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Variance Advancing the Science of Risk

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2010 V

OLU

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E 0

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2010121 Bootstrap Estimation of the Predictive

Distributions of Reserves Using Paid and Incurred Claims by Huijuan Liu and Richard Verrall

136 Robustifying Reserving by Gary G. Venter and Dumaria R. Tampubolon

155 Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach by Jackie Li

170 The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium by Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg

191 Optimal Layers for Catastrophe Reinsurance by Luyang Fu and C. K. “Stan” Khury

VOLUME 04 ISSUE 02

VARIANCE MissionVariance is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submission by non-CAS members is encouraged.

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Dates to Remember

Table of Contents115 A Note from the Editor by Roger W. Bovard

116 Contributors to this Issue

121 Bootstrap Estimation of the Predictive Distributions of Reserves Using

Paid and Incurred Claims by Huijuan Liu and Richard Verrall

This paper presents a bootstrap approach to estimate the prediction distributions of reserves produced by the Munich chain ladder (MCL) model. The MCL model was introduced by Quarg and Mack (2004) and takes into account both paid and incurred claims information. In order to produce bootstrap distributions, this paper addresses the application of bootstrapping methods to dependent data, with the consequence that correlations are considered. Numerical examples are provided to illustrate the algorithm and the prediction errors are compared for the new bootstrapping method applied to MCL and a more standard bootstrapping method applied to the chain-ladder technique.


Robust statistical procedures have a growing body of literature and have been applied to loss severity fitting in actuarial applications. An introduction of robust methods for loss reserving is presented in this paper. In particular, following Tampubolon (2008), reserving models for a development triangle are compared based on the sensitivity of the reserve estimates to changes in individual data points. This measure of sensitivity is then related to the generalized degrees of freedom used by the model at each point.

155 Prediction Error of the Future Claims Component of Premium Liabilities

under the Loss Ratio Approach by Jackie Li

In this paper we construct a stochastic model and derive approximation formulae to estimate the standard error of prediction under the loss ratio approach of assessing premium liabilities. We focus on the future claims component of premium liabilities and examine the weighted and simple average loss ratio estimators. The resulting mean square error of prediction contains the process error component and the estimation error component, in which the former refers to future claims variability while the latter refers to the uncertainty in parameter estimation. We illustrate the application of our model to public liability data and simulated data.

114 CASUALTY ACTUARIAL SOCIETY VOLUME 4/ISSUE 2


170 The Economics of Insurance Fraud Investigation: Evidence of a Nash

Equilibrium by Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg

The behavior of competing insurance companies investigating insurance fraud follows one of several Nash Equilibria under which companies consider the claim savings, net of investigation cost, on a portion, or all, of the total claim. This behavior can reduce the effectiveness of investigations when two or more competing insurers are involved. Cost savings are reduced if the suboptimal equilibrium prevails, and may instead induce fraudulent claim behavior and lead to higher insurance premiums. Alternative cooperative and noncooperative arrangements are examined that could reduce or eliminate this potential inefficiency.


Insurers purchase catastrophe reinsurance primarily to reduce underwriting risk in any one experience period and thus enhance the stability of their income stream over time. Reinsurance comes at a cost and therefore it is important to maintain a balance between the perceived benefit of buying catastrophe reinsurance and its cost. This study presents a methodology for determining the optimal catastrophe reinsurance layer by maximizing the risk-adjusted underwriting profit within a classical mean-variance framework.

VOLUME 4/ISSUE 2 CASUALTY ACTUARIAL SOCIETY 115


When I started off as an actuarial student, my mentors in life insurance introduced me to the technique known as “general reasoning.” One of the first examples I encountered was a derivation for the present value (P) of an ordinary annuity certain. The general reasoning derivation goes as follows: The sum of $1 invested at rate i will yield annual income of i per year for n years (present value i*P) with the $1 principal still intact at the end of n years (present value vn). Thus 1=iP+vn. From this expression, the well-known formula for P follows by simple algebra. I recall being impressed with the way this technique gets to the heart of the matter while bypassing the somewhat messier algebra associated with the mathematical derivation.

There are a number of such general reasoning demonstrations in actuarial mathematics on the life side, but I have yet to encounter any in property and casualty insurance. However, I do occasionally encounter another technique for deriving useful results in an elegant manner while avoiding tedious algebraic manipulation. The technique I have in mind is transition to a higher level of abstraction. An example of this technique can be found in the very readable paper “Credibility Formulas of the Updating Type” by Jones and Gerber. The derivation in this paper is somewhat abstract, but accessible to most actuaries. The benefit of abstraction is a smoother flow resulting in increased understanding of the subject matter. A reduction in the level of abstraction would probably force a substantial increase in algebraic manipulation, which is actually more distracting than illuminating. In an appendix to the paper, the authors present an alternate derivation that is even shorter, but more abstract. Readers familiar with linear algebra concepts are able to appreciate the shorter derivation.

These observations are intended for readers as well as authors. It is the na-ture of those who author technical papers to build on other technical papers previously published. Thus, knowledge is advancing as well as increasing. As a practical matter, I encourage readers to invest time mastering some of the more advanced tools required to stay abreast.

Jones, D.A., and H.U. Gerber, “Credibility Formulas of the Updating Type,” Transactions of the

Society of Actuaries 27, 1975, pp. 31-46. Available for download from the Society of Actuaries

Web Site: http://www.soa.org/library/research/transactions-of-society-of-actuaries/1975/

january/tsa75v274.pdf

Roger Bovard

A Note from the Editor



Contributors to this Issue

Stephen P. D’Arcy

Stephen P. D’Arcy, FCAS, MAAA, Ph.D., is professor emeritus of fi-nance at the University of Illinois, a visiting lecturer at California State University Fullerton’s Mihaylo Col-lege of Business and Economics, and President of D’Arcy Risk Consult-ing, Inc. He is a past president of the Casualty Actuarial Society and of the American Risk and Insurance Asso-ciation.

Luyang Fu

Luyang Fu, FCAS, MAAA, is the di-rector of predictive modeling for State Auto Insurance Companies, where he leads the developments of personal lines pricing models, com-mercial lines underwriting models, and corporate DFA and ERM mod-els. Prior to joining State Auto, he served in various actuarial roles with both Grange Insurance and Bristol West Insurance. He holds a B.S. and an M.S. in economics from Renmin University of China, and a M.S. in finance and a doctorate in agricul-tural and consumer economics from University of Illinois at Urbana-Champaign.

Richard A. Derrig

Richard Derrig, Ph.D., is president of OPAL Consulting LLC, which provides research and regulatory support to the P&C insurance indus-try. For over 27 years, Dr. Derrig held various posts at the Massachu-setts Bureau of Automobile Insurers and the Massachusetts Bureau of In-surance Fraud. He has won several prizes, including the CAS Ratemak-ing Prize (1993 co-winner), the ARIA Prize (2003), the RIMS Edith F. Li-chota Award (1998), and ARIA’s Mehr Award (2005). Dr. Derrig has coedited three books on solvency and coauthored papers applying fuzzy set theory to insurance.

Roger W. Bovard Editor in Chief

Editorial BoardEDITORS: Shawna S. Ackerman Avraham Adler Todd Bault Morgan Haire Bugbee Daniel A. Crifo Susan L. Cross Stephen P. D’Arcy Enrique de Alba Ryan M. Diehl Robert J. Finger

Steven A. Gapp Emily Gilde Annette J. Goodreau Richard W. Gorvett David Handschke Philip E. Heckman Daniel D. Heyer John Huddleston Ali Ishaq Eric R. Keen Ravi Kumar

ASSISTANT EDITORS: Joel E. Atkins Gary Blumsohn Frank H. Chang Clive L. Keatinge Dmitry E. Papush Christopher M. Steinbach

Richard Fein Associate Editor— Peer Review

Dale R. Edlefson Associate Editor—Copyediting

Gary G. Venter Associate Editor—Development




Jackie Li

Dr. Jackie Li is currently an assistant professor in the Division of Banking and Finance at Nanyang Business School (NBS), Nanyang Technologi-cal University (NTU), Singapore. Dr. Li obtained his Ph.D. degree in actu-arial studies from the University of Melbourne, Australia, and is a Fel-low of the Institute of Actuaries of Australia (FIAA). He has been lec-turing and tutoring various actuarial courses and his main research areas are stochastic loss reserving for prop-erty/casualty insurance and mortality projections. Before joining NBS, Dr. Li worked as an actuary in the areas of property/casualty insurance and superannuation.

Huijuan Liu

Dr. Huijuan Liu earned a Ph.D. at Cass Business School, London, in 2008. Her research was sponsored by Lloyd’s of London and she was su-pervised by Professor Richard Ver-rall. Together, they have recently published two papers in the ASTIN Bulletin. Dr. Liu now works for the Financial Services Authority in Lon-don.

Yin Lawn Pierre Lepage Martin A. Lewis Xin Li Cunbo Liu Kevin Mahoney Donald F. Mango Leslie Marlo Stephen J. Mildenhall Christopher J. Monsour Roosevelt C. Mosley Jr.

Mark W. Mulvaney Prakash Narayan Adam Niebrugge Darci Z. Noonan Jonathan Norton A. Scott Romito David Ruhm Theodore R. Shalack John Sopkowicz John Su James Tanser

Neeza Thandi George W. Turner Jr. Trent R. Vaughn Cheng-Sheng Peter Wu Satoru Yonetani Navid Zarinejad Yingjie Zhang Alexandros Zimbidis

COPY EDITORS: Nathan J. Babcock Laura Carstensen Hsiu-Mei Chang Andrew Samuel Golfin Jr. Mark Komiskey William E. Vogan

C. K. “Stan” Khury

C. K. ‘Stan’ Khury, FCAS, MAAA, CLU is a principal with Bass & Khury, an independent actuarial con-sulting firm located in Las Vegas, Nevada. Stan is a past president of the CAS and has written numerous papers and article in CAS publica-tions over a period spanning nearly 40 years. He provides a wide range of actuarial consulting services to in-surers, reinsurers, intermediaries, regulators, and law firms.

CAS STAFF: Elizabeth A. Smith Manager of Publications

Donna Royston Publication Production Coordinator

Sonja Uyenco Desktop Publisher



Dumaria R. Tampubolon

Dumaria Rulina Tampubolon com-pleted her S1 degree (similar to an honors degree) in mathematics, ma-joring in statistics, at Institut Teknologi Bandung in Bandung, In-donesia. She holds a master of sci-ence degree in statistics from Monash University, Melbourne, Australia; and a Ph.D. degree in actuarial stud-ies from Macquarie University, Syd-ney, Australia. Her research interest is in general insurance and applied statistics. Currently she teaches at the Faculty of Mathematics and Nat-ural Sciences, Institut Teknologi Bandung. She can be reached at [email protected] and [email protected].

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© 2010 Casualty Actuarial Society.

Gary G. Venter

Gary G. Venter, FCAS, CERA, ASA, MAAA, is head of economic capital modeling at Chartis and teaches graduate courses in actuarial science at Columbia University. His 35+ years in the insurance and reinsur-ance industry included stints at the Instrat group, which migrated from EW Payne through Sedgwick to Guy Carpenter; the Workers Compensa-tion Reinsurance Bureau; the Na-tional Council on Compensation In-surance; Prudential Reinsurance; and Fireman’s Fund. Gary has an under-graduate degree in mathematics and philosophy from UC-Berkeley and a master’s degree in mathematics from Stanford University. He has served on a number of CAS committees and is on the editorial team of several ac-tuarial journals.


Richard Verrall

Richard Verrall has been at City Uni-versity since 1987. He is an Honor-ary Fellow of the Institute of Actuar-ies (1999), an Associate Editor of the British Actuarial Journal, the North American Actuarial Journal, and In-surance: Mathematics and Econom-ics, and a principle examiner for The Actuarial Profession (U.K.). Courses for industry include “Statistics for Insurance,” an introductory course aimed at non-specialists, such as un-derwriters, in the uses of statistics in risk assessment; “Stochastic Claims Reserving,” a specialist course for actuaries and statisticians on how to apply statistical methods to reserving for non-life companies; and “Bayes-ian Actuarial Models,” an introduc-tory course in Bayesian methods for premium rating and reserving.




Herbert I. Weisberg

Herbert I. Weisberg received his Ph.D. in statistics from Harvard in 1970. He is the president of Correla-tion Research, Inc., a consulting firm specializing in litigation support and analysis of insurance fraud. He has published numerous articles and re-ports related to application and de-velopment of statistical methodolo-gy, and is a co-author of Statistical Methods for Comparative Studies: Techniques for Bias Reduction. Re-cently, his research has related to causal inference in statistics, draw-ing on and extending the burgeoning literature in this area. He has recently written a new book titled Bias and Causation: Models and Judgment for Valid Comparisons.

Bootstrap Estimation of thePredictive Distributions of ReservesUsing Paid and Incurred Claims

by Huijuan Liu and Richard Verrall

ABSTRACT

This paper presents a bootstrap approach to estimate the

prediction distributions of reserves produced by the Mu-

nich chain ladder (MCL) model. The MCL model was in-

troduced by Quarg and Mack (2004) and takes into account

both paid and incurred claims information. In order to pro-

duce bootstrap distributions, this paper addresses the appli-

cation of bootstrapping methods to dependent data, with

the consequence that correlations are considered. Numeri-

cal examples are provided to illustrate the algorithm and the

prediction errors are compared for the new bootstrapping

method applied to MCL and a more standard bootstrapping

method applied to the chain ladder technique.

KEYWORDS

Bootstrap, Munich chain ladder, correlation, simulation



1. IntroductionBootstrapping has become very popular in sto-

chastic claims reserving because of the simplic-

ity and flexibility of the approach. One of the

main reasons for this is the ease with which it

can be implemented in a spreadsheet in order to

obtain an approximation to the estimation error

of a fitted model in a statistical context. Further-

more, it is also straightforward to extend it to

obtain the approximation to the prediction error

and the predictive distribution of a statistical pro-

cess by including simulations from the underly-

ing distributions. Therefore, bootstrapping is a

powerful tool for the most popular subject for

reserving purposes in general insurance, the pre-

diction error of the reserve estimates. It should

be emphasized that to obtain the predictive dis-

tribution, rather than just the estimation error, it

is necessary to extend the bootstrap procedure by

simulating the process error. It is also important

to realize that bootstrapping is not a “model,”

and therefore it is important to ensure that the

underlying reserving models are correctly cali-

brated to the observed data. In this paper, we do

not address the issue of model checking, but sim-

ply show how a bootstrapping procedure can be

applied to the Munich chain ladder model.

In the area of non-life insurance reserving,

there are primarily two types of data used. In

addition to the paid claims triangle, there is fre-

quently a triangle of incurred data also available.

The traditional approach is to fit a model to either

paid or incurred claims data separately, and one

of the most popular methods in this context is the

chain ladder technique. While we do not believe

that this is the most appropriate approach for all

data sets, it has retained its popularity for a num-

ber of reasons. For example, the parameters are

understood in a practical context; it is flexible;

and it is easy to apply. This paper concentrates on

methods which have a chain ladder structure, and

in this context, two types of approaches exist:

deterministic methods such as chain ladder, and

the recently developed stochastic chain ladder re-

serving models. When the chain ladder technique

is used (either as a deterministic approach or us-

ing a stochastic model), one set of data will be

omitted–either the paid or the incurred data can

be used, but not both at the same time. Obviously,

this does not make full use of all the data avail-

able and results in the loss of some information

contained in those data.

This leads us to consider whether it is possible

to construct a model for both data sets, and to a

consideration of the dependency between the two

run-off triangles. A related issue also arises when

traditional methods are applied separately to each

triangle, which produces inconsistent predicted

ultimate losses. In response, Quarg and Mack

(2004) proposed a different approach within a

regression framework, considering the likely cor-

relations between paid and incurred data. Quarg

and Mack (2004) called this new method the

Munich chain ladder (MCL) model. It is this

model that is the subject of this paper, and we

show how the predictive distribution may be es-

timated using bootstrapping. Thus, in this paper

an adapted bootstrap approach is described, com-

bined with simulation for two dependent data

sets. The spreadsheets used in this paper can be

used in practice for any data sets, and are avail-

able on request from the authors.

The paper is set out as follows. Section 2

briefly describes the MCL model using a nota-

tion appropriate for this paper. In Section 3, the

basic algorithm and methodology of bootstrap-

ping is explained. Section 4 shows how to obtain

the estimates of the prediction errors and the em-

pirical predictive distribution using the adapted

bootstrapping and simulation methods. In Sec-

tion 5, two numerical examples are provided, in-

cluding the data from Mack (1993) as well as

some real London market data. Finally, Section 6

contains a discussion and conclusion.


Bootstrap Estimation of the Predictive Distributions of Reserves Using Paid and Incurred Claims

2. The Munich chain laddermethodThe MCL model aims to produce a more con-

sistent prediction of ultimate claims when model-

ing both paid and incurred claim data. It is spe-

cially designed to deal with the correlation be-

tween paid and incurred claims. The traditional

models, such as the chain ladder model, some-

times produce unsatisfactory results by ignoring

this dependence. It should be emphasized that

the paid and incurred claims from the same cal-

endar years are not correlated. It is that the paid

claims (incurred claims) are correlated with the

incurred claims (paid claims) from the next (pre-

vious) calendar year.

The fundamental structure of the MCL model

is the same as Mack’s distribution-free chain lad-

der model (Mack 1993). In other words, the chain

ladder development factors in the MCL model

are obtained by Mack’s distribution-free ap-

proach. However, the MCL model adjusts the

chain ladder development factors using the corre-

lations between the observed paid and incurred

claims. The adjusted chain ladder development

factors therefore become individual not only for

different development years but also for differ-

ent accident years. The adjustment is explained

in more detail in Sections 2.1 and 2.2.

2.1. Notation and assumptions

For ease of notation, we assume that we have

a triangle of data. Although the data could be

classified in different ways, we refer to the rows

as “accident years” and the columns as “devel-

opment years.”

Denote CPij as cumulative paid claims and CIij

as cumulative incurred claims occurred in acci-

dent year i, development year j, where 1· i · nand 1· j · n¡ i+1 for the observed data. Theaim of the chain ladder technique and of MCL

is to estimate the data up to development year n.

This produces estimates for CPij and CIij , where

1· i · n and n¡ i+2· j · n, and we therefore

refer to the complete rectangle of data in the as-

sumptions: 1· i, j · n.Mack’s distribution-free chain ladder method

models the pattern of the development factors,

which are defined as FPij = CPi,j+1=C

Pij , for paid

claims and FIij = CIi,j+1=C

Iij , for incurred claims.

Also the ratios of paid divided by incurred claims

and the inverse are introduced as Qij = CPij =C

Iij

and Q¡1ij = CIij=C

Pij , respectively.

Furthermore, define the observed data for

accident year i, up to development year k as Pi(k)

= fCPij : j · kg, Ii(k) = fCIij : j · kg and Bi(k) =fCPij ,CIij : j · kg, for paid claims, incurred claimsand both, respectively.

The following assumptions are taken from

Quarg and Mack (2004), Section 2.1.2.

Assumption A (Expectations)

(A1) For 1· j · n there exists a constant fPjsuch that (for i= 1, : : : ,n)

E[FPij j Pi(j)] = fPj :This assumption is for paid claims. It is neces-

sary to make another analogous assumption for

incurred claims since both data sets are taken into

account.

(A2) For 1· j · n, there exists a constant fIjsuch that (for i= 1, : : : ,n)

E[FIij j Ii(j)] = fIj :In order to analyze the two run-off triangles

dependently, the following assumptions are also

required.

(A3) For 1· j · n, there exists a constant q¡1jsuch that (for i= 1, : : : ,n)

E[Q¡1ij j Pi(j)] = q¡1j :(A4) For 1· j · n, there exists a constant qj

such that (for i= 1, : : : ,n)

E[Qij j Ii(j)] = qj:Note that assumptions (A3) and (A4) will ap-

ply that imply that Qij is constant, which is con-

tradictory–see Section 3.1.2 of Mack and Quarg

(2004) for a discussion of this point.



Assumption B (Variances)

(B1) For 1· j · n, there exists a constant ¾Pjsuch that (for i= 1, : : : ,n)

Var[FPij j Pi(j)] =(¾Pj )

2

CPij:

Again, the analogous assumption for the in-

curred claims is made as follows.

(B2) For 1· j · n, there exists a constant ¾Ijsuch that (for i= 1, : : : ,n)

Var[FIij j Ii(j)] =(¾Ij )

2

CIij:

Also, for the ratios of incurred to paid and

vice versa, the following variance assumptions

are made.

(B3) For 1· j · n, there exists a constant ¿Pjsuch that (for i= 1, : : : ,n)

Var[Q¡1ij j Pi(j)] =(¿Pj )

2

CPij:

(B4) For 1· j · n, there exists a constant ¿ Ijsuch that for (i= 1, : : : ,n)

Var[Qij j Ii(j)] =(¿ Ij )

2

CIij:

Assumption C (Independence) The usual as-

sumptions for individual triangles are as follows:

(C1) The random variables pertaining to dif-

ferent accident years for paid claims, i.e., fCP1j jj = 1,2, : : : ,ng, : : : ,fCPnj j j = 1,2, : : : ,ng, are sto-chastically independent.

(C2) The random variables pertaining to dif-

ferent accident years for incurred claims, i.e.,

fCI1j j j = 1,2, : : : ,ng, : : : ,fCInj j j = 1,2, : : : ,ng,are stochastically independent.

In fact, a stronger assumption is used (see Sec-

tion 3.2 of Quarg and Mack 2004), which is in-

dependence of accident years across both paid

and incurred claims:

(C3) The random sets fCP1j ,CI1j j j =1,2, : : : ,ng, : : : ,fCPnj ,CInj j j = 1,2, : : : ,ng, are stochasticallyindependent.

Using assumptions A to C, the Pearson resid-

uals used in the MCL model can be defined as

shown in Equations (2.1) to (2.4). These residu-

als are a crucial part of the bootstrapping proce-

dures described in Section 4.

rPij =FPij ¡E[FPij j Pi(j)]qVar[FPij j Pi(j)]

, (2.1)

rQ¡1

ij =Q¡1ij ¡E[Q¡1ij j Pi(j)]q

Var[Q¡1ij j Pi(j)], (2.2)

rIij =FIij ¡E[FIij j Ii(j)]qVar[FIij j Ii(j)]

, (2.3)

rQij =Qij ¡E[Qij j Ii(j)]qVar[Qij j Ii(j)]

: (2.4)

Assumption D (Correlations)

(D1) There exists a constant ½P such that (for

1· i, j · n)E[rPij j Bi(j)] = ½PrQ

¡1ij : (2.5)

The following equation states that the constant

½P is in fact the correlation coefficient between

the residuals. Note that since the residuals have

variance 1, the correlation is equal to the covari-

ance. The proof can be found in Quarg and Mack

(2004).

Cov[rPij ,rQ¡1ij j Pi(j)] = Corr[rPij ,rQ

¡1ij j Pi(j)]

= Corr[FPij ,Q¡1ij j Pi(j)] = ½P

(2.6)

Quarg and Mack (2004) derives expected MCL

paid development factors adjusted by the corre-

lation as shown in Equation (2.7).

E[FPij j Bi(j)]

= E[FPij j Pi(j)] +vuut Var[FPij j Pi(j)]Var[Q¡1ij j Pi(j)]

£Corr[FPij ,Q¡1ij j Bi(j)](Q¡1ij ¡E[Q¡1ij j Pi(j)]):(2.7)



(D2) Analogous to assumption (D1), for the

incurred claims it is assumed that there exists a

constant ½I such that (for 1· i, j · n)E[rIij j Bi(j)] = ½IrQij : (2.8)

Similarly, the constant ½I measures the corre-

lation coefficient or the covariance between the

residuals. i.e.,

Cov[rIij ,rQij j Bi(j)] = Corr[rIij ,rQij j Bi(j)]

= Corr[FIij ,Qij j Bi(j)] = ½I

(2.9)

Hence, the expected MCL incurred develop-

ment factors adjusted by the correlation can be

derived as follows:

E[FIij j Bi(j)]

= E[FIij j Ii(j)]+vuut Var[FIij j Ii(j)]Var[Qij j Ii(j)]

£Cov[FIij ,Qij j Bi(j)](Qij ¡E[Qij j Ii(j)]):(2.10)

2.2. Unbiased estimates of theparameters

In this section, we summarize the unbiased es-

timates of the parameters derived by Quarg and

Mack (2004). For the paid data, estimates are

required for the parameters of the development

factors, the variances and also the correlation co-

efficient.

The estimates of the paid development factor

parameters can be interpreted as weighted aver-

ages of the observed development factors FPij or

Q¡1ij , using CPij as the weights

fPj =

Pn¡ji=1 C

Pi,j+1Pn¡j

i=1 CPij

=

n¡jXi=1

CPijPn¡ji=1 C

Pij

FPij

(2.11)and

q¡1j =

Pn¡j+1i=1 CIijPn¡j+1i=1 CPij

=

n¡j+1Xi=1

CPijPn¡j+1i=1 CPij

Q¡1ij :

(2.12)

The unbiased estimates of the variances are as

follows:

(¾Pj )2 =

1

n¡ j¡ 1n¡jXi=1

CPij (FPij ¡ fPj )2

(2.13)and

(¿ Pj )2 =

1

n¡ jn¡j+1Xi=1

CPij (Q¡1ij ¡ q¡1j )2

(2.14)Hence the Pearson residuals are

rPij =FPij ¡ fPj¾Pj

qCPij (2.15)

and

rQ¡1

ij =Q¡1ij ¡ q¡1j

¿Pj

qCPij : (2.16)

Finally, the estimate of the correlation coeffi-

cient is given in Equation (2.17).

½P =

Pi,j r

Q¡1ij rPijP

i,j(rQ¡1ij )2

: (2.17)

Similarly, for incurred data, the estimates of

the development factor parameters can be inter-

preted as weighted averages of the development

factors FIij or Qij , using CIij as the weights:

fIj =

Pn¡ji=1 C

Ii,j+1Pn¡j

i=1 CIij

=

n¡jXi=1

CIijPn¡ji=1 C

Iij

FIij

(2.18)and

qj =

Pn¡j+1i=1 CPijPn¡j+1i=1 CIij

=

n¡j+1Xi=1

CIijPn¡j+1i=1 CIij

Qij:

(2.19)

Also, the unbiased estimates for the variance

parameters are as follows:

(¾Ij )2 =

1

n¡ j¡ 1n¡jXi=1

CIij(FIij ¡ fIj )2:

(2.20)and

(¿ Ij )2 =

1

n¡ jn¡j+1Xi=1

CIij(Qij ¡ qj)2:

(2.21)



Hence the Pearson residuals are

rIij =FIij ¡ fIj¾Ij

qCIij (2.22)

and

rQij =Qij ¡ qj¿ Ij

qCIij: (2.23)

And finally, the estimate of the correlation co-

efficient is given in Equation (2.24).

½I =

Pi,j r

Qij r

IijP

i,j(rQij )2: (2.24)

Assumptions A in Section 2.1 have defined the

expectations of the development factors, given

just the data in the respective triangles. In or-

der to produce a single estimate based on the

data from both paid and incurred data, Quarg

and Mack (2004) also considers the expectations

of the development factors given both triangles,

and defines E[FPij j Bi(j)] = ¸Pij and E[FIij j Bi(j)]= ¸Iij . Using plug-in estimates from Equations

(2.11) to (2.17), the estimates of the paid MCL

development factors are calculated using Equa-

tion (2.7):

ˆPij = f

Pj + ½

P¾Pj¿Pj(Q¡1ij ¡ q¡1j ): (2.25)

Similarly, plug-in estimates from Equations

(2.18) to (2.24) are used in Equation (2.10) so

that the estimates of the incurred development

factors are

ˆ Iij = f

Ij + ½

I¾Ij¿ Ij(Qij ¡ qj): (2.26)

3. Bootstrapping and claimsreserving

Bootstrapping is a simulation-based approach

to statistical inference. It is a method for produc-

ing sampling distributions for statistical quan-

tities of interest by generating pseudo samples,

which are obtained by randomly drawing, with

replacement, from observed data. It should be

emphasized that bootstrapping is a method rather

than a model. Bootstrapping is useful only when

the underlying model is correctly fitted to the

data, and bootstrapping is applied to data which

are required to be independent and identically

distributed. The bootstrapping method was first

introduced by Efron (1979) and a good introduc-

tion to the algorithm can be found in Efron and

Tibshirani (1993).

For the purpose of clarity we begin by giving

a general bootstrapping algorithm and briefly re-

viewing previous applications of bootstrapping

to claims reserving. In Section 4, we show how

an algorithm of this type can be applied to the

MCL. Suppose we have a sample ~X and we re-

quire the distribution of a statistic μ. The follow-

ing three steps comprise the simplest bootstrap-

ping process:

1. Draw a bootstrap sample ~XB1 = fXB1 ,XB2 , : : : ,XBn g1 from the observed data ~X = fX1,X2, : : : ,Xng.

2. Calculate the statistic of interest μB1 for the

first bootstrap sample ~XB1 = fXB1 ,XB2 , : : : ,XBn g1.3. Repeat steps 1 and 2 N times.

By repeating steps 1 and 2 N times, we obtain

a sample of the unknown statistic μ, calculated

fromN pseudo samples, i.e.,~μB = fμB1 , μB2 , : : : , μBNg.When N ¸ 1000, the empirical distribution con-structed from ~μB = fμB1 , μB2 , : : : , μBNg can be takenas the approximation to the distribution for the

statistic of interest μ. Hence all the quantities of

the statistic of interest μ can be obtained, since

such a distribution contains all the information

related to μ.

The above bootstrapping algorithm can be ap-

plied to the prediction distributions for the best

estimates in stochastic claims reserving. England

and Verrall (2007) contains an excellent review

of the application. In addition, Lowe (1994),

England and Verrall (1999) and Pinheiro (2003)

are also good resources for more details. Eng-

land and Verrall (2007) showed how bootstrap-

ping can be used for recursive models, follow-



ing from the earlier papers (England and Verrall

1999; England 2002) which applied bootstrap-

ping to the over-dispersed Poisson model.

It should be noted here that the Pearson resid-

uals are commonly used rather than the origi-

nal data in the generalized linear model (GLM)

framework. The Pearson residuals are required

in order to scale the response variables in the

GLM so that they are identically distributed. This

is necessary because the bootstrap algorithm re-

quires that the response variables are indepen-

dent and identically distributed.

Other papers in the actuarial literature that con-

sider triangles of dependent data include Taylor

and McGuire (2007) and Kirschner et al. (2008).

It should be noted here that a model taking ac-

count of all information available could be very

valuable, even when the data is dependent in

practice. The dependence makes it difficult to

calculate the prediction error theoretically. For

these reasons, we believe that adopting bootstrap

method for these models is worthy of investiga-

tion, particularly in order to obtain the predictive

distribution of the estimates of outstanding lia-

bilities.

4. Bootstrapping the Munich chainladder modelThis section considers bootstrapping the MCL

model. In Section 4.1 we describe the method-

ology and in Section 4.2 we give the algorithm

that is used.

