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On January 1, 2006 a new mandatory basic health insurance will be introduced in the Netherlands. One aspect of the new scheme is that the insured can choose to have a deductible. This option should increase the individual responsibility and reduce moral hazard. In the new scheme, a risk equalization system is aimed at avoiding preferred risk selection and insolvency of insurance companies with a relatively high-risk pool. A crucial issue with respect to a voluntary deductible in this type of social health insurance is whether the premium rebate should be community rated or risk rated. The Dutch government has chosen the former, which means that the premium rebate will be independent of health status and risk. Our analysis shows that, in a situation with "accurate" risk equalization, a community-rated premium rebate could lead to an adverse selection spiral. Over time, this spiral results in none of the insured taking a deductible and thus no reduction in moral hazard. INTRODUCTION On January 1, 2006 a new mandatory basic health insurance will be introduced in the Netherlands for the whole population. One aspect of the new scheme is that the insured can choose to have a deductible. (1) One of the issues related to the practical implementation of a voluntary deductible is whether the premium rebate should be community rated or risk rated. The Dutch government has chosen the former, implying that the premium rebate will be independent of health status and risk. In this article we analyze the consequences of a community-rated premium rebate in combination with the risk equalization system that will be functioning in new basic health insurance. Risk equalization is aimed at avoiding subtle forms of preferred risk selection and upward premium spirals for insurers with a relatively high-risk pool. The need for risk equalization comes from the ban on premium differentiation and the obligation for insurers to accept every new applicant. As part of risk equalization, the insurers receive a compensation for the relatively high-risk individuals in their insurance pool and pay a contribution for the relatively low risks. On the one hand risk equalization among insurers can reduce incentives for cream skimming. On the other hand it could be problematic with regard to the introduction of a voluntary deductible. Without risk equalization, the premium rebate for a deductible does not only reflect cost sharing, a reduction in moral hazard, and a reduction in administration costs, but also the effect of adverse selection by insured. Adverse selection is the phenomenon that within each risk group of insured who pay the same premium for full coverage, the relatively healthy will opt for a deductible whereas the relatively unhealthy opt for full insurance. Adverse selection thus results in market segmentation, which allows the insurer to ask different premiums from the two groups of insured. Competition will force the insurers to ask costs-based premiums. This implies that they increase the premium for the (relatively unhealthy) insured choosing full coverage and decrease the premium (i.e., further increase the premium rebate) for the insured taking the deductible. So, without a risk equalization system the difference in expected expenditures between the two groups of insured, which results from their different health status, can be reflected in the premium rebate for the deductible. However, with an accurate risk equalization system the insurer receives a compensation for the unhealthy insured and has to pay a contribution for the healthy insured. So the difference in expected expenditures between the two groups, which is the result of their different health status, is then fully reflected in these compensations and contributions. This holds true both in the case without a voluntary deductible and in the case with a voluntary deductible. So the differences in health status between the two groups cannot be reflected in the premium rebate for a deductible. …

The theory of optimal insurance with noninsurable background risk has previously been examined under the assumption that the background risk and insurable loss random variables are additively separable within the agent's utility function. In this study, these two variables are nonseparable because marginal insurable loss is positively dependent on the outcome of the background risk variable. This positive dependence implies that background risk has a smaller marginal impact on profits at higher levels of insurable loss. The optimal contract is shown to require coinsurance above a deductible minimum when the agent is prudent. This result is opposite Gollier's (1996) finding that a "disappearing deductible" is optimal when a higher level of insurable loss implies a more risky distribution of the background risk variable. With relatively weak restrictions on the various functions, the optimal indemnity schedule is a convex function of loss and the slope of this function is inversely related to certainty income.

Rothschild and Stiglitz (1970, 1971) pioneered the study of how an increase in risk affects the demand for a risky asset. Gollier (1995) first identified a necessary and sufficient condition for unambiguous comparative statics for demand under transformations of the asset's probability distribution function. In this article, the authors examine the necessary and sufficient conditions for unambiguous comparative statics for insurance demand for coverage deductibles and policy limits, when the loss variable undergoes certain classes of transformations. The authors show that, if a transformation is both mean-preserving above (below) the optimal deductible (policy limit), and probability-preserving above (below) the optimal deductible (policy limit), then the necessary and sufficient condition is given by first-order stochastic dominance below (above) the optimal point.

INTRODUCTION In recent years, profitability has returned to the workers' compensation insurance market. Insurers have experienced a dramatic improvement in their combined loss ratios and the frequency of workers' compensation indemnity claims has been on the decline.(1) Nationally, combined ratios after dividends were greater than 110 percent for every year in the 1983-1992 period. They dropped to 109 percent in 1993, 101 percent in 1994, and 98 percent in 1995. At the same time, the use of deductibles for workers' compensation insurance has more than doubled since 1992, accounting for approximately $8 billion in premium credits in 1994.(2) Though state workers' compensation reforms are often cited as the major reason for the improved underwriting results, this article explores the possibility that deductibles, through the creation of incentives for employers to prevent injuries and control the cost of claims, also have contributed to the downward turn in loss ratios. A number of safety and loss control incentives are inherent in the state workers' compensation systems. These incentives include statutory waiting periods for income benefits, income replacement rates, the workers' compensation experience rating system, self-insurance, retrospective rating,(3) the deregulation of insurance rates, and insurance deductible options, and they can potentially impact the behavior of both employees and employers. For instance, an increase in the real income replacement rate creates an incentive for employees to stay off work longer and it also creates an incentive for the employer to invest more resources in safety to avoid these higher benefit payments. To date, few empirical studies have involved risk-sharing insurance arrangements and their impact on employer costs. Although some studies found no evidence that experience rating workers' compensation premiums reduces injury rates (Chelius and Smith 1983 and 1993) or the duration of lost time from workplace injuries (Krueger 1990, Kralj 1995), most studies support the notion that loss control incentives reduce employer costs. Ruser (1985 and 1991) and Worrall and Butler (1988) found that the closer firms came to being fully experience rated(4) (i.e., their premiums are based solely on their own experience as opposed to their class experience), the lower their workers' compensation claim rates. Thomason (1993) found the probability of permanent partial disability to be lower for experience-rated employers obtaining insurance through private-sector insurance carriers. A study that tested the effects on fatality rates in Canada of a transition from a flat or manual premium rating system to an experience rating system provides strong evidence to support the hypothesis that experience rating improves workplace safety (Bruce and Atkins 1993). Moore and Viscusi (1990) found that as benefit levels increased, the risk of work-related deaths decreased. They concluded that the use of death rates, as opposed to injury rates, eliminates moral hazard problems from the analysis, and the findings of their study provide strong evidence that employers do invest in safety when faced with the risk of higher costs. Victor (1985), utilizing a simulation model to measure the injury prevention and loss control incentives associated with the experience-rating system, provides compelling arguments that experience rating may provide greater incentives than self insurance, which is not subject to experience rating. Among the loss-control incentives cited, loss-sensitive insurance arrangements such as insurance deductibles seem to offer employers the greatest direct benefit from investing in injury prevention programs and improved claims management strategies. Large deductible policies, which incorporate elements of self insurance and experience rating, may prove to have the most powerful safety/loss control incentives yet. In theory, a firm which selects a large deductible policy (typically ranging from $100,000 to $5 million per accident)(5) takes on the majority of the policy risk and should have a strong incentive both to prevent injuries from occurring and to manage claims in a cost-effective manner. …

This article examines the interaction between the demand for deductible insurance and the demand for the asset that can be insured. A model where the decision-maker can choose both the insurance level and the amount of the risky asset to hold in the portfolio is used. Allowing simultaneous determination of the demand for an asset and the level of the deductible alters the comparative statics of demand for both the deductible level and for the asset. This is illustrated by determining the impact of shifts in nonrandom initial wealth, the price of the risky asset, and the price of insurance.

