Equilibrium in Competitive Insurance Markets Under Adverse Selection and Yaari's Dual Theory of Risk

Under Yaari's dual theory of risk, we determine the equilibrium separating contracts for high and low risks in a competitive insurance market, in which risks are defined only by their expected losses, that is, a high risk is a risk that has a greater expected loss than a low risk. Also, we determine the pooling equilibrium contract when insurers are assumed non-myopic. Expected utility theory generally predicts that optimal insurance indemnity payments are nonlinear functions of the underlying loss due to the nonlinearity of agents' utility functions. Under Yaari's dual theory, we show that under mild technical conditions the indemnity payment is a piecewise linear function of the loss, a common property of insurance coverages.     

Background Risk and the Performance of Insurance Markets under Adverse Selection

Background risk can influence the performance of insurance markets that must deal with adverse selection when applicants are risk vulnerable, since they are more averse to bearing the insurable risk as a result of their exposures to background risk. We show that background risk always results in a lower deductible for the incentive constrained contract, and that a broader range of markets attains the stable sequential equilibrium cross-subsidized pair of separating contracts. We conclude that background risk always improves the performance of markets for coverage against (insurable) foreground risks that must deal with adverse selection. We also find, however, that these improvements are never sufficient to offset the cost to insureds of bearing the background risk.     

New Developments in the Theory of Adverse Selection in Competitive Insurance

We provide an overview of the paths taken to understand existence and efficiency of equilibrium in competitive insurance markets with adverse selection since the seminal work by Rothschild and Stiglitz (1976). A stream of recent work reconsiders the strategic foundations of competitive equilibrium by carefully modelling the market game.     

Adverse Selection in Reinsurance Markets

This paper looks for evidence of adverse selection in the relationship between primary insurers and reinsurers. We test the implications of a model in which informational asymmetry—and therefore, its negative consequences—decline over time. Our tests involve a data panel consisting of U.S. property-liability insurance firms that reported to the National Association of Insurance Commissioners during the period 1993–2012. We find that the amount of reinsurance, insurer profitability, and insurer credit quality all increase with the tenure of the insurer–reinsurer relationship.     

逆向选择条件下带甄别期的最优保险设计

为了改善信息不对称对保险市场交易效率的影响,分投保人为两种及两种以上风险类型建立了带甄别期的保险契约模型,指出可以利用投保人在甄别期的风险发生情况来推断投保人的风险类型.带甄别期的保险契约是指:自保险合同生效之日起的一段时间内(甄别期),如果投保人发生风险,保险公司将给予一定的赔偿,甄别期过后,如果投保人再次发生风险,保险公司将不再给予任何赔偿;如果投保人在甄别期未发生风险,而在甄别期之后的剩余保险期发生风险,保险公司仍然给予与上述情况相同的赔偿.证明指出效用最优时带甄别期的保险契约不比R-S传统部分保险契约差,并给出了前者是后者严格帕累托改进的充分条件.此外,对于两种以上风险类型情形,证明了满足对次低风险投保人的激励相容约束是满足对其余高风险投保人激励相容约束的一个充分不必要条件,并给出了相应的充分条件,进一步指出该充分条件的集合恰是带甄别期的保险契约能够产生分离均衡的一个充分条件.最后,以一个算例说明确实存在效用最优时带甄别期的保险契约是R-S传统部分保险契约的严格帕累托改进情形.     

Adverse Selection and Competing Deposit Insurance Systems in Pre-Depression Texas

In 1910, Texas instituted a unique deposit insurance program for its state chartered banks by providing a choice between two separate plans: the depositors guaranty fund, similar to insurance schemes in several other states, and the depositors bond security system, which required the procurement of a privately issued guarantee of indemnity. While, under most deposit insurance schemes, the incentive to monitor the financial condition of individual banks simply devolves from depositors to regulators, the bond security system established in Texas distinguished itself by attempting to reintroduce market discipline through the indemnity requirement. Using a probit model with heteroscedasticity, we find evidence that the choice of insurance coverage led to risk-sorting among the banks, with relatively conservative and financially secure institutions opting for the comparatively rigorous bond security system. In addition, the bank failure record indicates the risk differentials between banks in the two plans persisted over time and even possibly grew, suggesting the bond security system at least partially avoided the moral hazard incentives associated with the fixed-rate depositors guaranty plan. These findings support the general view that market discipline is effective in banking.     

