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A STATISTICAL ANALYSIS OF MANDATORY POOLING ACROSS HEALTH INSURERS.

ABSTRACT

Risk-adjusted capitation payments from a central fund to competing health insurers or health plans form an essential feature of market-oriented healthcare reforms in many countries. If these premium-replacing payments do not adequately reflect risk, it may lead to solvency problems for plans with relatively many high-risk members and to selection against such individuals. Mandatory pooling across insurers may alleviate both problems by making the market as a whole financially responsible for high-cost or high-risk individuals. To finance the pool, every insurer would be obliged to pay, for each of its members, a uniform contribution that would depend on the projected size of the pool. The statistical analysis indicates that mandatory pooling may substantially mitigate solvency and selection problems while retaining incentives for efficiency and cost containment.

INTRODUCTION

Systems of risk-adjusted capitation payments (RACPs) to competing health insurers or health plans [1] form an essential feature of market-oriented healthcare reforms that are currently taking place in many countries. [2] The idea of RACPs is that a (government) agency finances the respective insurers for a standard benefits package on the basis of their risk profiles, so that there are little or no direct premium payments from the individual members. [3] The intention of the capitation payments is to increase incentives for efficiency and cost containment in the provision of healthcare. However, if the RACPs do not adequately reflect expected costs of insured individuals, insurers also have an incentive to select against those whose healthcare costs are predictably above their capitation payment, thus endangering another objective of healthcare reforms, that is, access to affordable health insurance for everyone. Moreover, insurers that are--possibly as a result of adverse selection (Price and Mays, 1985; Bro wne, 1992; Browne and Doerpinghaus, 1993)--confronted with a relatively unhealthy portfolio not compensated for by the RACPs may experience severe solvency problems.

Governments in countries that use RACPs currently fail to use all potentially available information in setting capitation payments. The failure of insurers to insist that they do so can be seen as a form of bounded rationality that limits solutions to other than the first best. Therefore, this article deals with second best solutions to the selection and solvency problems, which can be appropriately implemented in the political realities in the world of health insurance.

The inadequacy of currently employed RACP-formulae can be illustrated by the fact that the AAPCC system used in the U.S. to finance HMOs for Medicare members accounts for at most 1 or 2 percent of the variation among individuals in annual healthcare costs in this population (Beebe, 1992), while perhaps as much as 20 percent of this variation may be predictable (Newhouse et al., 1989; van Vliet, 1992). The consequence is that it may be very simple to identify groups of people for whom the capitation payments are either much too low or much too high. For example, van de Ven et al. (1994) concluded that in a given year the 10 percent of the population with the highest healthcare expenditures have expenses in (at least) the next four years that are on average roughly double those of the per capita expenditures within their age and sex category.

Another illustration of the inadequacy of currently employed RACP-formulae is the age/sex-based capitation payment system used as of 1993 to finance the sickness funds that are operating in the public health insurance market in the Netherlands. The costs actually incurred by some of these sickness funds in 1993 were 10 percent below their capitation payments, while for others incurred costs were up to 25 percent above capitation payments. (Both figures are for funds with more than 100,000 members.) Without the present system of substantial ex-post equalizations of profits and losses, the latter sickness funds could have already gone bankrupt. However, the Dutch government intends to decrease this equalization to zero in the near future. There are clear indications that at least part of the observed discrepancies are caused by differences in health status between sickness funds (van Vliet, 1994).

An obvious solution to the preceding problems seems to be to include more and better predictors of healthcare costs into the RACP-formula. This has been the main approach followed by many researchers, for instance, Lubitz et al. (1985), Ash et al. (1989), Newhouse et al. (1989), and van de Ven et al. (1994). Although it appears technically possible to develop an adequate RACP-system, the application in practice is not straightforward, mainly because the proposed risk adjusters are either not available in the short run or fail to meet one or more of the criteria for ideal risk adjusters, i.e., validity, reliability, invulnerability to manipulation, obtainability, no perverse incentives, and no conflict with the right to privacy (van de Ven and van Vliet, 1992). In fact, in all of the countries that have implemented RACPs (at least ten), only very crude risk-adjusters are used, such as age, sex, and region.

Therefore, this article explores the statistical and empirical merits of another approach that alleviates the selection and solvency problems when crude RACP-formulae are implemented. The idea is to set up a system of mandatory pooling whereby each health insurer would be allowed to periodically designate a small fraction of its members whose costs would be (partially) pooled. To finance the pool, each insurer would be obliged to pay a uniform contribution for each of its members. The level of this contribution would be predetermined at the macro level as the quotient of the projected size of the pool in monetary terms and the total number of insured in the RACP system. Designation of members to the pool could be done either at the start or at the end of the pool period (for example, for one year). The first case refers to high-risk pooling or prospective reinsurance. Obviously the group of individuals for whom an insurer expects to be highly underpaid by the RACP-formula will be pooled. In the second case--h igh-cost pooling or retrospective reinsurance--the insurer naturally pools those of its members who have had the highest costs. In both cases, different groups of members could be pooled each year. It is clear that insurers with relatively many unhealthy members, who are very likely to be underpaid by any crude RACP system, will benefit more than others from such pooling systems.

There are three reasons that mandatory high-risk or high-cost pooling will reduce the insurers' incentives for selection. First, the potential profits (or avoided losses) of selection will be lower and thus selection will become less attractive from a financial perspective. Second, pooling will make it more difficult for insurers to point out exactly those persons who are likely to be (very) (un)profitable. Of course, when enrollees with the highest (expected) costs are pooled, selection may switch to those just below the pooling threshold. However, it is much harder to predict with any degree of certainty who will be in this precise risk segment. Third, pooling will make the selection process itself more costly to operate.

If improving the RACP-formula is characterized as adjusting the RACPs to better reflect the risk faced by insurers, then mandatory pooling may be regarded as the opposite: the insurer is allowed to adjust its risk in order to better reflect the RACPs. Thus, mandatory pooling simulates to some extent the practice of medical underwriting, which is not allowed in situations where health insurers are faced with open enrollment periods. The latter requirement is part of most healthcare reform proposals.

There are clear analogies with high-risk health insurance pools (Bovbjerg and Koller, 1986; Zellner et al., 1993) and with both reinsurance (either voluntary [Borch, 1990] or mandatory as in the Clinton proposals), and the outlier pool for Medicare HMO payments proposed by Beebe (1992). A major difference, however, is that the contribution of an insurer towards the financing of the pool does not depend on the risk profile of the insurer's portfolio or the risk profile of its pooled members: for each of its members the insurer has to pay a uniform contribution, predetermined at the macro level. This way, the financial burden of high-cost or high-risk individuals is carried (in part) by the market as a whole, thereby alleviating the selection and solvency problems inherent to crude capitation payment systems. Moreover, the system could be set up in such a way that those members whose costs are pooled need not be aware of this. In particular, their benefit package and premium need not reflect the fact that they are pooled. [4]

The purpose of this article is to analyze, both empirically and statistically, the consequences of various types of mandatory high-risk and high-cost pooling. [5] The context is a system of RACPs for financing competing health insurers from a central fund, with an open enrollment requirement and standardized benefits packages, such as the at-risk HMOs in the U.S. and the sickness funds in the Netherlands. The direct motivation of this article were the worries of Dutch sickness funds about the consequences of introducing high-risk or high-cost pooling, which was announced in 1996. The author compares the pooling variants on the basis of the part of total healthcare costs financed through the pool, the correlation between total revenue for an enrollee and its predictable costs, the reduction in the insolvency probability for the insurers, and the degree of correspondence between total revenue and incurred costs at the level of insurers.

