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The valuation of lifetime health insurance policies with limited coverage.

INTRODUCTION

The populations of many developed countries are aging significantly, due to lower mortality and fertility rates. Continued increases in longevity contribute to rising medical costs and greater demand in health insurance markets. Health insurance reduces individual financial burdens in the event of an illness or injury that requires hospitalization or results in the loss of income. For countries without National Health Insurance (NHI), such as the United States, the private health insurance market provides most coverage of medical expenses. The U.S. Health Insurance Portability and Accountability Act (HIPAA) of 1996 also mandates guaranteed renewability for almost all individual insurance policies (Herring and Pauly, 2006; see also Cochrane, 1995; Pauly, Kunreuther, and Hirth, 1995).

In contrast, in countries with NHI, private health insurance products differ, depending on whether they adopt partial or full NHI systems, that is, on how many citizens join the systems. For example, Germany is a partial NHI country, (1) and citizens choose between NHI or private health insurance systems. (2) If they choose the latter, they receive a comprehensive plan that is similar to those provided by insurance industries in non-NHI countries. For full NHI countries such as Japan, Korea, and Taiwan, NHI compensates for some portion of medical costs, such as 75 percent, and commercial health insurance exists mainly to reimburse customers whose expenses exceed these limits. Payments for these medical insurance policies can be categorized into two types: those in excess of NHI payments and fixed hospitalization payments. Although the former tend to invoke much smaller claim amounts than the latter, they can grow higher than the fixed payments, such as if patients receive new and expensive medicines or treatments not covered by NHI. A fixed hospitalization benefit instead pays a particular amount for each day of a hospital stay, rather than depending on the actual costs incurred. The fixed amount can be spent freely by the insured patient, such as for extra nursing care or the family's everyday living expenses.

Taiwan has been ranked as one of the healthiest countries in the world, largely due to its implementation of NHI in 1995. Taiwan's NHI covers almost 100 percent of the population and has attracted worldwide attention for its low, stable costs (from 5.1 to 6.3 percent of gross national product over 14 years) and short patient waiting times. The NHI's benefit package is comprehensive, including inpatient and outpatient care, dental care, traditional Chinese medicine, and prescription drugs, though not long-term care. National Long-Term Care Insurance is poised to be implemented. In addition, the NHI features a cost-sharing system, with co-insurance and co-payments for covered services. These co-payments, as well as income-based individual premiums, are waived for some people, such as the very poor, veterans, and natives of the region. Out-of-pocket payments are required for services not covered by the NHI, such as prosthodontics, orthodontics, extra charges for private and semiprivate rooms, and special nurses. Under the current NHI plan, patients generally pay NT$100-$200 (about US$3-$6) out of pocket for a physician visit. There is a 10 percent co-payment for inpatient services and 20 percent for outpatient services. On average, out-of-pocket spending by households accounts for 25-30 percent of total health care spending in Taiwan. Consequently, Taiwan's commercial health insurance market mainly serves to reimburse customers whose expenses exceed the NHI limits. A fixed hospitalization benefit is particularly popular.

Taiwanese insurance companies offer a variety of medical insurance plans to suit the needs of individual customers; hospitalization insurance (3) gives policyholders hospitalization benefits, which provide fixed cash benefits for each day of their stay in a hospital. Policyholders can spend the daily cash benefits freely, such as to cover their family's everyday living expenses or to obtain extra nursing care. Most insurance companies provide policies with fixed daily payments for these hospitalization benefits. As a result, without loss of generality, we focus on pricing health insurance policies with fixed daily hospitalization payments.

Popular fixed payment products generally provide lifetime coverage, (4) with limited or unlimited coverage plans, for which the premium rates remain fixed. The premiums are based on traditional actuarial calculations, a common pricing approach that creates significant overpricing when the coverage limit is neither very low nor very high, such that it contributes to excess premiums. (5) In this article, we propose an alternative, more effective pricing model for lifetime health insurance policies with limited coverage.

Currently, the aggregate claims anticipated under the terms of a health insurance contract are the primary information used to calculate health insurance premium rates. The aggregate claim amount generally is obtained by multiplying the claim frequency rate per insured person by the average dollar amount per claim by the number of insured. Two-part models (TPMs), sample selection, and hurdle models all constitute tactics for dealing with demand for medical care. The TPM has been used extensively to estimate demand responses to health insurance prices. For example, to model aggregated medical expenditures, the TPM can determine the frequency and cost of medical services (Duan et al., 1983; Keeler and Rolph, 1988; Mullahy, 1998; Silva and Windmeijer, 2001; Deb and Trivedi, 2002; Frees, Gao, and Rosenberg, 2011). In the actuarial and insurance literature, a model similar to the TPM, called the collective risk model, has been applied widely to model aggregate losses in nonhealthcare insurance markets (Klugman, Panjer, and Willmot 2012). This collective risk model constructs predictions for both the claim number and the claim size in actuarial mathematics, using compound Poisson processes, mixed Poisson processes, and so forth (Willmot, 1987; Ruohonen, 1988; Ambagaspitiya and Balakrishnan, 1994; Rolski et al., 2009).

