Optimal Insurance Under the Insurer's VaR Constraint

In this paper, we impose the insurer's Value at Risk (VaR) constraint on Arrow's optimal insurance model. The insured aims to maximize his expected utility of terminal wealth, under the constraint that the insurer wishes to control the VaR of his terminal wealth to be maintained below a prespecified level. It is shown that when the insurer's VaR constraint is binding, the solution to the problem is not linear, but piecewise linear deductible, and the insured's optimal expected utility will increase as the insurer becomes more risk-tolerant. Basak and Shapiro (2001) showed that VaR risk managers often choose larger risk exposures to risky assets. We draw a similar conclusion in this paper. It is shown that when the insured has an exponential utility function, optimal insurance based on VaR constraint causes the insurer to suffer larger losses than optimal insurance without insurer's risk constraint.     

First-Best Equilibrium in Insurance Markets With Transaction Costs and Heterogeneity

We investigate extensions of the classic Rothschild and Stiglitz (1976) (RS) model of adverse selection under asymmetric information. In RS, low‐risk customers are worse off owing to an externality created by high‐risk buyers in the market. We find critical changes in insurance buyers' behavior under the joint assumptions of transaction costs and buyer heterogeneity with respect to either risk aversion or wealth. Combining transaction costs and heterogeneity, we find a separating equilibrium in which neither high‐risk nor low‐risk individuals are penalized due to information asymmetry.     

Information and the hold‐up problem

We examine situations in which a party must make a sunk investment prior to contracting with a second party to purchase an essential complementary input. We study how the resulting hold‐up problem is affected by the seller's information about the investing party's likely returns from its investment. Our principal focus is on the effects of the investment's being observable by the noninvesting party. We establish conditions under which the seller's ability to observe the buyer's investment harms the seller, benefits the buyer, and reduces equilibrium investment and total surplus. We also note conditions under which investment and welfare rise when investment is observable.     

Pricing and Capital Allocation for Multiline Insurance Firms

We study multiline insurance companies with limited liability. Insurance premiums are determined by no‐arbitrage principles. The results are developed under the realistic assumption that the losses created by insurer default are allocated among policyholders following an ex post, pro rata, sharing rule. In general, the ratio of default costs to expected claims, and thus the ratio of premiums to expected claims, vary across insurance lines. Moreover, capital and related costs are allocated across lines in proportion to each line's share of a digital default option on the insurer. Our results expand and generalize those derived elsewhere in the literature.     

Insurance Contract Design When the Insurer Has Private Information on Loss Size

This article examines the optimal indemnity contract in an insurance market, when the insurer has private information about the size of an insurable loss. Both parties know whether or not a loss occurred, but only the insurer knows the true value of the loss and/or to what extent the losses are covered under the policy. The insured may verify the insurer's loss estimate for a fixed auditing cost. The optimal contract reimburses the auditing costs in addition to full insurance for losses less than some endogenous limit. For losses exceeding this limit, the contract pays a fixed indemnity and requires no monitoring. The optimal contract is compared with the contracts obtained in cases where it is only the insured who can observe the loss size.
     

Procurement when price and quality matter

A buyer seeks to procure a good characterized by its price and its quality from suppliers who have private information about their cost structure (fixed cost and marginal cost of providing quality). We characterize the buyer's optimal buying mechanism. We then use the optimal mechanism as a theoretical and numerical benchmark to study simpler buying procedures such as scoring auctions and bargaining. Scoring auctions can extract a significant proportion of the buyer's strategic surplus (the difference between the expected utility from the optimal mechanism and the efficient auction). Bargaining does less well and often does worse than the efficient auction.     

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.     

Insurance Market Effects of Risk Management Metrics

We extend the classical analysis on optimal insurance design to the case when the insurer implements regulatory requirements (Value-at-Risk). Presumably, regulators impose some risk management requirement such as VaR to reduce the insurers’ insolvency risk, as well as to improve the insurance market stability. We show that VaR requirements may better protect the insured and improve economic efficiency, but have stringent negative effects on the insurance market. Our analysis reveals that the insured are better protected in the event of greater loss irrespective of the optimal design from either the insured or the insurer perspective. However, in the presence of the VaR requirement on the insurer, the insurer's insolvency risk might be increased and there are moral hazard issues in the insurance market because the optimal contract is discontinuous.     

