expenses of the actual enrollee and the chronological age of the enrollee.  Thus, if an enrollee was actually 53 but had the health expenses of an average 57 year old, their health age differential would be 4.  If they had the health expenses of a 50 year old, their health age differential would be negative 3. The yellow line shows insurer losses as a function of the health age differential assuming that the joint distribution of gender and age stays the way it was when HHS last released data.  The blue line shows insurer losses as a function of the health age differential assuming that the joint distribution of gender and age ends up the way it was originally projected to be.  As enrollment under the ACA increases and the proportion of younger enrollees increases, one might expect the ultimate relationship to head from the yellow line down to the blue line.  My assertion that losses could well be 10% is based on the assumption that the joint distribution of gender and age stays the way it is now but that the health of enrollees is equivalent, on average, to those 2 years older than their chronological age.  An assumption that enrollees could have health equivalent, on average, to those 4 years older than their chronological age, yields insurer losses of greater than 15% assuming the current joint distribution stays in place and about 10% assuming the original distribution ends up being correct.

The key graphic for this entry
 

The graphic above is useful because it gives what hitherto had been missing in discussions of problems in the individual Exchanges: some sense of the relative magnitude of problems created by age-based adverse selection (older people enrolling disproportionately) and health-based adverse selection (sicker people enrolling disproportionately). Roughly speaking, the degree of price increases induced by the current age and gender imbalances is roughly equivalent to what would occur if the health of the enrollees was, on average, equivalent to those of persons 2.5 years old than they actually are.

So what does it all mean?

At some point,  a journalist is likely to ask me what this all means?  Is there going to be a death spiral?  I would say we are right on the cusp.  Losses of 10% by insurers relative to expectations, coupled with whatever increase results from medical inflation, isn’t so enormous that I could say, yes, for sure we are heading into a death spiral. But neither is it such a small number that the risk can be ignored.  Moreover, as noted above, the 10% figure is a national average and we need to reduce it because of risk corridors.  In some states, however, where the age and gender figures may be worse or the health of enrollees is particularly problematic or where insurers just bid too low and the winner’s curse overtakes them, I still believe there is a substantial risk of a serious problem. In other states, where age and gender figures are better or insurers more accurately forecast the health of their enrollees, the risk of a death spiral is minimal. And, of course, the more people that actually end up purchasing policies in the Exchanges over the next few months, regardless of whether they come from the ranks of the previously uninsured or those who find that they can not keep their current policies, the more stable the system of insurance created by the ACA is likely to be.

So, after a lot of research, I feel more confident than ever in giving a lawyer’s answer —  it all depends — and a cliche — we’re not out of the woods yet.

Computation details

The results obtained here are based on essentially the same data as user by the Kaiser Family Foundation, which includes data on the relation between age and premium under typical plans, data from the Society of Actuaries (SOA), also used by Kaiser, on the relation between gender, age and expected medical expenses, and my own prior work attempting, based on data from the Department of Health and Human Services released earlier this month, to derive a joint distribution of enrollment in the individual Exchanges based on age and gender.  And, although the math can get a little complicated, the basic idea behind the computations is not all that difficult. It is essentially the computation of some complicated weighted averages.  Each combination of gender and age has some expected level of insurance cost (computed by the Society of Actuaries based on commercial insurance data) and some expected premium (computed by Kaiser based on a study of the ACA). Thus, if we know the joint distribution of gender and age, we can weight each of those costs and each of those premiums properly.

There are three areas of the computation that prove most challenging.  First, because HHS has not released all of the needed data, one must develop a plausible method of moving from the marginal distributions that were provided by HHS on enrollment by age and enrollment by gender into a joint distribution by gender and age. Second, one must calibrate the SOA cost data and the Kaiser premium data, which are expressed in somewhat different units,  such that, if the joint distribution of gender and age was as was originally expected an insurer would just break even.  And, third, one must develop a reasonable method of modeling insured populations that are drawn disproportionately from persons who have higher medical expenses. I believe I have now come up with reasonable solutions to all three issues.

Solution #1

The solution to the first issue, moving from a marginal distribution to a joint distribution, was detailed in my prior blog entry. In short, one finds a large sample of possible joint distributions that match the marginal distributions and scores them according to how well they match the property that people who are subsidized more likely to enroll.  One takes an average of a set of solutions that score best. There is an element of judgment in this process on the degree to which individuals respond to subsidization incentives and, all I can say, is that I believe my methodology is reasonable, avoiding the pitfall of thinking that subsidization is irrelevant or of thinking that it is the only factor that matters in determining enrollment rates. I present again what I believe to be the most likely joint distribution of enrollment by gender and age.

Plausible age/gender distribution of ACA enrollees
Plausible age/gender distribution of ACA enrollees

Solution #2

The solution to the second problem is obtained using calculus and numeric integration. One computes the expected costs and expected premiums given the original joint distribution of enrollees, which is taken to be a product distribution of which one distribution is a “Bernoulli Distribution” in which the probability of being a male or female is equal and the other is a “Mixture Distribution” in which the weights are those shown below (and taken from the  Kaiser Family Foundation web site) and the components are discrete uniform distributions over the associated age ranges.

Original estimate of age distribution of enrollees
Original estimate of age distribution of enrollees

The Society of Actuary data on the relationship between age, gender and medical costs is shown here.

Society of Actuaries data on gender, age and commercial insured expense
Society of Actuaries data on gender, age and commercial insured expense

The premiums under the ACA are shown here.

ACA Premiums
ACA Premiums

These two plots combined can give us a subsidization rate plot by gender and age.  It is shown below along with an associated plot showing the distribution of enrollees by age as was originally assumed and as appears to be the case.

Subsidization rates by gender and age along with anticipated and current age distribution of enrollees
Subsidization rates by gender and age along with anticipated and current age distribution of enrollees

Solution #3

To model adverse selection based on expensive medical conditions, I simply added a health age differential to the insureds.  That is, in computing expected medical costs, I assumed that people were their actual age plus or minus some factor.  (Ages after this addition were constrained to lie between 0 and 64). The graphic above showed insurer losses as a function of this “health age differential” under two scenarios.

Technical Note

A Mathematica notebook containing the computations used in this blog entry is available . here on Dropbox. I’m also adding a PDF version  of the notebook here. I want to thank Sjoerd C. de Vries for coming up with an elegant method within Mathematica of describing the joint distribution used in the computations of various integrals.  I am responsible for any mistakes in implementation of this method and my use of Mr. de Vries idea implies nothing about whether he agrees, disagrees or does not care about any of the analyses or opinions in this post.

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Gender equity and the Affordable Care Act

Much has been made here and elsewhere about how young people are subsidizing older people under the Affordable Care Act. While there is a substantial element of truth to this contention, at least young people generally get to become older people.  So, if the ACA were to last for decades, one could drive a small bit of comfort by viewing the arguable inequity as instead amounting to younger purchasers under the ACA just financing the health care they will receive at subsidized rates as they enter their 50s and beyond. The analogy doesn’t work terribly well because unlike something like a long term life insurance policy in which a similar “subsidy” exists, there is nothing that forces those insured later in life to have insured earlier on.  But at least youth is a “burden” that most of us share.

A closer look at the evidence, however, shows that the major determinant of whether someone is subsidizing another or being subsidized under the ACA is gender.  As shown here, gender is more important than age for purposes of ACA subsidization. And, for most of their adult lives males subsidize women under the ACA. Since gender is largely immutable, males never get the money back. While there are many factors that bear on whether this system is fair, the extent of subsidization is large enough to be worth considering.

