cassandra cole and kathleen mccullough co-editors · 2018-09-04 · cassandra cole and kathleen...

Journal of Insurance Regulation

Cassandra Cole and Kathleen McCulloughCo-Editors

Vol. 35, No. 3

Optimal Cap on Claim Settlements Based on Social Benefit Maximation

Hong MaoJames M. Carson

Krzysztof M. Ostaszewski, FSA, CFA, CERA, MAAAYuling Wang

JIR-ZA-35-03

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Editorial Staff of the Journal of Insurance Regulation

Co-Editors Case Law Review Editor Cassandra Cole and Kathleen McCullough Jennifer McAdam, J.D.

Florida State University NAIC Legal Counsel II Tallahassee, FL

Editorial Review Board

Cassandra Cole, Florida State University, Tallahassee, FL

Lee Covington, Insured Retirement Institute, Arlington, VA

Brenda Cude, University of Georgia, Athens, GA

Robert Detlefsen, National Association of Mutual Insurance Companies, Indianapolis, IN

Bruce Ferguson, American Council of Life Insurers, Washington, DC

Stephen Fier, University of Mississippi, University, MS

Kevin Fitzgerald, Foley & Lardner, Milwaukee, WI

Robert Hoyt, University of Georgia, Athens, GA

Alessandro Iuppa, Zurich North America, Washington, DC

Robert Klein, Georgia State University, Atlanta, GA

J. Tyler Leverty, University of Iowa, Iowa City, IA

Andre Liebenberg, University of Mississippi, Oxford, MS

David Marlett, Appalachian State University, Boone, NC

Kathleen McCullough, Florida State University, Tallahassee, FL

Charles Nyce, Florida State University, Tallahassee, FL

Mike Pickens, The Goldwater Taplin Group, Little Rock, AR

David Sommer, St. Mary’s University, San Antonio, TX

Sharon Tennyson, Cornell University, Ithaca, NY

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* Shanghai Second Polytechnic University; [email protected]. ** Daniel P. Amos Distinguished Professor of Insurance, Terry College of Business, University

of Georgia; [email protected]. *** Actuarial Program Director and Professor of Mathematics, Illinois State University;

[email protected]. ****School of Finance, Shanghai University of Finance and Economics, Shanghai, 200433,

China; [email protected].

© 2016 National Association of Insurance Commissioners

Optimal Cap on Claim Settlements

Based on Social Benefit Maximization

Hong Mao* James M. Carson**

Krzysztof M. Ostaszewski, FSA, CFA, CERA, MAAA*** Yuling Wang****

Abstract

Caps on damages constitute one of the proposed reforms of the tort system, especially in the U.S. In this paper, we establish models to capture both the claimant’s and the insurer’s behaviors under a system whereby damages are capped. We also establish an objective function of maximizing the social benefit, expressed as the sum of claimant and insurer benefits. We examine the optimal upper limit that balances both claimant and insurer benefits. Our model implies that the optimal value of the upper limit should be the expected value of the random claim payment independent of the type of probability distribution the claim payment follows. Finally, we apply empirical data to our theory and arrive at an estimate of the optimal damage cap.

1



Introduction In the ongoing debate concerning tort reform, caps on damages have been

proposed as a possible solution to address the problem of increasing claims costs. One of the key purposes of such caps is to control insured losses, making insurance more available and affordable (Feldman, 2003; Gfell, 2003; and Liang, 2004). Supporters assert that caps would inhibit fraudulent claims and eventually help to reduce insurance premiums. Many states in the U.S. are considering passing, or have passed, legislation creating caps on damages, especially for non-economic damages. As an example, the crisis in medical malpractice insurance markets can be especially pronounced, and caps have been proposed as a method to help militate against such crises.

