Insurer commitment and dynamic pricing pattern

Abstract

A central issue in dynamic contracting is the type of inter-temporal pricing pattern. Some insurance products exhibit a highballing (front-loaded) pattern and others a lowballing (back-loaded) pattern, while still others are flat. We develop a unified competitive dynamic insurance model with asymmetric learning to investigate the impact of insurer commitment on the equilibrium inter-temporal pricing pattern. The model predicts that the equilibrium contract exhibits highballing under one-sided commitment and lowballing under no commitment. We then use a unique empirical setting of two products from one insurer, eliminating heterogeneity in firm, market, time horizon, and learning environment, to isolate the role of insurer commitment in determining the pricing pattern. Consistent with our theoretical predictions, we find that (i) the dynamic contracts exhibit a highballing pattern in loaner’s personal accident insurance, a one-sided commitment scenario, and (ii) a lowballing pattern in group critical illness insurance, a no-commitment scenario.

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Fig. 1
Fig. 2

Notes

  1. 1.

    For example, Swiss insurers usually give a premium discount to policyholders who accept 3- to 5-year contracts, indicating a strong preference for long-term coverage.

  2. 2.

    It is less common to observe single-period insurance relationships in practice. Even for project-based coverage, such as protections for construction projects or satellite launches, the project owner tends to continuously work with the same insurer on one project after another; hence, it is essentially a multi-period relationship.

  3. 3.

    The dynamic pricing pattern is in essence an insurer’s pricing strategy to charge upfront the value of its own commitment (highballing), or to pay for the opportunity of learning the risk type (lowballing). See Definition 1 in the next section for formal definitions of the highballing, lowballing, and flat pricing patterns. It is useful to point out that insurers will take into consideration the policyholder’s possible action (a choice of contracts or a claim) when they make pricing decisions at the beginning. For instance, a policyholder in our theoretical model can lapse his/her contract at the beginning of the second period and select an alternative one from the competing insurers. The insurers expect such policyholder action when they offer insurance contracts. Therefore, such a pricing strategy can coexist in parallel with the actuarial pricing based on risk discrimination (e.g., a bonus-malus system) and with the policyholders’ self-selection process (e.g., the design of menu contracts). See Cohen (2012) and Shi and Zhang (2016) for a detailed discussion on how the lowballing strategy works in a bonus-malus system. See also Cooper and Hayes (1987) for how the highballing strategy works when menu contracts are offered by the insurer. In our theoretical model below, the insurers can adjust their insurance prices based on policyholders’ claim histories, and thus a bonus-malus system or other form of experience rating is allowed. Like de Garidel-Thoron (2005) and Hendel and Lizzeri (2003), we assume away period-1 adverse selection in our model. Therefore, the insurers have no incentive to design menu contracts to screen policyholders. See footnotes 11 and 15 for discussions of the presence of adverse selection.

  4. 4.

    All models discussed in this paper (implicitly) assume away moral hazard. A separate stream of dynamic contracting studies investigates moral hazard issue (see e.g., Rubinstein and Yaari 1983; Rogerson 1985).

  5. 5.

    Learning partly reflects the updates in the initial (but unknown) differences in risks, and partly the signal (and real) changes in risks (Pauly 2003).

  6. 6.

    Also related, Kunreuther and Pauly (1985) and Nilssen (2000) develop models of multi-period insurance markets with asymmetric learning and no commitment. Unlike our framework, they assume the presence of adverse selection in the first period and predict a lowballing pricing pattern, as in our Proposition 2. See Footnote 15 for discussions in detail on the connections of these two papers with Proposition 2 in this paper.

  7. 7.

    This exception may be due to different interpretations of Cox and Ge’s (2004) results. Their empirical model includes the policy age and its square term as the key explanatory variables, and the loss ratio of each policy as the dependent variable. They document a positive coefficient for policy age and a negative coefficient for its square term. We believe that this result should be interpreted as a highballing profit pattern with a decreasing speed of profit increase. In this sense, their empirical results also confirm the highballing strategy.

  8. 8.

    In “Appendix 5,” we relax the no-risk-change assumption and again show that the results derived in Proposition 1 and Remark 1 (i.e., highballing under one-sided commitment) and Proposition 2 (i.e., lowballing under no commitment) are robust.

  9. 9.

    The presence of adverse selection introduces the possibility of a separating equilibrium. In a separating equilibrium, both the incumbent insurer and the competing insurers learn policyholders’ risk type immediately from policyholders’ contract choices. As a result, learning is of no value and there is no information asymmetry between the incumbent and its rivals in the second period. We discuss in footnotes 11 and 15 the scenarios incorporating adverse selection, which do not change our predictions.

  10. 10.

    The reclassification risk in the model refers to the period-2 risk change and the premium adjustment based on the past claim experience.

  11. 11.

    Proposition 1 is robust to period-1 adverse selection. Focusing on the separating equilibrium, Cooper and Hayes (1987) extend Rothschild and Stiglitz’s (1976) single-period adverse selection model to multi-periods and discuss the case of one-sided commitment. They assume that the competing insurers in period 2 learn neither the policyholders’ histories nor their choices of contract in period 1. They show that the incumbent insurer offers contracts that are independent of histories and are actuarially fair in both periods to the high risks, and experience-rated contracts to the low risks. Specifically, in the first period, the incumbent insurer charges the low-risk policyholder a higher premium than they would pay in a standard Rothschild and Stiglitz (1976) model; in the second period, the incumbent insurer gives those low risks without any period-1 claims a heavy discount, and thus charges them a lower premium than they would pay in a standard Rothschild and Stiglitz (1976) model. To summarize, the pricing pattern is again highballing for the low risks and flat for the high risks, with one-sided commitment.

  12. 12.

    Almost all empirical work uses some members of HARA utility functions. Specifically, the utility function exhibits constant absolute risk aversion (CARA, i.e., \(u(c) = - { \exp }( - ac)\)) if \(b = 1\) and \(\eta \to \infty\); the utility function exhibits constant relative risk aversion (CRRA, i.e., \(u(c) = \frac{1 - \eta }{\eta }\left( {\frac{ac}{1 - \eta }} \right)^{\eta }\)) if \(b = 0\). The commonly used utility functions, such as \(\ln (c)\) and \(\sqrt c\), are all members of HARA.

  13. 13.

    Specifically, the condition of a sufficiently large ratio of \(p_{2}^{A} /p_{2}^{N}\) is used to establish the inequality (9) in the proof of Proposition 1. Our numerical results in “Appendix 4” indicate that (9) holds for all values of \(p_{2}^{A} /p_{2}^{N} > 1\), under the expo-power (EP) utility functions (Saha 1993), the power risk aversion (PRA) utility functions (Xie 2000), and the flexible three parameter (FTP) utility functions (Conniffe 2007).

  14. 14.

    Note that reneging differs from Dionne and Doherty’s (1994) renegotiation. When reneging, the insurer can change or cancel the contract unilaterally in the second period; whereas with renegotiation, the contract can be changed if—and only if—this modification is mutually agreed by both the insurer and the policyholder. Therefore, reneging is a scenario of no commitment, and renegotiation can be considered as a weaker form of one-sided commitment.

  15. 15.

    The above intuition applies, and Proposition 2 remains valid, if adverse selection is present at the beginning of the first period. Indeed, Kunreuther and Pauly (1985) and Nilssen (2000) investigate the equilibrium contracts in a similar setup except that they assume adverse selection is present in the first period. They show that the equilibrium contract is lowballing in a pooling equilibrium where all risk types are provided with the same contract in the first period. Again, lack of insurer commitment implies directly that the incumbent insurer will not suffer a loss on any type of policyholder in the second period because it can simply withdraw the contract to avoid losses. Moreover, asymmetric learning endows the incumbent insurer with some market power from which positive profits can be earned on low risks. As a result, Kunreuther and Pauly (1985) and Nilssen (2000) establish the same lowballing pricing pattern.

  16. 16.

    Our theoretical model assumes a competitive market. We maintain the hypothesis of competition in the empirical analyses. The A&H lines, including the two products concerned, are the most open product markets in China in the sense that all L&H and P&C insurers are allowed to sell these products. The spatial segmentation in the Chinese market may reduce the degree of competition. Some insurers are only licensed to operate in one or a few provinces (e.g., AIG) but not nationwide. Although we are not able to empirically conclude for a fully competitive market, we cautiously maintain the competition hypothesis following the extant literature (Chan 2009; Lu et al. 2014).

