Optimal Insurance Coverage Research Articles

Correction to: Optimal insurance coverage of low-probability catastrophic risks

Correction to: Optimal insurance coverage of low-probability catastrophic risks

Optimal health insurance

AbstractI formulate expected‐utility‐maximizing models for health insurance with a single optimal coinsurance (C*) and (separately) a single optimal deductible (D*). While so‐doing, I formalize Nyman's challenge to standard welfare‐loss models, clarifying when and by how much this alters unadjusted models. Using MEPS‐calibrated lognormal distributions and incorporating skewness and kurtosis measures of financial risk, I show how C* shifts as various economic parameters change. For reasonable parameter values, C* < 0.1, much lower than variance‐only estimates would conclude. Omitting higher‐order risk parameters importantly understates risk and hence understates optimal insurance coverage. I separately develop methods to determine D*, showing that it is approximately a fixed percentage of income that falls as the distribution of financial risks rise. This finding contrasts with existing US public policy regarding high‐deductible health plans, which employ fixed deductibles, independent of income.

Optimal insurance coverage of low-probability catastrophic risks

Catastrophic risks are often characterised by a low probability, a high severity and a large number of affected individuals. Taking these specificities into account, we analyse the capacity of insurance contracts to provide coverage for those risks, independently from the market failures frequently observed in practice. On the demand side, we characterise individual preferences under which the willingness to pay for the coverage of large losses remains significant, although their occurrence probability is very small. On the supply side, the correlation between individual losses affects the insurance pricing through the insurers’ cost of capital. Analysing the interaction between demand and supply yields the key determinants of insurability and of a socially optimal risk sharing strategy.

The Optimal Insurance Coverage With Default Risk

This articles re-examines a standard result on the demand for insurance (full coverage with a fair premium and partial coverage with a loaded premium') in the presence of a default risk. It is established that the optimal insurance coverage is always partial coverage if the default is total, irrespective of whether the premium is fair or loaded. For partial default, it is demonstrated in both the fair premium case and the loaded premium case that the optimal insurance coverage might be either full coverage or partial coverage for coinsurance, deductible insurance and upper-limit insurance.

Optimal Investment-Consumption-Insurance with Durable and Perishable Consumption Goods in a Jump Diffusion Market

We investigate an optimal investment-consumption and optimal level of insurance on durable consumption goods with a positive loading in a continuous-time economy. We assume that the economic agent invests in the financial market and in durable as well as perishable consumption goods to derive utilities from consumption over time in a jump-diffusion market. Assuming that the financial assets and durable consumption goods can be traded without transaction costs, we provide a semi-explicit solution for the optimal insurance coverage for durable goods and financial asset. With transaction costs for trading the durable good proportional to the total value of the durable good, we formulate the agent's optimization problem as a combined stochastic and impulse control problem, with an implicit intervention value function. We solve this problem numerically using stopping time iteration, and analyze the numerical results using illustrative examples.

Optimal Forbearance of Bank Resolution

We analyze optimal strategic delay of bank resolution ('grq forbearance') and deposit insurance coverage. After bad news on the bank's assets, depositors fear for the uninsured part of their deposit and withdraw while the regulator observes withdrawals and needs to decide when to intervene. Optimal policy maximizes the joint value of the demand deposit contract and the insurance fund to avoid inefficient risk-shifting towards the fund while also preventing inefficient runs. Under low insurance coverage, the optimal intervention policy is never to intervene (laissez-faire). Optimal deposit insurance coverage is always interior. The paper sheds light on the differences between the U.S. and the European Monetary Union concerning their bank resolution policies.

Loss Risk Management: Perspectives from Corporate Finance

We develop a theoretical framework to analyze the relationship between loss risk management and firm value by focusing on insurance. In the model, the option analogy for equity implies that a higher insurance coverage decreases equity value, which further increases endogenous bankruptcy threshold chosen by shareholders. This plays a key role in the trade-off between tax benefits and bankruptcy costs. We derive that insurance can significantly increase firm value. Moreover, firm value shows monotonically increasing or U-shaped relationship with respect to insurance coverage, which implies optimal insurance coverage is full or none. By considering frictions of loss risk management (e.g., premium loadings in insurance practice), we can explain many phenomena in insurance practice, such as, incomplete insurance coverage, risk retention for small losses and subsidies for catastrophe insurance.

Optimal insurance design of ambiguous risks

We examine the characteristics of the optimal insurance contract under linear transaction costs and an ambiguous distribution of losses. Under the standard expected utility model, we know from Arrow (1965) that it contains a straight deductible. In this paper, we assume that the policyholder is ambiguity averse in the sense of Klibanoff et al. (Econometrica 73(6):1849–1892, 2005). The optimal contract depends upon the structure of the ambiguity. For example, if the set of possible priors can be ranked according to the monotone likelihood ratio order, the optimal contract contains a disappearing deductible. We also show that the policyholder’s ambiguity aversion may have the counterintuitive effect to reduce the optimal insurance coverage of an ambiguous risk.