INTRODUCTION When an individual faces a single source of risk, the optimal insurance contract contains full insurance above a deductible when the premium only depends upon the actuarial value of the contract. This result was established not only under expected utility (Arrow 1965; Raviv 1968) but also for any non-expected utility framework that satisfies the second order stochastic dominance (SSD) property (see Machina 1995). Besides Gollier and Schlesinger (1996) have shown that the superiority of the deductible policy results from its better ability to reduce the variability of final wealth for a given cost of insurance. More specifically they prove that any insurance contract of equal cost will be dominated, in the second order sense, by one with a deductible policy. Although this result turned out to be very important in many respects, its practical relevance is limited by the assumption that there is only one source of uncertainty. In practice, people and firms face many simultaneous sources of risk and they have to decide both on the amount of their total insurance budget and on its allocation among the different sources of risk. Recently, Gollier and Schlesinger (1995) showed that people would like to cover multiple risks by purchasing an "umbrella" contract covering all risks at the same time, with full insurance above a deductible on the aggregate loss. Unfortunately, such an umbrella contract is rarely offered in reality. As a consequence, decision-makers have to consider the optimal level of coverage for each risk separately. In this context, Gollier and Schlesinger prove that it is optimal to purchase contracts with full insurance above a deductible for each peril. In this paper, we extend all these results by considering rules that should be followed in choosing the deductible level for each risk in terms of its characteristics. To be more specific, we consider an individual who faces two pure risks, [Mathematical Expression Omitted] and [Mathematical Expression Omitted] which differ both in terms of the probability of the loss occurrence ([p.sub.1] and [p.sub.2]) and in terms of the level of potential loss ([L.sub.1] and [L.sub.2], respectively).(1) For instance, [Mathematical Expression Omitted] might be a low frequency risk ([p.sub.1] low, e.g. and earthquake risk or the risk of cancer in the year to come for a young healthy individual) with a high potential loss ([L.sub.1]). On the contrary, [Mathematical Expression Omitted] may stand for a risk likely to materialize ([p.sub.2] is high) but with moderate or mild severity ([L.sub.2] is not too large). Given that insurance is offered separately for each risk, how should the optimal deductibles ([[D.sub.1].sup.*] and [[D.sub.2].sup.*]) compare to each other? Intuition suggests that one should cut down the large losses and select deductible levels so that the net loss (the loss minus the indemnity) should be equal for each risk. After all, the Arrow's deductible result suggests that it is optimal not to indemnify any loss below a limit that is the same for any source of risk affecting the wealth of the insured person. If this extension of the Arrow's result were correct, one would select equal levels of deductible for each risk. In this paper we show that such a policy is not, in general, optimal despite it's a priori intuitive appeal. Reducing the deductible has two contradictory effects on expected utility. First, it increases the premium. Second, it increases the indemnity in case of loss. It is a priori not clear how a change in the probability of loss affects the optimal deductible which is a best compromise between these two adverse effects. In fact, in most cases, a strong "probability effect" will be at work inducing the individual to choose low deductibles (hence, high coverage) for those risks that are very unlikely. Hence, our paper tends to support the sometimes held view that insurance is the most appropriate risk management tool for low frequency risks. …

MORAL HAZARD AND WORKPLACE ABSENTEEISM Moral hazard arises whenever the difficult-to-monitor activities of workplace participants, who are protected by disability systems, increase the insurance liability of firms' owners. The owners of profit-maximizing firms would like to write enforceable contracts that make the whole disability process "objective" in the sense that it provides only the appropriate level of medical care. Of course, such contracts are difficult and therefore costly to monitor. Hence, the costs of monitoring workers, administrators, and health care providers reduce the amount of monitoring that firms would undertake in a full information world. Those participating in the disability process realize that this informational asymmetry exists and can turn the situation to their advantage. Moral hazard increases workplace absenteeism when such informational asymmetries are used to increase the frequency or duration of workers' compensation claims. These additional claims are known as "moral hazard" because the costs of extra usage (above that in a full information world) are borne by all the workers and owners of the company indirectly through higher cost for health and insurance benefits (with offsetting reductions in wages(1) and possibly company profitability), while the benefits are captured by individual participants. Unfortunately, since all face the same situation, individuals have some incentive to use the system more than they would have if they were paying with their own money, or if their behavior could be monitored without cost. Overwhelmingly, the extant literature indicates that workers respond to economic incentives provided by workers' compensation disability payments.(2) In the absence of moral hazard, a change in benefits would not cause a systematic change in the claims. However, claim severity and frequency tend to increase as benefits increase and as the waiting period prior to the receipt of wage benefits (a de facto deductible) falls.(3) Despite this evidence, the potential impact of moral hazard on real productivity has been ignored because moral hazard is primarily seen as a change in the incentive to report an accident, shifting costs from one type of employee benefit to another. For example, an employee may claim that a given condition arose from a job injury and seek workers' compensation benefits because their real health condition (broadly defined) does not qualify for short-term disability compensation. An extreme case of "claims reporting" moral hazard is overt fraud in which a worker facing a pending layoff claims injury benefits when no injury was incurred, either on or off the job. In both of these examples, employee claims shift costs: in the first case, costs are shifted between group health/paid sick time and workers' compensation medical/indemnity; and in the second case, costs are shifted between unemployment insurance and workers' compensation indemnity. While such benefits-induced cost-shifting may not have much impact on real output, increased workplace absenteeism will lower output to the extent that the firm specific capital is important. As statutory benefits increase, absenteeism rises with a concomitant loss of firm specific capital. This loss in firm specific capital causes output to fall. While firm-specific capital can not be directly observed and estimated, the impact of lost firm-specific capital on output can be inferred indirectly. Since higher statutory benefits increase absenteeism (see prior research cited in footnote 3), evidence that higher statutory benefits also lower output provides indirect evidence on the importance of lost, firm-specific capital. This paper measures changes in output as statutory benefits rise by embedding the benefits replacement rate in a Cobb-Douglas production function. In the next section, prior research on the importance of firm specific capital is discussed. In the third section, the effect of higher disability pay on value-added in manufacturing, using an industry longitudinal sample, is estimated. …

Medical savings accounts (MSAs) belong to a larger class of incentive-based health care plans. Using a model that allows the consumer to invest in healthy activities, we examine the efficiency properties of incentive plans and compare them to traditional full- coverage and deductible plans, under both experience rating and community rating. The model also is extended to include utilization of preventive health care. Properly constructed incentive plans have the capacity to induce socially efficient levels of healthy activities and preventive care, raising the expected wealth of consumers without reducing insurers' profits.

This article focuses on two cornerstone results in insurance economics: Mossin's Theorem on the optimality of full versus partial coverage, and Arrow's Theorem on the optimality of straight deductible policies. Both of these results are examined in a model assuming only risk aversion, and not necessarily expected-utility maximization. The results also are examined with the inclusion of a noninsurable background risk. Arrow's result is robust enough to hold in all of these situations. Mossin's result is shown to hold with a slight weakening, to account for possible "nonsmoothness" of preferences in non-expected-utility models.

This article considers the problem of the optimal form of insurance contract when the indemnity can be made contingent only upon an imperfect signal of the final wealth of the insured person. This is the case, for example, when some risks affecting wealth cannot be insured by the market. The optimal contract is shown to contain a straight deductible when the distribution of the uninsurable risk is independent of the size of the loss of the insurable asset. Under a plausible condition on the utility function, the existence of an uninsurable risk reduces the optimal deductible. If the risk on the uninsurable asset increases with the size of the loss of the insurable asset, the optimal contract contains a "disappearing deductible" if the policyholder is prudent (u"' > 0). This model also is useful for analyzing the design of optimal insurance when losses are observed by the insurer with an error.

It is known that if random prospect Y stochastically dominates prospect X by the second degree, then all risk-averse agents prefer to hold Y and some random initial wealth Z rather than X and the same initial wealth Z, if Z is statistically independent of both X and Y. We give more general conditions for the invariance of the stochastic dominance ordering to initial wealth to hold: that X be more positively dependent on Z than Y is, and that either X and Z or Y and Z be positively dependent. Applications to the choice of an insurance deductible and of insurance coverage are given.