Subsidization by gender and age
Subsidization by gender and age

The graphic above shows the extent of subsidization.  For each adult age (21-64) and each gender, I show the subsidy (positive or negative) the person receives under the ACA. The pink line shows the subsidy for women; the blue line shows the subsidy for men. Subsidization is the difference between the expected costs the person incurs and the person’s premiums under the ACA (without consideration of any government premium subsidies) normalized by dividing the difference by the person’s premiums.  Expected costs are calculated based on research by the Society of Actuaries and available in Excel data format from this web site. Premiums are calculated based on data provided by the Kaiser Family Foundation following its study of the ACA. To make sure that the units of of cost used by Kaiser and the Society of Actuaries match up, I apply a multiplicative correction factor to the premiums to ensure that the total level of subsidization is zero assuming that the estimated distribution of uninsured all enroll in ACA plans at an age-independent rate.  Use of more complicated assumptions about enrollment patterns, such as incorporation of the apparent fact that most of those purchasing policies in the individual Exchanges already had insurance, would result in a different correction factor but should not alter the basic conclusions of this post about cross-gender subsidization.

When one adds children into the mix, the picture becomes a bit more complex. As shown in the graphic below, insurers under the ACA appear heavily to subsidize children of both genders, although male children are subsidized somewhat more. The calculations here are based on an assumption that child-only policies cost 65.2% of the price for policies sold to 21 year olds.  (The 3:1 constraint on the ratio of premiums under the ACA applies only to adults (42 U.S.C. § 300gg(a)(1)(A)(iii)). This assumption was based on my sampling actual policies sold in the individual Exchanges under the ACA.

Subsidization by gender and age for all ages
Subsidization by gender and age for all ages

What is curious and perhaps somewhat comforting to those wanting to see the ACA succeed is the fact that, notwithstanding the significant differences in subsidization, women have not enrolled at rates way higher than men.  Overall, government statistics show that 54% of the enrollees are women and only 46% are men.  Nor are children forming a large part of the group enrolling in the individual Exchanges notwithstanding the high subsidization rates; they amount to just 6% of the total enrollees as of January 1, 2014. Now, part of this relative equality in enrollment rates by gender could be due to the masking effects of aggregation. It might be  that the female/male ratio is considerably higher among those ages 25-35, where the subsidization differential is quite large and the female/male ratio is much lower among those over age 60.  Thus, even if the overall ratio of enrollees was quite even, we could conceivably be seeing unequal enrollment patterns within age brackets.  As noted in an earlier post, neither the federal government nor any of the states have released data with the degree of detail that would be needed to confirm or refute this possibility and thus the actual joint distribution of enrollment by age and gender remains a matter for estimation using algebra and numeric methods rather than actual data. Still, it certainly appears that the rate of subsidization can not be the only factor affecting enrollment patterns; matters such as income, savings, risk aversion, as well as political, cultural and social factors are likely to be playing a role as well. How else can one, after all, explain the enormous differences in rates of enrollment across various states?

Now, is this “fair”?  That’s a difficult question. Most serious questions about insurance underwriting justice are difficult. (I’m going to include a short bibliography at the end of this post).  A large chunk of the difference between male and female healthcare expenses are based on the attribution of costs arising out of joint sexual activity to the female only.  It is, after all, the female’s body that is primarily affected by pregnancy. That attribution is based mostly on convenience, however, and, in many cases, the difficulty that would be created in trying to collect from a biological father. Moreover, it may be that subsidization in this area is compensatory, addressing countervailing subsidies of men in other government programs.

Even if it is fair, however, to the extent potential enrollees are responding to the extent of subsidization, we need to be concerned that unisex rating is reducing the efficacy of the ACA in shrinking the number of uninsureds.  Remember all the ills created by lack of insurance that substantially motivated the ACA? Charging men “too much” leaves many of those ills untreated. If men are not signing up because they are being asked to pay too high a price, the goals of the ACA in reducing the number of uninsureds and improving individual health are compromised. Let us not forget as various politicians attempt to diminish expectations about the achievements of the ACA that it was heavily advertised as a program to reduce the number of uninsureds. Don’t believe me? Look here (32 million), here (34 million by 2019) and here for examples.

There are two additional pictures that may be helpful to those graphically minded in considering this issue. The first, shown below, shows the expected costs of males (blue) by age, the expected costs of females (pink) by age, and the unisex ACA premium (green)(normalized so that the overall subsidization rate would be zero if enrollment rates were age-independent).

Comparison of expected costs by gender and unisex premium
Comparison of expected costs by gender and unisex premium

The second graphic lets one compare the degree of age subsidization under the ACA.  The purple line (kind of a blend of blue and pink) shows the expected costs of enrollees assuming that 50% are male and 50% are female. The green line shows the unisex ACA premium, again normalized so that the overall subsidization rate would be zero if enrollment rates were age-independent among the previously uninsured population. (A different normalization metric should not dramatically change the picture). As one can see although there is a zone between ages 20 and 32 in which premiums are exceeding cost and a zone between ages 60 and 64 where costs are exceeding premiums, and, although as mentioned above, children are heavily subsidized, for most of adulthood, premiums track expected costs pretty closely.  This may help explain why neither under my analysis nor under that of the Kaiser Family Foundation do departures of the age distribution from those originally foreseen have a gigantic affect on the profitability of the system.  What might have a larger effect, if it were to occur, would be departures of the gender distribution of enrollees from those originally foreseen; but, as mentioned above, thus far this does not seem to be occurring.

Comparison of blended expected costs and ACA premiums
Comparison of blended expected costs and ACA premiums

I do need to add one critical note.  All of this assumes that the expected costs for each age come in as predicted.  This is hardly known for sure.  There are many reasons, including adverse selection, moral hazard, and others why those costs might depart seriously from that which was projected.

A “starter set” bibliography on insurance underwriting justice

Kenneth S. Abraham, Distributing Risk (1986) (the starting point for thinking about this issue)

Tom Baker, Containing the Promise of Insurance: Adverse Selection and Risk Classification, 9 Conn. Ins. L.J. 371 (2002-2003), available online here.

Seth J. Chandler, Insurance Underwriting with Two Dimensional Justice, available here.

Seth J. Chandler, Insurance Regulation, in the Encyclopedia of Law and Economics, available here.

City of Los Angeles Department of Water & Power v. Manhart, 435 U.S. 702 (1978) (available here)

Technical Note

The Mathematica notebook that underlies the analysis and graphics presented in this blog entry is available on Dropbox here.

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The distribution of individual market enrollees by age and gender combined

Earlier this month, the Department of Health and Human Services released more detailed information than it had before on the age distribution and gender distribution of enrollees in the individual markets serviced by the various Exchanges. What it did not do, however, and what needs to be done in order to better predict the likelihood of significant insurer losses in the Exchanges and, thus, greater pressure on premiums is to release data on the combined age and gender distribution of the enrollees. We don’t know, for example, how many woman aged 35-44 are enrolled in the Exchanges. This finer look at the data is important because, as discussed in a previous post, it is the combination of age and gender that bears a stronger statistical relationship to expected medical expenses.  And, while the ACA incompletely compensates for age in its premium rating scheme through dampened age rating, it does not compensate at all for gender.

With the help of Mathematica, I have combined some algebra and some numeric methods to try and reverse engineer out combined distributions of age and enrollment that meet various constraints. I believe I have succeeded in finding a plausible combined distribution that can be used in developing plausible models of the likely extent of adverse selection in the individual health insurance markets under the ACA. I present the result in the table below and the chart below. I then have a “how it was done” technical appendix.   My work involves creation of a high dimensional polytope that satisfies the existing data and then a search for points on that polytope that appear most plausible. I have also posted a Mathematica notebook on Dropbox that shows the computation.

Plausible age/gender distribution of ACA enrollees
Plausible age/gender distribution of ACA enrollees

The pie chart above first groups the enrollees by gender. The inner ring shows males and the outer ring shows females. It then groups the enrollees by age bracket. As one can see, women outnumber men significantly in the 18-45 group, are about equal among minors and those between age 45 to 55, and are outnumbered by men in the 55-65 age group.

