Risk adjustment in aging societies

von Wyl, Viktor; Beck, Konstantin Article Risk adjustment in aging societies Health Economics Review Provided in Cooperation with: Springer Nature Suggested Citation: von Wyl, Viktor; Beck, Konstantin (2014) : Risk adjustment in aging societies, Health Economics Review, ISSN 2191-1991, Springer, Heidelberg, Vol. 4, Iss. 7, pp. 1-14, https://doi.org/10.1186/s13561-014-0007-5 This Version is available at: https://hdl.handle.net/10419/150461 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/2.0/ RESEARCH Open Access Risk adjustment in aging societies Viktor von Wyl 1,2* and Konstantin Beck 1,3 Abstract Background: In Switzerland, age is the predominant driver of solidarity transfers in risk adjustment (RA). Concerns have been voiced regarding growing imbalances in cost sharing between young and old insured due to demographic changes (larger fraction of elderly >65 years and rise in average age). Particularly young adults aged 19–25 with limited incomes have to shoulder increasing solidarity burdens. Between 1996 and 2011, monthly intergenerational solidarity payments for young adults have doubled from CHF 87 to CHF 182, which corresponds to the highest absolute transfer increase of all age groups. Results: By constructing models for age-specific RA growth and for calculating the lifetime sum of RA transfers we investigated the causes and consequences of demographic changes on RA payments. The models suggest that the main driver for RA increases in the past was below average health care expenditure (HCE) growth in young adults, which was only half as high (average 2% per year) compared with older adults (average 4% per year). Shifts in age group distributions were only accountable for 2% of the CHF 95 rise in RA payments. Despite rising risk adjustment debts for young insured the balance of lifetime transfers remains positive as long as HCE growth rates are greater than the discount rate used in this model (3%). Moreover, the life-cycle model predicts that the lifetime rate of return on RA payments may even be further increased by demographic changes. Nevertheless, continued growth of RA contributions may overwhelm vulnerable age groups such as young adults. We therefore propose methods to limit the burden of social health insurance for specific age groups (e.g. young adults in Switzerland) by capping solidarity payments. Conclusions: Taken together, our mathematical modelling framework helps to gain a better understanding of how demographic changes interact with risk adjustment and how redistribution of funds between age groups can be controlled without inducing further selection incentives. Those methods can help to construct more equitable systems of health financing in light of population aging. Keywords: Risk adjustment; Demography; Health insurance; Intergenerational solidarity JEL codes: I13; J11 Background Societies in highly industrialized countries in Western Europe, North America or Japan have undergone profound demographic changes over the past decades. Lifeexpectancy has increased substantially, owing to reductions of mortality, better life-styles and greater medical possibilities [1,2]. It is estimated that the average lifespan in OECD countries rose by more than 6 years between 1970 and 2000 [3]. Although this growing life expectancy is commonly perceived as positive, it puts strains on the welfare systems of industrialized societies. For example, the ratio of retired individuals to active workers is increasingly shifting towards the elderly, and fewer active workers have to support more retired persons [4]. Apart from pension systems, health insurance systems are also affected by those demographic trends. The impact of aging societies –increasing average age and a rising share of elderly (>65 years) –on health care expenditures (HCE) has long been recognized. For example, recent projections of Swiss health care costs adjusted for expected demographic changes predict substantial overall increases in HCE owing to a higher proportion of elderly in the population and higher cost growth for older insured [5]. As for the latter, the higher * Correspondence: [email protected] 1 CSS-Institute for Empirical Health Economics, Tribschenstrasse 21, 6002 Luzern, Switzerland 2 Institute for Social and Preventive Medicine, University of Bern, Finkenhubelweg 11, 3012 Bern, Switzerland Full list of author information is available at the end of the article © 2014 von Wyl and Beck; licensee Springer. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. von Wyl and Beck Health Economics Review 2014, 4:7 http://www.healtheconomicsreview.com/content/4/1/7 cost increase for older age groups is well documented by long-term observations. For example, Mendelson and Schwarz analyzed Health Care Financing Administration Data from 1977 through 1987 and noticed a disproportionally high cost growth among the elderly aged 65 and more [6], later termed “steepening”by Buchner & Wasem [7]. Such longitudinal analyses of age-stratified costprofiles were also performed for Switzerland and reached similar results [8,9]. However, the reasons for the accelerated cost growth among the elderly are still debated. In particular, the effects of age, medical progress, and interactions thereof are not fully understood, and their disentanglement in statistical models is very challenging. One prominent explanation termed “red herring hypothesis”was put forth by Zweifel et al. [10,11], which states that the cost increases are not linked to age per se but rather to proximity of death (which is more likely at older age), although opposing studies found significant age effects on HCE increases [12-14]. Despite this unresolved debate, there is unanimous agreement that health care expenditures will rise further in the future, which in turn will have implications for health insurance and premium financing. In settings with competing health insurers and community-rated premiums (e.g. Belgium, The Netherlands, and Switzerland), many of the effects of population aging on health financing are mediated through risk adjustment (RA). Risk adjustment (or risk equalization) is a necessary means to prevent risk selection because individual health care expenditures can vary greatly and in part even predictably, whereas health insurance premiums do not [15]. Therefore, premiums systematically do not match costs for certain age groups (e.g. elderly), which leads to incentives for “cream skimming”and discrimination of insured. Risk selection is also harmful from a societal perspective, because it can create losses in welfare and efficiency [15]. Risk adjustment reduces incentives for risk selection. It operates by estimating the difference between groupspecific average health care expenditures and the overall average. The difference between these two amounts is then taxed from groups with below average costs and passed on as a subsidy to groups with greater than average costs. Thus, risk adjustment should equal out risk differences in portfolios of insurers to eliminate “cream skimming”. In general, risk adjustment leads to re-distribution of money from younger, healthier individuals to older, sicker insured, thereby establishing an intergenerational solidarity because age is one of the main drivers of risk adjustment transfers (especially in the context of the Swiss risk adjustment formula). Hence, if the share of elderly in a population grows over time (and thus average health care expenditures increase), this means that younger, healthier individuals have to contribute more to risk adjustment in order to achieve risk equalization. Solidarity across age groups is also established by other transfer schemes in mandatory health insurance. In particular, young adults also benefit from tax-financed premium subsidies. But quantitatively risk adjustment is by far the largest solidarity component in mandatory health insurance: In 2012 young adults contributed CHF 1.37 billion to risk adjustment, but received only CHF 0.52 billion in premium subsidies [16]. As a consequence risk adjustment should play a key role in any attempt to re-distribute the burden of rising HCE in aging societies. Nevertheless, the knowledge on interactions between population aging and risk adjustment is still partial. What is more, frameworks for the implementation of fair (as defined normatively by society) and stable intergenerational solidarity transfers within risk adjustment are, to our knowledge, still lacking. In this paper, we aim to address two questions. First, we aim to investigate how the Swiss risk adjustment scheme (or any scheme) responds to population aging. We will analyze possible effects both from a cross-sectional (i.e. different age groups at single time-points) and from a lifetime perspective (i.e. following an age-cohort of insured over time). Second, we aim to seek ways how solidarity enforced by risk adjustment can be maintained in long term without financially overwhelming especially vulnerable age groups. The remainder of this paper is structured as follows. First