INSURANCE FRAUD AND OPTIMAL CLAIMS
SETTLEMENT STRATEGIES*
KEITH J. CROCKER
University of Michigan
and
SHARON TENNYSON
Cornell University
Abstract
We examine the optimal claims settlement strategy for a liability insurer when
claimants can permanently misrepresent their losses by engaging in costly claims
falsification. In this environment, claims auditing is not a possible deterrent to fraud,
and the settlement strategy consists of an indemnification profile that relates the
insurance payment to the claimed amount of loss. The optimal indemnification profile
is shown to involve systematic underpayment of claims at the margin as a means to
deter loss exaggeration, with the extent of underpayment limited by expected litigation
costs and potential bad-faith claims. The key testable implication of the theory is
that the extent of underpayment should be greater for classes of claims for which
loss exaggeration is easier. Empirical analysis of insurance settlements for bodily
injury liability in automobile accidents confirms this prediction. This suggests that
liability insurers optimally choose claims payment strategies to lessen a claimant’s
incentive to exaggerate losses.
I.
Introduction
S
ix months after accidentally setting his house on fire while attempting to
remove a beehive, an Indiana homeowner was still sleeping in his dining
room while awaiting his insurer’s initial settlement offer.1 A Pennsylvania
driver whose car was struck from behind three times by a tractor-trailer,
which resulted in permanent shoulder injuries that left him unable to put on
a shirt without assistance, was offered $500 by the trucker’s insurance company.2 In Manhattan, an insurer contested a $1 million claim by an antique
dealer involving artwork allegedly stolen from his brother-in-law’s parked,
* The authors gratefully acknowledge support from the National Science Foundation under
grants SBR-9507768 and SBR-9696260. We are also grateful to Jiangdong Ju for advice on
proving our main result.
1
In addition, the homeowner was required to defend, in a 6-hour deposition, the values of
personal items lost in the fire. “They even asked me for a receipt for a $40 Buck knife and a
$5 jock strap,” the homeowner recalled. (What If the Company Won’t Pay? Kiplinger’s Pers.
Fin. Mag., June 1, 1999, at 70.)
2
Personal Injury Claims Split Insurers, Lawyers, Sunday Patriot-News Harrisburg, March
26, 1995, at D1.
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䉷 2002 by The University of Chicago. All rights reserved. 0022-2186/2002/4502-0007$01.50
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the journal of law and economics
and unattended, station wagon.3 And, finally, a jury awarded $1,057,224 to
a New Mexico woman after a credit life insurer refused to pay a $10,000
claim after the death of her husband.4
The miserly proclivities of insurers when settling claims is legendary and
occupies a place in the pantheon of business stereotypes along with the sharp
horse trader and the obdurate banker. And while it may appear that a dollar
saved through reduced claims payments amounts to a dollar earned,5 insurers
engaging in a strategy of systematic underpayment of claims do so at their
peril. At the very least, underpayment is likely to generate administrative
costs to the insurer forced to deal with aggrieved claimants, and the most
egregious shortfalls may spawn protracted episodes of litigation, raising the
prospect of both attorneys’ fees and adverse judgments. Furthermore, in the
extreme, bad faith by an insurer in the denial or underpayment of an insurance
claim can result in substantial penalties, as in the case of the Pennsylvania
driver noted above, who was eventually awarded $40,000, or in the extraordinary judgment delivered to the woman from New Mexico. Thus, a strategy of aggressive claims settlement practices has costs that the insurer must
weigh against the anticipated benefits. One such benefit is to reduce the filing
of fraudulent or inflated claims.
There are, in principle, two distinct types of strategies that may be adopted
by insurers to reduce the incidence of fraudulent claiming. The first is to
audit claims that have observable characteristics that are associated with a
potential for fraud and then to deny those that are found to be invalid. When
the mishap exhibits physical manifestations that can be exploited to identify
fraud, the use of costly audits and investigations, coupled with the subsequent
denial of payment, can be an effective tool to reduce the costs associated
with illegitimate claims. But when accidents are devoid of observable physical markers from which fraudulent behavior may be definitively inferred,
auditing may be ineffectual as a deterrent. This leads to the second strategy
by which insurers may reduce the incentives of claimants to commit fraud:
by systematically underpaying claims, the insurer erodes the returns to the
claimant of investing in privately costly activities designed to inflate claims,
which thereby reduces the incidence of fraudulent claiming.
This paper examines the role of claims underpayment as a mechanism to
deter fraud when claims auditing is not a practical deterrent. We consider an
3
The insurer ultimately prevailed after a photograph emerged of the paintings sitting in a
closet of the claimant’s antique store—next to a copy of the New York Times that was dated
6 days after the supposed theft. (Low Risk Crime, Forbes, September 27, 1993, at 56.)
4
A Vehicle, Insurance Firm, and Bad Faith, Albuquerque J., October 5, 1998, at 10.
5
For example, “[a] 1998 Legg Mason report on Allstate . . . shows how profitable it can
be to put the squeeze on claims, which typically account for three-fourths of an insurance
company’s expenses. If Allstate could cut its loss ratio . . . by just one percentage point, the
report says, it would save $186 million and add 28 cents to its after-tax earnings per share.”
See What If the Company Won’t Pay? supra note 1, at 71.
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environment in which insurers face claimants who may expend resources to
inflate their apparent losses and conclude that an optimal settlement strategy
mitigates claims falsification by systematically underpaying those categories
of claims where the cost to claimants of falsification is the lowest. We then
turn to an analysis of data on actual insurance settlements for bodily injury
liability in automobile accidents and find that categories of insurance claims
that are traditionally viewed as being easy to falsify and, hence, susceptible
to inflation—such as sprain injuries or claims involving lost wages—are
systematically undercompensated in comparison with those where falsification is perceived to be less of a problem.
Our work draws on several antecedents in both the litigation and the
insurance literatures. The relationship between anticipated litigation costs
and settlements is a topic that has received substantial attention in law and
economics scholarship.6 While the details of the various litigation models
differ in terms of the specific informational structures and sequences of
actions that are assumed, a general lesson provided by this body of work is
that settlements are used to avoid costly litigation7 and that the larger the
proposed settlement, the lower the expected litigation costs.8 In the context
6
The 1989 survey article by Robert Cooter and Daniel Rubinfeld provides a comprehensive
overview of the various contributions to this literature. More recent empirical studies of legal
settlements include the 1991 article by Henry Farber and Michelle White, which examines the
effect of the quality of medical care actually provided on a defendant’s medical malpractice
liability, and the 1995 article by Daniel Kessler, whose concern is with the effect of the
negligence regime on the settlement amount. Using data on automobile liability settlements,
Kessler finds that settlements are higher under comparative negligence regimes (where damages
are assigned in proportion to fault) than under contributory negligence regimes (where any
negligence by the plaintiff is a complete bar to recovery). Since the negligence regime affects
the magnitude of settlements, we include indicators of negligence regime in our empirical
results reported in Section III. See Robert D. Cooter & Daniel L. Rubinfeld, Economic Analysis
of Legal Disputes and Their Resolution, 27 J. Econ. Lit. 1067 (1989); Henry S. Farber &
Michelle J. White, Medical Malpractice: An Empirical Examination of the Litigation Process,
22 Rand J. Econ. 199 (1991); and Daniel Kessler, Fault, Settlement and Negligence Law, 26
Rand J. Econ. 296 (1995).
7
For example, in their 1985 article, David Rosenberg and Steven Shavell consider a model
in which a plaintiff may file a “nuisance suit” at a nominal expense, after which the defendant
must mount a (costly) defense in order to avoid an even larger default judgement. Since, by
filing, the plaintiff can force the defendant to incur the cost of a defense, the defendant would
always be willing to settle for any amount below that cost. This result holds even if the
defendant knew that it would never pay the plaintiff to actually go to trial. Lucian Bebchuk’s
1988 article also examines the role of nuisance suits in extracting settlement offers. In contrast
to Rosenberg and Shavell, where it is the sequence of moves in the litigation game that generates
the settlement, in Bebchuk’s model it is uncertainty about whether or not the plaintiff would
actually go to trial that provides the defendant with the incentive to settle. See David Rosenberg
& Steven Shavell, A Model in Which Suits Are Brought for Their Nuisance Value, 5 Int’l
Rev. L. & Econ. 3 (1985); and Lucian Arye Bebchuk, Suing Solely to Extract a Settlement
Offer, 17 J. Legal Stud. 437 (1988).
8
For example, in his 1984 article, Bebchuk examines settlements in an environment where
the defendant possesses private information about her degree of fault and, hence, the plaintiff’s
likelihood of prevailing in litigation. In this setting, the optimal settlement offer reflects a
trade-off between the benefits accruing to the plaintiff of making a higher settlement offer (a
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of the insurance settlements that are the focus of this paper, one potential
cost arising from litigation is the penalties that may be imposed on an insurer
for bad faith in failure to settle an insurance claim.
In addition to the usual duties imposed on the parties to a contract, insurers
are held to an implied covenant of good faith and fair dealing,9 the violation
of which is considered to be a tort that exposes the insurer to extracontractual
liability.10 The most common application of this principle has occurred in
the context of third-party liability claims in which an insurer’s failure to
settle for an amount within the policy limits ultimately exposes the insured
to a judgement in excess of those limits.11 If found to be acting in bad faith,
such an insurer may not only incur liability for the entire judgement12 but
may also face liability for consequential13 and punitive14 damages as well.
higher payoff) and the costs (a higher probability of costly litigation). The efficient settlement
offer is accepted by the defendant, whose private information indicates that she is likely to
lose in litigation, and is rejected by the defendant, who expects to prevail. See Lucian Arye
Bebchuk, Litigation and Settlement under Imperfect Information, 15 Rand J. Econ. 404 (1984).
9
This higher standard is generally justified by the superior bargaining position of the insurer
and by the existence of a “special relationship” between the parties to an insurance agreement.
For example, “As the champion of the insured, [the insurer] must consider as paramount [the
insured’s] interests, rather than its own. . . . Its relationship is somewhat of a fiduciary
one. . . . ” (see John Alan Appleman, Insurance Law and Practice § 4711 (1941)) or, as put
somewhat differently by Guy Kornblum in his 1988 article, “a duty is imposed that ‘[An]
indemnity company is held to that degree of care and diligence which a man of ordinary care
and diligence would exercise in the management of his own business.’” See Guy O. Kornblum,
The Current State of Bad Faith and Punitive Damage Litigation in the United States, 23 Tort
& Ins. L. J. 812, 814 (1998).
10
Extracontractual damages refer to “the imposition of damages against these insurers which
are in addition to those usually anticipated in cases of breach of contract.” See Kornblum,
supra note 9, at 813.
11
The precedential case is Crisci v. Security Insurance Company, 426 P.2d 173 (1967). Kent
Syverud’s 1990 article provides an extended discussion of the state of the law regarding an
insurer’s “duty to settle.” It is important to note that the failure to settle with a third-party
claimant in a liability setting (say, as a result of aggressive claims settlement practices) may
expose the insurer to sanctions for bad faith that compensate the insured (that is, the purchaser
of the liability insurance policy) for judgments in excess of the stated policy limits of the
insurance policy and perhaps for other associated costs (see discussion below). See Kent D.
Syverud, The Duty to Settle, 76 Va. L. Rev. 1113 (1990).