There are different views about the efficacy of caps. On one hand, many believe that caps reduce losses for insurers, pointing out reductions in losses in jurisdictions that enacted a cap, when compared with those states that did not. For example, Sloan, Mergenhagen and Bovbjerg (1989) found that, based on closed claim data for 1975 through 1978 (and through 1984), insurers paid 31% less in states with caps on non-economic damages, and 38% less in states with caps on total damages. Kessler and McClellan (1997) found that premiums were reduced by 8.4% in the first three years after a tort reform. Tort reform also reduced the likelihood that a physician would be sued. The U.S. Congressional Budget Office (1998) summarized that caps on non-economic damages “have been found extremely effective in reducing the amount of claims paid and medical liability premiums.” Viscusi and Born (2005) used NAIC data for the period 1984 to 1991 to show that caps on non-economic damages reduced losses by 17% and premiums by 6%. Encinosa and Hellinger (2005) found evidence that states with caps on non-economic damages increased the supply of physicians by 2.2% per capita, as compared to states without caps, after observing the impact of non-economic damage caps from 1985 to 2000.

On the other hand, critics contend that an arbitrary cap amount cannot work for every case. For small damages, the award may still be excessive. For greater damages, the compensation may not be equitable. Also, caps may not necessarily lead to reduction of insurance premium rates, as relatively high insurance premiums on medical malpractice insurance may be caused by other factors such as insurance cycles, weak competition or declines in investment returns earned by insurers (Zuckerman et al., 1990; Richard, 2012; and Paik et al., 2012).

Critics of tort reform often cite the case of California to argue that the non-economic caps do not work to reduce premium rates. When California passed the legislation on caps on non-economic damages at $250,000 in 1975, premium rates were reduced only for a short time and then continued to rise. By 1988, premiums on medical malpractice were up 450% in comparison with 1975. California then enacted California Insurance Code 1861.01, Proposition 103, which required every insurer to reduce its rate by at least 20%. Subsequently, malpractice rates decreased quickly and premium rates decreased by 30% or more within three years

2



(Glassman, 2004). Many states have passed legislation to impose caps on claim payments. At the

national level, the Help Efficient, Accessible, Low-Cost, Timely Healthcare (HEALTH) Act of 2003, which would have capped non-economic damage awards in medical malpractice actions at $250,000, was passed by the U.S. House of Representatives in May 2004 but was not passed in the Senate (Govtrack, 2004).

The literature on caps on claims and/or damages is dominated by legal analysis. Crocker and Morgan (1998) and Crocker and Tennyson (2002) were unique contributions by the virtue of creating a quantitative model. They study optimal insurance contracts and indemnification profile, when controlling for fraudulent claims for both first-party and third-party claim settlements from the insurer’s perspective. They optimize the insurer’s response to a potentially exaggerated or fraudulent claim. However, determining an optimal cap should also consider the benefit to consumers. Regulators, of course, should attempt to balance both claimants’ and insurers’ benefits, and consider all relevant risks. Shoben (2005) proposes a societal (and by implication, regulatory) perspective on the issue of caps on claims and/or damages by proposing the following three elements of a tort reform to address the issues of measured degree of wrongfulness, measured severity of harm and connectedness between the defendant’s wrong and the plaintiff’s injury. Shoben (2005) also proposes replacement of contingent attorney’s fee by an hourly rate for work done.

In our opinion, whether the cap on damages helps to decrease the insolvency probability of insurers, protect consumers, promote the demand and supply of liability insurance, and improve the social fairness depends on how to rationally determine the cap on damages by scientific methods and from the perspective of maximizing the social benefit. Caps that are too high, as well as caps that are too low, are not optimally beneficial to the realization of maximizing social welfare.1

In our paper, we quantitatively study the optimal cap on claims and/or damages by analyzing both the behavior of claimants and insurers under tort reform law that puts caps on claims and/or damages paid by insurance. We analyze the problem from the regulators’ perspective. We study the following problem: How do we set up an optimal upper limit on claim damages if we want to balance benefits between insurers and claimants? Consumer protection regulators aim for damage awards large enough to properly compensate victims of negligence or of malfeasance. However, insurance regulators also have genuine interest in limiting the awards to levels that would not endanger insurer solvency. Hence, the issue at hand must be cognizant of consumer protection and the prudential regulatory perspectives.