follows a brief explanation of the Swiss health care setting. Second, to gain a better understanding of the processes leading to an increasing premium burden for young insured we develop a simple model of risk adjustment payments over time from the perspective of young adults, in which we include variables for health care growth and demographic changes. Although the model is generic and can accommodate any age splits we will center these calculations around young adults aged 19 to 25 years for reasons that will be explained in the methods section. The third section outlines the construction of a mathematical model to assess the balance of risk adjustment over a lifecycle. By use of those models from the second and the third section we assess the importance of demographic change for the increase of solidarity transfers from young to old using data from the Swiss risk adjustment statistics. Moreover, we will sketch out ideas on how to reduce and stabilize the levels of risk adjustment payments for specific age groups, again using the young adults as an example. The results section describes applications of the mathematical models within stochastic simulations and tests different reform suggestions for how they reduce the premium burden for 19 to 25 year old insured. The paper concludes with a discussion of the findings. von Wyl and Beck Health Economics Review 2014, 4:7 Page 2 of 14 http://www.healtheconomicsreview.com/content/4/1/7 Methods Setting The Swiss system of social health insurance is influenced by Enthoven’s concept of managed competition [17]. A large number of health insurers (124 in 1996; 62 in 2011) compete for customers and are obliged to accept any person willing to enroll, independent of age or health status. Mandatory health insurance is organized on a pay-as-you-go basis. Benefit packages are strictly defined and comprehensive. Insurance is not linked to employment, and each insured has to pay the premiums directly to the insurer, although state-funded premium subsidies are granted to individuals in need on the basis of taxable income. Currently 30% of all insured receive such financial assistance [16]. Premiums for mandatory insurance are charged as age-independent community rates with two exceptions. Children between 0 to 18 years of age are granted riskrated premiums with regard to age on a mandatory basis. Moreover, the health insurance law states that insurance companies can grant premium reductions to young adults aged between 19 to 25 years. An internal risk adjustment scheme (i.e. without supplemental funds from the government), defined by retrospective redistribution of premiums across sex and 15 age groups, was introduced in 1993 and left unchanged until 2011 [18]. Starting in 2012, the Swiss risk adjustment was reformed to be based on prospective payments and to additionally include prior hospitalization as a crude morbidity indicator [19]. Further reform steps are currently discussed in the Swiss parliament and will likely include the introduction of pharmaceutical cost groups. Variable notations for modeling analyses In the following, we will develop a simple model with only two groups of insured, which we term young adults and adults. We are using the following notation: C = Average health care expenditures x = Number of individuals p = Proportion in the general population of insured older than 18 years a = Risk adjustment payment Y = Indicator for young adults (e.g. 19–25 year olds) A = Indicator for adults (e.g. >25 year olds) i = Indicator variable for the 30 risk groups (ordered by age) that were included in the Swiss risk adjustment scheme until 2011 (15 age groups, male/female). The indicator i = 1,..,k corresponds to risk groups for young adults with the cut-off denoted by k, and the remaining k + 1 to 30 risk groups represent adults. Definition of risk adjustment equations We define average health care expenditures for young adults Y, adults Aand overall as  CY¼Xk i¼1Cixi Xk i¼1xi  CA¼X30 i¼kþ1Cixi X30 i¼kþ1xi and  C¼X30 i¼1Cixi X30 i¼1xi ð1Þ The proportion of young adults and of adults is defined, respectively, by pY¼Xk i¼1xi=X30 i¼1xi,andp A =1−p Y . Thus, we can also define average health care expenditures  Cas  C¼pY CYþ1−pY   CAð2Þ Risk adjustment transfers for young adults and adults can then be written as aH¼  CH−  Cð3Þ for H=Y or H=A. We will use equation (3)asstarting point for the development of a model of changes in the amount of risk adjustment transfers for young adults. Cross-sectional analysis of impact of demographic