12
“If the insurer . . . was guilty of bad faith or negligence in failing to negotiate and bring
about a settlement, the damage to the insured generally is the amount for which the insured
becomes charged in excess of his policy coverage. In fact, it may even be liable for consequential
damages.” See Appleman, supra note 9, § 4711, p. 414.
13
“The compensatory damages recoverable for breach of the implied covenant . . . include
not only contract damages but other consequential damages as well. The fact that breach of
this contract constitutes a tort enables the insured to recover damages for injuries proximately
caused by the insurer’s conduct whether those injuries should have been anticipated or not.
Thus, compensatory damages may include damages for emotional distress, attorney fees generated in an attempt to obtain the policy benefits, and other economic losses.” See Kornblum,
supra note 9, at 831.
14
“[To] award punitive damages in such cases there must not only be a breach of the implied
covenant but also a finding that there has been malice, oppression or fraud.” See Kornblum,
supra note 9, at 819.
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Similar sanctions have been imposed in first-party indemnity settings,15 and
for a period of time in California, even third-party claimants could sue an
insurer for damages resulting from settlement delays or the underpayment
of claims.16 Thus, an insurer choosing to deny, delay, or underpay a claim
assumes a risk that it will be held liable for any resulting damages, even
those in excess of the relevant policy limits.17
Of particular relevance is a recent article by Alan Sykes that examines the
impact of the bad-faith doctrine in first-party insurance settings. In this article,
Sykes also argues that costly litigation may serve as a screening device by
which insurers can sort valid claims from fraudulent ones.18 The environment
considered is one in which insurance adjusters may suspect fraud on the
basis of the specific characteristics of a claim, but where the cost of identifying
fraudulent claims with certainty is prohibitive, so auditing is not practical.
Assuming that the insurer is constrained either to pay the full claim or to
reject the claim in its entirety, he argues that the optimal strategy may be
for the insurer to reject suspicious claims with positive probability and that
the prospect of costly litigation by valid claimants wrongly denied will constrain the proportion of rejected claims.19
We develop a formal model of optimal third-party settlements between a
risk-neutral insurer and claimant, which is in the spirit of this approach.20
15
In contrast to third-party cases, the record in first-party claims has been mixed, although
a majority of the states have awarded extracontractual damages for bad faith. The reason for
the difference is that, in a first-party setting where the insured is also the claimant, the relationship between the insured and insurer is necessarily adversarial. Nonetheless, extracontractual damages have been awarded, and for substantial sums, in many first-party cases. See
Kornblum, supra note 9.
16
The decision in Royal Globe Ins. Co. v. Superior Court, 23 Cal. 3d 880 (1979), permitted
a third-party claimant to sue the insurer for violations of the California Fair Practices Act and
to recover damages. This decision was prospectively reversed in Moradi-Shalal v. Fireman’s
Fund Ins. Co., 46 Cal. 3d 287 (1988).
17
While we have no formal evidence on the extent to which insurers face sanctions for bad
faith as a consequence of underpaying claims, anecdotal evidence suggests this to be a nontrivial
risk exposure. An Internet search using the term “insurance bad faith” returns over 167,000
matches, a casual perusal of which indicates a remarkable number of references to tort attorneys
who handle these types of cases. There are also a number of regular publications (see, for
example, Mealey’s Litigation Report (2001)) that track the developments in the area. Clearly,
suits for bad faith comprise a very profitable line of business for enterprising attorneys.
18
This point is a critical part of the analysis of Alan Sykes in his 1996 article on bad faith
by first-party insurers, particularly at 423–29. See Alan O. Sykes, “Bad Faith” Breach of
Contract by First-Party Insurers, 25 J. Legal Stud. 405 (1996).
19
The benefit of this approach is that claimants perpetrating fraud may drop their claims,
since they are likely to have higher costs of pursuing a denied claim through litigation, while
valid claimants have every reason to proceed. In the context of our terminology, filers of
fraudulent claims must incur significant falsification costs to support their invalid claims.
20
Specifically, Sykes envisions a pure screening environment in which an insurer faces a
pool of claimants consisting of those who either have valid claims or have already exaggerated
their claims through fraud. Both types of claimants face the prospect of (random) denial of
their claims. Since fraudulent claimants are assumed to have higher costs of pressing their
claims through litigation, after denial, only those with valid claims find it worthwhile to proceed
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Our modeling strategy follows that of Keith Crocker and John Morgan, who
characterize efficient first-party insurance contracts in a setting where the
insured may engage in costly state falsification to inflate the apparent magnitude of a claim.21 In contrast, we consider a third-party environment in
which the claimant has suffered an insured loss, the magnitude of which is
private information.22 The insurer can observe only the size of the claim,
which may be larger than the actual loss if the claimant chooses to engage
in claims inflation by expending resources on costly state falsification. Faced
with the prospects of fraudulent claiming, the insurer must select a settlement
strategy, which in the context of this environment consists of an indemnification profile relating the size of the claim to the amount of indemnification
provided. An optimal settlement strategy reflects a tension between the into the litigation stage, while the fraudulent claimants drop out of the process. The costly state
falsification approach that we adopt is consistent with that of Sykes in the sense that one may
view the differential costs of pursuing valid and fraudulent claims as being formally embodied
in the falsification cost function, where truthful claims are costless to file and pursue and
fraudulent claims face a cost that is increasing in the amount of fraud committed. Moreover,
our approach recognizes that the decision to commit fraud is a (privately costly) economic
choice made by the claimant, so the reduced return to fraud in claiming, either through explicit
underpayment of a claim or by denial with positive probability, will reduce the incentive to
commit fraud ex ante. See Sykes, supra note 18.
21
The distinction between the first-party costly state falsification model developed by Keith
Crocker and John Morgan in their 1998 article and the one we consider here is more than one
of semantics. In their environment, the optimal indemnity contract results in underpayment of
claims at the margin and reflects a tension between the benefits of underpayment (reduced
incentives of the insured to expend resources in claims inflation) and the costs (less income
smoothing for the risk-averse insured). Interestingly, fraud is shown to have value in the optimal
insurance contract because it provides a mechanism by which the informationally constrained
insurer can sort the claimants on the basis of their underlying private information (that is, the
magnitude of the loss actually suffered). In the third-party environment that we consider in
this paper, the relationship analyzed is between the liable insurer and the claimant to which
the loss has already occurred. The optimal settlement strategy reflects a trade-off between the
benefits of claims underpayment (reduced incentives of the claimant to falsify) and the costs
(increased exposure by the insurer to the administrative and litigation costs associated with
claims underpayment and the sanctions for bad faith discussed above). In this setting, the
insurer harnesses the differential costs of filing falsified and valid claims to sort the claimants
who have private information about their losses, so those suffering larger losses will receive
higher compensation. See Keith J. Crocker & John Morgan, Is Honesty the Best Policy?
Curtailing Insurance Fraud through Optimal Incentive Contracts, 106 J. Pol. Econ. 355 (1998).
22
Note that the costly state falsification approach that we adopt assumes that the informational
asymmetry (the magnitude of the actual loss) is immutable, in contrast to an auditing approach
(generically referred to as “costly state verification”), which permits the uninformed agent to
obtain the private information, albeit after bearing a monitoring cost. Influential work in this
area includes Robert Townsend’s 1979 original analysis of the problem and, in insurance
settings, the 1994 article by Louis Kaplow, and the 1997 article by Eric Bond and Keith
Crocker. See Robert M. Townsend, Optimal Contracts and Competitive Markets with Costly
State Verification, 21 J. Econ. Theory 265 (1979); Louis Kaplow, Optimal Insurance Contracts
When Establishing the Amount of Losses Is Costly, 19 Geneva Papers on Risk & Ins. Theory
139 (1994); and Eric W. Bond & Keith J. Crocker, Hardball and the Soft Touch: The Economics
of Optimal Insurance Contracts with Costly State Verification and Endogenous Monitoring
Costs, 63 J. Pub. Econ. 239 (1997).
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475
surer’s attempts, through underindemnification,23 to mitigate the claimant’s
incentives to falsify on the one hand and the insurer’s desire to avoid the
expected litigation costs associated with systematic claims underpayment on
the other.24 The primary implication of the theoretical model is that an optimal
settlement strategy entails systematic underpayment of claims and that the
degree of underpayment should be larger for classes of claims where falsification is easiest.
The empirical analysis employs a large data set of liability insurance claims
for bodily injuries experienced in automobile accidents, which provides a
particularly appropriate venue for examining the implications of the theory.
Automobile insurance claims is an area in which fraud is legendary,25 and
automobile liability insurance claims are thought to be particularly prone to
exaggeration due to the possibility of compensation for pain and suffering
in addition to economic losses. Moreover, the severity of the injuries often
encountered in automobile accidents, such as sprains or other soft-tissue
injuries, may be inherently unverifiable, and some of the types of claims
filed, such as those for lost wages, may be manipulable by the claimant.26
In such a setting, ex post monitoring is likely to be an ineffectual method
of uncovering and deterring fraud, and insurers should utilize all of the tools
at their disposal—including indemnification strategies—to mitigate the incentives of claimants to engage in claims inflation.
23
As we show in the theoretical model below, when claimants face falsification costs that
are increasing in the amount of claims inflation, the incentives for fraud may be mitigated by
reducing the indemnification paid at the margin as claim size increases.
24
Note that Sykes requires an all-or-nothing approach to settlement in which claims are fully
paid or rejected in their entirety. But this is really a distinction without a difference in our
model with risk-neutral insurers and claimants, since what really matters is the expected value
of making a fraudulent claim. In other words, a claim underpaid with certainty is equivalent
to one fully paid with some positive probability. See Sykes, supra note 18.
25
In their 1990 article, Sean Mooney and Jeanne Salvatore find that consumers rank automobile insurance as the area in which fraud is most common. The 1991 analysis of Herbert
Weisberg and Richard Derrig and the 1995 study by Allan Abrahamse and Stephen Carroll,
which examine automobile liability insurance claims, have estimated that 10–30 percent of
claims are suspicious or exaggerated. See Sean Mooney & Jeanne Salvatore, Insurance Fraud
Project: Report on Research, New York: Insurance Information Institute (1990); Herbert I.
Weisberg & Richard A. Derrig, Fraud and Automobile Insurance: A Report on Bodily Injury
Claims in Massachusetts, 10 J. Ins. Reg. 497 (1991); and Alan F. Abrahamse & Stephen J.
Carroll, The Frequency of Excess Claims for Automobile Personal Injuries, in Automobile
Insurance: Road Safety, New Drivers, Risks, Insurance Fraud and Regulation (Georges Dionne
& Claire Laberge-Nadeau eds. 1999).
26
When the magnitude of a loss is verifiable after the fact, as in the case of property damage
from fire or flood, then ex post monitoring may be an effective strategy to deter fraud. The
efficacy of such monitoring, however, is less pronounced in settings where the nature of the
injury cannot be diagnosed with certainty, such as in the case of pain resulting from back or
neck injuries, or when the claimant can take actions that inflate the loss, such as malingering
to inflate claims for lost wages.
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II.