Introduction The underinvestment problem has been well known in finance since the appearance of Myers's (1977) seminal work on "The Determinants of Corporate Borrowing." Myers considers a situation in which the firm has an outstanding bond issue now and may invest in a positive net present value project then.(1) The payoff on the investment project is risky now but not when the decision is made then. The underinvestment occurs because there are realized project payoffs that cover the investment expenditure then but not the investment expenditure and the promised payment on the bond issue. If the firm did invest in the project under these circumstances, then all net proceeds would go to bondholders. Hence, management that acts in the best interests of its shareholders never makes such an investment decision. Given rational expectations, the stock market value of the firm is larger now because of management's decision then not to invest under the conditions just described. The Mayers and Smith (1987) article, "Corporate Insurance and the Underinvestment Problem," builds on Myers's work. Mayers and Smith have provided an interesting and intuitively appealing discussion of the role played by insurance in bonding the corporate investment decision. Schnabel and Roumi (1989) have extended the model proposed by Mayers and Smith to a consideration of the effect of a safety, or premium loading. Although Mayers and Smith assert that the gains associated with resolving the underinvestment problem are enjoyed by shareholders, their analysis does not explicitly provide the mechanism to show how this is accomplished.(2) This article provides two routes out of this difficulty. Because the Mayers and Smith and Schnabel and Roumi models do not allow the promised debt payment to change after insurance is introduced, it follows that the debt issue raises more money with the insurance covenant than without it. Hence, current shareholders get the additional value if the manager sets a dividend payment now equal to the difference between the debt values with and without the insurance covenant net of the insurance premium. We will refer to this later as the "cum dividend" interpretation of the Mayers and Smith and Schnabel and Roumi models. However, rather than confound financing and dividend decisions unnecessarily, we propose an alternative solution that requires the imposition of a financing condition. This allows us to show that an insurance deductible may be chosen that enables the manager to also reduce the promised debt payment and restore full value for the current shareholders. Our analysis explicitly shows that an insurance covenant can be designed that allows current shareholders to capture the gain in value and that the gain in value equals the agency cost of the underinvestment problem. We believe that the introduction of a financing condition rather than a dividend policy is more instructive, since it allows us to focus more neatly on the structure of the bond/insurance financing package. This difference becomes particularly important when premium loading is introduced, because the ability to adjust the promised debt payment allows the manager to reduce the cost due to premium loading. The Underinvestment Problem Although limited liability confers a number of important economic benefits, it also has its costs (see Easterbrook and Fischel, 1985). By creating an asymmetry between the costs and benefits of risky activities, limited liability causes bondholders and shareholders to have incompatible incentives whenever the corporation's debt is subject to the risk of bankruptcy. Mayers and Smith identify a situation in which the conflict of interests is so severe that an underinvestment problem occurs. In their model, underinvestment occurs in the sense that shareholders choose to forego a positive net present value investment that, in the absence of bankruptcy risk, would be undertaken. In order to focus their analysis on the role of insurance in resolving this problem, Mayers and Smith assume that the only source of uncertainty for the firm is whether it will suffer a property loss. …

Introduction Both theoretical and empirical studies have found that adverse selection in an insurance market reduces consumption of insurance by low risk insureds. The theoretical works of Akerlof (1970), Rothschild and Stiglitz (1976), Miyazaki (1977), and Wilson (1977) describe separating equilibria, where high risks purchase a policy with higher coverage than the policy that is purchased by low risks. Miyazaki extends the separating model to allow cross-policy subsidization, resulting in a wealth transfer from low risks to high risks. In addition to a separating equilibrium, Wilson describes a pooling model where high and low risks purchase the same policy so that low risks actually subsidize the insurance purchases of high risks. The empirical evidence generally supports the predictions of these theoretical models. Beliveau (1981) found that adverse selection leads to reduced insurance consumption by low risks in the life insurance market, and Dahlby (1983) and Puelz (1990) found similar results in the automobile insurance market. Browne and Doerpinghaus (1993) found evidence of adverse selection in the market for private supplemental medical insurance for the elderly. In another study, Browne (1992a) found that low risk individuals had less complete coverage in the individual medical insurance market than in the group market, where adverse selection is believed to be less problematic. Additionally, Browne found evidence that low risks subsidize the insurance consumption of high risks in the individual medical insurance market. His results could be explained by the presence of adverse selection or by low risks receiving more coverage in the group market than they would choose if they were purchasing insurance on their own in the individual market. His study does not address whether Miyazaki's separating model or Wilson's pooling model better characterizes the individual medical insurance market. This study extends empirical investigation of the individual medical insurance market and directly tests whether there is reduced consumption of insurance by low risks, whether a separating or pooling model best characterizes the market, and whether cross-subsidization from low to high risks occurs. The Data, the Model and Hypotheses The Data and Potential Tests The data used in the study are from the National Medical Care Expenditure Survey (NMCES) conducted during 1977 and 1978.(1) Only families for which the primary insured had nongroup, individually purchased medical expense insurance and for which complete insurance policy data are available are included for analysis. The data contain extensive demographic information on those insured, including age, sex, marital status, income, and geographic location. In addition, the data contain information on any activity limitations that the insured may have as well as a measure of self-reported health status.(2) In the study, insureds are classified as being low risk if the self-reported health status of family members is excellent, good, or fair, and they are classified as high risk if the self-reported health status of any family member is poor.(3) The health status data were collected ex ante in the first of the five rounds of interviews conducted during the two-year period covered by the study so that health status information does not reflect the insured's revised expectations based on actual claims history during the covered period.(4) Using the demographic data, the activity limitation variable, and the self-reported health status variable allows for not only a test of risk-induced insurance purchase but isolation of the component of risk (self-reported health status) about which insurers are by definition asymmetrically informed.(5) Demographic and health status information from the sample are summarized in Table 1. TABULAR DATA OMITTED Detailed data on the policies purchased by individuals (such as policy premiums, deductible amounts, out-of-pocket maximum limits, and policy coverage limits) allow for measurement of the completeness of coverage provided by each insurance policy. …

Using Arrow's (1971, 1974) model of the optimal (endogenously determined) deductible, this article examines the impact of increases in risk (Rothschild-Stiglitz mean-preserving spreads and Diamond-Stiglitz mean-utility-preserving spreads) and first-order stochastic dominance shifts in the distribution function of losses on the optimal insurance contract between a risk-averse decision-maker and a risk-neutral insurer. In particular, it is found that a first-order stochastic dominance shift in the distribution of losses has generally an ambiguous impact on the deductible. This result raises some doubt about the asymmetric information (signalling) literature's identification of low (high) risk with smaller (greater) demand for insurance.

The Tax Deductibility of Premiums Paid to Captive Insurers: A Risk Reduction Approach ABSTRACT This article analyzes the tax deductibility of the insurance premiums paid to a captive insurer by its parent company. A risk reduction approach is used and differs substantially from those proposed in the previous literature and from what has been upheld by the government. Some of the results of this study are conistent with the government's position on this issue or the results of the previous literature and some are not. The analysis suggests that the degree of tax deductibility be determined by the relative contribution of the parent's own risks to the parent's total risks, including the parent's own risks and the outside risks underwritten by the captive. It is shown that the higher the relative contribution by the parent's own risks, the lower the degree of tax deductibility. About one-third of the largest 500 companies in the nation have formed captive insurers. While some of the corporations established captive insurers to secure liability coverages that were not provided by conventional insurers, many did so to save premium costs or to help stabilize the parents' earnings. As a result, the captive insurance industry experienced substantial growth in the 1970s. This growth, however, was halted at the beginning of the 1980s. One of the factors contributing to the setback was the Internal Revenue Service's (IRS) and the Tax Court's tough rulings against deductibility of premiums paid to captives by parent companies. Premiums paid to unaffiliated insurers are tax deductible, but those to captives are often questioned. In general, the Tax Court and the IRS have taken the same position with regard to the non-deductibility of premiums paid to captives which do not write unrelated business. However, the IRS's recent ruling 88-72, on premiums paid to captives writing substantial outside risks, conflicts with the Tax Court's decisions (the Gulf Oil case of 1987) and even the IRS's own General Counsel memorandums (GCMs 35483 and 38136). In Revenue Ruling 88-72, the IRS held that no level of outside risk will result in risk shifting, therefore premiums paid to captives writing outside risks are not tax deductible.(1) On the other hand, the IRS in GCM 35483 and the Tax Court in the Gulf Oil case ruled that premiums paid to the captive could be deductible, if the captive underwrote substantial external risks, indicating that the parent could thus pass its risk to unrelated policyholders insured with the captive. Smith (1986) investigates the issue of risk shifting and provides a quantitative analysis utilizing a portfolio approach. He maintains that the effect of risk shifting due to the captive underwriting outside risks should be gauged by the ratio of the parent's total risk with and without outside risks. Following this rationale, he concludes that the parent's total risk could increase by underwriting outside business which "has a much larger variance than parent's risk." He therefore supports the IRS's comments that writing outside risks does not constitute a risk shifting from the parent to outside policyholders. Hofflander and Nye (1984) examine the taxation of self-insurance and premiums paid to captives. They propose that if a firm's expected net income and the variance of the net income do not vary under different risk management strategies, then the IRS should hold the same position with regard to tax deductibility. They show, among other cases, that the following two arrangements satisfy their criterion: (1) a firm self-insures; and (2) the firm insures its own risk with a conventional insurer and its captive insures outside risks with the same loss distribution as itself. They conclude that if the IRS allows deductibility of premiums paid to a conventional insurer in arrangement (2), then the firm's self-insurance fund should also be deductible. …