  17. 17.

    We note that many other insurance products feature no-commitment and asymmetric learning, for example, automobile insurance used by D'Arcy and Doherty (1990), Cohen (2012), Kofman and Nini (2013), and Shi and Zhang (2016). This paper first-time presents a non-automobile short-term health insurance portfolio to show the lowballing premium pattern.

  18. 18.

    In critical illness insurance, the insurance compensation paid after the occurrence of the insurance event is always equal to the insurance amount (i.e., the sum insured). In loaner’s personal accident insurance, the insurance compensation paid is less than the insurance amount (i.e., the sum insured) when partial disability occurs or equal to the insurance amount when death or complete disability occurs.

  19. 19.

    Borrowers may prefer bank channels for other reasons. For instance, shopping for products from other channels requires additional effort and professional knowledge. In addition, products from other channels may be more expensive because individual borrowers may not enjoy a group discount from being pooled together with all borrowers from the bank.

  20. 20.

    The claim information is electronically recorded in real time but only retrieved and organized by the actuarial team once per year. At the time the data for sample A were obtained, the claim information from September 2012 to June 2013 was not yet available. In the subsequent analysis, in order to avoid the potential truncation problem, the claim status of polices expiring after August 2012 is coded as missing values.

  21. 21.

    For Sample A, group policies with a 1-year duration account for 62% of all policy-year observations; for Sample B, individual policies with a 1-year duration account for 82% of all policy-year observations.

  22. 22.

    The original dataset A has 11,185 group-year observations from 4516 groups. The original dataset B has 1.6 million policyholder-year observations from 1.1 million individual policyholders.

  23. 23.

    The individual insureds buy this product at the request of their bank and thus they do not care very much about whether the insurer commits to multiple periods or not. In practice, when we discuss with the insurance company, it acknowledged that the renewal process is rather automatic; the insurer essentially delegates the underwriting authority to the bank; and thus hardly declines any renewal requests from the bank’s clients.

  24. 24.

    These two products are managed by a single A&H team. The size of the two product portfolios is similar and large to that team. The sales teams for these two products approach the groups (CI) or the banks (PA) to compete for business with other insurers. On the individual-risk level, the group (or the bank) usually enrolls most of its employees (borrowers) in the insurance coverage. We are not aware that the insurer or the A&H team has different commercial or marketing considerations to weigh the importance of these two products differently.

  25. 25.

    We choose the OLS as our core model because one insured may buy two or more policies in the nth year. All these policies have the same policy age of n; however, only one of them can be incorporated in the panel regressions. Thus, 14.8% of Sample A and 7.9% of Sample B have to be dropped if using the panel regressions, reducing the estimation efficiency.

  26. 26.

    Consistent with the full sample analysis, OLS regressions are used for sub-sample analyses with consecutive renewal policies. Alternatively, we conduct the Bayesian information updating regressions for the sub-sample analyses, with the prior assumption that the coefficient of policy age follows \(N(\mu , \sigma^{2} )\), where \(\mu\) and \(\sigma^{2}\) are estimated based on the OLS coefficient of a prior period. For example, in the sub-sample regression with first- and second-time renewed policies, the prior coefficient distribution is assumed to follow the corresponding coefficient in the sub-sample of the new and first-time renewed policies. The results of Bayesian regressions are consistent with the OLS results and are available from the authors upon request.

  27. 27.

    Additionally, we conduct the 2SLS regressions to address the endogeneity concern. The results are consistent with those of the OLS and are available from the authors upon request.

  28. 28.

    As shown in Tables 5 and 6, we note that in each sub-sample in Table 6, the variation of policy age is large enough to yield significant results. For instance, the ratio of new policies to first-time renewed policies is 3:2 in Sample A and 5:1 in Sample B.

  29. 29.

    The practice including experience rating factor is not applicable to Sample B because the loaner’s PA is experience-rated at the bank level instead of at the individual policyholder level.

  30. 30.

    The panel is set up with the (group) insured as \(i\) and with the renewal counts as \(t\). For example, suppose that insured X and Y have their first policy with the insurer in 2009 and 2010, respectively. Then these two policies are (X, 0) and (Y, 0) although they are in different years. Therefore, \(t\) varies from 0 to 5 in Sample A and from 0 to 4 in Sample B. The year fixed-effects are then controlled by year dummies in the regressions. See Zhang and Wang (2008) for a detailed discussion on why and how to apply random-effects models to dynamic insurance markets.

  31. 31.

    We thank one reviewer for suggesting this robustness check.

  32. 32.

    Guaranteed renewability not only guarantees renewals but also guarantees to only change a premium to the same extent for all in the initial rating class or following a pre-agreed premium schedule in subsequent periods. Experience rating based on an individual’s risk change or claim experience is not allowed.

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Acknowledgements

The authors thank Christian Biener, Alexander Braun, Martin Brown, Richard Butler, Martin Eling, Roland Füss, Felix Irresberger, Andreas Landmann, Tore Nilssen, Joan Schmit, Ruilin Tian, and Wei Zheng, as well as participants in the 2016 ARIA and APRIA conferences, and the 2017 Risk Theory Society Annual Seminar for useful comments and discussions. We thank Shuyan Liu and Linsheng Zhuang for their assistance. We thank the financial support from National Natural Science Foundation of China (Nos. 71703003 and 71803003), and the seeds fund of School of Economics, Peking University. All remaining errors are our own.

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Correspondence to Zenan Wu.

Appendices

Appendices

In “Appendix 1,” we review the theoretical literature on competitive dynamic insurance contracts. In “Appendix 2,” we provide a review of empirical literature on dynamic insurance pricing pattern. In “Appendix 3,” we present the detailed proof of Proposition 1 and Remark 1. In “Appendix 4,” we provide the simulation results and show that Proposition 1 holds for a wide array of non-canonical utility functions and for all values of \(p_{2}^{A}/p_{2}^{N}>1\). In “Appendix 5,” we assume policyholder’s risk type deteriorates over time and show that our theoretical results in the main text are robust. In “Appendix 6,” we redraw Fig. 2 in the contingent wealth space as in Rothschild and Stiglitz (1976).

Appendix 1: Review of theoretical literature on competitive dynamic insurance contracts

The dynamic insurance contracting models can be traceable to, among others, contract theories in labor (Harris and Holmstrom 1982), procurement (Laffont and Tirole 1990), and credit (Sharpe 1990) markets. The development of contract theory leads modern insurance economics in three directions (D’Arcy and Doherty 1990; Chiappori and Salanié 2013). The first is the classic single-period contracting in competitive insurance markets (e.g., Rothschild and Stiglitz 1976; Miyazaki 1977; Wilson 1977; Spence 1978). The second direction is the multi-period contracting in a monopolistic insurance market, where the role of experience rating is highlighted to solve the adverse selection problem (e.g., Dionne 1983; Dionne and Lasserre 1985, 1987; Hosios and Peters 1989). The third direction is the dynamic insurance contracting in competitive markets, on which this paper will focus. Recently, a growing literature has focused on the underwriting cycles, i.e., the mid- to long-term pricing dynamics in the insurance market (see, e.g., Henriet et al. 2016). A detailed discussion of papers is provided in Table 1.

Panel A: adverse selection is present in period 1

Cooper and Hayes (1987) and Kunreuther and Pauly (1985) initiate the modeling of multi-period contracting in the insurance context, in which an important feature is the presence of adverse selection. Assuming no commitment, Kunreuther and Pauly (1985) and Nilssen (2000) model the scenario of asymmetric insurer learning; Watt and Vazquez (1997) model symmetric learning. Kunreuther and Pauly (1985) focus on the equilibrium that involves pooling in the first period. In equilibrium, the insurer offers the same contract to both risk types with a premium reflecting the average of low and high risks. In the second period, risks who had claim(s) in the first period (high risks) switch to competing insurers. This is because the incumbent insurer can increase the premium for the period-1 claimant; however, the competing insurers cannot differentiate who had a claim in period 1. Risks who did not have a claim (low risks) stay with the incumbent insurer in equilibrium because the incumbent insurer will keep their premium unchanged. Therefore, in period 2, the insurer can earn positive profits by over-charging the staying (low) risks. Under the zero-profit constraint, the insurer must suffer a loss in the first period to attract new customers, which is considered as the cost of acquiring knowledge about the insured’s risk type. The incumbent insurer earns an informational quasi rent in period 2 while the competing insurers do not. This pricing pattern for a sequence of the contracts is hence lowballing.