Trade credit insurance, capital constraint, and the behavior of manufacturers and banks

The manufacturer who is a supplier of trade credit may face non-payment risk from customers and a capital shortage problem simultaneously. Trade credit insurance, as one of the most important risk management tools, has been widely used in companies’ daily operation. In this study, the manufacturer who allows customers to delay payment for goods already delivered purchases trade credit insurance to transfer and reduce non-payment risk and borrows money from a bank to accommodate the capital constraint problem. The Stackelberg game and loss-averse theory are used to establish a newsboy model including trade credit insurance, and the optimal insurance coverage and total sales of the manufacturer are thereby investigated. Subsequently, the interest rate decision of the bank under different risk-averse situations is also characterized. We find that the interest rate set by a loss-averse bank is equal to or greater than that given by a risk-neutral bank. The use of trade credit insurance can help the manufacturer expand sales and dramatically reduce its default risk. Both the bank and the manufacturer are better off due to the use of trade credit insurance, but contrary to what one might expect, the bank prefers giving a higher interest rate to the manufacturer when the premium rate is in a reasonable region, which indicates that the manufacturer cannot use the insurance to negotiate better financing terms.

Managing Disruption Risk: The Interplay Between Operations and Insurance

Disruptive events that halt production can have severe business consequences if not appropriately managed. Business interruption (BI) insurance offers firms a financial mechanism for managing their exposure to disruption risk. Firms can also avail of operational measures to manage the risk. In this paper, we explore the relationship between BI insurance and operational measures. We model a manufacturing firm that can purchase BI insurance, invest in inventory, and avail of emergency sourcing. Allowing the insurance premium to depend on the firm's insurance and operational decisions, we characterize the optimal insurance deductible and coverage limit as well as the optimal inventory level. We prove that insurance and operational measures are not always substitutes, and we establish conditions under which they can be complements; that is, insurance can increase the marginal value of inventory and can increase the overall value of emergency sourcing. We also find that the value of insurance is higher for those firms less able to absorb financially significant disruptions. As disruptions become longer but rarer, the value of emergency sourcing increases, and the value of inventory and the value of insurance increase before eventually decreasing. This paper was accepted by Martin Lariviere, operations management.

Management Insights

Management Insights

Optimal insurance contract and coverage levels under loss aversion utility preference

This study develops an optimal insurance contract endogenously and determines the optimal coverage levels with respect to deductible insurance, upper-limit insurance, and proportional coinsurance, and, by assuming that the insured has an S-shaped loss aversion utility, the insured would retain the enormous losses entirely. The representative optimal insurance form is the truncated deductible insurance, where the insured retains all losses once losses exceed a critical level and adopts a particular deductible otherwise. Additionally, the effects of the optimal coverage levels are also examined with respect to benchmark wealth and loss aversion coefficient. Moreover, the efficiencies among various insurances are compared via numerical analysis by assuming that the loss obeys a uniform or log-normal distribution. In addition to optimal insurance, deductible insurance is the most efficient if the benchmark wealth is small and upper-limit insurance if large. In the case of a uniform distribution that has an upper bound, deductible insurance and optimal insurance coincide if benchmark wealth is small. Conversely, deductible insurance is never optimal for an unbounded loss such as a log-normal distribution.

Optimal Insurance Coverage under Bonus-Malus Contracts

AbstractThe paper analyses the questions: Should – or should not – an individual buy insurance? And if so, what insurance coverage should he or she prefer? Unlike classical studies of optimal insurance coverage, this paper analyses these questions from a bonus-malus point of view, that is, for insurance contracts with individual bonus-malus (experience rating or no-claim) adjustments. The paper outlines a set of new statements for bonus-malus contracts and compares them with corresponding classical statements for standard insurance contracts. The theoretical framework is an expected utility model, and both optimal coverage for a fixed premium function and Pareto optimal coverage are analyzed. The paper is an extension of another paper by the author, see Holtan (2001), where the necessary insight to – and concepts of – bonus-malus contracts are outlined.

Duality and equilibrium prices in economics of uncertainty

A random variable (RV) X is given a minimum selling price (S) SU(X) := sup x {x + EU(X x)} and a maximum buying price (B) BP(X) := inf x {x + EP(X x)} where U(·) and P(·) are appropriate functions. These prices are derived from considerations of stochastic optimization with recourse, and are called recourse certainty equivalents (RCE's) of X. Both RCE's compute the "value" of a RV as an optimization problem, and both problems (S) and (B) have meaningful dual problems, stated in terms of the Csiszar -divergence I (p,q) := n X i=1 qi pi qi a generalized entropy function, measuring the distance between RV's with probability vectors p and q. The RCE SU was introduced in (5), studied further in (3), (4) and (6), and applied to production, investment and insurance problems. Here we study the RCE BP, and apply it to problems of inventory control (where the attitude towards risk determines the stock levels and order sizes) and optimal insurance coverage, a problem stated as a game between the insurance company (setting the premiums) and the buyer of insurance, maximizing the RCE of his coverage.