1. In the notes from Gfell (2004), he points out that while a national cap on non-economic

damages in medical malpractice actions may pass constitutional muster, Congress would be wise to attempt less invasive limitations first. That is, Gfell suggests that Congress should seek better economic data upon which to base its decisions, and thus help to assure the crisis would be covered with minimal negative consequences.

3



As a preview to our results, we get the intuitive theoretical result that the optimal cap on damages is the expected value of random loss. It implies that a cap on damages is not exogenous to the distribution of random loss. The frequency and severity distribution of liability injuries does affect the caps on damages.2 Therefore, the injured claimant can obtain fair claim payment. Based on our empirical data, our analysis indicates that the optimal cap on damages is far less than caps found in reality, such as caps for damages related to general liability insurance. For some other types of liability insurance (e.g., medical malpractice liability insurance), our analysis suggests that the insurer’s total claim payment without caps is much greater than that with caps. As more than one-half of U.S. states did not have caps on damages, the absence of caps may be one reason for the medical malpractice insurance crisis (besides long-tail liability and exaggerated losses to obtain higher judgment awards).

Our study is organized as follows: In the next section, we establish models used for the third-party claimant. Then, in the third section, we discuss the determination of optimal caps, based on the criterion of maximizing the social benefit. In the fourth section, we provide two important examples to illustrate the application of our models to the problem. We then present sensitivity analysis, and in the final section, we provide our conclusions.

Models of Third-Party Claims Settlements

We consider third-party liability insurance in which we assume the claimant and insurer to be risk-neutral. The claimant has a real loss of X , which is only fully known to the claimant. However, the claimant files a claim of s (≥ X ). The jurisdiction of the claim has an upper limit of claim damage of θ. The claim amount of s may still exceed the amount of θ, but the indemnification amount of I(s) paid by insurer must be less than or equal to θ. If insurer can identify the real amount of damage, then I(s) = X when X θ. When the indemnification amount is greater than the real amount of loss, I(s) – X is the realized exaggerated claim amount for claimant. When the real loss is greater than θ, the highest claim possible is the amount of θ, and X – θ is the loss for claimant after compensation.

Let E Y1

be the expected terminal wealth of the claimant, and E Y

2 be

the expected terminal wealth of the insurer.3 Using that notation, we can write:

2. Gfell (2004) points out (in the notes to the article) that reforms that protect the public are

what are needed, not reforms that blame the injured, the disabled, and victims of medical ineptitude and neglect. It is in this light that caps, based on the expected loss incurred to the injured, may help maximize social benefit.

3. When claimant and insurer are risk-neutral, their expected benefit functions are exactly the same as their utility functions. In the context of the theory of the firm, a risk-neutral firm facing risk about the market price of its product, and caring only about profit, would maximize

4

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8



only data for cases with caps already applied are available, the data are right-censored. Datta (2005) shows a possible statistical methodology to be applied in such a situation. Below we give an example of estimation of the expected value parameter for grouped data, when only ranges of data are available, and not all individual cases. (This situation may be encountered in practice by both insurance actuaries and insurance regulators.)

Examples to Illustrate Our Results on Optimal Caps

Example 1: General Liability Insurance

Table 1 lists the average claim payment of 217 general liability insurance

policies. The claim payment is divided into several intervals (the first column) within the minimum value zero and the maximum value $300,000, which is the policy upper limit. The second and third column, respectively, list the number of claims and the average claim payment in which the claim payment for each time falling in each interval (the data are used by Xiao, 2008, and originates from Klugman et al., 1998).5

We will now illustrate our result by calculating an estimate of the parameter of the mean economic loss, which in the model proposed here is equal to the optimal level of damages cap. We could estimate it from the sample mean, but given that the data are grouped, it is more reasonable to attempt to fit a probability distribution and then use an estimation of that distribution’s mean. By examining the data, we hypothesize that the lognormal distribution is a good fit. In the Appendix, we perform a test of this hypothesis (the chi-square goodness of fit test) and find that the distribution can be used to model this data. We also find (in the technical calculation) the estimates of the distribution’s parameters, and we use them in the analysis that follows.