Kunreuther and Pauly (1985) assume that policyholders are completely myopic when making the initial purchase decision, and that insurers are not allowed to offer menu contracts to screen policyholders. Nilssen (2000) relaxes these two assumptions and shows that: (i) a separating equilibrium is less likely to be sustained in a two-period model than in a one-period one; and (ii) a lowballing pricing pattern is possible but not necessary to emerge in equilibrium.

Watt and Vazquez (1997) consider a multi-period model with the presence of period-1 adverse selection. They assume that all insurers learn the insured’s risk type over time, and that learning is thus symmetric. They show that a pooling equilibrium with full coverage exists if low risks are patient enough, or a semi-pooling equilibrium exists where a portion of impatient low risks choose a sequence of Rothschild and Stiglitz’s (1976) partial coverages and all high risks and patient low risks choose a sequence of full coverages. In equilibrium, no inter-temporal subsidization is inferred. In period 1, the insurer undercharges high risks and overcharges low risks, generating zero profits in aggregate. In future periods, full information contracts are in place and thus all risks are charged at an actuarially fair rate. Therefore, the equilibrium pricing pattern is flat.

Assuming one-sided and/or full commitment, Cooper and Hayes (1987) and Dionne and Doherty (1994) model the scenario of asymmetric learning. The scenario of symmetric learning has not yet been covered by the literature. Focusing on the separating equilibrium, Cooper and Hayes (1987) extend Rothschild and Stiglitz’s (1976) single-period adverse selection model to multi-periods and discuss both the cases of one-sided and full commitment. They make an extreme assumption on insurers’ learning: the competing insurers in period 2 learn neither the policyholders’ histories nor their choices of contract in period 1. With one-sided commitment, the incumbent insurer offers contracts that are independent of histories and actuarially fair in both periods to the high risks, and experience-rated contracts to the low risks. Specifically, in the first period, the incumbent insurer charges the low-risk policyholder a higher premium than they would pay in a standard Rothschild and Stiglitz (1976) model; in the second period, the incumbent insurer gives those low risks without any period-1 claims a heavy discount, and thus charges them a lower premium than they would pay in a standard Rothschild and Stiglitz (1976) model. To summarize, the pricing pattern is highballing for the low risks and flat for the high risks, with one-sided commitment. Intuitively, the insurer tilts payoffs towards the future to provide an incentive for the insureds to remain with the incumbent insurer. With full commitment, it shows that the high-risk policyholder receives the same contract as s/he does in the case of one-sided commitment, whereas the low-risk policyholder is offered an experience-rated contract. Their model does not provide predictions on the dynamic pricing pattern for low risks in the case of full commitment.

Dionne and Doherty (1994) introduce the possibility of renegotiation into the one-sided commitment model by allowing the insurer to commit to a long-term contract in period 1 and to offer a revised short-term contract in period 2. The insured may stick to the long-term contract, accept the revised period-2 short-term contract, or switch to competing insurers in period 2. They characterize the equilibrium involving semi-pooling in period 1 followed by separation in period 2. In such an equilibrium, a fraction of the high-risk policyholders chooses full coverage and the rest of the high risks and all the low risks receive partial coverage in period 1. In period 2, the high risks switch to a short-term contract with full coverage, either offered by the incumbent insurer as a result of renegotiation or by a competing insurer, and the low risks stay with the long-term partial coverage. Dionne and Doherty (1994) show that Cooper and Hayes’s (1987) result is robust to the introduction of renegotiation. In equilibrium, the insurer earns positive period-1 expected profits and suffers period-2 expected losses from the low risks, implying a highballing pattern.

Panel B: adverse selection is not present in period 1

Assuming symmetric learning, Pauly et al. (1995) and Hendel and Lizzeri (2003) develop models with one-sided commitment. Pauly et al. (1995) assume that the insurer pre-commits to the long-term contract. To simplify the analysis, the coverage of the risk in their model is assumed to be exogenous. Therefore, insurers only engage in price competition on premiums. They construct a guaranteed renewabilityFootnote 32 insurance with a sequence of premiums that survives in a competitive equilibrium and show that such a contract is Pareto optimal. By their construction, this guaranteed renewability insurance exhibits a highballing premium pattern. Specifically, the insurer charges all risks a premium that is higher than the actuarially fair one in period 1 so as to upfront charge for its commitment and to lock in the low risks. Although the insured is allowed to cancel the contract, s/he would not do so in equilibrium because leaving for a spot market would not make him/her strictly better off.

Hendel and Lizzeri (2003) extend Pauly et al. (1995) by allowing insurers to choose both the premium and the coverage of a contract. The pricing pattern is again highballing, which is an important device to lock in low risks. In equilibrium, consumers with low risks tend to depart from the incumbent insurer. They show that more front-loaded contracts provide more insurance against reclassification risk and are selected by consumers with a lower income growth.

Assuming asymmetric learning, Pauly et al. (2011) relax the assumption of symmetric learning in Pauly et al. (1995) and focus on contracts with guaranteed renewability. With asymmetric learning, the result of highballing in Pauly et al. (1995) may not hold because the fact that the competing insurers cannot identify low risks directly provides incentives to the incumbent insurer to obtain information rents in later periods and to reduce the degree of frontloading. Again, they show that the highballing pricing pattern will not be overturned. They first argue that the optimal guaranteed renewable contract, which is essentially the same as in Pauly et al. (1995), is indeed immune to the information problem due to asymmetric learning. They then argue that the attempt to alter frontloading cannot be sustained in equilibrium because otherwise the competing insurers can craft a policy to attract only the low-risk consumers and earn profits by the standard Rothschild–Stiglitz argument.

De Garidel-Thoron (2005) presents a two-period model in which both the insured and the incumbent insurer learn the policyholder’s risk type from her/his period-1 claim history, but competing insurers do not. He derives the equilibrium contract and shows that information-sharing between insurers is detrimental to consumer welfare, independent of an insurer’s ability to commit to a long-term contract. The intuition is as follows. Although asymmetric information between incumbent and competing insurers generates information frictions and welfare loss to the policyholders, it also weakens the competing insurer’s ability to exercise a cream-off strategy and thus improves the long-term commitment of both parties. In equilibrium, this welfare gain outweighs the welfare loss due to information asymmetry. As a result, consumer welfare is higher when information-sharing is prevented than when information-sharing is allowed. In terms of the dynamic pricing pattern, assuming asymmetric learning, de Garidel-Thoron (2005) shows that the equilibrium contracts exhibit lowballing when insurers lack commitment power a bonus-malus pattern when insurers are able to commit. However, de Garidel-Thoron (2005) does not conclude whether the contract is highballing or lowballing in the latter case.

Recently, Hendel (2016) has proposed a simple framework in the spirit of Harris and Holmstrom (1982) to survey the theory and evidence on dynamic contracting under different environments of learning and commitment. Unlike our review in Appendices 1 and 2 that focuses on the dynamic pricing pattern, his survey focuses on other issues, such as the relationship between dynamic selection and policyholder’s lapsation behavior and the welfare consequences due to lack of insurer commitment.

Panel C: discussions connecting Panels A and B

Pauly (2003) provides an informal discussion on three scenarios. First, he takes de Garidel-Thoron’s (2005) no-commitment scenario and concludes a pooling equilibrium in period 1 and lowballing pricing pattern. Second, he discusses the one-sided commitment scenario in Pauly et al. (1995) and Hendel and Lizzeri (2003). A pooling equilibrium with a highballing pricing pattern is expected. Third, he attempts to build the connection between models with the adverse selection and those without. He argues that the period-1 adverse selection should not change the highballing implication embedded in insurer commitment (guaranteed renewability).

Appendix 2: Review of empirical literature on dynamic insurance pricing pattern

A detailed discussion of papers is provided in Table 2.

Testing dynamic pricing pattern in markets with no commitment

D’Arcy and Doherty (1990) pioneered empirical investigation of multi-period insurance contracting. They presented the aggregate loss ratios by policy age cohorts of automobile insurance from seven US insurers. All seven portfolios showed that loss ratios decline almost monotonically with policy age, suggesting a lowballing pricing pattern that supports Proposition 2 in this paper. They also looked at three new market entrants, who have only new policies but no private information. They found that these three new insurers’ loss ratios were indeed high (low profit) in the beginning and gradually converged with those of other matured firms, supporting the lowballing pricing pattern predicted in Proposition 2.