Effect of Ambulatory Surgery Policy Provisions on Medical Expense Insurance claims Introduction From 1965 to 1987 national medical expenditures rose at an annual rate of approximately 10 percent. (1) The proportion of Gross National Product spent on national medical care increased from about 6 percent to more than 11 percent. In 1987 private health insurance funds paid for 32 percent of national medical care; 39 percent of national health dollars went to hospital care alone (U.S. Department of Health and Human Services, 1988). These figures demonstrate the continued real growth of health care costs in the United States, the large absolute dollar amounts associated with providing medical services, the importance of the private sector's role in funding care, and the effect of hospitalization costs on total medical spending. Economic theory and empirical evidence indicate that costsharing by insureds decreases total medical claim costs. (2) Traditionally insurance plans have required insureds to share the cost of medical services through policywide deductible and coinsurance provisions, up to a designated out-of-pocket maximum amount. Since the early 1980's insurers also have offered policies with specific surgical costsharing provisions in addition to those in effect policywide. Higher deductible or coinsurance requirements for inpatient surgery relative to outpatient surgery are expected to induce the substitution of outpatient for inpatient surgery, and decrease surgical claim costs and in turn total medical costs. This study examines policies which encourage use of ambulatory surgery and investigates the effectiveness of policy design in reducing surgical and total medical care claim costs. The Rand Health Insurance Experiment offers the most comprehensive evidence to date on the effect of insurance policy design on the demand for medical services (Manning et al., 1987). The Rand experiment finds that policywide costsharing decreases demand for total medical services. Evidence also indicates that lower costsharing requirements for outpatient care, surgical and nonsurgical, does not decrease total medical expenditures by reducing inpatient expenditures. However, neither the Rand experiment nor nonexperimental demand studies address the effect of policy design which encourages usage of ambulatory surgery on insurer claim costs. Anecdotal evidence is not conclusive on the effect of policy design which encourages use of ambulatory surgery on insurer claim costs. Most reports exhibit little economic and statistical rigor. Many use firm or area specific data insufficient for producing results which may be generalized to a larger population. The study here improves on previous non-experimental demand analysis through use of a large data base covering 197,911 insureds from across the nation. The article begins with the theoretical model that provides the basis for the empirical work. The model considers the insured's demand for medical and surgical services, both outpatient and inpatient, as a function of policy design and demographic characteristics. The econometric model and the data set used for testing are described, and the empirical results are reported. The study concludes with a brief summary of the major findings. Research Methods Theoretical Model of Demand In Grossman's (1972) model of the demand for health, individuals inherit an initial stock of health which depreciates over time. Health stock is treated as a durable commodity produced by consumers which may be increased through investment of monetary resources and time in medical services. The model implies that individuals partly determine the length and quality of their lives through their health investment decisions. That is, the insured chooses or demands quantities of health (and medical care) in order to maximize utility, subject to budget constraints. …

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

A policyowner who wishes to select a deductible faces a problem in decision making under uncertainty. A question that the policyowner needs to answer is whether s/he is willing to assume the risk of possible large uncovered losses in return for the premium savings associated with a larger deductible. Proposed in this article is a method for applying risk-return analysis to deductible selection problems. A theoretical approach to selecting optimum deductibles is developed in this article. The approach consists of analyzing both risk and expected cost to select an optimum deductible amount. The method proposed is similar to a type of risk-return analysis used as a corner-stone of financial theory. Limitations of the method are noted and distinguishing characteristics of deductible selection problems are outlined.

Corporate Insurance and the Underinvestment Problem: An Extension In an insightful paper in this journal, Mayers and Smith (MS)[1] posit corporate insurance as a device to control the underinvestment problem which occurs when a firm experiences a casualty loss. Such a loss produces a Myers-type[2] option feature in affected assets because the assets' value depends on further discretionary investment. MS show that in the presence of risky debt, shareholders can have incentives to forego the investment required to rebuild the affected assets even though the investment may have a positive net present value. No such incentive exists in the absence of risky debt. Thus, the stipulation of corporate insurance coverage as a bond covenant is a way of controlling this type of perverse incentive. In their model, MS assume actuarially fair insurance policies. Given this assumption, they show that if the firm's debt is risky, it is always optimal to take out insurance coverage. In terms of their symbols, if F, the promised payment to the bondholders, exceeds the minimum possible value of [V.sup.*] - I(S) (where [V.sup.*] is the terminal value of the firm with no loss and I(S) is the investment required to restore the value of the firm conditional upon the state of nature, S) thus rendering the firm's debt risky, then it is optimal to take out coverage. On the other hand, if F is less than the minimum possible value of [V.sup.*] - I(S), the firm's debt is riskless and the firm is indifferent between obtaining and foregoing actuarially fair insurance coverage. Thus, the critical value of F equals the minimum of [V.sup.*] - I(S). This extension focuses on the implications of assuming insurance policies whose premiums incorporate a safety loading. It is shown that even if the firm's debt is risky, it may be optimal to forego insurance coverage. There is a new critical value of F, which exceeds the minimum value of [V.sup.*] - I(S), below which it is optimal to forego coverage and above which it is optimal to obtain coverage. In keeping with the development in MS, this extension assumes a binary choice situation where the firm decides to either insure or not to insure. Thus, the elements of the insurance contract, e.g. deductible, are not decision variables. This extension does not change the basic result of MS regarding the use of insurance policies to control the underinvestment problem occasioned by a casualty loss. The aim here is merely to formalize and more fully articulate the summary comments in MS regarding the impact of a positive loading fee, to wit, "a loading fee for the insurance policy reduces the benefits of additional debt in the capital structure ... as long as the benefit is greater than the loading fee, the additional debt should be issued and an insurance policy purchased". Following the development in MS, consider an insurance policy whose indemnity is I(S) with a deductible of I(Sa). Sa is the state of nature where [V.sup.*] - I(Sa) = F. Assuming a safety loading of [Lambda], the premium on such a policy would be given by (1) [Mathematical Expression Omitted] where g(S) is the state contingent price paid now for delivery of a dollar at the terminal date conditional on state S. Define [V.sub.u], [V.sub.o] and [V.sub.Lambda] as the present values of the unlevered firm, the levered firm with no insurance coverage and the levered firm with insurance coverage which incorporates a safety loading of [Lambda]. Define L(S) as the amount of the casualty loss or the reduction in the terminal value of the firm conditional on state S. Consistent with MS, assume that rebuilding is a positive net present value proposition, i.e. L(S) - I(S) [is greater than] O for all states S. Thus, (2) [Mathematical Expression Omitted] where Sc is the critical state of nature below which casualty losses are incurred and above which no such losses are incurred. (3) [Mathematical Expression Omitted] and (4) [Mathematical Expression Omitted] The first two terms of equation (4) constitute the payoff to the bondholders when S [is less than] Sa, i. …

This paper examines the choice of an insurance contract when insurance is of the deductible type. This optimal choice is shown to be directly related to the insured's degree of risk aversion. Under certain assumptions, it is shown that an individual with a higher loss probability, a higher degree of risk aversion, or a lower level of initial wealth will purchase more insurance. Conditions for the purchase of full coverage or no coverage are also examined.

The press recently has devoted some space to automobile insurance rates.' Attention has been focused upon increases in insurance premiums. These have been necessitated by rising automobile repair costs and larger injury settlements awarded by the courts. In order to reduce insurance costs to the consumer, some insurance companies have expanded the variety of deductible coverages from $50 and $100 to $200 and $250. The main purposes of this study were to make an estimation of differences in rates for automobile insurance and to make some comparisons among different types of firms. Secondary purposes were to examine any systematic differences in rates when insurance companies were grouped into different categories and to ascertain whether dealers in one brand of automobile offered insurance charges substantially different from dealers in other brands. Comparisons were to be made with a similar study conducted in 1962. The current study found that there were substantial differences in rates for automobile insurance. Insurance companies quoted rates that were significantly lower than those quoted by insurance brokers and automobile dealers. There were no significant differences in rates when insurance companies were classified by size, method of selling, location of home office or type of ownership. The Chrysler-Plymouth dealers and American Motors dealers contacted did not offer automobile insurance. There were no significant differences in rates quoted by Ford and Chevrolet dealers. The range of automobile insurance rates was greater in 1976 than it was in 1962.

