Journal of Risk and Uncertainty, 7:117-139 (1993) 1993 Kluwer Academic PublishersThe Risky Business of Insurance PricingW. KIP VISCUSI*Department of Economics, Duke University,Dtirttam, NC 27706AbstractThe factors influencing insurance pricing decisions are assessed using the ISO product liability ratemaking filesfor 1980-1984. The mean loss level has a strong positive effect on manual rates and premium rates/exposure.Evidence on a variety of ambiguity measures is more mixed. As a broad generalization, risk ambiguity lowersmanual rates, which may reffect exclusion of large loss outliers as being unrepresentative. Risk ambiguity tendsto have a positive effect on actual pricing decisions for particular policies, especially bodily injury lines and theinteractive risk-ambiguity model.Key words: insurance pricing, decision making, risk, ambiguityThe role of risk ambiguity has been of increasing interest in the literature on the character of choice under uncertainty.' Although the Ellsberg paradox and the assoeiatedaversion to imprecisely understood probabilities has long been a major feature of theliterature on the rationality of choice under uneertainty, it has been only recently thatinvestigators have generalized this phenomenon to actual instances of individual andsocietal decisions.A prominent example of such an extension consists of the work on the effect ofambiguity on insurance pricing. A recent paper by Hogarth and Kunreuther (1991)extended their research on this topic using a sample of professional aetuaries. Theirdata, derived from a mail survey of actuaries, implied that aetuaries would price ambiguous risks at an amount greater than the expected value. Another noteworthy feature oftheir results is that even in situations in which the professional actuaries make an adjustment for ambiguity, the expected loss associated with the policy is the main anchorrespondents use in assessing the level of insurance rates.Rather than focusing on stated questionnaire responses to various risky scenarios, thisarticle will address the influence of risk ambiguity using four sets of data on productliability insurance. In particular, for each ofthe industry groups and for each of the statesrepresented in the sample, the effect of the expected loss will be assessed as well as theambiguity of this loss on two measures of insurance prices—the manual rate for a partieular policy as well as the actual premium rate charged per unit exposure. Premium rates*This article was prepared for The Wharton School Conferenee, Making Decisions Abotit Liability and Insurance, Deeember 6-7,1991. Richard Zeckhauser, who was the discussant of this article at the Wharton conference, and Sharon Tennyson provided a variety of insightful comments. Patrieia Born provided superb researehassistance. This research was supported by NSF grant SES #3321057.
118W, KIP VISCUSIreflect the market prices actually charged by insurers, whereas manual rates are thepricing guidelines written in a book or manual. Manual rates are intended to provideguidance to underwriters in setting premium rates in specific contexts. In the usualinstance, the manual rate serves as the reference point for the average price appropriatefor a particular risk.The primary intent of this article is to assess the ramifications of risk-ambiguity aversion for actual insurance pricing decisions. The experimental evidence in the extensiveliterature on risk ambiguity is quite strong. Moreover, there are occasional references tothe role of risk ambiguity in the insurance ratemaking literature. At the same time,however, the dominant focus ofthe ratemaking literature is on expected losses and otheraspects of insurance pricing decisions, such as fixed costs and the return on the investedpremiums. Risk ambiguity does not play a central role in these basic rate making formulas. The role of uncertainty has, however, been a prominent concern in the literature onthe liability insurance crisis, which suggests that risk ambiguity may potentially influenceinsurance pricing decisions, The empirical evidence considered here is based on four very large sets of data pertaining to produce liability coverage. The sample consists of the insurance ratemakingfiles of the Insurance Services Oflice (ISO)—the industry group that pools this riskinformation—for four different lines of product liability coverage. The research findingssuggest that the expected loss associated with the policy has an extremely strong effect oninsurance pricing. This result is what one would expect. In contrast, the risk-ambiguityaversion terms have a less clearcut influence. Although the empirical evidence suggeststhat there are mixed effects, there is very little evidence suggesting that risk-ambiguityaversion affects the setting of manual rates. An influence of greater consequence appears to be the desire of insurers to trim oudiers from consideration so as to have a morerepresentative sample for rating projections. Actual pricing decisions appear to be morestrongly influenced by risk ambiguity, particularly if one postulates a mechanism bywhich the influence of risk ambiguity is through an interaction with the mean risk level.Section 2 of this article outlines the fundamentals of insurance pricing as well as therelationship of these pricing decisions to the potential role of risk ambiguity. Section 3explores the substantial volatility of the expected loses that insurers face under thesepolicies. Despite this variability, there is not clearcut evidence of ambiguity aversioneither in the additive risk ambiguity models explored in section 4 or in the interactivemodels that are the subject of section 5, The evidence in support of the risk-ambiguityaversion hypothesis is, however, stronger for the actual pricing of insurance than for thesetting of manual rates. The presence of this influence also varies by insurance line.Section 6 summarizes the findings and the relationships to the risk-ambiguity aversionhypothesis.1. The rate-setting processBefore considering the potential role of risk ambiguity, it is helpful to consider the basicaspects ofthe insurance ratemaking process. If we let K equal the variable costs associated with writing an insurance policy and T7 be an allowance for a normal rate of profits