4.1. MethodologyThe method of bootstrapping stochastic chain

ladder models can be seen in a number of dif-

ferent contexts. England and Verrall (2007) cat-

egorize the models as recursive and nonrecur-

sive and show how bootstrapping methods can

be applied in either case. Since we are dealing

with recursive models here, we follow England

and Verrall and consider the observed develop-

ment link ratios rather than the claims data them-

selves. In other words, for Mack’s distribution-

free chain ladder model the link ratios Fij are

randomly drawn against Xij , noting that

E[Fij j Xij] = E"Xi,j+1Xij

¯¯ Xij

#= fj:

Here, Xij is used to represent observed claims

data in general. Note that the bootstrap estimates

of the development factors fBj which are obtained

by taking weighted averages of the bootstrapped

observed link ratios FBij , use Xij rather than XBij

as the weights.

However, this method cannot be simply ex-

tended to the MCL method, since this model is

designed for dealing with two sets of correlated

data, the paid and incurred claims. This means

that it is not possible to use the normal bootstrap

approach because the independence assumption

cannot be met.

In order to address the problem of how to

adapt the existing bootstrap approach in order to

cope with the MCL method for dependent data

sets, the consideration of the correlation is cru-

cial. It should be noted that the correlation which

is observed in the data represents real depen-

dence between the paid and incurred claims, and

the model is specifically designed for this depen-

dence. Therefore, it should remain unchanged

within any resampling procedure. The straight-

forward solution is to draw samples pairwise so

that the correlation between the two dependent

original data sets will not be broken when gen-

erating a sampling distribution for a statistic of

interest.

Obviously, when bootstrapping the recursive

MCL method, the residuals of the paid and in-

curred link ratios are required instead of the

raw data. The question arises of how to deal

with these residuals in order to meet the require-

ment of not breaking the observed dependence

between paid and incurred claims. The answer

is to group all four sets of residuals calculated

in the MCL method, i.e., the paid and incurred



development link ratios, the ratios of incurred

over paid claims from the previous years, and its

inverse, individually. There are two reasons for

this. First, the paid claims (incurred claims) are

correlated to the incurred to the paid claims ratio

(paid to incurred claims ratio) from the previous

year, and doing this will preserve the required

dependence. Second, the correlation coefficient

of paid and incurred claims is equal to the corre-

lation coefficient of those residuals, as stated in

Equations (2.6) and (2.10).

Thus, in the case of the paid claims data, the

triangles (which have the same dimensions) con-

taining the residuals of the observed paid link

ratios and the residuals of the ratios of incurred

over paid (except the first column), are paired

together. The same procedure is used for the in-

curred claims data. We do this for convenience,

even though the ratios of the paid over incurred

claims and the inverse give the same information.

Note that these ratios should remain unchanged

when pairing them with paid and incurred claims

with the same dimensions. The consequence of

this is that all four sets of residuals for paid, in-

curred link ratios and the ratios of incurred over

paid claims and the inverse are all grouped to-

gether. (Note here that an alternative approach

would be to group three sets of residuals: the

residuals of the paid and incurred link ratios and

either the residuals of the paid over incurred ra-

tios or the inverse. This would produce the same

results as grouping four sets of residuals, as the

residuals of paid over incurred ratios and the in-

verse can always be calculated from each other.

However, it is simpler to group the four sets, as

the calculation of the fourth set of residuals is

naturally skipped in this case.)

This combines the four residuals triangles into

one new triangle that consists of these grouped

residuals and we name it as the grouped resid-

ual triangle. In each unit from this triangle of

quadruples, the residuals are from the same ac-

cident and development year and correspond to

paid and incurred claims. Therefore, the new tri-

angle of quadruples contains all the information

available and meanwhile maintains the observed

dependence.

When applying bootstrapping, this triangle is

considered as the observed sample. The new gen-

erated pseudo samples are obtained by random

drawing, with replacement, from the triangle of

quadruples.

The resampled incurred and paid triangles can

be obtained by separating the pairs in the pseudo

sample generated as above and backing out the

residual definition. The MCL approach can then

be applied to calculate all the statistics of inter-

est for the resampled paid and incurred triangles,

i.e., the correlation coefficient for paid and in-

curred, the paid and incurred development fac-

tors, the ratios of paid over incurred or the in-

verse, and the variances. Finally, adjusting the

paid and incurred development factors by the

correlation coefficient using the MCL approach,

the bootstrapped MCL reserve estimates are ob-

tained. This completes a single bootstrap itera-

tion.

Again, the bootstrap method provides only

the estimation error of the MCL method. In

order to include the prediction error and esti-

mate the predictive distribution for the MCL esti-

mates of outstanding liabilities, an additional step

is added at the end of each bootstrap iteration,

which is to add the process variance to the esti-

mation error.

Note that we apply the final simulation for the

process variance to paid and incurred claims, in-

dependently. This is because, for a particular ac-

cident and development year, paid and incurred

claims are actually independent. Under the as-

sumptions of the MCL model, paid (incurred)

claims are only correlated with previous incurred

(paid) claims, and the forecasts produced by the

bootstrapping will pick up this dependency.

In order to obtain a reasonable approximation

to the predictive distribution, at least 1000 pseudo

samples are required. For each of the pseudo



samples, the row totals and overall total of out-

standing liabilities are stored so that the sample

means, sample variances and the empirical dis-

tributions can be calculated and plotted. They

are taken as the approximations to the best es-

timates of outstanding liabilities, the prediction

errors and the predictive distributions of the out-

standing liabilities. Also, an estimate of any re-

quired percentile and confidence interval can be

calculated from the predictive distribution.

In order to satisfy the assumption that the sam-

ple is identically distributed in the bootstrapping

procedure, the Pearson residuals are calculated

and used. As in England and Verrall (2007), we

use the Pearson residuals of the observed de-

velopment factors rather than those for the ac-

tual claims, since we are using recursive mod-

els. Note that a bootstrap bias correction is also

needed, and the simplest way to do this is to mul-

tiply the residuals byp(n¡ j)=(n¡ j¡ 1).

In addition to drawing the grouped sample for

bootstrapping correlated data sets, there are also

two other practical points that should be men-

tioned. The first is to note that the fitted values

are obtained by starting from the final diagonal

in each triangle and working backwards, by di-

viding by the development factors. The second

is that the zero residuals which appear in both

triangles are also left out.

4.2. Algorithm

This section provides the algorithm, step by

step, which is needed in order to implement the

bootstrap process introduced in Section 4.1.

– Apply the MCL method to both the cumula-

tive paid and incurred claims data to obtain

the residuals for all four sets of ratios: the

paid, incurred link ratios, the paid over in-

curred ratios and the reverse. They can be

obtained from following equations:

rPij =FPij ¡ fPj¾Pj

qCPij ,

rQ¡1

ij =Q¡1ij ¡ q¡1j

¿ Pj

qCPij ,

rIij =FIij ¡ fIj¾Ij

qCIij and

rQij =Qij ¡ qj¿ Ij

qCIij :

– Adjust the Pearson residual estimates by mul-

tiplying byp(n¡ j)=(n¡ j¡ 1) to correct the

bootstrap bias.

– Group all four residuals together, i.e., rPij , rQ¡1ij ,

rIij and rQij . We write this as Uij = f(rPij ), (rQ

¡1ij ),

(rIij), (rQij )g.

– Start the iterative loop to be repeated N times

(N ¸ 1000). This consists of the followingsteps:

1. Randomly sample from the grouped resid-

uals with replacement, denoted as UBij =

f(rPij )B, (rQ¡1

ij )B , (rIij)B , (rQij )

Bg, from the group-

ed triangle so that a pseudo sample of the

grouped residuals is created.

2. Calculate the pseudo samples of the four

triangles for the paid, incurred link ratios, the

ratios of paid over incurred and the inverse by

inverting the Pearson residuals definition as

follows:

(FPij )B =

(rPij )B¾PjqCPi,j

+ fPj ,

(Q¡1ij )B =

(rQ¡1

ij )B¿PjqCPi,j

+ q¡1j ,

and

(FIij)B =

(rIij)B¾IjqCIi,j

+ fIj ,

(Qij)B =

(rQij )B¿ IjqCIi,j

+ qj :

3. Calculate the CPi,j-weighted and CIi,j-

weighted average of the bootstrap paid and



incurred development factors as follows:

(fPj )B =

n¡jXi=1

CPi,jPn¡ji=1 C

Pi,j

(FPij )B ,

(q¡1j )B =

n¡j+1Xi=1

CPijPn¡j+1i=1 CPij

(Q¡1ij )B

and

(fIj )B =

n¡jXi=1

CIi,jPn¡ji=1 C

Ii,j

(FIij)B ,

(qj)B =

n¡j+1Xi=1

CIijPn¡j+1i=1 CIij

(Qij)B:

Note that the weights used here are from the

original data sets and not from the pseudo

samples.

4. Calculate the corresponding correlation

coefficient for the resampled data using the

pseudo residuals (rPij )B , (rQ

¡1ij )B, (rIij)

B and

(rQij )B as follows,

(½P)B =

Pi,j(r

Q¡1ij )B(rPij )

BPi,j((r

Q¡1ij )B)2

and

(½I)B =

Pi,j(r

Qij )B(rIij)

BPi,j((r

Qij )B)2

:

5. Calculate the variances for the bootstrap

data as follows:

((¾Pj )2)B =

1

n¡ j¡ 1n¡jXi=1

CPij ((FPij )

B ¡ (fPj )B)2

((¿ Pj )2)B =

1

n¡ j¡ 1n¡jXi=1

CPij ((Q¡1ij )

B ¡ (q¡1j )B)2

((¾Ij )2)B =

1

n¡ j¡ 1n¡jXi=1

CIij((FIij)B ¡ (fIj )B)2

((¿ Ij )2)B =

1

n¡ j¡ 1n¡jXi=1

CIij((Qij)B ¡ (qj)B)2:

Note that all the sums here are from 1 to n¡ jbecause the last diagonals of paid to incurred

(and incurred to paid) are not included in the

resampling procedure.

6. Calculate the bootstrap development fac-

tors adjusted by the correlation coefficient

between the pseudo samples as follows:

( ˆPij)B = (fPj )

B +(½P)B(¾Pj )

B

(¿ Pj )B((Q¡1ij )

B ¡ (q¡1j )B)

and

( ˆ Iij)B = (fIj )

B +(½I)B(¾Ij )

B

(¿ Ij )B((Qij)

B ¡ (qj)B),

for the resampled bootstrap paid and incurred

run-off triangles, respectively.

7. Simulate a future payment for each cell in

the lower triangle for both paid and incurred

claims, from the process distribution with the

mean and variance calculated from the previ-

ous step. To do this, the following steps are

required:

² For the one-step-ahead predictions fromthe leading diagonal, a normal distribution

is assumed, i.e., for 2· i · n,XPi,n¡i+2 »Normal(( ˆPi,n¡i+1)BXPi,n¡i+1,

((¾Pn¡i+1)2)BXPi,n¡i+1)

for paid claims and

XIi,n¡i+2 »Normal(( ˆ Ii,n¡i+1)BXIi,n¡i+1,((¾In¡i+1)

2)BXIi,n¡i+1)

for incurred claims.

² For the two-step-ahead predictions up tothe n-step-ahead predictions, normal dis-

tributions are still assumed, but with the

mean and variance calculated from previ-

ous predictions instead of from the observ-

ed data, i.e., for 3· k · n and n¡ k+3·j · n,XPkl »Normal(( ˆPi,l¡1)BXPk,l¡1, ((¾Pl¡1)2)BXPk,l¡1)

for paid claims, and

XIkl »Normal(( ˆ Ii,l¡1)BCIk,l¡1, ((¾Il¡1)2)BXIk,l¡1)

for incurred claims.



Table 1. Paid claims from Quarg and Mack (2004)

j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7

i = 1 576 1804 1970 2024 2074 2102 2131i = 2 866 1948 2162 2232 2284 2348i = 3 1412 3758 4252 4416 4494i = 4 2286 5292 5724 5850i = 5 1868 3778 4648i = 6 1442 4010i = 7 2044

Table 2. Incurred claims from Quarg and Mack (2004)

j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7

i = 1 978 2104 2134 2144 2174 2182 2174i = 2 1844 2552 2466 2480 2508 2454i = 3 2904 4354 4698 4600 4644i = 4 3502 5958 6070 6142i = 5 2812 4882 4852i = 6 2642 4406i = 7 5022

8. Sum the simulated payments in the future

triangle by origin year and overall, to give

the origin year and total reserve estimates re-

spectively.

9. Store the results, and return to the start of

the iterative loop.

5. Examples

This section illustrates the bootstrapping ap-

proach to the MCL and uses two numerical ex-

amples to assess the results. The first example

uses the data from Quarg and Mack (2004). Ex-

ample 2 uses market data from Lloyd’s which

have been scaled for confidentiality reasons.

These data relate to aggregated paid and incurred

claims for two Lloyd’s syndicates, categorized at

risk level.

Example 1 is included in order to illustrate the

results for the original set of data used by Quarg

and Mack (2004). The purpose of example 2 is

to illustrate that the MCL model does not neces-

sarily provide better results in all situations. Our

results indicate that it performs better when the

data have less inherent variability and are less

“jumpy.”

Table 3. A Comparison of methods for reserves projected onpaid and incurred claims

Bootstrap MCL Mack

Paid Incurred Paid Incurred Paid Incurred

i = 1 0 43 0 43 0 43i = 2 37 94 35 96 32 97i = 3 109 131 103 135 158 88i = 4 277 321 269 326 331 277i = 5 299 296 289 302 407 191i = 6 657 651 646 655 919 465i = 7 5492 5646 5505 5606 4063 6380

Overall Total 6871 7182 6846 7163 5911 7540

Table 4. Comparison of bootstrap prediction errors for MCLand CL Mack chain ladder methods

MCL Mack

Paid Incurred Paid Incurred

i = 1 0 0 0 0i = 2 5 5 15 9i = 3 48 70 53 82i = 4 61 86 68 105i = 5 72 104 72 117i = 6 215 208 289 216i = 7 735 716 897 869

Overall Total 776 782 991 980

EXAMPLE 1. In this section, we apply the boot-

strapping methodology with 10,000 bootstrap

simulations to the data from Quarg and Mack

(2004).

Tables 1 and 2 show the data. In order to il-

lustrate the nature of the run-off of the data, Fig-

ures 1 and 2 are the plots of the data from Ta-

bles 1 and 2, respectively. From Figures 1 and

2, it can be seen that the data are stable and not

excessively spread out.

The results of applying the bootstrap method-

ology described in this paper are shown be-

low, and are compared with the results from

the straightforward chain ladder technique and

Mack’s method for the prediction errors. Table 3

shows that the theoretical MCL reserves (from

Quarg and Mack 2004) and the mean of the boot-

strap distributions, together with the chain ladder

reserves when the triangles are considered sep-

arately. It can be seen that the bootstrap means

are close to the theoretical values, for both the

paid and incurred claims.



Figure 1. Paid claims

Figure 2. Incurred claims

Figure 3. Comparison of predictive distributions of overall reservesfor CL and MCL reserves for paid and incurred claims

Table 4 displays the bootstrap prediction error

of the MCL reserves projected by both paid and

incurred claims. Also shown are the prediction

errors for the Mack method. It can be seen that

the MCL prediction errors are lower than those

of the Mack method.

Since the purpose of the MCL method is to

use more data to improve the estimation of the

reserves, it is expected that the prediction errors

should be lower than the Mack model. This is

confirmed for these data by Table 5, which shows

that the prediction error, as a percentage of the re-

serve, is lower for the MCL reserves than the pre-

diction error of CL the reserves from the Mack

model.

In Figure 3, the distributions of the MCL and

CL reserve projections for paid and incurred

claims are plotted. Figure 3 shows that the paid

and incurred best reserve estimates are very close

when using the MCL approach. In contrast, the



Table 5. Comparison of bootstrap prediction errors % forMCL and CL methods

MCL Mack


i = 1 — 0% — 0%i = 2 14% 5% 45% 9%i = 3 44% 53% 33% 93%i = 4 22% 27% 21% 38%i = 5 24% 35% 18% 61%i = 6 33% 32% 31% 46%i = 7 13% 13% 22% 14%

Total 11% 11% 17% 13%

paid and incurred best reserve estimates projected

by the chain ladder method are much further

apart. Furthermore, the CL method provides a

much more spread-out distribution than the MCL

approach, in the case of both paid and incurred

claims.

EXAMPLE 2. In this section, a set of aggregate

data from Lloyd’s syndicates is considered. In

this case, the data are not as stable or well-be-

haved and the results are quite different. Tables 6

and 7 show the data, which are plotted in Fig-

Table 6. Scaled aggregate paid claims at risk level from Lloyd’s Market

j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7 j = 8 j = 9 j = 10

i = 1 1139 5680 6906 7069 7205 7350 7421 7487 7506 7518i = 2 1101 6223 8038 8652 9064 9249 9343 9421 9455i = 3 1215 8058 10593 11638 12346 12784 12978 13161i = 4 949 5324 7608 8257 8719 8972 9103i = 5 638 4107 6367 7099 7489 7586i = 6 647 4166 6231 7029 7335i = 7 1198 4660 7303 7791i = 8 1194 6540 9251i = 9 1248 6062i = 10 1083

Table 7. Scaled aggregate incurred claims at risk level from Lloyd’s Market

j = 1 j = 2 j = 3 j = 4 j = 5 j = 6 j = 7 j = 8 j = 9 j = 10

i = 1 2170 6941 7709 7403 7452 7508 7514 7547 7555 7563i = 2 2184 7822 9182 9368 9445 9520 9508 9547 9585i = 3 2759 10947 12649 12947 13090 13283 13328 13360i = 4 1958 8398 9814 9800 9306 9370 9272i = 5 1376 6177 7699 7799 7984 7904i = 6 1464 5861 7546 7679 7687i = 7 2405 6385 8151 8234i = 8 3128 8772 10265i = 9 2980 8045i = 10 2722

ures 4 and 5. It can be seen from these figures

that the data are much more unstable and more

spread out compared with the previous two ex-

amples.

The MCL method still produces consistent ul-

timate loss predictions for this data set, as shown

in Table 8. However, the prediction error con-

tained in Table 9, estimated by the bootstrap

MCL approach, appears not to offer such an im-

provement as was seen in Example 1.

Table 10 shows a comparison of the predic-

tion errors as a percentage of the reserve, and

again it can be seen that the results do not in-

dicate that the MCL method is a significant im-

provement over the CL model. The conclusion

from this is that although the MCL method uses

more data, and should be expected to produce

lower prediction errors, this is not always the

case in practice. We believe that the reason for

this is that the assumptions made by the MCL

method–the specific dependencies assumed–

are not as strong as expected in this case. A

conclusion from this is that the data have to be



Figure 4. Scaled paid claims from Lloyd’s Market

Figure 5. Scaled incurred claims from Lloyd’s Market

Table 8. Comparison of methods for reserves projected onpaid and incurred claims

Bootstrap MCL CL

Paid Incurred Paid Incurred Paid Incurred

i = 1 0 45 0 45 0 0i = 2 24 138 19 139 15 15i = 3 78 245 71 245 63 62i = 4 146 236 139 237 143 142i = 5 252 357 246 354 220 215i = 6 383 455 373 454 400 395i = 7 590 614 579 624 829 825i = 8 1345 1355 1318 1366 1850 1820i = 9 3758 3811 3707 3787 4081 4042i = 10 9874 9962 9740 9840 8765 8698

Overall Total 16451 17218 16192 17092 16367 16214

examined carefully before the MCL method is

applied.

This conclusion is reinforced by Figure 6,

which shows the predictive distributions.

5. ConclusionThis paper has shown how a bootstrapping ap-

proach can be used to estimate the predictive

distribution of outstanding claims for the MCL

model. The model deals with two dependent data

sets, the paid and incurred claims triangles, for

Table 9. Comparison of bootstrap prediction errors for MCLand CL methods

MCL CL


i = 1 0 0 0 0i = 2 10 10 2 2i = 3 16 32 11 11i = 4 32 27 39 38i = 5 47 62 43 44i = 6 78 97 92 96i = 7 204 249 166 168i = 8 324 372 391 382i = 9 573 592 987 973i = 10 1762 1818 1940 1963

Overall Total 1911 1994 2277 2305

general insurance reserving purposes. We believe

that bootstrapping is well-suited for these pur-

poses from a practical point of view, since it

avoids complicated theoretical calculations and

is easily implemented in a simple spreadsheet.

This paper adapts the method by taking account

of the dependence observed in the data and re-

sampling pairwise.

A number of examples have been given, which

show that the MCL model does not always pro-

duce superior results to the straightforward chain



Figure 6. Comparison of predictive distributions of CL andMCL reserves predicted on paid and incurred claims

Table 10. Comparison of bootstrap prediction errors for MCLand CL methods using scaled data

MCL CL


i = 1 — — — —i = 2 43% 7% 69% 14%i = 3 21% 13% 27% 18%i = 4 22% 11% 28% 27%i = 5 19% 17% 20% 20%i = 6 20% 21% 23% 24%i = 7 35% 41% 20% 20%i = 8 24% 27% 21% 21%i = 9 15% 16% 23% 24%i = 10 18% 18% 22% 23%

Overall Total 12% 12% 14% 14%

ladder model. As a consequence, we believe that

it is important for the data to be carefully checked

to test whether the dependency assumptions of

the MCL model are valid for each data set be-

fore it is applied.

AcknowledgmentFunding for this project from Lloyd’s of Lon-

don is gratefully acknowledged.

ReferencesEfron, B., “Bootstrap Methods: Another Look at the Jack-

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Robustifying Reserving

by Gary G. Venter and Dumaria R. Tampubolon

ABSTRACT

Robust statistical procedures have a growing body of lit-

erature and have been applied to loss severity fitting in

actuarial applications. An introduction of robust methods

for loss reserving is presented in this paper. In particu-

lar, following Tampubolon (2008), reserving models for a

development triangle are compared based on the sensitiv-

ity of the reserve estimates to changes in individual data

points. This measure of sensitivity is then related to the

generalized degrees of freedom used by the model at each

point.

KEYWORDS

Loss reserving; regression modeling; robust, generalized degrees of freedom



All models are wrong but some are useful.

–Christian Dior (or maybe George E. P. Box)

1. Introduction

The idea of this paper is simple. For models

using a loss development triangle, the robustness

of the model can be evaluated by comparing the

derivative of the loss reserve with respect to each

data point. All else being equal, models that are

highly sensitive to a few particular observations

are less preferred than ones that are not. This is

supported by the fact that individual cells can be

highly volatile. This general approach, based on

Tampubolon (2008), is along the lines of robust

statistics, so some background into robust statis-

tics will be the starting point. Published models

on three data sets will be tested by this method-

ology. For two of them, unsuspected problems

with the previously best-fitting models are found,

leading to improved models.

The sensitivity of the reserve estimate to in-

dividual points is related to the power of those

points to draw the fitted model towards them.

This can be measured by what Ye (1998) calls

generalized degrees of freedom (GDF). For a

model and fitting procedure, the GDF at each

point is defined as the derivative of the fitted

point with respect to the observed point. If any

change in a sample point is matched by the same

change in the fitted, the model and fitting proce-

dure are giving that point full control over its fit,

so a full degree of freedom is used. GDF does

not fully explain the sensitivity of the reserve to

a point, as the position of the point in the triangle

also gives it more or less power to change the re-

serve estimate, but it adds some insight into that

sensitivity.

Section 2 provides some background of robust

analysis and Section 3 shows some previous ap-

plications to actuarial problems. These help to

place the current proposal into perspective in that

literature. Sections 4, 5, and 6 apply this ap-

proach to some published loss development mod-

els. Section 7 concludes.

2. Robust methods in general

Classical statistics takes a model structure and

tries to optimize the fit of data to the model un-

der the assumption that the data is in fact gener-

ated by the process postulated in the model. But

in many applied situations, the model is a con-

venient simplification of a more complex pro-

cess. In this case, the optimality of estimation

methods such as maximum likelihood estimation

(MLE) may no longer hold. In fact, a few obser-

vations that do not arise from the model assump-

tions can sometimes significantly distort the esti-

mated parameters when standard techniques are

used. For instance, Tukey (1960) gives examples

where even small deviations from the assumed

model can greatly reduce the optimality proper-

ties. Robust statistics looks for estimation meth-

ods that in one way or another can insulate the

estimates from such distortions.

Perhaps the simplest such procedure is to iden-

tify and exclude outliers. Sometimes outliers

clearly arise from some other process than the

model being estimated, and it may even be clear

when current conditions are likely to generate

such outliers, so that the model can then be ad-

justed. If the parameter estimates are strongly in-

fluenced by such outliers, and the majority of the

observations are not consistent with those esti-

mates, it is reasonable to exclude the outliers and

just be cautious about when to use the model.

An example is provided by models of the U.S.

one-month treasury bill rates at monthly inter-

vals. Typical models postulate that the volatility

of the rate is higher when the rate itself is higher.

Often the volatility is proposed to be proportional

to the pth power of the rate. The question is–

what is p? One model, the CIR or Cox, Inger-

soll, Ross model, assumes a p value of 0.5. Other

models postulate p as 1 or even 1.5, and others

try to estimate p as a parameter. An analysis by

Dell’Aquila, Ronchetti, and Troiani (2003) found

that when using traditional methods, the estimate



of p is very sensitive to a few observations in the

1979—82 period, when the U.S. Federal Reserve

bank was experimenting with monetary policy.

Including that period in the data, models with

p = 1:5 cannot be rejected, but excluding that

period finds that p = 0:5 works just fine. That

period also experienced very high values of the

interest rate itself, so their analysis suggests that

using p = 0:5 would make sense unless the inter-

est rate is unusually high.

A key tool in robust statistics is the identifica-

tion of influential observations, using the influ-

ence function defined by Hampel (1968). This

procedure looks at statistics calculated from a

sample, such as estimated parameters, as func-

tionals of the random variables that are sampled.

The influence function for the statistic at any ob-

servation is a functional derivative of the statis-

tic with respect to the observed point. In practice,

analysts often use what is called the empirical in-

fluence. For instance, Bilodeau (2001) suggests

calculating the empirical influence at each sam-

ple point as the sample size times the decrease

(which may be negative) in the statistic from ex-

cluding the point from the sample. That is, the

influence is n times [statistic with full sample mi-

nus statistic excluding the point]. If the statistic is

particularly sensitive to a single or a few obser-

vations, its accuracy is called into question. The

gross error sensitivity (GES) is defined as the

maximum absolute value of the influence func-

tion across the sample.

The effect on the statistic of small changes in

the influential observations is also a part of ro-

bust analysis, as these effects should not be too

large either. If each observation has substantial

randomness, the random component of influen-

tial observations has a disproportionate impact

on the statistic. The approach used below in the

loss reserving case is to identify observations for

which small changes have large impacts on the

reserve estimate.

Exclusion is not the only option for dealing

with outliers. Estimation procedures that use but

limit the influence of the outliers are also an im-

portant element of robust statistics. Also, find-

ing alternative models which are not dominated

by a few influential points and estimating them

by traditional means can be an outcome of a ro-

bust analysis. In the interest rate case, a model

with one p parameter for October 1979 through

September 1982 and another elsewhere does this.

Finding alternative models with less influence

from a few points is what we will be attempt-

ing in the reserve analysis.

3. Robust methods in insuranceSeveral papers on applying robust analysis to

fitting loss severity distributions have appeared

in recent years. For instance, Brazauskas and Ser-

fling (2000a) focus on estimation of the simple

Pareto tail parameter ® assuming that the scale

parameter b is known. In this notation the sur-

vival function is S(x) = (b=x)®. They compare

several estimators of ®, such as MLE, matching

moments or percentiles, etc. One of their tests is

the asymptotic relative efficiency (ARE) of the

estimate compared to MLE, which is the factor

which when applied to the sample size would

give the sample size needed for MLE to give

the same asymptotic estimation error. Due to the

asymptotic efficiency of MLE, these factors are

never greater than unity, assuming the sample is

really from that Pareto distribution.

The problem is that the sample might not be

from a simple Pareto distribution. Even then,

however, you would not want to identify and

eliminate outliers. Whatever process is generat-

ing the losses would be expected to continue, so

no losses can be ignored.1 The usual approach to

this problem is to find alternative estimators that

have low values of the GES and high values of

ARE. Brazauskas and Serfling (2000a) suggest

1A related problem is contamination of large losses by a non-

recurring process. The papers on robust severity also address this,

but it is a somewhat different topic than fitting a simple model to

a complex process.



estimators they call generalized medians (GM).

The kth generalized median is the median of all

MLE estimators of subsets of size k of the origi-

nal data. That can be fairly calculation-intensive,

however, even with small k of 3, 4, or 5.

Finkelstein, Tucker, and Veeh (2006) define an

estimator they call the probability integral trans-

form statistic (PITS) which is quite a bit easier

to calculate but not quite as robust as the GM.

It has a tuning parameter t in (0,1) to control

the trade-off between efficiency and robustness.

Since (b=x)® is a probability and so a number be-

tween zero and one, it should be distributed uni-

formly [0,1]. Thus (b=x)t® should be distributed

like a uniformly raised to the t power. The av-

erage of these over a sample is known to have

expected value 1=(t+1), so the PITS estimator

is the value of ¯ for which the average of (b=x)t¯

over the sample is 1=(t+1). This is a single-

variable root finding exercise. Finklestein,

Tucker, and Veeh give values of the ARE and

GES for the GM and PITS estimators, shown in

Table 1. A simulation suggests that the GES for

MLE for ®= 1 is about 3.9, and since its ARE

is 1.0 by definition, PITS at 0.94 ARE is not

worthwhile in this context. In general the gener-

alized median estimators are more robust by this

measure.

Other robust severity studies include Brazau-

skas and Serfling (2000b), who use GM estima-

tion for both parameters of the simple Pareto;

Gather and Schultze (1999), who show that the

best GES for the exponential is the median scaled

to be unbiased (but this has low ARE); and Ser-

fling (2002), who applies GM to the lognormal

distribution.

4. Robust approach to lossdevelopmentOmitting points from loss development trian-

gles can sometimes lead to strange results, and

not every development model can be automat-

ically extended to deal with this, so instead of

calculating the influence function for develop-

Table 1. Comparative efficiency and robustness of tworobust estimators of Pareto ®

ARE GM-k PITS-t GM-GES PITS-GES

0.88 3 0.531 2:27® 2:88®0.92 4 0.394 2:60® 3:54®0.94 5 0.324 2:88® 4:08®

ment models, we look at the sensitivity of the

reserve estimate to changes in the cells of the

development triangle, as in Tampubolon (2008).

In particular, we define the impact of a cell on

the reserve estimate under a particular develop-

ment methodology as the derivative of the esti-

mate with respect to the value in the cell. We

do this for the incremental triangle, so a small

change in a cell affects all subsequent cumula-

tive values for the accident year. This seems to

make more sense than looking at the derivative

with respect to cumulative cells, whose changes

would not continue into the rest of the triangle.

If you think of a number in the triangle as its

mean plus a random innovation, the derivative

with respect to the random innovation would be

the same as that with respect to the total, so a

high impact of a cell would imply a high impact

of its random component as well. Thus models

with some cells having high impacts would be

less desirable. One measure of this is the maxi-

mum impact of any cell, which would be anal-

ogous to the GES, but we will also look at the

number of cells with impacts above various

thresholds in absolute value.

This is just a toe in the water of robust analysis

of loss development. We are not proposing any

robust estimators, and will stick with MLE or

possibly quasi-likelihood estimation. Rather we

are looking at the impact function as a model

selection and refinement tool. It can be used to

compare competing models of the same develop-

ment triangle, and it can identify problems with

models that can guide a search for more robust

alternatives. This is similar to finding models that

work for the entire history of interest rate changes

and are not too sensitive to any particular points.