A Model of Claims Settlements
We consider an environment consisting of a risk-neutral claimant and
insurer.27 The claimant has suffered an injury, the actual magnitude of which,
x, is private information. The insurer who bears the liability knows only that
an accident has occurred and that the actual loss amount is drawn from the
interval [0, ⬁] according to the probability distribution function F. An individual suffering the loss x may file the claim y (≥ x), but in doing so must
bear the resource cost y(x ⫺ y) 2, where y 1 0 is a falsification cost parameter.
For simplicity, we assume that the filing of an honest claim (y p x) is
costless and that the costs of falsification are strictly increasing in the amount
of fraud, y ⫺ x. This accords with common intuition, as small amounts of
fraud tend to be relatively easy to accomplish, while more substantial attempts
are more costly to implement successfully. These falsification costs include
the direct and indirect expenditures that are associated with building up the
claim, such as visits to doctors or chiropractors, malingering, or the hiring
of attorneys and expert witnesses. Also relevant are the potential legal costs
and court sanctions associated with the criminality of filing fraudulent
claims.28
The preferences of a claimant who suffers the loss x and files the claim
y may be written as
U (W ⫺ x ⫹ I(y) ⫺ y(x ⫺ y) 2) ,
(1)
where U and W are the claimant’s utility function and initial wealth, respectively, and I(y) is the level of indemnification received from the insurer
when a claim of y is filed. Given x and the indemnification schedule I(y),
the utility-maximizing claim of the injured party satisfies the first-order condition dU/dy p 0, which implies
y p x ⫹ I (y)/2y,
(2)
27
Note that we are considering an environment of third-party insurance in which the individual suffering the loss is not the one who purchased the insurance coverage. This is the case
in liability settings, where an individual purchases insurance to provide indemnification against
claims levied for injuries suffered by third parties as a consequence of the insured’s actions.
In contrast, first-party insurance, such as property, health, or life insurance, indemnifies the
insured against his own losses.
28
The legal penalties associated with fraudulent claiming are not trivial. In their 2000 study,
Richard Derrig and Valerie Zicko find that of the approximately 3,100 cases of suspected
automobile and workers’ compensation fraud that were referred to the Insurance Fraud Bureau
of Massachusetts from 1991 to 1999, approximately 1,900 have reached final disposition. Of
these cases, 502 were referred for prosecution and 87 percent resulted in guilty or equivalent
verdicts. Jail sentences were given in 44 percent of the cases, and probation and restitution in
60 percent. See Richard A. Derrig & Valerie A. Zicko, Prosecution Outcomes 1991 through
1999 (2000).
insurance claims strategies
477
Figure 1.—Claims settlement strategy sequence of events
so the claimant will engage in claims inflation as long as the marginal return
on increased claiming (I ) is positive.29
When contemplating the settling of a claim, the insurer recognizes that
the claimant’s incentives to engage in claims inflation depend on the form
of the indemnification schedule, I(y). A settlement strategy that results in an
indemnification profile that reduces the marginal return associated with filing
a larger claim will erode the incentives of the claimant to invest resources
in embellishment. Such an approach, however, necessarily underpays some
claims, which generates settlement costs to the insurer through either litigation initiated by the aggrieved claimant or the prospect of formal court
sanctions for bad-faith failure to settle.
To formalize the economic trade-off faced by insurers in the crafting of a
claims settlement strategy, we consider the sequence of events depicted in
Figure 1. At date 1, the claimant suffers an injury x, the magnitude of which
is private information. The insurer observes only that an accident has occurred
and the falsification cost parameter, y, which is associated with either the
type of injury suffered or the category of claim made by the injured party.30
After observing the class of injury or claim as indicated by y, the insurer
selects at date 2 the settlement strategy, S { {a, b}, which is the specification
29
Note that the indemnification profile I(y) results in a sorting of the claimant types on the
basis of their underlying private information, x. In other words, the claimants signal types by
their selection of the claim signal, y.
30
For example, an accident may involve an automobile collision in which the injured party
has suffered neck trauma (“whiplash”), a class of injury that is both difficult to quantify precisely (since levels of pain are not readily observable) as well as notoriously easy to embellish
(so y is low). Similarly, it is fairly easy for a claimant to generate past wage losses and the
expectation of future wage losses by malingering. (Consider the common anecdotes of insurance
investigators surreptitiously photographing purportedly injured claimants engaging in yard work
or golfing.) Alternatively, injuries with discernable physical manifestations, such as fractures,
are easier to diagnose and more difficult to inflate (so y is high), and generating claims for
medical losses requires actual bills for hospital stays and visits to doctors.
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478
of a linear indemnification profile to govern the claim settlement.31 The
indemnification schedule resulting from the strategy S may be written as
I(a, b, y) { (1 ⫺ b)a ⫹ by.
(3)
After observing the insurer’s settlement strategy, at date 3 the claimant
selects the claim, y, that maximizes her postaccident utility (see equation
(1)). Finally, at date 4 the claim is settled, at which time the claimant receives
the indemnification I(y) and the insurer bears the expected litigation and
settlement costs of c(y ⫺ I), which are assumed to be increasing in the extent
to which claims are underpaid, y ⫺ I. In the analysis that follows, we will
assume that c 1 0, c 1 0, c(0) p 0, and c (0) p 1.32 Note that our specification of settlement costs is consistent with the implications of the litigation
and settlements literature previously discussed, in the sense that, ceteris paribus, lower settlement offers result in higher expected litigation and administrative costs.
The objective of the insurer is to design a settlement strategy that minimizes
the sum of the expected indemnification and settlement costs. Formally, the
insurer’s problem may be written as
min
{a, b}
冕
⬁
冕
⬁
I(a, b, y)f (x)dx ⫹
c (y ⫺ I(y)) f (x)dx,
(4)
xˆ
0
where y is determined by the claimant’s optimization (equation (2)) and
[x̂, ⬁] is the range of losses that result in claims underpayment (y 1 I) given
the settlement strategy S. The relationship between I and y is depicted in
Figure 2, and it is straightforward to demonstrate that x̂ p a ⫺ b/2y.33 The
following lemma characterizes an optimal settlement strategy, S*.
Lemma. The cost-minimizing settlement strategy S* { {a*, b*} satisfies
冕
⬁
(i)
ˆ ) f (x)dx p 1
c ((1 ⫺ b)(x ⫺ x)
x̂
31
Since our goal in developing this model is only to derive theoretical implications that are
to be examined in the context of data on actual bodily injury liability claims, we have simplified
the analysis by restricting our attention to linear indemnification schedules and quadratic falsification costs. A more general approach would be to permit nonlinear indemnification schedules and to use the Revelation Principle first discussed by Roger Myerson in his 1979 article
to characterize an optimal settlement contract, which was the technique adopted by Crocker
and Morgan in their 1998 analysis of first-party insurance contracts in environments with
claims falsification. See Roger B. Myerson, Incentive Compatibility and the Bargaining Problem, 47 Econometrica 61 (1979); and Crocker & Morgan, supra note 21.
32
The assumption that c (0) p 1 is simply a normalization and is chosen so that, in the
absence of claims fraud (that is, y p ⬁ ), insurers have no incentive to undercompensate claims.
33
Note that we are assuming that settlement costs are positive when y 1 I and zero otherwise.
insurance claims strategies
479
Figure 2.—Relationship between indemnification (I ) and claim ( y)
and
冕
⬁
(ii)
ˆ ) f (x)dx p m ⫹ 1/2y,
xc ((1 ⫺ b)(x ⫺ x)
x̂
where m { ∫ 0 xf (x)dx.
Proof. Using the definitions of x̂ and y and integrating, we may rewrite
the cost of indemnification and settlement as
⬁
冕
⬁
R { (1 ⫺ b)a ⫹ b /2y ⫹ bm ⫹
2
ˆ ) f (x)dx,
c ((1 ⫺ b)(x ⫺ x)
x̂
and the first-order conditions for an interior maximum are
冕
⬁
⭸R/⭸a p (1 ⫺ b) ⫺
ˆ )(dx/da
ˆ )
c ((1 ⫺ b)(x ⫺ x)
x̂
ˆ ˆ
# (1 ⫺ b)f (x)dx ⫺ c(xˆ ⫺ x)dx/da
p0
and
冕
(5)
⬁
⭸R/⭸b p ⫺a ⫹ m ⫹ b/y ⫹
ˆ)
c ((1 ⫺ b)(x ⫺ x)
x̂
ˆ ⫺ (1 ⫺ b)dx/da
ˆ ) f (x)dx
# (⫺(x ⫺ x)
(6)
ˆ p 0.
ˆ
⫺ c(xˆ ⫺ x)dx/db
Condition (i) of the lemma follows directly from equation (5) and the fact
480
the journal of law and economics
that c(0) p 0. To obtain condition (ii), substitute condition (i) into equation
(6) and simplify the result. Q.E.D.
As a point of departure, consider for the moment the implications of the
lemma for the shape of the cost-minimizing indemnity profile were claims
falsification to be prohibitively costly, so y p ⬁. Then, xˆ p a and, since
c (0) p 1 by assumption, the settlement strategy {a p 0, b p 1} solves (i)
and (ii) so, in the absence of claims falsification, the best strategy of the
insurer is to pay fully all claims. Alternatively, when y ! ⬁, claimants will
invest in fraud (y 1 x) as long as b 1 0, so the return to claims inflation is
positive. Even though the insurer could, in principle, completely eliminate
fraud by choosing a flat indemnification profile (so b p 0), such a strategy
is never optimal as long as the settlement costs associated with underpayment
of claims are positive.34
The insurer can mitigate the claimant’s incentives to falsify by flattening
the indemnification profile, but this will be at the cost of the increased
settlement costs that result from underpaid claims. Since deterring embellishment in this fashion is costly to the insurer, such a strategy is more
attractive in cases where falsification is less expensive to the claimant and,
hence, more likely. The following result, the proof of which may be obtained
from the authors upon request, will form the basis for our empirical work.35
Theorem. The cost-minimizing settlement strategy of the insurer satisfies db/dy 1 0.
The result of the theorem is that, as the cost to the claimant of committing
fraud increases, the optimal settlement strategy pays a higher proportion of
the filed claim. The implication is that if claims can be dichotomized into
categories with distinctly different falsification costs so y is large for one
class of claims and small for the other, then the former group should receive
more generous compensation, at the margin, than the latter. In particular, the
classes of injuries or claims that are characterized by lower falsification costs
should be associated with settlement strategies in which the marginal indemnification, as a function of the claimed amount, is lower.36 We now turn to
34
It is also worth noting that it is through the use of falsified claims that the insurer is able
to sort the claimants on the basis of their underlying (privately known) losses. With an indemnification profile that generates no fraud (so b p 0), all of the claimants are treated the
same and receive the same settlement amount. By exploiting the costly state falsification
environment, the insurer is able to pay more to claimants who experience higher losses, that
is, to treat the claimants differently on the basis of their underlying types. Put differently, the
claimants credibly signal their losses, x, though their choice of claim, y.
35
We express our appreciation to Jiangdong Ju for his assistance in the formal proof of the
theorem.
36
The effect of changes in the falsification cost parameter, y, on the intercept, (1 ⫺ b)a, is
less clear. To the extent that flattening of the settlement profile (that is, lower b) results in an
increase in claims underpayment (y ⫺ I ) and higher settlement costs (c), then an increase in
a serves to mitigate those costs. (The increase in a, however, increases the amount by which
small claims are overpaid.) While this intuition is correct, it turns out that the amount of
overpayment, y ⫺ I, is not necessarily monotonic as a function of b. From equations (2) and
insurance claims strategies
481
an investigation of data on automobile insurance settlements in order to
determine whether the structure of the settlements reflects the predicted relationship with the underlying costs of committing fraud.