THE RISKY BUSINESS OF INSURANCE PRICING119on the policy, then the basic formula for determining the rate that will be charged on aninsurance policy is given by(Loss/Exposure) - (Fixed Costs/Exposure)where Rate is the dollar premium charged per unit of exposure. The dominant unit forexposure, and the one that is employed in the data set below, is dollar sales of theproduct group being insured.The two major risk-related factors that govern how the risk associated with the policyaffects the insurance premium rate charged are the loss per unit of exposure and thefixed costs per unit of exposure, where the latter influence should generally be independent of the extent of the risks. All policies are also influenced by the variable costsassociated with writing a policy as well as the need to earn a profit on the policy, which inturn will be influenced by the level of interest rates. The basic formula that governs therate-setting process by insurance companies consequently does not include a measure ofrisk ambiguity.'' Indeed, the only component that is likely to be strongly related to theloss experience under the policy is the average loss/exposure amount.This is not to say that there may not be some adjustments for risk-ambiguity aversion.For example, the allowance for profitability -IT could potentially be influenced by theriskiness of the policy and the precision with which the loss is known. More generally,insurance companies are reluctant to write coverage in situations in which there is not agood statistical basis for setting the rates. The overriding hypothesis is that firms willreflect this reluctance in higher insurance prices when the likely performance of thepolicy in unclear.The econometric formulation implied by equation (1) isRate a -I- 3i (Loss/Exposure)-I- 32 (Fixed Costs/Exposure) e,(2)where 31, 32 1/(1 - K - IT).Thus, if there is no role for risk ambiguity, one can formulate the rate-setting processas a simple linear function of the loss per unit of exposure and the fixed cost per unit ofexposure, where the coeflicients of these variables reflect the influence of the variablecost and profitability terms.Risk ambiguity could enter this formulation in two different ways. It could alTect theweights placed on the difl erent components, or it could enter additively. Each of thesewill be explored in turn in this article.The empirical measures of risk ambiguity will all involve some measure of loss dispersion for policies written for a particular class of products. These measures of loss dispersion may capture a variety of influences. Situations in which the risk performance ofpolicies is less well understood should be associated with less precise risk assessments onthe part of insurers. If insurers display risk-ambiguity aversion, in the Ellsberg paradoxsense, as is hypothesized, then greater dispersion will lead to higher insurance prices.Observed insurance prices will, of course, reflect the joint influence of decisions by theinsured and by insurance companies, and these generally are mutually reinforcing.
120W, KJP VISCUSIAversion to ambiguous risks, for example, will raise the amount that those exposed torisk are willing to pay for coverage and will also boost the price at which insurance isoffered,A wide range of other economic influences linked to variability of losses may also be atwork, where these effects for the most part also should raise insurance prices. Morevariable risk outcomes will be less attractive to parties who are risk averse. From thestandpoint of the insurer, it is the variability of the entire portfolio of insured risks that isof consequence rather than the variability of the losses for each narrowly defined productgroup. The role of risk aversion should consequently be small except in the case ofpolicies with extremely large stakes.Highly variable outcomes are also of concern to insurers from the standpoint of theirsolvency. Large losses may threaten the viability of the firm and may lead to violation ofregulatory constraints imposed on insurers. Insolvency ratings by agencies such as A.M.Best will also be influenced by large losses. Hogarth and Kunreuther (1991), for example,note that the variance of losses also may have tax implications for firms facing a nonlineartax schedule.In situations in which there is substantial underlying variation in the performance of firmsmarketing products within a similar group, insurers are also subject to problems of adverseselection. Sellers of risky products that generate the highest product risks will have thegreatest desire to purchase insurance, and low-risk producers will tend to forgo insurance,thus boosting the mean level of risks above the average for all producers in the industry.Similarly, insurance companies that are successful in marketing policies in a high lossvariance situation may be subject to a variant ofthe "winner's curse" that might be moreaptly termed the "seller's curse," Insurers who most underprice particular risks will bemost likely to sell such policies in a market in which there is variability in the underlyingrisks posed by the insured.Sellers of insurance who are trying to predict insurance performance may excludeoudiers as being unrepresentative. In much the same way that statistical analysts ofsurvey data often trim extreme responses to maintain the sample's reliability, firms mayattempt to focus on the more representative policies, (The conference discussant of thisarticle, Richard Zeckhauser, once held a summer job with an insurance company, wherehis main task was to identiiy and discard such outliers.) This influence will tend togenerate a negative effect of risk ambiguity on insurance prices.1.1. Additive modelsThe simplest way in which risk ambiguity could enter the estimating equation is throughan additive ambiguity term. Thus, rate setting could proceed along the lines indicated byequation (2) except that the insurance adjustor appends a risk-ambiguity term to accountfor the precision with which the risk is understood, thus giving rise toRate a -I- Pi (Loss/Exposure) -I- (32 (Fixed Costs/Exposure) P3 Ambiguity -I- e,(3)
THE RISKY BUSINESS OF INSURANCE PRICING121This kind of formulation is not unprecedented in the insurance ratemaking literature.Lemaire (1986), for example, suggests that insurance companies should adjust the valueof the premium, setting the premium equal to the expected loss JL under the policy plus alinear term that is a function of the estimated variance or standard deviation, orR a t e jJL CT .orRate iJL -I- cd. This is a standard mean variance model of insurance ratemaking. Similarly, constrainedrisk-of-ruin models suggested by Stone (1973) may also lead to an adjustment for riskambiguity, although the functional form is less clear-cut in this case. Firms interested inkeeping the probability of depleting their reserves will be concerned with risk ambiguity,leading to a form of ambiguous belief aversion.If insurance companies exhibit risk-ambiguity aversion, as in the case of the classicEllsberg paradox, we would expect the coefficient 33 to be positive. Controlling for theexpected loss/exposure, the rate charged for insurance should increase as the risks beinginsured become less well understood.1.2. Interactive ambiguity modelsAn alternative possibility is that instead of entering as an additive ambiguity adjustmentterm, risk ambiguity could affect the operation of the rate-setting process in an interactive manner. For example, if risk ambiguity influences the profit allowance factor TT inequation (1), then both 3] and 32 would be a function of risk ambiguity.Similarly, the subjective ambiguity a