Cohen (2012) presented the first policy-level analysis of repeated short-term insurance contracts using an Israeli automobile insurance portfolio. During the entire sample period, no information-sharing platform among insurers was available and asymmetric learning thus best captured the nature of the market. He showed that (i) profits from the repeat insureds were higher than those from the new insureds (i.e., a lowballing pattern); and (ii) the insurer reduced the price charged to repeated insureds with a good claim history by less than the reduction in expected costs associated with such insureds (i.e., premium downward stickiness). The evidence was obtained after controlling for all risk classification factors and hence directly supports Proposition 2. Cohen’s (2012) sample matches Kunreuther and Pauly’s (1985) and Nilssen’s (2000) assumptions, and he points out the role of asymmetric learning in determining the lowballing pricing pattern accordingly.

Kofman and Nini (2013) examined the Australian automobile insurance market, where a claim information-sharing platform is in place to support a bonus-malus rating system. They believe that the publicly available data in the Australian system captures all relevant information regarding the policyholders’ risk types, with the only exception being brand-new policies. Thus, the market matches Watt and Vazquez’s (1997) assumptions (i.e., adverse selection in period 1, symmetric learning in later periods, and no insurer commitment). They documented evidence of lowballing pricing patterns and support our Proposition 2. Their evidence challenges Cohen’s (2012) argument regarding the important role of asymmetric learning in determining pricing pattern.

Shi and Zhang (2016) investigated an intermediate learning scenario in between the no information-sharing market in Cohen (2012) and the complete information-sharing market in Kofman and Nini (2013). The Singapore automobile insurance market has a no-claim-discount (NCD) system and a public information-sharing platform. However, the platform contains only information regarding the NCD status and contains no information regarding either the insureds’ claim history or policy choice. This implies a partial information-sharing system among insurers (Shi and Zhang, 2016). Their conclusions again support Proposition 2 and suggest that pricing pattern may not depend on insurer’s learning type.

Testing dynamic pricing pattern in markets with one-sided commitment

Dionne and Doherty (1994) examined a special automobile liability insurance portfolio from California, where two types of policies are offered: a long-term policy (one-sided commitment with renegotiation) and a short-term policy (no commitment). They approximated the average policy age in a portfolio by the premium growth of that portfolio (i.e., a high/low premium growth indicates an average younger/older policy age) and found a positive correlation between average policy age and loss ratio in the sub-sample of the low risks, which is in line with the highballing prediction of the one-sided commitment model (Proposition 1).

Hendel and Lizzeri (2003) presented the first piece of evidence controlling for underlying risk differences (i.e., risk classification) in multi-period insurance contracting. They looked at three products of life insurance from 55 US life insurers: 20-year term life insurance with level premium each year (TL), annual renewable term life insurance with premiums that depend only on age (ART), and annual renewable term life insurance with premiums that depend on age and time elapsed since last medical examination (Selection & Ultimate ART). They found that TL and ART are significantly front-loaded, where the insurer pre-commits to a long-term contract (LT) or to the guaranteed renewability with a determined rating schedule (ART). The relative price to the risk almost monotonically decreased through the 20 years, supporting Proposition 1. However, for Selection & Ultimate ART, where the premiums in later periods depended on whether the insured passed the medical re-examination (a weakened commitment with re-underwriting elements), frontloading existed only in the first year but not in the following years. They approximated the risk by the current values of premiums and found a negative correlation between the degree of frontloading and the policyholder’s risk: more front-loaded contracts insured a higher fraction of low-risk policyholders. They also showed that a more front-loaded contract (LT) has a lower lapse rate than a less front-loaded contract (ART), indicating that the low-risk lock-in effects are associated with the highballing pricing pattern.

Cox and Ge (2004) presented a panel data from the US long-term care (LTC) insurance market with cohort-specific information. They found a positive correlation between policy age and loss ratio but argued that this reflected the risk changes over time and was thus not a highballing pricing pattern. They found a negative coefficient for policy age2, indicating that the loss ratio increases became slower and slower over time. As previously mentioned, we, as well as other empirical works, have interpreted the above findings differently to Cox and Ge (2004). The positive coefficient of policy age is a direct indicator of a highballing pattern and the negative coefficient of policy age2 suggests that the price decrease in the highballing pattern becomes milder over time. This result is consistent with our Sample B results.

One way to solve this apparent theoretical (highballing, Proposition 1) and empirical (lowballing in Cox and Ge, 2004) contradiction is to use the policy-level data and control for risk classification, so that the coefficient between policy age and price/profit can directly reveal the pricing pattern. Finkelstein et al. (2005) examine the US LTC market in just such a direct way. They documented pricing highballing evidence consistent with Hendel and Lizzeri (2003), thus supporting Proposition 1.

Herring and Pauly (2006) numerically developed an ideal/optimal incentive compatible premium schedule for individual health insurance with guaranteed renewability, based on Pauly et al.’s (1995) one-sided commitment model. In addition, they estimated the actual market premiums for individual health insurance using a Medical Expenditure Panel Survey, Community Tracking Study Household Survey, and National Health Interview Survey. They found that the actual premium schedule and the estimated ideal premium schedule do “appear to be surprisingly consistent,” thus supporting the highballing prediction in Proposition 1. They concluded that the front-loaded premium is necessary for health insurers to provide guaranteed renewability and to insure the reclassification risk.

Pinquet et al. (2011) examine the dynamic lapsation behavior in a long-term package of coverages of health, life, and LTC from the Spanish market. Premiums were paid annually and experience rating on an individual basis was not allowed. They found a highballing pricing pattern (Proposition 1) in all three coverages, evidenced by the increased benefit ratios from younger to older groups.

Hofmann and Browne (2013) presented evidence from the German long-term private health insurance (one-sided commitment), where insurers commit to offer renewal at a premium rate that does not reflect the revealed future information about the insured’s risk. They support the theoretical predictions in Hendel and Lizzeri (2003): a highballing pricing pattern generates the effect of insured lock-in. The evidence from the German private health market demonstrates the robustness of the correlation between insurer commitment and the inter-temporal pricing pattern, which is immunized from strict regulation and from the existence and possible domination of social insurance programs. Their work also contributes to the debate on how private health solutions can insure the reclassification risk. The empirical evidence demonstrates the viability of the front-loaded premium schedule with guaranteed renewability (Pauly et al. 1995), at least in a strictly regulated and social insurance dominated market. In such a market, the accessibility of health coverage is much less a problem than in a private market.

Appendix 3: Proof of Proposition 1 and Remark 1

It is useful to first prove a lemma. Denote \(\left( {Q_{2}^{A\dag } ,Q_{2}^{N\dag } ,Q^{\dag } } \right)\) as the solution to the following non-linear system of equations:

$$p_{1} Q_{2}^{A\dag } + \left( {1 - p_{1} } \right)Q_{2}^{N\dag } = p_{1} L,$$
(6)
$$u\left( {W - Q_{2}^{N\dag } } \right) = \left( {1 - p_{2}^{N} } \right)u\left( {W - Q^{\dag } } \right) + p_{2}^{N} u\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{\dag } } \right),$$
(7)
$$u\left( {W - Q_{2}^{A\dag } } \right) = \left( {1 - p_{2}^{A} } \right)u\left( {W - Q^{\dag } } \right) + p_{2}^{A} u\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{\dag } } \right),$$
(8)

with \(Q^{\dag } \in [0,p_{2}^{N} L]\). In words, \((Q_{2}^{A\dag } ,Q_{2}^{N\dag } )\) refers to the premium profile that generates zero profits in the second period and satisfies constraint (3). It is clear that \(Q_{2}^{A\dag } > Q_{2}^{N\dag }\). Moreover, it can be verified \(Q_{2}^{N\dag }\) and \(Q_{2}^{A\dag }\) are both strictly decreasing in \(Q^{\dag }\) from (7) and (8) for \(Q\in[0, p_{2}^{N}L]\). Therefore, there exists a unique solution to the above non-linear system of equations.