III.
Characteristics of the Data
A.
The Insurance Claims
We analyze data on individual insurance claims for automobile-related
injuries. The data are from a nationwide study of automobile injury compensation, in which 34 automobile insurance companies completed extensive
reports on automobile injury claims that were closed in a 2-week period.37
The data include insurance claims from accidents in all 50 states and the
District of Columbia.
The data involve injury claims reported under bodily injury liability (BIL)
coverage. Bodily injury liability claims are filed by individuals who are
injured in an accident that is the fault of another driver. The claims are filed
against the insurance coverage of the at-fault driver, and claimants are eligible
for compensation of all financial losses due to injury (including medical bills,
lost wages, and rehabilitation expenses) without deductibles or copayments.38
In addition, claimants are eligible for general damages awards, which are
intended to compensate the victim for other losses associated with the injury,
such as “pain and suffering,” which are by their nature not documentable.
Since the amount of general damages awarded is usually linked to the amount
of direct financial loss experienced by the claimant,39 there is an incentive
for claimants to exaggerate the amount of their financial loss. General damage
(3), it follows that d(y ⫺ I )/da p ⫺(1 ⫺ b) and that d(y ⫺ I )/db p (xˆ ⫺ x) ⫹ (1 ⫺ b)/2y, which
is positive for values of x sufficiently close to x̂ and negative for larger values of x. Accordingly, there are no definitive theoretical predictions with respect to the effect on the intercept
of changes in y and, hence, on the degree by which smaller claims are overpaid. It is straightforward to demonstrate, however, by totally differentiating part (i) of the lemma, that
ˆ
dx/db
! 0. This, in conjunction with the result of the theorem, implies that the number of
overpaid claims increases as the falsification cost parameter y declines.
37
The survey was undertaken in 1987 by the Insurance Research Council, an industry
research and advisory group, which makes the data available for purchase by researchers. In
order to maximize participation, each insurance company participating in the study was allowed
to choose the 2-week period for which it reported.
38
The payment amount is limited, of course, by the maximum coverage limit of the policy.
As we shall see below, the coverage limits of liability insurance policies are generally high in
relation to the claimed amounts, and so this type of censoring of the payment amount is
uncommon.
39
See Herbert I. Weisberg & Richard A. Derrig, Modelling the Payment of General Damages
for Massachusetts Automobile Bodily Injury Liability Claims (1991); and James K. Hammitt,
Payments by Auto Insurers (1985).
482
the journal of law and economics
awards are often argued to be the primary motivating factor for liability
claims fraud.40
Bodily injury liability claims have a number of desirable characteristics
for use in our empirical analysis. The absence of deductibles or copayments
lessens concerns about biases in the sample of filed insurance claims. Full
compensation of losses, along with the possibility of general damages awards,
implies that even small amounts of losses are worth claiming. In addition,
since there are no prespecified limits to the amounts of general damages for
which the insured can be held liable, insureds tend to choose relatively high
policy coverage limits. As a result, we observe in the data BIL claims of
greatly varying sizes, for which the amount paid to the claimant is rarely
censored by the policy coverage limit.
Our data include extensive information on each claim that make them
uniquely suited to the task of examining the settlement responses to the
problem of claims falsification. Both the claimed amount and the amount
paid by the insurer are reported in some detail, including amounts of medical
expenses and lost wages, and also reported is the amount of general damages
paid by the insurer. Details are provided regarding the circumstances of each
accident, including the location (by state and size of city), the number of
vehicles involved, and whether the accident involved traffic offenses or resulted in traffic citations. The data also contain information regarding the
nature of the injuries incurred by the claimant and the extent of trauma
suffered. Finally, there are some data on the personal characteristics of the
claimant, including age, sex, and marital and employment status.
B.
Sample Selection
To preserve the homogeneity of the insurance claiming and settlement
environment, we eliminate from our sample all claims filed under residual
market insurance policies and those filed under no-fault insurance regimes.
Residual market policies are made available in all states to high-risk drivers
who are unable to obtain insurance in the private market. In many states,
the profits or losses incurred from residual market policies are shared across
all automobile insurers in the state. This implies that the handling and payment of claims under these policies may differ from those filed under voluntary market policies.41 Claims subject to no-fault laws are also excluded
40
See J. David Cummins & Sharon Tennyson, Controlling Automobile Insurance Costs, 6
J. Econ. Persp. 95 (1992); J. David Cummins & Sharon Tennyson, Moral Hazard in Insurance
Claiming: Evidence from Automobile Insurance, 12 J. Risk & Uncertainty 29 (1996); and
Weisberg & Derrig, supra note 25.
41
See Glenn B. Blackmon & Richard Zeckhauser, Mispriced Equity: Regulated Rates for
Auto Insurance in Massachusetts, 81 Am. Econ. Rev. 65 (1991).
insurance claims strategies
483
from the sample because the incentives for falsification are different for these
claims.42
Following previous empirical studies of liability insurance claims, we eliminate claims involving fatality or permanent total disability because the claim
and payment amounts for these claims exhibit different characteristics than
those for other injuries.43 We also omit all sample observations for which
claims were filed with more than one insurer. From the data available, it is
unclear in these cases the extent to which the payment strategies of the various
insurers involved were coordinated, and the survey coder was encouraged
to estimate the amounts paid under other policies if it was unknown. Finally,
we omit from the sample claims for which the data are not usable. Included
in this category are claims for which no loss amount is reported, those for
which the reported loss is zero, and those that demonstrate obvious inconsistencies. Since we are interested in knowing whether the payment amount
was constrained by the policy coverage limit, we also eliminate claims for
which the policy limit is missing or recorded as zero. Making these adjustments to the data leaves a sample of 12,866 insurance claims.44
C.
Variable Measurement
The data provide a number of possible measures of the amount paid by
the insurer (I ) relative to the loss amount reported by the claimant (y). We
measure the payment amount (I ) in two alternative ways. The first measure
we employ is the amount of economic damages paid by the insurer to the
claimant, including the economic damages paid under BIL coverage and any
such amounts paid under other automobile insurance coverages with the same
42
No-fault laws limit the ability of those injured in automobile accidents to file BIL claims.
Only claims exceeding a threshold level of severity, as specified in each state’s law, are eligible
for such awards. These thresholds create a greater incentive for the exaggeration of small
claims, which does not exist in states that use a traditional tort system of compensation. Each
claim in the survey is coded with respect to whether a tort threshold was relevant for the
claim. This coding is the basis for the elimination of claims. See Weisberg & Derrig, supra
note 25; and Cummins and Tennyson, supra note 40.
43
For fatalities, the reported economic loss amounts tend to be artificially low relative to
the severity of the injury because medical expenses and rehabilitation costs are lower. For both
fatalities and permanent total disability claims, the proportion of wage losses claimed under
economic damages is also much larger than for other claims. The claims are also much more
likely than other claims to report losses greater than the insurance policy limits. The disproportionately large amounts of losses claimed for these observations, and the distinct patterns
of claim amounts, led us to be concerned about the potential for these claims to unduly influence
our findings. See Weisberg & Derrig, supra note 39; and Hammitt, supra note 39.
44
The original sample included 21,688 BIL claims. Since many omitted claims possessed
more than one of the omission criteria, the exact numbers of claims eliminated by each of the
selection criteria depend on the order of elimination. Using the following order, 2,785 claims
were omitted because they were filed against residual market policies, 1,370 had missing data
for the claimed amount or the policy limit, 2,991 involved payments by more than one insurer,
1,584 were settled under no-fault regimes, 4 had miscellaneous reporting inconsistencies, and
88 involved claimant fatality or permanent total disability.
484
the journal of law and economics
insurer. Our second measure is the total amount paid to the claimant by the
insurer, which is the sum of economic damages payments and any payments
for general damages. This latter measure is of course the most comprehensive
measure of the amount paid. However, the amount of general damages sought
by the claimant is not reported in our data set, which limits our ability to
examine payment amounts relative to claimed amounts when using this measure. Hence, we examine both the economic damages payment amount and
the total damages payment amount in relation to the economic damages
amount claimed. We measure the amount of economic damages claimed, y,
as the sum of all medical expenses45 and wage losses.46
We utilize two alternative sets of proxy variables for the costs of falsification (y).47 Previous empirical studies of moral hazard and fraud in insurance
claiming suggest that one method of segregating claims by level of falsification costs is through the category of injury. In their study of workers’
compensation claims, Georges Dionne and Pierre St-Michel cite medical
experts who maintain that injuries can be classified according to the degree
of diagnostic difficulty. Perhaps not surprisingly, injuries involving spinal
disorders or lower back pain are categorized as very difficult to diagnose,
while injuries such as contusions, amputations, fractures, and burns are the
easiest.48 On the basis of these studies, we partition the sample of BIL claims
into two injury categories: those involving sprain injuries and those involving
no sprains. We hypothesize that claims involving sprain injuries are less
costly to falsify than those not involving soft-tissue injuries, and hence the
theory predicts that the slope of the claims settlement schedule should be
flatter for claims involving sprains.
We also investigate a measure of falsification costs that uses the data
available on the category of economic losses claimed. Medical expense
45
Our medical expense category also includes amounts for rehabilitation, replacement, and
other service expenses.
46
Note that, while medical payments do not accrue to the injured party, they may serve as
a basis for awarding general damages, which is a monetary payment to the claimant. Accordingly, individuals have the incentive to inflate all categories of economic damages.
47
Although not reported here, we also analyzed the data by partitioning claims into accidents
occurring in urban areas and accidents occurring in rural areas. Costs of falsification are often
argued to be lower in urban areas, where attorneys and doctors may take advantage of economies
of scale in producing documentation for exaggerated injury claims and the costs of information
transmission about such activities may be lower. The results of our analysis are consistent with
this view and consistent with the theory under the maintained hypothesis that falsification costs
are lower in urban areas. That is, settlement schedules vary systematically across urban and
rural claims, with flatter payment profiles observed for urban claims than for rural claims. See
Cummins & Tennyson, supra note 40.
48
In their 1991 analysis of fraud in the context of workers’ compensation claims, Georges
Dionne and Pierre St-Michel hypothesize that difficult-to-diagnose injuries should be more
prone to falsification. Consistent with this hypothesis, they find that worker injuries involving
lower back pain exhibit greater increases in recovery time in response to increases in the
generosity of compensation benefits. See Georges Dionne & Pierre St-Michel, Workers’ Compensation and Moral Hazard, 73 Rev. Econ. & Stat. 236 (1991).
insurance claims strategies
485
claims require documentation of the specific services and the date and cost
of each service received from each licensed practitioner who provides medical
treatments. Wage losses must also be documented, but the documentation
involves a simple form filled out by the claimant’s employer, which reports
the dates of claimant employment, absences following the accident, gross
earnings, and eligibility for other insurance payments.49 Thus, the documentation required to falsify wage losses is much lower than that for medical
expenses. Moreover, even documented wage losses may contain falsification
if the claimant extended his or her work absence by malingering. Such moral
hazard problems have been identified in previous empirical studies of injured
workers’ behavior in the presence of wage replacement insurance.50 The
extent of true wage losses may also be exaggerated by neglecting to report
earnings in the casual labor market. Finally, it may be particularly easy to
exaggerate future wage losses (for which BIL claimants are eligible), which
must by definition be only estimates of a claimant’s future earnings possibilities. On the basis of this reasoning, we partition the data into two reported
loss categories: claims reporting medical losses only and claims reporting
both medical and wage losses. To the extent that wage losses are easier to
falsify than claims for medical expenses, the theory predicts that the slope
of the payment schedule should be flatter for claims involving wage losses
than for claims involving only medical losses.