The graphic below shows the same data, but now age is the first grouping mechanism.

PlausibleGenderAgeDistributionTranspose

I also attempted to find the combined distribution that would satisfy the observed marginal distributions of age and gender but that would greatly reduce adverse selection. The graphic below thus presents pretty much of a  “best case” for the combined age-gender distribution in the Exchanges. Notice that now it is only in the 18-35 year old age brackets that there are substantial variations in the rates of male and female enrollment. I very much doubt that the actual statistics are as promising for ACA success as depicted in the graphic below, but I present them here to show the sensitivity of my methodology to various assumptions.

 

Distribution of enrollees by age and gender that would substantially reduce adverse selection
Distribution of enrollees by age and gender that would substantially reduce adverse selection

The next step

The next step in this process is to try to compute the difference between premiums and expenses based on these  combined age-gender distributions.  I will then compare it to the difference between premiums and expenses based on an age-gender distribution that might have been expected by those who earlier modeled the effects of the ACA.  The result should provide some insight into the magnitude of combined age-based and gender-based adverse selection.  It should be similar in spirit to the work I showed earlier on this blog here. I hope to have that analysis posted later this week or, I suppose more realistically given my ever pressing day job, early next week.

How it was done

I have essentially 12 variables we are trying to compute: the number of enrollees in the combination of two genders and six age brackets. I know 9 facts about the distribution based on data released by HHS. I know the total number of males and females and I know the total number of persons in each age bracket.  And I have 12 constraints on the values: they must all be positive. Using Mathematica’s “Reduce” command, I can use linear algebra to find the polytope that satisfies these equations and inequalities. I get an ugly expression, but it is one Mathematica can work with.

I can then sample 12-dimensions points on the polytope using Mathematica’s “FindInstance” command. I found 2400 points. Each of these points represents an allocation of enrollees among age and gender that satisfies the known constraints. I can then score each point based on its “distance” from my intuition about the strength of adverse selection. That intuition is expressed by “guesstimating” likely ratios between males and females for each of the six age groups.  I use a “p-Norm”  and Mathematica’s “Norm” command to measure the distance between the six male/female ratios generated by each of the 2400 points and my intuition.  I then take the 10 best 12-dimension points and thus obtain a 10x2x6 array. I take the average value of each of the 12 values over all 10 sample points.  It is that average that I show in the first two graphics above.

I then permitted the strength of adverse selection to vary by exponentiating the ratios in my intuition. By setting the exponent to zero, I basically try to minimize gender-based adverse selection and keep the gender ratios as close to each other as possible. The results of this effort are shown in the final graphic.

 

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The Kaiser analysis of ACA enrollment has problems

On December 17, 2013, the Kaiser Family Foundation published an influential study that comforted many supporters of the Affordable Care Act who had been made nervous by early reports that the proportion of younger persons enrolling in Exchanges was significantly less than expected.  If true, such a disproportion could have created major stress on future premiums in the Exchanges because the private Exchange system under the ACA depends — or so it was thought — on younger persons subsidizing older persons. The Kaiser study asserted, however, that even if one cut the number of younger persons by 50%, insurer expenses would exceed insurer premiums by “only” 2.4%.  This finding under what it thought was a “worst case scenario” underpinned Kaiser’s conclusion that a “premium death spiral was highly unlikely.”

This post evaluates the Kaiser analysis. I do so in part because it disagreed a bit with my own prior findings, in part because it has gotten a lot of press, and because I have had a great deal of respect for Kaiser’s analyses in general.  I conclude that this Kaiser analysis rests, however, on an implausible assumption about the behavior of insurance purchasers and lacks much of a theoretical foundation. Once one eliminates this implausible assumption and employs a better theory of insurance purchasing, the threat of a death spiral becomes larger.

The reason for all this is a little complicated but try to bear with me and I will do my best to explain the problem.  Essentially, what Kaiser did was to run its simulation simply by lopping off people under the age of 34 and assuming that, for some reason, the disinclination of people to purchase health insurance on an Exchange would magically stop at age 34.  Thus, if an enrollment of, say, 2 million had been projected to come 800,000 from people age 18-34, 600,000 from middle aged people and 600,000 from the oldest group of enrollees, the “worst case” scenario Kaiser created (Scenario 2) would reduce enrollment to 1.6 million by having 400,000 come from people age 18-34, 600,000 from middle aged people, and 600,000 from the oldest group of enrollees. Thus, the youngest group would now constitute 25% of enrollees rather than 40%, and the other groups would constitute 37.5% of enrollees rather than 30%.

Although there is often nothing automatically wrong with this sort of “back of the envelope computation” — I have done many of them myself —  sometimes they give answers that are wrong in a meaningful way. And sometimes “meaningful” means a difference of just a few percentage points. Thus,  although the difference between 0.045 and 0.024 is not large on an absolute scale, this is one of these instances in which there could be a big difference between predicting premium increases augmented by 2.4% due to this particular form of adverse selection and predicting a premium increases augmented by 4.5% due to this particular form of adverse selection.   The first might be too small to lead to a quick adverse selection death spiral; the second, particularly if it combined with other factors increasing premiums, might be enough to start a problem. Death spirals are  a non-linear phenomenon a little like the “butterfly effect” in which small changes at one point in time can cascade into very large changes later on. What I feel comfortable saying is that the additional risk of a death spiral created by disproportionate enrollment of the an older demographic is greater than Kaiser asserts.

By simply lopping off the number of people under 35 who would enroll, the Kaiser model lacked a good theoretical foundation.  The model Kaiser should have run — “Scenario 3” —  is one in which the rate of enrollment is a sensible function of the degree of age-related subsidy (or anti-subsidy). Their two other scenarios could then be seen as special cases of that concept. Had they run such a “Scenario 3”, as I will show in a few paragraphs, the result is somewhat different.

Let me give you the idea behind what I think is a better model. I’m going to present the issue without the complications created by the messiness of data in this field.  We need, at the outset to know at least two things: (1)  the number of people of each age who might reasonably purchase health insurance if the subsidy were large enough (the age distribution of the purchasing pool); and (2) the subsidy (or negative subsidy) each person receives for purchasing health insurance as a function of age. By subsidy, I mean the ratio between the expected profit the insurer makes on the person divided by the expected expenses under the policy, all multiplied by negative one. The bigger the subsidy, the more money the insurer loses and the more likely the person is to purchase insurance.

Suppose, then, that the probability that a person will purchase health insurance is an “enrollment response function” of this subsidy. For any such enrollment response function, we can calculate at least three items: (1) the total number of people who will purchase insurance; (2) the age distribution of purchasers (including the “young invincible percentage” of purchasers between ages 18 and 35); and (3) — this is the biggie — the aggregate return on expenses made by the insurer.  Thus, some enrollment response function might result in 6.6 million adults purchasing insurance of whom 40% were “young invincibles” that generated a 1% profit for the insurer on adults while another enrollment response function might result in 2.9 million adults purchasing insurance of whom 20% were “young invincibles” that generated a 3% loss for the insurer on adults.

What we can then do is to create a family of possible enrollment response functions drawn from a reasonable functional form and find the member of that family that generates values matching the “baseline assumptions” made by both Kaiser and, apparently, by HHS about total enrollment and about the “young invincible percentage.” We can then calculate the aggregate return of the insurer on adults and call this the baseline return. What we can then do is assume different total enrollments and different young invincible percentages, find the member of the enrollment response function family that corresponds to that assumption, and then calculate the new revised return on adults. The difference between the baseline return and the new revised return on adults can be thought of as the loss resulting from this form of adverse selection. There are a lot of simplifications made in this analysis, but it is better, I believe, than either the back of the envelope computation by Kaiser that has gotten so much press and, frankly, the back of the envelope computation I did earlier on this blog.