The next three sections describe the data, the statistical model, and the results of applying this model to the data. With the statistical model, it is possible to estimate beforehand the consequences for individual insurers of introducing mandatory pooling when only limited information is available for each portfolio. Subsequently, the empirical effects of pooling are explored. The article concludes with a summary and discussion of the main findings.

DATA AND METHODS

The empirical analysis of this article uses three data sets: a micro data set with information from one insurance portfolio; a data set with simulated portfolios; and a data set with information at the macro level for all Dutch sickness funds.

Micro Data

The analysis is largely based on administrative data on the annual healthcare expenditures over 1992 and 1993 of a random sample of 68,891 individuals who were both years enrolled with a large sickness fund in the Netherlands. The expenditures refer to short-term care and include the costs of inpatient room and board and of inpatient specialist care. Estimated costs of outpatient specialist care and physical therapy are included as well. Excluded are the costs of prescribed drugs because these were covered by a separate insurance scheme at the time, and the costs of care provided by family doctors, as they are paid on a flat-rate capitation basis (other healthcare suppliers are paid fee-for-service). Sickness fund members face virtually no deductibles or copayments. Costs in both 1992 and 1993 are inflated so that the means equal 1,640 Dutch forms (Dfl), [6] being the overall mean for all sickness fund members in 1993.

The database also contains information on age, sex, whether one receives an unemployment or disability allowance and data on costs of hospital care from 1988 through 1991, and on diseases that were diagnosed during hospital treatment from 1988 through 1990. The latter type of information, which is highly predictive for future healthcare expenses of individuals (see Ash et al., 1989; van Vliet and van de Ven, 1993), is used for estimating the best possible prediction of expenditures in 1993 for insured individuals. Table 1 gives some statistics about the concentration of 1993 costs, age/sex-expected costs (with an [R.sup.2] at the individual level of 0.026; see Appendix) and costs predicted on all the before-mentioned information from 1988 through 1992 ([R.sup.2] = 0.135). The table illustrates the well-known phenomenon of the heavily skewed distribution of healthcare expenditures: in a given year, the top 1 percent of people accounts for 32.9 percent of expenditures, while the bottom 50 percent is responsible for less than 2 percent. This is comparable with the findings of studies in the United States, for example, Berk and Monheit (1992) and Russell and Chaudhuri (1992). The last column indicates that only about one-seventh of the expenses of the top 1 percent could have been predicted beforehand, underlining again the fact that the larger part of the variation among individuals in annual expenditures is unpredictable.

Simulated Portfolios

To assess the ability of the statistical model presented in the next section to correctly estimate, on the basis of limited information, the consequences for individual insurers when various types of pooling are introduced into an age/sex-based capitation system, eleven portfolios with 100,000 members each were simulated. The members were randomly drawn, with replacement, from the database described above in such a way that within each age/sex category, people with relatively high inpatient costs in the years 1988 through 1991 were given a relatively high probability of being put in an "unhealthy" portfolio while those for whom the opposite held were given a high probability of being assigned to a "healthy" portfolio. This resulted in portfolios with exactly the same age/sex composition, leading also to the same age/sex-based capitation payments. This is important because the effects of some types of pooling correlate more strongly than others with the age composition of a portfolio. The maximum difference in average costs between the simulated portfolios is about Dfl 600, which approximately equals the maximum difference between Dutch sickness funds in 1993. The reason for simulating such large portfolios is to limit the effect of random variation and because 80 percent of Dutch sickness funds, which currently receive RACPs, have more than 100,000 members.

Macro Data

After assessing the performance of the statistical model on the simulated portfolios, it will be applied to actual data on Dutch sickness funds. These funds provide compulsory health insurance with a uniform, comprehensive benefit package for the about 60 percent of the population in the lowest income brackets (in total approximately 9.5 million people), who pay an income-dependent premium. [7] This premium is collected into a central fund from which, until 1993, the sickness funds were fully reimbursed for all of the healthcare expenditures of their members. As of 1993 each sickness fund was financed from the central fund on the basis of the age/sex composition of its portfolio. This capitation payment system was supplemented, for the time being, with a substantial ex-post equalization of profits and losses among sickness funds. This equalization has been lowered from 97 percent in 1993 to approximately 70 percent in 1998, thus greatly increasing the incentives for both efficiency and risk selection. Despite open enrollment, subtle ways of risk selection may occur (see van de Ven et al. [1994] for an overview of the relevant literature, as well as for the experience in the Netherlands with RACPs). For each of the twenty-two largest sickness funds, covering 99.5 percent of publicly insured individuals, 1993 data on capitation payments and incurred costs are available.

STATISTICAL MODEL

The starting point of the statistical analysis is the assumption that annual healthcare expenses of individuals follow a lognormal distribution with parameters [micro] and [sigma]. This assumption is used in many studies of healthcare expenses (for example, de Wit and Kastelijn, 1977; van der Laan, 1988; Newhouse et al., 1989) and can be justified by the fact that over a period of one year, many people have zero or very low healthcare costs, while few people have very high costs. The lognormal distribution closely follows this pattern (Crow and Shimizu, 1988). [8] With this assumption, the average costs for insurer j (denoted by [y.sub.j]) are given by: [9]

[y.sub.j] = exp([[micro].sub.j] + 0.5 . [[[sigma].sup.2].sub.j]) (1)

with:

[[[sigma].sup.2].sub.j] = log([[[cv.sup.2].sub.j] + 1) (2)

where [[micro].sub.j] and [[sigma].sub.j] are the parameters of the lognormaly distributed costs of the members of insurer j and with [cv.sub.j] the coefficient of variation (= standard deviation divided by the average) of costs. With these equations the values of [[micro].sub.j] and [[sigma].sub.j] can be estimated when [y.sub.j] and [cv.sub.j] are available. These parameters are assumed to be insurer specific and time independent, which seems plausible, at least in the short run.