In contrast, Rosenberg and Farrell (2008) use Bayesian models to predict expenditures and usage by individuals and the group. They assume that the number of hospitalizations follows a Poisson distribution, with a lognormal prior for the mean number of hospitalizations. Cost per hospitalization also may have a gamma distribution. Migon and Moura (2005) instead adopt a compound Poisson process and a hierarchical Bayesian collective risk model to construct an insurance plan. To evaluate premiums, they adopt a theoretical decision framework. Tessera (2007) builds two probabilistic models to capture claim sizes for families versus individuals.

A Markov process model is another technique to price a lifetime disability or medical insurance (Hoem, 1969, 1972, 1988; Consael and Sonnenscheim, 1978; Waters, 1984; Moller, 1992; Norberg, 1993; Wolthuis, 1994, 2003; Jones, 1995; Haberman and Pitacco, 1998; Stenberg, Silvestrov, and Manca, 2006; Stenberg, Manca, and Silvestrov, 2007; D'Amico, Guillen, and Manca, 2009); it is popular because it allows for more than two decision-making steps. Despite these important contributions though, no previous studies consider the problem of medical expenses with a coverage limit. In response, we use the collective risk model to estimate aggregated medical expenses when insurance coverage has a limit. For this type of insurance pricing, we use compound Poison processes to calculate the accumulated benefits for medical expenses.

For example, the accumulated benefit process for hospitalization medical insurance is a stochastic jump process, with uncertainty surrounding the payment time and payment value. Classical actuarial mathematics rely mainly on deterministic approaches, though uncertainty is a fundamental characteristic of the insurance business. Therefore, the insurance industry needs to study stochastic features to support classical deterministic techniques (Pai, 1997). In practice, actuaries use the expected values of aggregate claims, equal to the product of the expected value of claim frequency and the claim amount, to evaluate premiums for health insurance policies with unlimited coverage. An accumulated benefit process for hospitalization medical insurance with limited coverage also entails a stochastic jump process that involves a stop time problem. When they adopt a traditional actuarial perspective, insurance companies use these expected values to evaluate the premiums for health insurance policies with limited and unlimited coverage. Technically, it is possible to price policies with unlimited coverage using this traditional actuarial method, but in practice, it results in overpricing problem for health insurance policies with unlimited coverage. However, little existing research investigates pricing models for limited coverage health insurance policies or addresses the excess premiums that arise from calculations based on traditional actuarial pricing models. In one exception, Huang, Wang, and Liu (2012) provide a numerical example and use Monte Carlo simulation to demonstrate the overpricing problem in practical pricing methods.

Unfair pricing methods demand that insurance companies or policymakers find more accurate models to protect insured clients' rights. This research proposes a pricing model that can accurately evaluate fair premiums for limited coverage policies. We thus investigate, for the first time, the issue of overestimated premiums using traditional expectation principles in practice, and we construct an accurate pricing method for limited coverage policies.

To support our research objectives, we illustrate the overpricing problem for lifetime health insurance policies with limited coverage by Taiwanese insurance companies in the next section. In presenting a more accurate pricing method for lifetime health insurance contracts with limited coverage, we derive their closed-form formulas, then offer numerical analyses. Finally, the "Conclusion" section summarizes our findings and conclusions.

MODEL SETTING

In this article, we consider a lifetime health insurance policy with limited benefits with b x L, where b and L denote fixed daily payment for hospitalization and the limited hospitalization days that the lifetime health insurance policy coverage applies, respectively. Let [M.sup.x.sub.t] be the total number of days for hospitalization in the tth policy year for an insured aged x. When an x-aged insured is alive at time t, the payoff in the tth policy year of a lifetime health insurance policy with limited coverage is equal to the minimum of b x [M.sup.x.sub.t] and the rest of limited benefits.

Existing Pricing Model for Lifetime Health Insurance Policies

In practice, for an x-aged insured in the tth policy year, let y be the average age at which total expected hospitalization days first reach L and be determined as follows:

min {y| [y-x.summation over (t=1)]E([M.sup.x.sub.t]) [greater than or equal to] L}, y > x. (1)

Let [HB.sup.p.sub.x](t) denote the amount of hospitalization benefits received from the policy with limited benefits bL at time t, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Taking into account the probability of death and interest discount factors, the total present value of payments [P.sub.x](0) of a lifetime health insurance policy with limited coverage is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3)

where [v.sup.t] = [(1 + r).sup.-t]; r denotes the riskless interest rate; [T.sub.x] is a continuous random variable of the future lifetime of a life age x; and [1.sub.(.)] is the indicator function.