Lemma 2

Suppose that (i) \(p_{2}^{A} /p_{2}^{N}\)is large enough; or (ii) consumer’s preference exhibits HARA. Then the following inequality holds:

$$\frac{1}{{u^{\prime}\left( {W - p_{1} L} \right)}} > \frac{{p_{2}^{A} }}{{u^{\prime}\left( {W - Q_{2}^{A\dag } } \right)}} + \frac{{1 - p_{2}^{A} }}{{u^{\prime}\left( {W - Q_{2}^{N\dag } } \right)}}.$$
(9)

Proof

We first prove part (i) of the lemma. Holding fixed \((W, \, L, \, p_{1})\), consider \(p_{2}^{N}\) and \(\left( {Q_{2}^{A\dag } ,Q_{2}^{N\dag } ,Q^{\dag } } \right)\) as functions of \(p_{2}^{A}\). By the martingale property, we have that \(p_{2}^{N} = \frac{{p_{1} }}{{1 - p_{1} }} \times \left( {1 - p_{2}^{A} } \right)\), which is strictly decreasing in \(p_{2}^{A}\). Therefore, it suffices to show that (9) holds as \(p_{2}^{A}\) becomes sufficiently large.

Equations (6)–(8) imply that

$$\mathop {\lim }\limits_{{p_{2}^{A} \to 1}} Q_{2}^{A\dag } (p_{2}^{A} ) = L\;{\text{and}}\;\mathop {\lim }\limits_{{p_{2}^{A} \to 1}} Q_{2}^{N\dag } \left( {p_{2}^{A} } \right) = 0.$$

Therefore, we have that

$$\mathop {\lim }\limits_{{p_{2}^{A} \to 1}} \left[ {\frac{{p_{2}^{A} }}{{u'\left( {W - Q_{2}^{A\dag } } \right)}} + \frac{{1 - p_{2}^{A} }}{{u'\left( {W - Q_{2}^{N\dag } } \right)}}} \right] = \frac{1}{{u'\left( {W - L} \right)}} < \frac{1}{{u'\left( {W - p_{1} L} \right)}} = \mathop {\lim }\limits_{{p_{2}^{A} \to 1}} \frac{1}{{u'\left( {W - p_{1} L} \right)}},$$

which indicates immediately that (9) holds if \(p_{2}^{A}/p_{2}^{N}\) is large enough.

Next, we show that (9) holds under HARA preferences. Substituting (6) into (9) yields

$$p_{2}^{A} \frac{{u'\left( {W - p_{1} Q_{2}^{A\dag } - \left( {1 - p_{1} } \right)Q_{2}^{N\dag } } \right)}}{{u'\left( {W - Q_{2}^{A\dag } } \right)}} + \left( {1 - p_{2}^{A} } \right)\frac{{u'\left( {W - p_{1} Q_{2}^{A\dag } - \left( {1 - p_{1} } \right)Q_{2}^{N\dag } } \right)}}{{u'\left( {W - Q_{2}^{N\dag } } \right)}} < 1.$$

With HARA utility, the marginal utility is \(u^{\prime}\left( c \right) = a\left( {\frac{ac}{1 - \eta } + b} \right)^{\eta - 1} ,\) and the above inequality can be simplified as

$$p_{2}^{A} \left[ {p_{1} + \left( {1 - p_{1} } \right)\frac{{\frac{a}{1 - \eta }\left( {W - Q_{2}^{N\dag } } \right) + b}}{{\frac{a}{1 - \eta }\left( {W - Q_{2}^{A\dag } } \right) + b}}} \right]^{\eta - 1} + \left( {1 - p_{2}^{A} } \right)\left[ {p_{1} \frac{{\frac{a}{1 - \eta }\left( {W - Q_{2}^{A\dag } } \right) + b}}{{\frac{a}{1 - \eta }\left( {W - Q_{2}^{N\dag } } \right) + b}} + \left( {1 - p_{1} } \right)} \right]^{\eta - 1} < 1.$$

Denote \({{\left[ {\frac{a}{{1 - \eta }}\left( {W - Q_{2}^{{N^{\dag } }} } \right) + b} \right]} \mathord{\left/ {\vphantom {{\left[ {\frac{a}{{1 - \eta }}\left( {W - Q_{2}^{{N^{\dag } }} } \right) + b} \right]} {\left[ {\frac{a}{{1 - \eta }}\left( {W - Q_{2}^{{A^{\dag } }} } \right) + b} \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\frac{a}{{1 - \eta }}\left( {W - Q_{2}^{{A^{\dag } }} } \right) + b} \right]}}\) by μ. It remains to show that

$$g(\mu ):= p_{2}^{A} [p_{1} + (1 - p_{1} )\mu ]^{\eta - 1} + \left( {1 - p_{2}^{A} } \right)\left[ {\frac{{p_{1} }}{\mu } + \left( {1 - p_{1} } \right)} \right]^{\eta - 1} < 1.$$

Carrying out the algebra, \(g^{\prime}(\mu)<0\) is equivalent to

$$\left( {\eta - 1} \right)\left[ {\mu^{\eta } - \frac{{p_{1} }}{{1 - p_{1} }}\frac{{1 - p_{2}^{A} }}{{p_{2}^{A} }}} \right] < 0 \Leftrightarrow \left( {\eta - 1} \right)\left[ {\mu^{\eta } - \frac{{p_{2}^{N} }}{{p_{2}^{A} }}} \right] < 0 .$$

We consider the following three cases depending on the value of \(\eta\).

Case I: \(0 < \eta < 1\). It is clear that \(\mu > 1\) under such a scenario, and hence \(\mu^{\eta} > 1>p_{2}^{N}/p_{2}^{A}\). Therefore, \(g(\mu)\) is strictly decreasing in \(\mu\) for \(\mu > 1\), and hence \(g(\mu)<g(1)=1\).

Case II: \(\eta<0\). Again, \(\mu>1\). Moreover, \(\mu^{\eta}\) can be bounded from below by,

$$\mu^{\eta } = \left[ {\frac{{\frac{a}{1 - \eta }\left( {W - Q_{2}^{N\dag } } \right) + b}}{{\frac{a}{1 - \eta }\left( {W - Q_{2}^{A\dag } } \right) + b}}} \right]^{\eta } = \frac{{u\left( {W - Q_{2}^{N\dag } } \right)}}{{u\left( {W - Q_{2}^{A\dag } } \right)}} = \frac{{\left( {1 - p_{2}^{N} } \right)u\left( {W - Q^{\dag } } \right) + p_{2}^{N} u\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{\dag } } \right)}}{{\left( {1 - p_{2}^{A} } \right)u\left( {W - Q^{\dag } } \right) + p_{2}^{A} u\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{\dag } } \right)}} > \frac{{p_{2}^{N} }}{{p_{2}^{A} }},$$

where the second equality follows directly from the assumption of HARA utility, the third equality follows from (7) and (8), and the inequality follows from the fact that \(p_{2}^{A}>p_{2}^{N}\). Therefore, \(g(\mu)\) is strictly decreasing in \(\mu\) for \(\mu \in (1,(p_{2}^{N}/p_{2}^{A})^{1/\eta})\), which implies that \(g(\mu)<g(1)=1\).

Case III: \(\eta>1\). It is clear that \(\mu<1\). As with Case II, we must have that \(\mu^{\eta}>p_{2}^{N}/p_{2}^{A}\), which in turn implies immediately that \(\mu > (p_{2}^{N}/p_{2}^{A})^{1/\eta})\). Therefore, \(g(\mu)\) is strictly increasing in \(\mu\) for \(\mu \in ((p_{2}^{N}/p_{2}^{A})^{1/\eta}, 1)\), and hence \(g(\mu)<g(1)=1\).

