Unisex-Calculation and Secondary Premium Differentiation in Private Health Insurance. Oliver Riedel

Transcription

1 Unisex-Calculation and Secondary Preiu Differentiation in Private Health Insurance Oliver Riedel University of Giessen Risk Manageent & Insurance Licher Strasse 74, D Giessen, Gerany Eail: oliver.t.riedel@wirtschaft.uni-giessen.de

2 Unisex-Calculation and Secondary Preiu Differentiation in Private Health Insurance The de facto paid preiu in the (Geran private health insurance depends on the insurance payents in the past (clai s experience. In case the insured does not ake use of the insurance contract and accepts his health costs, he receives a defined preiu refund based on the onthly preiu the following year. The insured should therefore trade off between the preiu refund and the health costs he pays by her own, which can be calculated with the stochastic dynaic optiization, to iniize the total financial burden of the health costs (su of insurance contract and out-of-pocket payent. Priary and secondary preiu differentiation have to be analyzed siultaneously. If there is a tendency in Geran or European legislation to prohibit gender based (priary preiu differentiation (like in Gerany for Riester -contracts (as fro 2006 or private copulsory nursing care insurance, the insurance industry has still the opportunity to adjust the secondary preiu differentiation especially in health insurance with high clai probability. In this paper various preiu refund systes which differ e.g. in the aount of the repaid onthly preius and their results on the total financial burden of the insured are discussed under realistic assuptions (regarding interest rate, inflation of health costs, gender based probability of edical treatent. It is shown that under these assuptions strongly differentiating preiu refund systes result in total financial burdens of the insured that are close to the values achieved by gender based preius. Siulation studies lead to the conclusion that the strongly differentiating preiu refund systes can be accepted even if they have a wider range of the stochastic financial burden. The reason is that woen will pay less than in actual preiu refund systes with differentiated preius and en will be better off than in actual preiu refund systes with unisex-calculation. Classification: unisex-tariffs, out-of-pocket-payent, experience rating, siulation study, stochastic dynaic optiization

3 Unisex-Calculation and Secondary Preiu Differentiation in Private Health Insurance INTRODUCTION... 2 PRIMARY AND SECONDARY PREMIUM DIFFERENTIATION IN HEALTH INSURANCE PRIMARY PREMIUM DIFFERENTIATION SECONDARY PREMIUM DIFFERENTIATION OPTIMAL OUT OF POCKET EXPENSES IN DIFFERENT REFUND SYSTEMS CONSTANT PREMIUM REFUND SYSTEMS VARIABLE PREMIUM REFUND SYSTEMS DIFFERENT REFUND SYSTEMS EXPECTED FINANCIAL BURDEN SECONDARY PREMIUM DIFFERENTIATION UNISEX-TARIFFS IN HEALTH-INSURANCE EUROPEAN AND GERMAN LAW ECONOMIC EFFECTS OF UNISEX-TARIFFS UNISEX-TARIFFS AND EXPERIENCE RATING CONCLUSION... 9 Introduction On Deceber 3 th 2004 the Council of the European Union has adopted a directive to ipleent the principle of equal treatent between en and woen in the access to and supply of goods and services. This directive represents a further stage of the proposal for a directive ade by Anna Diaantopoulou in For the insurance business the ain focus is on unisex-tariffs which ust be the basis of calculation if there is no relevant and accurate actuarial and statistic data. The Geran governent has also established unisex-tariffs in different class of insurance (for exaple in copulsory nursing care insurance. Therefore it does see ipossible that there ay be an obligation to calculate unisex-tariffs in Geran insurance arkets even if the insurers do have relevant and accurate actuarial and statistic data.

