MODELLING ADVERSE SELECTION IN THE PRESENCE OF A

MODELLING ADVERSE SELECTION IN THE PRESENCE OF ACOMMON GENETIC DISORDER: THE BREAST CANCER POLYGENE

BY

ANGUS S. MACDONALD AND KENNETH R. MCIVOR

ABSTRACT

The cost of adverse selection in the life and critical illness (CI) insurance markets,brought about by restrictions on insurers’ use of genetic test information, hasbeen studied for a variety of rare single-gene disorders. Only now do wehave a study of a common disorder (breast cancer) that accounts for the riskassociated with multiple genes. Such a collection of genes is called a polygene.We take two approaches to modelling the severity of adverse selection whichmay result from insurers being unable to take account of tests for polygenesas well as major genes. First, we look at several genetic testing scenarios, witha corresponding range of possible insurance-buying behaviours, in a marketmodel for CI insurance. Because a relatively large proportion of the populationis exposed to adverse polygenic risk, the costs of adverse selection are poten-tially much greater than have been associated with rare single genes. Second,we use utility models to map out when adverse selection will appear, and whichrisk groups will cause it. Levels of risk aversion consistent with some empiri-cal studies do not lead to significant adverse selection in our model, but lowerlevels of risk aversion could effectively eliminate the market.

KEYWORDS

Adverse Selection; BRCA1; BRCA2; Breast Cancer; Critical Illness Insurance;Polygene.

1. INTRODUCTION

1.1. Major Genes and Polygenes

The link between high risks of breast cancer (BC) and ovarian cancer (OC),and rare mutations in either of the BRCA1 and BRCA2 genes, is well-estab-lished. Several actuarial studies (Lemaire et al., 1999; Subramanian et al., 2000;Macdonald, Waters & Wekwete, 2003a, 2003b; Gui et al., 2006) have consideredthe implications for the markets in life insurance and critical illness (CI) insurance

Astin Bulletin 39(2), 373-402. doi: 10.2143/AST.39.2.2044640 © 2009 by Astin Bulletin. All rights reserved.

(known as dread disease insurance in continental Europe). Single genes asso-ciated with greatly increased disease risk are called ‘major genes’. On the whole,these studies have found that insurance premiums for mutation carriers maybe greatly elevated, but that banning the use of genetic test results would notlead to significant costs arising from adverse selection, because mutations areso rare. Moreover, insurers may often still have regard to family history.

More recently a different type of genetic risk called ‘polygenic’ has beenassociated with BC. What we will call a ‘polygene’ is a collection of genes, eachwith several variants, called ‘alleles’, not necessarily rare. Adverse configura-tions of a polygene might confer susceptibility to a particular disease, or beneficialconfigurations might protect against it. These outcomes could also be stronglyinfluenced by the environment. It is likely that we all carry some ‘good’ andsome ‘bad’ polygene configurations, but this is quite speculative at this stage.

Genetic science is moving on, from understanding single-gene disorders tobeginning to understand polygenic disorders. We highlight some key differ-ences between the two, that could affect questions of insurance.

(a) Mutations in major genes are rare, but polygenes will be present (in alltheir varieties of configurations) in every person. So, instead of tiny numbersof people with very high risks, larger numbers with modestly increased risksmay contribute to the cost of adverse selection.

(b) Genetic testing for major gene mutations is, in many countries, controlledby public health services and is only offered if family history suggests amutation is present. Polygenic risk, however, need not be associated withany clear family history, so genetic testing for risky polygenes will be initi-ated in some other way, such as screening programmes or selling direct tothe public.

That said, research is at a very early stage and genetic testing for risky polygenesis not yet feasible.

1.2. The Polygenic Breast Cancer Model of Antoniou et al. (2002)

Antoniou et al. (2002) estimated rates of onset of BC and OC in a modelassuming the presence of: (a) rare mutations in the major genes BRCA1 andBRCA2; and (b) a polygene affecting BC risk only. The polygene was modelledas a collection of three genes, each with a protective and a deleterious allele.Since everyone possesses two copies of every nuclear gene (ignoring sex chro-mosomes) each person carries six of these alleles. The quantitative effect of thepolygene was governed by a number called the ‘polygenotype’, denoted P,which was the sum of the contributions of each allele: an adverse allele con-tributed +1/2, and a protective allele contributed –1/2. Thus P was an integerbetween –3 (beneficial) and +3 (adverse). P was assumed to act multiplicativelyon the onset rate of BC as follows:

Onset Rate for Polygenotype P = Baseline Onset Rate ≈ exp(cP) (1)

374 A.S. MACDONALD AND K.R. MCIVOR

where the baseline onset rate is that for P = 0, and the constant c is just a scalefactor. Assuming each allele to be equally common, and inherited independentlyof the others, the distribution in the population of the quantity (P + 3) is Bino-mial(6,1/2).

Macdonald & McIvor (2006) used this polygenic model to estimate criticalillness (CI) insurance prices. They considered the influence of the polygene onthe development of family histories of BC/OC, and their use in underwriting.