This completes the proof of Lemma 2.□

Now we can prove Proposition 1 and Remark 1. Suppose, to the contrary, that the equilibrium contract \((Q_{1}^{*}, Q_{2}^{A*}, Q_{2}^{N*})\) maximizes a consumer’s expected utility and is not front-loaded (i.e., \(Q_{1}^{*} \le p_{1}L\)), then we can construct an alternative contract that yields a strictly higher expected utility and satisfies constraints (1)–(4) by the following two steps:

Step 1 For notational convenience, denote the solution to \(IC_{2}^{N}\,(Q; C_{2}^{N*})=IC_{2}^{A}\,(Q; C_{2}^{A*})\) by \(Q^{*}\). For a sufficiently small \(\varepsilon > 0\), let \(\hat{Q}_{2}^{A} (\varepsilon )\) and \(\hat{Q}_{2}^{N} (\varepsilon )\) be the solution to

$$u\left( {W - \hat{Q}_{2}^{N} } \right) = \left( {1 - p_{2}^{N} } \right)u\left( {W - Q^{*} } \right) + p_{2}^{N} u\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{*} + \varepsilon } \right),$$
(10)

and

$$u\left( {W - \hat{Q}_{2}^{A} } \right) = \left( {1 - p_{2}^{A} } \right)u\left( {W - Q^{*} } \right) + p_{2}^{A} u\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{*} + \varepsilon } \right).$$
(11)

Let \(\hat{Q}_{1} \left( \varepsilon \right)\) be the solution to

$$\left( {\hat{Q}_{1} - p_{1} L} \right) + \delta \left[ {p_{1} \left( {\hat{Q}_{2}^{A} - p_{2}^{A} L} \right) + \left( {1 - p_{1} } \right)\left( {\hat{Q}_{2}^{N} - p_{2}^{N} L} \right)} \right] = 0.$$
(12)

It is obvious that the sequence of premiums \((\hat{Q}_{1} (\varepsilon ),\hat{Q}_{2}^{A} (\varepsilon ),\hat{Q}_{2}^{N} (\varepsilon ))\) satisfies constraints (1), (2), and (4). Next, we show that the constructed contract generates a strictly higher expected utility than that under contract \((Q_{1}^{*}, Q_{2}^{A*}, Q_{2}^{N*})\). Note that \((\hat{Q}_{1} (0),\hat{Q}_{2}^{A} (0),\hat{Q}_{2}^{N} (0)) = (Q_{1}^{*} ,Q_{2}^{A*} ,Q_{2}^{N*} )\). Differentiating Eqs. (10) and (11) with respect to \(\varepsilon\) and evaluating at \(\varepsilon=0\), we can obtain

$$\left. {\frac{{d\hat{Q}_{2}^{N} \left( \varepsilon \right)}}{d\varepsilon }} \right|_{\varepsilon = 0} = - p_{2}^{N} \frac{{u'\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{*} } \right)}}{{u'\left( {W - Q_{2}^{N*} } \right)}},$$

and

$$\left. {\frac{{d\hat{Q}_{2}^{A} (\varepsilon )}}{d\varepsilon }} \right|_{\varepsilon = 0} = - p_{2}^{A} \frac{{u'\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{*} } \right)}}{{u'\left( {W - Q_{2}^{A*} } \right)}}.$$

Together with (12), we have that

$$\left. {\frac{{d\hat{Q}_{1} \left( \varepsilon \right)}}{d\varepsilon }} \right|_{\varepsilon = 0} = \delta p_{1} p_{2}^{A} \frac{{u'\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{*} } \right)}}{{u'\left( {W - Q_{2}^{A*} } \right)}} + \delta \left( {1 - p_{1} } \right)p_{2}^{N} \frac{{u'\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{*} } \right)}}{{u'\left( {W - Q_{2}^{N*} } \right)}}.$$

Denote consumer’s expected utility under the premium profile \((\hat{Q}_{1} (\varepsilon ),\hat{Q}_{2}^{A} (\varepsilon ),\hat{Q}_{2}^{N} (\varepsilon ))\) by \(\hat{U}(\varepsilon )\). It is straightforward to verify that

$$\begin{aligned} \left. {\frac{{d\hat{U}\left( \varepsilon \right)}}{d\varepsilon }} \right|_{\varepsilon = 0} & = - u'\left( {W - Q_{1}^{*} } \right)\left. {\frac{{d\hat{Q}_{1} }}{d\varepsilon }} \right|_{\varepsilon = 0} - \delta p_{1} u'\left( {W - Q_{2}^{A*} } \right)\left. {\frac{{d\hat{Q}_{2}^{A} }}{d\varepsilon }} \right|_{\varepsilon = 0} - \delta \left( {1 - p_{1} } \right)u'\left( {W - Q_{2}^{N*} } \right)\left. {\frac{{d\hat{Q}_{2}^{N} }}{d\varepsilon }} \right|_{\varepsilon = 0} \\ & = \delta p_{1} u'\left( {W - L + \frac{{1 - p_{2}^{N} }}{{p_{2}^{N} }}Q^{*} } \right) \times \left[ {1 - p_{2}^{A} \times \frac{{u'\left( {W - Q_{1}^{*} } \right)}}{{u'\left( {W - Q_{2}^{A*} } \right)}} - \left( {1 - p_{2}^{A} } \right) \times \frac{{u'\left( {W - Q_{1}^{*} } \right)}}{{u'\left( {W - Q_{2}^{N*} } \right)}}} \right]. \\ \end{aligned}$$

It remains to show that \(\left. {\frac{{d\hat{U}\left( \varepsilon \right)}}{d\varepsilon }} \right|_{\varepsilon = 0} > 0\), which is equivalent to

$$\frac{1}{{u'\left( {W - Q_{1}^{*} } \right)}} > \frac{{p_{2}^{A} }}{{u'\left( {W - Q_{2}^{A*} } \right)}} + \frac{{1 - p_{2}^{A} }}{{u'\left( {W - Q_{2}^{N*} } \right)}}.$$

From the postulated assumption that the equilibrium contract \((Q_{1}^{*}, Q_{2}^{A*}, Q_{2}^{N*})\) is not front-loaded, we must have \(Q_{2}^{A*} \ge Q_{2}^{A\dag }\), \(Q_{2}^{N*} \ge Q_{2}^{N\dag }\), and \(Q_{1}^{*} \le {p_{1}}L\). Therefore, we have that

$$\frac{1}{{u'\left( {W - Q_{1}^{*} } \right)}} \ge \frac{1}{{u'\left( {W - p_{1} L} \right)}} > \frac{{p_{2}^{A} }}{{u'\left( {W - Q_{2}^{A\dag } } \right)}} + \frac{{1 - p_{2}^{A} }}{{u'\left( {W - Q_{2}^{N\dag } } \right)}} \ge \frac{{p_{2}^{A} }}{{u'\left( {W - Q_{2}^{A*} } \right)}} + \frac{{1 - p_{2}^{A} }}{{u'\left( {W - Q_{2}^{N*} } \right)}},$$

where the first and third inequalities follow from the monotonicity of \(u^{\prime}(\cdot)\), and the second inequality follows directly from Lemma 2. This completes the proof of Step 1.

Step 2 Fixing \(\hat{Q}_{1}\), we increase \(\hat{Q}_{2}^{N}\) and decrease \(\hat{Q}_{2}^{A}\) to satisfy constraints (1) and (3) simultaneously. Specifically, constraint (1) is always satisfied if we increase \(\hat{Q}_{2}^{N}\) by x > 0 and decrease \(\hat{Q}_{2}^{A}\) by \(\frac{{1 - p_{1} }}{{p_{1} }}x\). From the continuity of \(u(\cdot)\), there exists \(\tilde{x} > 0\) such that the contract \(\left( {\hat{Q}_{1} ,\hat{Q}_{2}^{A} - \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x},\hat{Q}_{2}^{N} + \tilde{x}} \right)\) satisfies constraint (3). Therefore, the constructed contract satisfies constraint (1)–(4). To complete the proof, it suffices to show that the contract \(\left( {\hat{Q}_{1} ,\hat{Q}_{2}^{A} - \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x},\hat{Q}_{2}^{N} + \tilde{x}} \right)\) generates a strictly higher expected utility than does the contract \(\left( {\hat{Q}_{1} ,\hat{Q}_{2}^{A} ,\hat{Q}_{2}^{N} } \right)\), which is equivalent to show

$$p_{1} u\left( {W - \hat{Q}_{2}^{A} } \right) + \left( {1 - p_{1} } \right)u\left( {W - \hat{Q}_{2}^{N} } \right) < p_{1} u\left( {W - \hat{Q}_{2}^{A} + \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x}} \right) + \left( {1 - p_{1} } \right)u\left( {W - \hat{Q}_{2}^{N} - \tilde{x}} \right).$$
(13)

Note that

$$W - \hat{Q}_{2}^{A} + \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x} = \frac{{\frac{{1 - p_{1} }}{{p_{1} }}\tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}\left( {W - \hat{Q}_{2}^{N} } \right) + \frac{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} - \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}\left( {W - \hat{Q}_{2}^{A} } \right),$$

and

$$W - \hat{Q}_{2}^{N} - \tilde{x} = \frac{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} - \tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}\left( {W - \hat{Q}_{2}^{N} } \right) + \frac{{\tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}\left( {W - \hat{Q}_{2}^{A} } \right).$$