According to the actuarial equivalent principle, because the fair upfront premium [C.sup.p.sub.x](0) for a lifetime health insurance policy with limited coverage is determined when the actuarial present value of the expected losses is equal to that of the expected revenue, the equilibrium condition takes the form: (6)

[C.sup.P.sub.x](0) = (1 + q)E([P.sub.x](0)), (4)

where q is a premium loading factor for the expense and profit of the insurance company. Assuming that the values of the future lifetime ([T.sub.x]) and total hospitalization days ([M.sup.x.sub.t]) are independent, the fair premium of a lifetime health insurance policy with limited coverage can be given by

[C.sup.P.sub.x](0) = (1 + q)[y - x.summation over (t=1)] ([v.sup.t] x [sub.t][P.sub.x] x E([HB.sup.P.sub.x](t))) (5)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the survival probability of the x-aged insured after t years.

Pricing Problem for Lifetime Health Insurance Policies With Limited Benefits

Many insurance companies in Taiwan employ the mean hospitalization benefits to determine the average age y as in Equation (1) and fair upfront premium for a lifetime health insurance policy as in Equation (5). This pricing method in Equation (5) is correct only if the hospitalization days of the policy coverage, or hospitalization benefits, are unlimited. (7) In the limited coverage case, insurance companies use the mean of hospitalization benefits as the pricing basis for all insured, such that hospitalization benefits for insured with more hospitalization days reach the coverage limit prior to the average age y defined in Equation (1). However, the total hospitalization benefits for insured with fewer hospitalization days probably do not reach the coverage limit prior to y, so their entitlements to benefits are underprivileged, creating an overpricing problem in the upfront premium of a lifetime health insurance policy with limited coverage.

To handle this overpricing problem, we need a more rational pricing method that reflects the distribution of hospitalization benefits, instead of just their mean. Accordingly, we propose a method that employs the distribution of lifetime hospitalization benefits to price a fair upfront premium of a lifetime health insurance policy with coverage limit.

Pricing Model for Lifetime Health Insurance Policies With Limited Benefits

To modify the overpricing problem, we suggest a simple, comprehensible method that can capture the distribution of hospitalization benefits and provide closed-form solutions for pricing lifetime health insurance policies with limited coverage. Assume an x-aged insured buys a lifetime health insurance policy with limited benefits bL. For an x-aged insured at inception (t = 0), two important variables for pricing lifetime health insurance policies with limited coverage are the number of hospitalizations (frequency) in the tth policy year, denoted by [N.sup.x.sub.t], and the number of days per hospital stay (severity) for the ith hospitalization in the tth policy year, denoted by [R.sup.x.sub.t,i]. For a x + t-aged insured, [N.sup.x.sub.t] reflects a Poisson distribution with time-varying intensity [[theta].sub.t]; in addition, [R.sup.x.sub.t,0] = 0 and [R.sup.x.sub.t,i] = [Y.sup.x.sub.t,i] + 1, (8) where [Y.sup.x.sub.t,i], independent of [N.sup.x.sub.t], follows a Poisson distribution with time-varying intensity [[lambda].sub.t]. Accordingly, [[lambda].sub.t] +1 is the average number of days per hospital stay in the tth policy year. For an insured of age x +1, the total number of hospitalization days in the tth policy year, denoted [M.sup.x.sub.t], takes the form:

[M.sup.x.sub.t] = [[N.sup.x.sub.i].summation over (i=0)] [R.sup.x.sub.t,i], (6)

which means that {[M.sup.x.sub.t]; t = 1, ..., [infinity]} follows a compound Poisson process in which both the number of jumps and jump size follow Poisson distributions.

For a lifetime health insurance policy with limited benefits, an x-aged insured receives a hospitalization payment b[M.sup.x.sub.t] in the tth policy year if the accumulated hospitalization payment in the tth policy year does not surpass the limited benefits bL; otherwise, the insured only receives the remainder of the benefit account. Specifically, for a person of age x +1 in the tth policy year, the amount of hospitalization benefits received from a policy with limited benefits bL at time t, denoted by [HB.sub.x](t), is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where t = 1, ..., [infinity] and [H.sub.x](t) denotes the total number of hospitalization days for the x-aged insured prior to the tth policy year, calculated as:

[H.sub.x](t) = [t.summation over (u=1)] [M.sup.x.sub.u] = [t.summation over (u=1)] [[N.sup.x.sub.u].summation over (i=0)] [R.sup.x.sub.i,i]. (8)

Assuming that future lifetime ([T.sub.x]) and total hospitalization days ([M.sup.x.sub.t]) are independent, the fair upfront premium [C.sub.x](0) of a lifetime health insurance policy with limited benefits therefore takes the form:

[C.sub.x](0) = (1 + q) [[infinity].summation over (t=1)] ([v.sup.t] x [sub.t][p.sub.x] x E([HB.sub.x] (t))), (9)

where v is the discounting rate, and [sub.t][p.sub.x] is the probability that the x-aged insured is alive after t years. The closed-form solution of [C.sub.x](0) then can be summarized in the following proposition.