Let us first consider high-risk pooling of x . 100 percent and assume that the insurer pools those members who had the highest costs in the preceding year (year t). This assumption is based on the fact that many studies have shown that the single best predictor of next year's healthcare expenditures of an individual are its expenditures in the previous year (e.g., van Vliet, 1992). The top x . 100 percent of members had costs above the following level:

[g.sub.j] = exp[[[phi].sup.-1](1 - x).[[sigma].sub.j] + [[micro].sub.j]], (3)

where [[phi].sup.-1] is the inverse of the cumulative density function of the standard normal distribution. The expected costs in the pool year (t + 1) for those who exceed this threshold in year t are given by:

E([Y.sub.j,t+1] \ [Y.sub.j,t] [greater than] [g.sub.j]) = ([y.sub.j] / x).{1 - [phi][[[phi].sup.-1](1 - X) - [p.sub.j].[[sigma].sub.j]]}, (4)

where [p.sub.j] denotes the correlation, at the level of insured individuals, between the logarithm of costs in the two years. With high-risk pooling the insurer would not receive a capitation payment for its pooled members because it is reimbursed afterwards for all its costs by the pool. So, an estimate of the reduction of the insurer's RACP because of pooling is needed as well. The capitation payment for the pooled members is:

E([Y.sub.j,t+1] \ [Y.sub.j,t] [greater than] [g.sub.1]) = ([y.sub.j] / x).{1 - [phi][[[phi].sup.-1](1 - x) - [R.sup.2].[[sigma].sub.j]]}, (5)

with [R.sup.2] the proportion of variance in log costs predicted by the RACP-formula. Equations (4) and (5) are based on the formulae for truncated lognormal distributions given by Johnson and Kotz (1970). [10]

The expected overall result (profit or loss) per member after high-risk pooling consists for insurer j of three parts: (a) the balance of RACPs ([Y.sub.j]) and incurred costs ([y.sub.j]) for its whole portfolio; plus (b) the balance of reimbursed costs [equation (4)] and RACPs [equation (5)] for the pooled members; minus (c) the uniform contribution for each of the members, which is used to finance the pool. Thus, the expected overall result per member for insurer j([r.sub.j]) is:

[r.sub.j] = ([Y.sub.j] - [y.sub.j]) + ([a.sub.j] . [y.sub.1]) - ([a.sub.N] . [y.sub.N]) (6)

with:

[a.sub.j] = [phi][[[phi].sup.-1](1 - x) - [R.sup.2] . [[sigma].sub.j]] - [phi][[[phi].sup.-1](1 - x) - [[rho].sub.j] . [[sigma].sub.j]]. (7)

The expression [a.sub.j] . [y.sub.j] is the total amount by which the actual costs of insurer j's pooled members exceed its RACPs, divided by the total number of members in j's portfolio. The parameter [a.sub.j] measures the excess costs of pooled members as a fraction of total costs and will be termed the "excess-cost fraction"; the greater this parameter, the more the insurer will benefit from pooling. The interpretation of the expression [a.sub.N] . [y.sub.N] is analogous to that of [a.sub.j] . [y.sub.j], except that it refers to the entire population that falls under the capitation and pooling systems. Therefore, the value of [a.sub.N] can be estimated similarly to (7), provided that the parameters [[rho].sub.N] and [[sigma].sub.N], which refer to the distribution of costs in the entire population, are available. Note that both [a.sub.j] and [a.sub.N] are independent of the average costs of insurer j, but dependent on the shape of its cost distribution. If the parameters that determine this shape--[[rho].sub.j] and [[sigma].sub.j] (so: [cv.sub.j])--are insurer specific and time independent, then [a.sub.j] is also insurer specific and time independent.

The term [a.sub.N] . [y.sub.N] is the uniform contribution necessary to finance the pool, with [a.sub.N] the monetary size of the pool expressed as a fraction of all healthcare expenditures covered through the capitation system. There is no guarantee that the amount [a.sub.N] . [y.sub.N] as calculated via the macro analog of equation (7) will be sufficient to finance the pool. However, this seems a reasonable first approximation, which appeared to suffice in all the analyses performed with this model. When in practice the actual payments from the pool turn out to be bigger (or smaller) than projected, it would seem reasonable to increase (decrease) the mandatory contributions of the respective insurers in proportion (this is exactly the procedure in the pooling system for Dutch sickness funds).

As the effect of pooling on the final result of an insurer is determined to some extent by the excess-cost fraction [a.sub.j], it is interesting to analyze how this is affected by the other parameters. First note that the expressions in equations (4) and (5) both increase with increasing values of respectively [[rho].sub.j] . [[sigma].sub.j] and [R.sup.2] . [[sigma].sub.j]. Furthermore, the results of van Vliet (1992) imply that the autocorrelation is always bigger than the [R.sup.2] of any RACP-formula. Based on these observations, it can be shown that higher values of [cv.sub.j] and [[rho].sub.j] and a lower value of [R.sup.2] will lead to increases of [a.sub.j]. The latter indicates that improving the RACP, resulting in a higher [R.sup.2], will reduce the effect of pooling, as one would expect.

To get a first indication of the value of [a.sub.j], this parameter was estimated on the data set with micro data described in the previous section. For a high-risk pool of 1 percent the excess-cost fraction appeared to amount to 0.0872, using equation (6), while empirically this was 0.0865, which seems acceptably close. [11] The interpretation of these figures is that an estimated 8.7 percent of all healthcare costs would be financed through the pool when this particular high-risk pool would be implemented.

A variation on the model for high-risk pooling described above would be that for the pooled members only those costs are reimbursed by the pool that exceed a certain threshold. If this threshold is set above the highest capitation payment that could be paid for any individual, it is not necessary to deduct the capitation payment for the pooled members (equation [5]) from their reimbursement (equation [4]). Another advantage of this variant is that an incentive for efficiency and cost containment remains with respect to the pooled members: the insurer will make a profit if the actual costs of a pooled member turn out to be below his or her capitation payment and it will suffer a (limited) loss if the opposite holds. In the latter case the loss is always smaller than the difference between threshold and capitation payment. Note that these profits and losses are taken by the pool when no threshold applies. The second row of Table 2 gives the formula for the excess-cost fraction that appears to hold for this vari ant of mandatory high-risk pooling.

To this point, the author has looked at pooling systems in which members must be designated to the pool before the start of the pool year. An alternative is that this happens after the pool year. This is comparable with reinsurance as often used in the insurance industry to cover extreme claims, with the difference being that reinsurance is voluntary (and not mandatory) and that reinsurance premiums are generally risk rated (and not uniform). Two variants of high-cost pooling will be analyzed, namely pooling of the costs of a fixed percentage of members and pooling of all costs above a certain threshold. The former variant is comparable with the high-risk pooiing of a fixed percentage described above, where again the capitation payment for those who are pooled is to be returned to the pool. This is not necessary for the second variant-which may be termed stop-loss pooling--making the insurer financially responsible for the difference between capitation payment and threshold. (Assuming again that for no indivi dual the capitation payment could exceed the threshold.) Inherent to stop-loss pooling is that, in contrast to pooling of a fixed percentage, the percentage of pooled members will vary from one insurer to the next. This seems fair if we accept the reasonable assumption that the tail end of the cost distribution is composed to a large extent of the costs of people with chronic conditions.

An important difference between high-risk and high-cost pooling is that the former will lessen mainly discrepancies between total revenue (RACP plus pool reimbursement minus pool contribution) and predictable costs whereas the latter will lessen mainly discrepancies between total revenue and actual costs.

On the basis of the lognormal distribution of costs, it is possible to derive the formulae for the excess-cost fraction also for high-cost pooling. These are presented in the third and fourth rows of Table 2.

An obvious variation to the types of pooling discussed so far is partial pooling whereby the insurer remains responsible for part of the costs of pooled members. Obviously, pooling above a fixed threshold already achieves this to some extent, but it may be introduced in all pooling variants by multiplying the parameters [a.sub.j] and [a.sub.N] inequation (6) with a factor f, f = 0 implying no pooling at all and f = 1 leaving equation (6) intact.