changes on risk adjustment transfers For this analysis, we consider the intergenerational part of risk adjustment transfers defined in equation (3). Given equation (2), we can write the intergenerational transfer per person for young adults at time point t = 0 as aY 0¼  CY−  C¼  CY−pY CY−1−pY   CAð4Þ Note that because C −Y<C −risk adjustment transfers for young adults are also negative (aY 0<0), meaning that they have to make payments into the fund. For time points t > 0 the intergenerational part of risk adjustment transfers becomes aY t¼  CY1þΔY ðÞ t−pY1þdY ðÞ t CY1þΔY ðÞ t −1−pY1þdY ðÞ t   CA1þΔA ðÞ t; ð5Þ whereby (1 + Δ Y ) t and (1 + Δ A ) t stand for average health care expenditure (HCE) growth for young adults and adults with average growth rates of Δ Y and Δ A ,respectively. The expression (1 + d Y ) t denotes average changes in the fraction of young adults in the population of all insured older than 18 years (decrease if d Y <0).  CY,  CAand p Y denote starting point values at time 0 (time index is left away for the sake of simplicity). We now combine equations (4) and (5) into a difference equation. In addition, we separate the terms into von Wyl and Beck Health Economics Review 2014, 4:7 Page 3 of 14 http://www.healtheconomicsreview.com/content/4/1/7 those which are independent of the demographic change d Y (first row) and those that are dependent on d Y (second row). aY t−aY 0¼h CY1þΔY ðÞ t−  CA1þΔA ðÞ t i −1−pY ðÞ  CY−  CA hi þpY1þdY ðÞ t CA1þΔA ðÞ t−  CY1þΔY ðÞ t hi ð6Þ If t= 1, then equation (6) simplifies to aY 1−aY 0¼1−pY   CYΔY−  CAΔA  þdYpY CA1þΔA ðÞ−  CY1þΔY ðÞ  ð7Þ This equation (7) can be transformed into a change rate Δ aY through division by aY 0. ΔaY ¼aY 1−aY 0  =aY 0ð8Þ which is important in the context of lifetime redistribution described in the next section. Equation (7) allows for comparative static analyses of the impact of different growth parameter combinations on changes of risk adjustment transfers over time. For example, if there was only a change in the composition of the population (i.e. d Y ≠0andΔ Y ,Δ A = 0), then expression (7) simplifies to dYpY ðÞ  CA−  CY  . The sign of this simplified expression depends on d Y only (change in proportion of insured aged 19 to 25 years), which for Switzerland turns out to be negative (<0, c.f. Results). For young adults with negative transfers (a Y <0) this means that the amount to be paid increases, because equation (7)becomes even more negative. Alternatively, if the only change present was high cost growth among adults (i.e. Δ A > 0andΔ Y ,d Y = 0), then expression (7) becomes pY−1ðÞ  CAΔA  , which is negative because (p Y −1) < 0. Finally, high cost growth for adults and an increase in their share of the population (i.e. d Y <0, Δ A > 0 and Δ Y = 0) leads to elevated risk adjustment contributions for young adults by the amount of dYpYþpY−1ðÞ  CAΔA  þdYpY ðÞ  CA−  CY  , because this expression is negative. A more formal analysis of partial derivatives of equation (7)with respect to different growth parameters is given in Appendix A.1. To summarize, from our two generation model we can derive the following conclusions with respect to population aging. All other things equal, the risk adjustment debt for young adults increases if HCE growth rates are higher for adults than for young adults and/or if the proportion of young adults in the population is shrinking. Basic model for lifetime transfers in risk adjustment (cohort analysis) Given that risk adjustment contributions made by specific age groups can change (as demonstrated in the previous section), what are possible implications for the lifetime balance of risk adjustment transfers? In particular, will the current young generations pay more into risk adjustment than they will ever receive back when they grow old? In the following we will address those questions by developing a discrete-time overlapping generations model, similar in spirit to those developed for the analysis of pension systems (e.g. [20]). Several features of the Swiss risk adjustment make such an analogy quite fitting. First, Swiss social health insurance, of which risk adjustment is an integral part, is a pay-as-you-go system. Because Switzerland only has an internal risk adjustment system (without any tax-financed contributions as for example in Germany) this means that in any given year the contributions made into the fund must equal the benefits paid (henceforth: “symmetry property”). In addition, risk adjustment transfers follow an age-gradient similar to pension schemes, with