Proposition 1. Assuming that future lifetime ([T.sub.x]) and total hospitalization days ([M.sup.x.sub.t]) are independent, for an x-aged insured, the closed-form formula of the upfront premium [C.sub.x](0) is given by

[C.sub.x](0) = b(1 + q) [[infinity].summation over (t=1)] ([v.sup.t] x [sub.t][p.sub.x] x ([U.sub.1](t) + [U.sub.2](t))), (10)

where [U.sub.2](t)([U.sub.1](t)) represents expected hospitalization days in the tth policy year if the accumulative hospitalization days are less than the limit days in the (t - 1)th policy year and (do not) hit the limit days in the tth policy year and can be expressed in the following:

[U.sub.1](t) = [L-1.summation over (n=1)] [L-1-n.summation over (m=0)] n x P ([H.sub.x](t - 1) = m) x P([M.sup.x.sub.t] = n) (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Furthermore, P([H.sub.x](t - 1) = m) is the probability that total accumulated hospitalization days prior to the (t - 1)th policy year are equal to m, and P([M.sup.x.sub.t] = n) is the probability that the total hospitalization days in the tth policy year are equal to n.

The proof of Proposition 1 is in Appendix A.

In Proposition 1, [C.sub.x](0) consists of two probability mass functions, P([H.sub.x](t - 1) = m) and P([M.sup.x.sub.t] = n). We start by providing the closed-form solution of P([M.sup.x.sub.t] = n) in Proposition 2.

Proposition 2. For an x-aged insured, the probability that total hospitalization days in the tth policy year equal n is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

The proof of Proposition 2 is in Appendix B.

According to Equation (13), we can obtain the first key component P([M.sup.x.sub.t] = n) of the closed-form solution for the upfront premium of a lifetime health insurance policy with limited benefits directly. However, the closed-form expression of P([H.sub.x](t - 1) = m) is difficult to derive. As one of our main contributions, we adopt two methods--the characteristic function approach and probability generating function approach--to derive the analytic solution of P([H.sub.x](t - 1) = m). We first provide the analytic formula of P([H.sub.x](t - 1) = m) through the characteristic function [H.sub.x]([tau]), or the sum of compound Poisson process, in Proposition 3.

Proposition 3. The characteristic function of the total hospitaZization days of an x-aged insured prior to the [tau]th policy year for [tau] = 1, ..., [infinity], denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (14)

where [[LAMBDA].sub.[tau]] = [[tau].summation over (t=1)] [[theta].sub.t] is the average hospitalization days of an x-aged insured prior to the tth policy year, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

The proof of Proposition 3 is in Appendix C.

Because a characteristic function is defined as the Fourier transform of the probability density function, the probability mass function P([H.sub.x](t - 1) = m) can be obtained by an inverse Fourier transform of the characteristic function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], using the following equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Equation (16) is a kind of discrete-time Fourier transform, similar to that found in Moler (2004).

Because [H.sub.x]([tau]) is a discrete random variable whose values can include nonnegative integers, we also can derive the analytic formula of P([H.sub.x](t - 1) = m) with the probability-generating function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this case, the probability mass function P([H.sub.x]([tau]) = m) for [tau] = 1, ..., [infinity] can be recovered:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the mth derivative of the probability generating function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In turn, we provide the explicit expression for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (z) and its mth derivative, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 4. The probability generating function of total hospitalization days of an x-aged insured prior to the tth policy year for [tau] = 1, ... [infinity], is of the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In addition, the mth derivative of the probability-generating function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (z) is given by (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (19)

where [a.sub.1], ..., [a.sub.m] are nonnegative integers and satisfy two conditions: [m.summation over (j=1)][a.sub.j] and [m.summation over (j=1)] [a.sub.j] = m. Finally, [f.sup.(m)] (z) is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

The proof of Proposition 4 is in Appendix D.

We can find P([H.sub.x](t - 1) = m) using Propositions 3 and 4. Therefore, using empirical data, we next discuss the pricing problem for lifetime health insurance policies with limited benefits.

NUMERICAL EXAMPLE OF TAIWAN HEALTH INSURANCE MARKET

In this section, we compare the premiums derived from our theoretical method with those offered by the practical pricing method for a fixed payment hospitalization health insurance product. The premiums of the theoretical method come from Proposition 1; those for the practical pricing method are derived using traditional expectation methods (Equation (5)). As basic assumptions for our numerical example, we establish that

1. Riskless interest r is 2 percent.

2. The fixed daily payments for hospitalization (b) are set at 1,000 New Taiwan dollars (NTD).

3. For both male and female insured ages range from 20 to 80 years.

4. The limited hospitalization days (L) are set to range from 5 to 300. Or equivalently, the limit benefits are from 5,000 NTD to 300,000 NTD.