D.
The Distribution of Claimed Amounts
Table 1 reports the summary characteristics of the distribution of the claims
for economic losses in our sample of observations. The first column of the
table shows that the minimum claim in the sample is $1 and the maximum
is $267,095. The mean claim amount is $2,145.96, but the median claim is
49
Richard A. Derrig, personal correspondence with the author, February 9, 2001. Consistent
with this, the data set of injury claims that we utilize in the study contains extensive separate
data entries on each medical and diagnostic procedure undergone by each claimant. In contrast,
there is a single variable entered for the wage loss amount and a single question in the survey
regarding whether the wage loss was verified.
50
In a 1995 study, Bruce Meyer, W. Kip Viscusi, and David Durbin compare the duration
of work absence before and after an increase in benefits for some workers and find significant
increases in duration for workers affected by the benefit increase but no changes in duration
for workers unaffected by the increase. Richard Butler’s 1996 analysis compares duration of
work injuries for workers in Texas firms that offer workers’ compensation benefits and those
that do not and found evidence that workers receiving benefits take significantly more time
off work than workers not receiving benefits. Also, Richard Butler and John Worrall’s 1991
article finds that increases in workers’ compensation benefits caused moral hazard, which was
evidenced by an increase in claims filed for lost wages but not for medical expenses. See
Bruce D. Meyer, W. Kip Viscusi, & David L. Durbin, Workers’ Compensation and Injury
Duration: Evidence from a Natural Experiment, 85 Am. Econ. Rev. 322 (1995); Richard J.
Butler, Lost Injury Days: Moral Hazard Differences between Tort and Workers’ Compensation,
63 J. Risk & Ins. 405 (1996); and Richard J. Butler & John D. Worrall, Claims Reporting and
Risk Bearing Moral Hazard in Workers’ Compensation, 58 J. Risk & Ins. 191 (1991).
the journal of law and economics
486
TABLE 1
Distribution of Claimed Amounts
Mean
SD
Minimum
25th percentile
Median
75th percentile
Maximum
Skewness
Full Sample
(N p 12,866)
Sprain Claims
(N p 9,714)
Nonsprain
Claims
(N p 3,152)
Wage Claims
(N p 4,649)
Nonwage
Claims
(N p 8,217)
2,145.96
6,008.22
1.00
236.00
722.00
2,166.00
267,095.00
18.08
2,108.89
5,407.62
1.00
309.00
903.00
2,330.00
267,095.00
20.19
2,260.22
7,564.91
1.00
125.00
318.50
1,174.50
239,000.00
14.24
3,494.31
8,673.76
10.00
546.00
1,451.00
3,455.00
267,095.00
14.58
1,383.10
3,514.61
1.00
160.00
461.00
1,429.00
134,069.00
14.13
Note.—Claimed amounts are economic damages claimed. SD p standard deviation.
only $722, suggesting that the claims distribution is significantly skewed
toward zero. Indeed, over 25 percent of claims are for amounts less than
$236, and more than 75 percent of claims are for amounts less than $2,166.
Hence, our claims distribution exhibits the characteristics observed of most
accident loss distributions, which consist of many small claims with decreasing numbers of claims for larger dollar amounts.
The remaining columns of the table show the distribution of claimed
amounts for sprain versus nonsprain claims and wage versus nonwage claims.
Over three-quarters of the claims in the sample involve a sprain injury (9,714
of 12,866 claims), while only about one-third of the claims include a claim
for lost wages (4,649 of 12,866 claims). The characteristics of the sprain and
wage claims distributions are also quite different, which indicates that the
two alternative characterizations of falsification cost differences achieve distinct sample partitions.
IV.
Econometric Evidence on Settlement Strategies
In a regression setting, the testable empirical implication of the theory is
that the size of the insurance payment, I, will depend on not just the amount
of the claim, y, but also the cost of falsification, y. Thus, we may write the
basic empirical relationship as
Ii p bo ⫹ bi yi ⫹ b 2 yi yi ⫹ g X i ⫹ ␧i ,
(7)
where Xi denotes a vector of control variables and ␧i is an independent and
identically distributed error term. The insurance payment is predicted to
increase in the size of the claimed loss, so b1 is greater than zero. Increases
in falsification costs should result in a steeper I(y) schedule, so b2 is also
greater than zero. These two hypotheses are tested in the analysis that follows.
insurance claims strategies
A.
487
Empirical Specification
We regress the claims payment amount on the claimed amount of loss,
allowing for different intercept and slope coefficients for claims with high
falsification costs and claims with low falsification costs. One set of models
uses the sprain versus nonsprain categorization, and a separate set of models
uses the wage versus nonwage categorization. We test the hypothesis that
the slope of the payment schedule for sprain claims is flatter than that for
nonsprain claims and that the payment schedule slope for wage claims is
flatter than that for nonwage claims.
Because many features of a claim and its settlement environment will
affect the amount paid to an individual claimant, we include a number of
control variables in the empirical model. The first set of controls involves
the severity of injury suffered by the claimant as measured by the length of
the claimant’s hospital stay, the extent of the claimant’s disability, the degree
of trauma experienced, and the weeks of work lost by the claimant. The
length of hospital stay is incorporated by the use of two dummy variables,
the first indicating a hospital stay of 2–7 days and the second indicating a
hospital stay of more than 7 days; the omitted category is no hospitalization.
The claimant’s disability status is given by two dummy variables, the first
indicating temporary disability and the second indicating permanent disability; the omitted category is no disability. The extent of trauma is controlled
through several dummy variables that indicate minor, moderate, severe, or
catastrophic trauma; the omitted category is no trauma. The weeks of work
lost by the claimant are reported directly in the data set and include full
weeks and partial weeks lost.
The second set of control variables involves the legal environment for
claims settlement. These include the insured’s degree of fault in the accident,
the negligence regime in the state where the accident occurred, and whether
joint and several liability applied to the claim. A measure of fault is included
because a lesser degree of fault reduces the liability of the injurer and hence
will reduce the damages paid by his insurer.51 Accordingly, we include in
our model an indicator variable equal to one if a traffic citation was issued
in the accident, as a proxy for the insured’s degree of fault.52 However, since
negligence regimes differ across states with regard to how the degree of fault
affects liability for damages, we also include indicator variables for the
51
Although the data include an estimate of the insured’s degree of fault, this is the insurer’s
estimate and hence may be determined jointly with the payment amount. Previous research
using these data has shown that this is a valid concern but also finds that traffic citations issued
in the accident provide valid instruments for fault. See Kessler, supra note 6.
52
Results are quite similar when the estimated degree of fault is included directly in the
regression models.
TABLE 2
Means and Standard Deviations of Key Variables
488
Economic damages claimed
Economic damages paid
Total damages paid
Insurance policy limit
Claimant age
Female
Married
Hospitalized 2–7 days
Hospitalized 17 days
Full Sample
Sprain Claims
Nonsprain Claims
Wage Claims
Nonwage Claims
2,145.963
(6,008.221)
2,017.032
(4,512.547)
5,099.898
(9,985.049)
87,414.190
(138,529.200)
31.720
(16.101)
.529
(.499)
.351
(.477)
.038
(.191)
.020
(.141)
2,108.889
(5,407.616)
2,000.023
(4,217.709)
4,944.954
(8,517.747)
89,465.200
(141,585.200)
33.027
(14.777)
.542
(.498)
.374
(.484)
.025
(.156)
.011
(.103)
2,260.221
(7,564.908)
2,069.451
(5,319.804)
5,577.411
(13,532.130)
81,093.270
(128,452.500)
27.620
(19.111)
.488
(.500)
.277
(.448)
.077
(.266)
.050
(.218)
3,494.310
(8,673.760)
3,168.580
(6,030.580)
7,274.290
(12,491.660)
88,810.930
(137,427.200)
32.507
(11.876)
.493
(.500)
.392
(.488)
.048
(.214)
.027
(.161)
1,383.097
(3,514.608)
1,365.512
(3,183.836)
3,869.673
(7,978.171)
86,623.950
(139,150.900)
31.266
(18.076)
.549
(.498)
.327
(.469)
.032
(.175)
.017
(.129)
Weeks of work lost
Temporarily disabled
Permanently disabled
Minor trauma
Moderate trauma
Severe trauma
Catastrophictrauma
Citation issued
489
% insured degree of fault
Joint and several liability
2.113
(16.331)
.429
(.495)
.043
(.202)
.541
(.498)
.355
(.479)
.055
(.228)
.001
(.025)
.815
(.389)
94.611
(16.147)
.004
(.054)
2.116
(14.424)
.046
(.499)
.033
(.178)
.535
(.499)
.393
(.488)
.040
(.196)
.000
(.010)
.813
(.390)
95.281
(15.042)
.004
(.065)
Note.—Standard deviations are in parentheses. All damages are in dollars.