The Impact of a Probationary Period on the Demand for Insurance

The literature devoted to the determination of optimal insurance coverage has always considered the effect of a monetary deductible on the demand for insurance. This paper examines the theory of optimal insurance purchasing in the presence of a probationary period. It is shown that contrary to intuition, most of the basic properties attached to the monetary deductible do not carry over to a time deductible such as the probationary period.

On the Theory of Rational Insurance Purchasing in a Continuous-Time Model

The determination of the optimal insurance coverage has raised a growing academic interest. The seminal works by Arrow [1963], Smith [1968] and Mossin [1968] contributed to establish the cornerstone results of insurance economics. Indeed, it is now a well-known result that a risk averse expected utility maximizer will never purchase full insurance coverage if the insurance premium is actuarially unfair A positive amount deductible is always optimal from the insured's point of view if the premium is based on expected insurance payments and includes a positive proportional loading. Arrow [1974] extended this result to the case of state-dependent utility functions. Some authors have however provided results which stand in sharp contrast with the above mentiond rule. Razin [1976], for instance, argued that the nature of results obtained very much depends on the decision framework. Using a binominal distribution of losses and Savage's minimax regret criterion as an alternative to the expected utility framework, he showed that it is always optimal to purchase insurance with a positive amount deductible even if the insurance premium is fair. In the same vein, Briys and Louberge [1985] obtained that, under the Hurwicz criterion, full insurance is very often optimal even under a non zero insurance loading. These results may seem conflicting. They nevertheless have a common basis: they focus on insurance in isolation. The optimal insurance coverage is determined in a single risk setting where consumption and portfolio decicions are not considered. In other words, previous studies assume either explicitly or implicitly that the insurance decision is perfectly separable from any other decisions. This strong assumption is obviously questionable. As recently pointed out by Doherty and Schlesinger [1983], the result by Smith and Mossin only holds if insurable and non insurable losses are non positively correlated. For a positive correlation between these losses more than full coverage is requested by risk averse individuals operating in incomplete markets.

On Optimal Insurance Purchasing

Briys and Louberge indicate that individuals who behave according to the Hurwicz criterion may prefer full insurance to partial insurance even when a deductible arrangement is available. We show that once the analysis is extended beyond risk neutral individuals, a partial insurance decision is not ruled out, while full or no insurance are other possible outcomes. Determination of optimal insurance coverage has long been a central issue in insurance economics. A well-known proposition on this issue is that a risk-averse individual will prefer a policy with a deductible to a policy with full coverage (Arrow (1963) and Mossin (1968) to name a few). This long established result, however, appears to be in conflict with everyday observation. In reality, people seldom ask for deductibles in their insurance policies. Recently, Briys and Louberge (BL) (1985) resolved this theory-reality conflict by showing that individuals who behave according to the Hurwicz criterion will not automatically purchase an insurance policy with a deductible. In many cases these individuals will choose insurance policies with full coverage. However, the BL (1985) result is limited due to their assumption about individual insureds' attitude toward risk. They deal only with risk neutral individuals. We re-examine the same problem in BL (1985) after risk aversion is incorporated into the Hurwicz criterion. Section I considers the general choice problem of a deductible on an insurance policy. In Section II an example in a simple two-state world is presented. A brief summary is given in Section III. I. Optimal Insurance Coverage Consider an individual endowed with two types of assets; non-risky assets denoted by A and risky assets denoted by L. The individual incurs a random *Hyong J. Lee is Senior Research Fellow with the Korea Institute for Defense Analyses. **Professor, School of Business, University of Kansas, Lawrence, KS. This research was completed when Hyong J. Lee was Assistant Professor of Finance at Texas Tech University. This content downloaded from 207.46.13.57 on Mon, 08 Aug 2016 06:26:32 UTC All use subject to http://about.jstor.org/terms 146 The Journal of Risk and Insurance loss X, 0 s X s L with probability density f(x) so that the probability of the occurrence of loss is L ir =ff(x)dx. 0 If the risky assets are not insured, the distribution of terminal wealth, W, over the states (loss, no loss) will be (A+L-X, A+L). In this case the terminal wealth, W, over the states (minimum wealth, maximum wealth) will be (A + L L, A + L). If the individual purchases insurance with a deductible D, 0 < D < L, an insurance premium P(D) is paid from the initial wealth. The distribution of terminal wealth over (loss, no loss) will be (A + L P(D) min(X,D), A + L P(D)). Assuming an insurance company asks the insurance premium P(D) based on the expected insurance payments and a factor loading to cover necessary expenses and profits, X, we have L P(D) = (1 + X) (x-D)f(x)dx. D The terminal wealth vector (A + L P(D) D, A + L P(D)) over the outcomes (minimum wealth, maximum wealth), in the uninsured state is (A + L P(L) L, A + L P(L)) = (A + L L, A + L), since P(L) = 0. In sum, the terminal wealth W is determined by the amount of the deductible and a random loss,