5. Both the average number of hospital visits ([[theta].sub.t]) and average days of a hospital stay ([[lambda].sub.t]) come from data obtained from a single Taiwanese life insurance company. (10)

6. The life table reflects actual Taiwanese life tables.

Figure 1 illustrates the premiums ([C.sub.x](0)) calculated by our proposed method (11) for different levels of limited hospitalization days (L). Premiums increase with limited hospitalization days, consistent with our intuition. Except for people older than 70 years, insurance premiums increase with age. The parameter ([[theta].sub.t] and [[lambda].sub.t]) settings in the Taiwanese life insurance industry result in downward trends in the premiums for people older than 70 years. For example, when the limit benefit is 100,000 NTD, the premium increases from 39,890 NTD at age 20 to 52,760 NTD at age 70, then back down to 49,030 NTD at age 80. As a rule of thumb, insurance companies in Taiwan set premiums for lifetime health insurance policies with limited coverage as an increasing function of age, though the decreasing trend among people older than 70 years contests this general rule. This phenomenon reflects the flat assumptions adopted by Taiwanese life insurance companies related to hospitalization data, average number of hospitalizations ([[theta].sub.t]), and average number of days per hospital stay ([[lambda].sub.t]) for people older than 95 years. However, it likely is not a concern for our study, because insurance companies typically set an upper age limit (e.g., 65 years), beyond which consumers may no longer purchase health insurance policies with limited coverage. Thus, premiums are an increasing function of age prior to age 65 years.

We define coverage limit effect (CIE) of a lifetime health insurance policy with limited coverage as follows:

CIE = b(1 + q)[[infinity].summation over (t=1)] ([v.sup.t] x [sub.t][p.sub.x] x [U.sub.2](t))/[C.sub.x](0). (21)

In view of Proposition 1 and Equation (21), a higher CIE represents a larger contribution of an embedded limited coverage clause to fair premium of a lifetime health insurance policy. In Figure 2, we depict the relationship between CIE and limited hospitalization days (L). Figure 2 clearly indicates a negative relationship between limited hospitalization days and CIE, which means that when limit hospitalization days are larger, a health insurance policy with limited coverage acts more like one with unlimited coverage and hence a limited coverage clause is much less economically important.

To demonstrate the relationship between excess premiums ([C.sup.p.sub.x](0) - [C.sub.x](0)) and limited hospitalization days (L), in Figure 3 we depict the premium difference between the theoretical method and the practical pricing method for men; it shows excess premiums at different level of limited hospitalization days. Figure 3 clearly indicates the overestimated pricing problem: excess premiums exhibit a humped curve for all ages. They initially increase with limited hospitalization days, then decrease. When limited hospitalization days fall between 100 and 150, the excess premiums are highest for all ages. In general, excess premiums also increase with age. These excess premiums represent the mispricing problem that we seek to address with this study.

To facilitate understanding of the percentage of overcharge caused by using the practical pricing method, we define the excess premium ratio (EPR) as

EPR = [C.sup.p.sub.x](0) - [C.sub.x](0)/[C.sub.x](0). (22)

In Figure 4, we depict the relationship between the EPR and limited hospitalization days (L). When limited hospitalization days increase, the ratio of the premium overestimated decreases. However, the overcharge ratio is slightly greater for younger customers. According to the data from the insurance industry, the ratios of these overcharged prices can be as high as 20 percent for some younger insured people.

In Figures 5-7, we also illustrate the CIE, excess premiums, and EPR for women; there are no major differences compared with the results for men, though both the excesses and ratios are slightly smaller.

CONCLUSION

Aging populations require increased attention to healthcare systems, as well as demand in the lifetime health insurance market. The high-risk exposure associated with lifetime health insurance policies also causes insurance companies to set coverage limits. Although pricing methods can accurately appraise policies with unlimited coverage, they produce incorrect evaluations for policies with limited coverage. We highlight the pricing mistakes that insurance companies make with regard to lifetime health insurance with coverage limits and demonstrate how these incorrect pricing methods compromise the rights of insurance consumers.

As we show, coverage limits have significant impacts on the amount of premiums charged; incorrect pricing methods may lead to premium excesses of as much as 20 percent. Our theoretical framework provides a means to evaluate fair premiums for lifetime health insurance policies with limited coverage more accurately. In turn, we hope that insurance companies take the initiative to avoid pricing mistakes and charge their customers more reasonable, fair, and accurate premiums.

In practice, hospitalization days may relate to the remaining lifetimes for each insured person. Other rating variables, such as education level and income/ wealth, also could affect hospitalization stays. It would be interesting to consider the impact of other rating variables on hospitalization stays and model the dependence structure between hospitalization days and remaining lifetime. In addition, actuaries use several methods to determine premiums, such as variance or standard deviation premium principles. Further research might examine pricing methods according to these different premium principles, to assess the risk and solvency controls. Finally, both single and level payments are available in the Taiwanese health insurance market. By adopting our single-payment case, further research could investigate the pricing of level premiums for a lifetime health insurance policy with limited coverage, including the effects of missing payment and disabilities.