RESULTS

This section will first assess the ability of the above-derived formulae to correctly estimate the consequences for individual insurers when an age/sex-based capitation system is supplemented with various types of pooling. The idea here is that we would like to get an indication of the effects of pooling before it is implemented, when at that point only limited information per insurer will be available in general. (This in fact was a major concern of the Dutch sickness funds when high-cost pooling was implemented in 1997.) The four variants of pooling were applied to the data set with micro data described above, where the pooling percentages and thresholds were chosen so that the excess-cost fractions are approximately equal. This is done to facilitate meaningful quantitative comparisons between the variants. For this purpose the excess-cost fraction seems a good criterion because the higher this fraction, the more costs are financed through the pool and thus the less financial responsibility the insurers fac e. The second column of Table 3 presents the results of this exercise. The third column gives the results of applying the formulae of Table 2, using only three parameters of the cost distribution, namely, mean, coefficient of variation and autocorrelation, plus the [R.sup.2] for the age/sex-based capitation payment (cf. Footnote 11).

The results of Table 3 show that even with a small proportion of pooled individuals, the fraction of total cost reimbursed by the pool may be substantial. This is an indication of the skewness of the cost distribution, with many small claims and few large ones. The difference between high-risk and high-cost pooling also becomes more clear: with the former type of pooling many more members can be pooled than with the latter, given a certain monetary size of the pool. The findings of Table 3 indicate that the excess-cost fractions estimated with the formulae of Table 2 closely resemble the empirical fractions. This conclusion appeared to hold true also for more extensive pooling variants.

Of course, Table 3 refers to only one portfolio and one level of the excess-cost fraction. Therefore, Table 4 presents results for eleven simulated portfolios and three levels of the excess-cost fraction. Columns 2 and 3 confirm that the theoretically derived fractions closely approximate the values that would actually apply for these portfolios. For the simulated portfolios together the excess-cost fraction indicates that the percentage of total costs that would be financed through high-risk pooling would increase from about 9 percent in case of a 1 percent pool to about 16 or 17 percent in case of a 3 percent pool. Note that the excess-cost fractions for the comparable variants of pooling are, by design, roughly the same.

Columns 4 and 5 provide an indication of the effect of pooling on the difference between actual costs and total revenues (i.e., the financial results) for the eleven portfolios. Without pooling, the average, over these portfolios, of the absolute values of the results per member amounts to Dfl 178.9. This mean absolute result is reduced by (higher levels of) pooling, the relatively biggest reductions occurring for the least extensive levels of pooling. This indicates that at insurer level, pooling brings closer together actual costs and revenues, with revenues calculated as the sum of capitation payments, costs reimbursed by the pool, and contributions to the pool. Judged by the reduction of the mean absolute result, high-cost pooling of a fixed percentage of members appears to be the least effective type of pooling, while the others are about equally effective. The reason for the relative ineffectiveness of high-cost pooling of a fixed percentage as compared with high-cost pooling above a threshold is that i nsurers with high average costs per member will almost automatically benefit more from high-cost pooling because they are likely to have a relatively high percentage of members who exceed the threshold.

The similarity between the empirical and theoretical findings supports the confidence in the statistical model from which the latter are derived. This confidence is further increased by the fact that the correlations between the empirical and the theoretical losses and profits for the eleven portfolios appeared to be higher than 0.998 for all the pooling variants analyzed. This encouraged the author to apply the statistical model to the data on actual costs and capitation payments for the sickness funds described above (see Table 5). The findings are roughly similar to those of Table 4, where the effectiveness, in terms of lowering the mean absolute result, of high-cost pooling is now somewhat bigger. The reason is that variation exists among the sickness funds in age/sex composition and thus some will pool more of their costs than others simply because their portfolio is older. This does not happen with the simulated portfolios because the age/sex compositions are all the same by design.

Note that with high-risk pooling the percentage of members whose costs are (partly) reimbursed by the pool is on average about 2.5 times as big as with high-cost pooling, suggesting that selection problems are reduced to a greater extent in case of high-risk pooling. Furthermore, with high-cost pooling this percentage appeared to vary over the sickness funds by a factor of almost 2.

In summary, the effects of various types of pooling on the outcome of a capitation payment system can be predicted reasonably well. Even restricted variants, involving only limited numbers of members, may reduce the discrepancies between costs and revenue per sickness fund considerably. Whether this is desirable depends also on other outcome measures, to be explored in the next section.

EMPIRICAL EFFECTS OF POOLING VARIANTS

The outcome measures analyzed so far did not indicate explicitly which type of pooling would be preferable on what criterion. This section extends the analysis by exploring some additional outcome measures.

Excess-Cost Fraction When Pooling increases

For a given monetary size of the pool, Table 3 showed that with high-risk pooling of a fixed percentage eight times as many members can be pooled compared to high-cost pooling. This proportional difference appears to vary with pool size, as Figure 1 reveals. This figure, which is based on the

formulae of Table 2 in combination with the parameter estimates for the data set with micro data (Footnote 11), displays that the excess-cost fraction in case of both high-risk and high-cost pooling appears to reach a maximum, after which it drops again to reach zero when all members would be pooled. The reason is that when ever-increasing numbers of members are pooled, there comes a point when the capitation payments for additionally pooled individuals are higher than their (expected) costs, implying that insurers are obliged to pool individuals on whom they would (expect to) make a profit. Though this maximum is reached at roughly 20 percent for both pooling variants, pooling more than 10 percent of members would seem useless because the excess-cost fraction continues to rise only very slowly after this point.

Figure 1 makes clear that the effectiveness of high-risk pooling quickly drops with increasing numbers of pooled members: the first 1 percent have costs in excess of their RACPs that amount to 9 percent of total costs, for the second this is 4 percent and for the third, 3 percent.

Predictability of Pooled Costs

If it is the intention of the RACP system to compensate health insurers for predictable costs only, then the question arises to what extent a pooling system could contribute to this aim. An indication can be obtained by comparing the total payment an insurer receives for each member--capitation payment plus reimbursed costs in case of pooling minus the pool contribution--with the best prediction of costs attainable in the present data set. The latter was estimated by means of a regression model that related costs in 1993 to all available information in the micro data: age, sex, unemployment and disability allowances, costs in each of the years 1988 through 1992 and diseases diagnosed during hospital treatment from 1988 through 1990. This yielded an [R.sup.2]-value of 0.135 (see Appendix). Subsequently, two statistics were calculated for the various types of pooling: first, the correlation between total payments and predicted costs at the level of insured individuals; and second, the sum of predicted costs for the group of pooled individuals expressed as a percentage of the sum of total payments for this group (see Table 6).

When comparing high-risk with high-cost pooling in Table 6, the results are very conclusive with respect to the predictability issue: total payments correlate most strongly with predicted costs in case of high-risk pooling, a correlation that reduces slightly with an increase in the pooling percentage. Moreover, with high-risk pooling total payments for the group of pooled individuals roughly equal predicted costs. In contrast, with high-cost pooling an insurer would be excessively overpaid for this group: three times (the threshold variant) or even eight times (the percentage variant) as much as would seem reasonable. This implies that there must be groups for which the insurer is substantially underpaid, judged on actuarial principles. It is furthermore remarkable that for each of the six high-risk variants the correlation between total payments and predicted costs is bigger than that for an RACP system without pooling, whereas the opposite holds for the high-cost variants.