To help interpret the impact function, we will

also look at the generalized degrees of freedom

(GDF) at each point. This is defined as the deriva-

tive of the fitted value with respect to the ob-

served value. If this is near 1, the point’s initial

degree of freedom has essentially been used up

by the model. The GDF is a measure of how

much a point is able to pull the fitted value to-

wards itself. Part of the impact of a point is this

power to influence the model, but its position in

the triangle also can influence the estimated re-

serve. Just like with the impact function, high

values of the GDF would be a detriment.

For the chain-ladder (CL) model, some obser-

vations can be made in general. All three corners

of the triangle have high impact. The lower left

corner is the initial value of the latest accident

year, and the full cumulative development applies

to it. Since this point does not affect any other

calculations, its impact is the development factor,

which can sometimes be substantial. The upper

right corner usually produces a development fac-

tor which, though small, applies to all subsequent

accident years, so its impact can also be substan-

tial. When there is only one year at ultimate, this

impact is the ratio of the sum of all accident years

not yet at ultimate, developed to the penultimate

lag, to the penultimate cumulative value for the

oldest accident year. The upper left corner is a

bit strange in that its impact is usually negative.

Increasing it will increase the cumulative loss at

every lag, without affecting future incrementals,

so every incremental-to-previous-cumulative ra-

tio will be reduced. The points near the upper

right corner also tend to have high impact, and

those near the upper left tend to have negative

impact, but the lower left point often stands alone

in its high impact.

The GDFs for CL are readily calculated when

factors are sums of incrementals over sums of

previous cumulatives. The fitted value at a cell

is the factor applied to the previous cumulative,

so its derivative is the product of its previous

cumulative and the derivative of the factor with

respect to the cell value. But that derivative is just

the reciprocal of the sum of the previous cumu-

latives, so the GDF for the cell is the quotient

of its previous cumulative and the sum. Thus

these GDFs sum down a column to unity, so

each development factor uses up a total GDF of

1.0. Essentially each factor uses 1 degree of free-

dom, agreeing with standard analysis. The aver-

age GDF in a column is thus the reciprocal of

the number of observations in that column. Thus

the upper right cell uses 1 GDF, the previous

column’s cells use 12each on average, etc. Thus

the upper right cells have high GDFs and high

impact.

We will use ODP, for over-dispersed Poisson,

to refer to the cross-classified development mod-

el in which each cell mean is modeled as a prod-

uct of a row parameter and a column parame-

ter, the variance of the cell is proportional to its

mean, and the parameters are estimated by the

quasi-likelihood method. It is well known that

this model gives the same reserve estimate as CL.

Thus if you change a cell slightly, the changed

triangle will give the same reserve under ODP

and CL. Thus the impacts of each cell under

ODP will be the same as those of CL. The GDFs

will not be the same, however, as the fitted val-

ues are not the same for the two models. The

CL fitted value is the product of the factor and

the previous cumulative, whereas the ODP cu-

mulative fitted values are backed down from the

latest diagonal by the development factors, and

then differenced to get the incremental fitted. It

is possible to write down the resulting GDFs ex-

plicitly, but it is probably easier to calculate them

numerically.

It may be fairly easy to find models that re-

duce the impact of the upper right cells. Usually

the development factors at those points are not

statistically significant. Often the development is

small and random, and is not correlated with the

previous cumulative values. In such cases, it may

be reasonable to model a number of such cells as

a simple additive constant. Since several cells go



Table 2. Incremental loss development triangle

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

11,305 18,904 17,474 10,221 3,331 2,671 693 1,145 744 112 40 138,828 13,953 11,505 7,668 2,943 1,084 690 179 1,014 226 16 6168,271 15,324 9,373 11,716 5,634 2,623 850 381 16 28 5587,888 11,942 11,799 6,815 4,843 2,745 1,379 266 809 128,529 15,306 11,943 9,460 6,097 2,238 493 136 11

10,459 16,873 12,668 9,199 3,524 1,027 924 1,1908,178 12,027 12,150 6,238 4,631 919 435

10,364 17,515 13,065 12,451 6,165 1,38111,855 20,650 23,253 9,175 10,31217,133 28,759 20,184 12,87419,373 31,091 25,12018,433 29,13120,640

Table 3. Impact of CL

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

AY0 ¡1:21 ¡0:34 0.04 0.39 0.73 1.10 1.48 1.85 2.46 3.35 4.61 7.31AY1 ¡1:21 ¡0:34 0.04 0.39 0.73 1.10 1.48 1.85 2.46 3.35 4.61 7.31AY2 ¡1:17 ¡0:29 0.08 0.44 0.78 1.14 1.53 1.89 2.51 3.39 4.66AY3 ¡1:15 ¡0:27 0.10 0.46 0.80 1.16 1.55 1.91 2.53 3.41AY4 ¡1:14 ¡0:27 0.11 0.46 0.80 1.17 1.56 1.92 2.54AY5 ¡1:10 ¡0:23 0.15 0.50 0.84 1.21 1.59 1.96AY6 ¡1:07 ¡0:20 0.18 0.53 0.87 1.24 1.62AY7 ¡1:03 ¡0:16 0.22 0.57 0.91 1.28AY8 ¡0:95 ¡0:08 0.30 0.65 0.99AY9 ¡0:73 0.14 0.52 0.87AY10 ¡0:31 0.57 0.95AY11 0.70 1.58AY12 4.95

into the estimation of this constant, the impact of

some of them is reduced. Alternatively, the fac-

tors in that region may follow some trends, linear

or not, that can be used to express them with a

small number of parameters. Again, this would

limit the impact of some of the cells.

The lower left point is more difficult to deal

with in a CL-like model. One alternative is a

Cape-Cod type model, where every accident year

has the same mean level. This can arise, for in-

stance, if there is no growth in the business, but

also can be seen when the development triangle

consists of on-level loss ratios, which have been

adjusted to eliminate known differences among

the accident years. In this type of model, all the

cells go into estimating the level of the last ac-

cident year, so the lower left cell has much less

impact. This reduction in the impact of the ran-

dom component of this cell is a reason for using

on-level triangles.

The next three sections illustrate these con-

cepts using development triangles from the ac-

tuarial literature. The impacts and GDFs are cal-

culated for various models fit to these triangles.

The impacts are calculated by numerical deriva-

tives, as are the GDFs except for those for the

CL, which have been derived above.

5. A development-factor example5.1. Chain ladder

Table 2 is a development triangle used in Ven-

ter (2007a). Note that the first two accident years

are developed all the way to the end of the tri-

angle, at lag 11. Table 3 shows the impact of

each cell on the reserve estimate using the usual



Figure 1. Impact of chain ladder by diagonal

Table 4. GDFs of CL

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

AY0 1.0 0.080 0.093 0.114 0.133 0.151 0.177 0.201 0.245 0.306 0.394 0.581AY1 1.0 0.063 0.070 0.082 0.097 0.110 0.128 0.145 0.174 0.221 0.285 0.419AY2 1.0 0.059 0.073 0.079 0.103 0.124 0.147 0.167 0.202 0.250 0.321AY3 1.0 0.056 0.061 0.076 0.089 0.106 0.128 0.148 0.177 0.223AY4 1.0 0.061 0.073 0.086 0.104 0.126 0.149 0.168 0.202AY5 1.0 0.074 0.084 0.096 0.113 0.130 0.149 0.170AY6 1.0 0.058 0.062 0.077 0.089 0.106 0.123AY7 1.0 0.074 0.086 0.098 0.123 0.146AY8 1.0 0.084 0.100 0.134 0.149AY9 1.0 0.122 0.141 0.158AY10 1.0 0.138 0.156AY11 1.0 0.131AY12 1.0

sum/sum development factors. In the CL model

an explicit formula can be derived for these im-

pacts, but it is easier to do the derivatives numeri-

cally, simply by adding a small value to each cell

separately and recalculating the estimated reserve

to get the change in reserve for the derivative.

As discussed, the impacts are highest in the up-

per right and lower left corners, and the upper left

has negative impact. The impacts increase mov-

ing to the right and down. The last four columns

and the lower left point have impacts greater

than 2, and six points have impacts greater than

4. Table 4 shows the GDFs for the chain ladder

using the formula previous cumulative/sum pre-

vious cumulatives derived in Section 4. L0’s

GDFs are shown as identically 1.0. Like the im-

pact function, except for lag 0, these increase

from one column to the next. Within each col-

umn the sizes depend on the volume of the year.

Figure 1 graphs the impacts by lag along the

diagonals of the triangle. After the first four lags,

the impacts are almost constant across diagonals.



Table 5. Impact of regression model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

AY0 ¡1:36 0.02 0.42 0.67 0.10 0.87 1.35 1.35 0.97 1.35 0.97 1.73AY1 ¡1:56 0.22 0.66 ¡0:04 0.67 1.28 1.35 0.97 1.35 0.97 1.73 1.35AY2 ¡1:53 0.52 ¡0:39 0.38 1.02 1.27 0.97 1.35 0.97 1.73 1.35AY3 ¡0:51 ¡0:64 0.15 0.78 1.07 0.90 1.35 0.97 1.73 1.35AY4 ¡1:24 ¡0:31 0.45 0.76 0.64 1.27 0.97 1.73 1.35AY5 ¡1:38 0.11 0.47 0.32 1.00 0.89 1.73 1.35AY6 ¡1:61 0.22 0.18 0.80 0.68 1.66 1.35AY7 ¡0:89 ¡0:36 0.35 0.24 1.34 1.25AY8 ¡1:34 0.00 ¡0:12 0.87 0.94AY9 0.29 ¡0:44 0.61 0.57AY10 ¡0:18 0.66 0.43AY11 1.11 1.04AY12 4.31

Table 6. GDFs of regression model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11


5.2. Regression model

Venter (2007a) fit a regression model to this

triangle, keeping the first 5 development factors

but including an additive constant. The constant

also represents development beyond lag 5. By

stretching out the incremental cells to be fitted

into a single column Y, this was put into the formof a linear model Y=X¯+ ", which assumes anormal distribution of residuals with equal vari-

ance (homoscedasticity) across cells. X has the

previous cumulative for the corresponding incre-

mentals, with zeros to pad out the columns, a

column of 1s for the constant. There were also

diagonal (calendar year) effects in the triangle.

Two diagonal dummy variables were included in

X, one with 1s for observations on the 4th di-agonal and 0 elsewhere, and one equal to 1 on

the 5th, 8th, and 10th diagonals, ¡1 on the 11th

diagonal, and 0 elsewhere. The diagonals are

numbered starting at 0, so the 4th is the one

beginning with 8,529 and the 10th starts with

19,373. The variance calculation used a hetero-

scedasticity correction. This model with 8 param-

eters fit the data better than the development

factor model with 11 parameters. Here we are

only addressing the robustness properties, how-

ever.

Table 5 gives the impact function for this mod-

el. It is clear that the large impacts on the right

side have been eliminated by using the constant

instead of factors to represent late development.

The effects of the diagonal dummies can also be

seen, especially in the right of the triangle. Now

only one point has impact greater than 2, and one

greater than 4.

Table 6 shows the GDFs for the regression

model. For regression models the GDFs for the



Figure 2. Impact of regression model by diagonal

observations in the Y vector are known to be

calculable as the diagonal of the “hat” matrix,

where hat =X(X0X)¡1X0, e.g., see Ye [2]. How-ever in development triangles, changing an in-

cremental value also changes subsequent cumu-

latives, so the X matrix is a function of lags of

Y. This requires the derivatives to be done nu-merically. The total of these, excluding lag 0, is

8.02, which is a bit above the usual number of

parameters, due to the exceptions to normal lin-

ear models. Compared to the CL, the GDFs are

lower for lag 6 onward, but are somewhat higher

along the modeled diagonals. They are especially

high for diagonal 4, which is short and gets its

own parameter.

Figure 2 graphs the impacts. Note that due to

the diagonal effects, diagonal 11 has higher im-

pact than diagonal 12 after the first two lags.

5.3. Square root regression modelAs a correction for heteroscedasticity, regres-

sion courses sometimes advise dividing both Yand X by the square root of Y, row by row. This

makes the model Y1=2 = (X=Y1=2)¯+ ", wherethe " are IID mean zero normals. Then Y=X¯+Y1=2", so now the variance of the residuals is pro-portional to Y. This sounds like a fine idea, but itis a catastrophe from a robust viewpoint. Table 7

shows the impact function. There are 12 points

with impact over 2, seven with impact over 4,

five with impact over 10, and three with impact

over 25.

Part of the problem is that the equation Y=X¯+Y1=2" is not what you would really want.The residual variance should be proportional to

the mean, not the observations. This setup gives

the small observations small variance, and so the

ability to pull the model towards them. But the

observations might be small because of a neg-

ative residual, with a higher expected value. So

this formulation gives the small values too much

influence.

Table 8 shows the related GDFs. It is unusual

here that some points have GDFs greater than 1.

A small change in the original value can make a



Table 7. Impact of square root regression model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

AY0 ¡0:94 ¡0:08 0.16 0.68 0.15 0.56 0.01 0.00 0.01 0.38 4.57 15.61AY1 ¡1:06 ¡0:10 0.28 ¡0:30 2.19 1.86 0.01 0.15 0.00 0.15 10.21 0.01AY2 ¡0:58 0.12 ¡0:09 0.20 0.68 0.39 0.01 0.03 28.26 3.09 0.02AY3 ¡0:20 ¡0:50 0.13 0.66 0.69 0.27 0.00 0.11 0.00 32.67AY4 ¡0:90 ¡0:15 0.33 0.41 0.59 0.56 0.03 0.14 37.14AY5 ¡1:28 ¡0:36 0.17 0.37 2.05 2.87 0.00 0.00AY6 ¡1:20 ¡0:09 0.01 0.77 0.71 2.34 0.02AY7 ¡1:02 ¡0:18 0.36 0.23 0.76 1.97AY8 ¡0:86 ¡0:07 ¡0:01 1.23 0.46AY9 ¡0:91 ¡0:06 0.59 1.02AY10 ¡0:45 0.48 0.89AY11 0.50 1.46AY12 4.56

Table 8. GDFs of square root regression model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11


greater change in the fitted value, but due to the

non-linearity the fitted value is still not exactly

equal to the data point. The sum of the GDFs

is 13.0, which is sometimes interpreted as the

implicit number of parameters.

5.4. Gamma-p residuals

Venter (2007b) fits the same regression model

using the maximum likelihood with gamma-p

residuals. The gamma-p is a gamma distribution,

but each cell is modeled to have the variance pro-

portional to the same power p of the mean. This

models the cells with smaller means as having

smaller variances. However, the effect is not as

extreme as in the square root regression, where

the variance is proportional to the observation,

not its expected value.

In this case, p was found to be 0.71. The im-

pacts are shown in Table 9 and graphed in Fig-

ure 3. It is clear that these are not nearly as dra-

matic as the square root regression, but worse

than the regular regression, and perhaps compa-

rable to the chain ladder. Diagonals 10 and 11

can be seen to have a few significant impacts.

These are at points with small observations that

are also on modeled diagonals. Even with the

variance proportional to a power of the expected

value, these points still have a strong pull. The

GDFs are in Table 10.

Again this is less dramatic than for the square

root regression, but the small points on the mod-

eled diagonals still have high GDFs. The total of

these is 11.3, which is still fairly high. This is

somewhat troublesome, as the gamma-p model

fit the residuals quite a bit better than did the

standard regression. The fact that the problems

center on small observations on the modeled di-

agonals suggests that additive diagonal effects



Figure 3. Impact of gamma-p residual model

Table 9. Impact of gamma-p residual model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

AY0 ¡0:59 ¡0:07 0.24 0.59 ¡0:03 1.47 1.37 1.37 1.23 1.25 ¡1:45 7.97AY1 ¡0:90 ¡0:05 0.28 0.10 0.90 1.11 1.30 0.77 1.37 0.91 6.73 1.36AY2 ¡0:46 0.08 ¡0:07 0.56 0.94 1.43 1.22 1.45 ¡5:62 4.33 1.35AY3 ¡0:29 ¡0:58 0.21 0.47 1.31 1.37 1.21 0.98 1.47 0.10AY4 ¡0:68 ¡0:15 0.19 0.51 0.94 1.48 1.24 1.96 0.02AY5 ¡1:04 ¡0:18 0.20 0.49 0.96 1.07 1.43 1.38AY6 ¡1:00 0.09 0.22 0.45 1.28 1.13 1.41AY7 ¡1:02 ¡0:18 0.50 0.50 0.95 1.17AY8 ¡0:71 ¡0:12 0.12 0.66 0.96AY9 ¡0:85 ¡0:02 0.80 0.86AY10 ¡0:44 0.48 0.88AY11 0.46 1.45AY12 4.43

Table 10. GDFs of gamma-p residual model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11




Table 11. Impact of gamma-p multiplicative model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

AY0 ¡0:94 ¡0:03 0.22 0.58 0.09 1.16 1.43 1.43 1.42 1.36 0.55 2.31AY1 ¡1:02 0.00 0.32 0.17 0.56 1.02 1.43 1.26 1.43 1.30 2.14 1.43AY2 ¡0:74 0.15 ¡0:46 0.39 0.98 1.30 1.42 1.42 ¡0:78 1.82 1.42AY3 ¡0:25 ¡0:50 ¡0:02 0.46 0.97 1.26 1.43 1.33 1.43 0.69AY4 ¡0:68 ¡0:39 0.23 0.51 0.83 1.26 1.39 1.50 0.64AY5 ¡1:09 ¡0:10 0.33 0.26 0.93 0.69 1.43 1.43AY6 ¡1:02 0.05 0.00 0.45 0.79 1.12 1.42AY7 ¡0:72 ¡0:37 0.31 0.29 1.11 1.07AY8 ¡0:81 ¡0:01 ¡0:21 0.92 0.99AY9 ¡0:76 ¡0:25 0.85 0.88AY10 ¡0:58 0.56 0.94AY11 0.35 1.50AY12 4.34

Table 12. GDFs of gamma-p multiplicative model

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11


may not be appropriate for this data. They do

fit into the mold of a generalized linear model,

but that is not too important when fitting by MLE

anyway. As an alternative, the same model but

with the diagonal effects as multiplicative factors

was fit. The multiplicative diagonal model can be

written:

EY=X[,1 : 6]¯[1 : 6] ¤¯[7]X[,7] ¤¯[8]X[,8],which means that the first six columns of X aremultiplied by the first six parameters, which in-

cludes the constant term, and then the last two di-

agonal parameters are factors raised to the power

of the last two columns of X. These are nowthe diagonal dummies, which are 0, 1, or ¡1.Thus the same diagonals are higher and the same

lower, but now proportionally instead of by an

additive constant. It turns out that this model

actually fits better, with a negative loglikelihood

of 625, compared to 630 for the generalized lin-

ear model. This solves the robustness problems

as well. The impacts are in Table 11, the GDFs

in Table 12, and the impacts are graphed in

Figure 4.

Diagonal 11 still has more impact than the oth-

ers, but this barely exceeds 2.0 at the maximum.

The sum of the GDFs is 8.67. There are eight

parameters for the cell means but two more for

the gamma-p. It has been a question whether or

not to count those two in determining the number

of parameter used in the fitting. The answer to

that from the GDF analysis is basically to count

each of those as 1/3 in this case. Here the robust

analysis has uncovered a previously unobserved

problem with the generalized linear model, and

lead to an improvement.



Figure 4. Impact of gamma-p multiplicative model

Table 13. Incremental triangle (Taylor and Ashe 1983)

Lag 0 L1 L2 L3 L4 L5 L6 L7 L8 L9

357,848 766,940 610,542 482,940 527,326 574,398 146,342 139,950 227,229 67,948352,118 884,021 933,894 1,183,289 445,745 320,996 527,804 266,172 425,046290,507 1,001,799 926,219 1,016,654 750,816 146,923 495,992 280,405310,608 1,108,250 776,189 1,562,400 272,482 352,053 206,286443,160 693,190 991,983 769,488 504,851 470,639396,132 937,085 847,498 805,037 705,960440,832 847,631 1,131,398 1,063,269359,480 1,061,648 1,443,370376,686 986,608344,014

6. A multiplicative fixed-effectsexample

A multiplicative fixed-effects model is one

where the cell means are products of fixed factors

from rows, columns, and perhaps diagonals. The

most well-known is the ODP model discussed in

Section 4, where there is a factor for each row, in-

terpreted as estimated ultimate, a factor for each

column, interpreted as fraction of ultimate for

that column, and the variance of each cell is a

fixed factor times its mean. This model if esti-

mated by MLE gives the same reserve estimates

as the chain ladder and so the same impacts for

each cell, but the GDFs are different, due to the

different fitted values.

The triangle for this example comes from Tay-

lor and Ashe (hereafter TA; 1983) and is shown

in Table 13. The CL = ODP impacts are in Ta-

ble 14 and are graphed in Figure 5.

Because the development factors are higher,

the impacts are higher than in the previous ex-

ample. Even though it is a smaller triangle, 14

points have impacts with absolute values greater



Figure 5. Impact of CL = ODP on TA

Table 14. Impact of CL = ODP on TA

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9

AY0 ¡3:11 ¡1:62 ¡1:01 ¡0:45 0.01 0.51 1.16 2.27 4.54 12.59AY1 ¡2:87 ¡1:38 ¡0:77 ¡0:20 0.25 0.76 1.40 2.51 4.78AY2 ¡2:43 ¡0:93 ¡0:33 0.24 0.69 1.20 1.85 2.95AY3 ¡2:21 ¡0:72 ¡0:11 0.45 0.91 1.41 2.06AY4 ¡1:95 ¡0:46 0.15 0.71 1.17 1.67AY5 ¡1:67 ¡0:18 0.43 0.99 1.45AY6 ¡1:25 0.25 0.85 1.42AY7 ¡0:14 1.35 1.96AY8 2.07 3.57AY9 13.45

Table 15. GDFs of CL on TA

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9

AY0 1.0 0.108 0.110 0.115 0.120 0.153 0.208 0.272 0.423 1.0AY1 1.0 0.106 0.121 0.144 0.182 0.211 0.258 0.365 0.577AY2 1.0 0.087 0.126 0.147 0.175 0.222 0.259 0.363AY3 1.0 0.093 0.138 0.146 0.204 0.224 0.275AY4 1.0 0.133 0.111 0.141 0.157 0.189AY5 1.0 0.119 0.130 0.145 0.162AY6 1.0 0.132 0.126 0.161AY7 1.0 0.108 0.139AY8 1.0 0.113AY9 1.0



Table 16. GDFs of ODP on TA

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9


Table 17. Impact of 6-parameter gamma- 12 on TA

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9

AY0 0.65 ¡0:82 ¡1:08 ¡2:07 ¡0:87 0.97 ¡0:32 0.33 0.53 12.06AY1 1.45 ¡0:02 0.68 0.60 ¡0:25 1.90 1.40 1.61 1.57AY2 1.64 0.75 ¡0:19 0.84 0.90 1.93 1.66 1.36AY3 1.26 0.43 ¡0:21 0.97 ¡0:36 1.70 1.71AY4 1.62 0.08 0.67 0.37 0.63 1.35AY5 1.19 ¡0:11 0.57 0.51 1.17AY6 2.56 1.19 0.91 1.13AY7 2.18 1.27 1.49AY8 1.72 0.92AY9 1.59

Table 18. GDFs of 6-parameter gamma- 12 on TA

L0 L1 L2 L3 L4 L5 L6 L7 L8 L9


than 2, four are greater than 4, and two are great-

er than 12. The CL GDFs are in Table 15. These

sum to 9, excluding the first column, and are

fairly high on the right where there are few ob-

servations per column. The ODP GDFs are in

Table 16. These sum to 19, and are fairly high

near the upper right and lower left corners.

The GDFs can be used to allocate the total

degrees of freedom of the residuals of n¡ p. Then is allocated equally to each observation, and the

p can be set to the GDF of each observation. This

would give a residual degree of freedom to each

observation which could be used in calculating a

standardized residual that takes into account how

the degrees of freedom vary among observations.

Venter (2007a) looked at reducing the number

of parameters in this model by setting parameters

equal if they are not significantly different, and

using trends, such as linear trends, between pa-

rameters. Also, diagonal effects were introduced.

The result was a model where each cell mean is a

product of its row, column, and diagonal factors.

There are six parameters overall. For the rows

there are three parameters, for high, medium, and



Figure 6. Impact of gamma- 12 on TA

low accident years. Accident year 0 is low, year

7 is high, year 6 is the average of the medium

and high levels, and all other years are medium.

There are two column factors: high and low. Lags

1, 2, and 3 are high, lag 4 is an average of high

and low, lag 0 and lags 5 to 8 are low, and lag

9 is 1 minus the sum of the other lags. Finally

there is one diagonal parameter c. Diagonals 4

and 6 have factors 1+ c, lag 7 has factor 1¡ c,and all the other diagonals have factor 1.

With just six parameters this model actually

provides a better fit to the data than the 19 param-

eter model. The combining of parameters does

not degrade the fit much, and adding diagonal ef-

fects improves the fit. An improved fit over that

in Venter (2007a) was found by using a gamma-

p distribution with p = 12so the variance of each

cell is proportional to the square root of its mean.

The impacts and GDFs of this model are shown

in Tables 17 and 18, and the impacts are graphed

in Figure 6, this time along accident years.

The impacts are now all quite well contained

except for one point–the last point in AY0. Pos-

sibly because AY0 gets its own parameter, lag 9

influences the level of the other lags’ parameters,

and this is a small point with a small variance,

this model only slightly reduces the high level

of impact that point has in ODP. The same thing

can be seen in the GDFs as well, where this point

has slightly less than a whole GDF. The points

on AY7 and the modeled diagonals also have rel-

atively high GDFs, as do some small cells. The

total of the GDFs is 6.14. There are six param-

eters affecting the means, plus one for the vari-

ance of the gamma. That one can affect the fit

slightly, so counting it as 1/7th of a parameter

seems reasonable.

In an attempt to solve the problem of the

upper-right point, an altered model was fit: lag 9

gets half of the paid in the low years. This can

be considered a trend to 0 for lag 10. Making the

lags sum to 1.0 now eliminates a parameter, so

there are five. The negative loglikelihood (NLL)

is slightly worse, at 722.40 vs. 722.36, but that is

worth saving a parameter. The robustness is now

much better, with only two impacts greater than

2.0, the largest being 2.35.

7. Paid and incurred exampleVenter (2008), following Quarg and Mack

(2004), builds a model for simultaneously es-



Table 19. Quarg-Mack paid increments

L0 L1 L2 L3 L4 L5 L6

AY0 576 1228 166 54 50 28 29AY1 866 1082 214 70 52 64AY2 1412 2346 494 164 78AY3 2286 3006 432 126AY4 1868 1910 870AY5 1442 2568AY6 2044

Table 20. Quarg-Mack unpaid

L0 L1 L2 L3 L4 L5 L6

AY0 402 300 164 120 100 80 43AY1 978 604 304 248 224 106AY2 1492 596 446 184 150AY3 1216 666 346 292AY4 944 1104 204AY5 1200 396AY6 2978

Table 21. Average reserve impact of paid

L0 L1 L2 L3 L4 L5 L6

A0 ¡0:68 ¡:02 0.32 0.86 2.32 5.95 13.99A1 ¡0:45 0.20 0.54 1.08 2.54 6.17A2 ¡0:41 0.24 0.58 1.12 2.59A3 ¡0:36 0.30 0.64 1.18A4 ¡0:32 0.33 0.67A5 ¡0:20 0.46A6 1.37

timating paid and incurred development, where

each influences the other. The paid losses are part

of the incurred losses, so the separate effects are

from the paid and unpaid triangles, shown in Ta-

bles 19 and 20.

First, the impacts on the reserve (7059.47) from

the average of the paid and incurred chain ladder

reserves are calculated, where the paids at the last

lag are increased by the incurred-to-paid ratio at

that lag. Tables 21 and 22 show the impacts of

the paid and unpaid triangles, and Tables 23 and

24 show the GDFs.

The impacts of the lower left are not great,

mostly because the development factors are fairly

low in this example. The impacts on the upper

right of both paid and unpaid losses are quite

high, however. The unpaid losses other than the

last diagonal have a negative impact, because

Table 22. Average reserve impact unpaid

L0 L1 L2 L3 L4 L5 L6

A0 ¡0:29 ¡0:15 ¡0:26 ¡0:72 ¡1:76 ¡4:01 14.99A1 ¡0:29 ¡0:15 ¡0:26 ¡0:72 ¡1:76 3.57A2 ¡0:29 ¡0:15 ¡0:26 ¡0:72 1.77A3 ¡0:29 ¡0:15 ¡0:26 1.08A4 ¡0:29 ¡0:15 0.82A5 ¡0:29 0.68A6 0.84

Table 23. Average reserve GDF of paid

0 L1 L2 L3 L4 L5 L6

A0 1 0.068 0.109 0.140 0.233 0.476 1A1 1 0.102 0.117 0.153 0.257 0.524A2 1 0.167 0.227 0.301 0.509A3 1 0.271 0.319 0.406A4 1 0.221 0.228A5 1 0.171A6 1

Table 24. Average reserve GDF unpaid

0 L1 L2 L3 L4 L5 L6

A0 1 0.067 0.106 0.139 0.232 0.464 1A1 1 0.126 0.129 0.160 0.269 0.536A2 1 0.198 0.219 0.306 0.499A3 1 0.239 0.300 0.395A4 1 0.192 0.246A5 1 0.180A6 1

they lower subsequent incurred development fac-

tors, but do not have factors applied to them. The

GDFs are similar to CL in general.

The model in Venter (2008) used generalized

regression for both the paid and unpaid triangles,

where regressors could be from either the paid

and unpaid triangles or from the cumulative paid

and incurred triangles. Except for the first couple

of columns, the previous unpaid losses provided

reasonable explanations of both the current paid

increment and the current remaining unpaid. The

paid and unpaid at lags 3 and on were just mul-

tiples of the previous unpaid, with a single fac-

tor for each. That is, expected paids were 33.1%,

and unpaids 72.3%, of the previous unpaid. Since

these sum to more than 1, there is a slight upward

drift in the incurred. The lag 2 expected paid was

68.5% of the lag 1 unpaid. The best fit to the lag



Table 25. Weibull model impact of paid

L0 L1 L2 L3 L4 L5 L6

A0 0.09 ¡0:18 ¡1:58 4.38 0.38 7.67 5.45A1 0.04 0.26 0.59 1.90 2.75 2.32A2 ¡:37 0.33 0.42 0.57 ¡0:28A3 ¡:13 0.17 0.67 1.26A4 ¡:02 0.20 0.31A5 ¡:94 0.70A6 1.25

Table 26. Weibull model impact unpaid

L0 L1 L2 L3 L4 L5 L6

A0 0.06 0.67 ¡1:02 ¡1:45 ¡1:82 0.51 4.14A1 ¡0:17 ¡0:44 ¡1:80 ¡0:73 0.52 2.56A2 ¡0:20 ¡0:16 0.47 ¡1:17 3.63A3 ¡0:09 ¡0:32 ¡1:17 2.51A4 ¡0:10 ¡0:34 1.89A5 ¡0:32 1.47A6 0.65

Table 27. Weibull model GDF of paid

0 L1 L2 L3 L4 L5 L6

A0 1 0.938 0.725 0.235 0.268 0.125 .143A1 1 0.451 0.057 0.052 0.066 0.065A2 1 0.192 0.347 0.377 0.290A3 1 0.137 0.250 0.145A4 1 0.094 0.277A5 1 0.269A6 1

Table 28. Weibull model GDF unpaid

0 L1 L2 L3 L4 L5 L6

A0 1 0.824 0.172 ¡0:058 0.015 .072 .054A1 1 0.357 0.700 ¡0:044 0.115 .052A2 1 0.152 0.465 ¡0:173 0.113A3 1 0.050 0.136 0.123A4 1 0.507 0.089A5 1 0.687A6 1

2 expected unpaid was 9.1% of the lag 1 cumu-

lative paid. For lag 1 paid, 78.1% of the lag 0

incurred was a reasonable fit. Lag 1 unpaid was

more complicated, with the best fit being a re-

gression, with constant, on lag 0 and lag 1 paids.