5. We use general liability insurance as an example since firms generally purchase this

coverage as one of the first steps to protect their assets. This safety net is critical in a society in which the number of lawsuits and the value of judgment awards have increased over time (Howell, Commercial General Liability Insurance Definition |: www.ehow.com/about_5184654_ commercial-general-liability-insurance-definition.html).

9

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n us p

ms of t. y

ut 6

g ul

ez he he ns ge nd te ur re

11

negligmedicinsuramarkeKane insurathe opnumbin Co

* Sincenumbethis pointerna

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1 statist

gence lawsuitscal liability insance crisis. Seets (e.g., Karl and Emmons,

ance from a diptimal cap wit

ber of claims bonnecticut from

Siz

e the range of $1 mr and percentages rtion of the data in

als.

e: Connecticut In2008

Figure 1 desbution (CDF) bution quite wbution by m

.4 7.57 We usetic STAT=0.26

s (Amon and surance by inseveral empiricand Nyce, 201, 2006). In thisfferent perspecth the empiricaby size of indemm the 4th quarte

ze of Indemnit

million and over iof claims falling i

nto four intervals a

nsurance Departm

cribes the eusing the datawell. We calc

maximum like

e the Kolmogor682. Since the

Jou

© 2016 Nation

Winn, 2004).surers may be cal papers stud14; Born et al.,s second examctive. We use al data from thmnity paymenter of 2005 thro

Table 2: ty Payments (

in the last line of Tin this interval. Weand assume the nu

ment, Connecticu

empirical funca in Table 2. Tculate the valelihood estim

rov-Smirnov te critical value

urnal of Insu

nal Association of

. Thus, large one importan

dy medical m, 2009; Lei and

mple, we study our theoretical

he “real world.t of medical m

ough 2007.

(2005 4th Q–20

Table 2 is so largee address this aspe

umber of claims un

ut Medical Malp

ction of cumThe data clearlues of param

mation and o

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rance Regul

Insurance Commi

claim paymennt reason for limalpractice insu

d Schimit, 200medical malpr

l model to dete” Table 2 show

malpractice insu

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practice Annual

mulative probrly fit the logn

meters of lognobtain ˆ 12

the value of thng whether to

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12

Optimal

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the null hywe do not r

E

Thus,

damages c

We kn

on damageare relative$521,230, payment w

In Apfrom 2005 analysis hNorth Dakand it is Georgia, IdVirginia. Wthis may bin the U.Slosses to ob

Cap on Clai

onal Association o

ypothesis is 0.3reject the null

Empirical Prob

our theoreticalap should equa

now from Avrae for medical mely large. Settinwould reduce

with this cap is pendix Table to 2007. Com

ere is close tokota and Utah. higher than cdaho, Kansas,

We note that mbe one of the imS., along with btain higher ju

im Settlemen

f Insurance Comm

3912, which ishypothesis.

Figbability Distri

Using the D

l model calculaal:

aham (2014) thmalpractice insung the damagee the total claabout 24% of tA2, we list thparing the datao those set by The cap we c

caps set by thMissouri, Nev

more than one-hmportant reasoother reasons

udgment award

nts

missioners

s greater than

gure 2: ibution of ClaData in Table

ates that, based

hat the state of urance. Thus, t

e cap at the amim loss paid that without th

he caps of 20 a in Table A2, y five states: Fcalculate is lowhe states of Avada, Oklahomhalf of the U.S

ons related to tsuch as long-s.

the value of th

aim Payment F2

d on the data, th

Connecticut hthe total claimount suggestedby insurers. T

he cap. states set by sthe cap of $52Florida, Illinoi

wer than that seAlaska, Califorma, South CaroS. states did nothe medical matail liability an

he test statistic

Fitted

he optimal

has not set a capm payments herd by our modelThe total claim

state legislation21,230 from ouis, Mississippi

et by Marylandrnia, Coloradoolina and Wesot set caps, andalpractice crisind exaggerated

c,

p re l, m

n ur i, d, o, st d is d

13

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distribvariabmodedistribthe clparam

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mption that thetion (12) by equ