1.3. Multi-State Market Models for Adverse Selection

Our aim is to study the impact of the polygenic model on adverse selection inthe CI insurance market. Adverse selection arises when an individual is ableto obtain insurance without having to disclose all the information that theinsurer requires to price the contract accurately, so the applicant obtains coverbelow its full cost. An obvious first approach is to adapt the models used toaddress this question in respect of single-gene disorders. These are Markovmulti-state models of the life history of a person who starts healthy and unin-sured, and who may pass through a series of states representing relevant events,such as acquiring a family history, taking a genetic test, buying CI insurance,and suffering onset of BC/OC. By allowing some transition intensities to dependon the person’s current knowledge, we can model how that influences variousdecisions. By basing premium rates on the insurer’s lack of knowledge, we canmodel their exposure to adverse selection. The outcomes translate directly intothe uniform increase in premiums that would be needed to recoup the cost ofthe adverse selection, a simple and relevant measure. We do this in Section 2,where a new consideration is the different routes to genetic testing that maycome to be associated with polygenes.

1.4. An Economic Framework for Adverse Selection

An acknowledged weakness of the multi-state market model described aboveis its lack of economic rationale. That is, it posits certain behaviour (adverseselection) on the part of market participants. This behaviour changes the priceof CI policies. There it ends, in our model, but an economist would iterate theprocess, price and behaviour changing each other, until (or if) equilibrium wasreached.

It is hard to incorporate equilibrium prices into the Markov market modeldirectly. Other approaches exist, considering the insurance-buying decision ata fixed time, setting the applicant’s knowledge (and utility function) against theinsurer’s ignorance (and pooling instincts), and seeing whether insurance wouldbe purchased, by whom, and perhaps also how much (Hoy & Witt, 2007; Mac-donald & Tapadar, 2008). Using the polygenic BC model, we implement thisapproach in Section 3. We discuss the different insights that the two approachesbring, and draw conclusions, in Section 4.

THE BREAST CANCER POLYGENE 375

2. MULTI-STATE MARKET MODELS FOR ADVERSE SELECTION

2.1. The Basic Market Model

We begin with the model in Figure 1. Each genotype is represented by a versionof this model, with different rates of onset of BC and OC. The model representsthe life history of a person, as yet uninsured, who may buy insurance beforeor after having a genetic test. Premiums are payable while in either of the insuredstates, and the benefits are payable on transition from either of these statesinto a ‘critical illness’ state (which represents the onset of BC, OC, or anothercritical illness).

Some of the assumptions we make are as follows:

(a) Large and small markets are represented by ‘normal’ rates of insurance pur-chase of 0.05 or 0.01 per annum, respectively.

(b) In both markets, low risk polygenotype carriers may buy less insurancethan the ‘normal’ rate. These carriers purchase at the normal rate, half ofthe normal rate, or not at all.

(c) Genetic testing occurs at three possible rates per annum: 0.02972 (low),0.04458 (medium), or 0.08916 (high), based on an uptake proportion of 59%(Ropka et al., 2006) over a period of 30, 20, or 10 years of testing respec-tively. Also, testing may only occur between ages 20 and 40 (when it has highpriority).

(d) ‘Severe’ adverse selection means that high-risk polygenotype carriers willpurchase insurance at rate 0.25 per annum.


1UninsuredUntested

2Insured

Untested

5Critical Illness

6Dead

4Uninsured

Tested

3InsuredTested

FIGURE 1: A model of the behaviour of a genetic subpopulation with respect to the purchasing of CIinsurance. Genetic testing is available at an equal rate to all subpopulations.

All other intensities, governing transitions into the ‘Dead’ and ‘Critical Illness’states, are as in Macdonald & McIvor (2006) and are omitted for brevity.NPEPVs of benefits and premiums are found by solving Thiele’s differentialequations backwards numerically with force of interest d = 0.05. Occupancyprobabilities are found by solving Kolmogorov’s Forward Equations. For both,we use a step-size of 0.0005 years.

2.2. A Genetic Screening Programme for the Polygene Only

For simplicity, the first possibility we consider is that a genetic screening pro-gramme exists for the polygenotype only, not extending to the BRCA1/2 geno-types. There are seven polygenotypes, therefore a 42-state model. We assumethat the distribution of new-born persons in the seven sub-populations is Bino-mial(6,1/2), and that mortality and morbidity before age 20 does not dependon genotype (so that the expected proportions in each starting state at age 20have not changed). Since the rate of BC onset is negligible before about age 30,this assumption seems reasonable.


Low Risk High Risk

(a) –3 –2 –1 0 +1 +2 +3

Buy Less Insurance Buy More Insurance

(b) –3 –2 –1 0 +1 +2 +3


(c) –3 –2 –1 0 +1 +2 +3


FIGURE 2: Three possible behaviours of tested polygenotype carriers in the adverse selection model,labelled (a), (b) and (c).

Tested carriers may alter their insurance-buying habits, in one of two ways:carriers of deleterious polygenotypes may buy more insurance, or carriers ofprotective polygenotypes may buy less insurance. This latter behaviour isuncommon in adverse selection studies; it is usually assumed that individualswho receive negative test results for rare, severe, mutations will purchase insur-ance at the normal market rate. One study (Subramanian et al., 1999) performeda sensitivity analysis where tested non-carriers could reduce their coverage.It is plausible that this makes more sense from an economic point of view.Figure 2 shows three scenarios of differing severity.