2.105
(21.156)
.319
(.466)
.073
(.261)
.561
(.496)
.237
(.426)
.100
(.300)
.002
(.047)
.819
(.385)
92.536
(19.027)
.004
(.064)
5.724
(26.047)
.697
(.460)
.064
(.244)
.446
(.497)
.446
(.497)
.080
(.271)
.000
(.021)
.841
(.365)
95.612
(13.742)
.002
(.049)
.070
(4.714)
.277
(.448)
.031
(.173)
.595
(.491)
.304
(.460)
.041
(.197)
.001
(.027)
.799
(.400)
94.044
(17.338)
.005
(.072)
TABLE 3
Ordinary Least Squares Estimates of Payment Schedules by Sample Partition
Single-Slope Estimates
(1)
Constant
490
⫺68.7294
(151.035)
(2)
⫺239.5556
(1,829.354)
Sprain claim
Sprain-Nonsprain Slopes
(3)
⫺207.2575
(152.3772)
282.286*
(45.0566)
(4)
⫺366.8231
(1,823.225)
247.1461*
(45.795)
Wage claim
Claim amount
.7368*
(.0045)
.7231*
(.0047)
.7942*
(.0081)
Nonsprain # claim amount
Nonwage # claim amount
Hospitalized 2–7 days
⫺175.4122
(150.6152)
(6)
⫺128.6141
(1,818.097)
372.9495*
(42.8626)
392.075*
(43.4046)
.7149*
(.0082)
.8164*
(.0082)
⫺3.1161*
(1.0262)
514.9167*
(91.6217)
.7066*
(.0049)
.7989*
(.0084)
⫺3.0547*
(1.0264)
608.3637*
(94.0177)
.7135*
(.0049)
.7837*
(.0082)
Wage # claim amount
⫺4.3575*
(1.0179)
591.9398*
(91.7952)
(5)
.7271*
(.0046)
Sprain # claim amount
Weeks of work lost
Wage-Nonwage Slopes
⫺4.0288*
(1.0183)
676.0466*
(94.2210)
⫺4.1841*
(1.0143)
553.4826*
(92.4333)
⫺3.8628*
(1.0150)
623.4591*
(94.8457)
Hospitalized 17 days
Minor trauma
Moderate trauma
Severe trauma
Catastrophic trauma
Temporarily disabled
Permanently disabled
Citation issued
491
Joint and several liability
Fixed effects
Adjusted R2
N
F: equal slopes
1,474.8930*
(143.233)
25.4117
(83.5661)
321.6278*
(88.0750)
1,223.3840*
(123.0869)
2,008.9890*
(747.0139)
25.4117
(83.5661)
321.6278*
(88.0750)
146.1411
(103.3573)
⫺208.7214
(252.0608)
No
.8245
11,827
1,631.939*
(145.2054)
4.4130
(85.1900)
236.4378*
(90.4760)
1,160.7960*
(125.8385)
1,939.7770*
(745.0064)
102.2415*
(38.6142)
1,086.0920*
(102.5053)
160.3382
(108.6720)
⫺220.1609
(255.0938)
Yes
.8284
11,661
1,159.794*
(149.2121)
⫺31.9928
(83.7715)
258.1474*
(88.5988)
1,107.95*
(123.2402)
1,587.692**
(745.9995)
94.5846*
(37.6655)
989.7583*
(100.0541)
147.8136
(102.9896)
⫺167.4144
(251.215)
No
.8253
11,827
75.15*
1,302.583*
(151.3223)
⫺41.8042
(85.3388)
188.3856*
(90.8185)
1,061.025*
(125.8991)
1,540.347*
(744.124)
93.8480*
(38.5924)
1,013.701*
(102.5667)
163.6789
(108.3293)
⫺179.4337
(254.3027)
Yes
.8276
11,661
72.12*
1,157.9167*
(145.5745)
16.0350
(83.0035)
280.2052*
(87.5695)
1,165.474*
(122.4118)
2,151.335*
(742.0434)
⫺3.4952
(40.7195)
927.0988*
(100.3684)
145.0021
(102.6581)
⫺151.2166
(250.4027)
No
.8264
11,827
145.09*
1,350.774*
(147.9116)
⫺3.2533
(84.6672)
204.9565*
(89.9703)
1,116.669*
(125.1674)
2,090.785*
(740.5353)
⫺19.6524
(41.8302)
948.9038*
(102.9082)
158.0728
(108.0150)
⫺151.8322
(253.6031)
Yes
.8286
11,661
117.24*
Note.—Standard errors are in parentheses. The dependent variable is economic damages paid. In addition to variables reported in the table, all models include claimant
demographics, claimant role in the accident, state negligence regime, citation interacted with state negligence regime, and accident city size indicator. Models with fixed
effects also include accident state, insurance firm, and accident year dummy variables.
* Significant at the .05 level, two-sided test.
the journal of law and economics
492
accident state’s negligence regime and interact our measure of the insured’s
degree of fault with the negligence regime indicator.53
Joint and several liability rules may apply to accidents in which more than
one party caused injury to the claimant. In such cases, the insurer of one of
the liable parties may be forced to pay a greater fraction of the claim than
that which reflects his degree of fault in the accident. We control for higher
payments due to joint and several liability by including an indicator variable
in the regression models of economic damages payments, and, in the regression models that examine the total damages payments, we subtract any
additional amount paid because of joint and several liability in our construction of the dependent variable.
A final set of controls involves characteristics of the claimant, the insurer,
and the accident location. The variables include the claimant’s role in the
accident (driver, passenger, cyclist, or pedestrian), age, gender, and marital
status. Differences in payment amounts across firms, accident locations, and
accident dates are controlled by including firm, accident state, city size (major
urban, suburban, medium city, or small town), and accident year fixed effects
in many of the estimated models.54 The summary statistics for these and
other key variables in the data set are reported in Table 2.
B.
Results for Economic Damages Payments
Table 3 reports ordinary least squares regression estimates of the claims
payment schedule for economic damages payments, allowing the intercept
and slope coefficients to differ for claims hypothesized to have different
falsification costs.55 Two model specifications are presented. The first includes
53
We include indicator variables for pure comparative negligence and modified comparative
negligence statutes in the accident state as did Kessler in his 1995 study of settlements. The
omitted category in our analysis is contributory negligence. See Kessler, supra note 6.
54
The data are obtained from claim files closed in 1987 and hence result from accidents
that occurred in several different years. Accidents reported in the data set span the time period
1975–87. The vast majority of the accidents (95.3 percent) occurred in years 1985–87, with
85.7 percent occurring in either 1986 or 1987, and hence the extent of time variation in the
sample is not great. One concern that arises because of variation in the time dimension is the
extent to which the reported data reflect nominal or real values of payments. Insurers in this
data set appear to report claims in nominal amounts only (or the claimed amounts are themselves
adjusted for inflation), since preliminary analysis of the data revealed no systematic underpayment or overpayment of claims with earlier accident dates. A constraint on making our
own adjustments to the reported values is that we have very limited data on the timing of
losses relative to payments made. We know only the date of the accident, the date the claim
was filed, and the dates of the first and last payments made by the insurer. Hence, we use the
dollar amounts as reported in the data set and control for potential time variation via the fixed
effects.
55
The sample used for estimation omits the 22 claims whose payments were censored by
the policy limit. We also estimated the models using tobit regression methods, but because of
the small numbers of censored claims, only the ordinary least squares estimates are presented
in the table. The economic damages payment on a claim is defined to be censored if the amount
of economic loss claimed exceeds the value of the BIL policy limit plus any payments made
insurance claims strategies
493
all control variables, and the second includes the control variables plus firm,
location, and accident year fixed effects.56
The estimation results strongly support the predictions of the theory. The
proportion of claimed losses paid by the insurer is lower at higher levels of
reported loss, which is consistent with the notion that insurers flatten the
claims payment schedule to reduce incentives to exaggerate loss amounts.
Also consistent with theory, the payment schedules for claims with low
falsification costs display a flatter slope than those for which falsification
costs are higher. In particular, the payment schedule for injuries involving
sprains displays a flatter slope than that for nonsprain injuries. The results
of the model that includes both control variables and fixed effects indicates
that nonsprain claims are paid at a marginal rate of $.78 on the dollar, while
those involving a sprain injury receive about $.71 on the margin. Moreover,
the estimated payment profile for claims including wage losses has a flatter
slope than that for claims with medical losses only. In the model that includes
all control variables and fixed effects, nonwage claims are compensated at
a marginal rate of $.80 on the dollar, while claims that involve wage losses
are compensated at a marginal rate of $.71. In all models, the slope coefficient
estimates are statistically significant, and the reported F-statistics demonstrate
that the slope coefficients are significantly different across the falsification
cost categories.
It is also notable that classes of injury for which falsification is less costly
(those involving sprains and wage claims) have larger estimated intercepts
than do the types of claims for which falsification is more difficult (those
involving no sprains or medical expenses). This result is consistent with the
intuition that the flattening of the settlement profile to reduce falsification
also increases settlement costs, c(y ⫺ I ), which are partially mitigated by a
concomitant increase in the fixed component of the settlements paid to all
claimants. This latter effect increases the degree of overpayment for classes
of small claims in which falsification is easiest.
The coefficient estimates for several of the control variables are also significant and generally consistent with expectations. The greater the severity
of trauma experienced by the claimant, the larger the insurance payment.
The length of time spent in the hospital and the level of claimant disability
also positively affect the payment amount. Contrary to expectations, but
possibly reflecting the falsification issues involved, a greater number of weeks
lost from work decreases the payment amount. Neither joint and several
liability nor traffic citations are significantly related to the size of the insurance
under other auto insurance coverages and the value of total economic losses paid equals that
limit value.
56
Models with no control variables yielded higher slope estimates but identical conclusions.
The estimated payment schedule slope for sprain claims (.77) was significantly flatter than that
for nonsprain claims (.91), and that for wage claims (.76) was significantly flatter than that
for nonwage claims (.92).
TABLE 4
Ordinary Least Squares (OLS) and Censored Normal Estimates of Payment Schedules
Constant
Single-Slope Estimates
Sprain-Nonsprain Slopes
Wage-Nonwage Slopes
OLS
OLS
OLS
494
⫺341.0217
(467.1543)
Tobit
⫺456.1391
(501.5336)
Sprain claim
. . .
. . .
. . .
.9989*
(.0162)
. . .
1.1165*
(.0152)
. . .
. . .
. . .
. . .
⫺757.2965
(467.9342)
908.2464*
(139.5599)
Tobit
⫺847.5321
(501.2848)
862.0772*
(148.6873)
Wage claim
Claim amount
. . .
. . .
. . .
. . .
.9140*
(.0169)
1.3918*
(.0295)
. . .
1.0238*
(.0161)
1.5356*
(.0246)
. . .
. . .
. . .
Sprain # claim amount
Nonsprain # claim amount
Wage # claim amount
Nonwage # claim amount
Weeks of work lost
Hospitalized 2–7 days
. . .
⫺14.9092*
(3.1776)
4,116.684*
(296.5547)
. . .
⫺19.8148*
(3.4023)
3,998.392*
(262.88)
. . .
⫺14.0344*
(3.1436)
3,618.089*
(297.0027)
. . .
⫺18.0547*
(3.3582)
3,464.616**
(263.708)
Tobit
⫺648.7377
(462.9888)
⫺780.6848
(496.5019)
. . .
1,075.656*
(134.4953)
. . .
1,084.212
(143.1282)
. . .
. . .
. . .
. . .
. . .
.9123*
(.0168)
1.4839*
(.0320)
⫺8.3038*
(3.1736)
3,523.029*
(294.8406)
. . .
1.0255
(.0158)
1.6004
(.0263)
⫺12.0433
(3.3988)
3,425.971
(262.9027)
Hospitalized 17 days
Minor trauma
Moderate trauma
Severe trauma
Catastrophic trauma
Temporarily disabled
Permanently disabled
Citation issued
495
Adjusted R2
Log likelihood
N
F: equal slopes
5,707.605*
(485.1781)
166.0135
(257.809)
2,110.478**
(272.068)
8,808.369*
(395.4417)
18,706.53*
(2,819.48)
639.0915*
(116.5406)
7,795.375*
(325.3362)
74.6010
(318.9748)
.5967
. . .
11,592
. . .
5,898.663*
(488.7259)
125.0541
(278.1121)
2,062.196*
(292.9706)
9,801.173*
(386.8514)
18,191.56*
(2,670.07)
617.8253*
(125.2872)
8,501.206*
(281.4152)
219.1535
(342.6172)
. . .
⫺11,7518.62
11,849
. . .
3,691.15*
(498.5207)
⫺8.5003
(256.6016)
1,955.328*
(271.7419)
8,235.347*
(393.0543)
14,212.89*
(2,802.862)
628.2099*
(115.6521)
7,439.47*
(322.5979)
93.6023
(315.5345)
.6054
. . .