4 2 In contrast to life insurance, private health insurers cannot only use priary (ex ante preiu differentiation but also secondary (experience-based preiu differentiation. In this paper the possibility to copensate the loss of priary preiu differentiation for unisextariffs through an extended preiu refund syste is discussed. 2 Priary and secondary preiu differentiation in health insurance 2. Priary preiu differentiation Private insurance copanies calculate their preius according to the actuarial equivalent principle. The net risk preiu (P equals the expected payents by the insurer E(S, where S represents the rando variable of the insurer s benefit: P = E(S. If this equation is applied to a group of individual contracts we use the ter collective equivalent principle. The ter individual equivalent principle is used, if the equation holds for each risk and policy holder. Individual equivalent preius are also called (risk- differentiated, risk-adequate or fair preius. Fair preius only refer to actuarial fairness but can be socially undesirable. In these cases the governent is in a position to prohibit such tariffs. For exaple in Gerany 8e VAG (Insurance Supervisory Law prohibits to use the ethnic group or the nationality of the applicant to calculate different preius for these groups. Preiu differentiation can be divided into priary and secondary fors. Priary preiu differentiation is based on risk factors which can be easily observed ex ante like gender and age in life insurance or type of vehicle and iles travelled in car insurance. All these risk factors are soe kind of objective in ters of verifiable. In Geran private health insurance these fundaental principles have to be adapted for longter insurance contracts. The eployer does only have to pay his 50%-share of the preiu if the substitutional health insurance is calculated along the lines of life insurance. In the end alost every Geran private health insurance contract is calculated this way. So the private health insurance copany has to calculate preius which on soe assuptions do not 2 ( VAG states that a substitutive health insurance is a private which can replace the public health insurance.

5 3 rise during the duration of the contract especially when the insured ages even if the expected edical costs rise with age. The adapted expected individual equivalent principle for long-ter insurance contracts can be forulated as follows: expected discounted paid preius by the insured ust equal expected discounted payents by the insurer. Geran private health insurers use apart fro the extend of the insurance coverage three tariff factors to differentiate preius ex ante: state of health and age at the begin of the contract as well as gender of the insured. Let x be the age of the insured at the coenceent of the contract, C i the estiated edical costs (differentiated by gender resulting fro the contract and paid by the insurer, i the interest rate (set to by the calculation regulation and P x the (constant preiu paid by the insured, using the expected individual equivalent principle the preiu can be calculated as follows: P x ω C j j= x = ω j= x l l j j ( ( + i + i j x j x where l j represents the reaining portfolio of policyholders of age j (l 20 is set to 00,000. This paraeter is calculated using ortality rate (q j and probability of cancellation at age j (w j which are published by the insurance supervisors (VerBaFin according to the forula: l j+ =l j (-w j -q j In conclusion, as in live insurance the health insurance preiu will be too high copared to the expected edical costs at the beginning of the contract and the systeatic surplus will be transferred to a special reserve (called ageing reserve. After soe years the preiu will be too low to cover the expected edical costs and the issing aount will be taken fro the ageing reserve. 2.2 Secondary preiu differentiation Even in a priary differentiated collective of policyholders there are often still different types of risks which should pay different preius but are calculated identically. The resulting clai histories depend in ex ante equal groups of insured risks on risk factors which can not be used as tariff-paraeters although they have a significant influence. Such risk factors

6 4 are usually called subjective and cannot be observed ex ante because they depend strongly on the acts of the insured and her attitude towards daages. To include subjective risk factors at least ex post in calculation, the insurer can use secondary preiu differentiation as a part of experience-based rating where the preiu depends on the insurance payents in the past (clai s experience. But experienced-based rating has ore effects because it can also be used in hoogeneous groups where secondary preiu differentiation (aspect of rating does not ake any sense. The ai in such groups is not an ex post preiu differentiation but creating incentives for the insured to pay for sall daages on her own (aspect of co-payent / own contribution. 2 The last ai sees surprising because the contract could have ipleented deductibles with sae incentives for the insured to pay for sall aounts of edical costs by her own. However in Gerany the eployer usually pays 50% of the preiu but often only 0% of the deductible (although a payent by the eployer for the deductible is possible, so the insured gets 50% of preiu reduction but pays 00% deductible. Therefore significant deductibles are rare in the Geran health insurance arket but can be indirectly achieved through refund systes where the insured gets a 00% reduction of the de facto paid preiu for 00% of out-of-pocket payent. A high probability of loss is necessary to be able to apply experience rating to aspects of secondary preiu differentiation. 3 Health insurance contracts eet the requireents because of the high probability of edical treatent. Though, existing preiu refund systes in health insurance are far less pronounced than in autoobile insurance where the proportion between the highest and the lowest preiu is about ten. 4 Usually the insurer refunds a nuber of onthly preius (depending on the insurance copany these ay be up to six resulting in a spread no higher than two if the insured does not get payents fro the contract in the previous year with significant influence to current years de facto paid preiu. The nuber ay be constant (considering only the previous year or ay depend on the nuber of years passed since the last insurance copensation. During the last years soe insurance copanies have changed fro a constant to a variable refund syste. At first sight, we should assue that this iplies a shift of systes with focus on co-payents to those with focus on secondary preiu differentiation. We will discuss this intuitive conclusion in chapter See Farny (2000, p See Karten (993, p. 60 ff. 4 See Morlock/Zipleann (999, p. 7