The fact that \(\hat{Q}_{2}^{N} < \hat{Q}_{2}^{N} + \tilde{x} < \hat{Q}_{2}^{A} - \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x} < \hat{Q}_{2}^{A}\) together with the strict concavity of \(u(\cdot)\) implies instantly that

$$u\left( {W - \hat{Q}_{2}^{A} + \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x}} \right) > \frac{{\frac{{1 - p_{1} }}{{p_{1} }}\tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}u\left( {W - \hat{Q}_{2}^{N} } \right) + \frac{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} - \frac{{1 - p_{1} }}{{p_{1} }}\tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}u\left( {W - \hat{Q}_{2}^{A} } \right),$$
(14)

and

$$u\left( {W - \hat{Q}_{2}^{N} - \tilde{x}} \right) > \frac{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} - \tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}u\left( {W - \hat{Q}_{2}^{N} } \right) + \frac{{\tilde{x}}}{{\hat{Q}_{2}^{A} - \hat{Q}_{2}^{N} }}u\left( {W - \hat{Q}_{2}^{A} } \right).$$
(15)

Multiplying (14) by \(p_{1}\), (15) by \(1-p_{1}\), and summing them yields (13). This completes the proof of Step 2. Therefore, the equilibrium contract must be highballing under one-sided commitment and asymmetric learning.□

Appendix 4: Simulation results

In this part, we provide some numerical results, which suggest that the requirement of a sufficiently large ratio between \(p_{2}^{A}\) and \(p_{2}^{N}\) in Proposition 1 is not crucial to drive the highballing pricing pattern.

Fixing \((W, \, L, \, p_{1})\), consider \(p_{2}^{A}\) and \(\left( {Q_{2}^{A\dag } ,Q_{2}^{N\dag } ,Q^{\dag } } \right)\) as functions of \(p_{2}^{N}\in (0,p_{1})\). From the martingale property, we have that

$$p_{2}^{A} = 1 - \frac{{1 - p_{1} }}{{p_{1} }} \times p_{2}^{N} .$$

Moreover, \(\left( {Q_{2}^{A\dag } ,Q_{2}^{N\dag } ,Q^{\dag } } \right)\) solves Eqs. (6)–(8). For Proposition 1 to hold, it suffices to show that condition (9) is satisfied for all \(p_{2}^{N}\in (0,p_{1})\), which is equivalent to

$$\begin{aligned}\psi \left( {p_{2}^{N} } \right)& := \left( {1 - \frac{{1 - p_{1} }}{{p_{1} }} \times p_{2}^{N} } \right) \times \frac{{u'\left( {W - p_{1} L} \right)}}{{u'\left( {W - Q_{2}^{A\dag } \left( {p_{2}^{N} } \right)} \right)}} + \left( {\frac{{1 - p_{1} }}{{p_{1} }} \times p_{2}^{N} } \right) \times \frac{{u'\left( {W - p_{1} L} \right)}}{{u'\left( {W - Q_{2}^{N\dag } \left( {p_{2}^{N} } \right)} \right)}} < 1. \end{aligned}$$

To proceed, we set \((W,L)\) = (1, 0.5), and consider the following families of utility functions:

  1. (a)

    The expo-power (EP) utility functions (Saha 1993):

    $$u\left( c \right) = - { \exp }\left( { - \beta \cdot c^{\alpha } } \right), \,\,{\text{with}}\,\, \alpha \ne 0,\,\, \beta \ne 0,\,\, {\text{and}}\,\, \alpha \beta > 0.$$
  2. (b)

    The power risk aversion (PRA) utility functions (Xie 2000):

    $$u\left( c \right) = \frac{1}{\gamma }\left\{ {1 - { \exp }\left[ { - \gamma \left( {\frac{{c^{1 - \sigma } - 1}}{1 - \sigma }} \right)} \right]} \right\},\,\, {\text{with}}\,\,\sigma \ge 0,\,\, {\text{and}}\,\,\gamma \ge 0.$$
  3. (c)

    The flexible three parameter (FTP) utility functions (Conniffe 2007):

    $$u\left( c \right) = \frac{1}{\theta }\left\{ {1 - \left[ {1 - \lambda \theta \left( {\frac{{c^{1 - \tau } - 1}}{1 - \tau }} \right)^{{\frac{1}{\lambda }}} } \right]} \right\},\,\,{\text{with}}\,\,\tau > 0,\,\, \theta > 0,\,\,{\text{and}}\,\,\lambda \le 1.$$

Note that the EP and the PRA forms are equivalent. To translate parameters, let σ = 1 − α and γ = αβ. Figure 3a, b plot \(\psi(p_{2}^{N})\) with different parameters of the above utility functions, assuming the first-period loss probability is small \((p_{1}=0.2)\) and large \((p_{1}=0.8)\), respectively. It is clear that \(\psi(p_{2}^{N})<1\) is satisfied for all \(p_{2}^{N}\in (0,p_{1})\), suggesting that Proposition 1 holds regardless of the size of the ratio \(p_{2}^{A}/p_{2}^{N}\).

Fig. 3
figure3

a\(\left( {W,L,p_{1} } \right) = \left( {1, 0.5, 0.2} \right)\), b\(\left( {W,L,p_{1} } \right) = \left( {1, 0.5, 0.8} \right)\)

Appendix 5: Robustness of Propositions 12 under an alternative assumption of risk dynamics

In the main text, we assume that policyholder’s risk type does not change (e.g., accident insurance). In this part, we relax this assumption and assume that policyholder’s type worsens over time (e.g., life insurance) as in Hendel and Lizzeri (2003). Again, we can show that the equilibrium pricing patterns predicted in Propositions 1 and 2 hold under this alternative assumption of risk changes.

More formally, we assume that the second-period loss probability is \(p_{2}\in \{ p_{2}^{A}, p_{2}^{N}\}\), with \(p_{2}^{A}>p_{2}^{N}>p_{1}\) and \({ \Pr }(p_{2} = p_{2}^{A} ) = 1 - { \Pr }\left( {p_{2} = p_{2}^{N} } \right) = \tilde{p} \in (0,1)\). The second-period loss probability \(p_{2}\) is learned by the incumbent insurer and the policyholder, but not observed by the entrants.

Proposition 3

Suppose that policyholder’s risk type worsens over time and learning is asymmetric. Then the equilibrium contract exhibits:

  1. (i)

    highballing (front-loaded) pricing pattern under one-sided commitment;

  2. (ii)

    lowballing (back-loaded) pattern under no commitment.

Proof

Denote \(\tilde{p} \times p_{2}^{A} + (1 - \tilde{p}) \times p_{2}^{N}\) by \(\bar{p}_{2} .\) It is clear that \(\bar{p}_{2} \equiv \tilde{p} \times p_{2}^{A} + (1 - \tilde{p}) \times p_{2}^{N} > \tilde{p} \times p_{1} + (1 - \tilde{p}) \times p_{1} = p_{1}\). Note that the proof of our Proposition 2 is exactly the same as that of Proposition 2 in de Garidel-Thoron (2005), which does not rely on the martingale property \(p_{1} = p_{1} p_{2}^{A} + \left( {1 - p_{1} } \right) p_{2}^{N}\) and the distribution of \(p_{2}\). Therefore, the equilibrium contract still exhibits lowballing with the alternative assumption \(p_{2}^{A}>p_{2}^{N}>p_{1}\); and it remains to prove part (i) of the proposition. The equilibrium contract solves the following maximization problem:

$${ \hbox{max} }_{{\left\{ {Q_{1} ,Q_{2}^{A} ,Q_{2}^{N} } \right\}}} u\left( {W - Q_{1} } \right) + \delta \left[ {\tilde{p}u\left( {W - Q_{2}^{A} } \right) + \left( {1 - \tilde{p}} \right)u\left( {W - Q_{2}^{N} } \right)} \right],$$

subject to

$$\left( {Q_{1} - p_{1} L} \right) + \delta \left[ {\tilde{p}\left( {Q_{2}^{A} - p_{2}^{A} L} \right) + \left( {1 - \tilde{p}} \right)\left( {Q_{2}^{N} - p_{2}^{N} L} \right)} \right] = 0,$$
(16)
$$Q_{2}^{A} \le p_{2}^{A} L,$$
(17)
$$IC_{2}^{N} \left( {Q;C_{2}^{N} } \right)\,\,{\text{and}}\,\,IC_{2}^{A} \left( {Q;C_{2}^{A} } \right)\,\,{\text{cross}}\,\,{\text{on}}\,\,{\text{the}}\,\,{\text{line }}\left( {1 - p_{2}^{N} } \right)Q - p_{2}^{N} R = 0,$$
(18)
$$IC_{2}^{N} \left( {Q;C_{2}^{N} } \right)\,\,{\text{and}}\,\, \left( {1 - \bar{p}_{2} } \right)Q - \bar{p}_{2} R = 0\,\,{\text{do }}\,\,{\text{not}}\,\,{\text{intersect}} .$$
(19)