These findings suggest that with high-cost pooling some 70 to 90 percent--depending on the variant--of all pooled cost would never have been predictable beforehand whereas this percentage is practically zero for high-risk pooling.

Reduction of Insolvency Probability

An important issue for any insurer is whether its (premium and other) revenues in a particular year will be sufficient to cover all claims, even the occasional very high claims. The probability that an insurer will not be able to fulfill the commitments under the contracts in its portfolio is, under certain regularity conditions, given by:

[alpha] = [phi][-RF/(S.[square root]N)], (8)

with RF the reserve fund of the insurer, S the standard deviation over the contracts of premium minus reimbursed cost, and N the number of contracts. Thus, given the size of the reserve fund and the number of contracts, this so-called probability of ruin (Borch, 1990) decreases with decreasing S. In the present context the statistic S is the standard deviation of the financial result--that is, loss or profit--per insured individual, where result is defined as total payments minus actual costs. Obviously, all types of pooling offer some protection against the eventuality of insolvency problems.

The last column of Table 6 presents S for the types of pooling explored here. An RACP-model based on age/sex and without pooling would yield S = 6,705. If this model were to be improved and reach an [R.sup.2]-value of 0.20, S would become:

S = 4.134 * 1,640 * [square rout of](1- 0.20) = 6,064 [cf. Footnote 12].

The findings indicate that more pooling leads to less variability in financial results and that high-cost pooling above a threshold scores best on this criterion, as could have been expected. Note that the marginal improvement of the standard deviation decreases rapidly with increased pooling. So, here again the first slice of pooling appears to be the most important.

Equation (8) shows that the size of a portfolio is another important factor with respect to (in)solvency. The background of the present study is the Dutch situation with rather large sickness funds, 80 percent with more than 100,000 members. In case of much smaller health plans, for example with 1,000 members, and assuming the reserve fund to be proportional to N, the expression in square brackets in (8) would be ten times greater, resulting in a rise of [alpha], for example from 0.05 to 0.43. Even with the best RACP-formula theoretically attainable, with an [R.sup.2] of 0.20, more extensive types of risk sharing are necessary to keep the insolvency probability within acceptable limits in that situation.

Combination of High-Risk and High-Cost Pooling

A major effect of high-risk pooling appears to be the reduction in (predictable) risk faced by the insurer, whereas a major effect of high-cost pooling is the reduction of the insolvency probability. These positive effects may be combined by using a mix of the two pooling variants, for instance, high-risk pooling of 2 percent and high-cost pooling above Dfl 45,000. Of course, for enrollees falling into both categories, an insurer would be reimbursed only once from the pool. In the micro data set this variant leads to an excess-cost fraction of 0.177, which is comparable with the most extensive pooling variants of Table 4. With respect to the three outcome measures of Table 6, the findings were, respectively: correlation = 0.444; percentage of costs predictable = 84 percent; standard deviation 4,335. The first two outcomes are clear improvements on the findings for all high-cost variants while the standard deviation is a clear improvement on the findings for all high-risk variants.

The conclusion is that some mix of high-risk and high-cost pooling could have advantages over either pooling variants separately.

SUMMARY AND DISCUSSION

Governments in countries that use RACPs currently fail to use all potentially available information in setting capitation payments. The failure of insurers to insist that they do so can be seen as a form of bounded rationality that limits solutions to other than the first best. Therefore, this article dealt with second best solutions that can be appropriately implemented in the political realities in the world of health insurance. In particular, this article analyzed statistical and empirical properties of various types of mandatory high-risk and high-cost pooling schemes that could be used to mitigate the selection and solvency problems inherent to crude RACP systems. An important effect of high-risk pooling is that it reduces the (predictable) risk faced by insurers and by that the insurer's incentive to select against high-risk individuals. The main effect of high-cost pooling is that it lowers the insolvency probability for insurers and by that improves fair competition. The price of these desirable effec ts is reduced incentives for efficiency and cost containment on the part of the insurers. If we are prepared to give up part of these incentives, then the question is which type of pooling gives the biggest reduction in incentives for risk selection and in the insolvency probability Of course, improving the RACPs themselves would be preferable, as better RACPs reduce both risk selection and (non-random) insolvency without reducing incentives for efficiency and cost containment. Unfortunately, this is often impossible because of the lack of good risk adjusters. Much of the information available in administrative data of insurers that can be used for cost prediction is not directly suitable for use in capitation payments because it does not meet the criteria for ideal risk-adjusters mentioned in the Introduction. In particular, the reliability of such data would require constant monitoring and checking by the agency responsible for the RACPs.

For all of the analyzed pooling variants, the amount of costs financed through the pool increases and the discrepancies for insurers between their revenues and incurred costs decrease when the percent of members that they may pool is raised or the threshold above which costs are reimbursed via the pool is lowered.

The pooling variants were compared on the basis of five quantitative criteria (cf. Table 7). First, the monetary size of the pool as a percentage of total cost: in general, the higher this percentage, the smaller is the incentive for efficiency and cost containment. However, if the number of pooled members is relatively small, pooling is unlikely to discourage insurers' managed care activities. Pooled members may be seen as free riders in this respect. The second criterion is the percent of pooled cost that would have been predictable; and the third, the correlation between total revenue (RACP plus--for pooled members--reimbursements from the pool minus pool contributions) received for a member and his or her predictable cost. A pooling variant should score high on both these criteria if one maintains that insurers should be compensated as little as possible for unpredictable high costs since it is the insurer's job to deal with such uncertainties. The fourth criterion for comparing pooling variants was the r eduction in the insolvency probability for an insurer; and the fifth the discrepancies between total revenue and incurred costs at the level of insurers. The latter two criteria are especially important in case of RACPs to small health insurers, although it would seem that in that situation, even the most refined capitation payment systems will have to be combined with extensive forms of risk sharing between health insurers and central fund.

Given a certain monetary size of the pool, the summary findings of Table 7 clearly suggest that of the two high-risk variants, the one without threshold is to be preferred because it scores best on the percentage of pooled cost that is predictable. Of the two high-cost variants, the threshold variant is obviously preferable as it scores better on all quantitative criteria. The choice between either high-risk or high-cost pooling depends to a large extent on the intention of the basic capitation payment system: if its aim is to compensate insurers for predictable costs and by that reduce incentives for risk selection, then high-risk pooling is clearly superior while it also lowers the insolvency probability with about 20 percent. In contrast, the high-cost variant lowers this probability about 40 percent, mainly by pooling costs that are 70 percent unpredictable. A combination of high-risk and high-cost pooling could have advantages over either pooling variants separately.

It is not possible on the basis of these analyses to determine the "optimal" amount of money financed through the pool. The results do indicate, however, that the effectiveness of pooling quickly drops with increasing pool size: pooling is clearly useful for covering high (expected) costs only.

In sum, the analyses of this article indicate that the solvency and selection problems inherent to crude RACP models may be alleviated to a considerable extent by various types of mandatory pooling across insurers with uniform contributions to finance the pool.