Here’s a summary of the results.  When I (1) use the Kaiser/HHS age binning of the uninsured and indulge the simplifying assumption that the age distribution is uniform within each bin; (2) use Kaiser’s own estimate of the subsidy received by each age, (3) assume 7 million total purchasers ; and (4) assume 40% young invincibles with uniform age distribution within age bins, I find that the baseline return on adults is 1.0%. When I modify assumption (3) to have 3 million total purchasers and, as Kaiser did in Scenario 2, modify assumption (4) to have 20% young invincibles, the baseline return on adults is -3.5%.  Thus, a better computation of Kaiser’s worst case scenario is not a reduction in insurer profits of 2.4%, but rather a reduction of 4.5%.  

The graphics here compare enrollment rates, the age distribution of enrollees and various statistics for the baseline scenario and the scenario in which there are 3 million total purchasers and approximately 20% young invincibles.

Comparison of baseline scenario v. worst case using better assumptions
Comparison of baseline scenario v. worst case using better assumptions

We can use this methodology to run a variety of scenarios. I present them in the table below. A Mathematica notebook available here shows the computations underlying this blog entry in more detail. I am also making available a CDF version of the notebook and a PDF version of the notebook.

Various scenarios showing changes in insurer profits due to different enrollment response functions
Various scenarios showing changes in insurer profits due to different enrollment response functions

Please note that the computations engaged in here essentially ignore those under the age of 18.  This is unfortunate, but I do not have the data on the expected premiums and expenses of  children. It does not look as if Kaiser had that data either. Since children are expected to comprise only a small fraction of insured persons in the individual Exchanges, however, this omission probably does not change the results in a major way.

A humbling thought

The more I engage in this analysis, the more I realize how difficult it is.  There are data issues and, more fundamentally, behavioral issues that we do not yet have a good handle on.  Neither my model nor Kaiser’s model can really explain, for example, why, as has recently been noted, enrollment rates are so much higher in states that support the ACA by having their own Exchange and with Medicaid expansion than in states that more greatly oppose the ACA.  As I have suggested before, there is a social aspect and political aspect to the ACA that is difficult for simple models to capture.  Moreover, as I noted above, this is an area where getting a number “close to right” may not be good enough.  Premium increases of, say, 9% might not trigger a death spiral; premium increases of 10% might be enough.  And neither my nor anyone else’s social science, I dare say, is precise enough to distinguish between 9% and 10% with much confidence.

So, longer though it makes sentences, and less dramatic as it makes analyses and headlines, the humbling truth is that we can and probably should engage in informed rough estimates as to the future course of the Affordable Care Act, but it is hard to do much more as to many of its features. I wish everyone engaged in this discussion would periodically concede that point.

Other Problems with Kaiser

There are  other issues with the Kaiser analysis. Let me list some of them here.

Even accepting Kaiser’s analysis premium hikes would likely be more than 2%

Kaiser’s discussion of insurer responses to losing money is inconsistent. Look, for example, at this sentence in the report: “[i]f this more extreme assumption of low enrollment among young adults holds, overall costs in individual market plans would be about 2.4% higher than premium revenues.”  Kaiser further reports “Insurers typically set their premiums to achieve a 3-4% profit margin, so a shortfall due to skewed enrollment by age could reduce the profit margin of insurers substantially in 2014.” I don’t have a quarrel with this sentence.  But then look at what the Kaiser report says. “But, even in the worst case, insurers would still be expected to earn profits, and would then likely raise premiums in 2015 to make up the shortfall,” No! According to Kaiser’s own work, “even in the worst case,” insurer costs would be 2.4% greater than premium revenues.  Since there is little float in health insurance and investment return rates are low these days, insurers would likely not earn profits.  Then it gets worse. “However, a one to two percent premium increase would be well below the level that would trigger a “death spiral.” Perhaps so, but if insurers need to earn 3-4% to keep their shareholders happy and they are losing 1-2%, a more logical response would not be a 1-2% increase in premiums but a 4-6% increase. And, as Kaiser points out, larger premium increases could trigger a premium death spiral in part because death spirals are like avalanches: they start out small, only a little snow moves, but once the process starts it can become very difficult to abort.

Logical Fallacies

The first paragraph of Kaiser’s report asserts:  “Enrollment of young adults is important, but not as important as conventional wisdom suggests since premiums are still permitted to vary substantially by age. Because of this, a premium “death spiral” is highly unlikely.” Even if the first sentence of this quote were correct — a point on which this entry has cast serious doubt — the second sentence does not follow.  To use a sports analogy, it would be like saying that,  the role of a baseball “closer” is important but not as important as conventional wisdom suggests. Therefore the Houston Astros, who lack a good closer, are highly unlikely to lose.  No!  There are multiple factors that could cause an adverse selection death spiral.  Just because one of them is not as strong as others make out, that does not mean that a death spiral is unlikely. That’s either sloppy writing or just a pure error in logic.

Other Factors

And, in fact, if we start to look at some of those other factors, the threat is very real.  As discussed here in more depth, I would not be surprised if adverse selection based on completely unrated gender places as much pressure on premiums as adverse selection based on imperfectly rated age. And, as I have discussed in an earlier blog entry, the transitional reinsurance that somewhat insulates insurers from the effects of adverse selection will be reduced in 2015. This will place additional pressure on premiums.

And, on the other hand, the individual mandate, assuming it is enforced, will triple in 2015 and risk adjustment measures in 42 U.S.C. § 18063, will likely provide greater protection for insurers.  These two factors are likely to dampen adverse selection pressures.

Notes on Methodology

There are a number of simplifying assumptions made in my analysis.  Some of them are based on data limitations. Here are a few of what I believe are the critical assumptions.

1. Functional form: I experimented with two functional forms, one based on the cumulative distribution function of the logistic distribution and the other based on the cumulative distribution function of the normal distribution.  These are both pretty conventional assumptions and make sure that the enrollment rate stays bounded between 0 and 1. The results did not vary greatly depending on which family of functions the enrollment response functions were drawn from.

2. Uniform distribution of ages within each age bin of potential purchasers. I believe this is the same assumption made by Kaiser and it results from the absence of any more granular data on the age distribution of the uninsured that I was able to find.

3. The enrollment rate depends on the subsidy rate standing alone and not other possibilities such as subsidy rate and age. The data on enrollment rates is very sparse and so it is difficult to use very complex functions.  Perhaps a more complex analysis would assert that enrollment depends on both subsidy rate and age, since age may be a bit of a proxy for the variability of health expenses and thus of risk.

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Middle class not selecting plans in the individual Exchanges

No more than 1% of those with household incomes between 139% and 400% and eligible to select a plan on the individual Exchanges have thus far done so. This is the information about those with middle incomes and lower-middle incomes one can derive from statistics released this week by Health and Human Services.   The rate of plan selection among those with incomes over 400% of the federal poverty level is at least 3 times higher than that of persons with middle and lower-middle incomes. It could well be 4 times greater.

No matter how you fine tune the computations, I believe it is fair to say that the middle class is finding the carrots too small and the sticks too small. Some of this may be due to difficulties with the enrollment process rather than the underlying architecture of incentives under the ACA, but either way, most of those eligible to do so, are, at least for now, rejecting the benefit theoretically available to them on the individual Exchanges under the Affordable Care Act.

Visualization

Here are two figures showing the results of my calculations in more detail.