younger age groups contributing and older generations profiting from risk adjustment (although the addition of further morbidity-criteria may weaken the age dependency). The overall idea for the model is as follows. The curve of risk adjustment transfers ordered by amount and weighted by group size roughly resembles two triangles: One below the zero line (net payers aged 19 to 60 years) and one above (net beneficiaries, aged 61 years and older, Figure 1a). As mentioned above, we will make use of the fact that for internal risk adjustment schemes the two areas defined by the zero-line and the riskadjustment curve are of equal size (symmetry property). Initially, we assume that the shape of those triangles in a given year resembles the pattern of risk adjustment transfers for a single person over a lifetime. In other words, the x-axis interpretation in Figure 1a changes from “age”to “time”. In the simplest model we further assume that a person only lives for two generations: one in which contributions are made and the second in which payments are received. Subsequently we will expand the model to more generations and by modeling growth of risk adjustment payments (Figure 1a and b). The model notation is the same as for the previous section. Two overlapping generations, no population change, undiscounted payments As a convention the superscripts Nand Pdenote negative transfers (contributions into fund) and positive transfers (payments from the fund), respectively. von Wyl and Beck Health Economics Review 2014, 4:7 Page 4 of 14 http://www.healtheconomicsreview.com/content/4/1/7 The area of the dotted triangle below the 0 line in Figure 1a (AN 0at baseline 0) can be approximated by the area of a triangle 0;pN 0;aY 0 ) in Figure 1b. AN 0¼aY 0pN 0=2¼AP 0ð9Þ The notation is similar as in the previous crosssectional model, with aY 0denoting the risk adjustment of the youngest age group (young adults) and pN 0represent- ing the proportion of net-payers (aged 19 to 60 years) at baseline (time 0). Because of the symmetry property AN 0¼AP 0must hold. We now turn to the case of two time periods with a cost increase of (1 + Δ aY ) between the two periods. The variable Δ aY stands for the increase of risk adjustment contributions to be made by young adults in the second time period (cf. Equation 8). Again, there is one time period with payments made into and one with payments received out of the fund. Turning to Figure 1b the young age period with negative transfers is defined by the triangle 0;pN 0;aY 0  and the old age period by triangle pN 0;1;amax 3  .Becauseofthesymmetrypropertyofinternal risk adjustment we can define both triangles in terms of AN 0. AN 1¼AP 1¼aY 01þΔaY ðÞpN 0=2¼1þΔaY ðÞAN 0ð10Þ If Δ aY ≥0 then it is straightforward to show that the inequality AN 0≤AN 1holds. This means that the payments made into the fund at time 0 are surpassed by the payments received out of the fund at time point 1. More than two net-payer generations, no population change, present-value perspective In order to enhance realism of the model we split the life phase of payments to the fund into several separate time periods and allow cost growth between periods. This is shown in Figure 1b for three phases. As a person ages she transits two periods with decreasing contributions into the fund, i.e. the blue area including the points 0;pN 0=2;aY 0=2;aY 0  and the red triangle pN 0=2;pN 0;aY 1=2  , as well as a third life phase with payments out of the fund as shown by the green triangle pN 0;1;amax 3  . The term aY 0=2 follows from the second intercept theorem: If we divide the line (0,pN 0)oftriangle(0;pN 0; aY 0) into halves, then the length of the vertical downward line 0;aY 0  also reduces to 0;aY 0=2  at point pN 0=2. This implies that the area of triangle (0;pN 0=2; aY 0=2) is one fourth of the larger triangle (0;pN 0;aY 0), a property used in expression (11). The steepening dotted black lines symbolize cost growth between different time periods (also note that the area of the red triangle is larger than what the corresponding blue segment for the same period would be). Moreover, we now discount all payments at a rate of r. Let’s first focus on the blue area 0;pN 0=2;aY 0=2;aY 0  and the red area pN 0=2;pN 0;aY 1=2  in Figure 1b. We denote the sum of those negative transfers over n(here two) time periods by TN n. TN n¼AN 0X n−1 i¼0 1þΔaY ðÞ i1 1þrðÞ i n−i n  2 −n−iþ1ðÞ n  2 "# ð11Þ Figure 1 Schematic drawing outlining the concept for modelling lifetime risk adjustment transfers. a shows