APPENDIX A

This appendix illustrates the detailed proof of Proposition 1. In view of (9), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A1)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A2)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A3)

Because [H.sub.x]([tau]), [tau] = 1, ..., [infinity], is a nondecreasing function that takes nonnegative integer values on the nonnegative integers, we can rewrite Equation (A2) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A4)

where we use the fact that n x P([H.sub.x](t - 1) = m) x P([M.sup.x.sub.t] = n) = 0 if n = 0. Similarly, Equation (A3) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (A5)

where we use the fact that [1.sub.(L-n [less than or equal to] m [less than or equal to] L-1)] = 0 if n = 0. Accordingly, we can derive the closed-form expression of [U.sub.2](t) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A6)

This completes the proof of Proposition 1.

APPENDIX B

This appendix illustrates the detailed proof of Proposition 2. For an x-aged insured, because [R.sup.x.sub.t,0] = 0 and [R.sup.x.sub.t,j] = [Y.sup.x.sub.t,j] + 1 [greater than or equal to] 1 for j [greater than or equal to] 1, total hospitalization days in the tth policy year ([M.sup.x.sub.t]) equals 0 is only when the number of hospitalizations (frequency) in the tth policy year ([N.sup.x.sub.t]) equals 0. As a result, the probability that total hospitalization days in the tth policy year equal zero can be expressed as follows:

P([M.sup.x.sub.t] = 0) = P([N.sup.x.sub.t] = 0) = exp(-[[theta].sub.t]). (B1)

When [M.sup.x.sub.t] is greater than 1, [N.sup.x.sub.t] is also greater than 1. In addition, because [Y.sup.x.sub.t,i], i = 1, 2, ..., are independent identical Poisson distributions with intensity [[lambda].sub.t], we know that [j.summation over (i =1)] [Y.sup.x.sub.t,i] follows a Poisson distribution with intensity j[[lambda].sub.t]. Therefore, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (B2)

where we use the fact that [j.summation over (i=0)] [R.sup.x.sub.t,i] = [j.summation over (i=1)][R.sup.x.sub.t,i] and p([R.sup.x.sub.t,0] = 0 for n [greater than or equal to] 1. Because the value of [j.summation over (i=1)][Y.sup.x.sub.t,i] can include nonnegative integers, n - j [greater than or equal to] 0, that is, j [greater than or equal to] n. As a result, Equation (B2) can be rewritten as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B3)

This completes the proof of Proposition 2.

APPENDIX C

This appendix illustrates the detailed proof of Proposition 3. Because [N.sup.x.sub.t] is independent of [R.sup.x.sub.t,i], the characteristic function of [H.sub.x]([tau]) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (C1)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because [R.sup.x.sub.t,i] = [Y.sup.x.sub.t,i] + 1 and [Y.sup.x.sub.t,i] follows a Poisson distribution with intensity [[lambda].sub.t], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (C2)

This completes the proof of Proposition 3.

APPENDIX D

This appendix illustrates the detailed proof of Proposition 4. Using Equations (C1) and (C2), the probability generating function of [H.sub.x]([tau]), for [tau] = 1, ..., [varies], is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (D1)

Before calculating [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (z), we introduce the Faa di Bruno formula in Johnson (2002), which is a chain rule of the functional derivative:

Lemma D1 (Faa di Bruno formula): If g and f are functions with a sufficient number of derivatives, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (D2)

where the sum over [a.sub.1], ..., [a.sub.m] are nonnegative integers and satisfy two conditions: [m.summation over (j=1)][a.sub.j] = k and [m.summation over (j=1)] j[a.sub.j] = m. In addition, [g.sup.(k)](f (t)) = [d.sup.k]g(f(t))/df[(t).sup.k] According to the Faa di Bruno formula, let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (D3)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. As a result, g(u) is an exponential function of u and satisfies [g.sup.(m)] (f(z)) = g(f (z)). In addition, the mth derivative off, denoted by [f.sup.(m)] (z), is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (D4)

We can prove Equation (D4) through mathematical induction. For the case in which m = 1, the first derivative of f is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (D5)

so Equation (D4) holds when n = 1. Assume that Equation (D5) is valid for m = k, such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (D6)

Then by virtue of Equation (D6), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (D7)

Consequently, Equation (D5) is also valid for m = k + 1. This completes the proof of Proposition 4.