Rene C. J. A. van Vliet is associate professor of health economics and statistics in the Department of Health Policy and Management, Erasmus University, Rotterdam, the Netherlands. He thanks Wynand van de Ven, Erik van Barneveld, and Leida Lamers for helpful comments on a previous version of this article.

(1.) The phrases "health insurer," "health plan," and "sickness fund" will be used synonymously in this article.

(2.) Examples of countries that have already implemented RACPs are: the United States (at-risk HMOs), Belgium, the Czech Republic, Germany, the Netherlands, Ireland, Israel, Switzerland, the United Kingdom, and Taiwan.

(3.) The author focuses here on situations in which the RACPs are non-negotiable, which is the case in the countries mentioned in Footnote 2.

(4.) For the same reasons, the mandatory pooling arrangements studied in this article differ from assigned risk pools and joined underwriting associations. Particularly in the European context, it is often deemed unacceptable in health insurance when high-risk individuals would have to pay substantially higher premiums than low-risks, or could obtain less coverage.

(5.) One variant of mandatory high-risk pooling has been analyzed by van Barneveld et al. (1996).

(6.) In November 1999, one U.S. dollar was worth about Dfl 2.09.

(7.) The remaining 40 percent either have a compulsory insurance because they are civil servants (6 percent) or insure themselves with any of the about 60 private health insurance companies (34 percent).

(8.) In theory a delta-lognormal (Crow and Shimizu, 1988), which puts a positive probability mass at zero, would be better because a significant proportion of people will have no costs at all (in the present data base some 20 percent). However, this would lead to seemingly insurmounbale problems in the subsequent derivations while, moreover, the lower end of the distribution is of little importance in the anyalysis.

(9.) To simplify the exposition, this article assumes that at the insurer level expected costs-- E(y)--equals average actual costs. With large insurers (more than 100,000 members in the Dutch situation) and a correction for inflation, this seems a plausible assimption.

(10.) The calculation of [R.sup.2] and [[rho].sub.j], as defined here, is not straightforward because information on [R.sup.2]-values and autocorrelations are generally available at linear, not logarithmic level. It appears, however, that when log(X) and log(Y) follow a bivariate normal distribution with standard deviations [s.sub.x] and [s.sub.y] and correlation (at the logarithmic level) of [[rho].sub.log], then the square of the correlation between X and Y is given by (Johnson and Kotz, 1972, p. 20):

[[rho].sup.2]lin = [[exp([[rho].sub.log].[s.sub.x].[s.sub.y])-1].sup.2] / {[exp([[s.sup.2].sub.x])-1].[exp([[s.sup.2].sub.y])-1]}.

Rewriting this equation, using the fact that [s.sub.x] = [s.sub.y] = s and that [cv.sup.2] = exp([s.sup.2]) -1, gives:

[[rho].sub.log] = log([[rho].sub.lin].[cv.sup.2] + 1) / log([cv.sup.2] + 1).

When log(X) is an OLS-estimate of log(Y) derived from a regression with [R.sup.2]log = [[s.sup.2].sub.x]/[[s.sup.2].sub.y], then: [[rho].sup.2]lin = [R.sup.2]lin, [[rho].sub.log] . [s.sub.x].[s.sub.y] = [R.sub.log] . ([s.sub.y].[R.sub.log]) . [s.sub.y] = [R.sup.2]log . [[s.sup.2].sub.y] and [[s.sup.2].sub.x] = [R.sup.2]log . [[s.sup.2].sub.y] and thus the formula of Johnson and Kotz changes in: [R.sup.2]lin = [exp([R.sup.2]log . [[s.sup.2].sub.y]) - 1] / [exp([[s.sup.2].sub.y]) - 1]. Rewriting gives: [R.sup.2]log = log([R.sup.2]lin . [cv.sup.2] + 1) / log([cv.sup.2] + 1).

(11.) The moment estimators of the relevant parameters for this database have the following values: y = 1,640, cv = 4.134, [rho] = 0.304, and [R.sup.2] = 0.0262. The last two parameters are calculated at the linear level.

REFERENCES

Ash, A., et al., 1989, Adjusting Medicare Capitation Payments Using Prior Hospitalization Data, Health Care Financing Review, 11: 17-30.

Beebe, J. C., 1992, An Outlier Pool for Medicare HMO Payments, Health Care Financing Review, 14: 59-63.

Berk, M. L., and A. C. Monheit, 1992, The concentration of Health Expenditures: An Update, Health Affairs, Winter: 145-149.

Borch, K. H., 1990, Economics of Insurance (Amsterdam, North-Holland).

Bovbjerg, R. R., and C. F. Koller, 1986, State Health Insurance Pools: Current Performance, Future Prospects, Inquiry, 23: 111-121.

Browne, M. J., 1992, Evidence of Adverse Selection in the Individual Health Insurance Market, Journal of Risk and Insurance, 59: 13-33.

Browne, M. J., and H. I. Doerpinghaus, 1993, Information Asymmetries and Adverse Selection in the Market for Individual Medical Expense Insurance, Journal of Risk and Insurance, 60: 300-312.

Crow, E. L., and K. Shimizu, eds., 1988, Lognormal Distributions, Theory and Applications (New York: Marcel Dekker, Inc.)

de Wit, G. W., and W. M. Kastelijn, 1977, An Analysis of Claim Experience in Private Health Insurance To Establish a Relation Between Deductibles and Premium Rebates, Austin Bulletin, 9: 257-266.

Johnson, N. L., and S. Kotz, 1970, Continuous Multivariate Distributions (New York: John Wiley and Sons).

Lubitz, J., J. Beebe, and G. Riley, 1985, Improving the Medicare HMO Payment Formula To Deal With Biased Selection, in: R. M. Scheffler and L. F. Rossiter, eds., Advances in Health Economics and Health Services Research 6 (Greenwich: JAI Press).

Newhouse, J. P., et al., 1989, Adjusting Capitation Rates Using Objective Health Measures and Prior Utilization, Health Care Financing Review, 11: 41-54.

Price, J. R., and J. W. Mays, 1985, Biased Selection in the Federal Employees Health Benefits Program, Inquiry, 22: 67-77.

Russell, L. B., and A. Chaudhuri, 1992, The Inequality of Medical Expenditures for Several Years in a Healthy, Nonelderly Population, Medical Care, 30-10: 908-916.

van Barneveld, E. M., R. C. J. A. van Vliet, and W. P. M. M. van de Ven, 1996, Mandatory High-Risk Pooling: An Approach To Reducing Incentives for Cream Skimming, Inquiry, 33: 133-143.

van der Laan, B. S., 1988, Modeling Total Costs of Claims of Non-Life Insurances, unpublished doctoral dissertation, Erasmus University Rotterdam, The Netherlands.

van de Ven, W. P. M. M., and R. C. J. A. van Vliet, 1992, How Can We Prevent Cream Skimming in a Competitive Health Insurance Market? The Great Challenge for the 90's, in P. Zweifel and T. Frech, eds.: Health Economics Worldwide (Amsterdam: Kluwer).

van de Ven, W. P. M. M., et al., 1994, Risk-Adjusted Capitation: Recent Experiences in the Netherlands, Health Affairs, 13: 118-136.

van Vliet, R. C. J. A., 1992, Predictability of Individual Health Care Expenditures, Journal of Risk and Insurance, 59: 443-460.

van Vliet, R. C. J. A., 1994, Regio-Component in Normuitkeringen (Incorporating a Regional Factor in Risk-Adjusted Capitation Payments to Sickness Funds), unpublished report, Department of Health Policy and Management, Erasmus University, Rotterdam.

van Vliet, R. C.J.A., and W.P.M.M. van de Ven, 1993, Capitation Payments Based on Prior Hospitalizations, Health Economics, 2: 177-188.