An Examination of Adverse Selection in the Public Provision of Insurance

Using a unique data set from Florida's residual property insurer, we test for adverse selection in the public provision of homeowners’ insurance in Florida. We find a significant relationship between the losses and deductible choices of insureds in Florida's residual homeowners’ insurance market. This relationship provides strong evidence of the existence of an adverse selection problem in Florida's residual property insurance market. While this relationship is important to Florida regulators (and taxpayers) specifically, a finding of an adverse selection problem in residual markets in general has implications more broadly for government providers of insurance as an adverse selection problem in these settings will impact the public policy debates and decisions involving these markets.     

Cream Skimming: Innovations in Insurance Risk Classification and Adverse Selection

We demonstrate how innovations in insurance risk classification can lead to adverse selection, or cream skimming, against insurers that are slow to adopt such pricing innovations. Using a model in which insurers with insufficient pricing data cannot differentiate between low‐ and high‐risk policyholders and therefore charge both the same premium, we show how innovative insurers develop new risk classification data to identify overcharged low‐risk policyholders and attract them from rival insurers with reduced prices. Less innovative insurers thus insure a growing percentage of high‐risk customers, resulting in adverse selection attributable to their informational disadvantage. Next, we examine two cases in which “Big Data” innovations in risk classification led to concerns about cream skimming among U.S. auto insurers. First, we track the rapid adoption of credit‐based insurance scores as pricing variables in personal auto insurance markets. Second, we examine the growing popularity of usage‐based insurance programs like telematics, plans in which insurers use data on policyholders’ actual driving behavior to set prices that attract low‐risk customers. Issues associated with the execution of such pricing strategies are discussed. In both cases, we document how rival insurers quickly adopt successful innovations to reduce their exposure to adverse selection.     

中国健康险市场逆向选择和正向选择并存现象的研究

本文以中国的健康险市场为例,考察不对称信息的影响。通过考察投保人投保金额以及附加险选择和索赔情况的相关关系,论文发现事后出现索赔的投保人,事前往往会选择购买附加险,但是投保金额却相对较低。结合理论模型分析,论文认为投保人在财富、风险偏好等方面的异质性以及信息不对称的存在是导致市场同时出现逆向选择和正向选择的主要原因。     

An Insurance Market Simulation With Both Adverse and Advantageous Selection

The theory of adverse selection predicts that high‐risk individuals are more likely to buy insurance than low‐risk individuals if asymmetric information regarding individuals’ risk type is present in the market. The theory of advantageous selection predicts the opposite—a negative relationship between insurance coverage and risk type can be obtained when hidden knowledge in other dimensions (e.g., the degree of risk aversion) is present in addition to the risk type. Using the heterogeneity of insurance buyers in either risk type or risk aversion, we first introduce a classroom‐based insurance market simulation game to show that adverse selection and advantageous selection can coexist. We then explain the underlying concepts using two methods: a mathematical framework based on expected utility theory and an empirical framework based on the results of the game itself. The game is easy to implement, reinforces textbook concepts by providing students a hands‐on experience, and supplements current textbooks by bringing their content up to date with current research.     

Full Insurance, Bayesian Updated Premiums, and Adverse Selection

In the classic Rothschild-Stiglitz model of adverse selection in a competitive environment, we analyse a no-claims bonus type contract (bonus-malus). We show that, under full insurance coverage, if the insurance company applies Bayes's rule to learn about client probability types over time and uses this information in premium calculations for contract renewals, then there exist conditions under which all client types strictly prefer the Bayesian updating contract to the classic Rothschild-Stiglitz separating equilibrium.     

Unravelling vs Unravelling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that insurance markets may “unravel”. This memo clarifies the distinction between these two notions of unravelling in the context of a binary loss model of insurance. I show that the two concepts are mutually exclusive occurrences. Moreover, I provide a regularity condition under which the two concepts are exhaustive of the set of possible occurrences in the model. Akerlof unravelling characterises when there are no gains to trade; Rothschild and Stiglitz unravelling shows that the standard notion of competition (pure strategy Nash equilibrium) is inadequate to describe the workings of insurance markets when there are gains to trade.     

Interventions in Markets with Adverse Selection: Implications for Discount Window Stigma

I study the implications for central bank discount window stigma of a workhorse model of adverse selection in financial markets. In the model, firms (banks) need to borrow to finance a productive project. There is limited liability and firms have private information about their ability to repay their debts, which gives rise to the possibility of adverse selection. The central bank can ameliorate the impact of adverse selection by lending to firms. Discount window borrowing is observable and it may be taken as a signal of firms' credit worthiness. Under some conditions, firms borrowing from the discount window may pay higher interest rates to borrow in the market, a phenomenon often associated with the presence of stigma. I discuss these and other outcomes in detail and what they suggest about the relevance of stigma as an empirical phenomenon.     