Specification and negotiation in incomplete contracts

We investigate contractors' bargaining power and holdup on buyers in procurement auctions of incomplete contracts held by California Department of Transportation. Using a model where contractors bid competitively in response to a buyer's choice of initial contract design, we infer the contractors' costs and bargaining power from the bids and transfers negotiated after the auction. We find that the contract winners have substantial bargaining power in post‐auction negotiation. The average holdup on the buyer is about 20% of project costs. Counterfactual cost‐plus contracts would reduce the buyer's surplus in 72% of the projects, with an average reduction over $382,000.     

Strategic ex ante contracts: rent extraction and opportunity costs

This article cosiders the possibility that a seller can contract with one uninformed buyer prior to an auction involving two potential buyers. The seller's optimal strategic ex ante contract more accurately reflects joint opportunity costs of the seller and the contracted buyer, and therefore extracts more rent from the entrant. Moreover, this ex ante contract mitigates the seller's ex post rent seeking vis‐à‐vis the contracted buyer. Accordingly, it may create more social welfare than the absence of ex ante contracts, depending upon the contracted buyer's financial constraint and the distributions of trade surplus. Implementation of the optimal strategic ex ante contract and policy implications are discussed.     

Optimal Loss Mitigation and Contract Design

This work examines the interaction between the premium rates set by an insurer and the incentives of an individual to purchase market insurance and undertake mitigation to reduce the size of a potential loss. A risk‐neutral monopolistic insurer prices insurance according to the price‐elasticity of demand for coverage. The elasticity of demand is affected by the presence of both mitigation and government intervention. The availability of loss reduction activities increases the consumer's elasticity of demand and lowers the optimal rate charged by the monopolist. Government intervention reduces both expenditures on mitigation and the rate charged by the monopolistic insurer.     

Adverse Selection With Frequency and Severity Risk: Alternative Risk‐Sharing Provisions

The analysis considers an insurance market with adverse selection where individuals' loss distributions may differ with respect to both the frequency and severity of loss. We show that the combination of deductibles and coinsurance can be used to sort rationed policyholders. Because of their screening properties, coinsurance and deductibles may both be equilibrium forms of risk sharing for a particular insurer facing asymmetric information, with different rationed consumers choosing different risk‐sharing provisions.     

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.     

Robust pricing with refunds

Before purchase, a buyer of an experience good learns about the product's fit using various information sources, including some of which the seller may be unaware of. The buyer, however, can conclusively learn the fit only after purchasing and trying out the product. We show that the seller can use a simple mechanism to take best advantage of the buyer's post-purchase learning to maximize his guaranteed-profit. We show that this mechanism combines a generous refund, which performs well when the buyer is relatively informed, with non-refundable random discounts, which work well when the buyer is relatively uninformed.     

Arrow's Theorem of the Deductible with Heterogeneous Beliefs

In Arrow's classical problem of demand for insurance indemnity schedules, it is well-known that the optimal insurance indemnification for an insurance buyer—or decision maker (DM)—is a deductible contract when the insurer is a risk-neutral Expected-Utility (EU) maximizer and when the DM is a risk-averse EU maximizer. In Arrow's framework, however, both parties share the same probabilistic beliefs about the realizations of the underlying insurable loss. This article reexamines Arrow's problem in a setting where the DM and the insurer have different subjective beliefs. Under a requirement of compatibility between the insurer's and the DM's subjective beliefs, we show the existence and monotonicity of optimal indemnity schedules for the DM. The belief compatibility condition is shown to be a weakening of the assumption of a monotone likelihood ratio. In the latter case, we show that the optimal indemnity schedule is a variable deductible schedule, with a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow's classical result is then obtained as a special case.     