The percentage by which all premiums must be raised in order to negatethe adverse selection costs is:

(EPV[Loss|Adverse Selection] – EPV[Loss|No Adverse Selection]100 ≈ ( ) . (2)

EPV[Premium Income|Adverse Selection]

Table 1 shows the premium increases needed to absorb the costs of the adverseselection under each scenario in Figure 2. Compared with previous results basedon major genes only (Gui et al., 2006; Gutierrez & Macdonald, 2007) these arevery high. This is because deleterious polygenotypes are more common thanmajor gene mutations, and also because these authors did not have to considerthe possibility that carriers of beneficial genotypes would buy less insurance.Note the large fall in costs between Scenarios (b) and (c). In Scenario (c), adverseselection is confined to the tails of the Binomial distribution of polygenotypes.

Curiously, in a small market the cost of adverse selection is always higherin Scenario (b) than in (a). This is because premium increases are relative to abaseline ‘ordinary’ rate (OR rate), which is higher in Scenario (a) than in (b).


TABLE 1

COSTS OF ADVERSE SELECTION RESULTING FROM HIGH RISK POLYGENOTYPE CARRIERS

BUYING MORE INSURANCE THAN LOW RISK POLYGENOTYPE CARRIERS IN A CRITICAL ILLNESS

INSURANCE MARKET OPEN TO FEMALES BETWEEN AGES 20-60.SCREENING AVAILABLE FOR THE POLYGENE ONLY.

Genetic Market Insurance Purchasing of Premium Increase in ScenarioTesting Size Low Risk Polygenotypes (a) (b) (c)

% % %

Low Large Normal 1.05975 0.90051 0.26447Half 1.69947 1.42748 0.30825Nil 2.86994 2.36206 0.38349

Small Normal 6.81382 7.03421 1.03701Half 7.72964 7.90872 1.10420Nil 8.86328 8.94001 1.17995

Medium Large Normal 1.39994 1.20315 0.29909Half 2.27895 1.93093 0.35897Nil 3.95151 3.25381 0.46294


High Large Normal 2.01615 1.77037 0.36453Half 3.40261 2.92799 0.45793Nil 6.32401 5.15661 0.62418


We repeated the experiment for the case where high-risk polygenotype car-riers purchase insurance at the normal rate, hence they do not contribute toadverse selection. However low-risk polygenotype carriers may still modifytheir purchasing behaviour by purchasing at half the normal rate or not pur-chasing at all. Table 2 shows the effect of such behaviour on the cost of adverseselection. (When the low-risk polygenotype carriers purchase at the normalrate, there is no adverse selection.)

2.3. A Genetic Screening Programme for the Polygene and Major Genes

Now we consider the possibility that screening is available for the major BRCA1and BRCA2 genotypes, as well as for the polygenotype. We have 3 ≈ 7 = 21 dis-tinct genotypes, and 126 states in the model. We assume population frequenciesof BRCA1 and BRCA2 mutations of 0.0010181 and 0.0013577 respectively (Anto-niou et al., 2002).

We consider the same adverse selection scenarios as in Figure 2, but addi-tionally those who carry adverse BRCA1/2 mutations exercise selection (that


TABLE 2

COSTS OF ADVERSE SELECTION RESULTING FROM LOW RISK POLYGENOTYPE CARRIERS BUYING LESS INSURANCE

THAN NORMAL IN A CRITICAL ILLNESS INSURANCE MARKET OPEN TO FEMALES BETWEEN AGES 20-60.HIGH RISK POLYGENOTYPE CARRIERS BUY INSURANCE AT NORMAL RATE.

SCREENING AVAILABLE FOR THE POLYGENE ONLY.


% % %







is to say, they decide to insure, without sharing their knowledge of their ele-vated risk) regardless of polygenotype. The resulting premium increases are shownin Table 3. They are not much larger than those in Table 1, the greatest increasebeing in Scenario (c). Compared with screening for the polygene alone, theadverse selection costs if screening is extended to BRCA1/2 mutations are nothigh.

We also considered the possibility that some BRCA1/2 mutation carriers donot buy more insurance, if they carry a protective polygenotype which ‘voids’the BRCA1/2 risk. It was shown in Macdonald & McIvor (2006) that BRCA1/2mutation carriers with P = –3 could plausibly obtain CI insurance at ordinaryrates. The effect was very small, and we omit the results.

2.4. More Limited Genetic Testing for the Polygene and Major Genes

Only about 25% of BC cases are hereditary and only about 20% of these arecaused by identifiable (major) genes, so mass screening programmes for thesewould be ineffective. Much more likely is that testing will continue to be offered


TABLE 3

COSTS OF ADVERSE SELECTION RESULTING FROM HIGH RISK POLYGENOTYPE CARRIERS BUYING MORE

INSURANCE THAN LOW RISK POLYGENOTYPE CARRIERS IN A CRITICAL ILLNESS INSURANCE MARKET OPEN TO

FEMALES BETWEEN AGES 20-60. SCREENING AVAILABLE FOR MAJOR GENES AND THE POLYGENE.


% % %







only to women who present a relevant family history of BC/OC, who aremore likely to carry deleterious genes. Therefore we adjust our original modelas shown in Figure 3, to include the development of a relevant family history,which is now taken to be a prerequisite for genetic testing.


FIGURE 3: A model of the behaviour of a genetic subpopulation with respect to the purchasing of CIinsurance. Genetic testing is available only after the appearance of a relevant family history (FH) of BC/OC.