A coinsurance clause is an agreement to own an amount of insurance equal to at least a specified percentage of the value of the covered property. Conventional wisdom and established economic theory hold that the required amount of insurance usually should be purchased, but situations in which other strategies prove useful are easily found. Potential interaction between coinsurance and deductible clauses makes the selection problem more complicated than commonly assumed, and pricing anomalies may cause some insureds to fail to meet coinsurance requirements. Implications for pricing and product development are discussed and a possible solution is proposed.

As a risk analysis consultant, I read Professor Schott's article with great interest. While my own work in this area leads me to agree that alternative approaches to Monte Carlo simulation can prove useful, I must disagree with Schott's claim that the Gaussian model is viable. Furthermore, some questionable arguments are given to substantiate the case for the normal approximation. In particular, employing the Kolmogorov-Smirnoff (K-S) statistic as a measure of departure from normality can lead to very misleading conclusions. For example, in Table 1 on page 626, the K-S statistic at the $2,500 deductible level is less than the K-S statistic for either the $500 or $1,000 deductible. That the distribution of annual loss becomes less and less like the normal as the deductible level increases from $500 to $2,500 can be seen definitively by calculating the kurtosis and skewness of the annual loss distribution. These calculations show that the kurtosis and positive skewness both increase as the deductible level increases from $500 to $2,500. Given the parameters of the annual loss distribution, the kurtosis and skewness can be precisely calculated, and provide easily interpreted measurements of departures from normality. As a matter of fact, combining the Poisson frequency with any severity distribution results in an annual loss distribution with larger kurtosis than the normal distribution. Furthermore, if the severity distribution is a lognormal distribution with o in the range (1,2), then any deductible greater than 202 percent of the median loss results in an annual loss distribution which is positively skewed (in the severity distribution underlying Table 1, any deductible greater than $290 leads to a positively skewed annual loss distribution). Thus, any realistic application of the normal approximation to obtaining upper percentiles of the annual loss distribution is bound to lead to underestimation. The normal approximation provides no information regarding the amount of undQrestimation, which can be substantial. Finally, simulations were run using the frequency and severity distributions underlying Table 1, assuming a $2,500 deductible. The first simulation

Optimal insurance coverage has been analyzed by numerous researchers. Arrow [1963], Mossin [1968] and Smith [1968] describe insurance buying behavior of utility maximizing individuals. Their results, however, as well as those of later authors (e.g., Schlesinger [1981] and Turnbull [1983]) are qualitative in their nature. One of the important findings, for example, is that coverage decreases when the risk premium increases. To what degree coverage decreases is, however, not specified. Raviv [1979] concludes that Pareto optimal insurance policies involve a deductible and coinsurance above the deductible, but specific expressions as to the magnitudes are not treated. This paper extends these analyses by presenting explicit, albeit approximate, expressions for the amount of insurance coverage. Section 2 derives an expression for the expected utility maximizing individual who is free to choose any degree of coverage. Section 3 examines two other types of insurance contracts, and Section 4 summarizes the findings. Throughout, the results will be illustrated numerically for the "average" American household.

This paper considers the second-order condition for the choice of an optimal deductible in insurance contracts. An equivalent condition is derived in terms of comparing growth rates of the net price of insurance and a relative utility weighting reflecting the degree of risk aversion. Mathematical models of economic choice often examine the properties of the best economic choice without adequately ruling out the possibility that they may actually be examining the worst choice instead. The choice of a best insurance contract by an individual is no exception. In an earlier paper in this journal, Schlesinger [7] showed that the necessary conditions for the choice of an optimal deductible for a given insurance policy can lead to a worst (minimal-expected-utility) outcome if one is not careful. A sufficient condition for a best (expected-utility-maximizing) choice of deductibles was presented, but its interpretation seems rather artificial (Schlesinger [ 198 1; p. 476 eqn. (18)]). This sufficient condition is hardly trivial as is evidenced by a recent paper of Boyle and Mao [1] in which more than half the paper is an appendix, the sole purpose of which is to verify this sufficient condition in a relatively simple model. This note reexamines this sufficient condition, showing it depends upon the growth rate of the net price of insurance coverage and a utility weight that reflects a measure of risk aversion. The paper begins with a streamlined presentation of the necessary firstorder condition for a best deductible choice. This necessary condition also Harris Schlesinger is Assistant Professor of Economics at Vanderbilt University and is currently a Research Fellow at the International Institute of Management in Berlin, West Germany. This content downloaded from 157.55.39.92 on Wed, 22 Jun 2016 07:06:04 UTC All use subject to http://about.jstor.org/terms Deductible for Insurance Contracts 523 receives a more canonical interpretation than in previous literature (cf. Gould [3], Pashigian, et al. [5] and Schlesinger [7]). The second order condition sufficient for a best deductible is then discussed and interpreted. For this purpose, consider a risk averter in a simple one period model in which a loss x occurs. This loss is the realization of a random variable with density function f(x) and corresponding cumulative density F(x). For simplicity, assume F(x) is differentiable for x > 0. A straight deductible insurance policy fully indemnifies the individual for losses in excess of some predetermined deductible level. The model allows for the purchase of an insurance policy with any desired deductible level D at a price of P(D) called the insurance premium. At the individual's choice of a deductible level D, it must follow that -1 < P'(D) < 0.1 The actuarial value of a policy with deductible D is given by A(D) J (x-D)dF(x). (1) D

This paper illustrates the use of a quantitative decision analysis approach for the evaluation of diverse risk treating alternatives. The decision to invest in fire protection and/or insurance for two sizes of airport facilities is investigated. The high dependence of the optimum choice on the cost of fire protection and its relation to insurance is illustrated. The strong influence on the final decision of parameters such as the structure of the insurance companies' deductible schedule and the degree of risk aversion of the decision maker is also addressed. The problem of choosing the best level of fire protection for any facility is central to sensible planning. Whatever the advances in fire technology, this question remains: how much protection is enough? Increased protection is always desirable from the point of view of reducing losses from fire. But increased protection costs money: large capital sums may be required as well as substantial annual cost. There is, thus, a tradeoff between the cost of the protection and the actual value of the protection. Conceptually, the determination of the best choice for the level of fire protection is simple: one should spend money on additional protection while the extra cost of more protection is less than or equal to the value of the extra protection received. In practice, however, this criterion is difficult to apply. It may be difficult, first of all, to evaluate the cost of each of the large number of possible choices, especially when there is so much David Shpilberg, M.S., is a Doctoral Candidate, Massachusetts Institute of Technology and is Research Scientist at Factory Mutual Research Corporation. He was project leader for a two year research study, financed by Factory Mutual Research, on "Risk, Insurance and Fire Protection, a Systems Approach." Richard de Neufville, Ph.D., is Associate Professor, Civil Engineering Systems Laboratory, Massachusetts Institute of Technology. He was a Guggenheim Fellow, 197374 and was Director of the MIT Civil Engineering Systems Laboratory, 1970-73. Dr. de Neufville is author of Systems Analysis for Engineers and Managers (1971) and Systems Planning and Design-A Case Study (1974). This paper was submitted in December, 1973. The authors appreciated the constructive reviews of an earlier draft of this paper by P. Byer, at M.I.T., and two members of this Journal's editorial staff. ( 133 ) This content downloaded from 207.46.13.149 on Thu, 20 Oct 2016 04:10:27 UTC All use subject to http://about.jstor.org/terms 134 The Journal of Risk and Insurance uncertainty about whether a fire will occur and how big it might be. Secondly, it is often difficult to determine what, precisely, is the value of preventing or minimizing the losses from a fire. The expected value of losses cannot be used as a direct basis for decision since the decision makers usually prefer the certain loss of a small sum (the insurance premium) to the combination of a small chance of a much larger loss and a larger chance of no loss. This preference for small fixed losses over a risk of large losses originates not only from an aversion to the psychological state of uncertainty but also from a decreasing marginal utility of wealth (1,2). In the case of large entities (such as airlines or airport operators) they are willing to take risks for losses up to an amount that will not seriously affect their financial position, but their disutility for losses increases sharply when contemplating the possibility that these losses might affect their ability to continue in business. Generally speaking, the value of preventing catastrophic losses, which can seriously disnipt or bankrupt a business, is greater than the direct monetary losses. But how much greater is it? The two issues of how to evaluate complicated choices in a probabilistic environment, and how to determine the value (disutility) of losses, can both be answered by procedures that have been recently developed (1,3). Collectively, these procedures are known as decision analysis. The purpose of this paper is to show how decision analysis can be used to determine the best choice for the level of fire protection, first, in theory and then, by application to a real problem. Theory Within certain legal and financial limitations, a planner has two basic choices in planning fire protection. First, there is the degree to which one invests in physical protection systems, such as sprinklers, fire stations, hydrants and the like. Second, there is the choice of insurance coverage: full coverage, self-insurance, or something in between. All together, a large number of alternative protection strategies exist for any property. The possible decisions can be conveniently represented in a decision tree, as shown in Figure 1. This is a mechanism for systematically organizing complex alternatives, and is now becoming widely used in modem management. Any choice of fire protection still involves some risk of losses. These can occur in a variety of ways, ranging from small fires to total losses. The risk of losses of any particular size, vi, depends upon the level of physical protection, k, as well as on the choice of insurance option, 1. This risk can be determined from past experience and in conveniently defined terms of an annual probability of occurrence, pikl. All the possible outcomes can be illustrated by the decision tree. With this information, it is possible to calculate the expected annual losses, Lkl, that would be associated with the choice of a particular level of protection. This content downloaded from 207.46.13.149 on Thu, 20 Oct 2016 04:10:27 UTC All use subject to http://about.jstor.org/terms An Airport Study 135 This is simply