7 5 3 Optial out-of-pocket payent in different refund systes 3. Constant preiu refund systes The preiu refund in these systes only depends on the previous year s clais: If the insurer has not paid for the insured s edical treatent and the insurance contract continues to exist until the end of June, the insured receives a fixed nuber of onthly preius, for exaple three. Let N be the nuber of annual payents for edical costs, NCB i the no clais bonus class i, OP i the out-of-pocket payent of the insured and C the edical cost in NCB i. The insured should ake her own decision using the following figure. Figure : Out-of-pocket payent in constant preiu refund systes NCB 0 NCB P(C>OP 0 P(C>OP P(C OP 0 P(C OP OP 0 OP P(N= P(N=0 P(N= P(N=0 NCB 0 NCB If there has not been any edical treatent in the previous year (N=0, the insured is located in NBC with the probability P(N=0 and ay for exaple receive a preiu refund of three onthly preius. If there was at least one edical treatent (N= the insured has to decide whether she pays on her own to reach NBC and get the preiu refund or to shift the edical costs to the insurer an get no preiu refund in NCB 0. Since only the previous year is considered in the constant preiu refund syste, there are no effects of the total edical financial burden in following years. Therefore the insured only copares edical costs and preiu refund. The optial out-of-pocket payent equals the

8 6 preiu refund neglecting interest rates 5 and is identical in NCB 0 and NBC because each no clai bonus class can be reached every year. Therefore the insured should pay any edical costs below three onthly preius on her own. 6 The insured should act as if they had an insurance contract with a deductible of three onthly preius (aspect of copayent. On the other hand, this iplicit deductible is the lowest aount paid out to the insured. Therefore the probability of clais in the interval [0, OP equals zero and the insurance copany has to anticipate this insured s reaction and increases the preiu Variable preiu refund systes To deterine the extend of preiu refund when using variable preiu refund systes, the insurer does not only consider the previous year but all of the since the last insurance copensation. Usually the insured receives one onthly preiu for every continuous year without indenification and the axiu nuber of preiu refunds being liited to three or four. If the insurer pays for edical costs, the insured does not receive a preiu refund independent fro her current no clai bonus class. The insured s decision for a refund syste with no clai bonus classes 0 to 3 (representing a preiu refund of 0 to 3 onthly preius can be explained using Figure 2: 5 If interest rates are taken into account, optial out-of-pocket payent equals the discounted preiu refund. 6 We should assue optial out-of-pocket payent to be slightly higher than the preiu refund due to transaction costs (like postage for letters but epirical studies about clais show clais of 0,56 EUR in health insurance. Therefore transaction costs can be neglected. 7 The refund syste should not depend on the annual balance, because only these refund systes can be applied without alterations for the whole duration of the contract and for this reason trusted for optiization by the insureds.

9 7 Figure 2: Out-of-pocket payent in variable preiu refund systes NCB 0 NCB NCB 2 NCB 3 P(C>OP 0 P(C>OP P(C>OP 2 P(C OP 0 P(C>OP 3 P(C OP P(C OP 2 P(C OP 3 OP 0 OP OP 2 OP 3 P(N= P(N=0 P(N= P(N=0 P(N= P(N=0 P(N= P(N=0 NCB 0 NCB NCB 2 NCB 3 For variable refund systes the optial out-of-pocket payent can not be calculated as easily as in systes with constant refunds because only coparing the costs of one year neglects the fact that with exception of NBC 0 each no clai bonus class i can only be reached if the insured was located in class i- during the previous year. Therefore a payent by the insurer does not only effect the preiu refund of the current year but also reduces the preiu refund for the following year. Assuing there will not be any edical treatent in the following years, an insured in NBC 3 has got to take into account that a payent by the insurance in 2004 will decrease the preiu refund by six onthly preius in the following years: Figure 3: Basis odel to calculate optial out-of-pocket payent Year NCB NCB Difference (payent by the insurer (payent by the insured