Financing and the demand for corporate insurance

In this paper we examine the insurance decision of a firm with private information regarding its cash flows and insurable losses. We show that, even in the absence of bankruptcy costs and information production by insurers, the firm's attempts to hedge its information risk can induce it to demand insurance. If higher operating revenues are accompanied by a lower insurance risk, the firm will choose to self-insure. In contrast, if higher operating revenues are accompanied by a higher insurance risk, the firm will demand insurance. In fact, if its insurable losses are relatively small, the firm will fully insure its losses. Further, if there exists considerable uncertainty regarding the firm's insurance risk, the level of coverage demanded by the firm is dependent on its private information, with higher levels of coverage signaling favorable information regarding the firm's future operations.     

Creating Customer Value in Participating Life Insurance

The value of a life insurance contract may differ depending on whether it is looked at from the customer's point of view or that of the insurance company. We assume that the insurer is able to replicate the life insurance contract's cash flows via assets traded on the capital market and can hence apply risk‐neutral valuation techniques. The policyholder, on the other hand, will take risk preferences and diversification opportunities into account when placing a value on that same contract. Customer value is represented by policyholder willingness to pay and depends on the contract parameters, that is, the guaranteed interest rate and the annual and terminal surplus participation rate. The aim of this article is to analyze and compare these two perspectives. In particular, we identify contract parameter combinations that—while keeping the contract value fixed for the insurer—maximize customer value. In addition, we derive explicit expressions for a selection of specific cases. Our results suggest that a customer segmentation in this sense, that is, based on the different ways customers evaluate life insurance contracts and embedded investment guarantees while ensuring fair values, is worthwhile for insurance companies as doing so can result in substantial increases in policyholder willingness to pay.     

Pricing Double-Trigger Reinsurance Contracts: Financial Versus Actuarial Approach

This article discusses various approaches to pricing double‐trigger reinsurance contracts—a new type of contract that has emerged in the area of ‘‘alternative risk transfer.’’ The potential coverage from this type of contract depends on both underwriting and financial risk. We determine the reinsurer's reservation price if it wants to retain the firm's same safety level after signing the contract, in which case the contract typically must be backed by large amounts of equity capital (if equity capital is the risk management measure to be taken). We contrast the financial insurance pricing models with an actuarial pricing model that has as its objective no lessening of the reinsurance company's expected profits and no worsening of its safety level. We show that actuarial pricing can lead the reinsurer into a trap that results in the failure to close reinsurance contracts that would have a positive net present value because typical actuarial pricing dictates the type of risk management measure that must be taken, namely, the insertion of additional capital. Additionally, this type of pricing structure forces the reinsurance buyer to provide this safety capital as a debtholder. Finally, we discuss conditions leading to a market for double‐trigger reinsurance contracts.     

Do Publicly Traded Property–Casualty Insurers Cater to the Stock Market?

This article examines the catering theory in the insurance industry. We investigate whether managers of publicly traded insurers pursue a growth strategy catering to the stock market's preference. Two hypotheses are tested in this study: (1) an insurer will devote more efforts to increasing premium growth when the stock market places greater values on growth, and (2) this catering effect will be more pronounced at firms where managers have greater incentives to maximize short‐term stock prices. We find evidence supporting both hypotheses. Our study discovers a new channel through which the stock market and executive compensation affect insurance companies’ business strategies and the insurance market. The implication of the interplay between insurers and the stock market is significant and deserves future research.     

Auctions versus negotiations: the effects of inefficient renegotiation

For the procurement of complex goods, the early exchange of information is important to avoid costly renegotiation. If the buyer can specify the main characteristics of possible design improvements in a complete contingent contract, scoring auctions implement the efficient allocation. If this is not feasible, the buyer must choose between a price‐only auction (discouraging early information exchange) and bilateral negotiations with a preselected seller (reducing competition). Bilateral negotiations are superior if potential design improvements are important, if renegotiation is very costly, and if the buyer's bargaining position is strong. Moreover, negotiations provide stronger incentives for sellers to investigate design improvements.