On the Cost of Adverse Selection in Individual Annuity Markets: Evidence From Singapore

New evidence is presented on the cost of adverse selection in individual annuity markets using Singapore data. The Singapore annuity market is an interesting setting to examine the cost of adverse selection for three reasons. First, unlike many Western countries, the Singapore government provides very limited public financial assistance for retirees. Second, while social security contributions mandated under the Central Provident Fund (CPF) result in a high forced savings rate, a large proportion of CPF savings, are used up for housing. Third, to ensure that retirees have sufficient funds to meet basic needs, individuals who reach age 55 are required to set aside a minimum amount of their CPF savings, which can be withdrawn at age 62. The CPF Board allows various options for investing the minimum sum, but the most attractive option is to purchase an annuity. The institutional setting in Singapore in effect provides insurers with a large captive market for annuities. It is conjectured that this should be reflected in a significantly lower cost of adverse selection for annuities sold in Singapore as compared with other countries. The results herein, using data for CPF‐approved insurers, are strongly consistent with this conjecture. On average, money's worth of annuities is higher than annuities sold to a similar age‐gender mix in the United States, United Kingdom, and Australia. Adverse selection accounts for less than 13 percent of the cost of longevity insurance compared to 30–50 per‐ cent documented in many previous studies. These results suggest that one way to resolve the adverse selection problem is to adopt a universal individual defined contribution pension scheme that mandates or provides strong incentives for retirees to purchase annuities.     

Complete Versus Incomplete Insurance Contracts under Adverse Selection with Multiple Risks

This article extends the standard adverse-selection model for competitive insurance markets, which assumes a single source of risk, to the case where individuals are subject to multiple risks. We compare the following market situations—the case where insurers can offer comprehensive policies against all sources or risks (complete contracts) and the case where different risks are covered by separate policies (incomplete contracts). In the latter case, we consider whether the insurer of a particular risk has perfect information regarding an individual's coverage against other sources of risks. The analysis emphasizes the informational role of bundling in multidimensional screening. When the market situation allows bundling, it is shown that in equilibrium the low-risk type with respect to a particular source of risk does not necessarily obtain partial coverage against that particular risk.     

Arbitrage and Viability in Insurance Markets

Insurance markets are subject to transaction costs and constraints on portfolio holdings. Therefore, unlike the frictionless asset markets case, viability is not equivalent to absence of arbitrage possibilities. We use the concept of unbounded arbitrage to characterize viable prices on a complete and an incomplete insurance market. In the complete market, there is an insurance contract for every possible event. In the incomplete market, risk can be insured through proportional and excess of loss like insurance contracts. We show how the the structure of viable prices is affected by the portfolio constraints, the transaction costs, and the structure of marketed contracts.     

Adverse Selection and Competitive Market Making: Empirical Evidence from a Limit Order Market

This article presents a new methodology for testing economicrestrictions on the price schedules offered in a limit orderbook that are based on (i) break-even conditions for marginallimit orders and (ii) rational updating conditions for orderbook revisions over time. Using order flow data from the StockholmStock Exchange, I find strong evidence of insufficient depthin the limit order books relative to the theoretical predictions.An extended model, which allows the model parameters to dependon market conditions, captures some of the systematic variationin the observed order book depth.     

Risk Mutualization and Competition in Insurance Markets

The aim of this paper is to analyze the impact of mutual firms on competition in the insurance market. We distinguish two actors in this market: mutual firms, which belong to their pooled members, and traditional companies, which belong to their shareholders. Our approach differs from the literature by one crucial assumption: the expected utility of the consumers depends on the size of their insurance firm, which generates network externalities in this market. Thus, the choice of a contract results in a trade-off between the premium level and the probability of that premium being ex-post adjusted. The optimal contract offered by a mutual firm involves a systematic ex-post adjustment (negative or positive), while the contracts a company offers imply a fixed premium that is possibly negatively adjusted at the end of the contractual period. In an oligopoly game, we show that three types of configurations are possible at equilibrium: either one mutual firm or insurance company is active, or a mixed structure emerges in which two or more companies share the market with or without a mutual firm.