Rates of Take Up

The first graph shows the absolute rates of take up (selection of a plan) among with lower-middle and middle incomes (the lower surface) and the wealthier (the higher surface).  The x-axis of the graph shows the assumption one makes about those reasonable eligible to purchase policies.  When x is low, one assumes the income distribution of the eligible pool most closely resembles that of persons currently without health insurance. When x is high, one assumes the income distribution of the eligible pool most closely resembles that of persons currently with health insurance form their employer. The y-axis of the graph shows the assumption one makes about the number of persons current with health insurance from their employer who might reasonably be considered eligible to purchase insurance on a health insurance Exchange. A low value of y means that very few of these people should be considered eligible. A high value of y means that 10% of these people should be considered eligible. The z-axis (vertical) shows the fraction of people eligible to do so who have to date selected a policy on an Exchange.

Take Up Rates among the Wealthier (top surface) and the Lower Middle and Middle Income Group (lower surface)
Take Up Rates among the Wealthier (top surface) and the Lower Middle and Middle Income Group (lower surface)

As one can see the values are always less than 1% for the lower-middle and middle incomes. The values for the wealthier depends on the assumptions made but for  all values are below 6% and is frequently below 4%. And these are values for selection of a plan, not for actual purchase of a policy. Those numbers are likely to be even smaller due to many people leaving items in their “shopping cart” without paying at the check out counter.

Take Up Ratios

The second graphic shows the ratio between the take up rates among the wealthier and the take up rates among the lower-middle and middle income group. The x and y axes are the same as before.  A value of 3.4 on the z-axis means that the take up rate among the wealthy is 3.4 times what it is among the lower-middle and middle income groups. As one can again see the ratio is above 3 for almost all assumptions one could make and is frequently above 4.

Take Up Ratios
Take Up Ratios

Show me the calculation

How do I get to these figures? Algebra. Some of it is very nasty algebra, but I have the world’s best computer algebra system, Mathematica, at my disposal to make the problem much easier. Rather than include the somewhat complex computations directly in this blog post, I’m going to include a PDF file showing the computations and a CDF file (a Mathematica file format). You can read the CDF file either with Mathematica itself or with the free CDF Player available here. The data, by the way comes from a combination of  this tidbit of information found on page 7 of the report released  by HHS on  December 11, 2013, and data from the Urban Institute and Kaiser Foundation.

Page 7 of the HHS Report
Page 7 of the HHS Report

Sources

Distribution of the Nonelderly Uninsured by Federal Poverty Level (FPL)

Distribution of the Nonelderly with Employer Coverage by Federal Poverty Level (FPL)

HEALTH INSURANCE MARKETPLACE: DECEMBER ENROLLMENT REPORT For the period: October 1 – November 30 (December 11, 2013)

 

 

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The new Exchange enrollment numbers are bad

The federal government announced today that 137,204 people have selected a healthcare plan through the federal Exchange as of November 30, 2013. The number is an increase over the 29,794 who had done so by the end of October, a month during which the website portal for enrollment, healthcare.gov, was in disarray. The government reports that 258,497 have now selected a plan through one of the state Exchanges, making a total of 364,682 enrolled. Asked by reporters whether the Obama administration stands by its estimate that 7 million will enroll in individual plans sold on the various Exchanges by March 31, 2013, the day necessary to do so in order to avoid a tax penalty,  Michael Hash, director of the office of health reform in the federal Health and Human Services Department, said that they were “on track, and we will reach the total that we thought.”

The pace of enrollment announced by the federal government today is inconsistent with the claim that its 7 million goal will be achieved. The claim rests on hopes of two surges, one taking place over the next 12 days before the December 23, 2013, deadline for coverage starting January 1, 2014 and a second surge taking place as we approach the end of March at which point, if coverage has not been obtained, many Americans will be hit with a tax penalty.

The magnitude of the surge required strains credulity.  A scenario in which most of  those who wanted coverage have already applied and in which the pace of enrollment stays the same or even sags for lengthy periods as we go forward would appear almost as likely. Plus, it seems unlikely that there will be major enrollment between December 23, 2013, the first deadline, and March 23, 2014, the second deadline. If someone wanted coverage, they would try to get it earlier. What does applying in the middle of February accomplish? Moreover, if, given the unpredictability of human behavior, the surge actually materialized, it might well strain the government’s computer systems.

Analysis

There are many disturbing aspects to today’s release of numbers. First, forget for the moment about the March 2014 projection date and the March deadline.  There are only 12 shopping days left before the pool will be closed for those who will have coverage as of January 1, 2014.  Even if the pace of enrollment surges by a factor of 10 over what it was for the last two weeks on which we have data and healthcare.gov enrolls people at 45,000 per day, that would still put only about 668,000 persons enrolled through the federal Exchange as of that deadline.  Even this rather cheery estimate would result in only 14% of the 4.8 million the Obama administration has projected will be enrolling in the federal Exchanges in 2014.  The original projections for enrollment on opening day, January 1, 2014, were considerably higher, 3.3 million.

The number enrolled as of December 23, 2014, matters greatly. While of  course there could be a second surge, in the mean time insurers are having to pay claims for three months on those first 14% to enroll. The initial enrollees are very likely to be comprised disproportionately of people with above average health care expenses. The result will be that, until that prayed-for second surge occurs, insurers will likely be losing large sums of money in the Exchanges and, ultimately, seeking reimbursement pursuant to the Risk Corridors program from the federal government and, derivatively, taxpayers.

Moreover, the aggregate numbers mask the fact that there are 50 different sets of Exchanges. While numbers are better in some, there are many jurisdictions in which there are huge problems.  It is not “OK” if the Exchanges succeed in California, New York and a few other states if insurers and insureds in many other states suffer severe adverse selection problems that result in rapidly rising prices or reductions in availability.

Let’s look at a few states. I start with Texas. There, out of 780,959 projected to be enrolled, there are 14,038 as of the end of November.  This is fewer than 2% of the ultimate projected amount.  Even if one assumes that enrollments in Texas surge to go 20 times faster in December than they did in November, which is a pretty heroic assumption, this would still result in only 183,425 being enrolled as of the December 23 deadline. This would be  only 23% of what needs to occur. It would be as if a football team were down 35-3 in the 3rd quarter and hoping to make a comeback. It could, I suppose, happen, and you shouldn’t turn off the TV set, but the probabilities are remote.

One might argue that Texas is an exceptional case due to the degree of hostility prevailing among many here about “Obamacare.” Take another fairly large state using the federal Exchange, Pennsylvania. There, we see 11,788 enrolled out of 268,858 ultimately projected, just 4.4%.  To get to even 1/3 of the ultimate projected number being enrolled by December 23, the pace for December would have to be 6 times greater than it was in the last two weeks of November. Not impossible given procrastination, but again, a major challenge.

The figures when one looks to the various state Exchanges are a mixed bag. The poster child for the Obama administration would appear to be California. It has 107,087 of the 691,016 it ultimately hopes to enroll, over 15%.  With a decent last minute kick, it is not unimaginable that California could make 1/3 of its total by the December 23, 2013 deadline and get closer to its ultimate goal by the end of March.  But even with these better-than-average numbers, there is the risk of at least some adverse selection in a pool substantially smaller than projected. Also doing better than many is New York. There, we see 45,513 enrolled. But even this is but 11% of the 411,304 projected. It will again take a major surge over the next 12 days if New York were to get to even 1/4 of the ultimate projected enrollment by the December 23 first deadline.

But for every California or New York running its own show, there is an Oregon or a Maryland. These are large states in which enrollment is lagging. In Oregon, owing substantially to the collapse of its computer system, only 44 people have enrolled in plans on their Exchange. It will take an unimaginable surge there to make the system functional. Officials there and in Washington, D.C. will soon need to start contemplating what to do about a failed system; will, for example, tax penalties be imposed for those in Oregon who do not have health care coverage? In Maryland, where the director of the program recently quit, just 3,758 have enrolled out of 91,528 projected, just 4.1%. It goes beyond hope and into the realm of fantasy to believe that Maryland is not going to have a serious adverse selection problem starting January 1, 2014, when those 3,758 who penetrated the state’s application system start filing claims.