APPENDIX E

In this appendix, we provide the expected values of [N.sup.x.sub.t] for hospitalization frequency in Table E1 and [R.sup.x.sub.t,i] for hospitalization severity in Table E2. It is lack of hospitalization data for people aged older than 95. Therefore, the Taiwan life insurance company adopted the flat assumption for people aged older than 95
Table E1
Average Hospitalization Frequency (Times)

Age    Male    Female

0     0.2784   0.2343
1     0.246    0.207
2     0.2136   0.1797
3     0.1812   0.1524
4     0.1487   0.125
5     0.1162   0.0977
6     0.0837   0.0704
7     0.0513   0.0431
8     0.0463   0.0383
9     0.0412   0.0334
10    0.0362   0.0285
11    0.0312   0.0236
12    0.0262   0.0187
13    0.0303   0.0216
14    0.0344   0.0245
15    0.0385   0.0273
16    0.0426   0.0302
17    0.0467   0.0331
18    0.0515   0.0384
19    0.0563   0.0437
20    0.0611   0.049
21    0.066    0.0543
22    0.0708   0.0596
23    0.069    0.0669
24    0.0673   0.0741
25    0.0655   0.0813
26    0.0637   0.0885
27    0.0619   0.0957
28    0.064    0.097
29    0.0662   0.0983
30    0.0683   0.0996
31    0.0704   0.1009
32    0.0725   0.1022
33    0.0747   0.0977
34    0.0769   0.0932
35    0.0791   0.0887
36    0.0813   0.0843
37    0.0835   0.0798
38    0.086    0.0798
39    0.0884   0.0799
40    0.0909   0.0799
41    0.0934   0.08
42    0.0959   0.08
43    0.0988   0.0827
44    0.1017   0.0854
45    0.1047   0.088
46    0.1076   0.0907
47    0.1105   0.0934
48    0.1156   0.0975
49    0.1206   0.1017
50    0.1257   0.1058
51    0.1307   0.11
52    0.1358   0.1141
53    0.144    0.1198
54    0.1523   0.1255
55    0.1605   0.1312
56    0.1687   0.1369
57    0.1769   0.1426
58    0.1878   0.1508
59    0.1986   0.1591
60    0.2094   0.1674
61    0.2202   0.1757
62    0.231    0.184
63    0.2454   0.1956
64    0.2598   0.2072
65    0.2742   0.2188
66    0.2886   0.2303
67    0.303    0.2419
68    0.325    0.2576
69    0.3469   0.2732
70    0.3689   0.2888
71    0.3909   0.3044
72    0.4129   0.32
73    0.4384   0.3386
74    0.4639   0.3572
75    0.4894   0.3758
76    0.5149   0.3943
77    0.5404   0.4129
78    0.5654   0.4358
79    0.5905   0.4587
80    0.6156   0.4816
81    0.6406   0.5045
82    0.6657   0.5274
83    0.6808   0.5469
84    0.6959   0.5664
85    0.711    0.5859
86    0.7261   0.6054
87    0.7412   0.6249
88    0.7563   0.6444
89    0.7717   0.6639
90    0.7871   0.6834
91    0.8025   0.7029
92    0.8179   0.7224
93    0.8333   0.7419
94    0.8487   0.7614
95    0.8641   0.7809
96    0.8641   0.7809
97    0.8641   0.7809
98    0.8641   0.7809
99    0.8641   0.7809
100   0.8641   0.7809
101   0.8641   0.7809
102   0.8641   0.7809
103   0.8641   0.7809
104   0.8641   0.7809
105   0.8641   0.7809
106   0.8641   0.7809
107   0.8641   0.7809
108   0.8641   0.7809
109   0.8641   0.7809
110   0.8641   0.7809

Source: Huang, Wang, and Liu (2012)

Table E2
Average Hospitalization Severity (Days)

Age    Male    Female

0      6.28    5.61
1      6.05    5.41
2      5.82    5.21
3      5.59    5.01
4      5.36    4.8
5      5.12    4.59
6      4.89    4.39
7      4.65    4.18
8      4.85    4.36
9      5.05    4.54
10     5.25    4.72
11     5.45    4.89
12     5.65    5.07
13     6.07    5.45
14     6.49    5.82
15     6.91    6.19
16     7.33    6.57
17     7.74    6.94
18     7.9     7.08
19     8.05    7.21
20     8.21    7.35
21     8.36    7.49
22     8.52    7.62
23     8.66    7.73
24     8.8     7.85
25     8.93    7.96
26     9.07    8.07
27     9.21    8.18
28     9.52    8.45
29     9.83    8.72
30     10.13   8.99
31     10.44   9.26
32     10.75   9.53
33     11.18   9.92
34     11.61   10.31
35     12.04   10.71
36     12.47   11.1
37     12.9    11.5
38     12.98   11.58
39     13.06   11.66
40     13.14   11.74
41     13.22   11.82
42     13.3    11.9
43     13.13   11.76
44     12.97   11.62
45     12.8    11.48
46     12.64   11.34
47     12.48   11.2
48     12.3    11.05
49     12.12   10.89
50     11.95   10.73
51     11.77   10.58
52     11.6    10.42
53     11.52   10.34
54     11.43   10.27
55     11.35   10.19
56     11.27   10.12
57     11.18   10.04
58     11.16   10.02
59     11.14   10
60     11.12   9.98
61     11.1    9.96
62     11.08   9.94
63     11.17   10.02
64     11.26   10.1
65     11.34   10.18
66     11.43   10.26
67     11.52   10.34
68     11.68   10.5
69     11.85   10.65
70     12.01   10.81
71     12.18   10.96
72     12.34   11.11
73     12.57   11.32
74     12.8    11.52
75     13.03   11.72
76     13.26   11.92
77     13.48   12.13
78     13.84   12.44
79     14.19   12.76
80     14.54   13.08
81     14.9    13.4
82     15.25   13.71
83     15.54   13.97
84     15.83   14.23
85     16.12   14.49
86     16.41   14.75
87     16.7    15.01
88     16.99   15.27
89     17.26   15.54
90     17.53   15.81
91     17.8    16.08
92     18.07   16.35
93     18.34   16.62
94     18.61   16.89
95     18.88   17.16
96     18.88   17.16
97     18.88   17.16
98     18.88   17.16
99     18.88   17.16
100    18.88   17.16
101    18.88   17.16
102    18.88   17.16
103    18.88   17.16
104    18.88   17.16
105    18.88   17.16
106    18.88   17.16
107    18.88   17.16
108    18.88   17.16
109    18.88   17.16
110    18.88   17.16