There were also diagonal effects in both mod-

els. The residuals were best fit with a Weibull

distribution. Tables 25—28 show the fits.

The two highest impacts for the average of

paid and incurred are 14 and 15. For the Weibull

they are 7.7 and 5.5. The average has two other

points with impacts greater than 5, whereas the

Weibull has none. Below 5 the impacts are rough-

ly comparable. Since the Weibull has variance

proportional to the mean squared, small obser-

vations have lower variance, and so a stronger

pull on the model and higher impacts. In total,

excluding the first column, the GDFs sum to 9.9,

but including the diagonals (see Venter 2008 for

details) there are 12 parameters plus two Weibull

shape parameters. The form of the model appar-

ently does not allow the parameters to be fully

expressed. The Weibull model still has more high

impacts than would be desirable, but it is a clear

improvement over the average of the paid and

incurred. The reserve is quite a bit lower for the

better fitting Weibull model as well: 6255 vs.

7059.

8. ConclusionRobust analysis has been introduced as an

additional testing method for loss development

models. It is able to identify points that have a

large influence on the reserve, and so whose ran-

dom components would also have a large influ-

ence. Through three examples, customized mod-

els were found to be more robust than standard

models like CL and ODP, and in two of the ex-

amples, even better models were found as a re-

sponse to the robust analysis.

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Prediction Error of the Future ClaimsComponent of Premium Liabilitiesunder the Loss Ratio Approach

by Jackie Li

ABSTRACT

In this paper we construct a stochastic model and derive

approximation formulae to estimate the standard error of

prediction under the loss ratio approach of assessing pre-

mium liabilities. We focus on the future claims compo-

nent of premium liabilities and examine the weighted and

simple average loss ratio estimators. The resulting mean

square error of prediction contains the process error com-

ponent and the estimation error component, in which the

former refers to future claims variability while the latter

refers to the uncertainty in parameter estimation. We illus-

trate the application of our model to public liability data

and simulated data.

KEYWORDS

Premium liabilities, loss ratio, standard error of prediction, mean square

error of prediction



1. IntroductionThere has been extensive literature on loss re-

serving models over the past 25 years, includ-

ing the Mack (1993) model. While the focus has

been largely on how to tackle outstanding claims

liabilities, relatively few materials have been pre-

sented for premium liabilities. Some references

include Cantin and Trahan (1999), Buchanan

(2002), Collins and Hu (2003), and Yan (2005),

which focus on the central estimate (i.e., the

mean) of premium liabilities but not on the un-

derlying variability.

As noted in Clark et al. (2003), the Interna-

tional Accounting Standards Board (IASB) has

proposed a new reporting regime for insurance

contracts, in which both outstanding claims li-

abilities and premium liabilities should be as-

sessed at their fair values. It is generally under-

stood that this fair value includes a “margin” al-

lowing for different types of variability for insur-

ance liabilities. Accordingly, the Australian Pru-

dential Regulation Authority (APRA) has pre-

scribed an approach similar to the fair value ap-

proach. Under Prudential Standard GPS 310, a

“risk margin” has to be explicitly calculated such

that outstanding claims liabilities and premium

liabilities are assessed at a sufficiency level of

75%, subject to a minimum of the mean plus one-

half the standard deviation. Australian Account-

ing Standard AASB 1023 also requires inclusion

of a risk margin, though there is no prescription

on the adequacy level. No matter what approach

one takes, it is obvious that urgency for develop-

ing proper tools to measure liability variability

exists not only for outstanding claims liabilities

but also for premium liabilities. In addition, ac-

cording to Yan (2005), premium liabilities ac-

count for around 30% of insurance liabilities for

direct insurers and 15% to 20% for reinsurers in

Australia from 2002 to 2004. Premium liabilities

represent a significant portion of an insurer’s li-

abilities and proper assessment of the underlying

variability should not be overlooked.

The definition of premium liabilities varies for

different countries. Broadly speaking, premium

liabilities refer to all future claim payments and

associated expenses arising from future events

after the valuation date which are insured under

the existing unexpired policies. Buchanan (2002)

notes that there are two main methods of de-

termining the central estimate of premium lia-

bilities. The first method is prospective in na-

ture and involves a full actuarial assessment from

first principles. Yan (2005) calls this method the

claims approach and differentiates it into the loss

ratio approach and historical claims approach.

The loss ratio approach is the most common one

for premium liability assessment in practice and

is essentially an extension of the outstanding

claims liability valuation. It applies a projected

loss ratio to the unearned premiums or number

of policies unexpired. The historical claims ap-

proach uses the number of claims and average

claim size and is more suitable for short-tailed

lines of business where data is sufficient. While

the historical claims approach has been studied

extensively under the classical risk theory, the

loss ratio approach has received relatively little

attention in the literature. In this paper we follow

the loss ratio approach and attempt to supplement

this knowledge gap.

On the other hand, the second method noted in

Buchanan (2002) is retrospective in nature and

involves an adjustment of the unearned premi-

ums to take out the profit margin. As discussed

in Cantin and Trahan (1999) and Yan (2005),

both Canadian and Australian accounting stan-

dards require a reporting of this unearned premi-

ums item, in which a premium deficiency reserve

is added if this item is less than the premium li-

ability estimate determined by the first method.

Obviously the first method above plays a key role

in premium liability assessment, and we focus

on the loss ratio approach under this prospective

method.


Prediction Error of the Future Claims Component of Premium Liabilities under the Loss Ratio Approach

In this paper we construct a stochastic model

to estimate the standard error of prediction under

the loss ratio approach of assessing premium lia-

bilities. We focus on modeling the future claims

which form the largest component in premium li-

abilities (about 85% according to Collins and Hu

2003). We look at the weighted average ultimate

loss ratio and simple average ultimate loss ratio,

and derive approximation formulae to estimate

the corresponding mean square error of predic-

tion with respect to the accident year following

the valuation date. As similarly reasoned in Tay-

lor (2000), the resulting mean square error of

prediction is composed of the process error com-

ponent and the estimation error component, and

no covariance term exists as one part is related

only to the future while the other only to the past.

We also illustrate the application of our model to

Australian private-sector direct insurers’ public

liability data and some hypothetical data simu-

lated from the compound Poisson model.

The outline of the paper is as follows. In Sec-

tion 2 we introduce the basic notation and as-

sumptions of our model. In Section 3 we present

the formulae for estimating the standard error of

prediction for premium liabilities. In Section 4

we apply the model to public liability data and

simulated data and analyze the results. In Sec-

tion 5 we set forth our concluding remarks. Ap-

pendices A to D furnish the proofs for the for-

mulae stated in this paper.

2. Notation and assumptions

Let Ci,j (for 1· i · n+1 and 1· j · n) bea random variable representing the cumulative

claim amount (either paid or incurred) of acci-

dent year i and development year j. Assuming

all claims are settled in n years, Ci,n represents

the ultimate claim amount of accident year i.

We consider the case where a run-off triangle

of Ci,j’s is available for i+ j · n+1. In effect,the valuation date is at the end of accident year

n, Ci,j’s for i+ j > n+1 and 1· i· n refer to

the future claims of outstanding claims liabili-

ties, and Cn+1,j’s refer to the future claims of

premium liabilities. Let Ei (for 1· i · n+1) bethe premiums of accident year i. The premiums

are assumed to be known. The term Ci,n=Ei then

becomes the ultimate loss ratio of accident year i.

It is also assumed that exposure is evenly dis-

tributed over each year, and the exposure dis-

tribution of accident year n+1 is the same as

that of the past accident years. In reality, the

future exposure relating to premium liabilities

would arise more from the earlier part of accident

year n+1, while the past exposure would spread

more uniformly across the whole year. Although

the timing of claims development is actually dif-

ferent between the two cases, the way that the

claims develop to ultimate remains basically the

same. As our focus is on the ultimate loss ratio,

this approximation is reasonable and represents

a convenient simplification for the model setting.

As mentioned in the Introduction, the loss ratio

approach for the premium liability valuation is

basically an extension of the outstanding claims

liability valuation. Hence we start with the struc-

ture of the chain ladder method, which is the

most common method for assessing outstanding

claims liabilities in practice and is linked to a

distribution-free model in Mack (1993). Incor-

porating Ei into the three basic assumptions of

the Mack (1993) model, we deduce the follow-

ing for 1· i · n+1:

E

ÃCi,j+1Ei

¯¯Ci,1,Ci,2, : : : ,Ci,j

!=Ci,jEifj ;

(for 1· j · n¡ 1) (2.1)

Var

ÃCi,j+1Ei

¯¯Ci,1,Ci,2, : : : ,Ci,j

!=Ci,j

E2i¾2j ;

(for 1· j · n¡ 1) (2.2)

Ci,j and Cg,h are independent.

(for i 6= g) (2.3)

The parameter fj is the development ratio and

the parameter ¾2j is related to the conditional vari-



ance of Ci,j+1. These two parameters are un-

known and need to be estimated from the claims

data.

As of the valuation date, there is no claims

data for accident year n+1. In order to model

the future claims in the first development year

of accident year n+1, we add the following two

assumptions for 1· i · n+1, which are anal-ogous to those for new claims in Schnieper

(1991):

E

μCi,1Ei

¶= u; (2.4)

Var

μCi,1Ei

¶=v2

Ei: (2.5)

Rearranging assumptions (2.4) and (2.5) into

E(Ci,1) = Eiu and Var(Ci,1) = Eiv2, we can see

that the mean and variance of the claim amount

of the first development year is effectively as-

sumed to be proportional to the premiums. This

is analogous to assumptions (2.1) and (2.2), in

which the conditional mean and variance of the

claim amount Ci,j for 2· j · n depends on theprevious development year’s claim amount

Ci,j¡1. The parameters u and v2 are unknown andcan be estimated from the claims and premiums

data.

Mack (1993) suggests the following unbiased

estimators for fj and ¾2j and proves that fj and

fh are uncorrelated for j 6= h:

fj =

Pn¡jr=1Cr,j+1Pn¡jr=1Cr,j

=

Pn¡jr=1Cr,j

Cr,j+1Cr,jPn¡j

r=1Cr,j;

(for 1· j · n¡ 1) (2.6)

¾2j =1

n¡ j¡ 1n¡jXr=1

Cr,j

ÃCr,j+1Cr,j

¡ fj!2;

(for 1· j · n¡ 2)

¾2n¡1 = minÃ¾4n¡2¾2n¡3

, ¾2n¡3

!: (2.7)

We now introduce two unbiased estimators for

u and v2 as follows, which are again based on

Schnieper (1991):

u=

Pnr=1Cr,1Pnr=1Er

=

Pnr=1Er

Cr,1ErPn

r=1Er; (2.8)

v2 =1

n¡ 1nXr=1

Er

μCr,1Er

¡ u¶2: (2.9)

It can be seen that both formulae (2.6) and

(2.8) are weighted averages and that both formu-

lae (2.7) and (2.9) use weighted sums of squares.

The proofs for unbiasedness of u and v2 are givenin Appendix A.

In effect, we integrate the model assumptions

in Mack (1993) with those of development year

one for new claims in Schnieper (1991). The

overall structure is based on the chain ladder

method. It then becomes possible to assess the

next accident year’s ultimate loss ratio using the

observed run-off triangle. As shown in the next

section, the results of the outstanding claims lia-

bility valuation (i.e., projected ultimate loss ratios

of the past accident years) are carried through

to the premium liability valuation (regarding the

expected ultimate loss ratio of the accident year

following the valuation date).

3. Standard error of predictionIn practice, actuaries often examine the pro-

jected ultimate loss ratios of past accident years

and compare these figures with target or bud-

get ratios or industry ratios to obtain an esti-

mate of the next accident year’s ultimate loss

ratio. Here we assume no such prior knowledge

or objective information is available and inves-

tigate the following two estimators for the next

accident year’s expected ultimate loss ratio q=E(Cn+1,n=En+1):

q=

Pni=1Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Pn

i=1Ei

=

Pni=1Ci,n+1¡iSn+1¡i,n¡1Pn

i=1Ei=

Pni=1 Ci,nPni=1Ei

=

Pni=1Ei

Ci,nEiPn

i=1Ei; (3.1)



q¤ =1

n

nXi=1

Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Ei

=1

n

nXi=1

Ci,n+1¡iSn+1¡i,n¡1Ei

=1

n

nXi=1

Ci,nEi:

(3.2)

Let Sj,h = fj fj+1 : : : fh (for j · h; equal to oneotherwise) and Ci,j = Ci,n+1¡iSn+1¡i,j¡1 (for i+j > n+1 and 1· i · n). C1,n is read as equalto C1,n. Formula (3.1) gives a weighted average

while formula (3.2) provides a simple average.

Both estimators are unbiased and the proofs are

set forth in Appendix B. For the expected future

claims component of premium liabilities of the

next accident year E(Cn+1,n), we define its esti-

mator as

Cn+1,n = En+1q: (3.3)

For now we deal with (3.1) and, as shown later,

the results of (3.1) can readily be extended to the

use of (3.2). We will also look at the effects of

excluding some accident years when calculating

q, as a practitioner may exclude or adjust a few

years’ projected loss ratios that are regarded as

inconsistent with the rest, out of date, or irrele-

vant. Such circumstances arise when there have

been past changes in, for example, the regula-

tions, underwriting procedures, claims manage-

ment, business mix, or reinsurance arrangements.

Using the idea in Chapter 6 of Taylor (2000),

we define the mean square error of prediction of

the estimator q as follows:

MSEP(q) = E

ÃμCn+1,nEn+1

¡ q¶2!

= E

ÃμCn+1,nEn+1

¡ q+ q¡ q¶2!

= E

ÃμCn+1,nEn+1

¡ q¶2!

+E((q¡ q)2)

(Cn+1,n and q are independent due to (2.3))

= Var

μCn+1,nEn+1

¶+Var(q): (q is unbiased)

(3.4)

The mean square error of prediction above

consists of two components: the first allows for

process error and the second for estimation er-

ror. The process error component refers to future

claims variability and the estimation error com-

ponent refers to the uncertainty in parameter esti-

mation due to sampling error. As similarly noted

in Taylor (2000), there is no covariance term in

(3.4) because at the valuation date, Cn+1,n is en-

tirely related to the future while q is completely

based on the past observations.

The corresponding standard error of prediction

can then be calculated as

SEP(q) =qMSEP(q): (3.5)

For the next accident year’s expected ultimate

claim amount, we compute the standard error of

prediction of its estimator as

SEP(Cn+1,n) = En+1 SEP(q) = En+1

qMSEP(q):

(3.6)

We derive the process error component as fol-

lows and the proof is given in Appendix C:

Var

μCn+1,nEn+1

¶=

1

En+1E

μCn+1,nEn+1

¶

£n¡1Xj=1

¾2jfjfj+1fj+2 : : :fn¡1

+v2

En+1f21 f

22 : : :f

2n¡1, (3.7)

which can be estimated by

dVarμCn+1,nEn+1

¶=

q

En+1

n¡1Xj=1

¾2j

fjSj+1,n¡1

+v2

En+1S21,n¡1: (3.8)

The estimation error component requires more

computation. We derive the following approxi-

mation for this component and the proof is pro-



vided in Appendix D:

Var(q)¼ 1¡Pn

i=1Ei¢2 n¡1X

j=1

ÃnX

i=n+1¡j

E(Ci,n)

fj

!2

Var(fj)

+1¡Pn

i=1Ei¢2 nX

i=1

f2n+1¡if2n+2¡i : : :f

2n¡1Var(Ci,n+1¡i)

+2¡Pn

i=1Ei¢2 n¡1X

j=1

n¡jXi=1

ÃnX

r=n+1¡j

E(Cr,n)

fj

!

£ (fn+1¡ifn+2¡i : : :fn¡1)Cov(fj ,Ci,n+1¡i), (3.9)


dVar(q) = 1¡Pni=1Ei

¢2 n¡1Xj=1

0@ nXi=n+1¡j

Ci,n

fj

1A2dVar(fj)+

1¡Pni=1Ei

¢2 nXi=1

S2n+1¡i,n¡1dVar(Ci,n+1¡i)+

2¡Pni=1Ei

¢2 n¡1Xj=1

n¡jXi=1

nXr=n+1¡j

Cr,n

fj

£ Sn+1¡i,n¡1dCov(fj ,Ci,n+1¡i),(3.10)

where the estimators of the variance and covari-

ance terms are derived as

dVar(fj) = ¾2jPn¡jr=1Cr,j

; (3.11)

dVar(Ci,n+1¡i) = Ci,n+1¡i n¡iXj=1

¾2j

fjSj+1,n¡i+Eiv

2S21,n¡i;

(3.12)

dCov(fj ,Ci,n+1¡i) = Ci,n+1¡iPn¡jr=1Cr,j

¾2j

fj: (3.13)

By now we have shown all the formulae that

are needed to calculate the standard error of pre-

diction of (3.1). Note that the term fjfj+1 : : :fn¡1for j > n¡ 1 is read as equal to one in the sum-mations. In the next section we will apply these

formulae to some real claims data and simulated

data.

4. Illustrative examples

We first apply the formulae shown previously

to some public liability data. We use the aggre-

gated claim payments and premiums (both gross

and net of reinsurance) of the private-sector di-

rect insurers from the “Selected Statistics on the

General Insurance Industry” (APRA) for acci-

dent years 1981 to 1991 (n= 10). Adopting the

approach as described in Hart et al. (1996), all

the figures have been converted to constant dol-

lar values in accordance with the average weekly

ordinary time earnings (AWOTE) before the cal-

culations. This procedure is common in practice

and is based on the assumption that wage infla-

tion is the ‘normal’ inflation for the claims.

The inflation-adjusted claims (incremental)

and premiums data are presented in Table 1 be-

low for gross of reinsurance and in Table 2 for

net of reinsurance. All the figures are in thou-

sands.

The two run-off triangles show that it takes

several years for public liability claims to de-

velop and this line of business is generally re-

garded as a long-tailed line of business. We use

formulae (2.6) to (2.9), (3.1), and (3.3) to es-

timate the parameters, accident year 1991’s ex-

pected ultimate loss ratio, and so the expected fu-

ture claims of premium liabilities. We then adopt

formulae (3.4) to (3.13) to compute the corre-

sponding standard error of prediction. Table 3

below presents our results both gross and net of

reinsurance.

As shown in Table 3, the estimated gross and

net expected ultimate loss ratios for accident year

1991 are 49.2% and 53.6%. The standard error of

prediction for the future claims of premium lia-

bilities, expressed as a percentage of the mean, is

greater for gross than for net. The gross and net

percentages are 47.1% and 33.1% respectively.

This feature is in line with the general percep-

tion that gross liability variability is greater than

its net counterpart. In both cases the process er-

ror component is larger than the estimation error

component.



Table 1. Public liability (gross of reinsurance)

Claims 1 2 3 4 5 6 7 8 9 10 Premiums

1981 15,898 20,406 17,189 19,627 35,034 12,418 8,922 12,555 8,965 6,693 289,7321982 16,207 21,518 17,753 18,780 19,113 18,634 15,857 13,050 9,362 319,2161983 14,141 20,315 16,458 25,473 16,427 92,888 18,698 15,295 314,6071984 14,649 21,162 19,084 23,857 20,171 15,098 17,637 344,4461985 21,949 26,455 23,285 25,251 22,286 23,424 418,3581986 18,989 28,741 32,754 30,240 28,443 535,6581987 19,367 36,420 31,204 27,487 639,1301988 26,860 39,550 33,852 751,8971989 23,738 52,683 780,6691990 34,567 719,1811991 334,566

Table 2. Public liability (net of reinsurance)

Claims 1 2 3 4 5 6 7 8 9 10 Premiums

1981 13,451 16,801 12,947 13,752 13,802 8,583 6,847 9,237 5,641 3,784 168,9751982 13,533 17,489 13,111 13,541 13,603 11,937 10,524 8,609 5,987 186,9901983 11,808 17,525 12,644 15,609 11,821 17,305 10,524 11,061 200,4751984 13,309 17,806 14,777 17,295 15,340 12,060 11,752 222,8431985 19,546 22,786 19,686 21,860 19,268 18,692 262,7481986 17,865 25,888 28,194 25,578 22,985 333,7161987 17,797 33,517 24,182 24,337 410,4291988 24,591 33,398 28,512 502,8691989 21,567 46,146 532,2981990 30,343 545,2181991 234,659

All accident years’ estimated ultimate loss ra-

tios are fairly consistent with one another ex-

cept the gross loss ratio of accident year 1983.

A closer look at the claims data reveals that the

gross claim payments made at accident year 1983

and development year 6 are $92,888 thousand,

the amount of which is much larger than the

other figures in the same development year. We

find that if the amount is changed to say $18,000

thousand, then the standard error of predic-

tion reduces significantly from 47.1% to 35.5%.

Whether to allow for this extra variability or ad-

just the data is a matter of judgment and in prac-

tice requires further investigation into the under-

lying features of those claims.

As mentioned in the previous section, one can

exclude some accident years’ loss ratios when

calculating q if those loss ratios are considered

inconsistent, out of date, or irrelevant. This com-

putation can readily be done by setting an indica-

tor variable for each accident year, in which the

indicator is one if the loss ratio of that accident

year is included and zero otherwise. Table 4 be-

low demonstrates some results of using different

numbers of accident years in computing q and

SEP(q) with (3.1), (3.8), and (3.10).

For each case of a particular number of acci-

dent years being included, Table 4 sets out the

average figures across all the possible combina-

tions of accident years in that case. It can be seen

that the estimation error component and so the

standard error of prediction decreases when more

accident years (i.e., more data) are used. The pro-

cess error component is stable because in our

analysis, the indicator adjustments are only

applied to (3.1) and (3.10) but not (2.6) to

(2.9).

Hitherto we have been focusing on the use of

(3.1). In many situations one may prefer using

a simple average of loss ratios as in (3.2). We

only need to replace (3.9) and (3.10) with the

following, the proof of which is analogous to



Table 3. Estimated results using weighted average loss ratio

Gross of Reinsurance Net of Reinsurance

Accident Ultimate Loss Accident Ultimate LossYear Premiums Claims Ratio Year Premiums Claims Ratio

1981 289,732 157,705 54.4% 1981 168,975 104,844 62.0%1982 319,216 156,934 49.2% 1982 186,990 112,391 60.1%1983 314,607 244,292 77.6% 1983 200,475 118,959 59.3%1984 344,446 159,365 46.3% 1984 222,843 124,138 55.7%1985 418,358 192,494 46.0% 1985 262,748 165,031 62.8%1986 535,658 247,328 46.2% 1986 333,716 191,706 57.4%1987 639,130 259,865 40.7% 1987 410,429 195,706 47.7%1988 751,897 313,187 41.7% 1988 502,869 227,747 45.3%1989 780,669 364,832 46.7% 1989 532,298 264,892 49.8%1990 719,181 421,727 58.6% 1990 545,218 297,641 54.6%1991 334,566 164,750 49.2% 1991 234,659 125,678 53.6%

Var

μCn+1,n

En+1

¶Var(q) SEP(q) SEP(Cn+1,n)

E(Cn+1,n)Var

μCn+1,n

En+1


E(Cn+1,n)

0.0481 0.0058 0.2322 47.1% 0.0292 0.0022 0.1773 33.1%

Gross 1 2 3 4 5 6 7 8 9

fj 2.5556 1.5283 1.3761 1.2773 1.3170 1.1148 1.0886 1.0648 1.0443

¾2j 2,227.06 242.72 235.27 720.66 13,377.69 166.44 35.49 0.78 0.02

u 0.0404v2 42.1016

Net 1 2 3 4 5 6 7 8 9

fj 2.5075 1.4858 1.3431 1.2323 1.1744 1.1167 1.1043 1.0588 1.0374

¾2j 1,992.25 206.88 36.77 11.43 157.84 32.84 11.97 0.02 0.00

u 0.0546v2 50.2089

Table 4. Average results with different number of accident years included

Gross of Reinsurance Net of ReinsuranceNo. ofAccidentYearsIncluded

q Var

μCn+1,n

En+1


E(Cn+1,n)q Var

μCn+1,n

En+1


E(Cn+1,n)

1 50.7% 0.0490 0.0340 0.2852 57.0% 55.5% 0.0295 0.0245 0.2311 41.7%2 49.9% 0.0485 0.0159 0.2532 51.2% 54.4% 0.0293 0.0110 0.2006 36.9%3 49.6% 0.0483 0.0112 0.2438 49.4% 54.1% 0.0293 0.0071 0.1906 35.3%4 49.4% 0.0482 0.0091 0.2394 48.6% 53.9% 0.0292 0.0053 0.1858 34.5%5 49.4% 0.0482 0.0080 0.2369 48.1% 53.8% 0.0292 0.0042 0.1829 34.1%6 49.3% 0.0482 0.0072 0.2353 47.8% 53.7% 0.0292 0.0036 0.1810 33.7%7 49.3% 0.0482 0.0067 0.2342 47.5% 53.6% 0.0292 0.0031 0.1797 33.5%8 49.3% 0.0481 0.0063 0.2333 47.4% 53.6% 0.0292 0.0027 0.1787 33.3%9 49.3% 0.0481 0.0060 0.2327 47.3% 53.6% 0.0292 0.0025 0.1779 33.2%

10 49.2% 0.0481 0.0058 0.2322 47.1% 53.6% 0.0292 0.0022 0.1773 33.1%



Table 5. Estimated results using simple average loss ratio

Gross of Reinsurance Net of Reinsurance

Accident Ultimate Loss Accident Ultimate LossYear Premiums Claims Ratio Year Premiums Claims Ratio

1991 334,566 169,752 50.7% 1991 234,659 130,184 55.5%

Var

μCn+1,n

En+1

¶Var(q¤) SEP(q¤) SEP(Cn+1,n)

E(Cn+1,n)Var

μCn+1,n

En+1


E(Cn+1,n)

0.0490 0.0063 0.2353 46.4% 0.0295 0.0027 0.1794 32.3%

Appendix D:

Var(q¤)¼ 1

n2

n¡1Xj=1

0@ nXi=n+1¡j

E(Ci,n)

Eifj

1A2

Var(fj)

+1

n2

nXi=1

f2n+1¡if2n+2¡i : : :f

2n¡1

E2iVar(Ci,n+1¡i)

+2

n2

n¡1Xj=1

n¡jXi=1

0@ nXr=n+1¡j

E(Cr,n)

Erfj

1A£μfn+1¡ifn+2¡i : : :fn¡1

Ei

¶Cov(fj ,Ci,n+1¡i),

(4.1)


dVar(q¤) = 1

n2

n¡1Xj=1

0@ nXi=n+1¡j

Ci,n

Eifj

1A2dVar(fj)+1

n2

nXi=1

S2n+1¡i,n¡1E2i

dVar(Ci,n+1¡i)+2

n2

n¡1Xj=1

n¡jXi=1

nXr=n+1¡j

Cr,n

Erfj

£ Sn+1¡i,n¡1Ei

dCov(fj ,Ci,n+1¡i):(4.2)

The estimated results using (3.2), (3.8), and

(4.2) are shown in Table 5. The resulting ulti-

mate loss ratios are slightly larger than previ-

ously while the standard error of prediction esti-

mates are slightly larger in magnitude but smaller

in percentage.

Finally we apply our formulae to some hypo-

thetical data simulated from the compound Pois-

son model Xi,j =PNi,jk=1Yi,j,k. Let Xi,j be a ran-

dom variable representing the incremental claim

amount of accident year i and development year

j and so Ci,j = Ci,j¡1 +Xi,j for 2· j · 10 andCi,1 = Xi,1. Let Ni,j and Yi,j,k be independent ran-

dom variables representing the number of claims

and the size of the kth claim of accident year i

and development year j. Let Ni,j » Pn(Ei¸j) andYi,j,k » LN(¹,¾) where ¸j’s are equal to 9£ 10¡6,8£ 10¡6, 7£ 10¡6, 6£ 10¡6, 5£ 10¡6, 5£ 10¡6,4£ 10¡6, 3£ 10¡6, 2£ 10¡6, 1£ 10¡6 respec-tively for 1· j · 10, ¹= 8:8638, and ¾ =

0:8326 (E(Yi,j,k) = 10,000 and SD(Yi,j,k) =

10,000). We assume Ei grows from 1,000,000 at

10% each year and the unearned premiums are

half of E11. Effectively, accident year 11’s ulti-

mate loss ratio has a mean of 50% and a variance

of 0.0077. We then simulate a run-off triangle

based on this compound Poisson model and ap-

ply our formulae (3.2), (3.8), and (4.2) to this

triangle.

Under the compound Poisson model above,

Xi,j’s are independent while under our model,

Ci,j+1 depends on Ci,j . Hence we expect our for-

mulae to produce a process error estimate larger

than the true variance underlying the simulated

data. The simulated run-off triangle and estimat-

ed results are presented in Tables 6 and 7.

As expected, the process error estimate of

0.0259 is larger than the underlying variance of

0.0077. In dealing with real claims data, one

should check the underlying assumptions thor-

oughly regarding the conditional relationships or

independence between different development

years.



Table 6. Simulated data (10% growth)

Claims 1 2 3 4 5 6 7 8 9 10 Premiums

1 80,946 97,396 43,469 40,208 52,068 19,518 14,644 1,692 12,429 1,964 1,000,0002 63,077 76,181 46,565 68,880 26,412 44,620 53,513 14,540 3,577 1,100,0003 93,688 112,399 87,149 133,804 17,549 14,814 91,392 35,367 1,210,0004 116,704 224,930 87,005 61,843 101,357 42,731 27,057 1,331,0005 192,542 147,366 85,361 61,776 90,964 103,829 1,464,1006 118,717 97,519 83,964 114,058 89,192 1,610,5107 156,966 172,695 139,843 156,225 1,771,5618 175,068 116,656 100,157 1,948,7179 164,691 112,805 2,143,589

10 239,127 2,357,94811 1,296,871

Table 7. Estimated results using simple average loss ratio

Accident Ultimate LossYear Premiums Claims Ratio

11 1,296,871 581,948 44.9%

Var

μCn+1,n

En+1


E(Cn+1,n)

0.0259 0.0030 0.1699 37.9%

5. Concluding remarksIn this paper we examine the weighted and

simple average loss ratio estimators and construct

a stochastic model to derive some simple ap-

proximation formulae to estimate the standard

error of prediction for the future claims compo-

nent of premium liabilities. Based on the idea in

Taylor (2000), we deduce the mean square er-

ror of prediction as comprising the process error

component and the estimation error component,

and no covariance term exists as the first part

is associated only with the future while the sec-

ond part only with the past observations. We ap-

ply these formulae to some public liability data

and simulated data and the results are reason-

able in general. Since the starting part of our

model follows the structure of the chain ladder

method, one may apply the various tests stated

in Mack (1994) to check whether the model as-

sumptions are valid for the claims data under in-

vestigation.