monthly risk adjustment payments in 2011, ordered by amount of payment (which corresponds to increasing age). Because the volume of payments into and out of the fund are symmetric in Switzerland, the area under the curves below and above the zero-line are identical. For the model of lifetime payments the volume of transfers into the fund (and hence out of the fund because of the symmetry) are approximated by a triangle confined by the most negative transfer to the point where the curve of transfers crosses the zero-line on the x-axis. bgeneralizes the model of lifetime payments for 3 time periods (two as a net-payer into and one as a beneficiary of risk adjustment). von Wyl and Beck Health Economics Review 2014, 4:7 Page 5 of 14 http://www.healtheconomicsreview.com/content/4/1/7 The mechanism of this equation is easy to demonstrate for the case of two periods (n= 2) with net contributions into and a third period (n+ 1) with payments out of the fund. When i= 0 then the first term in the square bracket defines the full triangle at time 0 0;pN 0;aY 0  .In order to obtain the blue segment 0;pN 0=2;aY 0=2;aY 0  we have to subtract the smaller triangle pN 0=2;pN 0;aY 0=2  .The area of the triangle in the second period pN 0=2;pN 0; aY 0 1þΔaY ðÞ=2Þtakes cost growth into account. Because of thesymmetrypropertyforinternalriskadjustment,the area of the green triangle pN 0;1;amax 3  representing insured with positive payments out of the fund at time point 3 (n+1) can be defined according to the following equation. TP nþ1¼AP nþ1¼AN 01þΔaY ðÞ n1 1þrðÞ nð12Þ In order to show that over a lifetime the present value of transfers received from risk adjustment are equal or greater than the amounts paid in we have to verify the following inequality. X n−1 i¼0 1þΔaY ðÞ i1 1þrðÞ i n−i n  2 −n−iþ1ðÞ n  2 "# ≤1þΔaY ðÞ n1 1þrðÞ n ð13Þ It is commonplace in the literature to discount payments at rates between 2% and 3%. In contrast, we estimated the increase of Δ aY at 4.5% p.a. between 1996 and 2011 using Swiss risk adjustment statistics ([21], not shown). Under those circumstances the expression 1þΔaY 1þr  ias a function of iis monotonically increasing, which is an important result on the way to proof inequality (13). In order to demonstrate that expression (13) is true we consider a triangle 0;pN 0;aY 01þΔaY ðÞ  . It is obvious from Figure 1b that the area of this triangle is an overestimate of the actual area of all colored segments, but the mathematical formulation is more tractable. If this triangle is still smaller in size than the one determined by the right hand side of the inequality (green triangle in Figure 1b), then inequality (13) must hold. The triangle 0;pN 0;aY 01þΔaY ðÞ  corresponds to a situation where the index iin (1 + Δ aY ) i and 1 1þrðÞ iof the left hand side of equation (13) is held fixed at i=n−1. This leads to (1 + Δ aY ) n−1 and 1 1þrðÞ n−1, which are the largest possible values in the iterations (provided that Δ aY . and rare both positive and the function is monotonically increasing). Equation (13) can then be rewritten as follows. 1þΔaY ðÞ n−11 1þrðÞ n−1X n−1 i¼0 n−i n  2 −n−iþ1ðÞ n  2 "# ≤1þΔaY ðÞ n1 1þrðÞ n ð14Þ It is easy to show that the sum of the square brackets is just 1 (an expansion of the sum cancels out all terms except for the first and the last, which are 1 and 0, respectively), leading after some re-arrangements to the following equation. 1≤1þΔaY ðÞ 1þrðÞ ð15Þ Given the observed value for Δ aY and an assumed discount rate of 3% inequality (15) holds true. Further note that the right hand side of this inequality can be interpreted as a return rate for the payments made into the fund. Moreover, inequality equation (15) fully integrates with equations (7) and (8) defining Δ aY , and the discussion of effects of different change parameters on risk adjustment payments for young adults applies. In particular, the expected greater increase in costs for adults (compared with young adults) and the decreasing fraction of young adults in the population (as a result of demographic changes) both increase Δ aY and therefore will enhance the return rate. More than two net-payer generations, population change, present-value perspective It is straightforward to show that the above reasoning can be generalized for any ngenerations of net payers (the return rate will actually remain the same). But what happens if, in addition, the population as a whole gets older in average age? We model this change by introducing an additional growth rate d N , which stands for changes in the