Source: Huang, Wang, and Liu (2012).


DOI: 10.1111/jori.12070

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Shang-Yin Yang is an Assistant Professor in the Department of Finance, Tunghai University, Taichung, Taiwan. Chou-Wen Wang is a Professor in the Department of Finance, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan and Fellow of Risk and Insurance Research Center, College of Commerce, National Chengchi University, Taipei, Taiwan. Hong-Chih Huang is a Professor in the Department of Risk Management and Insurance, Fellow of Risk and Insurance Research Center, College of Commerce, National Chengchi University, Taipei, Taiwan. The authors can be contacted via e-mail: shangyin@thu. edu.tw, chouwenwang@gmail.com, and jerry2@nccu.edu.tw, jerryhch68@gmail.com. The authors would like to acknowledge helpful comments from the JRI editor and two anonymous referees.

(1) For more information about the Gesetzliche Krankenversicherung (GKV), see http://www. gesetzlichekrankenkassen.de/.

(2) If a person's income is below a certain value, he or she must accept NHI through the GKV. If a person's income is above that amount, or if he or she is self-employed or a civil servant, he or she may apply for private health insurance but also may choose GKV.

(3) In Kenneth and Skipper's (2000) terminology, hospitalization insurance is a hospital confinement indemnity policy.

(4) Some insurance companies also provide 1-year health insurance policies, with or without guaranteed renewability. We exclude these cases from this study.

(5) We define the excess premium as the difference in the premiums determined by the practical pricing method and our pricing model.

(6) We consider the single upfront payment case, though both this and the annual payment cases suffer from overpricing problems.

(7) In an unlimited case, Equation (5) is revised as [C.sup.P.sub.x](0) = (1 + q) [[infinity].summation over (t=1)] ([v.sup.t] x [sub.t][p.sub.x] x E([HB.sup.p.sub.x](t))) and [HB.sup.P.sub.x](t) = b[M.sup.x.sub.t], [for all]t [member of] N.

(8) This setup ensures that when an x-aged insured stays at the hospital, the minimum number of days per hospital stay is greater than 1.

(9) We employ the numerical method introduced by Abate, Choudhury, and Whitt (2000) to obtain the mth derivatives [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](0). The approximation error is less than [10.sup.-8]. We also use inverse Fourier transforms to confirm that the results obtained from probability generation function are accurate.

(10) The data come from one Taiwanese life insurance company that uses them to price its health insurance offerings. Both are functions of age and gender. We use these data to examine the pricing problem. The data of ([[theta].sub.t] and [[lambda].sub.t]) are ultimate table (please see Appendix E).

(11) The shape of the premiums calculated by the practical pricing method is similar to that in Figure 1; we do not reprint it here.

Caption: FIGURE 1 Fair Premiums for Men, Varying Ages and Limited Hospitalization Days

Caption: FIGURE 2 Coverage Limit Effect for Men, Varying Ages and Limited Hospitalization Days

Caption: FIGURE 3 Excess Premiums for Men, Varying Ages and Limited Hospitalization Days

Caption: FIGURE 4 Excess Premium Ratios for Men, Varying Ages and Limited Hospitalization Days

Caption: FIGURE 5 Coverage Limit Effect for Women, Varying Ages and Limited Hospitalization Days

Caption: FIGURE 6 Excess Premiums for Women, Varying Ages and Limited Hospitalization Days

Caption: FIGURE 7 Excess Premium Ratios for Women, Varying Ages and Limited Hospitalization Days
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Date:Sep 1, 2016
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