The formulae derived in this paper appear to

serve as a good starting point for assessment of

premium liability variability in practice. Never-

theless, there are other practical considerations

in dealing with premium liabilities such as the

insurance cycle, claims development in the tail,

catastrophes, superimposed inflation, multi-year

policies, policy administration and claims han-

dling expenses, future recoveries, future reinsur-

ance costs, retrospectively rated policies, un-

closed business, refund claims, and future

changes in reinsurance, claims management, and

underwriting. To deal with these issues, a prac-

titioner needs to judgmentally adjust the data or

make an explicit allowance, based on managerial,

internal, and industry information.

Acknowledgments

The author thanks the editors and the anony-

mous referees for their helpful and constructive

comments. This research was partially support-

ed by Nanyang Technological University AcRF

Grant, Singapore.



Appendix A

In this appendix we prove that the estimators u and v2 of (2.8) and (2.9) are unbiased:

E(u) = E

μPnr=1Cr,1Pnr=1Er

¶=

Pnr=1E(Cr,1)Pnr=1Er

=

Pnr=1EruPnr=1Er

= u; (from (2.4))

Var(u) = Var

μPnr=1Cr,1Pnr=1Er

¶=

Pnr=1Var(Cr,1)¡Pn

r=1Er¢2 =

Pnr=1Erv

2¡Pnr=1Er

¢2 = v2Pnr=1Er

; (from (2.3) and (2.5))

E(v2) = E

"1

n¡ 1nXr=1

Er

μCr,1Er

¡ u¶2#

=1

n¡ 1nXr=1

ErE

"μCr,1Er

¡ u¶2#

=1

n¡1nXr=1

Er

(E

ÃC2r,1E2r

!¡ 2E

ÃCr,1Er

Png=1Cg,1Png=1Eg

!+E(u2)

)

=1

n¡1nXr=1

Er

(Var

μCr,1Er

¶+E

μCr,1Er

¶2¡ 2

Er

"Png=1E(Cr,1)E(Cg,1)Pn

g=1Eg+Var(Cr,1)Pn

g=1Eg

#+Var(u)+E(u)2

)

(from (2.3))

=1

n¡1nXr=1

Er

(v2

Er+ u2¡ 2

Er

"Png=1ErEgu

2Png=1Eg

+Erv

2Png=1Eg

#+

v2Png=1Eg

+ u2

)

(from (2.4), (2.5), and above)

=1

n¡1nXr=1

Ãv2¡ Erv

2Png=1Eg

!= v2:

Appendix B

We prove in this appendix that both (3.1) and (3.2) give unbiased estimators. To start with, we have

to show the following with repeated use of the law of total expectation:

E

μCn+1,nEn+1

¶= E

μE

μCn+1,nEn+1

¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡1

¶¶= E

μCn+1,n¡1En+1

fn¡1¶

(from (2.1))

= E

μE

μCn+1,n¡1En+1

fn¡1

¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡2

¶¶= E

μCn+1,n¡2En+1

fn¡2fn¡1¶

(from (2.1) again)

= ¢ ¢ ¢= EμCn+1,1En+1

f1f2 : : :fn¡1¶

(repeat above)

= uf1f2 : : :fn¡1: (from (2.4))

The above results can readily be extended to accident years and development years other than n+1

and n shown here.



Mack (1993) proves that fj is unbiased and that fj and fh are uncorrelated for j 6= h. First we lookat the expected value of the weighted average estimator of (3.1):

E(q) = E

0@Pni=1Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Pn

i=1Ei

1A=

Pni=1E(Ci,n+1¡i)fn+1¡ifn+2¡i : : :fn¡1Pn

i=1Ei(from Mack (1993))

=

Pni=1Eiuf1f2 : : :fn¡1Pn

i=1Ei= uf1f2 : : :fn¡1 = q: (from above)

Similarly, the expected value of the simple average estimator of (3.2) is as follows:

E(q¤) = E

0@1n

nXi=1

Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Ei

1A=1

n

nXi=1

E

μCi,n+1¡iEi

¶fn+1¡ifn+2¡i : : :fn¡1 (from Mack (1993))

=1

n

nXi=1

uf1f2 : : :fn¡1 = uf1f2 : : :fn¡1 = q: (from above)

Appendix CWe derive the process error component in (3.7) of the mean square error of prediction as follows,

with repeated use of the law of total variance:

Var

μCn+1,nEn+1

¶= E

μVar

μCn+1,nEn+1

¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡1

¶¶+Var

μE

μCn+1,nEn+1

¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡1

¶¶

= E

μCn+1,n¡1E2n+1

¾2n¡1

¶+Var

μCn+1,n¡1En+1

fn¡1

¶(from (2.2) and (2.1))

= E

μCn+1,n¡1E2n+1

¾2n¡1

¶+E

μVar

μCn+1,n¡1En+1

fn¡1

¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡2

¶¶

+Var

μE

μCn+1,n¡1En+1

fn¡1

¯Cn+1,1,Cn+1,2, : : : ,Cn+1,n¡2

¶¶= E

μCn+1,n¡1E2n+1

¾2n¡1

¶+E

μCn+1,n¡2E2n+1

¾2n¡2f2n¡1

¶+Var

μCn+1,n¡2En+1

fn¡2fn¡1

¶(from (2.2) and (2.1) again)

= ¢ ¢ ¢= EμCn+1,n¡1E2n+1

¾2n¡1

¶+E

μCn+1,n¡2E2n+1

¾2n¡2f2n¡1

¶+ ¢ ¢ ¢+E

μCn+1,1E2n+1

¾21f22 f

23 : : :f

2n¡1

¶

+Var

μCn+1,1En+1

f1f2 : : :fn¡1

¶(repeat above)

=1

En+1

·E

μCn+1,n¡1En+1

¶¾2n¡1 +E

μCn+1,n¡2En+1

¶¾2n¡2f

2n¡1 + ¢ ¢ ¢+E

μCn+1,1En+1

¶¾21f

22 f

23 : : :f

2n¡1

¸

+v2

En+1f21 f

22 : : :f

2n¡1 (from (2.5))



=1

En+1

"E

μCn+1,nEn+1

¶¾2n¡1fn¡1

+E

μCn+1,nEn+1

¶¾2n¡2fn¡2

fn¡1 + ¢ ¢ ¢+EμCn+1,nEn+1

¶¾21f1f2f3 : : :fn¡1

#

+v2

En+1f21 f

22 : : :f

2n¡1 (from Appendix B)

=1

En+1E

μCn+1,nEn+1

¶ n¡1Xj=1

¾2jfjfj+1fj+2 : : :fn¡1 +

v2

En+1f21 f

22 : : :f

2n¡1:

Appendix DIn the following we derive the estimation error component in (3.9) of the mean square error of

prediction. We first apply the Taylor series expansion to the estimator q of (3.1):

q=

Pni=1Ci,n+1¡ifn+1¡ifn+2¡i : : : fn¡1Pn

i=1Ei¼Pni=1E(Ci,n+1¡i)fn+1¡ifn+2¡i : : :fn¡1Pn

i=1Ei

+1Pni=1Ei

n¡1Xj=1

(fj ¡fj)nXi=1

@

@fjCi,n+1¡ifn+1¡ifn+2¡i : : : fn¡1

¯¯Ci,n+1¡i=E(Ci,n+1¡i); fj=fj

+1Pni=1Ei

nXi=1

(Ci,n+1¡i¡E(Ci,n+1¡i))nXr=1

@

@Ci,n+1¡iCr,n+1¡rfn+1¡rfn+2¡r : : : fn¡1

¯Ci,n+1¡i=E(Ci,n+1¡i); fj=fj

= E(q)+1Pni=1Ei

n¡1Xj=1

(fj ¡fj)nX

i=n+1¡j

E(Ci,n)

fj(from Appendix B)

+1Pni=1Ei

nXi=1

(Ci,n+1¡i¡E(Ci,n+1¡i))fn+1¡ifn+2¡i : : :fn¡1:

Moving the term E(q) to the left-hand side of the equation and then taking the expectation on the

square of the resulting equation, we deduce that:

Var(q)¼ 1¡Pni=1Ei

¢2 E2640@n¡1Xj=1

(fj ¡fj)nX

i=n+1¡j

E(Ci,n)

fj

1A2375

+1¡Pni=1Ei

¢2 E24Ã nX

i=1

(Ci,n+1¡i¡E(Ci,n+1¡i))fn+1¡ifn+2¡i : : :fn¡1!235

+2¡Pni=1Ei

¢2 E240@n¡1X

j=1

(fj ¡fj)nX

i=n+1¡j

E(Ci,n)

fj

1AÃ nXi=1

(Ci,n+1¡i¡E(Ci,n+1¡i))fn+1¡ifn+2¡i : : :fn¡1!35

=1¡Pni=1Ei

¢2 n¡1Xj=1

0@ nXi=n+1¡j

E(Ci,n)

fj

1A2

Var(fj) (fj’s are unbiased and uncorrelated)

+1¡Pni=1Ei

¢2 nXi=1

f2n+1¡if2n+2¡i : : :f

2n¡1Var(Ci,n+1¡i) (from (2.3))

+2¡Pni=1Ei

¢2 n¡1Xj=1

n¡jXi=1

0@ nXr=n+1¡j

E(Cr,n)

fj

1A (fn+1¡ifn+2¡i : : :fn¡1)Cov(fj ,Ci,n+1¡i):(fj is unbiased; fj and Ci,n+1¡i are independent for j > n¡ i due to (2.3)).



As shown in Mack (1993), E(fj j Bj) = fj and Var(fj j Bj) = ¾2j =Pn¡jr=1Cr,j where Bj represents all the

past claims data to development year j. We then deduce the following:

Var(fj) = E(Var(fj j Bj))+Var(E(fj j Bj))

= E

0@ ¾2jPn¡jr=1Cr,j

1A+Var(fj)= E

0@ ¾2jPn¡jr=1Cr,j

1A ,which can be approximated by dVar(fj) = ¾2j =Pn¡j

r=1Cr,j .

Repeatedly using the law of total variance as in Appendix C, we derive that Var(Ci,n+1¡i) = E(Ci,n+1¡i)¢Pn¡i

j=1(¾2j =fj)fj+1fj+2 : : :fn¡i+Eiv

2f21 ¢ f22 : : :f2n¡i, which can then be estimated by dVar(Ci,n+1¡i) =Ci,n+1¡i

Pn¡ij=1(¾

2j =fj)Sj+1,n¡i+Eiv

2 ¢ S21,n¡i.Finally, we derive the covariance between fj and Ci,n+1¡i for j · n¡ i as follows:

Cov(fj ,Ci,n+1¡i) = E(fjCi,n+1¡i)¡E(Ci,n+1¡i)fj (fj is unbiased)

= E(E(fjCi,n+1¡i j Bn¡i))¡E(Ci,n+1¡i)fj

= E(fjCi,n¡ifn¡i)¡E(Ci,n+1¡i)fj (from (2.1))

= ¢ ¢ ¢= E(fjCi,j+1fj+1fj+2 : : :fn¡i)¡E(Ci,n+1¡i)fj (repeat above)

= E

ÃPn¡jr=1Cr,j+1Ci,j+1Pn¡j

r=1Cr,jfj+1fj+2 : : :fn¡i

!¡E(Ci,n+1¡i)fj (from (2.6))

= E

ÃPn¡jr=1 E(Cr,j+1Ci,j+1 j Bj)Pn¡j


!¡E(Ci,n+1¡i)fj

= E

ÃPn¡jr=1 E(Cr,j+1 j Bj)E(Ci,j+1 j Bj) +Var(Ci,j+1 j Bj)Pn¡j


!¡E(Ci,n+1¡i)fj

(from (2.3))

= E

ÃPn¡jr=1Cr,jCi,jf

2j +Ci,j¾

2jPn¡j


!¡E(Ci,n+1¡i)fj (from (2.1) and (2.2))

= E

ÃCi,jf

2j fj+1fj+2 : : :fn¡i+

Ci,j¾2jPn¡j


!¡E(Ci,n+1¡i)fj

= E

ÃCi,n+1¡iPn¡jr=1Cr,j

!¾2jfj, (from Appendix B)

which can be approximated by dCov(fj ,Ci,n+1¡i) = (Ci,n+1¡i=Pn¡jr=1Cr,j)(¾

2j =fj).



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The Economics of Insurance FraudInvestigation: Evidence of a Nash

Equilibriumby Stephen P. D’Arcy, Richard A. Derrig, and Herbert I. Weisberg

ABSTRACT

The behavior of competing insurance companies investi-

gating insurance fraud follows one of several Nash Equilib-

ria under which companies consider the claim savings, net

of investigation cost, on a portion, or all, of the total claim.

This behavior can reduce the effectiveness of investigations

when two or more competing insurers are involved. Cost

savings are reduced if the suboptimal equilibrium prevails,

and may instead induce fraudulent claim behavior and lead

to higher insurance premiums. Alternative cooperative and

noncooperative arrangements are examined that could re-

duce or eliminate this potential inefficiency. Empirically, an

examination of Massachusetts no-fault auto bodily injury

liability claim data for independent medical examinations

shows that (1) investigation produces a net total savings

as high as eight percent; (2) the investigation frequency is

likely in excess of the theoretical optimal; and (3) predic-

tive modeling of claim suspicion scores can significantly

enhance the net savings arising from independent medical

examinations.

KEYWORDS

Insurance fraud, Nash equilibrium, automobile medical payments, liability

insurance, independent medical examination


The Economics of Insurance Fraud Investigation: Evidence of a Nash Equilibrium

1. IntroductionFraud is a major problem for the insurance in-

dustry. Although the true cost of fraud for the

industry, and subsequently for insurance policy-

holders who bear this cost through higher pre-

miums, cannot be known, the FBI estimates the

annual cost of fraud to be $40 billion (FBI 2009).

Insurance fraud comes in two varieties: hard and

soft fraud are the operational terms. Hard fraud

applies to claims for fictitious accidents and in-

juries, while soft fraud denotes the increase of

claimed loss through unnecessary and/or inflated

values of claimants’ loss costs. The former is

criminal and is the purview of the criminal jus-

tice system; the latter is generally a civil matter

that is the larger of the two in dollar terms and

is the purview of the insurers (Derrig 2002).

Insurers can take steps to reduce the amount

of fraud, especially soft fraud, but these steps

are costly and these costs have to be weighed

against the expected savings. Insurance fraud has

been the subject of considerable research from a

variety of angles. This paper examines how the

number of insurers in the market and how the

different laws regarding subrogation in liability

claims affect the incentives to investigate, and

therefore reduce, fraud.

A number of recent studies have examined

claim settlement behavior by insurers as it re-

lates to insurance fraud (Artis, Ayuso and Guillen

2002; Crocker and Tennyson 2002; Derrig 2002;

Derrig and Weisberg 2004; Loughran 2005;

Dionne, Giuliano and Picard 2009). Several stud-

ies have utilized a Nash Equilibrium framework

(Nash 1951) between insurers and policyhold-

ers to examine auditing strategies for fraudulent

claiming behavior by policyholders (Boyer 2000;

Boyer 2004; Shiller 2006). In this paper, a model

of insurance company behavior combining the

cost of claims, the cost of investigating claims

and the potential for reducing claim costs is de-

veloped and analyzed in a game theoretic ap-

proach in which the other players are insurers,

rather than policyholders. The presence of a Nash

Equilibrium, in which no player in a simulta-

neous noncooperative game can unilaterally im-

prove its position by shifting its strategy for in-

vestigating claims, is observed under a variety of

different market conditions.

For a simple example of a Nash Equilibrium

consider Jack and Jill, two very young entrepre-

neurs operating lemonade stands in front of ad-

jacent houses. They have an unlimited supply

of their product from their parents’ kitchen at

no cost (to Jack and Jill) and they consider the

time they spend staffing their stands to be fun,

so there are no labor costs involved. They know

the thirst level and financial position of each of

their potential customers, so they can determine

how the demand for lemonade is affected by the

price. Both sellers and buyers can see what each

competitor is charging, and buyers will get their

lemonade from the lower cost seller, so the mar-

ket is fully competitive. The battle of the sexes,

junior edition, prevents the sellers from pricing

their product cooperatively, so this is a classic

noncooperative game to which Nash’s work ap-

plies. The sellers’ decision is whether to charge

1¢ or 2¢ per cup. Demand is such that 10 cups

will be sold if the price is 1¢, but only 8 cups

will be sold if the price is 2¢. All sales will be

made by the lemonade stand charging the lower

price. If both stands charge the same price, each

will get 1/2 of the sales.

Game theory often utilizes payoff matricies to

illustrate the results from different strategies. In

the payoff matrix below, and the subsequent ones

in this paper, the choices of one competitor are

shown along the top of the matrix and the choices

of the other along the left side. To determine

the payoff from a particular strategy for the top

competitor, look at the column representing their

choice and move down to the row representing

its competitor’s choice. The top-right half of the

appropriate box is the payoff to the competitor

listed along the top, and the bottom-left half of



Table 1. Payoff matrix

that box shows the payoff to the competitor listed

along the side.

The payoff matrix that illustrates the lemonade

pricing decision facing Jack and Jill is shown in

Table 1.

If both children charge 2¢ per cup, each will

sell 4 cups of lemonade (half of the total de-

mand) and earn a profit of 8¢. However, this is

not a Nash Equilibrium since if either Jack or

Jill (but not both) lowered the price to 1¢, that

child would sell 10 cups and earn a total profit

of 10¢. In this example, the situation where each

child charges 1¢ a cup is a Nash Equilbrium, be-

cause neither child can unilaterally change the

price and earn a higher profit. By raising the

price alone to 2¢ a cup, they would not sell any

lemonade and their total profit would be 0. For a

more complete description of game theory eco-

nomics1 and Nash Equilibrium, see chapter 6 of

Miller (2003). In most insurance cases, the Nash

Equilibrium is not at the globally optimal claim

investigation strategy.

Claims presented to an insurance company for

payment may include a variety of different com-

ponents. One component is a valid expense that

should be paid in full by the insurer, since both

the amount is appropriate and the coverage is

applicable. Another component could be an ex-

cessive charge on a claim that would otherwise

1Miller characterizes a Nash Equilibrium as “a no regrets outcome

in which all the players are satisfied with their strategy given what

every other player has done.”

be covered. A charge is considered excessive if

it is judged by the insurer to be “unreasonable”;

most insurance policies cover only “reasonable”

charges with reasonability defined by context and

ultimately determined by negotiation, arbitration

or, if necessary, a court. A third component could

be a claim for a service that is not covered al-

though other services would be covered. A final

component could be for an incident that is not

covered in its entirety by the insurance policy.

Sorting out the different components of a claim

efficiently is a constant process within a claims

department.

For automobile insurance coverage in the

United States, bodily injury claims can consist

of two different insurance coverages. Medical

expenses incurred by the policyholder or any-

one else insured under the policy (family mem-

bers, anyone occupying the covered vehicle) as

the result of an automobile accident are covered,

subject to policy limits, by the insurance com-

pany providing medical payments (MedPay) or

personal injury protection (PIP) coverage with-

out regard to fault.2 If someone is injured as

the result of the fault of another person, then

that injured party could pursue a liability claim

against the responsible party, depending on the

tort threshold applicable under the policy (IRC

2003, chapter 2, 2004a, 2004b). The insurance

company of the responsible party would be liable

2PIP coverage in no-fault states includes compensation for wage

loss and other benefits.



for the damages incurred by the injured person,

subject to policy limits and degree of fault, under

the liability insurance policy. Bodily injury lia-

bility damages consist of such tangible expenses

as medical expenses and loss of income, which

are termed special damages, and intangible com-

ponents such as pain-and-suffering, loss of con-

sortium or hedonic damages, which are termed

general damages.3 The insurer that paid the med-

ical expenses under the medical payments or PIP

policy may also be able to recoup its payments

from the liability insurer under subrogation. Si-

mon (1989) examined rules for allocating loss

adjustment expenses between primary insurers

and reinsurers when subrogation was involved.

Similar complexities are generated for insurers

when determining the cost of claim investigation

when two policies are involved.

In some cases the same insurer is responsible

for both the medical expenses and the bodily in-

jury liability payment. This would occur when

the driver is responsible for an injury to a pas-

senger, or if the same insurer covered the injured

person under medical payments coverage and the

responsible party under a different liability insur-

ance policy. When a single insurer is responsi-

ble for all payments, determining the appropriate

level of fraud investigation considers the entire

cost of all claims.

2. Claim investigation for injuryclaimsSeveral types of claim investigation are com-

monly used by automobile insurers in addition

to the routine gathering and evaluation of the

circumstances of the accident and the cost of

the treatment for the injury. The most common

method is an Independent Medical Examination

(IME), in which a doctor selected by the insurer

examines the injured claimant and develops an

independent assessment of the injury and the ap-

3See Loughran 2005 for an extensive analysis of auto BI liability

general damage settlements.

propriate treatment. If the IME indicates a more

moderate level of injury or treatment than the

claimant has reported through his or her medi-

cal care provider, then the claims department has

a stronger case for denying some or all of the

medical expenses that have been, or are likely to

be, submitted. Another type of investigation is a

Medical Audit (MA), in which the medical ex-

penses are reviewed by a specialist or an expert

system. Unusual factors that appear in the medi-

cal audit may provide the claims department with

justification to reduce the claim payment. A third

alternative is to refer the claim to a Special In-

vestigation Unit (SIU), where specifically trained

personnel are assigned to investigate claims with

unusual questions in order to determine whether,

and how much of, the claim should be paid.4

Derrig and Francis (2008) examined a collection

of objective factors in a predictive model for re-

ferring Massachusetts auto injury claims for an

IME or a special investigation for fraud, along

with the likelihood of success at reducing the

claim amount. Such predictive models should al-

low for more efficient, i.e., less costly by reduc-

ing false positives, selection of claims to investi-

gate.

IMEs and MAs can be used to reduce the

amount of claim payments for medical expenses.

SIUs can also reduce these expenses, but can

also impact other expenses or even determine if

the claim is valid at all. One level of investiga-

tion would be to investigate each claim for which

the expected savings from the investigation ex-

ceed the cost of the investigation. We call that

approach “tactically optimal.” Another level of

investigation would vary according to the char-

acteristics of the claim so that the savings net of

costs for the entire portfolio of claims is optimal

in some way. We call this approach “strategically

optimal.” In order to measure the expected sav-

ings, the insurer needs to ascertain the chance of

4The Insurance Research Council provides countrywide claims han-

dling outcome data for these three techniques for a sample of 2002

bodily injury claims (IRC 2003, pp. 92—104).



finding unreasonable or fraudulent activity and

the potential savings if that activity is discov-

ered. We now turn to a formalization of the cost/

savings process when total claim payments con-

sist of first party PIP and, when applicable, third

party liability.5

3. Savings versus costThe following notation will be used:

Cost of claim without any investigation:

PIP claim = P

Liability claim (excess of PIP) = L

Total Compensation = P+L

Subscripts on P and L:

First subscript indicates company responsible for

PIP

Second subscript indicates company responsible

for Liability (0 if no liability)

P1,0 represents a PIP claim where company 1 has

the PIP coverage and there is no liability claim


the PIP coverage and the liability coverage


the PIP coverage and company 2 has the liability

coverage

P1,¢ represents the sum of all PIP claims where

company 1 has the PIP coverage

L1,1 represents a liability claim where company

1 has the PIP coverage and the liability coverage

L2,1 represents a liability claim where company

2 has the PIP coverage and company 1 has the

liability coverage

L¢,1 represents the sum of all liability claims

where company 1 has the liability coverage

Savings from investigations:

Savings on PIP claims = SP

Savings on Liability claims = SL

Savings on Total claim = ST = SP+SL

5The Insurance Research Council provides an analysis of their 2002

claim sample for four no-fault states: Colorado (now a full tort

state), Florida, New York, and Michigan (IRC 2004a).

Level of investigation:

No investigation = 0

Optimal investigation based upon information on

first party claims = A

Optimal investigation based upon information on

both first party and liability claims = B

Subscripts on SPA, SPB, SLA and SLB:


PIP


for Liability (0 if no liability claims)

SPA1,0 represents the savings on PIP claims from

an A level investigation where company 1 has the

PIP coverage and there is no liability claim

SPA1,1 represents the savings on PIP claims from

an A level investigation where company 1 has the

PIP coverage and the liability coverage

SLA2,1 represents the savings on liability claims

from an A level investigation where company 2

has the PIP coverage and company 1 has the li-

ability coverage

Investigation cost:

Cost of an A level investigation = IA

Cost of an B level investigation = IB

Subscripts on IA and IB:


PIP


for Liability (0 if no liability claims)

IA1,0 represents the cost of an A level investiga-

tion where company 1 has the PIP coverage and

there is no liability claim



the liability coverage



company 2 has the liability coverage

The relationships between the cost of investi-

gation and expected savings, as well as the de-



Figure 1. Optimal level of claim investigation

termination of the optimal levels of investigation

under different circumstances, are illustrated in

Figure 1. The x axis represents the number of

claims. The y axis indicates dollar values. The

claims are ordered in decreasing size of expected

savings from claim investigations. The use of ex-

ante expectations of savings from investigation

is important and differs strongly from the ex-

post ordering of claims with savings from inves-

tigation. In practice, actual investigations will be

taken from a random or targeted draw of claims

that yield the largest expected savings. The cost

of investigations (I) function is a straight line un-

der the assumption that each investigation has

the same expected cost.6 Two concave functions

represent the expected savings from an investi-

gation. The lower curve, labeled SP, represents

the savings on first party claims and the higher

curve, labeled ST, represents the savings on the

total claim including both PIP and Liability pay-

ments. Both SP and ST have positive slopes and

negative curvature. A point will be reached where

all the remaining claims are expected to be com-

pletely valid, so no additional savings are achiev-

ed by additional investigation.

6Insurers generally pay, for example, a fixed amount for an IME. If

the claimant does not appear for the examination, the fee is reduced,

but the insurer would not know, when requesting the IME, if the

claimant will appear for it or not. SIU investigations cost more than

IMEs and Medical audits cost less. The use of multiple techniques

is relatively small. Thus, the assumption is made that the expected

cost of an investigation is the same for each claim, and the function

is linear.

The optimal level of investigation is determin-

ed when the slopes of the cost of investigation

line and the savings are equal. The tactically op-

timal number of claims to investigate, based on

information in first party claims, is A. At this

point, SP¡ I is maximized. The cost of this in-vestigation is IA, the savings on first party claims

is SPA, and the savings on total claims is STA =

SPA+SLA.7 The strategically optimal number

of first party claims to investigate, based on total

claim savings is B, an amount in excess of A.

Some of the relationships that develop from

this approach are:

SPB> SPA

IB> IA

SPA> IA

STB> IB

SPB¡SPA< IB¡ IA:

4. Single insurer caseWhen a single insurer writes the entire auto-

mobile insurance market, this company will be

responsible for paying both the PIP expenses and

the liability award resulting from every automo-

bile accident. In this case, the company can weigh

the potential cost savings on the total claim

against the cost of this investigation. The tacti-

cally optimal level of investigation would be to

investigate all claims where the expected savings

from the investigation exceed the cost of the in-

vestigation. This is the situation we will consider

first.

The three choices a single insurer faces regard-

ing the level of claim investigation are displayed

in Table 2. The insurer can perform no investiga-

tions and simply pay the amount claimed. This

situation is displayed in the first box. Alterna-

tively, the insurer can investigate A claims. The

7Dionne, Guiliano, and Picard (2009) derive varying optimal levels

of investigation depending on a (fraud) risk class partition of the

set of claims.



Table 2. Single insurer case, net cost of claim and investigations

Level of Claim Investigation

None (0) PIP Based (A) Total Claim Based (B)

P1,0 +P1,1 +L1,1 P1,0 +P1,1 +L1,1¡SPA1,0¡SPA1,1¡SLA1,1 + IA1,0 + IA1,1 P1,0 +P1,1 +L1,1¡SPB1,0¡SPB1,1¡SLB1,1 + IB1,0 + IB1,1

additional cost is IA1,0 + IA1,1 and the associ-

ated savings are SPA1,0 +SPA1,1 +SLA1,1. Since

the savings on the PIP claims alone, SPA1,0 +

SPA1,1 exceed the cost of the investigations, the

insurer would prefer this option over the case

of no investigations. The third choice, though,

where the insurer investigates B claims, is the

optimal choice. The cost of this additional inves-

tigation is IB¡ IA. The additional savings areSPB+SLB¡SPA¡SLA. Since the slope of theTotal Savings curve exceeds the slope of the cost

of investigations curve over the range from A to

B, then the savings exceed the costs, and the in-

surer would minimize net claim costs by investi-

gating B claims.

This strategy will have the benefit of reduc-

ing the cost of unreasonable medical treatment

to the lowest feasible level considering the cost

to investigate claims. This strategy also reduces

liability awards and the cost of automobile insur-

ance to the lowest level feasible given the cost

of investigating these claims and the ability to

lower awards through negotiation (Derrig and

Weisberg 2004). Additional reductions in claims

costs could be obtained, but the additional inves-

tigation expenses would exceed the claim cost

savings, so insurance premiums would actually

increase. The other expenses of the insurer, in-

cluding underwriting expenses and normal loss

adjustment expenses (other than investigating for

fraud), are not included in this analysis, since

they will be the same regardless of the level of

investigation for claims fraud.

5. Two insurer case: NosubrogationAssume the market consists of two compet-

ing insurers of equal size, with similar claim dis-

tributions (the SP and ST curves are the same

for each insurer). Assume the claim settlement

system does not permit the recovery of the PIP

claim payment and adjustment expense from any

at-fault party through subrogation.8 Then they

would each face a decision about the appropri-

ate level of investigation of claims fraud, but

their net claim costs would depend both on their

own investigation level decision and the decision

of their competitor. The outcomes, in the case

where there is no subrogation, are shown on Ta-

ble 3. The upper segment of each cell denotes

the position of insurer 1; the lower segment that

of insurer 2.

If both insurers were to investigate optimally

based on aggregate claim costs, then each insurer

would bear the cost of investigating B claims,

and benefit from the savings in claim costs on

both PIP and Liability claims. This situation is

represented in cell (B,B) and resembles the op-

timal position for the single insurer case. Unrea-

sonable medical expenses are reduced to the low-

est economically efficient level, liability costs are

minimized, and the total cost of auto insurance

is kept at the lowest level.9

However, this is not a stable situation. Insur-

er 1 might be better off if it only investigated

8Medical payments excess of PIP in a no-fault state, for example.9This insurer might prefer to investigate the claims it knows it has

the liability insurance coverage on up to the aggregate level, and

only investigate the remaining claims on which there is either no

liability coverage or coverage provided by the other insurer, if it

could identify those claims. However, there are several problems

with this strategy. First, an insurer may not know if another com-

pany will be liable for a claim or not early enough in the claim

process to make this distinction. Second, adopting a claim process

that requires claims adjusters to have different strategies for investi-

gation can complicate the process and increase overall costs. Based

on discussions with claims personnel, such differential strategies

are not common.