Equation (14) se parameter site is true whe

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7. The distributionmmetric, non-negmodels with two ution (Xiao, 2008dual claim distrib

y AnalysiClaims D

tial derivative o

ormal distributi

s of (12) and (1of both and

e claim loss fuation (13), we

shows that whe

is smaller th

en 1. In f , which is, le

ed by taking tnormal distrib

distributions,7 oon claims will on, while for mominate.

ns of loss often usgative and with faparameters, such

8). We have testebution in our exam

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© 2016 Nation

is in theDistribut

of equation (11

ion used in that

13), we see thad parameter

follows the loge also obtain:

en 1, the han its sensitiv

fact, the ratio oet us recall, th

the natural logbution is a reaour work sugg be more sens

more stable cla

sed in actuarial scat tail, etc. Based h as: lognormal died that we can usemple.

urnal of Insu

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e Lognortion

1) with respect

t model, we ob

at the optimal vrs of the loss d

gnormal distri

sensitivity of tvity to the par

of sensitivity tohe standard dev

garithm of theasonable modegests that for msitive to the voaims distributio

cience generally hon these characteistribution, Weibe lognormal distr

rance Regul

Insurance Commi

mal

to the paramet

btain:

value of the capdistribution und

ibution. By di

the cap on payrameter , an

o to sensitivviation of the n

e lognormal rael for many remore volatile olatility parameons, sensitivity

have the characteeristics, we usualull distribution, Gibution to describ

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ividing

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14



Implications and Conclusion This paper develops models to analyze both claimant and insurer behaviors

with caps on claim damage for third-party insurance. We establish the optimal strategy for regulators to set the cap to balance both the claimant and insurer’s benefits, considering different risks, based on the criterion of maximizing social benefit. The cap and the cost parameter can be used to control the claimant’s claim inflation. As there is information asymmetry for the amount of damage, the regulator and insurer hardly capture the actual loss amount, especially for the non-economic loss. Our analysis suggests that the optimal value of the cap, considering the maximization of social benefit, should be the expected value of random loss amount independent of the kind of probability distribution it follows.

For losses that are relatively easy to evaluate in terms of their real amount (e.g., economic losses), regulators may consider capping the damage awards on a case-by-case basis, following the principles of equitable justice. But the principle developed here is a simple and practical guide even for such case-by-case evaluations. We acknowledge that our paper utilizes a static equilibrium model, and this is undoubtedly its limitation. There are significant dynamic processes involved in consideration of incentives of all parties involved in legal damages caps. But, such dynamic processes require a more complex multi-period model, and must include incentives faced by the injured party, insurance company and political decision-makers. We consider our work to be a first step toward such a complex model, and creation of such a model is reserved for a far larger and longer-term research project.

Our calculation with empirical data indicates that the cap determined by us in general liability policies is much less than caps observed in the real world. Our other example for medical malpractice insurance indicates that the insurer's total claim payment is much higher without caps than that with caps determining based on the expected loss. The fact that many states have not set caps on damage might be one of the main reasons for the crisis of medical practice insurance in the U.S. We believe that our work may be a valuable contribution in the ongoing debate on managing the cost of claims and the issue of imposing a legal cap on them.

Because insurance regulators are focused on the solvency of insurers, regulators may tend to favor damage award caps in order to reduce the likelihood of insurer insolvency. Similarly, insurance regulators are concerned with insurance pricing, and regulators also may tend to favor caps on damage awards as a way to reduce the expected costs of insurance. Our analysis on optimal damage caps provides insurance regulators with results that shed light on optimum cap levels. The analysis, thus, can help insurance regulators to better achieve the goals of insurer solvency and consumer protection. Of course, others may also need to consider factors such as trying to limit frivolous litigation, decreasing moral hazard and holding parties responsible for their behavior, which might suggest increases in the level of caps or even the removal of caps on damage awards.