Optimal Insurance Coverage: Comment

Do you believe them? Suppose we restrict both of the above propositions to cases where the probability of a loss is and where the amount at risk is too large? Both of these propositions follow easily from a recent communication by Szpiro (1985) in this Journal. If Szpiro's results are to be believed, we only need to know the price of insurance and the measure of absolute risk aversion at the individual's level of initial wealth to determine what level of coverage is optimal -or at least as an approximation without any of great magnitudes. One of the culprits in obtaining the above results turns out to be the author's mimicking of Pratt's approximation of his risk premium formula using Taylor expansions. Szpiro argues that reasonable levels of the amount at risk, L, and the loss probability, q, do not yield serious errors in calculating his results. For his examples, Szpiro provides the magnitudes of the errors in estimating utility at post-loss wealth levels when we expand utility around the pre-loss wealth level. Even when these errors are small, what we should really care about is how close our predicted optimal insurance level is to the true optimal level. If L is small in relation to the pre-loss wealth level, W, then any predictor will have a small error as a percent of W. If W is $1 million and L is $1 thousand, then a predictor of insuring half the amount at risk has a maximal error in coverage (as a fraction of W-L) of about 0.0005. Although this may be a good predictor of final total wealth, it is doubtful that it is a good predictor of the level of insurance coverage.

Optimal Insurance Coverage: Reply

The introduction to my original article notes that the results are explicit, albeit approximate, expressions for the amount of coverage (p. 704, emphasis added). Previously existing exact results sometimes offer meagre guidance for real-life decisions. Schlesinger is wary of approximations and cites a numerical example to support his position. Certainly most of us would prefer a better approximation; but this requires knowing either the utility function itself (in which case there would be no need for an approximation) or, at least, higher order terms in the Taylor expansionterms that relate to preference for skewness and higher moments. In this instance, however, 'multivariate' measures of risk aversion would have to be assessed (i.e., in terms of variance, skewness and higher moments) and the expressions would become much more complicated. The original article must be read in proper context. Consider, for instance, the following propositions: if a consumer becomes more risk averse, additional will be purchased, and [if absolute risk aversion decreases with wealth], an individual with less wealth . . . will purchase more insurance (Schlesinger pp. 469, 471). Such statements are very nearly tautological and cannot be put to operational use; they do not indicate by how much the insured amount increases for a given increase in wealth. One cannot even specify the order of magnitude. Does the insured amount increase by one hundred dollars, or by a million dollars? Theorems which simply restate the obvious do not add much to the understanding of complex phenomena. My original paper permits approximate calculations, thereby adding insight and operational value. Table 1, which applies my formula to Schlesinger's example, contrasts the approximations with the qualitative results, as well as with the exact numerical solutions. Note that in the qualitative approach no distinction can be drawn between the magnitudes of change for case 2 ($753) and for case 3 ($378,088). Hence, for example, no aid is given to companies who would like to know

Insurance and Consumption: The Continuous Time Case

During the last twenty years, a large body of literature devoted to the determination of the optimal insurance coverage has developed. The seminal works by Arrow (1963), Smith (1968) and Mossin (1968) provide the corner stone results of insurance economics. But surprisingly enough this literature only concentrated on the case of a single insurable asset, thus implicitly assuming a perfect separability of the insurance decision. Recent research has shown that this picture must be modified when insurance demand is jointly analyzed with portfolio or consumption decisions. In the portfolio case, insurance coexists with background risk. The optimal level of insurance coverage is affected by covariance effects between the potential loss and the socalled background risk. (see e.g. Doherty and Schlesinger [1983, Mayers and Smith (1983)1). The consumption case has been less intensively examined. As recently noted by Dionne and Eeckhoudt (1984) 'the literature on the demand for damage insurance and that on optimal saving seem to have followed rather independent paths. Moffet (1977), Falciglia (1980) and Dionne and Eeckhoudt (1984) have derived some propositions about the possible interactions between consumption and insurance decisions but their works constitute notable exceptions to the general course of research in optimal insurance. The specific purpose of this note is to extend the three last mentioned contributions and more precisely to analyze the consumption and insurance decisions in a continuous time setting using the methods of Merton (1969). Sufficient conditions for a separability of the insurance decision will be given. Separability means that decisions can be made sequentially without any feed back on each other.