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.

A casualty loss produces option-like characteristics in assets because their value depends on further discretionary investment. With risky debt in the firm's capital structure, the shareholders can have incentives to forgo the discretionary investment, even though it has a positive net present value. Thus a potential incentive conflict exists between the bondholders who want the investment made and the shareholders who do not. The incentive problem can be controlled by including a covenant in the bond contract requiring insurance coverage. Full coverage is generally not required. The maximum deductible depends on the amount of debt in the firm's capital structure and the feasible set of net casualty losses.

The problem of selecting a deductible when an asset is exposed to risk is analysed in the framework of consumption theory. The riskbearing budget is defined as the difference between the optimal consumption level under complete certainty and the optimal consumption level in a context of risk. How this risk-bearing budget should be divided between buying insurance and building up a contingency reserve is investigated. The wealth effect on both the risk-bearing budget and the deductible also is analysed. Insurance, perceived as an instrument of risk redistribution, undoubtedly plays a vital role in the theory of risk-bearing. Individuals as well as corporations are concerned about finding some optimal insurance arrangements when confronted with risk situations. At the beginning of the sixties, Borch introduced utility theory into contemporary actuarial thought. He was concerned particularly with the problem of risk redistribution through the reinsurance market. Borch [3] (reprinted in Borch [4]), then found that an insurer seeking to maximize the variance reduction of its claim distribution for a given net premium should seek to conclude a "Stop Loss" treaty. In his classical analysis of the economics of medical care, Arrow [1] generalized Borch's result by demonstrating the superiority of a deductible form of insurance for a risk averting individual facing the risk of an income reduction. The works of Borch and Arrow generated further research and a theory for the demand of insurance gradually took form. On the theoretical side, the contributions of Mossin [7] and Smith [9] are among the most interesting ones. Along the lines of Arrow, the approach commonly adopted in attempting to build a decision model to determine the optimal insurance coverage is based on the maximization of the expected utility of terminal wealth. The two major deficiencies in this approach are: First, it completely ignores saving as an alternative to insurance buying. Denis Moffet is Assistant Professor of actuarial science, Universite Laval in Qu6bec. This paper is based on a chapter of "Essays in the Economics of Insurance which was awarded the 1976 Ernst Meyer Prize by the Geneva Association. Comments by Karl Borch, Jan Mossin, Agnar Sandmo, Baruch Berliner and two referees have been most useful. An earlier version of this paper was presented at the 1975 Risk Theory Seminar. ( 669 ) This content downloaded from 157.55.39.255 on Mon, 01 Aug 2016 06:10:57 UTC All use subject to http://about.jstor.org/terms 670 The Journal of Risk and Insurance Second, it neglects the effect upon current consumption of any outlay intended specifically for the purpose of risk-bearing. By considering a simple risk situation, it was illustrated (Moffet [5]) how the insurance buying problem could be formulated in terms of consumption theory (see, for example, Sandmo [8]). This present study, however, concentrates on a more realistic problem, namely the one of choosing an appropriate deductible level when a given asset is subject to a risk of deterioration. Also attention is focused on some actuarial considerations about the premium for a given deductible. The analysis flows from the determination of an optimal consumption level, assuming an initial wealth. The risk-bearing budget is defined as the difference between the optimal consumption level under complete certainty and the optimal consumption level in a context of risk. How this risk-bearing budget should be divided between buying insurance and building up a contingency reserve fund then is investigated. The Model A single-period model is used. At the beginning of the period the consumer's net worth is W. One of the consumer's assets, the value of which is N < W, can suffer a loss amounting to x during the period, subject to the following probability law: p: probability that no loss will be incurred, i.e. x = 0 q: probability that a full loss will be incurred, i.e. x = N f(x): probability density function over the open interval N (o,N) such that p + f(x)dx + q 1 where 0 < p < 1 and 0 < q < 1 This risky situation can be confronted by buying insurance on the market and/or by building a contingency reserve fund. It is assumed that only one claim can occur during the period and that its settlement is made at the end of the period. Insurance can be bought with a deductible D, that is the insurer will pay the amount in excess of D if a loss is incurred. The insurance premium is denoted P(D) and when there is no danger of confusion it is written P instead of P(D). The consumption level for the period and the consumer's net worth at the end of the period are denoted by C and Y. Assume that the consumer's preferences are represented by a utility function V having the form

The application of deductibles to insurance claims yields a statistical sample of truncated loss amounts. The problem is compounded when a different deductible value may have been applied to each policy in the sample. This article develops a statistical procedure for fitting the Log-normal distribution to claims experience in which a different deductible amount may have been applied to each loss. Maximum likelihood estimators for the parameters and their asymptotic sampling distribution are derived. The method is illustrated by applying it to a sample of industrial fire insurance claims.

Three states (Maine, Minnesota, and Rhode Island) have had programs to assist any resident in meeting uncovered "catastrophic" expenses. The annual cost of these programs after 4-6 years of experience has been only a couple of dollars per state resident. A reasonable method of forecasting a priori the cost of such a program is shown to yield a much larger cost than actually has occurred. Some of the controllable factors affecting program growth (e.g., the deductible formula) are discussed, along with a variety of influences on the policy decisions at the state level. Finally, the paper warns against extrapolating the low cost of these state programs to federal initiatives in catastrophic coverage. Castastrophic health insurance (CHI) can be ideally defined as a guarantee that direct costs to eligible consumers for covered health services will be limited to a modest amount in relation to family resources. A variety of existing private and public programs approximate this essential criterion, while differing substantially in other dimensions. Within the private sector, contracts may vary in the type of services covered, the amount of cost sharing, and the choice of providers. A major This research was supported by HCFA grant # 18-P-97265/5/04 to the Center for Health Services and Policy Research. The authors thank Dr. Edward F.X. Hughes for helpful suggestions, Cathy Griffin, Ray Dow, and Harvey Zimmerman for data and insights on state programs. Anonymous referees for this Journal provided many useful criticisms and sugges

A practical mathematical development of a probabilistic model for annual loss amounts is developed. A simulation is used to validate the hypothesis that under general conditions, the Gaussian (normal) probability model is appropriate for annual loss amounts when straight deductible insurance is purchased. Formulae for deriving the parameters of the Gaussian model are provided and a numerical example is given. Summary simular output is provided. Straight deductible limits are a feature of many insurance coverages. Monte Carlo simulation has provided one of the few generalized approaches to expressing such coverages on a fiscal period basis instead of a per incident basis [3, p. 504]. In this paper an attractive alternative approach to Monte Carlo simulation is suggested for a class of frequently discussed models for loss frequency and loss amounts (the Poisson and lognormal distributions, respectively) [6, pp. 699-704]. The methodology discussed in the present paper provides a direct method of describing the mean and variance of annual loss amounts for given deductible limits. Furthermore, evidence is offered in support of the thesis that annual loss amounts can be reflected by the common Gaussian (normal) probability distribution when expected loss frequency is about eight or more a year. RANDOM VARIABLES AND THEIR PARAMETERS Definitions and Notation X = A random variable representing the amount of loss associated with a single fortuitous event. In the present paper the assumption is that X is lognormally distributed. Stated differently, it is assumed that logeX is normally (Gaussian) distributed. fX(x) = ( exP0(1/2a2)(1og x 2]for x > 0, 0 for x < 0. Note the parameters of f (x): 1 and a2. Brian Schott is Associate Professor of Quantitative Methods, Georgia State University. He holds the Ph.D. degree and the A.S.A. designation. He is coauthor of SIMQ: A Business Simulation Game for Decision Science Students and author of RISKM: A Risk Management! Financial Management Model. He has served as an advisor to Anistics, a subsidiary of Alexander and Alexander, Inc. Research support for this work was provided by the College of Business Administration Research Council, Georgia State University.