10 8 This basic exaple with strict assuptions shows that the optial out-of-pocket payent exceeds the current preiu refund. In conclusion, the out-of-pocket payent is not only the self payent of one year s edical costs but also an investent in future preiu refunds. It is onotonically increasing in relation to the actual NBC 8 not only because of the following year s higher preiu refund but also because of the extended period to reach (highest refund NBC 3. Under ore realistic assuptions further edical treatents during the following years have to be considered. If risk neutral behaviour is assued the optial out-of-pocket payent is one which iniizes the expected discounted health costs for the entire life as a su of insurance preius, out-of-pocket payents and preiu refunds. For each treatent, the insured has to decide whether she pays on her own or whether she lets the insurer pay. To deterine the optial out-of-pocket payent for the entire contract duration, the stochastic dynaic optiisation 9 is used. 0 This procedure uses a backward calculation with the expiration of the contract as starting point. Figure 4: Decision at age 00 NCB 0 NCB NCB 2 Expiration of the contract NCB 3 During the final year of life the insured should not pay edical costs of her own because she will not receive any preiu refund which depends on an existing contract until June the following year. If we assue that no insurance holder will live after her 00 th birthday, the 8 Optial out-of-pocket payent in NBC 2 equals 6 onthly preius, 5 onthly preius in NBC and 3 onthly preius in NBC 0. 9 The procedure of stochastic dynaic optiisation is extensively presented in Neuann/Morlock (2002, p Obviously this technique can also be used for constant preiu refund systes, but the previous approach is less coplex.

11 9 optial out-of-pocket payent at age 00 does not depend on the no clai bonus class and equals zero. We can conclude that the total financial burden at age 00 depends only on the NBC and equals for NBC i: Preiu (age 00 i onthly preius (age 99. At age 99 the optiisation can be described as in Figure 5: Figure 5: Decision at age 99 NCB 0 CR Reg. IC NCB 0 NCB NCB 2 CR CR Reg. IC Reg. IC Reg. IC NCB NCB 2 Expiration of the contract NCB 3 CR NCB 3 Now the insured has to take into account that her out-of-pocket payent influences the financial burden at age 00. If the insured is located in NBC i her expected financial burden only depends on her decision about her out-of-pocket payent ( OP, which as stated before depends on her current age ( and current NBC i. This su includes the expected financial burden of all ages till death and is coposed of annual preiu (age 99, preiu refund (age 98, expected out-of-pocket payent (age 99 and expected financial burden at age 00 (as calculated before. Siilar calculations can be ade for ages 98, 97, 96, Therefore, the expected financial burden till death at each age only depends on the out-ofpocket payent at this age, assuing optial insured s decisions during the following years till death and can be calculated i

12 0 for each age, and NBC i<3 according to: + ( ( OP = yearly preiu preiu refund + ( q P( N = 0 E( C E C + P i i i + following cos ts without edicial treatent no effect by out of pocket payent ( NBC 0 OPi + + ( N = xf ( x dx + P( C OPi E( Ci+ + P( C > OPi E( C0 0 exp ected out of pocket payent exp ected cos t reduction by out of pocket payent ( NBC i+ and for each age and NBC i = 3 due to restraining no clai bonus classes to three according to: + ( ( OP = yearly preiu preiu refund + ( q P( N = E( C E C + P where 3 following cos ts without edicial treatent no effect by out of pocket payent ( NBC 0 OP3 + + ( N = xf ( x dx + P( C OP3 E( C3 + P( C > OP3 E( C0 0 exp ected out of pocket payent exp ected cos t reduction by out of pocket payent ( NBC i+ OP i out-of-pocket payent, NBC i and age C rando variable of health costs at age f density function of health costs at age ( C ( i OP E expected financial burden till death as a function of + ( C i + i age OP i, NBC i and E expected financial burden fro age + till death assuing optial out-of-pocket payent fro age + till death q probability of death at age.

13 This equation is iniized for i<3 by OP i = E C + + ( E( C 0 i+ and for i=3 by ( E( C OP = E C Different refund systes 4. Expected Financial burden First of all considering the effects of changes in refund systes on the financial burden, these effects are calculated using following statistics: ortality rate (published in VerBaFin 6/2003 probability of cancellation (published in VerBAV 2/2002 estiated expected edical costs (published in VerBaFin /2003. To obtain ore detailed inforation about the distribution of edical costs and not only using an estiator for expected edical costs a health insurance collective of a Geran insurance copany has been analysed. The relevant interval [0, OP ax can be analysed for age using an exponential distribution with the paraeter /C and the probability of edical treatent being 80% per year throughout the collective. Furtherore, for siplification but not risking oversiplification it ust be assued that the annual preiu is paid on January st, edical treatent takes place on January st and people die on Deceber 3 st. Explicitly, only the net risk preiu is considered. In Gerany there is a 0% loading to prevent the insureds fro very high preius at retireent age which is no part of the preiu refund. Therefore, this loading is only iplicitly regarded: if there is a rise in health costs, it only affects the preius for people under E ( = 0 will be used as optiization s starting point. C i