Finally, nowhere in the release do I see an age distribution of those enrolling. Unquestionably, the administration has this information. It is required in the enrollment process. And, perhaps this is a bit cynical, but I have to think that if those numbers looked good, if the hoped-for proportion of younger persons were enrolling, the Obama administration would release the information.  I believe we are entitled to draw a negative inference from the fact that the information was not released that the pool is disproportionately elderly. If this is correct, what we are seeing is a small pool composed disproportionately of the elderly. That does not augur well for those who want to see the promises of the Affordable Care Act fulfilled.

An Experiment

HHS was kind enough to include a graphic in their report. Here it is.

Cumulative enrollment in the federal Exchange for various states
Cumulative enrollment in the federal Exchange for various states

The graphic plots time on the x-axis and cumulative enrollment on the y-axis. Recognizing all the enormous problems with doing so, I thought it would still be interesting to try to fit a curve to the data and extrapolate it out to see where we might end up.

The short version is that if we extrapolate the curve using quadratic and cubic models, we end up at between 278,000 to 383,000 enrolled in the federal system by the December 23, 2013 first deadline. This would represent fewer than 10% of the ultimate projected enrollment and will create substantial adverse selection problems for at least the first three months of the program, particularly in the less enthusiastic states. This all assumes, of course, that all people who have selected a plan actually pay the premiums. The numbers could be worse. Regardless, insurers are going to be very concerned if these are the sort of numbers that materialize; the federal government better get out its Risk Corridors checkbook to help relieve the pain.

By March 23, 2013, however, the same models show we could be at 1.35 million to 3.94 million, depending on the model chosen.   This would represent 28% to 82% of that originally projected and would cause serious adverse selection problems at 28% or mild adverse selection problems at 82%.  I appreciate fully that these are large error bars but we just don’t have the data or an a priori model that permits me to extrapolate with any confidence this far into the future.

Here’s a graphic showing these results.  The Mathematica notebook that generated them has been placed here on Dropbox.

Extrapolation of enrollment data
Extrapolation of enrollment data

 

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The incredible increase in pace that will be needed to meet ACA enrollment projections

Healthcare.gov appears to be working much better, at least in enabling individuals to select plans. And some of the state exchange web sites appear to be improving their functionality too. Some have heralded these advances as providing hope that the Exchanges will be able to meet the enrollment projections on which the economics of insurance without medical underwriting in part depend. But do these claims stand up to the cold light of mathematics?  Not very well.

Here’s the headline:

A close look at the numbers shows that the pace of enrollments from here to the close of open enrollment needed to meet projections is high in every state, even those touted as successful, and almost impossibly high in many.  Given the incredibly slow start in most jurisdictions, it will not just take a little pickup over the next few months to achieve the projected and needed number of persons in the Exchanges. It will take a miraculous last minute stampede. Since miracles seldom occur,  the result may be two different stories of the Affordable Care Act: a few states in which the Exchanges proved from the start to be a somewhat stable mechanism for providing health insurance without medical underwriting but a significant number of other states in which the results for at least the first year represent a large failure.

Today’s news

News appears to be breaking today that the federal exchanges enrolled about 100,000 in November.  This is being heralded as somewhat of a success compared to the 26,000 who enrolled in October. And, of course, enrollment figures from healthcare.gov are difficult to assess due to the actual and feared dysfunctionality of the web site. But one way to look at this is to consider what has to happen between December 1, 2013, and March 23, 2014, the close of open enrollment to make projections. The states that are dependent on healthcare.gov need about 4.84 million enrollees by the end of that period if the nation is to meet the goal of having 7 million enrolled in the Exchanges by the close of open enrollment.  If, right now, there are about 126,000 enrollees in those states, we are just 2.5% of the way there.  The pace of enrollment on healthcare.gov will need to increase by a factor of about 20 in order to meet goal.  In absolute terms, healthcare.gov needs to be enrolling about 42,000 people per day. And while perhaps not every single one of those people need to enroll for the system to succeed, the 7 million enrollment goal isn’t just a mere wish. There are, as I and many others have noted potentially serious consequences to the stability of insurance markets if the figures fall well short, even in several states.

Whether healthcare.gov can score the needed come back, however, will basically depend on two related factors: (1) whether healthcare.gov is truly fixed and can stand up to the increased pace that will be needed and (2) whether the requisite increase in pace is likely.  This latter factor depends in turn on where on the following spectrum the possibilities fall. On one end of the spectrum, there is the possibility that there is this pent up demand from procrastinators that will surge forward to access the web site in the coming weeks.  Perhaps March Madness for 2014 will constitute this huge surge — kind of like April 15 rushes to the post office to send in tax returns — as the March 23 “deadline” approaches. On the other end of the spectrum, there is the possibility that most people who wanted to and had the means to enroll — the wealthy sick — did so already and others have looked at the prices, the coverages and the penalties and decided that, for now, Exchange coverage is not for them.  The fact that a surprising 70% of current enrollees in the Exchange plans are unsubsidized gives some support to this gloomier hypothesis.

To get further insights, we can also take a closer look at some representative states.

Connecticut

First, let’s look at what has to happen in the most successful state, Connecticut. There, as of  November 14, 2013 (the date of the last report), 7,591 people had selected a plan.  That’s not all it will take finally to get coverage — among other things, people will have to start actually paying premiums — but it’s a solid start. This 7,591 figure represents 12.9% of the projected total of 58,637 for Connecticut.  A little math shows that in order to make projections, people in Connecticut will need to enroll at a pace 2.3 times faster than they had as of November 14 in order to make the projection by the March 23, 2014 date.

It hardly seems impossible that Connecticut could make the projections.  Whether they do so, however, will basically depend on the location of Connecticut on the spectrum discussed above.  We should have a better sense of where on the spectrum we are falling when Connecticut releases new numbers.

Texas

Connecticut is a small state.  The enrollment there was projected to constitute only about 1.4% of the total enrollment in the Exchanges.  Let’s take a look at a big state: my home state of Texas. With the largest uninsured population in the country and with no Medicaid expansion into which some Exchange eligible persons might otherwise “fudge into,” Texas was supposed to enroll 780,959. As of November 2, 2014, Texas had enrolled just 2,991. This means that Texas will have to enroll at a pace 59 times faster than it had as of that date in order to meet enrollment projections.  And, even if due to failure to the healthcare.gov website, Texas has enrolled, say, just 10,000 as of November 30, 2013, it will still need to up its pace by a factor of 41 in order to meet projections. Viewed in absolute terms, Texas will need to enroll at a pace of over 6,800 per day. Again, we will have a better sense of the plausibility of this increase when the federal government releases newer data.

Another way of thinking about the issue is to consider what would happen if Texas’ future enrollment relative to its prior enrollment is the same as Connecticut needs to be in order to meet the Connecticut projection. If Texas enrolls at a pace 2.3 times faster than it has thus far, Texas enrollment will be something like 29,000. That would be just 4% of what was originally projected, a shortfall of 752,000.  Connecticut could double its projected enrollment and it would barely make a dent in compensating for a shortfall of this magnitude. Even if Texas celebrates the rebirth of healthcare.gov by stepping up its enrollment by a factor of 10, that still gives it less than 150,000 enrollees, a shortfall of 630,000 over the projected value.

It would also help if the government could get the Spanish language version of its website, cuidadodesalud.gov, to accept applications the same way that healthcare.gov does.

California

A problem with projecting Texas numbers is that it has been hamstrung by its chosen dependency on healthcare.gov, which has been completely dysfunctional until recently. So, what about a large state that shares some demographic characteristics of Texas but that has a mostly functional web site?  Let’s look at California.