Zellner, B. B., D. K. Haugen, and B. Down, 1993, A Study of Minnesota's High-Risk Health Insurance Pool, Inquiry, 30: 170-179.
 Concentration of Healthcare Expenditures
 (1993), Overall Mean Costs = 1,640
 (Micro Data Set; N = 68,891) [a]
 % of total predicted
 % of members % of total costs based on
ranked by actual costs actual costs age/sex ([R.sup.2] = 0.026)
 Top 0.5 percent 22.3% 0.85%
 Top 1 percent 32.9 1.7
 Top 5 percent 68.0 7.6
 Top 10 percent 81.5 13.4
 Top 25 percent 92.8 28.0
 Top 50 percent 98.3 50.0
 % of total predicted
 % of members costs based on all risk-
ranked by actual costs adjusters ([R.sup.2] = 0.135)
 Top 0.5 percent 3.0%
 Top 1 percent 4.9
 Top 5 percent 14.9
 Top 10 percent 23.4
 Top 25 percent 41.3
 Top 50 percent 62.7


(a.)Predicted costs are based on a regression model that related costs in 1993 to all available information in the micro data set: age, sex, unemployment and disability allowances, costs in each of the years 1988 through 1991, and diseases diagnosed during hospital treatment from 1988 through 1990 (see Appendix).
 Four Variants of Mandatory Pooling:
 Formulae for Excess-Cost Fractions [a, b]
High-risk pooling (x .100%) [phi]{[[phi].sup.-1](1-x)-[R.sup.2].
 [[sigma].sub.j]}-[phi]{[[phi].sup.-1]
 (1-x)-[[rho].sub.j].[[sigma].sub.j]}
Idem, with threshold (d) [a] (1-d.[[pi].sub.j]/[y.sub.j])-
 [[[integral].sup.[infinity]].sub.
 [g.sup.i]]x.h(x).[delta](x)-[phi]
 {[[phi].sup.-1](1-x)-
 [[rho].sub.j].[[sigma].sub.j]}
High-cost pooling (x.100%) [phi]{[[phi].sup.-1](1-x)-[R.sup.2].
 [[sigma].sub.j]}-[phi]{[[phi].sup.-1]
 (1-x)-[[sigma].sub.j]}
High-cost pooling above threshold (d) (1-d/[y.sub.j]) + (d/[y.sub.j]).[phi]
 {(log(d)-[[micro].sub.j]}-[phi]{(log(d)
 -[[micro].sub.j])/[[sigma].sub.j]-
 [[sigma].sub.j]}


(a.)The excess-cost fraction is the part of total costs financed through the pool.

(b.) The parameter [[pi].sub.j] is the probability that a member of insurer j will be pooled and that his or her expenses in the pool year exceed the threshold d:

[[pi].sub.j] = P([Y.sub.j,t] [greater than] [g.sub.j] and [Y.sub.j,t+1] [greater than] d) = x-[phi][(log(d)-[[micro].sub.j])/[[sigma].sub.j]]-[phi]{[[phi].sup.-1 ](1-x);[(log(d)-[[micro].sub.j])/[[sigma].sub.j]]; [[rho].sub.j]}

with [phi]{.;.;.) the cumulative density function of the bivariate, standard normal distribution. The integral of the function h(x) in the second row has no analytical solution, so it has to be evaluated by numerical methods. It can be expressed as follows:

h(x) = [(2.[pi].[[sigma].sup.2]).sup.-0.5] .exp(-0.5.[z.sup.2]).[phi]{(log(d)-[micro]-[[sigma].sup.2]-[rho].z)/[ [sigma][square root](1-[[rho].sup.2])]}, with z = [log(d)-[micro]-[rho].[[sigma].sup.2]]/[sigma].
 Excess-Cost Fraction: Part of Costs Financed Through the Pool [a]
 Empirical results Theoretical results
High-risk pooling (1%) 0.086 0.087
Idem, with threshold (Dfl. 6,000) 0.082 0.078
High-cost pooling (0.125%) 0.088 0.091
High-cost pooling above threshold
 (Dfl. 45,000) 0.082 0.086


(a.)The empirical results are based on the micro data set (N = 68,891); the theoretical results are calculated from the formulae of Table 2 and the parameter estimates of Footnote 11.
 Empirical and Theoretical Effects of
 Pooling for Eleven Simulated Portfolios
 With 100,000 Insured Each
 Excess-cost fraction [a]
 No pooling Empirical Theoretical
High-risk, percentage:
 1% 0.086 0.092
 2% 0.132 0.137
 3% 0.161 0.169
High-risk, percentage and above
threshold = Dfl 6,000:
 1% 0.082 0.083
 2% 0.127 0.120
 3% 0.155 0.146
High-cost, percentage:
 0.125% 0.090 0.093
 0.25% 0.143 0.133
 0.375% 0.183 0.162
High-cost, above threshold:
 Dfl 45,000 0.089 0.089
 Dfl 32,500 0.136 0.135
 Dfl 27,500 0.165 0.166
 Mean Drop of mean
 result [b] Absolute absolute result
 No pooling Dfl.178.9 Dfl. 178.9 0.0% 0.0%
High-risk, percentage:
 1% 148.9 149.9 -17 -16
 2% 124.7 136.7 -30 -24
 3% 115.6 127.2 -35 -29
High-risk, percentage and above
threshold = Dfl 6,000:
 1% 151.1 148.3 -16 -17
 2% 130.3 134.3 -27 -25
 3% 121.9 124.1 -32 -31
High-cost, percentage:
 0.125% 164.6 156.6 -8 -12
 0.25% 154.2 146.9 -14 -18
 0.375% 147.1 139.8 -18 -22
High-cost, above threshold:
 Dfl 45,000 142.5 149.0 -20 -17
 Dfl 32,500 127.9 136.0 -29 -24
 Dfl 27,500 119.6 127.5 -33 -29


(a.)The excess-cost fraction is the part of total costs financed through the pool.