14 2 Using these paraeters, we obtain an annual preiu of 2, at the age of 30 for a feale and of,59.3 for a ale, assuing that the insured iniizes the financial burden and the insurer anticipates the preiu refund of a variable refund syste in the calculation. 2 The resulting optial out-of-pocket payents are as follows: Figure 6: Optial out-of-pocket payents Optial out-of-pocket payent - feales Optial out-of-pocket payent - ales 000,00 700,00 EUR 800,00 600,00 400,00 200,00 SF 0 SF SF 2 SF 3 EUR 600,00 500,00 400,00 300,00 200,00 SF 0 SF SF 2 SF 3 0, ,00 0, Alter Alter The figure shows a siilar schee for both sexes. At the beginning of the contract, the optial out-of-pocket payent is higher than during the duration or at the end of the contract. 3 The optial out-of-pocket payent is onotonically increasing with NBC i but differs between the sexes. The difference depends ainly on the higher absolute preiu refund for feales due to their higher preius (copared to ales preius. The resulting expected financial burden (preiu, preiu refund and out-of-pocket payent is 48,20.98 EUR for feales and 83,38.98 EUR for ales. Considering a constant preiu refund syste with a no clais bonus of three onthly preius, the insurer calculates a preiu of 2, for feales and,667.5 for ales. The optial out-of-pocket payent is a constant during the duration of the contract for feales and 46,88 for ales. Resulting expected financial burden is 48,55.99 for feales and 83,649.3 for ales, so a refund syste with anticipated refunds does not influence the expected financial burden. 4 2 The analysed syste refunds up to three onthly preius. 3 There are soe exceptions for woen up to the age of 42 due to edical treatent during pregnancy and costs of giving birth 4 The difference of 0,03% for feales and 0,5% for ales results fro the use of the exponential distribution for calculating the expected out-of-pocket payent.

15 3 4.2 Secondary preiu differentiation Since the expected financial burden does not differ between the analysed refund systes, it should be exained which syste has advantages with respect to secondary preiu differentiation. We consider a portfolio of policyholders with 50% high and 50% low risks. Three scenarios have been analysed, assuing differences between these two types of risks as follows: Variant expected edical costs equal 4/3 (high risk and 2/3 (low risk of expected costs of all policyholders [scenario ], probability of edical treatent for one year equals 0,9 (high risk or 0,7 (low risk with respect to 0,8 for an average risk [scenario 2] and both effects considered siultaneously [scenario 3]. Figure 7: Proportion of the expected financial burden between high risk and low risk, constant and variable ( preiu refund syste Proportion of the expected financial burden between high risk and low risk feale constant syste variable syste Scenario,50% 0,79% Scenario 2 4,85% 3,06% Scenario 3 6,37% 3,93% Variant ale constant syste variable syste Scenario,55% 0,98% Scenario 2 4,8% 3,09% Scenario 3 6,38% 4,4% Surprisingly, the ore differentiated variable refund syste is less differentiating for both sexes than the constant refund syste, if the insured s total financial burden of health costs is taken into consideration. Therefore an increase in the axiu nuber of refunded onthly preius has been analysed. It can be shown that, given the assuptions entioned above, the variable refund syste is ore differentiating if the nuber of refunded preius exceeds eight.