In California, as of November 19, 2013, there were 78,891 counted as enrolled relative to a projected enrollment as of March 23, 2014 of 691,016. Viewed one way, California is going to need to step up the pace of its enrollment by a factor of 3.07 in order to meet its target. In absolute terms, California needs a pace of about 4,936 per day (including weekends and including the busy holiday season) in order to meet target.

Whether viewed in relative or absolute terms, the pace needed in California is ambitious. Covered California, which brags of a recent tripling in the pace of enrollment, still enrolled just 2,700 per day for Exchange plans in the most recent period for which data currently exists. (It is only by counting Medicaid/Medical enrollments that the numbers get higher).  If California were to persist at that pace for the remainder of the enrollment period, it would have something like 414,000 enrolled by the March 23, 2014 date. Depending on the precise composition of the pool of insureds, such a figure would likely be enough to stave off severe adverse selection but probably not enough to do so entirely. And, again, for those focusing on the 7 million nationwide figure, shortfalls of 277,000 in California or 700,000 in Texas are just difficult to compensate for even if other states are considerably more successful.

New York

Let’s pick one more big state.  And, again, let’s pick one where the Exchange is generally said to be doing well: New York.  In New York, as of November 24, 2013, 41,021 are claimed to have enrolled in plans out of the 411,304 originally projected.  This means New York will have to quadruple the pace of enrollments (4.09) in order to meet projections. In absolute terms, the state needs to be enrolling 3,111 per day every day until March 23, 2014.  It is thus in roughly the same position as California. Whether it meets its goals depends on why there has thus far been a shortfall.  If it’s massive procrastination, perhaps New York can get pretty close.  If, on the other hand, many New Yorkers are rejecting the product, and the pace of enrollment just doubles from what it has been, expect New York to fall short by about 190,000 people (46%). Again, smaller states that are more successful will have difficulty compensating for such a large absolute shortfall.

Conclusion

There is no state in which a significant uptick in the pace of enrollments will not be needed in order to meet enrollment projections.  This is true in states that have their own Exchanges and states that do not.  It is true in states touted as a success as well, of course, of those seen as failing.  It is most definitely true for states that depend on the federal website, healthcare.gov.

In a few states, the burden may be met. Many people do indeed procrastinate, even perhaps when it comes to subsidized health insurance that they now lack. To meet enrollment projections in many other states,however,  we need for Exchange applications to be far more the province of procrastinators than even income tax returns.  After all,  only 25% or so wait until the last two weeks to file those. In these other states, the appropriate analogy may not be tax procrastination but the miracle of The Heidi Game in which two touchdowns were scored in the closing 9 seconds after the mainstream media (NBC Sports) assumed the game was lost.  But it’s been 45 years since that turnaround occurred.  It just might happen again with applications for health insurance on the Exchanges, but it seems unlikely.

Could I add one more point?

People are focusing on March 23, 2014 (earlier March 15, 2014) as the measuring date.  The ACA will presumably be deemed a success if enrollment figures meet projections by that date.  But this strikes me as an awfully generous measure. The problem for insurers is that there will be a smaller pool during the first three months of the policy year. And smaller pools generally have higher claims per person.  So, if I ran the world, I’d be looking at two dates to examine enrollments: January 1, 2014 when the plans kick in and March 23, 2014 when open enrollment ends.  Yes, great enrollment by March 23 will ultimately go a long way to reassuring insurers if enrollment is problematic on New Years Day. But if enrollment is really bad come Rose Bowl time, expect insurers to lose a lot of money on claims filed between then and the close of open enrollment.  If they can’t make that money back on the late filers, expect insurers to figure out some way of getting even for 2015.

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Shocking secrets of the actuarial value calculator revealed!

That might be how the National Enquirer would title this blog entry.  And, hey, if mimicking its headline usage attracts more readers than “Reconstructing mixture distributions  with a log normal component from compressed health insurance claims data,” why not just take a hint from a highly read journal?  But seriously, it’s time to continue delving into some of the math and science behind the issues with the Affordable Care Act. And, to do this, I’d like to take a glance at a valuable data source on modern American health care, the data embedded in the Actuarial Value Calculator created by our friends at the Center for Consumer Information and Insurance Oversight (CCIIO).

This will be the first in a series of posts taking another look at the Actuarial Value Calculator (AVC) and its implications on the future of the Affordable Care Act. (I looked at it briefly before in exploring the effects of reductions in the transitional reinsurance that will take effect in 2015).  I promise there are yet more important implications hidden in the data.  What I hope to show in my next post, for example, is how the data in the Actuarial Value Calculator exposes the fragility of the ACA to small variations in the composition of the risk pool.  If, for example, the pool of insureds purchasing Silver Plans has claims distributions similar to those that were anticipated to purchase Platinum Plans, the insurer might lose more than 30% before Risk Corridors were taken into account and something like 10% even after Risk Corridors were taken into account. And, yes, this takes account of transitional reinsurance. That’s potentially a major risk for the stability of the insurance markets.

What is the Actuarial Value Calculator?

The AVC is intended as a fairly elaborate Microsoft Excel spreadsheet that takes embedded data and macros (essentially programs) written in Visual Basic, and is intended to help insurers determine whether their proposed Exchange plans conform to the requirements for the various “metal tiers” created by the ACA. These metal tiers in turn attempt to quantify the ratio of the expected value of the benefits paid by the insurer to the expected value of claims covered by the policy and incurred by insureds. The programs, I will confess, are a bit inscrutable — and it would be quite an ambitious (and, I must confess, tempting) project to decrypt their underlying logic — but the data they contain is a more accessible goldmine. The AVC contains, for example, the approximate distribution of claims the government expects insurers writing plans in the various metal tiers to encounter.

There are serious limitations in the AVC, to be sure. The data exposed has been aggregated and compressed; rather than providing the amount of actual claims, the AVC has binned claims and then simply presented the average claim within each bin.  This space-saving compression is somewhat unfortunate, however, because real claims distributions are essentially continuous. Everyone with annual claims between $600 and $700 does not really have claims of $649. This distortion of the real claims distribution makes it more challenging to find analytic distributions (such as variations of log normal distributions or Weibull distributions) that can depend on the generosity of the plan and that can be extrapolated to consider implications of serious adverse selection. It’s going to take some high-powered math to unscramble the egg and create continuous distributions out of data that has had its “x-values” jiggled.  Moreover, there is no breakdown of claim distributions by age, gender, region or other factors that might be useful in trying to predict experience in the Exchanges.  (Can you say “FOIA Request”?)

This blog entry is going to make a first attempt, however, to see if there aren’t some good analytic approximations to the data that must have underlain the AVC. It undertakes this exercise in reverse engineering because once we have this data, we can make some reasonable extrapolations and examine the resilience — or fragility — of the system created by the Affordable Care Act. The math may be a little frightening to some, but either try to work with me and get it or just skip to the end where I try to include a plain English summary.

The Math Stuff

1. Reverse engineering approximate continuous approximations to the data underlying the Actuarial Value Calculator

Nothwithstanding the irritating compression of data used to produce the AVC, I can reconstruct a mixture distribution composed mostly of truncated exponential distributions that well approximates the data presented in the AVC.   I create one such mixture distribution for each metal tier. I use distributions from this family because they have been proven to be “maximum entropy distributions“, i.e. they contain the fewest assumptions about the actual shape of the data. The idea is to say that when the AVC says that there were 10,273 claims for silver-like policies between $800 and $900 and that they averaged $849.09, that average could well have been the result of an exponential distribution  that has been truncated to lie between $800 and $900.  With some heavy duty math, shown in the Mathematica notebook available here, we are able, however, to find the member of the truncated exponential family that would produce such an average. We can do this for each bin defined by the data, resorting to uniform distributions for lower values of claims.