1 UninsuredUntestedNo FH

2 InsuredUntestedNo FH

7 CriticalIllness

8Dead

4 InsuredUntested

FH

6 InsuredTested

FH

5 UninsuredTested

FH

3 UninsuredUntested

FH

We define a ‘relevant’ family history to mean that a healthy woman has twoor more female first-degree relatives (FDRs) who contracted BC/OC beforeage 50. (FDR normally means parents and siblings but in what follows welimit it to mother and sisters.) The incidence of a relevant family history wascalculated using the formula common in epidemiology:

Number of new cases arising in specified time periodIncidence Rate = . (3)

Number of individuals at risk during the time period

In order to be categorised as ‘at risk’ of a relevant family history developinga woman must be healthy, with either: (a) no FDRs affected before age 50 andat least two unaffected FDRs under age 50; or (b) one FDR affected beforeage 50 and at least one unaffected FDR under age 50. If a relevant family his-tory develops, each healthy daughter contributes as a ‘new case’ in the incidenceat that time. These rates were estimated by simulation in Macdonald & McIvor(2006). The incidence rate is shown in Figure 4 for the subpopulations with-out BRCA mutations in the family, and in Figure 5 for the BRCA1 and BRCA2carrier families. Since we assume all siblings to be the same age, no relevantfamily history can develop after age 50.

FIGURE 4: The incidence of a relevant family history for the subpopulationswithout BRCA mutations.

FIGURE 5: The incidence of a relevant family history for the subpopulationswith BRCA1/2 mutations in the family.


The rate of genetic testing per annum among women who have developed a rel-evant family history we take to be 0.04012 (low), 0.06020 (medium) or 0.12040(high), based on a proportion of 70% (Ropka et al., 2006) being tested over a30, 20 or 10 year period, respectively.

The results are in Table 4. The costs of adverse selection are greatly reducedwhen a relevant family history is a prerequisite for a genetic test. Once againa small insurance market suffers higher relative costs.

2.5. Separate Testing for the Polygene and Major Genes

Perhaps a more realistic situation is that testing for major genes is conductedthrough a public health service, once a relevant family history has signalled therisk, and testing for the polygene may be sought privately (by asymptomaticindividuals). Therefore, we have two different testing events: one for the BRCA1/2genes and one for the polygene, leading to the model shown in Figure 6. Boththe family-history related and the non-family-history related testing rates maybe at the low, medium or high levels.


TABLE 4


INSURANCE THAN LOW RISK POLYGENOTYPE CARRIERS IN A CRITICAL ILLNESS INSURANCE MARKET

OPEN TO FEMALES BETWEEN AGES 20-60. TESTING AVAILABLE FOR MAJOR GENES AND THE POLYGENE

AFTER THE ONSET OF A RELEVANT FAMILY HISTORY.


% % %








FIGURE 6: A model of the behaviour of a genetic subpopulation with respect to the purchasing of CIinsurance. Genetic testing for major genes (MG) is available only after the appearance of a relevant familyhistory (FH) of BC/OC. Testing for the polygene (P) is available before a relevant family history has appeared.

7 UninsuredTested (P)

No FH

1 UninsuredUntestedNo FH

3 UninsuredUntested

FH

5 UninsuredTested (MG)

FH

8 InsuredTested (P)

No FH

2 InsuredUntestedNo FH

4 InsuredUntested

FH

9 CriticalIllness

10Dead

6 InsuredTested (MG)

FH

TABLE 5


INSURANCE THAN LOW RISK POLYGENOTYPE CARRIERS IN A CRITICAL ILLNESS INSURANCE MARKET OPEN TO

FEMALES BETWEEN AGES 20-60. SEPARATE TESTING FOR POLYGENE AND MAJOR GENES.


% % %








Our results, in Table 5, show somewhat smaller costs than the ‘combined test-ing’ model (Table 3). In fact, the costs are close to those of the polygene-onlyscreening programme (Table 1).

We have so far assumed that ‘severe’ adverse selection takes place, definedas a rate of insurance purchase of 0.25 per annum by carriers of high-riskpolygenotypes and BRCA1/2 mutations. We assume a less severe rate of adverseselection to be 0.1 per annum. Table 6 gives the costs of adverse selection whenwe apply this more modest rate of adverse selection. Note that the costs arestill very high in the small market and in general the reduction in costs is small.

3. AN ECONOMIC FRAMEWORK FOR ADVERSE SELECTION

3.1. Utility Functions

A weakness of the models just described, shared by the models used by pre-vious authors, is the lack of any economic rationale for the insurance-buyingdecisions. In particular, adverse selection is assumed to cause premium rates

TABLE 6


INSURANCE THAN LOW RISK POLYGENOTYPE CARRIERS IN A CRITICAL ILLNESS INSURANCE MARKET

OPEN TO FEMALES BETWEEN AGES 20-60. SEPARATE TESTING FOR POLYGENE AND MAJOR GENES.MODEST ADVERSE SELECTION.


% % %







to change, which ought in turn to affect insurance-buying decisions — the startof the classic ‘adverse selection spiral’. Without pursuing this sequence of priceand behavioural changes, we cannot be sure that premiums will eventuallyreach an equilibrium close to the changes suggested above. The usual approachto studying market equilibria starts with utility functions. It is hard to intro-duce these fully in the Markov models just described, but we can, nevertheless,use them to describe limits on the behaviour of market participants.

The utility function U(w), with U�(w) > 0 and U�(w) < 0, can be interpretedas an increasing concave relation that describes the relative satisfaction gainedfrom holding wealth w. Macdonald & Tapadar (2009) parameterised four utilityfunctions, three from the Iso-Elastic family and one from the Negative Exponen-tial family. We use the same utility functions and will refer to them as Models 1,2, 3 and 4, as shown in Table 7. Models 1 and 2 have low risk-aversion.Models 3 and 4 were parameterised using data from a 1995 Italian thought-experiment (Eisenhauer & Ventura, 2003), adjusted for the sterling/lira exchangerate and UK price inflation up to 2006, and have higher risk-aversion.