Suppose, to the contrary, that the equilibrium contract, denoted by \((Q_{1}^{**}, Q_{2}^{A**}, Q_{2}^{N**})\), maximizes consumer’s expected utility and is not front-loaded (i.e., \(Q_{1}^{**} \le p_{1}L\). Then we can construct an alternative contract that yields a higher expected utility and satisfies constraints (16)–(19) by the following two steps:

Step 1 Fix \(Q_{2}^{A**} \), decrease \(Q_{2}^{N**} \) by \(\varepsilon^{\prime}\), and increase \(Q_{1}^{**} \) by \(\delta \left( {1 - \tilde{p}} \right)\varepsilon^{\prime}\) for a sufficiently small \(\varepsilon^{\prime}>0\). It is clear that constraints (16), (17) and (19) are still satisfied but constraint (18) is violated. Denote consumer’s expected utility with contract \(\left( {Q_{1}^{**} + \delta \left( {1 - \tilde{p}} \right)\varepsilon^{'} ,Q_{2}^{A**} ,Q_{2}^{N**} - \varepsilon^{'} } \right)\) by \(U(\varepsilon^{\prime})\). It can be verified that

$$\left. {\frac{{dU(\varepsilon^{\prime})}}{{d\varepsilon^{\prime}}}} \right|_{{\varepsilon^{'} = 0}} = \delta (1 - \tilde{p}) \times \left[ {u^{\prime}(W - Q_{2}^{N**} ) - u^{\prime}(W - Q_{1}^{**} )} \right].$$

From constraint (18) and the shape of \(IC_{2}^{N}\,(Q; C_{2}^{N})\), we have \(Q_{2}^{N**}\ge p_{2}^{N}L\). Together with the postulated \(Q_{1}^{**} \le p_{1}L\), we must have that

$$Q_{1}^{**} \le p_{1} L < p_{2}^{N} L \le Q_{2}^{N**} ,$$

which in turn implies that \(u^{\prime}(W-Q_{2}^{N**})>u^{\prime}(W-Q_{1}^{**})\) and \(\left. {\frac{{dU\left( {\varepsilon^{'} } \right)}}{{d\varepsilon^{'} }}} \right|_{{\varepsilon^{'} = 0}} > 0\). By the continuity of the utility function \(u(\cdot)\), there exists \(\varepsilon_{1}^{\prime}>0\) such that the contract with the premium profile \(\left( {Q_{1}^{**} + \delta \left( {1 - \tilde{p}} \right)\varepsilon_{1}^{'} , Q_{2}^{A**} ,Q_{2}^{N**} - \varepsilon_{1}^{'} } \right) \equiv \left( {\hat{\hat{Q}}_{1} ,\hat{\hat{Q}}_{2}^{A} ,\hat{\hat{Q}}_{2}^{N} } \right)\) generates a strictly higher expected utility than does the equilibrium premium profile \(\left( {Q_{1}^{**} , Q_{2}^{A**} ,Q_{2}^{N**} } \right).\)

Step 2 Fixing \(\hat{\hat{Q}}_{1}\), we increase \(\hat{\hat{Q}}_{2}^{N}\) and decrease \(\hat{\hat{Q}}_{2}^{A}\) to satisfy constraints (16) and (18) simultaneously. It is clear that the constructed contract satisfies constraints (16)–(19). By the same argument as in Step 2 in the proof of Proposition 1, this contract generates strictly higher utility to the policyholders than does the contract with the premium profile \(\left( {\hat{\hat{Q}}_{1} ,\hat{\hat{Q}}_{2}^{A} ,\hat{\hat{Q}}_{2}^{N} } \right)\).□

Proposition 3 complements Hendel and Lizzeri (2003) by showing that the highballing pricing pattern under one-sided commitment is robust across different learning environments. It should be noted that both symmetric learning and one-sided commitment play indispensable roles in shaping the highballing pricing pattern in Hendel and Lizzeri (2003). On the one hand, symmetric learning indicates that an insurer faces fierce competition with the entrant in every state in the second period, and, hence, the incumbent insurer cannot obtain information rents in the second period. To see this more clearly, suppose to the contrary that the incumbent insurer offers a contract that generates positive profits for some state in the second period. The entrant can then earn profits by providing a contract that yields less profit and a strictly higher expected utility to attract policyholders in that state. Therefore, the incumbent insurer must earn zero or negative profits in the second period. On the other hand, one-sided commitment implies that insurers have incentives to insure against the period-2 reclassification risk in terms of level premiums to maximize consumers’ inter-temporal expected utility. This implies directly that policyholders of low-risk types will lapse the contract in the second period, yielding zero profits to the incumbent insurer, while policyholders of high-risk types will stick with the contract, generating losses instead. As a result, a highballing pricing pattern emerges in the competitive equilibrium in Hendel and Lizzeri (2003). Note that their result cannot be directly generalized to the environment of asymmetric learning as is assumed in this paper because the incumbent insurer earns profits from type-N consumers [see constraint (8)] and suffers losses from type-A consumers [see constraint (7)].

Appendix 6: Figure 2 in the contingent wealth space

In this section, we redraw Fig. 2 in the contingent wealth space as in Rothschild and Stiglitz (1976). Before we proceed, it is useful to first introduce several notations. In each period, an insurance contract can be represented by \(c \equiv (\bar{x},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\), where \(\bar{x}\) specifies the consumption level when no accident occurs, and \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}\) refers to the consumption level when an accident occurs. Therefore, a two-period insurance contract can be indexed by \(\left( c_{1} ,c_{2}^{A} ,c_{2}^{N} \right) \equiv \left( \bar{x}_{1} , {\underline{x}}_1\right),\left( \bar{x}_{2}^{A} ,{\underline{x}}_{{2}}^{A} \right),\left( \bar{x}_{2}^{N} , {\underline{x}}_{{2}}^{N}\right)\), where \(c_{1} \equiv \left( {\bar{x}_{1} ,\underline{x}_{1} } \right)\) is the first-period contract and \(c_{2}^{k} \equiv \left( {\bar{x}_{2}^{k} ,\underline{x}_{2}^{k} } \right)\) is the second-period contract contingent on \(k\in\{ A, \, N \}\). Fixing \((R,Q)\) as specified in the main text of the paper, it is clear that \((\bar{x},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\) can be derived as the following:

$$\bar{x} = W - Q,\quad {\text{and}}\quad \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} = W - L + R.$$

Moreover, consumers’ indifference curve that crosses a contract \(c_{2}^{k} \equiv \left( {\bar{x}_{2}^{k} ,\underline{x}_{2}^{k} } \right)\) for type \(k\in\{ A, \, N \}\), which we denote by \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} = IC_{2}^{k} \left( {\bar{x};c_{2}^{k} } \right)\), is the solution to

$$\left( {1 - p_{2}^{k} } \right)u\left( {\bar{x}} \right) + p_{2}^{k} u\left( {IC_{2}^{k} \left( {\bar{x};c_{2}^{k} } \right)} \right) = \left( {1 - p_{2}^{k} } \right)u\left( {\bar{x}_{2}^{k} } \right) + p_{2}^{k} u\left( {\underline{x}_{2}^{k} } \right).$$

Then Fig. 2 can be redrawn as Fig. 4 in the \((\bar{x},\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x} )\) space as follows.

Fig. 4
figure4

Graphical illustration for proof of Proposition 1 in the contingent wealth space

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Jia, R., Wu, Z. Insurer commitment and dynamic pricing pattern. Geneva Risk Insur Rev 44, 87–135 (2019). https://doi.org/10.1057/s10713-018-0036-9

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Keywords

  • Dynamic contract
  • Commitment
  • Asymmetric learning
  • Information asymmetry
  • Inter-temporal pricing

JEL Classification

  • D86
  • G22