16 4 Figure 8: Proportion of the expected financial burden between high risk and low risk, constant and variable ( preiu refund syste Variant Proportion of the expected financial burden between high risk and low risk feale constant syste variable syste Scenario,50%,3% Scenario 2 4,85% 5,75% Scenario 3 6,37% 5,93% Variant ale constant syste variable syste Scenario,55% 8,09% Scenario 2 4,8% 6,87% Scenario 3 6,38% 8,66% Scenario 2 with only differences in the probability of edical treatent is the least differentiating. Scenario with differences in the expected edical costs is far ore differentiating is. The highest differentiation can be observed in scenario 3 with differences in the probability of edical treatent and differences in expected edical costs. 5 Unisex-Tariffs in Health-Insurance 5. European and Geran law On Deceber 3 th 2004 the Council of the European Union has adopted a directive (Council Directive 2004/3/EC to ipleent the principle of equal treatent between en and woen in the access to and supply of goods and services. For the insurance industry article 5 [actuarial factors] is iportant. All new contracts concluded after Deceber 2 st 2007 should be calculated without taken the insured s sex into consideration. In contrast to Anna Diaantopoulou s proposal of a directive the new directive grants the eber states the right to allow using the insured s sex as a risk paraeter if the assessent of risk is based on relevant and accurate actuarial and statistic data with exception of costs related to pregnancy and aternity. The Geran private insurance industry has to use unisex-tariffs in the copulsory nursing care insurance and fro 2006 on in state-aided annuity insurance (so-called Riester-Rente. Other restrictions of preiu calculation can be found in the Geran law against discriination which has yet to be ipleented by the governent and will not be ipleented before

17 5 the election in Septeber Nevertheless, it ust be taken into account that the Geran governent has not only adopted the European directive into the Geran law but went further by ipleenting ore restraining laws than the European directive deands. 5.2 Econoic Effects of Unisex-Tariffs Unisex-Tariffs would result in equal preius for different types of risks. This ay be socially desirable but iplies econoic reactions of the insureds with have to be taken into account. First of all, the effect of adverse selection should be considered. If high risks (feales, regarding unisex-tariffs pay the sae preiu as low risks (ales, regarding unisex-tariffs there should be an increase in feales insurance deand and a decrease in ales insurance deand. 5 Geran private health insurers therefore stated that unisex-tariffs will equal feales tariffs. Learning fro Rothschild/Stiglitz (976, such effects can be neglected if different insurance policies with different extents of coverage are offered. Under these conditions, high risks should buy full coverage and low risks only partial coverage. Therefore, there is a selfsignalling and both types of risks can be calculated according to the actuarial equivalent principle. As discussed in 2.2, deductibles are not very popular in Gerany due to eployer s 50% share of the preiu. Therefore, deductibles can not solve the proble and different preiu refund systes should be analysed. 5.3 Unisex-Tariffs and Experience Rating Technically unisex-tariffs are only a variation of priary preiu differentiation. As discussed in 4., the total financial burden of the insured does not only depend on priary preiu differentiation but also on secondary preiu differentiation. Morlock/Zipelann (999 analysed analogue effects on the Gerans car insurance arket after deregulation. After 994 any insurance copanies extended their priary preiu differentiation and 5 The adverse selection should occur especially in copleentary insurance. The effects for substitutive insurance ay be less strong due to the actual preiu syste of public health insurance with high preius for people who ay buy substitutive private health insurance instead of public health insurance. Unisex-Tariffs in private health insurance for these ales ay still be lower than public preius. But regarding the discussion about a fixed preiu for everyone in Gerany a changeover to this calculation principle would increase the effects of unisex-tariffs in private health insurance.

18 6 Morlock/Zipelann showed in their paper that those insurers should have adapted their preiu refund systes to calculate risk adequate preius and to avoid over-differentiation with high risks preius being to high and low risks preius being too low. The sae technique can be used for unisex-tariffs. In contrast to car insurance industry with its extended priary differentiation and the resulting need for less secondary preiu differentiation, unisex-tariffs stand for less priary preiu differentiation and should therefore result in extended secondary differentiation. Using the results of 4.2, secondary preiu differentiation can be used to obtain a risk adequate financial burden of the insureds: Statistics, published by Geran insurance supervisors, present significant differences in expected health costs between both genders with higher expected costs for feales. It can be derived fro gender studies that feales and ales have different probabilities of edical treatent with a higher probability for feales. 6 Therefore, unisex-tariffs are a type 3 scenario and an extensive secondary preiu differentiation can be expected. To quantify the ipact of secondary preiu differentiation various preiu refund systes have been analysed. To odel the differences between feales and ales the following results are based on the assuption that the probability of edical treatents is not 0.8 for both sexes but 0.9 for feales and 0.7 for ales. 7 Furtherore it can be assued that the insureds anticipate an increase of health costs by % per year and use an interest rate of 3% per year. The results are as follows: 6 See Klotz (998, p. 37 f. and p These probabilities correspond to epirical data for screening with the observation, that 35% of the feales but only 7-8% of the ales participate in these screenings.