proportion of net contributors (aged 19 to 60 years) into risk adjustment. Turning again to the scenario with multiple time steps on the left hand side and with one time step on the right hand side, we obtain the following inequality. X n−1 i¼0 1þΔaY ðÞ i1þdN ðÞ i1 1þrðÞ i n−i n  2 −n−iþ1ðÞ n  2 "# ð16Þ ≤1þΔaY ðÞ n1þdN ðÞ n1 1þrðÞ n Again, studying the combination of growth rates 1þΔaY ðÞ i1þdN ðÞ i1 1þrðÞ iis a key step in the analysis of inequality (16). Because d N was positive in the past (the fraction of net payers - corresponding to all age groups von Wyl and Beck Health Economics Review 2014, 4:7 Page 6 of 14 http://www.healtheconomicsreview.com/content/4/1/7 between 19 and 60 years - increased at 0.33% p.a., although not continuously [21]) it is reasonable to assume that all growth rates of net payer fractions combined as a function of iare also monotonically increasing. Proceeding like above we fix all growth rates on the left hand side at i=n−1 and at i=nfor the right hand side term of equation (16). After rearrangements and simplifications we obtain inequality (17). 1≤1þΔaY ðÞ1þdN ðÞ 1þrðÞ ð17Þ This inequality (17) holds true if Δ aY ,d N ,r≥0 and Δ aY ≥ror d N ≥r. Because demographic changes are likely to increase Δ aY , the return rate on the right-hand side of the equal sign is also expected to become larger, all other things equal. In summary, the models developed in this section suggest that, all other things equal, insured will on average receive more (discounted) payments out of the risk adjustment fund than they will have contributed over a lifetime, and that the expected demographic changes are likely to have an increasing effect on the return rate. How can the solidarity burden be redistributed? The calculations from the basic models suggest that, if unchecked, the solidarity burden for young adults will continue to grow at high rate, which is mainly due to health care expenditure growth among older generations. While those younger generations may still be net beneficiaries of risk adjustment payments over lifetime, their owed risk adjustment debt may nonetheless overwhelm their financial means (cf. [22]). In Switzerland, one such vulnerable group are the young adults aged 19–25 years, which are entitled to premium reductions by law. This relaxation from the community-rate principle was introduced to provide relief for young adults who have limited disposable income. However, by 2011 those rebates have all but disappeared [22]. As already observed by Beck in 2004 [23], the reason for those diminishing rebates is that the Swiss risk adjustment scheme ignores the possibility for premium reductions to young adults and overcharges this age group. This problem has been recognized by federal authorities (e.g. [24]), but no convincing solutions have been presented so far. In particular, many proposed solutions neglect that any reduction of solidarity transfers (be it as a premium reduction or a reduction of risk adjustment contributions) must be compatible with risk adjustment so as not to induce selection incentives for insurers. McGuire et al. [25] and Beck et al. (Beck K, Buchner F, van Kleef R, von Wyl V: Theory of risk equalization: Are we on the wrong track? submitted) have provided methodologies for how to limit solidarity transfers within risk adjustment systems. In line with their suggestions we develop a method that can correct for the expected shifts in demography and keep solidarity contributions for specific age groups stable over time. To this end, we introduce two additional parameters in the model described by equation (7). γ= Growth factor determining an upper limit of risk adjustment payment growth. For example, this parameter can be used to decouple risk adjustment contribution growth for young adults from the growth rate of the remaining adults, which is the main driver for the observed increases in younger age groups (c.f. Results). ρ= Factor for reduction of risk adjustment payments (e.g. a premium rebate for young adults). In addition to the stabilization by γa further reduction of nominal payments can be granted. For example, in Switzerland it is currently discussed to charge only 50% of the nominal risk adjustment contributions from 19 to 25 year old individuals (i.e. ρ= 0.5) [22]. Solution for two risk adjustment groups In general, the reductions are implemented by subtracting an amount u t from the nominal RA-payment aY t(defined in Equation 7) so that current RA-payments (at time t) for young adults after correction are equivalent