Table 3. Two insurer case, net cost of claim and investigations, no subrogation

Table 4. Two insurer case, no subrogation

Nash Equilibrium

Insurer 1 Insurer 2

(0,0) IA1,¢ > SPA1,¢ + SLA1,1 IA2,¢ > SPA2,¢ + SLA2,2

(A,A) IA1,¢ < SPA1,¢ + SLA1,1 IA2,¢ < SPA2,¢ + SLA2,2

IB1,¢ ¡ IA1,¢ > SPB1,¢ ¡SPA1,¢ + SLB1,1¡SLA1,1 IB2,¢ ¡ IA2,¢ > SPB2,¢ ¡SPA2,¢ + SLB2,2¡SLA2,2

(B,B) IB1,¢ ¡ IA1,¢ < SPB1,¢ ¡SPA1,¢ + SLB1,1¡SLA1,1 IB2,¢ ¡ IA2,¢ < SPB2,¢ ¡SPA2,¢ + SLB2,2¡SLA2,2

claims at the A level, which would lower its

cost of investigations by (IB¡ IA), and only in-crease claim costs by (SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1). If insurer 2 were to continue to investi-

gate claims at the B level, then insurer 1 would

benefit on its liability claims for which insurer

2 had the PIP coverage (SLB2,1). For the two

insurer example, the lower investigation costs

may or may not exceed the savings. Although

(IB¡ IA)> (SPB1,¢ ¡SPA1,¢), whether it also

exceeds (SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1) de-pends on the relationship between the SP and ST

curves and the cost of the claims where insurer 1

has both PIP and Liability. The cost savings on

liability claims must be included in the decision

of which level of investigation to pursue. How-

ever, if it is advantageous for insurer 1 to move

to a lower level of investigation, then it would

also benefit insurer 2 to move to that level, so

the resulting position would be that displayed in

cell (A,A).

If the insurers move to cell (A,A), that will

prove to be a Nash Equilibrium. Neither insurer

can move unilaterally to another position that

benefits itself. Insurer 1 will not stop investigat-

ing claims at the A level and move to the no

investigation level. If it were to do so, the sav-

ings would be IA and the cost would be SPA1,¢+SLA1,1. Since IA< SPA alone, this change

would increase the net cost of claims. Although

the overall optimal position would be cell (B,B),

that is not a stable equilibrium since one com-

pany might benefit by reducing the level of in-

vestigations.

Table 4 describes the conditions that lead to

each claim investigation strategy for the insur-

ers. Cell (B,B) is a Nash Equilibrium if IB1,¢ ¡IA1,¢ < SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1. Since



both insurers are assumed to be the same size and

have the same distribution of claims and costs,

then if this relationship holds for insurer 1, it

should also apply to insurer 2. This equilibrium

would apply if the cost savings for each insurer

on claims where it had both the PIP and the

liability coverage exceeded the additional cost

of investigating claims at the B level. Each in-

surer would not be assured of receiving the sav-

ings of a B level investigation on its liability

claims where the other insurer has the PIP claim,

since that insurer might elect a lower level of in-

vestigation. Alternatively, cell (A,A) would be

the Nash Equilibrium if IB1,¢ ¡ IA1,¢ > SPB1,¢ ¡SPA1,¢+SLB1,1¡SLA1,1 and IA1,¢ < SPA1,¢+SLA1,1. Since SPA1,¢ > IA1,¢ by itself, then cell(0,0) will never be the Nash Equilibrium if there

is no subrogation. Note also that the off-diagonal

investigation levels in Table 4 exhibit elements

of the free rider problem; namely, one insurer

reaps the liability benefit of the higher PIP in-

vestigation level of the other insurer without the

additional cost.

In this paper, we assume that insurers follow

the same approach for determining the level of

investigation for all claims, regardless of whether

they are providing the liability coverage or a

competitor is providing this coverage. There are

several reasons for this assumption. The most im-

portant reason is that asking claims personnel to

follow different approaches for PIP claims de-

pending on which insurer will bear the liability

costs would significantly complicate and poten-

tially delay the claims process. PIP claims de-

velop quickly and must be covered regardless of

fault, so PIP claim files may not contain enough

information to determine whether liability cover-

age will apply and, if it does apply, which insurer

will provide this coverage. Decisions about how

to investigate potential fraud cannot be delayed

until all the information is available, or it could

be too late to reduce total economic loss. Having

a single process in place allows for a more ef-

fective decision-making process. There are also

some cases in which it is unknown which insurer

will ultimately be held liable, such as when a pas-

senger is injured in a two car accident and it is

not known which driver will be held liable un-

til the claim is finally settled. Another reason is

that insurers know the levels of investigation that

other insurers adopt, through subrogation cases

and through hiring each other’s former employ-

ees. If an insurer followed a suboptimal claim

investigation process that put other insurers at a

disadvantage, it could trigger retaliation. Finally,

regulators may object, and even fine, a company

if the claims department had a policy of know-

ingly underinvestigating fraud in cases where the

BI liability lies with another carrier. Some regu-

lators have already disallowed a portion of rate

requests based on assumed inadequate fraud in-

vestigation; if a company had a specific policy in

place not to investigate claims where savings are

expected to exceed the costs, this approach could

provide a strong case for this type of regulatory

reaction.

6. Two insurer case: SubrogationThis situation differs from the no subrogation

case in several ways.10 First, each Liability in-

surer is responsible for paying the PIP or Med-

Pay claims of the other insurer when liability at-

taches and the PIP or MedPay insurer and the Li-

ability insurer are different (Pi,j where i 6= j). Onepossible situation is to allow subrogation for the

claim, but not for loss adjustment expense. The

rationale for this approach is that claim payments

are more easily verifiable than loss adjustment

expenses. In the case that only claim payments

are subrogated, if insurer 1 investigates claims at

the A level but insurer 2 does not investigate, in-

surer 1 does not benefit from the savings on the

PIP claims where insurer 2 has the PIP claim but

10Table 7 shows that 35 of the 51 jurisdictions allow subrogation

of PIP and/or MedPay to the liability carrier.



Table 5. Two insurer case, net cost of claim and investigations, subrogation

Table 6. Two insurer case, subrogation

Nash Equilibrium

Insurer 1 Insurer 2

(0,0) IA1,0 + IA1,1 > SPA1,0 + SPA1,1 + SLA1,1 IA2,0 + IA2,2 > SPA2,0 + SPA2,2 + SLA2,2

(A,A) IA1,0 + IA1,1 < SPA1,0 + SPA1,1 + SLA1,1 IA2,0 + IA2,2 < SPA2,0 + SPA2,2 + SLA2,2

IB1,0 + IB1,1¡ IA1,0¡ IA1,1 >

SPB1,0¡SPA1,0 + SPB1,1¡SPA1,1 + SLB1,1¡SLA1,1

IB2,0 + IB2,2¡ IA2,0¡ IA2,2 >


(B,B) IB1,0 + IB1,1¡ IA1,0¡ IA1,1 <


IB2,0 + IB2,2¡ IA2,0¡ IA2,2 <


insurer 1 has the liability (SPA2,1). Insurer 2 ben-

efits from the savings on PIP claims, however,

where insurer 1 has the PIP claim and insurer 2

has the liability (SPA1,2).11 Thus, the free rider

problem may be more severe when subrogation

is considered. In this situation, the Nash Equilib-

rium could be no claims investigation, since the

insurer bears the cost of investigating its own PIP

claims, but benefits only on those claims where

there is no liability or if the same company has

the liability coverage, unless the other insurer in-

vestigates all its own PIP claims. The outcomes,

11In cases where the liability insurer negotiates an overall fair set-

tlement independent of the PIP claim investigation result, and pays

that settlement less the PIP payment to the claimant, there would

be no effect from PIP levels of investigation on the liability insurer.

Generally, however, a favorable PIP investigation may curtail treat-

ment, limit both PIP and overall economic damages and, thus, lower

the (total) liability settlement. The latter is the situation we assume.

given this approach to subrogation, are shown in

Table 5.

Table 6 describes the conditions that lead to

each claim investigation strategy for the two in-

surers when subrogation is introduced. In this

case, cell (0,0) may be a Nash Equilibrium, since

each insurer only saves money on claims where

there either is no liability or it has the liability

claim as well. Insurers no longer save money

on PIP claims if another insurer has the liabil-

ity, since those payments would be reimbursed

under subrogation. Thus, subrogation introduces

a disincentive to investigating claims for fraud

unless the cost of investigation is also subject to

the subrogation recovery.

An alternative approach to subrogation would

be to allow subrogation for both the claim pay-

ment and any allocated loss adjustment expense,



which would include IMEs and MAs, but not

SIU costs which are internal. In this approach,

the insurer responsible for paying subrogation

would have to trust, or be able to audit, the cod-

ing of allocated loss adjustment expenses by the

other insurer to assure that the investigation costs

do apply to the appropriate claim. Subrogation of

allocated claim expense increases the incentive to

investigate PIP claims since the (allocated) costs

to investigate PIP claims would be reimbursed

if another insurer has the liability coverage. In

this case, if one insurer conducted an IME that

results in cost savings for a second insurer, the

second insurer would have reimbursed the first

insurer for the cost of the IME.

A third approach would be to allow subroga-

tion for unallocated loss adjustment expenses as

well as allocated loss adjustment expenses.12 Un-

allocated LAE (ULAE) are the claim expenses

that cannot be assigned to a particular claim,

which would consist of the cost of running a

company’s claim department, including salaries,

supplies and office expenses. In Massachusetts,

where this approach to subrogation is applied,

ULAE is calculated as 10 percent of the claim

cost. If a claim adjuster is considering investi-

gating a claim in which the expected savings will

exceed the cost of the investigation, but another

company is likely to be liable for the loss, the

insurer is saving the other insurer money and re-

ducing its ULAE reimbursement. For example,

assume that a claim on which one insurer had the

PIP coverage and the other insurer had the lia-

bility coverage generated $2200 in claimed med-

ical expenses. The PIP insurer could request an

IME that is expected to cost $300 and that would

reduce the medical expenses by $800, to $1400.

12Recent changes in annual statement reporting have two new cat-

egories: Defense and Cost Containment Expenses (DCCE), which

parallels the allocated expense category, and Other Adjusting Ex-

pense (OAE) that parallels the unallocated expense category. Our

paper continues to use the prior terminology of allocated and un-

allocated expense for expenses assignable to particular claims and

those expenses that are not, respectively.

The PIP insurer may not do this investigation un-

der a tactically optimal strategy. If the claim were

to qualify for subrogation, then the reimburse-

ment for ULAE declines from $220 (10 percent

of $2200) to $140 (10 percent of $1400) even

though the claim department puts in additional

effort to request and review the IME and then

negotiate with the claimant to reduce the claim

payment. On the other hand, under a strategi-

cally optimal strategy, the PIP insurer may well

investigate reimbursable PIP claims to reinforce

a “hard-line” attitude on unreasonable medical

charges in order to maximize savings on its own

claims.

Subrogation rules can have a significant effect

on the incentives for investigating claims. Table 7

summarizes the different subrogation regulations

by state, and also indicates the type of compensa-

tion system in effect in each state. In some states,

including California and Florida, no subrogation

is allowed for either Medical Payments or PIP.

In other states, both Medical Payments and PIP

are eligible for subrogation. Most states allow

subrogation for either Medical Payments or PIP,

but not both. Massachusetts allows subrogation

of PIP claims but not Medical Payments excess

of PIP.

7.1. Multiple insurer caseA more realistic situation arises where there

are many insurers in the market. Some insurers

may write a major share of the market within an

individual state, in a few cases in excess of 30

percent. However, in most states a large number

of insurers compete and the market share of most

companies represents only a small share of the

market. Thus, it is relatively rare that the same

insurer provides PIP coverage under one policy

involved in a claim and liability coverage under

another policy by covering both cars involved in

a two car accident.13 In this situation, the Nash

13In Massachusetts with only 18 active personal auto insurers, ap-

proximately 80 percent of liability claims have different PIP and

liability insurers.



Table 7. Tort type and subrogation laws by state

2006 Subrogation

State Tort Type Med Pay PIP

Alabama Tort Yes NoAlaska Tort Yes NoArizona Tort No NoArkansas Add-on Yes YesCalifornia Tort No NoColorado Tort No YesConnecticut Tort No NoDelaware Add-on No YesDC Add-on No YesFlorida No-Fault No NoGeorgia Tort No NoHawaii No-Fault No YesIdaho Tort Yes NoIllinois Tort Yes NoIndiana Tort Yes NoIowa Tort Yes NoKansas No-Fault No YesKentucky Choice No-Fault No YesLouisiana Tort Yes NoMaine Tort Yes NoMaryland Add-on No NoMassachusetts No-Fault No YesMichigan No-Fault No YesMinnesota No-Fault No NoMississippi Tort Yes NoMissouri Tort No NoMontana Tort No NoNebraska Tort Yes NoNevada Tort No NoNew Hampshire Tort No NoNew Jersey Choice No-Fault No NoNew Mexico Tort Yes NoNew York No-Fault No YesNorth Carolina Tort No NoNorth Dakota No-Fault Yes YesOhio Tort Yes NoOklahoma Tort Yes YesOregon Add-on No YesPennsylvania Choice No-Fault No NoRhode Island Tort Yes NoSouth Carolina Add-on No NoSouth Dakota Add-on Yes YesTennessee Tort Yes NoTexas Add-on Yes NoUtah No-Fault No YesVermont Tort Yes NoVirginia Add-on No NoWashington Add-on No YesWest Virginia Tort Yes NoWisconsin Add-on Yes NoWyoming Tort Yes No

Sources: IRC (2003),Insurance Information Institute (2006),Mattheisen, Wickert, and Lehrer (2006)

Table 8. Multiple insurer case, subrogation

Nash Equilibrium

Insurer k

(0,0) IAk,¢ > SPAk,0 + SPAk,k + SLAk,k

(A,A) IAk,¢ < SPAk,0 + SPAk,k + SLAk,k

IBk,¢ ¡ IAk,¢ >SPBk,0¡SPAk,0 + SPBk,k ¡SPAk,k + SLBk,k ¡SLAk,k

(B,B) IBk,¢ ¡ IAk,¢ <SPBk,0¡SPAk,0 + SPBk,k ¡SPAk,k + SLBk,k ¡SLAk,k

When the number of insurers, n, increases:The total investigation cost is IAk,¢ =

Pn

j=1 IAk,j + IAk,k + IAk,0 (k 6= j).The share of efficient part (IAk,k + IAk,0) in the investigation, whichis spent on SPAk,0 + SPAk,k + SLAk,k , is (IAk,k + IAk,0)=IAk,¢. Whenn!1,

Pn

j=1 IAk,j=IAk,¢ ! 1, (IAk,k + IAk,0)=IAk,¢ ! 0. That meansthat little of the investigation cost is spent to improve savings fromSPAk,0 + SPAk,k + SLAk,k . Thus, no insurer would be likely to investi-gate claims for fraud. The Nash Equilibrium would tend to be (0,0).

Equilibrium position is even more likely to be the

No Investigation level, since most of the benefits

of the investigations will accrue to other insur-

ers. The simple relationships for a market with

multiple insurers and subrogation are described

in Table 8.

7.2. Example

The decision process facing each insurer can

be illustrated by an example. A PIP claimant is

visiting a physical therapist for treatment. The

current cost of the claim is $2000 for medical

expenses. Another driver is expected to be held

liable for the accident. Based on past experience

for that type of injury with that physical thera-

pist, the PIP insurer expects the total claim for

medical treatment will be $2200. If the PIP in-

surer orders an IME, which costs $300, the in-

surer expects to be able to determine that no addi-

tional physical therapy is needed, limiting med-

ical expenses to $2000. The liability award for

noneconomic losses (pain and suffering) is ex-

pected to be $4000 if no additional treatment is

received, but $4360 if additional treatment is pro-

vided. Assume that the liability insurer is not in

a position to undertake this investigation and re-

duce its costs because at the time a determination



of liability is made the full treatment of physical

therapy has been be completed.14

The cost of the IME, $300, exceeds the PIP

savings of $200 on this claim, but is less than the

total of the PIP and liability savings ($560). In

the single insurer case, the insurer will request an

IME on this claim and curtail the additional costs.

In the two-insurer case, if there is no subroga-

tion, the PIP insurer spends $300, saves $200 on

the PIP, and has a 50% chance of saving $360

more on the liability claim (with only two insur-

ers, the PIP insurer has a 1 in 2 chance of writing

the responsible party’s liability insurance in sim-

ple two-car collisions). Therefore, the PIP insurer

would also request the IME on this claim. In the

two-insurer case where there is subrogation, the

PIP insurer faces a 50% chance of saving on the

PIP claim and on the noneconomic losses (if it

also has the liability), so the expected savings

would be $280 (half of the $560 total savings).

Thus, the PIP insurer would not investigate this

claim unless the allocated LAE is reimbursable.

If LAE is not reimbursable, the cost of investi-

gating the claim is $300. If LAE is reimbursable,

then the expected cost of the IME is reduced to

$150, which would encourage the PIP insurer to

undertake this investigation in order to save an

expected value of $280. If unallocated LAE is

covered by subrogation as a percentage of the

PIP claim, the insurer would be slightly less in-

clined to perform this investigation, as it would

reduce the PIP payment by $200, and the reim-

bursement from the other carrier by one-half of

the ULAE subrogation rate times $200 (one-half

since there is an equal chance that each insurer

will be the one responsible for the liability). In

Massachusetts, where the unallocated LAE reim-

bursement rate is 10% of a PIP claim, this would

reduce the value of investigating this claim by

$10 (:5£ 10%£ 200).14The liability insurer may not be able to reduce the claimed medi-

cal expenses but the negotiated award may be lower if the additional

treatment is known or suspected of being unnecessary (Derrig and

Weisberg 2004).

In the multiple insurer case, the PIP insurer

will have a lower chance of providing the lia-

bility coverage on this claim. In this example,

with no subrogation, if the chance of covering

both the PIP and liability is less than 28%, then

the expected savings on the noneconomic losses

would not be enough to compensate the PIP in-

surer to undertake this investigation. (The cost

of the investigation is $300, the savings on the

PIP are $200, and the expected savings on the

liability would be 28% of $360.) If there is sub-

rogation of losses, but not of LAE, then the PIP

insurer would never investigate this claim unless

the chance of covering both PIP and liability is

greater than 54% (300/560), as the expected sav-

ings would be the market share times $560 and

the cost of investigating would be $300. If al-

located LAE is also covered under subrogation,

then the PIP insurer would have the incentive

to investigate this claim, as the expected savings

would be the chance of having both PIP and li-

ability times $560 and the expected cost of in-

vestigating would be that chance times $300. As

long as the total expected savings exceeds the

expected cost of the investigation, the PIP in-

surer should perform the investigation. However,

reimbursement of unallocated LAE can change

the decision again. For example, if the chance

of covering both PIP and liability is only 5%,

then the expected savings from this investiga-

tion is $13 (5%£ (560¡ 300)), while the reduc-tion in expected ULAE reimbursement is $19

(95%£ 10%£ $200).Thus, the incentive for insurers to be strategi-

cally optimal is much lower when a large num-

ber of insurers compete. There is more room

for some insurers to exploit a free rider prob-

lem when more than two insurers are involved

in splitting the costs and benefits of adjusting

claims. This would be one disadvantage of hav-

ing a hybrid no-fault-limited tort system rather



than a simple tort or no-fault system of compen-

sation.15

8. Alternative arrangementsIncentives to underinvestigate claims can be

addressed in several ways. If the claim inves-

tigation strategy is viewed as a repeated game,

with each insurer monitoring the performance of

the other insurers for free riding and adapting

their own behavior based on what other insurers

are doing, then rules can be established to pro-

vide incentives to investigate claims more fully

to the mutual benefit of all, leading to the optimal

(B,B) equilibrium. The prior strategy described

in this paper assumes that insurers make only one

choice of investigation after considering the ex-

pected costs and savings. Alternatively, insurers

can switch levels of investigation depending on

the behavior of the other insurer, making this sit-

uation a repeated, noncooperative game. In this

situation, negotiation and monitoring might be

able to move the equilibrium position back to cell

(B,B). Liability insurers will know, when paying

the claim to the injured person and the subroga-

tion costs to the other insurer, whether the claim

has been investigated fully, especially if ALAE is

covered under subrogation. If a company is not

investigating an appropriate proportion of claims

(each insurer would know this, since the opti-

mal level of investigation is assumed to be the

same among all insurers), other insurers could

retaliate against the offending insurer by treating

that company’s PIP claims differently For exam-

ple, they might be less cooperative when deter-

mining subrogation payments or provoke regu-

latory oversight. Therefore, competing insurers

could investigate claims at the strategically op-

timal level in order to reduce claim costs, and

premiums, to the lowest feasible level and then

monitor competitors to make sure they are liv-

15IRC 2004a provides contrasting medical expense and total claim

cost in four no-fault states, one of which (Colorado) has subse-

quently changed to a system of tort liability only.

ing up to this standard. However, in the case of

an insurer that expects to become insolvent in

the near future, there is no expectation of the re-

peated game. Such an insurer may revert to no

investigation for at least those claims with sav-

ings accruing to other insurers without fear of

future retaliation.16 Thus, observing an insurer’s

claim investigation pattern could also prove to be

an early warning sign of financial problems.

A second approach to addressing the underin-

vestigation problem would be to develop a sys-

tem under which the claim investigation costs

are shared among all insurers. This approach is

similar to that recommended by Picard (1996)

for dealing with claim audit costs. One possi-

ble approach would be to handle claim investiga-

tions in a manner similar to a reinsurance pool,

where bills are submitted to the pool and any

market share adjustments necessary are made at

the pool level. Each company is required to pay a

proportionate cost of claim investigations, based

on market share, regardless of its own investi-

gation strategy. This strategy may introduce in-

creased overall system costs above those of the

market monitoring strategy and, thereby, be less

efficient. Another (partial) method of doing this

would be to establish a separate fraud investiga-

tion unit, with the costs shared by all insurers,

to decides which claims to investigate based on

the total cost savings impact, regardless of which

insurer would benefit from these savings.17

9.1. Empirical evidenceThere is evidence in Massachusetts auto injury

claims that insurers follow the strategy of using

independent medical examinations (IMEs) to in-

vestigate claims at least to the extent that they de-

16Additionally, a failing insurer will attempt to minimize the sub-

rogation payments to other insurers giving yet another sign of fi-

nancial weakness.17Separate insurance fraud bureaus in the United States are, how-

ever, chiefly concerned with reducing criminal fraud with the sav-

ings accruing to the policyholders of all insurers just as the costs

are shared among all policyholders.



rive (1) positive savings net of investigation costs

overall and (2) no net loss on PIP investigations

(Derrig and Weisberg 2003). Massachusetts is a

no-fault state, with all auto insurance companies

required to offer first-party PIP coverage to pol-

icyholders. This coverage provides up to $8000

of coverage for economic losses such as medical

expenses, loss of income, compensation for loss

of services, and other expenses related to an in-

jury caused by an automobile accident. These can

be the bulk of the expenses that typically serve as

the special damages in a tort claim, the remain-

der being general damages or pain and suffer-

ing. There is also a $2000 medical expense tort

threshold for liability claims in Massachusetts.18

This threshold can be met by eligible medical ex-

penses including ambulance, hospital, physician,

chiropractor, or physical therapy bills. An injured

person can only recover noneconomic losses if

the accident is the fault of another party and med-

ical expenses exceed $2000. Since medical ex-

penses are covered by the PIP insurance, there is

an incentive for a claimant to incur at least this

amount in medical bills (Weisberg, Derrig, and

Chen 1994).

If the PIP insurer can contain the medical ex-

penses below $2000, not only will the PIP claim

be lower but lower (or no) payments will be

made for any noneconomic losses. Even if med-

ical expenses exceed the threshold, limiting the

total claimed medical expenses can have an ad-

ditional impact on the liability claim, since the

noneconomic losses included in liability settle-

ments tend to be directly related to claimed med-

ical expenses. Although demonstrating that the

total liability settlement is not simply a multi-

ple of the medical expenses, Derrig and Weis-

18There is also a verbal threshold (Mass C351 s6D) listing particu-

lar injuries that can be compensated by general damages but those

injuries generally incur medical expenses in excess of $2,000, as

well. Those compensatory injuries are (1) cause of death, (2) con-

sists in whole or in part of loss of a body member, (3) consists in

whole or in part of permanent and serious disfigurement, (4) result

of loss of sight or hearing and (5) consists of a fracture.

berg (2004) and Loughran (2005) found that the

settlements for noneconomic losses do increase

with the cost of the medical expenses incurred

but are reduced in other circumstances (such as

high suspicion of fraud or positive findings from

a BI IME) by negotiation. Thus, any impact the

PIP insurer can have to restrain medical expenses

will have an additional cost savings on the non-

economic losses and level B investigation may

raise the return to investigation for all insurers.

9.2. Massachusetts independentmedical examinationsThe Automobile Insurers Bureau of Massachu-

setts conducted a study of three data sets of

claims involving IMEs, the primary tool used by

insurers to control auto injury costs (Derrig and

Weisberg 2003), that we summarize here. The

methodology is a “tabular” analysis that simply

compares the mean payments for four subgroups

of claims:

² IME not requested² IME requested but not completed (no-show)² IME completed and positive outcome (positiveIME)

² IME completed and negative outcome (nega-tive IME)

The estimated gross savings for each of the

first three subgroups above is the difference be-

tween the average payment for that category and

the average for the last subgroup (completed with

a negative outcome). An average IME cost is

then subtracted to obtain an estimate of net sav-

ings. The following results are taken from the

AIB Derrig and Weisberg (2003) report.

Table 9 displays the results for the three sets

of tabular analyses:

² 1993 AIB sample (claims from a prior AIB

study)

² 1996 DCD sample (claims from AY 1996 in

the AIB detailed claim database of all auto in-

jury claims19)

19A random sample of all reported claims.



Table 9. Summary results of Massachusetts IME Study

PIP

Sample: 1993 AIB 1996 DCD 1996 CSE

Total Net Savings (PIP) 0.2% ¡0:2% ¡0:8%Savings from IME Requested but not Completed 0.7% 0.3% 0.7%Savings from Positive IMEs 0.7% 0.4% ¡0:4%Cost of Negative IMEs ¡1:3% ¡0:9% ¡1:1%

PIP + BI

Sample: 1993 AIB 1996 DCD 1996 CSE

Total Net Savings (PIP + BI) 3.8% 5.7% 8.7%Savings from IME Requested but not Completed* 4.4% 2.8% 4.3%Savings from Positive IMEs 0.1% 3.2% 4.9%Cost of Negative IMEs ¡0:7% ¡0:3% ¡0:5%

¤Inclusion of All PIP claims with IME requested but not completed. 4.2% of savings for 1993 AIB comes from PIPs withno matching BIs where IME requested but not completed. 2.1% savings for 1996 DCD. 2.7% savings for 1996 CSE.

² 1996 CSE sample (claims from AY 1996 in

the claim screen experiment20)

Results are shown for both the PIP payment and

for the total payment (PIP+BI). The results sug-

gest that IMEs as currently employed represented

roughly a break-even proposition on PIP, al-

though for the CSE sample the cost of IMEs

slightly outweighed their benefit.

The bottom half, however, tells a different

story. Here the overall net savings for BI and PIP

payments are combined. These savings are based

on the outcome for the “best” IME, whether car-

ried out on the PIP or BI claim. The average

gross savings for the CSE sample was 9.2%, with

a net savings of 8.7%. Nearly half of the gross

savings (4.3%) is attributable to IMEs requested

but not actually completed. That is, the claimant

fails to show for the exam. In that case, savings

can result either if a potential BI claim is never

made, or if the BI settlement is reduced through

negotiation.

Somewhat more than half (4.9%) results from

a positive IME outcome (reduction of medical

20The Claim Screen Experiment (CSE) tracked about 3,000 PIP

claims arising at four large carriers during May—September, 1996.

Each carrier tracked the arrival of a preset collection of “red flags”

which in turn generated a running suspicion score for the adjusters

and their supervisors. Outcomes were recorded for all PIP claims

and for any associated BI tort liability claim.

Figure 2. Level of claim investigations inMassachusetts

expenses or curtailment of medical treatment) on

the PIP or BI IME. In the case of a BI IME or a

PIP IME used in the BI settlement, it may be too

late to have a meaningful impact on any ongoing

treatment. However, evidence of excess treatment

uncovered during the IME may provide leverage

to the adjuster in negotiating a lower settlement

by eliminating those medicals from any proposed

settlement.

Figure 2 illustrates the placement of the Mas-

sachusetts IME investigations relative to the the-

oretically optimal levels of Figure 1. PIP savings

equal to the costs, as shown in Table 9, would

place the Massachusetts investigation level, with

negative net PIP savings, to the right of both the

A and B optimal levels. In general, this would



Table 10. Net savings by suspicion level

imply that Massachusetts carriers were investi-

gating more claims than the B optimal level. As

we will see next, the judicious use of the suspi-

cion score could have resulted in fewer IMEs by

limiting investigations to only claims with mod-

erate scores.21

9.3. Net savings by suspicion level

The CSE PIP adjusters collected data that was

used to calculate and return a suspicion score

on a 10-point scale. The suspicion score was

based on a linear regression analysis (Weisberg

and Derrig 1998). The net savings effects of the

IMEs are analyzed separately by the level of sus-

picion in Table 10. A positive net saving on PIP

occurred only for claims with a moderate level

of suspicion (4—6). The results of the PIP IMEs

are shown in the top line of Table 10. These re-

21This, of course, is easier said from hindsight than done in real

time. The complication in making the decision to investigate is

the tension between the timing of the arrival of the red flags that

determine the risk class and the ongoing treatment.

sults, based on the all PIP claims with IMEs, in-

dicate a modest 2.6% net savings for moderately

suspicious claims, and a net negative for other

categories.

A subset of the PIP claims was found to re-

sult in BI claims. When attention is restricted to

these BI-bound PIP claims, we obtain the results

in line 3 (highlighted). For moderately suspicious

claims, the estimated savings remains relatively

unchanged at 3.4%. The other categories appear

to change, but it should be noted that the num-

bers of claims with zero suspicion or high suspi-

cion are fairly small in the BI sample. So, these

changes might in part be attributable to random

variation.

The bottom part of the table is similar, except

that the suspicion breakdown is based on a dif-

ferent measure of suspicion. Since there was no

suspicion score model for BI claims within the

CSE, an alternative external scoring model de-

veloped by National Healthcare Resources, Inc.,



Table 11. IME performance data

% of Claims with IME Requested

Strain/Sprain Other Injury

1993 AIB 1996 DCD 1996 CSE 1996 CSE 1996 CSE

PIP IME (PIP Claims) 18% 23% 26% 32% 14%PIP IME (BI Claims) 34% 35% 52% 54% 47%PIP or BI IME (BI Claims) 41% 40% 57% 58% 53%

% of Completed IMEs with Positive Outcomes

Strain/Sprain Other Injury

1993 AIB 1996 DCD 1996 CSE 1996 CSE 1996 CSE

PIP IME (PIP Claims) 34% 59% 58% 59% 55%PIP IME (BI Claims) 32% 60% 59% 59% 57%PIP or BI IME (BI Claims) 36% 60% 70% 71% 62%

(NHR) is used.22 This NHR model was used to

obtain a suspicion score for the BI claims, based

on the data extracted by coders from the BI claim

files.

The critical line is the last in Table 10, which

is highlighted. This line presents results for the

best outcome (PIP or BI) for the matched sample.