19 7 Figure 9: Proportion of the expected financial burden between feales and ales, different preiu refund systes NBC syste syste syste syste syste syste syste syste A B C D E F G H Proportion 45,64% 44,49% 7,66% 9,85% 9,5% 38,45% 9,65 3,0% Feales expected financial burden of current priary preiu differentiation is in syste C (reference refund syste 63,80% higher than ales expected financial burden. The results lead to the conclusion, that an alost risk adequate expected financial burden can be achieved even in unisex-tariffs as long as secondary preiu differentiation is considered. Except for different expected preiu refunds due to different probabilities of edical treatents the ain influence is based on the higher out-of-pocket payent of ales and the lower out-ofpocket payent of feales due to higher preius for ales and lower for feales in unisex-tariffs copared to actual differentiated preius. The extent of the differentiation of the expected financial burden between feales and ales depends on the preiu refund syste and is greater for refund systes with

20 8 ore no clais bonus classes and higher preiu refunds in each no clais bonus class. Apart fro differentiation between feales and ales an extended experience rating also differentiates between high and low risks of each gender. Therefore the expected financial burden is risk adequate, not only gender adequate. Associated with an increase in secondary preiu differentiation is the risk transfer back to the insured. Especially in long ter insurance the contract iplies a coverage of changes in the health status of the insured during the duration of the contract. To value this effect a Monte-Carlo-Siulation has been executed. The results for refund syste B (easured with the relation of 95% percentile to 5% percentile are as follows: Figure 0: Proportion of 95% percentile to 5% percentile (syste B age ales,60,46,5,53,49,43,37,34 feales,5,25,28,23,20,7,6,5 The differences of the expected financial burden between the 95% and 5% percentile are greater than in the actual syste with differentiated preius for feales and ales. This leads to the conclusion that the coverage for changes in the health status has decreased. But we should also consider that the ales 95% percentile of financial burden in ore differentiated refund systes is after soe years lower than the 50% percentile in less differentiated ones considering unisex-tariffs. Therefore, the additional uncertainty sees for ales acceptable. The sae arguent holds for feales. First of all, the differences between the 95% and 5% percentile are uch saller than the difference for ales but greater than in current systes. But if we consider that feales 95% percentile of financial burden in ore differentiated refund systes and unisex-tariffs is after soe years lower than the 50% percentile in less differentiated refund systes and current priary preiu differentiation, it can be concluded that the additional risk can and should be accepted by the insureds. This arguentation uses the following figure.

21 9 Figure : Percentiles of the insured s financial burden , ,00 Males, Unisex, Syste B, 95% percentile Males, Unisex, Syste C, 50% percentile EUR 0000,00 Feales, Unisex, Syste B, 95% percentile 000, Alter Feales, Priary differentiation, Syste C, 50% percentile 6 Conclusion The ipleentation of unisex-tariffs in health insurance is technically only a variation of priary preiu differentiation. Because the expected financial burden of the insured is deterined by priary and secondary preiu differentiation, the private health insurance has (in contrast to life insurance the opportunity to copensate less priary differentiation by ore secondary differentiation. More differentiated preiu refund systes lead, even in unisex-tariffs, to widely differentiated expected financial burdens of different types of risks. References Farny, D. (Farny 2000: Versicherungsbetriebslehre, 3. Aufl., Karlsruhe Karten, W. (Karten 993: Das Einzelrisiko und seine Kalkulation; Versicherungswirtschaftliches Studienwerk, Studientext 2, Versicherungsbetriebslehre; Wiesbaden 993. Klotz, T. (Klotz 998: Der frühe Tod des starken Geschlechts, Göttingen 998. Meyer, U. (Meyer 997: Langfristige Versicherungsverhältnisse in der Privaten Krankenversicherung, in: Langfristige Versicherungsverhältnisse Ökonoie, Technik, Institutionen, hrsg. von Leonhard Männer, Karlsruhe 997. Morlock, M., M. Zipelann (Morlock/Zipelann 999: Bonus-alus systes Influence of priary preiu differentation on bonus-alus systes, in: K-Foru /99, S

22 20 Neuann, K., M. Morlock (Neuann/Morlock 2002: Operations Research, 2. Aufl., München, Wien Rothschild, M., J. Stiglitz (Rothschild/Stiglitz 976: Equilibriu in copetetive insurance arkets: an essay on the econoics of iperfect inforation, in: Quarterly journal of econoics, 976, p