The result of this process is a  messy mixture distribution, one for each metal tier. The number of components in the distribution is essentially the same as the number of bins in the AVC data. This will be our first approximation of “the true distribution” from which the claims data presented in the AVC calculator derives. The graphic below shows the cumulative density functions (CDF) for this first approximation. (A cumulative density function shows, for each value on the x-axis the probability that the value of a random draw from that distribution will be less than the value on the x-axis).   I present the data in semi-log form: claim size is scaled logarithmically for better visibility on the x-axis and percentage of claims less than or equal to the value on the x-axis is shown on the y-axis.

CDF of the four tiers derived from the first approximation of the data in the AVC
CDF of the four tiers derived from the first approximation of the data in the AVC

There are two features of the claims distributions that are shown by these graphics.  The first is that the distributions are not radically different.  The model suggests that the government did not expect massive adverse selection as a result of people who anticipated higher medical expenses to disproportionately select gold and platinum plans while people who anticipated lower medical expenses to disproportionately select bronze and silver plans. The second is that, when viewed on a semi-logarithmic scale, the distributions for values greater than 100 look somewhat symmetric about a vertical axis.  They look as if they derive from some mixture distribution composed of a part that produces a value close to zero and something kind of log normalish. If this were the case, it would be a comforting result, both because such mixture distributions would be easy to parameterize and extrapolate to lesser and greater forms of adverse selection and because such mixture distributions with a log normal component are often discussed in the literature on health insurance.

2. Constructing a single Mixture Distribution (or Spliced Distribution) using random draws from the first approximation

One way of finding parameterizable analytic approximations of “the true distribution” is to use our first approximation to produce thousands of random draws and then to use mathematical  (and Mathematica) algorithms to find the member of various analytic distribution families that best approximate the random draws. When we do this, we find that the claims data underlying each of the metal tiers is indeed decently approximated by a three-component mixture distribution in which one component essentially produces zeros and the second component is a uniform distribution on the interval 0.1 to 100 and the third component is a truncated log normal distribution starting at 100.  (This mixture distribution is also a “spliced distribution” because the domains of each component do not overlap). This three component distribution is much simpler than our first approximation, which contains many more components.

We can see how good the second-stage distributions are by comparing their cumulative distributions (red) to histograms created from random data drawn from the actuarial value calculator (blue).  The graphic below show the fits to look excellent.

Note: I do not contend that a mixture distribution with a log normal distribution perfectly conforms to the data.  It is, however, pretty good for practical computation.

Actual v. Analytic distributions for various metal tiers
Actual v. Analytic distributions for various metal tiers

 

 3. Parameterizing health claim distributions based on the actuarial value

The final step here is to create a function that describes the distribution of health claims as a function of a number (v) greater than zero. The concept is that, when v assumes a value equal to the actuarial value of one of the metal tiers, the distribution that results mimics the distribution of AVC-anticipated claims for that tier.  By constructing such a function, instead of having just four distributions, I obtain an infinite number of possible distributions. These distributions collapse as special cases to the actual distribution of health care claims produced by the AVC. This process enables us to describe a health claim distribution and to extrapolate what can happen if the claims experience is either better (smaller) than that anticipated for bronze plans or worse (higher) than that anticipated for platinum plans. One can also use this process to compute statistics of the distribution as a function of v such as mean and standard deviation.

Here’s what I get.

Mixture distribution as a function of the actuarial value parameter v
Mixture distribution as a function of the actuarial value parameter v

Here is a animation showing, as a function of the actuarial value parameter v, the cumulative distribution function of this analytic approximation to the AVC distribution.  

Animated GIF showing Cumulative distribution of claims by "actuarial value
Cumulative distribution of claims by “actuarial value”

 

One can see the cumulative distribution function sweeping down and to the right as the actuarial value of the plan increases. This is as one would expect: people with higher claims distributions tend to separate themselves into more lavish plans.

Note: I permit the actuarial value of the plan to exceed 1. I do so recognizing full well that no plan would ever have such an actuarial value but allow myself to ignore this false constraint.  It is false because what one is really doing is showing a family of mixture distributions in which the parameter v can mathematically assume any positive value but calibrated such that (a)  at values of 0.6, 0.7, 0.8 and 0.9 they correspond respectively with the anticipated distribution of health care claims found in the AVC for bronze, silver, gold and platinum plans respectively and (b) they interpolate and extrapolate smoothly and, I think, sensibly from those values.

The animation below presents largely the same information but uses the probability density function (PDF) rather than the sigmoid cumulative distribution function. (If you don’t know the difference, you can read about it here.)  I do so via a log-log plot rather than a semi-log plot to enhance visualization.  Again, you can see that the right hand segment of the plot is rather symmetric when plotted using a logarithmic x-axis, which suggests that a log normal distribution is not a bad analytic candidate to emulate the true distribution.

Log Log plot of probability density function of claims for different actuarial values of plans

 

Some initial results

One useful computation we can do immediately with our parameterized mixture distribution is to see how the mean claim varies with this actuarial parameter v. The graphic below shows the result.  The blue line shows the mean claim as a function of “actuarial value” without consideration of any reinsurance under section 1341 (18 U.S.C. § 18061) of the ACA.  The red line shows the mean claim net of reinsurance (assuming 2014 rates of reinsurance) as a function of “actuarial value.” And the gold line shows the shows the mean claim net of reinsurance (assuming 2015 rates of reinsurance) as a function of “actuarial value.” One can see that the mean is sensitive to the actuarial value of the plan.  Small errors in assumptions about the pool can lead to significantly higher mean claims, even with reinsurance figured in.

Mean claims as a function of actuarial value parameter for various assumptions about reinsurance
Mean claims as a function of actuarial value parameter for various assumptions about reinsurance

I can also show how the claims experience of the insurer can vary as a result of differences between the anticipated actuarial value parameter v1 that might characterize the distribution of claims in the pool and the actual actuarial value parameter v2 that ends up best characterizing the distribution of claims in the pool.  This is done in the three dimensional graphic below. The x-axis shows the actuarial value anticipated to best characterize an insured pool. The y-axis shows the actuarial value that ends up best characterizing that pool.  The z-axis shows the ratio of mean actual claims to mean anticipated claims.  A value higher than 1 means that the insurer is going to lose money. Values higher than 2 mean that the insurer is going to lose a lot of money.  Contours on the graphic show combinations of anticipated and actual actuarial value parameters that yield ratios of 0.93, 1.0, 1.08, 1.5 and 2. This graphic does not take into account Risk Corridors under section 1342 of the ACA.

What one can see immediately is that there are a lot of combinations that cause the insurer to lose a lot of money.  There are also combinations that permit the insurer to profit greatly.

Ratio of mean actual claims to mean expected claims for different combinations of anticipated and actual actuarial value parameters
Ratio of mean actual claims to mean expected claims for different combinations of anticipated and actual actuarial value parameters

Plain English Summary

One can use data provided by the government inside its Actuarial Value Calculator to derive accurate analytic statistical distributions for claims expected to occur under the Affordable Care Act.  Not only can one derive such distributions for the pools anticipated to purchase policies in the various metal tiers (bronze, silver, gold, and platinum) but one can interpolate and extrapolate from that data to develop distributions for many plausible pools.  This ability to parameterize plausible claims distributions becomes useful in conducting a variety of experiments about the future of the Exchanges under the ACA and exploring their sensitivity to adverse selection problems.

Resources

You can read about the methodology used to create the calculator here.

You can get the actual spreadsheet here. You’ll need to “enable macros” in order to get the buttons to work.

The actuarial value calculator has a younger cousin, the Minimum Value Calculator.  If one looks at the data contained here, one can see the same pattern as one finds in the Actuarial Value Calculator.

Joke

Probably I should have made the title of this entry “Shocking sex secrets of the actuarial value calculator revealed!” and attracted yet more viewers.  I then could have noted that the actuarial value calculator ignores sex (gender) in showing claims data.  But that would have been going too far.

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