In effect, this analysis attempts to estimate the

overall impact of IMEs, taking into account both

the PIP payment and BI settlement. For moder-

ately suspicious claims, there is a 14.4% net sav-

ings, which accounts for most of the total sav-

ings. For claims with slight suspicion, IMEs rep-

resent effectively a break-even proposition, and

for the very low or very high suspicion a nega-

tive impact. It might appear counterintuitive that

IMEs do not have a positive value for claims with

high suspicion. Our explanation is that

such claims are not very amenable to reduction

through negotiation based on IME results. IMEs

are used primarily to constrain the total amount

of medical treatment, not to question the validity

of the injury itself or the circumstances of the

accident. To deal with “hard fraud” requires the

techniques of special investigation (e.g., exam-

ination under oath (EUO), accident reconstruc-

tion, surveillance).

22This scoring product was originally developed by Correlation Re-

search, Inc., while it was owned by National Healthcare Resources,

Inc. (NHR). NHR subsequently became part of Concentra, Inc.

9.4. Comparison across samples

Table 9 showed an overall 8.7% net savings for

the 1996 CSE claims, taking into account both

BI and PIP payments and IMEs. This outcome

is higher than the 5.7% for all claims (DCD) in

1996, which is in turn higher than the 3.8% reg-

istered in 1993. We now turn to the factors that

produce the IME savings.

In general, there are two factors that determine

the savings:

² Percent of claims on which an IME was re-quested

² Percent of completed IMEs with a positive out-come

Table 11 displays these percentages for each of

the three cohorts reported in Table 9. Compar-

ing first the 1993 sample versus the 1996 DCD,

we find the total number of IME requests has

remained essentially constant. However, the ap-

parent effectiveness of the IMEs performed has

increased dramatically. While this may be an ar-

tifact of the different coding patterns in the two

samples, it could reflect increased sophistication

on the part of adjusters regarding the selection

and/or utilization of the performed IMEs.

The results for the 1996 CSE claims pinpoint

where additional savings above the DCD esti-

mates were being derived. During the CSE, IMEs

were requested on a much higher percentage of



those PIP claims destined to result in BI claims.

For example, 40% of the DCD BI claims had ei-

ther a PIP or BI IME requested, compared with

57% of the CSE sample. Since this increase oc-

curred regardless of experimental or control sta-

tus, the feedback of suspicion scores to the ex-

perimental group cannot explain this increase.

Rather, we suspect that a “Hawthorne effect” may

have resulted from the awareness by adjusters

that a study was happening.23

Interestingly, the increased IMEs did not re-

sult in a diminution of IME effectiveness. The

IMEs still produced positive outcomes at effec-

tively the same rate, or perhaps somewhat higher.

So the adjusters may have been quite discrimi-

nating in their selection of claims. In any event,

it is encouraging that the CSE intervention may

have in some manner generated an improvement

in performance.

Table 11 also shows an IME requested (au-

dit) ratio for strains and sprains of 32%, more

than twice the 14% ratio for the remaining in-

juries. This large difference is indicative of a gen-

eral strategy of auditing riskier (for fraud and

buildup) classes of claims more often than less

risky claims (Dionne et al. 2009). The similar-

ity of positive outcome percentages for the two

classes (59%, 55%) indicates that this differen-

tial auditing strategy is playing a role in deterring

fraudulent and build-up claims as well as detect-

ing them (Tennyson and Salsas-Forn 2002, pp.

304—306).

A comparison with the national data on IME

use in auto injury coverages is instructive. The

IRC (2004b, pp. 93—98) study of 2002 claims

countrywide shows that IMEs are used on about

40 percent of the less than 20 percent of PIP

claims with an appearance of fraud or build-up

or about 10 percent overall. By way of contrast,

the Massachusetts claims in the IRC sample had

23“Paying attention to people, which occurs in placing them in

an experiment, changes their behavior. This rather unpredictable

change is called the Hawthorne effect.” (Kruskal, W. H. and

J. M. Tanur,, International Encyclopedia of Statistics, Free Press,

New York, vol. 4, p. 210, 1978).

about 50 percent more PIP claims (29%) with the

appearance of fraud or buildup than the overall

sample, suggesting an IME rate between 15 and

20 percent overall (IRC 2004b). The CSE anal-

ysis of Massachusetts PIP claims in Table 11 in-

dicates a somewhat higher 1996 IME rate of 26

percent. For BI claims, the countrywide rate of

IME use is less than 10 percent overall (IRC,

2003), consistent with the approximate 4 to 8

percent use of BI IMEs in Table 11 (compare

upper table rows 2 and 3). As noted previously,

the IMEs on PIP claims in Massachusetts pro-

duce a positive outcome (favorable to the insurer)

slightly less than 60 percent of the time. Coun-

trywide, over 80 percent of PIP claims are miti-

gated by the use of an IME, 90 percent on claims

with the appearance of fraud or buildup. These

comparative data reinforce the observation that

Massachusetts insurers may be conducting IMEs

at a rate in excess of the desired near-optimal

level, perhaps because of the higher or broader

levels of suspicious auto injury claims, and that

a more judicious choice of claims, based upon

suspicion scoring methods, would produce more

cost-efficient results.

10. ConclusionThe optimal level of claim investigation de-

pends on how many insurers are in the market

and what the subrogation rules are for loss ad-

justment expenses. Viewing claim investigation

strategy in a game theoretic framework demon-

strates the incentives and disincentives that cur-

rently exist to investigate automobile insurance

claims for excessive claim behavior. A frame-

work for establishing Nash equilibria was devel-

oped for the monopoly and two-insurer cases.

When insurers can choose two levels of inves-

tigation or none at all, the equilibrium is estab-

lished by the relationship of the savings to the

cost of investigation. Circumstances are identi-

fied for the case where Nash equilibrium may be

inefficient. In no-fault systems, when subrogation



of PIP claims exists, subrogation of allocated ex-

pense provides an incentive for investigation, but

a percentage reimbursement for unallocated ex-

pense provides a disincentive.

Empirical results from Massachusetts personal

auto injury claims were examined. Analyses of

three data sets show that carriers are generat-

ing substantial net savings from IME investiga-

tions, but those savings accrue mostly to the tort

carrier, indicating the workings of a noncoop-

erative game near equilibrium. A closer look at

the suspicion levels of the Massachusetts claims

shows that (1) insurers may have been conduct-

ing too many IMEs and (2) that a better selec-

tion (more toward the optimal equilibrium) could

be obtained with the use of a suspicion scoring

model.

Based on this analysis, additional cooperative

behavior should be encouraged in order to more

effectively reduce excessive medical treatment

and overall insurance costs. Subrogation rules

should cover allocated loss adjustment expenses.

If unallocated loss adjustment expenses are also

subject to subrogation, these payments should be

a set amount for each claim, and not a function

of claim size, as claim size adjustment provides a

disincentive to spend time and money to reduce

fraudulent claim costs. Other methods to encour-

age insurers to engage in strategically optimal ap-

proaches to investigating claims should be devel-

oped that consider the long-term, industry-wide

impact of reducing fraud.

Finally, the empirical data used above in the

study of Massachusetts claims was examined as

presented without optimizing beyond what indi-

vidual companies procedures produced for claim

investigation at that time. It is clear from Table 10

results that suspicion scores can be used to select

better candidates for investigation with higher net

savings and the application of so-called predic-

tive models can increase efficiency through bet-

ter claim selection methods (Derrig and Francis

2008). Many such procedures have been covered

in the annual CAS Ratemaking Seminars and

in the published literature, for example, Dionne,

Guilliani, and Picard (2009) and Artis, Ayuso,

and Guillen (2002).

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Optimal Layers for CatastropheReinsurance

by Luyang Fu and C. K. “Stan” Khury

ABSTRACT

Insurers purchase catastrophe reinsurance primarily to re-

duce underwriting risk in any one experience period and

thus enhance the stability of their income stream over time.

Reinsurance comes at a cost and therefore it is important to

maintain a balance between the perceived benefit of buying

catastrophe reinsurance and its cost. This study presents a

methodology for determining the optimal catastrophe rein-

surance layer by maximizing the risk-adjusted underwrit-

ing profit within a classical mean-variance framework.

From the perspective of enterprise risk management, this

paper improves the existing literature in two ways. First, it

considers catastrophe and noncatastrophe losses simultane-

ously. Previous studies focused on catastrophe losses only.

Second, risk is measured by lower partial moment which

we believe is a more reasonable and flexible measure of

risk compared to the traditional variance and Value at Risk

(VaR) approaches.

KEYWORDS

Catastrophe reinsurance layer, downside risk, lower partial moment,

semivariance, utility function, enterprise risk management



1. Introduction1

Catastrophe reinsurance serves to shield an in-

surer’s surplus against severe fluctuations aris-

ing from large catastrophe losses. By purchas-

ing catastrophe reinsurance, the reinsured trades

part of its profit to gain stability in its under-

writing and financial results.2 If catastrophe risks

are independent of other sources of risk and di-

versifiable in equilibrium, Froot (2001) argued

that, under the assumption of a perfect financial

market, the reinsurance premium for catastrophe

protection should equal the expected reinsurance

cost and that a risk-averse insurer would seek

the protection against large events over hedging

low-retention layers. Gajek and Zagrodny (2004)

concluded that if a reinsured has enough money

to buy full protection against bankruptcy (ruin),

the optimal reinsurance arrangement is the aggre-

gate stop-loss contract with the largest possible

deductible.

In practice, financial markets are not perfect,

reinsurance prices are significantly higher than

those indicated by the theory noted above, and

it does not make economic sense for insurers

to buy full protection against ruin. Froot (2001)

showed that the ratios of the catastrophe reinsur-

ance premium to the expected catastrophe loss

can be as high as 20 for certain high-retention

layers that have low penetration probabilities.3

Facing the reality of relatively high reinsurance

prices and the practical economic constraints on

the catastrophe reinsurance buyer, primary insur-

ers are reluctant to surrender a large portion of

their profit for limited catastrophe reinsurance

protections. Thus the reinsured often purchases

1This study is jointly sponsored by the Actuarial Foundation and

Casualty Actuarial Society.2While it is a minor point, if these concepts are extended to a line of

business that is not expected to be profitable (for whatever reason)

during the prospective reinsurance exposure period, the reduction

in profit referenced here becomes an increase in the projected loss.

The ideas are still the same.3Froot (2001), Figure 4, p. 540.

low reinsurance layers that are subject to a high

probability of being penetrated.

Economists have offered many explanations

for the inefficiency of the reinsurance market.

Borch (1962) investigated the equilibrium of the

reinsurance market and found that the reinsur-

ance market will, in general, not reach a Pareto

Optimum if each participant seeks to maximize

his utility. Froot (2001) identified eight theoreti-

cal reasons to explain why insurers buy relatively

little reinsurance against large catastrophe events.

He found the supply restrictions associated with

capital market imperfections (insufficient capital

in reinsurance, market power of reinsurers, and

inefficient corporate forms for reinsurance) pro-

vide the most powerful explanation. As practic-

ing actuaries, the authors believe there are signif-

icant limitations on reinsurers’ ability to diversify

risk. In the context of catastrophe reinsurance, no

reinsurer is big enough to fully diversify away

catastrophe risk. To support the risk, a reinsurer

needs to hold a prohibitively large amount of ad-

ditional capital and will, in turn, need to realize

an adequate return on such capital.

Froot (2007) found that product-market sen-

sitivity to risk and exposure asymmetry tend to

make insurers more conservative in accepting un-

systematic risks, more eager to diversify under-

writing exposures, and more aggressive in hedg-

ing. Even though reinsurance prices are high rel-

ative to expected loss, insurers are still willing to

seek reinsurance protections.

In this study, the authors do not explore the

level of reinsurance price and its reasonableness.

We also do not investigate why insurance firms

are risk averse to catastrophe losses which are

unsystematic and uncorrelated with aggregate

market return. Instead, we treat reinsurance price

as a predetermined variable in the overall strate-

gic decision-making process and develop an opti-

mal reinsurance strategy for insurers conditioned

by their risk appetite, prevailing reinsurance


Optimal Layers for Catastrophe Reinsurance

prices at the time a decision is made, and the

overall profitability of the enterprise.

This introductory discussion would not be

complete if we did not explicitly point out that

a reinsurance arrangement, in fact, is a combina-

tion of art and science. It often depends on gen-

eral economic conditions in the state or coun-

try of the reinsured and worldwide, the recent

history of catastrophe reinsurance in the state or

country of the reinsured and worldwide, and the

risk characteristics of the reinsured. Both rein-

surer and reinsured usually are well informed

and are free to negotiate in the spirit of open

competition. In the negotiation, it is quite com-

mon that certain terms of the treaties would be

modified, such as changing the retention level

and the size of the layer. In pursuing the opti-

mization process outlined in this paper, we are

not attempting to deny or ignore this general

art/science characteristic of the reinsurance ar-

rangement. Instead our hope is that knowledge

of the optimal balance between profit and risk,

as measured using the process outlined in this pa-

per, in the particular circumstances of a reinsured

vis-a-vis the then-prevailing market prices, will

serve to enhance the quality of the reinsurance

decision, ceteris paribus.

Reinsurance arrangements have been studied

extensively because of their strategic importance

to the financial condition of insurance compa-

nies. However, previous studies on optimal catas-

trophe reinsurance only utilized partial informa-

tion in the reinsurance decision-making process.

Gajek and Zagrodny (2000) and Kaluszka (2001)

investigated the optimal reinsurance arrangement

by way of minimizing the consequential variance

of an insurer’s portfolio. Gajek and Zagrodny

(2004) discussed the optimal aggregate stop-loss

contract from the perspective of minimizing the

probability of ruin. Those studies focus on the

risk component, but ignore the profit side of the

equation. Bu (2005) developed the optimal rein-

surance layer within a mean-variance framework.

Insurers are assumed to minimize the sum of the

price of reinsurance, the catastrophe loss net of

reinsurance recoveries, and the risk penalty. Bu

used both the profit and risk components in the

optimization. However, his method focused on

the catastrophe loss only and ignored the concur-

rent effect of noncatastrophe underwriting per-

formance on the financial results of the reinsured.

In practice, the overall profitability is an impor-

tant factor impacting the reinsurance strategy be-

cause, among other things, it can enhance an in-

surer’s capability to assume risk.

Lampaert and Walhin (2005) studied the opti-

mal proportional reinsurance that maximizes

RORAC (return on risk-adjusted capital). The ap-

proach requires the estimation of economic cap-

ital based on VaR or TVaR (tail value at risk)

at a small predetermined probability. VaR and

TVaR are popular in insurance generally and in

actuarial circles specifically. VaR is the point at

which a “bad” outcome can occur at a prede-

termined probability, say 1%. TVaR is the mean

of all outcomes that are “worse” than the pre-

determined “bad” outcome. VaR, and especially

TVaR, has some convenient features as a risk

measure.4 TVaR only contemplates severe losses

having a probability at or lower than a given

probability as the central risk drivers, and it treats

those losses linearly. For example, if an insurer

has a 5% probability of a loss of $3 million, a 4%

probability of a loss of $5 million, a 0.9% proba-

bility of a loss of $10 million, and a 0.1% proba-

bility of a loss of $100 million, VaR at 1% is $10

million and TVaR is $19 million (10 ¤ 0:9%+100 ¤ 0:1%)=1%. VaR and TVaR are not consis-tent with common risk perception from two per-

spectives: (1) fear is not just of severe losses, it is

also of smaller losses (Bodoff 2009). In the case

above, $3- and $5-million losses will not con-

tribute to VaR because VaR only considers 1%

probability at which risk is generated; (2) risk-

4For example, Meyers (2001) discussed that TVaR satisfies the four

criteria of coherent risk measures.



bearing entities do not weigh the risk of loss in

a linear manner and are more concerned about

the incidence of large losses than smaller ones.

In other words, risk perception is exponentially,

not linearly, increased with the size of loss.

In practice, the RORAC method has been pop-

ular in calculating the optimal catastrophe rein-

surance layer. In this study, we improve the pop-

ular mean-variance approach advocated in aca-

demic studies by using lower partial moment

(LPM) as the measure of risk, and provide an

alternative method for determining optimal rein-

surance layers. Compared with the RORAC ap-

proach, our method has three advantages. First, it

does not involve the calculation of the necessary

economic capital, which has no universally ac-

cepted definition. Second, by VaR or TVaR, true

risk exists only at the tail of the distribution. By

LPM, on the other hand, all the losses are con-

sidered as generating risk to the risk-bearer, but

severe losses contribute to LPM disproportion-

ately. Third, the estimation of variance and semi-

variance is relatively robust compared to VaR

and TVaR in the context of catastrophe losses.

The tail estimation of remote catastrophe losses

generally is not robust, and is very sensitive to

the assumptions about the underlying distribu-

tion, especially at high significance levels. The

limitations of the proposed method are the limita-

tions inherent to the mean-variance framework. It

can be difficult to estimate the risk-penalty coef-

ficient, as the parameter is often time-dependent

and subject to management’s specific risk

appetite.

This paper improves the previous mean-var-

iance optimal reinsurance studies from two per-

spectives. First, it considers noncatastrophe and

catastrophe losses simultaneously. Second, the

risk is measured by LPM (semivariance), which

is a more reasonable and appropriate risk mea-

sure than the traditional risk measures, such as

total variance, used in previous studies (i.e.,

Borch 1982; Lane 2000; Kaluszka 2001; Bu

2005). Even though the authors investigate the

optimal layers in the context of catastrophe rein-

surance, the proposed method can be easily ap-

plied to aggregate excess-of-loss (XOL) treaties

and occurrence XOL treaties that cover shock

losses at individual occurrence/claim levels.

2. Risk-adjusted profit modelInsurance companies buy catastrophe reinsur-

ance to reduce potential volatility in earnings and

to provide balance sheet protection from catas-

trophic events. However, reinsurance comes at a

cost, and therefore attaining an optimal balance

between profit levels after the effect of catastro-

phe reinsurance and the reduction in their risk

exposure is important. Buying unnecessary or

“excessive” reinsurance coverage would give up

more of the reinsured’s underwriting profit than

is necessary or desirable. Buying inadequate rein-

surance coverage would still expose the reinsured

to the volatility engendered by the risk of large

catastrophe events, the reinsurance cover not-

withstanding. The value of reinsurance is the sta-

bility gained or the risk reduced and the cost

is the premium paid less the loss recovered. As

Venter (2001) pointed out, the analysis of a rein-

surance arrangement is the process of quantify-

ing this cost/benefit relationship. It is self-evident

that in an insurance company’s decision-making

process, a relatively certain, but maximal, profit

is preferable over other, perhaps higher profit po-

tentials that are also exposed to the risk of large

catastrophic losses. Following the classic mean-

variance framework in financial economics, a

reinsured will buy reinsurance to maximize its

risk-adjusted profit, defined as

RAP = E(r)¡ μ ¤Var(r) (1)5

where r is the net underwriting profit rate, E(r) is

the mean of r and Var(r) is its variance. μ ¤Var(r)

5In financial economics, it is often referred to as the expected util-

ity function with the formula E(r)¡ 0:5A ¤Var(r), where A is theso-called risk aversion coefficient. In this paper, the risk-penalty

coefficient μ = 0:5A.



is the penalty on risk. μ is the risk-penalty coef-

ficient: the higher the risk-penalty μ, the greater

is the reinsured’s risk aversion. If μ = 0, the rein-

sured is risk-neutral. It will try to maximize profit

and not care about risk. In this scenario, it will

not give up any profit to purchase reinsurance.

The most common measurement of risk is the

variance associated with a particular outcome.

Variance reflects the magnitude of uncertainty

(variability) in underwriting results, and how

widely spread the values of the profit rate are

likely to be around the mean. Therefore, within

a variance framework, all the variations, both de-

sirable and undesirable, are viewed as manifes-

tations of risk. Large favorable swings will lead

to a large variance, but insurers certainly have

no problem with such favorable underwriting re-

sults. Markowitz (1959) pointed out the draw-

backs of using total variance as a measure of risk,

as there is implicitly and directly a cost to both

upside and downside movements.

Fishburn (1977) argued that risk should be

measured in terms of only below-target returns.

Hogan and Warren (1974) and Bawa and Lin-

denberg (1977) suggested using LPM to replace

total variance as the risk measure:

LPM(T,k) =

Z T

¡1(T¡ r)kdF(r) (2)

where T is the minimum acceptable profit rate, k

is the moment parameter which measures one’s

risk perception sensitivity to large loss, and F(r)

is the probability function of r. Unlike total vari-

ance, LPM only measures the unfavorable vari-

ation (e.g., when r < T) as risk. Because LPM

does not change with favorable deviations, it

would seem to be a superior measure of risk.

When T is triggered at the 1% probability level

and k = 1, LPM is equal to 0:01 ¤TVaR. Whenthe distribution is symmetric, T is the mean, and

k = 2, it is equal to 0:5 ¤ variance. LPM com-

bines the advantages of variance and TVaR. It

is superior to variance by not treating the favor-

able outcomes as risks. It is superior to TVaR be-

cause (1) it considers small and medium losses

as risk components, and (2) it provides nonlin-

ear increasing penalties on larger losses when

k > 1. In the example above, suppose a $100 mil-

lion loss will cause a financial rating downgrade

while a $10 million loss merely causes a bad

quarter. Management inevitably will perceive a

$100 million loss to be more than 10 times as

bad as a $10 million loss. By VaR, a $100 mil-

lion loss is 10 times as bad as a 10 million loss.

By LPM with k = 1:5, the risk of a $100 million

loss is 31.6 times that of a $10 million loss; and

by LPMwith k = 2, it is 100 times. The k value is

a direct measure of risk aversion to large losses.

When k = 2, LPM is often called “semivari-

ance” (it excludes the effects of variance associ-

ated with desirable outcomes in the measurement

of risk) and has been gaining greater acceptance.

By formula, semivariance is defined as

SV(T) =

Z T

¡1(T¡ r)2dF(r): (3)

A growing number of researchers and practi-

tioners are applying semivariance in various fi-

nancial applications. For example, Price, Price,

and Nantell (1982) showed that semivariance

helps to explain the puzzle of Black, Jensen, and

Scholes (1972) and Fama and MacBeth (1973)

that low-beta stocks appear systematically un-

derpriced and high-beta stocks appear systemat-

ically overpriced. However, to date, the casualty

actuarial literature has seldom used the semivari-

ance as a risk management tool and neither does

it appear much in practice.6

Generally, a decision-maker can be expected

to be more concerned with the semivariance than

with the total variance. Using downside risk in-

stead of total variance, the downside-risk-adjust-

ed profit (DRAP) becomes

DRAP =Mean(r)¡ μ ¤LPM(T,k), (4)

6Berliner (1977) studied a special case of semivariance with the

mean as the minimum acceptable value, against variance as risk

measures. He concluded that although the semi-variance is more

theoretically sound, variance provides a better risk measure.



where μ is the penalty coefficient on downside

risk.

Three parameters T, k, μ in the DRAP for-

mula interactively reflect risk perception and risk

aversion. With these three parameters, the DRAP

method provides a comprehensive and flexible

capacity to capture risk tolerance and appetite.

T is the benchmark point of profit below which

the performance would be considered as “down-

side” (lower than is minimally unacceptable). T

can be a target profit rate, the risk-free rate, zero,

or even negative, depending on one’s risk per-

ception. When T is at the very right tail of r,

only large losses contribute to the downside risk.

T can vary by the mix of lines of business. For

example, for long tail lines, negative underwrit-

ing profits may be tolerable because of antici-

pated investment income on loss reserves.

The moment parameter k reflects one’s risk

perception as the size of loss grows: k > 1 im-

plies exponentially increasing loss perception to

large losses; 0< k < 1 represents concavely in-

creasing loss perception to large losses; k = 1 im-

plies linearly increasing loss perception. In gen-

eral, k is larger than 1 since fear of extreme

events that can lead to a financial downgrade is

greater than the fear of multiple smaller losses.

Because semivariance is the most popular LPM,

we choose k = 2 to illustrate our approach in the

case study presented below.

The risk aversion level is represented by μ,

which is a function of T and varies according to

its values. For example, when T = 0, all the un-

derwriting losses contribute to LPM(0,k); when

T =¡10%, only losses exceeding the 10% loss

rate contribute to LPM(¡0:1,k). LPM(¡0:1,k)represents a much more severe risk than

LPM(0,k). μ is also a function of k. For example,

LPM(T,1) and LPM(T,2) are at different scales

because the former is at the first moment and the

latter is at the second moment. μ should vary with

k because k changes the scale of risk measure.

Note that μ may not be constant across loss

layers. For example, when k = 1, LPM is a lin-

ear function of loss. For a run of smaller losses

that cause a “bad” quarter, μ may be very small.

For losses that cause a “bad” year or eliminate

management’s annual bonus, μ may be larger.

For losses that lead to a financial downgrade or

the replacement of management, μ will be even

larger. Interested readers can expand the models

in this paper by adding a series of risk-penalties

upon predetermined loss layers with various

risk aversion coefficients. When k ¸ 2, becauseLPM increases exponentially with extreme

losses, it may not be necessary to impose higher

risk-penalty coefficients on higher layers.

In addition, k may not be constant across loss

layers. The scale of loss impacts the value of k.

If k is constant, say k = 2, it implies that $100

million loss is 100 times worse than $10 million

loss. It also indicates that $100 loss is 100 times

worse of $10 loss. The latter, in general, is not

true because of linear risk perception when view-

ing a smaller nonmaterial loss. In the context of

reinsurance, k might be closer to one at a working

layer (low retention with high probability of pen-

etration) and would increase for higher excess

layers. Interested readers can expand the models

in this paper by adding a series of risk-penalties

upon predetermined loss layers with various mo-

ment parameters.

The academic tradition in financial economics

has been to set μ as a constant and k = 2.7 Assum-

ing an individual has a negative exponential util-

ity function u(r) =¡exp(¡A ¤ r), where A > 0.If r is normally distributed, the expected utility

is E[u(r)] =¡exp[¡A ¤E(r) +0:5A2 ¤Var(r)].Maximizing E[u(r)] is equivalent to maximizing

E(r)¡ 0:5A ¤Var(r). Also, ¡u00(r)=u0(r), which isequal to A in this specific case, is often referred

as the “Arrow-Pratt measure of absolute risk

7In recent years, academics have found increasing evidences of

higher moments of risk aversion. For example, Harvey (2000)

showed that skewness (3rd moment) and kurtosis (4th moment)

are priced in emerging stock markets but not in developed markets.



aversion.” Constant μ and k = 2 are built-in fea-

tures under the assumptions of negative expo-

nential utility function and normality. Alterna-

tively, interested readers can use negative expo-

nential utility, logarithmic utility, or define their

own concavely increasing utility curves, and se-

lect the optimal reinsurance layer to maximize

the expected utility function.8 To simplify the il-

lustration and to be consistent with academic tra-

dition, we use a constant μ and k = 2 in the case

study.

An inherent difficulty in mean-variance type of

analysis is the need to estimate the risk-penalty

coefficient empirically. The key is to measure

how much risk premium one is willing to pay

for hedging risk. The flip counter-party ques-

tion is how much investors would require for

assuming that risk. For overall market risk pre-

mium, one can obtain the market risk premium

by subtracting the risk-free treasury rate from

market index return. For example, if the market

return is 10%9 and the risk-free rate is 5.5%, the

risk premium is 4.5%, or 45% of total “profit.”

For the risk premium in the insurance/reinsur-

ance market, one can use the market index for in-

surance/reinsurance companies. For the risk pre-

mium in catastrophe reinsurance, one can com-

pare catastrophe bond rates to the risk-free rate.

For example, if the catastrophe bond yield is

12%,10 the treasury rate yield is 5.5%, and the

expected loss from default is 0.5%, then the risk

premium11 is 6%, consisting of 50% of total

yield.

The methods above provide objective estima-

tion of μ assuming that management’s risk ap-

8Insurance profit is generally not normally distributed. It can be

positive or negative, but has a fatter tail on left side. Reinsurance

layers complicate the distribution of r. It may be very difficult

to derive an analytical solution for E[u(r)]. However, a numerical

solution maximizing E[u(r)] can be obtained easily.9According to Vanguard, its S&P 500 index returns 10.34% annu-

ally from its inception (08/31/1976) to 01/31/2010.10In 1997, USAA issued its one-year catastrophe bond at LIBOR

plus 576 basis points, which was close to 12%.11Risk premium is equal to cat bond yield¡risk-free rate¡expectedloss. For details, please refer to Bodoff and Gan (2009).

petite is consistent with the market. In reality, μ

varies by risk-bearing entity. Each risk-bearing

entity has its own risk perception and tolerance.

To measure management’s risk aversion, one can

obtain μ by asking senior executives, “In order to

reduce downside risks, how much of the other-

wise available underwriting profit per unit of risk

are you willing to pay?” The answer to a single

question may not be sufficient to pin down the

value of theta. Most likely management would

require information about expected results under

optimal reinsurance programs at various values

of theta to fully understand the implications of

the final theta value selected. To replicate the sen-

sitivity tests that management may perform when

determining the utility function, the case study

provides optimal insurance solutions at various

values of theta.

For the same management within the same in-

stitution, μ often is time-variant12 as the risk ap-

petite often changes to reflect macro economic

conditions or micro financial conditions. For ex-

ample, after a financial crisis, insurance compa-

nies may become more risk averse. μ also varies

by the mix of business. For lines with little catas-

trophe potential, such as personal auto, the tol-

erance on downside risk might be higher and μ

would be smaller. For lines with higher catas-

trophe potential, such as homeowners, μ can be

larger. μ is difficult to estimate because of its sub-

jective nature. Actuarial judgment plays an im-

portant role when determining the risk-penalty

coefficient.

In the context of catastrophe reinsurance, the

layers are bands of protection associated with

catastrophe-triggered loss amounts. Outside of

price, the main parameters of a catastrophe layer

are the retention, the coverage limit, and the ces-

sion percentage within the layer of protection.

Retention is the amount that the total loss from

12The authors tried to estimate a constant risk penalty coefficient

based on management past reinsurance decisions. The result clearly

indicates that the risk coefficient is time-variant.



a catastrophe event must exceed before reinsur-

ance coverage attaches. The limit is the size of

the band above the retention that is protected by

reinsurance. The cession percentage is the per-

centage of the catastrophe loss within the rein-

surance layer that will be covered by the rein-

surer. The limit multiplied by the cession per-

centage is the maximum reinsurance recovery

from a catastrophe event within that particular

band of loss. The coverage period of catastro-

phe reinsurance contracts is typically one year.

Let xi denote the gross incurred loss from the

ith catastrophe event within a year, and Y be the

total gross non-catastrophe loss of the year. Let

R be the retention level of the reinsurance, L be

the coverage layer of the reinsurance immedi-

ately above R,13 and Á be the coverage percent-

age within the layer.

The loss recovery from reinsurance for the ith

catastrophe event is

G(xi,R,L) =

8>><>>:0 if xi · R

(xi¡R) ¤Á if R < xi · R+LL ¤Á if xi > R+L

:

(5)

Let EP be the gross earned premium, EXP be

the expense of the reinsured, N be the total num-

ber of catastrophe events in the reinsurance con-

tract year, RP(R,L) be the reinsurance premium,

which is a decreasing function of R and an in-

creasing function of L, and RI be the reinstate-

ment premium.