Mortgage Rate Insurance Pricing under an Interest Rate Diffusion with Drift ACKNOWLEDGEMENTS This research was supported by a grant from the Financial Research Foundation of Canada. Computing resources were provided by Digital Equipment Corporation. The author thanks an anonymous referee for helpful comments. ABSTRACT The federal government of Canada has introduced a program whereby a mortgage borrower can purchase insurance which gives protection against interest rate rises at mortgage renewal. A diffusion model of interest rates incorporating a drift term is applied to the valuation of mortgages, and the resulting partial differential equation is solved numerically. The boundary conditions of this problem necessitate a novel solution method which is likely to have applications in other areas. Estimates of the appropriate net premium are given. INTRODUCTION As a result of pressure from homeowners renewing mortgages at the high interest rates which prevailed in the early 1980s, the federal government of Canada introduced on March 1, 1984 the Mortgage Rate Protection Program. The Program is administered through Canada Mortgage and Housing Corporation, a federal agency. In Canada, residential mortgages are usually amortized over 25 years, but the mortgage is periodically renewed at prevailing interest rates. The inter-renewal period is usually one, two, three or five years. Under the Mortgage Rate Protection Program, up to $70,000 of the principal at inception can be insured. For the rate rise a deductible of 2 percent per annum applies, and the portion of renewal mortgage payment resulting from an increase in rates of over 12 percent is not insured. The principal at the end of the insured period is calculated using the market rate rather than the rate reduced by the effect of the insurance. The insurance only pays 75 percent of the increase in monthly payments resulting from a rate rise and only covers payments over a period equal to the original inter-renewal term. As an example(1) of the operation of the insurance, consider the case of an insured mortgage with an amortization period of 25 years and repayments at the end of each month, when mortgage rates have risen 5 percent in the five year period from the date of mortgage inception and insurance purchase. The situation is illustrated(2) as Table 1. The single premium paid for the insurance is 1.5 percent of insured principal, regardless of the inter-renewal term. Thus the premium structure takes no account of either the longer period of potential payout under the longer inter-renewal periods, or the higher probability that substantial rate increases will have occurred over a longer inter-renewal period[14]. At mortgage renewal at time five years in the example it would be possible to again purchase the insurance to protect against higher monthly payments between times ten years and 15 years. However, the insurance would then be based on the 17.5 percent per annum market rate which prevails at time five years. It should be noted that the insurance can be transferred to the purchaser if the house changes ownership. Thus, ownership transfer would not affect the value of the insurance. The possibility exists, but is not included in the model below, that a fall in house value could cause the mortgagor to abandon the house and correspondingly to give up any claim to benefits under the insurance. Price volatility extensive enough to lead to house abandonment is unusual in Canada, but to the extent that it occurs the insurance values found below should be regarded as upper limits. This consideration does not affect the result found that the premium charged is too high. This study presents a method for calculating the value of the insurance. Work on the valuation of other types of mortgages by a different method without insurance has been reported by Dunn and McConnell[7], while the valuation of bond options has been studied by Dietrich-Campbell and Schwartz[6]. …

The risk retention decision is a classic example of decision making under conditions of uncertainty. A cost savings may be obtained by the firm in exchange for accepting additional risk through risk assumption, but little attention has been directed toward optimizing this tradeoff for the firm. In this paper, the authors develop a model which uses the probability of loss to evaluate deductible selection. The advantage of this model is that it leads to decisions that are optimal for the firm.

Properly pricing deductible may help insurers to meet business objectives such as minimizing adverse selection, eliminating small claims, maintaining risk charges, and minimizing policyowner dissatisfaction. Technical considerations-such as the nature of loss data, the effect of inflation, and the effect of deductibles on operating costs-impose constraints on pricing procedures. After analyzing these business objectives and technical considerations, this paper concludes that a deductible pricing decision may involve compromises among competing requirements and objectives. Furthermore, some of the accepted wisdom of deductibles-such as their effect in reducing "nuisance claims"-is opened to serious question. Although an insurer's problem of setting rate credits for deductibles is at least as complex and worthy of study as the insurance buyer's problem of selecting a deductible, the insurance seller's decision has not received the intensive analysis long focused on the buyer's problem. This paper seeks partially to correct this imbalance by setting forth guidelines insurers may follow rationally in pricing deductibles for property insurance coverages. Pricing guidelines emanate from two sources: an insurer's business objectives and technical considerations which underlie proper rate credits. To meet its business objectives, one of which presumably is an underwriting profit, an insurer should offer deductibles and set deductible credits which

Robert I. Mehr [3, pp. 428-431] has suggested that there is tax discrimination against cash value life insurance relative to installment debt. He argues that a consumer is entitled to deduct the service costs of installment debt, but is unable to deduct an interest component in life insurance (installment) premiums which "services" the cost of providing protection over the policy duration. Mehr, however, fails to identify or treat the very interesting question of how this interest component would be measured for tax purposes. He also alleges discrimination because gains over the net purchase price incurred on disposition of an asset acquired through installment debt are treated as capital gains, whereas gains realized at the surrender or maturity by endowment of a cash value life insurance policy are taxed as current income, subject to income averaging. Finally, he wonders why property taxes charged against some items acquired through installment purchases, such as a house, are deductible expenses although premium taxes (included in the expense loading of a life insurance policy) are not deductible. These arguments represent interesting counter-charges to conventional wisdom which holds the life insurance policy in a favored tax status that endows it with an economic subsidy-tax free inside interest buildup oln savings. However, Mehr's allegations do not seem to be consistent with the facts. Failure to pay installments due on debt would cause forfeiture of assets at least to the extent of the outstanding debt and reduce liabilities accordingly. Failure to pay life insurance premiums does not result in forfeiture of an asset. Furthermore, the use of installment debt involves the simultaneous creation of an asset (the item being financed) and a liability (the loan). The purchase of life insurance does not involve the formation of a liability. It is generally deemed desirable to allow tax deduction for the costs of servicing a liability while it is being extinguished. That is, the costs incurred in eliminating such liabilities, other than the principal sum of the liability itself, receive a tax exemption because these costs represent income earned by another factor and are taxed at that source. Savings, however, are distinguished as the accumulation of assets, where an asset, say cash, is