We assume that a risk-averse individual facing uncertainty will seek to max-imise his or her expected utility. For example, suppose an individual with totalwealth W faces a loss L with probability q. The actuarial value (or fair value)of insurance against this random event is qL, but the individual would be pre-pared to pay premium PL $ qL if:

U (W – P L) > qU (W – L) + (1 – q)U(W ). (4)

The premium per unit of loss P* at which an individual would no longer pur-chase insurance is found by converting the inequality in Equation (4) to anequality and solving it.

Now suppose the population is stratified into separate subpopulations,within each of which the loss probability is different. Suppose that individu-als are able to discover which stratum they are in, for example, by genetic test-ing. If the insurer charges everyone the same rate of premium, persons in eachstratum will decide whether or not to insure, using their own level of risk.If the premium rate is high enough, low-risk individuals will leave the market.This is the boundary at which adverse selection begins to affect the market.


TABLE 7

THE FOUR UTILITY FUNCTIONS PARAMETERISED BY MACDONALD & TAPADAR (2009).

Family Utility Function Parameter Model

Iso-Elasticl = 0.5 1!( ) ( ) / <

( )logU w w

wandl l l

l1 1 0

0

l

=-

=* l = 0 2

l = –8 3

Negative Exponential U(w) = – exp(–Aw) A = 9 ≈ 10–5 4

Macdonald & Tapadar (2009) explored this aspect of adverse selectionusing a hypothetical genetic model with two genotypes interacting with twolevels of an environmental factor. We can now extend their study, to the morerealistic setting of BC/OC and a real parameterised model.

3.2. Critical Illness Insurance Premiums

We will use the CI model in Figure 7 to calculate single premiums for stand-alone CI policies. We do not consider CI policies sold as riders to life insur-ance (so-called ‘accelerated’ benefits) because of the need to model survival afteronset; this would be useful future work.

In the CI model a unit sum assured is payable on transition from the Healthystate to any CI state (BC, OC or Other Critical Illness). For simplicity, and con-sistency with previous studies of insurance and utility (Hoy & Witt, 2007;Macdonald & Tapadar, 2009), let the force of interest be d = 0. This meansthat EPVs are equivalent to the probabilities of the CI event occurring.


FIGURE 7: A model of the life history of a critical illness insurance policyholder, beginning in theHealthy state. Transition to the non-Healthy state d at age x is governed by an intensity md(x) depending

on age x or, in the case of BC and OC, mgd(x) depending on genotype g as well.

2Breast Cancer

3Ovarian Cancer

1Healthy

4Other Critical Illness

5 Dead

mgBC(x)

mgOC(x)

mOCI(x)

mD(x)

For convenience, introduce genotype BRCA0 to represent non-carriers ofBRCA1/2 mutations. Let (P,M ) denote the genotype of a woman with poly-genotype P and major genotype BRCAM, and let P(P,M ) represent the CIsingle premium per unit benefit she would be charged, for a given entry age andpolicy term.

3.3. Threshold Premiums

For a given entry age and policy term, let P be the single premium per unitbenefit offered to all women when the insurer has no information regardinggenotype. Thus P is the weighted average premium:

P = , ,P M P Mw P,P M! ] ]g g (5)

where w(P, M) is the proportion of the population with genotype (P, M). Theseweighted average premiums are given in Table 8. Insurance will be bought bywomen with genotype (P, M) if P # P*(P, M) where P*(P, M) is the solution of:

U (W – P*(P, M )L) = P(P, M )U(W – L) + (1 – P (P, M ))U(W ). (6)

We call P*(P, M ) the threshold premium for the onset of adverse selection inrespect of genotype (P, M ). Clearly, adverse selection will appear first whenP > P*(–3,0). Tables 9 and 10 show the values of P*(–3,0) for a selection ofCI policies and a range of losses L when initial wealth W = £100,000. We cansee that as the ratio L/W of loss to wealth increases, the threshold premiumincreases, implying greater propensity to insure more serious losses.


TABLE 8

THE WEIGHTED AVERAGE SINGLE PREMIUM FOR VARIOUS CI POLICIES (SEE EQUATION (5)).

Age Term Rate of Premium PP

20 10 0.0067020 0.0308130 0.0954940 0.20315

30 10 0.0243620 0.0896930 0.19840

40 10 0.0678220 0.18009

50 10 0.12331

From Tables 9 and 10 we can roughly deduce what level of loss ratio L/W willinitiate adverse selection. These ratios are about 0.85 for Model 1 and 0.55 forModel 2. However for Models 3 and 4, P < P*(–3,0) for almost all levels ofloss L we have tabulated. By replacing P*(P, M) with P in Equation (6) andsolving for L with genotype (–3,0), we can find the levels of loss that wouldinitiate adverse selection. These are given in Table 11. We can see that adverseselection could occur under Models 3 and 4, but only for very low levels ofinsured loss relative to wealth (for which CI insurance is certainly unnecessary).


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10.

0327

60.

0367

80.

0428

30.

0540

730

0.06

652

0.07

018

0.07

447

0.07

960

0.08

591

0.09

398

0.10

490

0.12

115

0.15

079

400.