In future work, we hope to focus on the reasons that for some liability

15



insurance (e.g., general liability insurance) the value of optimal cap observed in reality, while for other types of liability insurance (e.g., medical malpractice), the value of the optimal cap is close to that observed in reality. Another further study may focus on determining optimal cap on damages and corresponding premium considering dynamic incentives of the potential injurers and corresponding injury phase of the environment.8

8. We thank an anonymous reviewer for this suggestion.

16

Optimal

© 2016 Natio

Techn 1. DerivaEquation

In ordwith equat

Setting

Note t

i.e., when tvalue obtaiequation (Abenefit. Fr

condition E

From

parameters

on the expequation (7

Cap on Clai

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nical App

ation of the On (7)

der to maximizeion (7) and obt

g equation (A1

that since

the first derivained by solvingA2), we can orom equation (

E X * 0

the equation s of claim cost

pected value o7), we obtain th

im Settlemen

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Optimal Solu

e the quantity tain the follow

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ative is zero, thg the equation btain the optim(A2), we see

0 so that

(A3), we see and settlement

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nts

missioners

ution to the

in equation (7)ing equation:

o, we obtain:

he second deriv(A2) will yield

mal cap of claithat the optim

that the cap t cost 1 and stribution. Subial benefit:

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), we use the L

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ive, the optima By solving thizing the socia* satisfies th

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al he al he

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17

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ment can be takeion is:

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meters. (They ahe normal disble.)

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tion function o

are not the meatribution equa

tribution functi

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Fit Test for th to Xiao, 200

st and based on

from a lognor

thesis test

H1 : F x F0 of the lognorm

an and varianceal to the natu

ion chosen to c

urnal of Insu

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he Lognorma08.)

n its outcome a

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with the

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mal distribution

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calculate the ex

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assuming that c

on. Its density

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:

18

Optimal

© 2016 Natio

Where

likelihood

where ~Q

Q

2 r

reject the

distribution

(4) Caobtain: following r

and simila

are listed b

Cap on Clai

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e n is sampl

estimation and

2~ ( 1)r k

k 1,

hypothesis o

n for the data.

alculate the val 9.29376, results are obta

arly E1

49.17below in Table

im Settlemen

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le size, and d consider the f

and k is the

f 0H and th

lues of parame1.62713.Usin

ained:

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following statis

number of unk

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ters by maximng the equation

, ..., etc. can a

ues obtained fr

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known paramet

n assumed is

mum likelihood ns (A6), (A7)

also be obtain

from maximum

ters. If

not a suitabl

d estimation and) and (A8), th

ned. The result

m

le

d he

ts

19

The CCalculation Re

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Table A1:

esults of 2 G

urnal of Insu

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Goodness of Fi

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© 2016 Natio

Since

Q 4

it is failed

takes the lclaims in th

The Val

9. The

(May 2014).

Cap on Clai

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4.51 0.052 12

d to reject the

ognormal distrhis context.

lues of Caps oU.S. in 2

data in this table The states that a

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ribution as a su

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nts

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21



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22



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23


Guidelines for Authors

Submissions should relate to the regulation of insurance. They may include

empirical work, theory, and institutional or policy analysis. We seek papers that advance research or analytical techniques, particularly papers that make new research more understandable to regulators.

Submissions must be original work and not being considered for publication elsewhere; papers from presentations should note the meeting. Discussion, opinions, and controversial matters are welcome, provided the paper clearly documents the sources of information and distinguishes opinions or judgment from empirical or factual information. The paper should recognize contrary views, rebuttals, and opposing positions.

References to published literature should be inserted into the text using the “author, date” format. Examples are: (1) “Manders et al. (1994) have shown. . .” and (2) “Interstate compacts have been researched extensively (Manders et al., 1994).” Cited literature should be shown in a “References” section, containing an alphabetical list of authors as shown below.

Cummins, J. David and Richard A. Derrig, eds., 1989. Financial Models of

Insurance Solvency, Norwell, Mass.: Kluwer Academic Publishers. Manders, John M., Therese M. Vaughan and Robert H. Myers, Jr., 1994.

“Insurance Regulation in the Public Interest: Where Do We Go from Here?” Journal of Insurance Regulation, 12: 285.

National Association of Insurance Commissioners, 1992. An Update of the NAIC

Solvency Agenda, Jan. 7, Kansas City, Mo.: NAIC. “Spreading Disaster Risk,” 1994. Business Insurance, Feb. 28, p. 1.

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