11,592
255.32*
3,510.875*
(488.1642)
⫺40.3295
(276.1825)
1,931.792*
(292.1301)
9,203.48*
(392.0314)
14,496.91*
(2,592.456)
619.948*
(124.0977)
8,064.841*
(284.4082)
217.9953
(338.3668)
⫺11,7369.76
11,849
396.78*
3,836.8*
(491.7014)
101.1383
(254.4729)
1,825.666*
(269.0448)
8,379.987*
(391.1619)
19,508.73*
(2,783.082)
409.2032*
(125.2819)
7,410.355*
(323.9569)
128.5311
(314.8423)
.6071
. . .
11,592
309.56*
3,910.443
(393.9027)
63.3529
(274.3247)
1,778.829
(289.456)
9,372.594
(382.5375)
18,711.69
(2,660.229)
392.5951
(134.4744)
8,072.158
(286.84)
264.7897
(337.7782)
. . .
⫺11,7366.54
11,849
442.35
Note.—Standard errors are in parentheses. The dependent variable is total damages paid less amount paid because of joint and several liability. In addition to variables
reported in the table, all models include claimant demographics, claimant role in the accident, state negligence regime, citation interacted with state negligence regime, and
accident city size indicator.
* Significant at the .05 level, two-sided test.
TABLE 5
Ordinary Least Squares (OLS) and Censored Normal Estimates of Payment Schedules
Single-Slope Estimates
OLS
Constant
496
⫺634.9181*
(372.8546)
Tobit
⫺659.8167
(403.431)
Sprain claim
. . .
. . .
. . .
1.9870*
(.0185)
. . .
2.0611*
(.0157)
Sprain-Nonsprain Slopes
OLS
⫺674.5664*
(373.5535)
263.6468*
(113.3572)
Tobit
⫺655.367
(404.2063)
189.3084
(122.0641)
Wage claim
Claim amount
Sprain # claim amount
. . .
. . .
. . .
. . .
. . .
. . .
Nonsprain # claim amount
. . .
1.8771*
(.0199)
2.3013*
(.0285)
. . .
1.9435*
(.0180)
2.3862*
(.0219)
Wage # claim amount
. . .
Nonwage # claim amount
Weeks of work lost
Hospitalized 2–7 days
. . .
⫺6.1474*
(2.4482)
2,341.832*
(254.3661)
. . .
⫺7.8778*
(2.6535)
2,544.356*
(215.1929)
⫺6.2836*
(2.4244)
1,922.587*
(254.367)
. . .
⫺7.7299*
(2.6254)
2,110.017*
(216.4518)
Wage-Nonwage Slopes
OLS
Tobit
⫺642.5203*
(372.1654)
⫺658.9157
(403.0003)
. . .
⫺18.2264
(113.1076)
. . .
⫺70.4599
(121.6025)
. . .
. . .
. . .
. . .
. . .
1.9318*
(.0198)
2.2538*
(.0311)
⫺2.8539
(2.4525)
1,977.375*
(254.8247)
. . .
2.0102*
(.0176)
2.3090*
(.0245)
⫺4.4342
(2.6631)
2,201.35*
(218.8171)
Hospitalized 17 days
Minor trauma
Moderate trauma
Severe trauma
Catastrophic trauma
Temporarily disabled
Permanently disabled
Citation issued
497
Adjusted R2
Log likelihood
N
F: equal slopes
2,261.96*
(425.1269)
94.7560
(202.0603)
1,043.35*
(214.6309)
5,628.126*
(322.6427)
7,204.952*
(2,139.621)
154.7261
(93.7277)
2,261.96*
(425.1269)
⫺36.0170
(255.0049)
.7572
. . .
10,362
. . .
2,246.152*
(327.7611)
74.8830
(219.7977)
1,005.215*
(232.9978)
6,516.569*
(311.8238)
8,070.654*
(2,101.748)
132.706
(101.4915)
6,227.897*
(230.2197)
43.9706
(275.8663)
. . .
⫺10,2200.88
10,567
. . .
953.4246*
(430.8317)
55.5990
(201.556)
1,102.431*
(214.7513)
5,357.056*
(320.4622)
3,319.16
(2,136.406)
197.0794**
(93.1205)
5,429.104*
(266.0359)
⫺57.6681
(252.5497)
.7619
. . .
10,362
200.00*
719.4656*
(343.6006)
50.9173
(219.0322)
1,097.353*
(233.1621)
6,259.876*
(320.5072)
4,754.396*
(2,082.177)
187.4106
(100.8444)
5,997.068*
(239.4485)
9.5778
(273.2246)
. . .
⫺10,2092.48
10,567
340.44*
1,349.922*
(430.1981)
50.8362
(200.8572)
873.7397*
(213.8445)
5,437.409*
(321.1176)
7,617.315*
(2,127.507)
274.0695*
(101.4257)
5,659.036*
(268.3362)
10.2744
(253.4720)
.7602
. . .
10,362
100.35*
1,402.313*
(326.4384)
34.6795
(218.6763)
846.322*
(232.2837)
6,342.326*
(314.0874)
8,278.56*
(2,094.152)
265.9261*
(109.9185)
6,254.138*
(240.8596)
85.1401
(274.4827)
. . .
⫺1,02146.01
10,567
130.84*
Note.—Standard errors are in parentheses. The dependent variable is total damages paid less amount paid because of joint and several liability. The sample includes
only those claims with fully paid economic damages. In addition to variables reported in the table, all models include claimant demographics, claimant role in the
accident, state negligence regime, citation interacted with state negligence regime, and accident city size indicator.
* Significant at the .05 level, two-sided test.
TABLE 6
Robustness Check
Sprain-Nonsprain Slopes
Constant
Sprain claim
Wage-Nonwage Slopes
Robust
Regression
Quantile
Regression
Trimmed
Sample OLS
Robust
Regression
Quantile
Regression
Trimmed
Sample OLS
⫺198.7506*
(79.9293)
203.6437*
(24.2551)
⫺209.8122*
(55.6776)
157.5922*
(16.9592)
⫺880.0307*
(345.6784)
212.547*
(107.5555)
⫺87.7535
(77.7423)
⫺116.0379*
(50.9840)
⫺735.5707*
(342.6904)
. . .
2.2537*
(.0043)
2.5247*
(.0061)
. . .
2.0677*
(.0030)
2.3958*
(.0042)
. . .
2.1130*
(.0224)
2.5144*
(.0377)
. . .
⫺39.8404
(23.6273)
. . .
⫺45.3666*
(15.5298)
. . .
⫺239.6938*
(107.4643)
. . .
. . .
. . .
. . .
. . .
. . .
Wage claim
Sprain # claim amount
Nonsprain # claim amount
Wage # claim amount
Nonwage # claim amount
F: entire model
Pseudo-R2
N
F: equal slopes
. . .
23,495.09*
10,362
1,783.23*
. . .
.6028
10,362
5,381.97*
. . .
918.04*
.7429
10,159
112.72*
. . .
1.9947*
(.0041)
2.4182*
(.0065)
19,241.24*
10,362
3,977.45*
. . .
1.9782*
(.0027)
2.3848*
(.0043)
.6073
10,362
8,490.83*
. . .
2.1059*
(.0253)
2.3604*
(.0297)
917.16*
.7427
10,159
57.09*
Note.—Standard errors are in parentheses. OLS p ordinary least squares. The dependent variable is total damages paid less amount paid because of joint and several
liability. The sample includes only those claims with fully paid economic damages. All control variables are included in the estimated models.
* Significant at the .05 level, two-sided test.
insurance claims strategies
499
TABLE 7
Linear Spline Estimates with Single-Slope Break Point
Estimated Sprain-Nonsprain Slopes
Claim amount ! break point:
Sprain # claim amount
Nonsprain # claim amount
F: equal slopes
Claim amount 1 break point:
Sprain # claim amount
Nonsprain # claim amount
F: equal slopes
$236
$722
$2,166
$4,588
5.1622*
(.9348)
3.7188*
(1.1851)
.97
3.4118*
(.2211)
3.4278*
(.3644)
.00
2.4581*
(.0711)
3.0368*
(.1471)
14.40*
2.2256*
(.0414)
2.9354*
(.0858)
66.14*
1.8665*
(.0203)
2.2969*
(.0290)
194.38*
1.8361*
(.0211)
2.2761*
(.0300)
180.61*
1.7851*
(.0237)
2.2230*
(.0327)
138.37*
1.7244*
(.0283)
2.1268*
(.0367)
83.87*
Estimated Wage-Nonwage Slopes
Claim amount ! break point:
Wage # claim amount
Nonwage # claim amount
F: equal slopes
Claim amount 1 break point:
Wage # claim amount
Nonwage # claim amount
F: equal slopes
$236
$722
$2,166
$4,588
4.7410*
(2.1018)
3.7877*
(.8021)
.19
3.1387*
(.3743)
3.1208*
(.2243)
.00
2.3375*
(.1012)
2.4961*
(.0823)
1.74
2.1612*
(.0529)
2.3982*
(.0526)
12.29*
1.9287
(.0200)
2.2407*
(.0320)
87.89*
1.9146**
(.0206)
2.2029*
(.0343)
64.51*
1.8889**
(.0224)
2.1820*
(.0406)
47.54*
1.8575**
(.0254)
2.1368*
(.0512)
26.88*
Note.—Standard errors are in parentheses. The dependent variable is total damages paid less amount
paid because of joint and several liability. The sample includes only those claims with fully paid economic
damages. All control variables are included in the estimated models.
* Significant at the .05 level, two-sided test.
payment. The interactions of the latter variable with the state liability regime,
although not reported in the table, are also not statistically significant.57 The
claimant’s characteristics and role in the accident, although not reported in
the table, are also not statistically significant.
C.
Results for Total Damages Payments
Having established that the predicted relationships from our theory hold
for economic damages payments, we turn to estimates of the total payment
schedule. Because of the data limitations that we noted earlier, we regress
57
The comparative negligence regime indicators, not reported in the table, have a positive
and often significant effect on settlement amounts (relative to the omitted regime of contributory
negligence).
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the journal of law and economics
the total payment amount on the economic damages claimed.58 As before,
we test the hypothesis that the slope of the payment schedule is flatter for
sprain claims than for nonsprain claims and for wage claims than for nonwage
claims.
The estimation results are presented in Table 4. The dependent variable
in the model is the sum of economic damages plus general damages payments
less any amounts paid because of joint and several liability. The table reports
the models and includes all control variables, which are estimated using both
ordinary least squares and tobit regression methods.59 The total claim payment
amount is defined to be censored by the policy limit if the total compensation
paid equals the value of the BIL policy limit, which results in 270 censored
claims. These claims are omitted from the data in the ordinary least squares
estimates and included as censored observations in the tobit estimates.
The estimation results are consistent with those obtained in the case of
economic damages payments. Focusing on the censored normal regression
estimates, in the case of sprain claims the total claims payment (including
general damages) increases only $1.02 for every $1 increase in economic
losses claimed, while for nonsprain claims the marginal increase in total
payment amount is $1.54. A similar pattern is observed across wage and
nonwage claims. The slope of the total payment schedule is 1.60 for nonwage
claims, but only 1.03 for the claims involving lost wages; F-tests show that
the estimated slope differences are statistically significant at the 5 percent
confidence level in both sets of models.
These estimates of the claims payment schedules demonstrate that the
marginal increase in the insurance payment as a function of the claimed loss
is significantly lower for categories of claims thought to have low costs of
falsification. This is true for both economic damages payments and for total
damages payments. This pattern is consistent with our theoretical prediction
that insurers will choose claims payment strategies to optimally balance a
claimant’s incentives to exaggerate losses on the one hand with the transactions and litigation costs associated with the underpayment of claims on
the other.