1545

60.

1622

60.

1712

20.

1818

50.

1948

00.

2111

50.

2329

40.

2646

90.

3205

930

100.

0168

10.

0177

90.

0189

30.

0203

10.

0220

20.

0242

10.

0272

10.

0317

20.

0401

220

0.06

093

0.06

430

0.06

826

0.07

299

0.07

882

0.08

627

0.09

636

0.11

141

0.13

892

300.

1497

70.

1572

70.

1660

10.

1763

80.

1890

10.

2049

80.

2262

70.

2573

40.

3121

440

100.

0450

60.

0475

90.

0505

70.

0541

40.

0585

50.

0641

90.

0718

50.

0833

20.

1044

220

0.13

579

0.14

270

0.15

076

0.16

034

0.17

203

0.18

684

0.20

662

0.23

560

0.28

698

5010

0.09

583

0.10

094

0.10

691

0.11

403

0.12

277

0.13

389

0.14

885

0.17

098

0.21

086

Model 1 Model 2


TA

BL

E 1

0

TH

RE

SHO

LD

PR

EM

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RA

TE

SP

* (–3

,0)

AT

WH

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W=

£100

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.

Los

s to

Wea

lth

Rat

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rm0.

10.

20.

30.

40.

50.

60.

70.

80.

9

2010

0.00

959

0.01

778

0.03

769

0.09

005

0.21

518

0.41

504

0.61

436

0.77

441

0.89

972

200.

0352

60.

0633

90.

1239

50.

2432

60.

4179

70.

5942

60.

7369

10.

8463

90.

9317

230

0.10

009

0.16

789

0.28

336

0.44

015

0.59

803

0.72

657

0.82

357

0.89

704

0.95

424

400.

2208

40.

3322

40.

4745

80.

6170

00.

7336

10.

8204

60.

8843

50.

9325

20.

9700

130

100.

0261

10.

0474

30.

0952

70.

1980

00.

3674

20.

5540

60.

7101

40.

8307

10.

9247

620

0.09

200

0.15

558

0.26

661

0.42

239

0.58

347

0.71

628

0.81

688

0.89

313

0.95

250

300.

2145

70.

3244

70.

4665

80.

6103

10.

7287

10.

8171

10.

8821

80.

9312

60.

9694

5 40

100.

0687

10.

1190

50.

2140

70.

3630

70.

5329

30.

6801

40.

7933

40.

8793

80.

9463

920

0.19

610

0.30

114

0.44

206

0.58

947

0.71

337

0.80

661

0.87

540

0.92

730

0.96

769

5010

0.14

167

0.22

829

0.35

984

0.51

576

0.65

780

0.76

832

0.85

064

0.91

285

0.96

127

2010

0.00

941

0.01

611

0.02

880

0.05

236

0.09

286

0.15

297

0.22

655

0.30

203

0.37

102

200.

0345

90.

0576

70.

0973

70.

1587

90.

2387

70.

3247

30.

4043

70.

4723

70.

5285

830

0.09

825

0.15

418

0.23

365

0.32

782

0.42

065

0.50

101

0.56

629

0.61

830

0.65

991

400.

2171

10.

3098

00.

4132

60.

5096

30.

5889

90.

6507

60.

6982

40.

7351

00.

7642

230

100.

0256

10.

0430

90.

0741

60.

1250

40.

1965

50.

2793

60.

3603

60.

4316

70.

4915

520

0.09

030

0.14

272

0.21

875

0.31

112

0.40

414

0.48

585

0.55

273

0.60

623

0.64

910

300.

2109

30.

3023

40.

4054

00.

5022

40.

5824

40.

6450

60.

6932

70.

7307

10.

7603

140

100.

0674

30.

1088

50.

1729

00.

2573

80.

3493

00.

4345

10.

5064

20.

5648

30.

6119

820

0.19

272

0.27

997

0.38

148

0.47

950

0.56

218

0.62

737

0.67

781

0.71

708

0.74

815

5010

0.13

913

0.21

081

0.30

338

0.40

215

0.49

159

0.56

506

0.62

310

0.66

872

0.70

499

Model 3 Model 4

3.4. Parameterisation of the Polygenic Model

The heterogeneity of risk introduced by the polygene, therefore the potentialfor adverse selection, is directly related to the standard deviation of the expo-nent cP in Equation (1). Denote this standard deviation sR, which Antoniouet al. (2002) estimated to be 1.291. Here we ask, what would be the effect ofa larger or smaller sR?

For larger sR, persons with the (–3,0) genotype have lower relative risk andhence value insurance less, so we expect adverse selection to appear more read-ily. On the other hand, if sR = 0 there would be no adverse selection on accountof the polygene, so as sR " 0 there may sometimes be a non-zero value of sR

at which adverse selection disappears. The major genes still play a role ofcourse, but here they constitute a fixed background.

We will call the levels of sR at which adverse selection appears or disappearsthreshold standard deviations and denote them s*

R. They will depend on theentry age, policy term, utility model, wealth W and loss L. Table 12 showsvalues of s*

R. Boldface indicates values that are less than the actual estimatesR = 1.291, meaning that we should expect adverse selection to appear, giventhe model of Antoniou et al. (2002). In agreement with Section 3.3, higherloss ratios L/W mean individuals with genotype (–3,0) would be less motivatedto insure, so s*

R increases with L.The missing figures in Table 12 are cases where the propensity to insure is

sufficiently strong, that persons with genotype (–3,0) will insure unless their rel-ative risk is very small, hence s*

R is very large. In fact we could not computethem because of numerical overflow, which is why they are missing, but for allpractical purposes these are cases where adverse selection will never appear. This


TABLE 11

MINIMUM LOSS AT WHICH ADVERSE SELECTION OCCURS WITH WEALTH W = £100,000.