This article develops a model which describes the expected costs of adopting an insurance program which has a deductible feature. Rules for choosing the least expected cost deductible level are developed for policies with aggregate deductibles and for those with straight deductibles. Also, it is noted that the decision maker may not want to move from a smaller deductible to a lower cost larger one if the expected cost benefits are small and the risk assumption is large. The size of the deductible to select when purchasing insurance is a decision for which rigorous decision rules are lacking." In spite of this deficiency, the use of deductibles by corporations of all sizes has been increasing markedly during the recent past. This paper develops models for analyzing the cost of deductible programs and develops rules, based on these models, for choosing the least expected Richard M. Duvall, Ph.D., is Professor of Finance at the University of Tennessee. Formerly on the faculties of the University of Texas and the University of Michigan, Dr. Duvall has authored articles which appeared in the Journal of Finance, Management Science, Journal of the American Statistical Association, and the Journal of Financial and Quantitative Analysis. Tom C. Allen, Ph.D., is Vice President and Director of Research and Planning of Ingram, Armstead, Wallace, Inc. He was formerly a member of the Finance Department in the University of Tennessee. Other publications of Dr. Allen have appeared in Risk Management, The Insurance Law Journal, Business and Economic Dimensions. This paper was submitted in September 1972. 'Two papers which have established rules based on analysis similar to that in this paper are Lawrence L. Schkade and George H. Menefee, "A Normative Model for Deductible Collision Insurance Selection", Journal of Risk and Insurance, Vol. 33 (September, 1966), p. 42735; and George L. Head, "Optimizing Property Insurance Deductibles, a Theoretical Model for the Corporate Buyer", Journal of Risk and Insurance, Vol. 32 (September, 1965). cost size of the deductible. The first model specified is for policies with aggregate deductibles, and then, a model for policies with straight deductibles is analyzed. Finally, the implementation of the rules is shown where there are several policies and a restriction is placed on the sum of the deductible amounts from all policies. Perhaps the principle reason offered for using deductibles is that they give the insured an opportunity to reduce insurance costs by avoiding "dollar trading" with the insurers in areas where losses are frequent and their normal aggregate dollar impact can be absorbed easily, while allowing the firm to continue to insure against rare but catastrophic losses. Deductibles are often used as a means of encouraging corporate personnel at all levels of responsibility tomaintain good housekeeping and to promote on-the-job safety. In a case such as this, both the employees and management of an operating division are informed of the fact that all first dollar losses are the responsibility of the firm and not the insurer. Thus, all such losses below the deductible level will subtract from the firm's earnings. When employee profit-sharing efficiency

The decision process involved in the selection of an optimum automobile collision insurance deductible provides an excellent opportunity to utilize the von NeumannMorgenstem utility development. In an extension of the theoretical formulations presented previously for this decision, a model was developed which was based upon a minimization of expected dis-utility. An intelview technique was used to develop dollar-dis-utility coordinates for each of ten subjects. A least squares program was invoked to fit a function to these data. These functions were then used in the model, along with both subjective and objective probabilities of loss plus premiums applicable to the individual, to arrive at an optimum deductible. Results of checking the model for reliability and validity are reported along with a comparison of actual with predicted deductibles. The model, as anticipated, was not descriptive. It is contended, however, that in a "qualified normative" sense it is a valuable decision tool. The selection of an optimum automobile collision insurance deductible by an individual is a decision exceptionally well suited to the application of a decision model incorporating von Neumann-Morgenstem utility. Previous efforts in this direction have been limited to a theoretical formulation. Herein is reported the results of a study designed to develop and utilize a model in an actual decision

The purchase of collision insurance for private passenger vehicles almost always includes the provision for some sort of deductible in the policy. Under the rules in the automobile physical damage manual, a policy may include a fixed deductible, the amount of which may vary from $25 to $1,000. Other types of deductible including 80/20 deductible, convertible, the homeowners disappearing deductible, and full coverage (no deductible) are available, but these classes combined constitute less than one per cent of all deductible coverage written and therefore are not considered in this study. The selection of the amount of deductible by vehicle owners may be influenced by any of a variety of factors, including the value

The expected utility method has been widely suggested for making decisions under uncertainty. This paper describes an experiment designed to determine the relative acceptability of the expected utility method and of two other normative methods (the worry and comparison methods) to practicing insurance agents and risk managers in deciding how much insurance to buy. It was found that the subjects ranked the comparison method first in terms of acceptability, the worry method second, and the expected utility method third. When the subjects rated the methods according to five specified criteria-confidence in the results, understanding of the method, simplicity, ability to explain, and ease of application-the results were basically the same as their usefulness-on-balance ratings. The subjects considered confidence in the results and understanding of the method to be most important in determining a method's usefulness on balance. None of the several personality and demographic characteristics of the subjects considered showed any clear association with the different ratings of the methods. The utility method usually led to the purchase of no insurance while the other two methods suggested complete insurance with or without a deductible. The best acts according to the comparison and worry methods were quite consistent with the subjects' actual preferred acts. Decision-making under risk has been the subject of numerous books and articles. One of the most popular tools, at least John Neter, Ph.D., is Professor of Quantitative Analysis in the School of Business Administration of the University of Minnesota. He is a Fellow of both the American Statistical Association and of the American Association for Advancement of Science and is co-author of Fundamental Statistics for Business and Economics, and of Statistical Sampling for Auditors and Accountants. C. Arthur Williams, Jr., is Professor of Economics and Insurance and Associate Dean of the School of Business Administration of the University of Minnesota. He is a member of the American Academy of Actuaries and was President of A.R.I.A. in 1965. Dr. Williams is author of Price Determination in Property and Liability Insurance and is co-author of Risk Management and Insurance, of Economic and Social Security, and of Insurance: Its Theory and Practice. This paper was submitted in July, 1970. The authors wish to acknowledge the financial support of the Graduate School and the School of Business Administration of the University of Minnesota in the conduct of this research. They also appreciate the financial support of the Insurance Conference of the Cooperative League of the U.S.A., which made possible a presentation of this paper at the American Risk and Insurance Association Risk Theory Seminar in New Orleans on 17-19 April 1970. The following members of the Conference contributed to this support: Cooperative Federee de Quebec, Cooperative Insurance Services, Ltd., Cooperators Insurance Association, CUNA International, Inc., Farmers Elevator Mutual Insurance Company, Farmland Insurance Agency, Inc., L'AssuranceVie Desjardins, La Federation des Caisses p. Desjardins, La Societe d'Assurance des Caisses Populaires, League Life Insurance Company, Members Mutual Insurance Company, MidAmerica Mutual Life Insurance Company, Mutual Service Insurance Companies, National Farmers Union Insurance Companies, National Rural Electric Cooperative Association, Nationwide Insurance Companies, and Urban Community Insurance Company. They would like also to acknowledge help received in carrying out the experiment and analyzing the results thereof from Stephen Barkin, Richard Cardozo, Amin Eldirghami, Robert Fiske, Ming te Lu, Ivan Ross, and Mark Walton.

In order to investigate the usefulness of the von Neumann-Morgenstern utility concept in an actual decision making context, 30 subjects were interviewed. Their responses to questions were used to acquire a utility function which was then incorporated in a deductible selection model. The utility function was found to have little external validity either when used to predict choices between two lotteries or in hypothetical insurance consumption decisions. Used in this model it did, however, appear to be slightly more useful in describing actual deductible selections when compared with an expected monetary value model. An effort was made to determine the influence of personal variables on the deductible selection decisions. Using chi-square tests it was determined that the level of deductible is related to the individual's subjective probability-of-loss estimate and his income tax bracket. Utilization of the variables in stepwise regression and discriminant analyses yielded insignificant results for all variables. The question of whether utility functions are amenable to use in actual decision-making contexts has been the subject of considerable debate. Reflecting those who disparage its value, Houston states:1 Modem decision theory represents an important breakthrough or contradiction of the traditional interpretation of utility. However, it is only fair to note that many regard the theory as non-operational in all but the most trivial cases. Representing a more optimistic viewpoint are Friedman and Savage who note:2 Michael L. Murray, D.B.A., is Assistant Professor in the College of Business Administration of The University of Iowa. This paper, funded by a research grant from A.R.I.A., was presented at the 1971 Annual Meeting of A.R.I.A. 1 Houston, David B., "Risk, Insurance and Sampling," Journal of Risk and Insurance, December, 1964, pp. 522. 2 Friedman, M. and L. J. Savage, "The Utility Analysis of Choices Involving Risk," The Journal of Political Economy, August 1948, reprinted in Whether a numerical utility function will in fact serve to rationalize any particular class of reactions to risk is an empirical question to be tested; there is no obvious contradiction as was once thought to exist. This paper reports the result of a continuing effort to determine empirically the usefulness of utility3 functions in an actual decision context. The decision which is being analyzed is the selection of an automobile collision insurance deductible. This decision is viewed in both a descriptive and a normative manner. In the descriptive vein, a number of personal variables of the respondents to the study are analyzed for their ability to predict the Hammond, J. D. (ed.), Essays in the Theory of Risk and Insurance, Scott, Foresman and Co., Glenview, Illinois, 1968, pp. 97. 3The utility concept which is used in this paper and which is referred to in the two previous quotations is that which is referred to as "4von Neumann-Morgenstern utility" as developed in their landmark work: von Neumann, John and Morgenstem, Oskar, The Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944, pp. 1-31.