V.
Robustness of the Results
One potential criticism of these results is that a flattened payment schedule
estimated through a linear regression model could simply reflect denial of a
few large claims and thus could result from insurer auditing rather than from
58
Recall that the data do not contain information regarding the amount of general damages
claimed.
59
The insurer, state, and time fixed effects are omitted here because of the additional computational complexity and the similarities in previous results with and without fixed effects.
Many of the models were also estimated with the fixed effects and produced very similar
results.
insurance claims strategies
501
TABLE 8
Linear Spline Estimates with Slope Breaks at Quartiles
Sprain-Nonsprain
Wage-Nonwage
Estimate
Claim amount ! $236:
Sprain # claim amount
Nonsprain # claim amount
F: equal slopes
Claim amount between $236
and $722:
Sprain # claim amount
Nonsprain # claim amount
F: equal slopes
Claim amount between $723
and $2,166:
Sprain # claim amount
Nonsprain # claim amount
F: equal slopes
Claim amount 1 $2,166:
Sprain # claim amount
Nonsprain # claim amount
F: equal slopes
1.7996 (1.1609)
2.3812 (1.4844)
.10
3.0673* (.3884)
2.4331* (.7224)
.61
2.3065* (.1154)
3.3625* (.2792)
12.58*
1.7925* (.0240)
2.2134* (.0334)
121.75*
Estimate
Claim amount ! $236:
Wage # claim amount
Nonwage # claim amount
F: equal slopes
Claim amount between $236
and $722:
Wage # claim amount
Nonwage # claim amount
F: equal slopes
Claim amount between $723
and $2,166:
Wage # claim amount
Nonwage # claim amount
F: equal slopes
Claim amount 1 $2,166:
Wage # claim amount
Nonwage # claim amount
F: equal slopes
1.6187 (2.5597)
1.8370 (.9970)
.01
2.5727 (.6225)
3.5861* (.4141)
1.86
2.3115** (.1602)
3.5861* (.4141)
.60
1.8901* (.0226)
2.2054* (.0415)
52.79*
Note.—Standard errors are in parentheses. The dependent variable is total damages paid less amount
paid because of joint and several liability. The sample includes only those claims with fully paid economic
damages. All control variables are included in the estimated models.
* Significant at the .05 level, two-sided test.
a strategy of systematic underpayment of claims. That is, our estimates are
also consistent with insurers auditing a few large claims for fraud but not
bothering to audit small claims owing to the fixed costs of undertaking an
audit.60
The adjusting of payment schedules by insurers to deter claims falsification
is more likely to occur in the context of general damages payments rather
than with payments for economic damages. Insurers may be unwilling to
declare openly that claimants are being routinely compensated less than the
claimed amount of documented losses. In cases where fraud is likely but
cannot be definitively identified or would require a costly investigation to
prevail against a court challenge, general damages payments are the area in
which insurers may implement underpayment strategies.
To provide a stronger test for the systematic flattening of payment schedules
in cases where fraud is likely, we estimate the total payment schedules after
excluding from the sample all claims that were not fully paid in the amount
60
In their 1998 article, Keith Crocker and Sharon Tennyson find that the payment patterns
observed in this data are not consistent with optimal auditing strategies. Models of optimal
auditing generally predict flat payments for small claims that are not audited and random
auditing of large claims to an extent that eliminates claims exaggeration; hence, the prediction
of the auditing models is that large claims should be fully paid in equilibrium. See Keith J.
Crocker & Sharon Tennyson, Costly State Falsification or Verification? Theory and Evidence
from Bodily Injury Liability Claims, in Dionne & Laberge-Nadeau eds., supra note 25.
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the journal of law and economics
of economic losses claimed. In this way, we can remove the effects of claims
auditing on the estimated payment schedule slopes, since these types of claims
denials are likely to have been a consequence of investigative actions by the
insurer. The ordinary least squares and censored normal estimation results
for this restricted sample are reported in Table 5. As would be expected, the
estimated slope coefficients for the payment schedules are larger after eliminating claims for which part of the economic loss claimed is unpaid.61
The important finding is that there remain significant slope differentials
across sprain and nonsprain claims and across wage and nonwage claims
and that these differences are in the direction predicted by the theory. At the
margin, nonsprain claims are compensated at a marginal rate of 2.39, while
sprain claims are compensated at a marginal rate of 1.94. Similarly, nonwage
claims are compensated at a marginal rate of 2.31, while wage claims are
compensated at a rate of only 2.01 at the margin; F-tests indicate that these
differences are statistically significant at the 5 percent confidence level. Thus,
the relative flattening of payment schedule slopes for claims with low costs
of falsification is systematic in the data and not driven by denial of a few
large claims.
As an additional robustness check, and to demonstrate that these findings
are not driven by a few outliers in the data, we also estimate the payment
schedules using methods that are less susceptible to influence from extreme
observations. The three methods used are robust regression, which iteratively
weights observations on the basis of the absolute values of residuals from
the previous iteration of the regression; median regression, which estimates
the relationship between the explanatory variables and the median of the
dependent variable rather than the mean; and ordinary least squares regression
on a trimmed sample, in which the smallest 1 percent and the largest 1
percent of claims were omitted. As in Table 5, all claims for which economic
damages were not fully paid are eliminated from the sample. To conserve
on space, and to highlight the impact of the different approaches on the
estimated slope coefficients, Table 6 presents only a summary of the estimation results.
Using all of the alternative estimation methods, the estimated slope parameters are moderately larger than those estimated previously, consistent
with the idea that claims far from the center of the distribution reduced the
previous slope estimates. In each case, however, the predicted relationship
between the slope estimates for different categories of claims is still observed.
The payment schedule slopes for sprain claims are significantly smaller than
those for nonsprain claims, and the payment schedule slopes for wage claims
are significantly smaller than those for nonwage claims.
61
So, for example, focusing on the single-slope tobit estimates, the marginal effect of claim
size on payment is 2.06 in the restricted sample, while it is 1.12 in the sample including claims
in which economic damages were underpaid.
insurance claims strategies
503
A final robustness check is to allow the slope parameters to vary with the
size of the claim, in addition to varying with the claim category. This allows
us to investigate whether the relative flattening of the low falsification cost
payment schedules occurs throughout the claims distribution or is driven only
by lowered payments for very large claims. We examine this issue by using
a linear spline model to estimate a piecewise linear payment schedule, allowing for different slopes below and above specified break points, with the
different segments constrained to join at the break. In our empirical setting,
the form of the estimated model with a single break point is
Ii p bo ⫹ bi yi ⫹ b 2 yi yi ⫹ D{b 3 yi ⫹ b 4 yi y}
i ⫹ g X i ⫹ ␧i ,
(8)
where D is an indicator variable equal to zero for claims amounts below the
break point and equal to one for claims amounts above the break point.
Recall that yi represents the claimed amount and yi is the falsification cost
parameter.
We have no a priori reason to suppose a specific break point in the payment
schedule slopes. We therefore estimate a number of models, choosing different break points, and compare the results. Table 7 reports a summary of
the estimation results for several models that include a single-slope break
point. The first column in each panel sets the slope break at $236, the 25th
percentile of the claims distribution; the second column sets the break point
at the median claim amount of $722; the third column sets the break point
at the 75th percentile claim amount of $2,166; and the fourth column sets
the break at $4,588, the 90th percentile. As previously, all claims for which
economic damages were not fully paid are eliminated from the sample.
The estimation results are consistent with systematic flattening of the payment schedule for easy-to-falsify claims. The estimated slopes of the payment
schedules for claims with low falsification costs (sprain and wage) are significantly smaller than those for claims with high falsification costs (nonsprain
and nonwage) above the break point in all cases, and in some cases below
the break point as well. When the break point is set at a small claim amount,
there are no significant differences in the slopes of the payment schedules
across claims categories below the break point. When the break point is set
at $2,166 or higher, the estimated slope of the sprain payment schedule is
significantly smaller than that for the nonsprain payment schedule in both
payment schedule segments. A similar result is observed for the wage versus
nonwage payment schedules, although somewhat higher values of the break
point are required in order to obtain significant differences between the estimated slopes in the first payment schedule segment.
Similar inferences are drawn from estimates of a model that allows multiple
break points, one at each of the claims quartiles. These estimation results
are summarized in Table 8. The estimated slope parameters are not monotonically decreasing across the claims quartiles. However, in every quartile
and for both partitions of the data, the payment schedule slope for claims
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the journal of law and economics
with low falsification costs is either significantly smaller than the slope for
claims with high falsification costs or not significantly different from it. The
sprain payment schedule becomes significantly flatter than the nonsprain
payment schedule for claims amounts above $722, while the wage payment
schedule becomes significantly flatter than the nonwage payment schedule
for claims amounts above $2,166. Thus, while the estimates suggest that the
relative flattening of the low-falsification-cost claims payment schedules is
more pronounced for larger-valued claims, it is clearly not limited to only
very large claims. This provides additional evidence of systematic flattening
of claims payment schedules for claims with low falsification costs.
VI.
Conclusion
This paper has developed and tested a theory of optimal payment strategies
for liability insurance settlements when claimants can inflate their apparent
magnitudes of losses. The theory predicts that when audits of claims cannot
uncover the true loss amount, the optimal strategy of an insurer is to reduce,
at the margin, the settlement payment as a function of the claimed amount,
thereby mitigating the incentives facing claimants to expend resources on
claims inflation. This underpayment of claims at the margin, however, may
generate costs to the insurer from claims negotiations and potential litigation
and can result in civil damages if the insurer is found to have engaged in
bad faith. As a result, insurers must balance optimally the effects of claims
underpayment on reducing the incentives for falsification on the one hand
against the costs of underpayment on the other.
We find empirical support for this economic trade-off in a large data set
of liability insurance settlements for bodily injuries received in automobile
accidents. Claims involving sprain injuries, an area in which the potential
for fraud is particularly acute, receive on average lower insurance payments
than do nonsprain claims of equal magnitudes. In a similar vein, claims
involving wage losses, which can be manipulated by malingering on the part
of the injured party, receive substantially less generous settlements at the
margin than claims entailing only medical expenses. In both cases, the results
hold even after including extensive controls for other claims characteristics
that might affect payment amounts and are robust to alternative estimation
techniques and sample definitions. Our findings thus provide convincing
support for the notion that insurers adopt claims payment strategies designed
to mitigate a claimant’s incentives to invest resources in inflating injury
claims.
These results also provide an insight into the economic rationale for permitting extracontractual damages in cases of bad-faith failure to settle. In
the absence of such sanctions, there is an underlying externality that affects
the selection of a settlement strategy: the insurer sees only the benefits of
underindemnification in terms of reduced claims falsification and does not
insurance claims strategies
505
appreciate the costs to other parties of such an underpayment strategy. To
the extent that the damages awarded in cases of insurer bad faith reflect actual
costs to the claimant or the insured of underindemnification, the awarding
of such damages causes the insurer to consider these factors when crafting
a settlement strategy. Accordingly, the insurer internalizes both the benefits
and the costs of aggressive claims settlement strategies, resulting in an efficient balance between the costs of claims underpayment and the benefits
of reduced falsification.
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