ModelAge Term

1 2 3 4£ £ £ £

20 10 45,600 25,100 3,100 3,10020 84,300 53,800 7,500 7,70030 100,000 61,600 9,100 9,40040 84,800 55,500 8,000 8,300

30 10 100,000 60,600 8,800 9,00020 100,000 63,800 9,500 9,90030 85,600 56,200 8,200 8,500

40 10 100,000 65,200 9,800 10,20020 85,300 55,800 8,100 8,300

50 10 80,300 50,600 7,000 7,200


TABLE 12

THRESHOLD STANDARD DEVIATIONS s*R AT WHICH ADVERSE SELECTION APPEARS, FOR WEALTH W = £100,000.FIGURES IN BOLD INDICATE VALUES OF s*R LOWER THAN THE ESTIMATE sR OF ANTONIOU ET AL. (2002).

MISSING VALUES INDICATE VALUES OF s*R TOO LARGE TO COMPUTE, IN PRACTICAL TERMS MEANING THAT

ADVERSE SELECTION WILL NEVER APPEAR.

Loss to Wealth Ratio L /WModel Age Term

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 20 10 0.193 0.485 0.844 1.152 1.393 1.592 1.770 1.942 2.13020 0.050 0.131 0.232 0.363 0.530 0.733 0.957 1.189 1.44130 0.039 0.101 0.175 0.268 0.387 0.542 0.738 0.974 1.25340 0.048 0.119 0.207 0.319 0.467 0.658 0.889 1.151 1.467

30 10 0.038 0.103 0.183 0.284 0.412 0.576 0.778 1.008 1.27020 0.036 0.094 0.163 0.249 0.359 0.501 0.686 0.914 1.19230 0.047 0.116 0.201 0.311 0.455 0.640 0.868 1.127 1.441

40 10 0.036 0.092 0.159 0.242 0.347 0.483 0.659 0.880 1.15320 0.049 0.119 0.207 0.319 0.466 0.655 0.883 1.140 1.445

50 10 0.060 0.146 0.256 0.399 0.583 0.803 1.040 1.286 1.5652 20 10 0.462 1.073 1.465 1.734 1.949 2.138 2.320 2.515 2.761

20 0.125 0.322 0.594 0.907 1.194 1.447 1.685 1.934 2.25930 0.096 0.238 0.432 0.685 0.970 1.248 1.523 1.832 2.29540 0.113 0.280 0.516 0.815 1.126 1.432 1.767 2.210 3.136

30 10 0.098 0.253 0.463 0.732 1.014 1.277 1.524 1.777 2.08920 0.089 0.222 0.400 0.636 0.911 1.188 1.463 1.766 2.20930 0.110 0.273 0.502 0.795 1.104 1.408 1.738 2.171 3.044

40 10 0.087 0.216 0.387 0.613 0.881 1.153 1.421 1.708 2.10120 0.113 0.281 0.516 0.813 1.120 1.418 1.736 2.149 2.915

50 10 0.139 0.351 0.647 0.976 1.276 1.551 1.837 2.196 2.7673 20 10 2.263 2.930 3.705 4.867

20 1.605 2.475 3.53230 1.412 2.536 4.60640 1.587 3.377

30 10 1.444 2.292 3.24520 1.355 2.449 4.42130 1.563 3.301

40 10 1.322 2.337 3.97920 1.574 3.204

50 10 1.708 3.0394 20 10 2.234 2.840 3.381 4.155 4.915 7.458

20 1.566 2.346 3.127 4.24430 1.366 2.345 3.67340 1.529 2.920

30 10 1.404 2.177 2.878 3.69920 1.309 2.266 3.45230 1.506 2.855 4.733

40 10 1.277 2.175 3.109 4.81020 1.519 2.792 4.637

50 10 1.659 2.768 4.548

observation has an exact analogue in Macdonald & Tapadar (2009). Note thatmost missing values appear under utility Models 3 and 4, which were the twomodels parameterised from real economic data; Models 1 and 2, in contrast,were chosen simply to illustrate a lower level of risk-aversion.

3.5. Adverse Selection by Multiple Subpopulations

Previously we considered the case where adverse selection is triggered when thelowest-risk subpopulation refuses to purchase insurance. However, given a pop-ulation with such a broad range of risks, it may also be of interest to considerthe prospect that more than one low-risk subpopulation will no longer purchaseinsurance. (In fact, our discrete subpopulations defined by a polygene with threebi-allelic loci, meaning that there are two variants for each of the three genes, isitself the result of discretising an underlying Normal distribution for the poly-genic risk in Equation (1)). Here we suppose that adverse selection extendsto both the (–3,0) and (–2,0) genotypes. Adverse selection first occurs withinthe (–3,0) genotype who, as a result, are removed from the risk pool. Ourattention is then directed on the point where persons with the (–2,0) genotypestop purchasing insurance, given a new actuarially fair premium rate P2 > Pwhich no longer includes the (–3,0) genotype.