The underwriting profit gross of reinsurance is

¼ = EP¡EXP¡Y¡NXi=1

xi: (6)

Reinstatement premium is a pro rata reinsur-

ance premium charged for the reinstatement of

the amount of reinsurance coverage that was

“consumed” as the result of a reinsurance loss

13In general, reinsurance is structured in multiple layers. The for-

mulation works for continuous layers. For disruptive layers, one

needs to introduce additional retentions and limits.

payment under a catastrophe cover. The stan-

dard practice is one or two reinstatements. The

number of reinstatements imposes an aggregate

limit on catastrophe reinsurance. The reinstate-

ment premium after the ith catastrophe event is

RI(xi,R,L) = RP(R,L) ¤G(xi,R,L)=L: (7)

The underwriting profit net of reinsurance is

¼ = EP¡EXP¡Y¡NXi=1

xi¡RP(R,L)

+NXi=1

G(xi,R,L)¡NXi=1

RI(xi,R,L): (8)

The underwriting profit rate net of reinsurance

is

r = 1¡ EXP+Y+RP(R,L)EP

¡PNi=1 xi¡G(xi,R,L) +RI(xi,R,L)

EP:

(9)

Thus the optimal layer is that combination of

R and L which maximizes DRAP:

MaxR,L

Mean(r)¡ μ ¤ r), subject to C:14

(10)

Capital asset pricing model suggests that firms in

a perfect market are risk neutral to unsystematic

risks. If reinsurers are adequately diversified and

price as though they are risk neutral, the reinsur-

ance premium would be equal to the expected

reinsurance cost.15 That is, the total premium

paid by reinsured, RP(R,L)+PNi=1RI(xi,R,L),

is equal to the expected reinsurer’s loss cost,

14C is the constraint for the optimization, which can be (a) a budget

constraint on reinsurance premium; (b) a risk tolerance constraint

such as the probability of downgrade is less than 1%, or the PML

will reduce surplus by 15% no more than 1 in 100 years; or (c) any

number of other possible constraints that are relevant to the partic-

ular reinsured. In the case study, to simplify the analysis, we do not

impose constraints on the optimization. The same framework can

be applied to optimal aggregate reinsurance strategies. Actuaries

would select four parameters: event deductible, event cap, aggre-

gate retention, and limit to maximize the risk-adjusted profit.15According to financial economics theory, risk premium only ap-

plies to nondiversifiable systematic risk. Catastrophe risk is diver-

sifiable in theory so that the risk premium of assuming catastrophe

risk is zero. The theory is extensively discussed in Froot (2001).


SV(


PNi=1G(xi,R,L) plus reinsurer’s expense. In this

case, reinsurancewill significantly reduce the vol-

atility of the underlying results of the insured

over time, but slightly reduce the expected profit

by the reinsurance expense over the same period

of time. Figure 1 shows reinsurance optimiza-

tion under the assumption of perfect market di-

versification. A is the combination of profit and

downside risk without any reinsurance. B is the

profit and risk with full reinsurance. B is not

downside-risk-free because noncatastrophe loss

could cause the profit to fall below the minimum

acceptable level. Line AB is the efficient frontier

with all possible reinsurance layers. Closer to B,

it represents buying a great deal of reinsurance

coverage. Closer to A, it represents buying mini-

mal reinsurance coverage. Under the assumption,

the reinsurance premium only covers reinsurer’s

costs, and the line is relatively flat.16 U1, U2, and

U3 are the utility curves. The slope of those lines

is the risk-penalty coefficient μ. The steeper the

curve, the more risk-averse. All the points on a

given curve provide the exact same utility. The

higher utility curve represents the higher utility.

The utilities on line U1 are higher than those on

lines U2 and U3. An insurance company gains

the highest utility at point B. Thus maximizing

the risk-adjusted profit is equivalent to minimiz-

ing the downside risk. The optimal solution oc-

curs when R = 0 and L=+1. A retention equalto zero coupled with an unlimited reinsurance

layer will completely remove the volatility from

catastrophe events with a low cost (reinsurance

premium-recovery). Under the perfect market di-

versification assumption, the proposed method

yields a solution consistent with Froot (2001).

In practice, however, because of the need to

reward reinsurers for taking risk, the reinsurance

price RP(R,L) is always larger than the expected

reinsurer’s loss cost. The expected loss/premium

ratio is generally a decreasing function of reten-

tion R. The higher the retention R, the lower the

16Line AB would be flat assuming zero reinsurer’s expense.

Figure 1. Reinsurance optimization under theassumption of perfect diversification

Figure 2. Reinsurance optimization in reality

expected reinsurance loss ratio. From the rela-

tionship between risk transfer and reinsurance

premium, a higher layer implies a higher level

of risk being transferred to the reinsurer. To sup-

port the risk associated with higher layers, the

reinsurer needs more capital and thus requires a

higher underwriting margin. Therefore, a rein-

sured has to pay a larger risk premium on higher

layers to hedge its catastrophe risk. In practice,

E(PNi=1G(xi,R,L))=RP(R,L) is often less than

40%, and even below 10% for high retention

treaties. The relatively high prices associated with

high retentions often deter the reinsured from

purchasing coverage at those levels. Subject to

the constraints imposed by reinsurance prices and

the willingness of the reinsured to pay, as Froot

(2001) discussed, the optimal solutions are of-

ten low reinsurance retentions at a relatively low

price and a high probability of being penetrated.

Figure 2 shows reinsurance optimization in real-



ity. As in Figure 1, curve AB represents the ef-

ficient frontier. Because reinsurance companies

cannot fully diversify catastrophe risk and re-

quire higher returns to assume the risk on higher

layers, AB is a concave curve: from A to B, the

slope becomes steeper to reflect higher risk pre-

miums associated with higher layers. Close to

point B, the slope is very steep to reflect the ex-

tra capital surcharge at the top layers. Of all the

possible reinsurance layers, point C provides the

highest utility (or downside-risk-adjusted profit)

to the reinsured.

3. A case study

3.1. Key parameters

Suppose an insurance company with $10 bil-

lion17 gross earned premium plans to purchase

catastrophe reinsurance. Within one year, the

number of covered catastrophe events is normally

distributed18 with a mean of 39.731 and a stan-

dard deviation of 4.450; and the gross loss from a

single catastrophe event is assumed to be lognor-

mally distributed. The logarithm of the catastro-

phe loss has a mean of 14.478 and a standard de-

viation of 1.812,19 which imply a mean of $10.02

million and a standard deviation of $50.77 mil-

lion for the catastrophe loss from one event. The

mean of the aggregate gross loss from all the

catastrophe events within a year is $397.94 mil-

17The premium is for all lines of business. The catastrophe losses

are from property lines. All the catastrophe loss parameters in this

case study are estimates drawn from Applied Insurance Research

(AIR) data simulated based on the property exposures of an insur-

ance company and are scaled accordingly to be consistent with $10

billion earned premium.18The number of catastrophe events (severe storm and hurricane)

at state level generally fits Poisson distributions better than normal.

At aggregate company level, the number of catastrophe events is

asymptotically normal by AIR data.19The frequency and severity are estimated using the data from

AIR. In this study we randomly generate the loss data to avoid

revealing proprietary information. The AIR data has a longer tail

than the fitted lognormal distribution. In practice, actuaries can use

catastrophe data directly from AIR, RMS (Risk Management So-

lutions), or EQECAT, or generate loss data based on proprietary

catastrophe models.

lion and the standard deviation is $322.92 mil-

lion. The expense ratio of the insurance company

is assumed to be 33.0%. The aggregate gross

noncatastrophe loss is also assumed to be log-

normally distributed with a logarithm of non-

catastrophe loss mean of 22.497 and standard de-

viation of 0.068. This implies that the mean of

gross noncatastrophe loss is $5.91 billion and the

standard deviation is $402.10 million. Assum-

ing catastrophe and noncatastrophe losses are in-

dependent,20 the mean of the aggregate gross

loss is $6.30 billion and the standard deviation

is $515.72 million. The mean underwriting profit

rate is 3.93% and the standard deviation is 5.16%.

The numerical study is based on this hypothetical

company using the simulated noncatastrophe and

catastrophe loss data. The simulation is repeated

10,000 times. In each simulation, the noncatas-

trophe loss and catastrophe losses within a year

are generated, the losses covered by the reinsur-

ance treaty are calculated by Equation (5), and

the total profit rate is calculated by Equation (9)

assuming two reinstatements. Let rm be the profit

rate from the mth round of simulation. The semi-

variance is

R,L) =1

10000

10000Xm=1

(min(rm¡T, 0))2:

(11)

Equation (11) is a discrete formula of semi-

variance, which is an approximation of Equa-

tion (3).

Let us assume that reinsurance will cover 95%

of the layer (R,L) and UL is the upper limit of

the covered reinsurance layer (R,L), UL = R+L.

For reinsurance prices, we fit a nonlinear curve

using actual reinsurance price quotes.21 The fit-

20This is a simplifying assumption for the case study. In practice,

the correlation is weakly positive from the perspective of the pri-

mary insurer because catastrophe events cause claims that are oc-

casionally categorized as noncatastrophe losses.21The reinsurance prices are proportionally scaled by the premium

adjustment.


SV(


Table 1. Distributional summaries of loss covered from reinsurance in a year for quoted reinsurance layers

Retention Upper Limit Recovery/reinsurance Penetration(million) (million) Mean Standard Deviation Premium Probability

305 420 8,859,074 29,491,239 42.59% 10.18%420 610 8,045,968 35,917,439 37.08% 6.04%610 915 6,496,494 41,009,356 32.81% 3.15%610 1,030 7,923,052 51,899,244 31.44% 3.15%

1,030 1,800 4,858,545 55,432,115 16.93% 1.11%1,800 3,050 2,573,573 48,827,021 6.58% 0.40%

ted reinsurance price of layer (R,L) is22

RP(R,L) = 1:2300 ¤ (UL¡R)+ 1:2978 ¤ 10¡4

¤ (UL2¡R2)¡ 1:3077 ¤ 10¡8

¤ (UL3¡R3)¡ 0:1835¤ (UL ¤ log(UL)¡R ¤ log(R))+45:4067 ¤ (log(UL)¡ log(R)):

Appendix contains both the quoted reinsur-

ance prices and the fitted reinsurance prices. The

actual prices below layer ($1,800 million, $3,050

million) are derived by combining the six lay-

ers with known quotes. Simon (1972) and Khury

(1973) discussed the importance of maintaining

the logical consistency among various alter-

natives, especially on pricing. The fitted price

curve is logically consistent in two ways: (1) the

rate-on-line is strictly decreasing with reten-

tion and consistent with actual observations; (2)

for two adjacent layers, the sum of prices is

equal to the price of the combined layer, that is,

RP(R,L1 +L2) = RP(R,L1)+RP(R+L1,L2).

The minimum acceptable profit rate T and risk-

penalty coefficient μ vary by business mix and

by risk-bearing entity. In this case study, 0% is

selected as the minimum acceptable profit rate

for illustrative purposes. So, only underwriting

losses contribute to the risk calculation. For μ, we

use three values, 16.71, 22.28, and 27.85. Those

coefficients represent management’s willingness

22The curve fitting of reinsurance price quotes is discussed in Ap-

pendix. In practice, actuaries can select rate-on-line by judgment,

or fit their own curves by regression or use interpolation. In the

case study included in this paper, the three options do not produce

significantly different results.

to pay 30%,23 40%, and 50% of underwriting

profits to hedge downside risk, respectively. The

risk-penalty coefficients in the case study are se-

lected solely for illustrative purposes.

3.2. Numerical results

In the simulation, we generate catastrophe loss

and noncatastrophe loss for 10,000 years. In the

instant case, 397,257 catastrophe events are gen-

erated. Table 1 summarizes the losses covered by

reinsurance for quoted layers; Table 2 reports the

distribution summary of underwriting profit rates

net of reinsurance and the risk-adjusted profit

rates for quoted layers and their continuous com-

binations.

As illustrated in Table 1, a catastrophe loss has

a 10.18% chance to penetrate the retention level

of $305 million within one year. So, roughly in 1

of 10 years, the reinsured will obtain recoveries

by purchasing the reinsurance for this layer. The

higher the retention level, the lower the prob-

ability that the catastrophe loss penetrates the

layer. For example, the catastrophe loss has only

a 0.40% chance of penetrating a retention level

of $1,800 million.24 This is expected because the

frequency of a very large catastrophe loss is rel-

atively small. For the layer ($305 million, $420

million), the reinsurance price is $20.8 million,

while the mean of loss recovered from the rein-

23The mean profit and semivariance without any reinsurance are

3.93% and 0.07%, respectively. If a primary insurer would like to

use 30% of its gross profit to hedge downside risk, the risk penalty

coefficient is 16:71 = 3:93% ¤ 0:3=0:07%.24By using the catastrophe losses from AIR, the probability is

higher because the AIR models produce catastrophe losses with

larger tails than the fitted lognormal distribution.



Table 2. Distributional summary of underwriting profit rates for selected reinsurance layers when μ = 22:28

Retention Upper Limit Risk-adjusted(million) (million) Probr < 0% Probr <¡15% Mean Variance Semivariance25 Profit

No Reinsurance 18.41% 0.48% 3.916% 0.263% 0.070% 2.350%305 420 19.02% 0.42% 3.781% 0.253% 0.067% 2.291%420 610 19.17% 0.35% 3.771% 0.249% 0.064% 2.341%610 915 19.31% 0.30% 3.779% 0.247% 0.061% 2.412%610 1030 19.53% 0.27% 3.739% 0.243% 0.059% 2.428%

1030 1800 19.95% 0.26% 3.676% 0.243% 0.057% 2.397%1800 3050 20.44% 0.41% 3.551% 0.247% 0.061% 2.186%

305 610 19.63% 0.33% 3.637% 0.241% 0.061% 2.268%305 915 20.50% 0.25% 3.503% 0.228% 0.055% 2.287%305 1,030 20.76% 0.22% 3.465% 0.224% 0.053% 2.293%305 1,800 22.31% 0.13% 3.231% 0.210% 0.045% 2.231%305 3,050 24.77% 0.04% 2.869% 0.200% 0.042% 1.934%420 915 19.85% 0.25% 3.634% 0.235% 0.057% 2.373%420 1,030 20.06% 0.22% 3.595% 0.232% 0.054% 2.382%420 1,800 21.79% 0.14% 3.358% 0.216% 0.046% 2.330%420 3,050 24.25% 0.05% 2.995% 0.206% 0.043% 2.038%610 1,800 21.05% 0.16% 3.500% 0.226% 0.049% 2.402%610 3,050 23.35% 0.11% 3.135% 0.215% 0.045% 2.124%915 1,030 18.63% 0.40% 3.877% 0.258% 0.067% 2.380%915 1,800 20.14% 0.21% 3.637% 0.239% 0.055% 2.407%915 3,050 22.44% 0.17% 3.272% 0.226% 0.050% 2.155%

1030 3,050 22.15% 0.20% 3.311% 0.230% 0.052% 2.156%

680 1,390 20.00% 0.21% 3.667% 0.237% 0.055% 2.451%

Layers are rounded to 5 million.Layers below (1800, 3050) and above (680, 1390) are all the continuous combinations of quoted layers.

Figure 3. Reinsurance efficient frontier

25Semivariance using the original AIR catastrophe losses are larger

because the AIR models produce catastrophe losses with larger tails

than the lognormal distribution.

surance is $8.9 million. The ratio of reinsurance

recovery to reinsurance premium is 42.59%. The

reinsurance is costly, especially for the higher



Table 3. Optimal reinsurance layers when μ = 16:71, 22.28, 27.85

Theta Retention Upper Limit Risk-Adjusted Profit Risk-Adjusted Profit Risk-Adjusted ProfitTheta (million) (million) Mean Semivariance theta = 16:71 theta = 22:28 theta = 27:85

16.71 795 1220 3.771% 0.060% 2:768% 2.434% 2.100%22.28 680 1390 3.667% 0.055% 2.755% 2:451% 2.147%27.85 615 1460 3.610% 0.052% 2.736% 2.445% 2:154%

The optimal layers are rounded to 5 million.

layers. For the top layer ($1,800 million, $3,050

million), the reinsurance price is $39.1 million

while the mean of loss recovered by reinsurance

is $2.6 million. The ratio of recovery to premium

is 6.58%. So, the capital charge on the top layer

of reinsurance tower is very high.

Table 2 reports the probability of net under-

writing loss, the probability of severe loss (de-

fined as more than 15% of net underwriting loss),

mean of profit, variance of profit, semivariance

of profit, and risk-adjusted profits at μ =

22:28. The scattered dots (except for A, C, D, and

E) in Figure 3 represent the quoted reinsurance

layers and all possible continuous combinations

of those layers. A represents the no reinsurance

scenario and B represents the maximal reinsur-

ance scenario of stacking all quoted layers. The

slope from A to B becomes steeper and reflects

the reality of reinsurance pricing. The concave

curve in Figure 3 represents the efficient frontier.

Not unexpectedly, some of the quoted reinsur-

ance layers are not at the frontier: one can find

another layer to produce a higher return at the

same downside risk or a lower risk at the same

return. For example, layer ($305 million, $1,030

million) is not efficient with a mean profit and a

semivariance 3.465% and 0.053%, respectively.

Layer ($610 million, $1,800 million) is clearly

superior because it increases average return

(3.500%) while reducing risk (0.049%).

As shown in Table 2, the reinsured will maxi-

mize its downside-risk-adjusted profit by select-

ing the layer ($680 million, $1,390 million) as-

suming a 22.28 risk-penalty coefficient, which

implies that the management would be willing

to pay 1.567% of gross premium (40% of gross

underwriting profit) to hedge downside risk. For

a lower layer, even though the layer has a greater

chance to be penetrated, the potential risk of

catastrophe loss is tolerable by the reinsured. For

a higher layer, the reinsurance price is too high

compared to the risk mitigation it provides. Point

C in Figure 3 represents this optimization op-

tion. The straight tangent line represents the util-

ity curve at μ = 22:28. All other possible layers

are below the line and therefore have lower util-

ity values.

It is also clear from Table 2 that catastrophe

reinsurance does not increase the probability of

being profitable in the instant case. Without rein-

surance, the probability of underwriting loss is

18.41%. With reinsurance of various layers, the

probabilities of underwriting loss are over 19%.

The purpose of reinsurance is to buy protections

against large events. Without reinsurance, the

chance of severe loss is 0.48%, or roughly one

in 200 years. With a minimal reinsurance layer

($305 million, 420 million), it reduces to 0.42%,

or roughly one in 250 years. With the optimal

reinsurance layer ($680 million, $1,390 million),

the chance of severe loss reduces to 0.21%, or

roughly one in 500 years.

If the reinsured is less risk-averse, the opti-

mal layer will be narrower and the retention level

will be higher. As shown in Table 3, when μ =

16:71, or when the management would like to

pay 1.175% of gross premium (30% of total un-

derwriting profit) to hedge its downside risk, the

optimal layer is ($795 million, $1,220 million).

On the contrary, if the reinsured is more risk-

averse, the optimal layer will be wider and the

retention level will be lower. For example, when

μ = 27:85, or the management would like to pay

1.958% of gross premium (50% of total under-

writing profit) to hedge its downside risk, the



optimal layer is ($615 million, $1,460 million).

Point D in Figure 3 represents the optimization

at reduced risk aversion while Point E represents

the optimization at higher risk aversion.

In practice, actuaries may not be able to choose

reinsurance layers from an unlimited pool of op-

tions. They often need to select a layer or a com-

bination of layers from a limited number of op-

tions. A simple method is to calculate the risk-

adjusted profit for the candidate layers using

Equation (9) and select a layer associated with

the highest score. Layer ($610 million, $1,030

million) is the best of the six quoted options. In

this case, actuaries do not need to fit a nonlin-

ear curve on reinsurance prices and to solve the

complicated optimization problem.

The underwriting performance may impact the

reinsurance selection from two perspectives: (1)

the more profitable the business, the more risk

the insurer can retain, and the less reinsurance the

insurer may be willing to buy; (2) the more prof-

itable the business, the more capital can be de-

ployed for reinsurance, and the more reinsurance

the insurer is able to buy. The optimal reinsur-

ance layer, assuming a 3.93% gross underwrit-

ing profit rate with μ = 22:28, is ($680 million,

$1,390 million). If the company could make 2%

more underwriting profit by lowering its non-

catastrophe loss ratio, the reinsurance optimiza-

tions could be formularized by the following pa-

rameters:26

1. The benchmark point for minimum accept-

able profit may increase to 2%. In this case, the

semivariance will not be impacted by a profitabil-

ity change; the optimal layer remains the same as

($680 million, $1,390 million).

2. The minimum acceptable profit rate re-

mains at 0%. In this case, the semivariance re-

duces with improved profitability. The semivari-

ance decreases from 0.07% to 0.05% and down-

side deviation from 2.652% to 2.240%. This is

26None of the three parameterizations (1, 2a, or 2b) can fully re-

flect risk perception and risk aversion associated with improved

profitability. The true parameters are probably somewhere among

1, 2a, and 2b.

because smaller catastrophe events no longer

produce underwriting losses, and larger events

would produce 2% less loss.

(a) If the penalty on the semivariance remains

at 22.28, the optimal layer becomes ($740 mil-

lion, $1,420 million). The insurer would max-

imize its risk-adjusted profit by retaining more

loss from relatively small events (higher reten-

tion). This is because the reinsured has more

underwriting profit to cover smaller catastrophe

events. And by increasing the retention level, it

could have additional capital to buy more protec-

tion from a higher layer ($1,390 million, $1,420

million). The limit is reduced from $710 mil-

lion to $680 million because the downside risk

is smaller.

(b) If the reinsured would like to use the same

level of profit, or 1.567% of gross premium, to

fully hedge downside risk, μ would be 31.22. In

this scenario, the optimal layer is ($630 million,

$1,555 million). The reinsured would like to buy

a wider layer with a lower retention due to in-

creased risk-aversion (willing to pay the same

amount of price to hedge a semivariance that is

28.6% smaller than before).

4. ConclusionsWhen selecting reinsurance layers for catastro-

phe loss, the reinsured weighs two dimensions

in the decision making process: profit and risk.

The reinsured would give up too much of its

underwriting profit if purchasing excessive rein-

surance. On the other hand, the reinsured would

still be under the risk of large catastrophe losses

if carrying little reinsurance. This study explores

the determination of the optimal reinsurance

layer for catastrophe loss. The reinsured is as-

sumed to be risk-averse and chooses the rein-

surance layer that maximizes the underwriting

profit net of reinsurance adjusted for downside

risk. It provides a theoretical and practical model

under classical mean-variance framework to esti-

mate the optimal reinsurance layer. Theoretically,



the paper improves previous studies by utilizing

both catastrophe and noncatastrophe loss infor-

mation simultaneously and using the lower par-

tial moment to measure risk. Practically, the op-

timal layer is determined numerically by the risk

appetite of the reinsured, the reinsurance price

quotes by layer, and the loss (frequency and

severity) distributions of the business written by

the reinsured. The proposed approach uses three

parameters to reflect the insurer’s risk percep-

tion and risk aversion. T is the minimum accept-

able profit and the benchmark point to define

“downside.” The moment that represents one’s

risk perception of larger losses is k; the higher

the k, the greater the fear of severe losses. μ

is the risk-penalty coefficient which represents

one’s risk aversion. The higher the μ, the greater

the risk aversion to downside risks. The DRAP

(downside-risk-adjusted profit) framework pro-

vides a flexible approach to capture the insurer’s

risk appetite comprehensively and precisely. All

the information required by the model should be

readily available from catastrophe modeling ven-

dors and the actuarial database of an insurance

company.

Additionally, we would like to make the fol-

lowing concluding remarks:

1. From the perspective of enterprise risk

management (ERM), catastrophe insurance is a

risk management tool to mitigate one of many

risks faced by insurance companies. Catastro-

phe reinsurance should not be arranged or eval-

uated solely upon the information of catastrophe

losses. Instead, its arrangement should be viewed

concurrently with other types of risks, such as

noncatastrophe underwriting risk and various in-

vestment risks. In this paper we consider both

catastrophe and noncatastrophe losses and the

analysis (simulation) is carried out with both sets

of variables operating concurrently. This particu-

lar path views the catastrophe reinsurance cover

as a part of the ERM process. We believe the

decision reached by viewing the transaction as a

step in the ERM process to be a superior deci-

sion. This idea may be extended to all other el-

ements of reinsurance considered and/or utilized

by an insurer. In effect, our suggestion is that

the reinsurance decision, for catastrophe reinsur-

ance and otherwise, is an important element of

the total ERM process.

2. The reinsurance purchase decision is sel-

dom, if ever, guided solely by the dry mechanics

of pricing a layer above a particular attachment

point to pay a certain percentage of the covered

layer. An aspect of the transaction that goes be-

yond the mechanical factors deals with “who”

the prospective reinsurer is. This is an important

input item, but it is always an intangible. The size

of the reinsurer, the size of its surplus, the finan-

cial rating of the reinsurer, the length and quality

of the relationship with the reinsurer, how much

of the reinsurance is retained for its own account,

and so forth form important intangibles that are

impossible to factor into any simulation. All the

same, these factors do operate and they can in-

fluence the final decision.

3. Another aspect of the reinsurance decision

is the way the ultimate decision maker may be

able to use the outputs of modeling such as those

proposed in this paper. The models and their out-

put in effect provide the ultimate decision maker

with some absolute points of reference that can

be factored into the final decision. For exam-

ple, if the model results show a clearly economi-

cally advantageous reinsurance proposition is be-

ing offered, the ultimate decision maker now has

some “elbow room” to fully capitalize on the ad-

vantage that is being offered: he may seek to ex-

pand layers of coverage, extend the terms of cov-

erage, add additional reinstatement provisions,

and so on. On the other hand, if the proposed

reinsurance is particularly disadvantageous, the

ultimate decision maker also is well-armed to

seek alternatives that are consistent with his ap-

petite for risk: change the point of attachment,

change the size of the reinsured layer, seek out-



Table 4. Regression statistics

Variable Coefficients Standard Error t Stat

x2¡ x1 1.2300 0.0995 12.37x2

2¡ x21 1:2978 ¤10¡4 6:6023 ¤10¡6 19.66

x32¡ x3

1 ¡1:3077 ¤10¡8 6:0976 ¤10¡10 ¡21:45x2 log(x2)¡ x1 log(x1) ¡0:1835 0.0135 ¡13:56log(x2)¡ log(x1) 45.4067 3.935 11.54

side bids for the same coverage, and so on. In

all cases, the knowledge that is imparted from

these simulations to the ultimate decision maker

enhances his level of comfort with what is being

offered as well as with any final decision.

4. The downside risk measure and utility func-

tion (downside-risk-adjusted profit) in this study

can be adapted to analyze whether an insurance

company should write catastrophe exposures and

the design of the catastrophe reinsurance pro-

gram would be one component of such an analy-

sis. For example, if the profit in a property line is

not high enough to cover reasonable reinsurance

costs, or even negative, it is better to not write

that property line. The risk-adjusted profit of

Table 5. Fitted vs. actual prices¤

Upper Bound ofRetention Layer Reinsurance Limit Reinsurance Price Rate-on-line Fitted Price Fitted Rate-on-line

305 420 115 20.8 18.09% 20.84 18.12%420 610 190 21.7 11.42% 21.69 11.41%610 915 305 19.8 6.50% 19.87 6.51%610 1,030 420 25.2 5.99% 25.18 6.00%

1,030 1,800 770 28.7 3.72% 28.73 3.73%1,800 3,050 1,250 39.1 3.13% 39.10 3.13%

305 610 305 42.5 13.93% 42.52 13.94%305 915 610 62.3 10.22% 62.39 10.23%305 1,030 725 67.7 9.33% 67.70 9.34%305 1,800 1,495 96.5 6.45% 96.43 6.45%305 3,050 2,745 135.6 4.94% 135.53 4.94%420 915 495 41.5 8.39% 41.55 8.39%420 1,030 610 46.9 7.68% 46.87 7.68%420 1,800 1,380 75.6 5.47% 75.60 5.48%420 3,050 2,630 114.7 4.36% 114.69 4.36%610 1,800 1,190 53.9 4.53% 53.91 4.53%610 3,050 2,440 93.0 3.81% 93.01 3.81%915 1,030 115 5.3 4.64% 5.32 4.62%915 1,800 885 34.0 3.85% 34.04 3.85%915 3,050 2,135 73.1 3.42% 73.14 3.43%

1,030 3,050 2,020 67.8 3.36% 67.83 3.36%

*The actual prices below layer (1800, 3050) are derived by combining the six layers with known prices.

the primary insurer without the property line will

be higher than that by adding the line and buy-

ing the optimal catastrophe reinsurance cover. In

reality, a property line may not be profitable and

it may not be a viable option to completely exit

or even shrink the line. Under the scenario of un-

profitable property lines with predetermined ex-

posures, the proposed method can still help the

primary insurer to find an optimal reinsurance

solution and to mitigate severe downside pains

from property lines by giving up a portion of

profit from other profitable lines.

5. Uncertainty in modeling and estimating net

underwriting profit is an important consideration.

Model and parameter risks inherent in catastro-

phe loss simulations can influence actual vs. per-

ceived costs as well as the optimal amount of

capacity companies choose to buy.

Finally, there is no question that, when all is

said and done, the ultimate decision maker has to

weigh many things, both objective and subject-

ive, on the way to finalizing the reinsurance de-



cision. Having the results of the simulations pre-

sented in this paper serves to improve the quality

of decision making.

Appendix: Fitting the ReinsurancePrice CurveIn calculating catastrophe reinsurance rates,

the premium and rate-on-line are associated with

two values: the starting and end points of a layer.

When the layer is infinitesimal, rate-on-line can

be thought as a function of the midpoint of the

layer. In other words, rate-on-line in a continu-

ous setting is f(x), a function with a single vari-

able. Let p(x1,x2) be the reinsurance rate with

retention at x1 and upper limit of the layer at x2,

then

p(x1,x2) =

Z x2

x1f(x)dx: (A.1)

Because the reinsurance rate is expressed as

the integral, it meets the addable requirement

of reinsurance pricing: p(x1,x3) = p(x1,x2)+

p(x2,x3). From actual quotes, it is clear that f(x)

is a decreasing nonlinear function of x. To cap-

ture the nonlinearity, we assume that f(x) con-

tains a quadratic term x2, and logarithm log(x),

and inverse term 1=x in addition to a linear

term x.

f(x) = ¯0 +¯1x+¯2x2 +¯3 log(x)+¯4x

¡1:

(A.2)27

Combining A.1 and A.2, the reinsurance rate is

p(x1,x2) = ¯0(x2¡ x1)+ 12¯1(x

22¡ x21)

+ 13¯2(x

32¡ x31)

+¯3(x2 log(x2)¡ x1 log(x1))

+¯4(log(x2)¡ log(x1)): (A.3)

By Equation (A.3), one could fit a linear re-

gression with quotes as observations of the de-

pendent variable, p, and x2¡ x1, x22¡ x21, x32¡x31, x2 log(x2)¡ x1 log(x1), and log(x2)¡ log(x1)

27One could also consider adding other polynomial terms.

as corresponding observations of the indepen-

dent variables.28 There are 21 observations in Ta-

ble 5 and five possible variables in the regression.

To avoid over-fitting problem associated with re-

gression, we selected the model with the lowest

value of BIC (Bayesian information criterion).

Another way to minimize over-fitting is to obtain

more quotes to increase the number of observa-

tions. Finally, actuaries should review the shape

of fitted reinsurance curve to check its reason-

ableness. The regression results are reported in

Table 4.

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