to a baseline payment times a pre-specified growth rate γ. aY 01þγðÞ t¼aY t−utð18Þ Solving for u t yields equation (19). ut¼aY t−aY 01þγðÞ tð19Þ Corrected payments as defined by (18) can be modified further by a second parameter ρthat defines the rebate (<1) on nominal RA payments for specific groups (e.g. 50% to 19 to 25 year olds). This reduction is applied to the stabilized risk adjustment contribution aY 01þγðÞ t for young adults defined in (18), and then equation (19) becomes u t¼aY t−ρaY 01þγðÞ t:ð20Þ Thus, the degree of fairness is determined by the parameters γand ρ. The full equation for reduced intergenerational risk adjustment transfers in young adults reads as a ˜ t Y¼aY t−u t¼ρaY 01þγðÞ tð21Þ For the transfers to sum to zero the payments benefitting older generations must also be shortened by a certain amount. We denote this deduction by v t . von Wyl and Beck Health Economics Review 2014, 4:7 Page 7 of 14 http://www.healtheconomicsreview.com/content/4/1/7 a ˜ t A¼aA t−vtð22Þ Moreover, we set the restriction that the sum of intergenerational transfers between young adults and adults must equal to zero according to equation (23) 1−pY t  aA t−vt  þpY taY t−u t  ¼0ð23Þ Solving for aA t−vt  and plugging into (22) yields expression (24) a ˜ t A¼−pY t 1−pY t ρaY 01þγðÞ t;ð24Þ which corresponds to the new intergenerational risk adjustment payment of young adults to adults. An extension of the reasoning in this section to several age groups is given in Appendix A.2. Additionally, in Section A.3 of the appendix we show that the lifetime balance of payments stays positive even after reductions of risk adjustment for specific young age groups. Results Retrospective analysis of intergenerational solidarity transfers in Switzerland Next, we illustrate the mathematical models defined by equations (7) and (16) from the methods sections by retrospectively analyzing solidarity transfers between generations over the period of 1996 to 2011 within Swiss risk adjustment. As an example, we center this and the following analysis on the group of young adults aged 19–25 years, and the rationale for that decision is detailed in the methods section and in a companion paper (von Wyl V, Beck K: Distribution of premium burden for mandatory health insurance in Switzerland, submitted). To inform the models we used data from the official Swiss risk adjustment statistics 1996 through 2011 [21], which are displayed in Table 1. The second column shows the proportion of young adults in the Swiss population of individuals older than 18 years. Over the 15 year observation period this proportion has decreased by 0.8% points, thus the share of young adults has shrunken slightly. Columns 3 and 4 show average costs for the two age groups, whereas columns 5 and 6 represent relative changes using 1996 as the base year (100%). Overall, health care expenditures for young adults have increased by 32% at an average growth rate of 1.88% and even by 78% for adults older than 25 years (average growth rate 3.94%). Over time, intergenerational risk adjustment payments for young adults have risen from CHF 87 per month in 1996 to CHF 182 in the year 2011 (Table 1, column 7). Yet the effect of those solidarity transfers on the adults’ side remained rather small. In 1996 each adult received CHF 11 per month in solidarity transfers from 19 to 25 year olds. Fifteen years later those payments have only risen by CHF 10 to a total of CHF 21 per month (Table 1, column 8). Overall the solidarity burden for young adults has experienced the highest absolute Table 1 Evolution of risk adjustment transfers between 1996 and 2011 Health care expenditures Cost increase Monthly risk adjustment payment into (<0) or from (>0) the fund (CHF per month) (base year 1996) Year Proportion of young adults Young adults Adults Young adults Adults Young adults Adults 1996 11.1% 61 159 100% 100% −87 11 1997 10.8% 61 166 101% 105% −94 11 1998 10.6% 61 175 101% 110% −101 12 1999 10.6% 62 181 101% 114% −107 13 2000 10.5% 65 192 106% 121% −114 13 2001 10.4% 68 203 111% 128% −121 14 2002 10.5% 69 210 113% 133% −127 15 2003 10.5% 71 220 117% 139% −133 16 2004 10.5% 73 234 121% 147% −143 17 2005 10.5% 75 246 123% 155% −153 18 2006 10.4% 72 247 118% 156% −157 18 2007 10.3% 73 257 120% 162% −165 19 2008 10.3% 76 267 125% 169% −172 20 2009 10.3% 79 275 129% 173% −176 20 2010 10.3% 80 280 131% 177% −180 21 2011 10.3% 80 283 132% 178% −182 21 von Wyl and Beck Health Economics Review 2014, 4:7 Page 8 of 14 http://www.healtheconomicsreview.com/content/4/1/7