(b.)The mean absolute result is the average over the simulated portfolios of the discrepancies--in absolute terms--between revenues and actual costs. The former is defined as the sum of capitation payments, reimbursements from the pool, and contributions to the pool.
 Theoretical Effects of Pooling for 22
 Sickness Funds (1993) [a]
 % of pooled Excess-cost Mean absolute
 members [a] fraction result [c]
No pooling 0% 0 Dfl. 83.6
High-risk, percentage:
 1% 1 0.091 70.9
 2% 2 0.135 65.4
 3% 3 0.167 62.0
High-cost, above threshold:
 Dfl 45,000 0.42[0.3-0.6] 0.090 66.6
 Dfl 32,500 0.86[0.7-1.2] 0.135 60.2
 Dfl 27,500 1.26[1.1-1.8] 0.167 56.2
 % drop relative
 to no pooling [d]
No pooling 0%
High-risk, percentage:
 1% -15
 2% -22
 3% -26
High-cost, above threshold:
 Dfl 45,000 -21
 Dfl 32,500 -28
 Dfl 27,500 -33


(a.)This table is based on the following assumptions: [R.sup.2] 0.0264, [cv.sub.j] 4.25 and [[rho].sub.j] = 0.167 + 0.0000858 [y.sub.j]. This last relation is estimated by OLS on the simulated portfolios. For each sickness fund both the actual mean costs and the capitation payments are available.

(b.)For high-cost pooling the minimum and maximum over the 22 sickness funds are given of the percentage of pooled members (in brackets).

(c.)Weighted by the number of members of each sickness fund.

(d.)The last column shows the drop in mean absolute result relative to the situation of no pooling (i.e., the first row of the table).
 Predictability of Pooled Costs and Standard Deviation of
 Total Payments Minus Actual Costs (1993; N = 68,891)
 Correlation between Predicted costs as
 total payment and % of total payments
 predicted costs for pooled members
No pooling 0.413 --
High-risk, percentage:
 1% 0.500 112%
 2% 0.479 105
 3% 0.474 103
High-risk, percentage and
above threshold (= Dfl 6,000):
 1% 0.499 116
 2% 0.485 109
 3% 0.483 106
High-cost, percentage:
 0.125% 0.280 12
 0.25% 0.287 12
 0.375% 0.293 12
High-cost, above threshold:
 Dfl 45,000 0.330 30
 Dfl 32,500 0.336 31
 Dfl 27,500 0.341 30
 Standard deviation
 of total payments
 minus actual costs
No pooling 6,705
High-risk, percentage:
 1% 5,855
 2% 5,460
 3% 5,272
High-risk, percentage and
above threshold (= Dfl 6,000):
 1% 5,863
 2% 5,475
 3% 5,296
High-cost, percentage:
 0.125% 5,164
 0.25% 4,591
 0.375% 4,203
High-cost, above threshold:
 Dfl 45,000 4,930
 Dfl 32,500 4,346
 Dfl 27,500 4,035
 Summary of Quantitative Measures for Comparing
 pooling Variants: Pool Finances Approximately 16
 Percent of All Costs (Abstracted From Tables
 4 Through 6)
 High-risk
 No 3% and
 pooling 3% [greater than] 6,000
(1) Excess-cost fraction -- 0.161 0.155
(2) % of pooled cost
 that is predictable -- 103 106
(3) Correlation between total
 revenue and predicted costs 0.413 0.474 0.483
(4) Standard deviation of total
 revenue minus actual costs 6,705 5,272 5,296
(5) Mean absolute result in
 simulated portfolios 178.9 115.6 121.9
 High-cost
 0.375% [greater than]27,500
(1) Excess-cost fraction 0.183 0.165
(2) % of pooled cost
 that is predictable 12 30
(3) Correlation between total
 revenue and predicted costs 0.293 0.341
(4) Standard deviation of total
 revenue minus actual costs 4,203 4,035
(5) Mean absolute result in
 simulated portfolios 147.1 119.6
 Regression Results (OLS): Dependent
 Variable Is Healthcare Expenditures
 in 1993 (N= 68,891)
Explanatory variables [a] Demographic model Extended model
 Intercept 807.26 [*] 378.47
 Men Women Men
 0-4 -- -244.48 --
 5-9 -249.68 -392.88 -156.04
 10-14 -168.73 -290.20 6.38
 15-19 9.07 130.08 160.63
 20-24 -211.72 293.13 -39.83
 25-29 -142.23 308.11 31.22
 30-34 -231.05 744.78 [*] -85.13
 35-39 92.22 487.20 168.88
Explanatory variables [a]
 Intercept
 Women
 0-4 -113.74
 5-9 -179.80
 10-14 -50.15
 15-19 202.51
 20-24 374.11
 25-29 260.50
 30-34 524.41 [*]
 35-39 316.63
 Men Women Men
 40 - 44 323.81 633.20 [*] 278.18
 45 - 49 544.99 [*] 791.57 [*] 365.97
 50 - 54 924.71 [*] 745.12 [*] 555.46 [*]
 55 - 59 1182.97 [*] 1127.32 [*] 543.16 [*]
 60 - 64 1561.12 [*] 1608.13 [*] 902.53 [*]
 65 - 69 2660.79 [*] 2289.53 [*] 1913.38 [*]
 70 - 74 3083.61 [*] 2067.20 [*] 2145.05 [*]
 75 - 79 3303.41 [*] 3032.13 [*] 2199.35 [*]
 80 - 84 3733.19 [*] 3106.00 [*] 2705.07 [*]
 85+ 3290.79 [*] 2809.07 [*] 2483.42 [*]
In institution (long-term) -1665.42 [*]
Widow(er) 108.56
Unemployed -179.27
Social security 32.00
Disabled 846.01 [*]
In 1992: hospitalized -353.91 [*]
In 1991: hospitalized 178.79
 1990 1989
DCG1 [b] -84.20 -724.55 [*]
DCG2 -628.57 [*] -637.91 [*]
DCG3 -522.63 -227.01
DCG4 310.44 171.60
DCG5 174.80 -222.19
DCG6 -1376.87 [*] 6193.95 [*]
DCG7 5865.43 [*] 4478.58 [*]
DCG8 4205.79 [*] -1110.19 [*]
Healthcare costs: 1992 0.288 [*]
Healthcare costs: 1991 0.023 [*]
Healthcare costs: 1990 0.030 [*]
Healthcare costs: 1989 0.019 [*]
Healthcare costs: 1988 0.016 [*]
[R.sup.2] 0.0262 0.1354
 Women
 40 - 44 435.63
 45 - 49 544.07 [*]
 50 - 54 458.62
 55 - 59 675.94 [*]
 60 - 64 1088.09 [*]
 65 - 69 1662.11 [*]
 70 - 74 1408.69 [*]
 75 - 79 2118.48 [*]
 80 - 84 2018.26 [*]
 85+ 2220.15 [*]
In institution (long-term)
Widow(er)
Unemployed
Social security
Disabled
In 1992: hospitalized
In 1991: hospitalized
 1988
DCG1 [b] -672.14 [*]
DCG2 -303.47
DCG3 -183.20
DCG4 77.61
DCG5 339.09
DCG6 305.50
DCG7 5443.19 [*]
DCG8 2049.57 [*]
Healthcare costs: 1992
Healthcare costs: 1991
Healthcare costs: 1990
Healthcare costs: 1989
Healthcare costs: 1988
[R.sup.2]


(a.)All explanatory variables are dummies, except healthcare costs in 1988 through 1992.

(b.)DCG = Diagnostic Cost Groups as developed by Ash et al. (1989). The reference categories for these dummy variable are DCG0, i.e., not hospitalized in the year in question.

(*.)p [less than] .05.
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Author:van Vliet, Rene C. J. A.
Publication:Journal of Risk and Insurance
Geographic Code:1USA
Date:Jun 1, 2000
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