The threshold premium is P*(–2,0), and values of this are given in Tables 13and 14. Table 15 shows the threshold standard deviations, as in Section 3.4.These latter values are higher than the values in Table 12, since the polygenicrisk must be greater to trigger adverse selection within a second subpopulation.

3.6. The Polygenotype as a Continuous Random Variable

We mentioned in Section 3.5 that the Binomial model of the polygenotypeis in fact a discretised version of a model in which the numerical value of thepolygenotype has a continuous distribution, usually assumed to be Normal(Strachan & Read, 2004). One reason to discretise the polygenotype is inorder to model the transmission of the polygenotype from parents to children,because the basis of inheritance is the transmission of discrete genes (Lange,1997).

Since P is the sum of six independent random variables, each taking values–1/2 and +1/2 with equal probability, we see it has mean 0 and variance 3/2.If we revert to the continuous Normal model of P, we equate moments andassume that P + N (0,3 /2). Denote its density fP( p), let the (discrete) distribu-tion of the major genotype M be fM(m), and let the (mixed) distribution of thecombined genotype be f(P,M)(p,m) = fP( p) fM (m). Then the insurer’s fair valuepremium, denoted Pc is:

Pc = ,p mPm 3

3

-#! ^ h f(P,M )( p,m)dp. (7)



TA

BL

E 1

3

TH

RE

SHO

LD

PR

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TE

SP

* (–2

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AT

WH

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10.

20.

30.

40.

50.

60.

70.

80.

9

2010

0.00

601

0.00

618

0.00

638

0.00

660

0.00

686

0.00

717

0.00

756

0.00

808

0.00

888

200.

0225

60.

0231

90.

0239

00.

0247

20.

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30.

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630

0.06

615

0.06

791

0.06

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0.07

223

0.07

492

0.07

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0.08

221

0.08

763

0.09

594

400.

1534

30.

1571

80.

1614

20.

1662

80.

1719

60.

1787

90.

1873

50.

1988

00.

2163

430

100.

0166

90.

0171

60.

0176

90.

0183

00.

0190

20.

0198

70.

0209

50.

0223

80.

0245

820

0.06

067

0.06

231

0.06

416

0.06

628

0.06

876

0.07

174

0.07

548

0.08

048

0.08

814

300.

1487

20.

1523

70.

1565

00.

1612

30.

1667

70.

1734

30.

1817

70.

1929

20.

2100

240

100.

0449

00.

0461

30.

0475

20.

0491

10.

0509

80.

0532

20.

0560

30.

0597

90.

0655

520

0.13

478

0.13

814

0.14

194

0.14

629

0.15

139

0.15

752

0.16

519

0.17

546

0.19

119

5010

0.09

482

0.09

729

0.10

008

0.10

328

0.10

703

0.11

153

0.11

717

0.12

471

0.13

626

2010

0.00

617

0.00

653

0.00

696

0.00

747

0.00

811

0.00

892

0.01

004

0.01

173

0.01

489

200.

0231

40.

0244

80.

0260

40.

0279

30.

0302

50.

0332

50.

0373

30.

0434

70.

0548

630

0.06

779

0.07

152

0.07

588

0.08

111

0.08

753

0.09

574

0.10

684

0.12

337

0.15

348

400.

1568

80.

1646

70.

1737

30.

1844

90.

1975

90.

2141

30.

2361

50.

2682

30.

3246

430

100.

0171

30.

0181

20.

0192

90.

0206

90.

0224

30.

0246

60.

0277

10.

0323

10.

0408

620

0.06

218

0.06

562

0.06

965

0.07

448

0.08

041

0.08

801

0.09

829

0.11

361

0.14

160

300.

1520

80.

1596

70.

1685

20.

1790

10.

1918

00.

2079

60.

2294

90.

2608

90.

3162

240

100.

0460

40.

0486

20.

0516

60.

0553

10.

0598

00.

0655

60.

0733

70.

0850

70.

1065

820

0.13

787

0.14

487

0.15

303

0.16

273

0.17

457

0.18

955

0.20

957

0.23

886

0.29

077

5010

0.09

710

0.10

227

0.10

831

0.11

551

0.12

435

0.13

559

0.15

073

0.17

309

0.21

338

Model 1 Model 2


TA

BL

E 1

4

TH

RE

SHO

LD

PR

EM

IUM

RA

TE

SP

* (–2

,0)

AT

WH

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AD

VE

RSE

SEL

EC

TIO

NB

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HT

HE

P=

–3 A

ND

P=

–2 P

OLY

GE

NO

TY

PE

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PO

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LA

TIO

NS

WIL

LT

AK

EP

LA

CE,

FO

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=£1

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00.

Los

s to

Wea

lth

Rat

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rm0.

10.

20.

30.

40.

50.

60.

70.

80.

9

2010

0.00

965

0.01

787

0.03

789

0.09

047

0.21

591

0.41

581

0.61

491

0.77

473

0.89

987

200.

0357

90.

0642

90.

1255

30.

2456

30.

4204

90.

5962

10.

7382

00.

8471

40.

9320

630

0.10

193

0.17

066

0.28

707

0.44

402

0.60

117

0.72

878

0.82

501

0.89

788

0.95

461

400.

2238

50.

3359

50.

4783

70.

6201

60.

7359

20.

8220

30.

8853

70.

9331

20.

9702

730

100.

0265

90.

0482

80.

0968

30.

2005

80.

3704

60.

5565

30.

7117

90.

8316

80.

9251

920

0.09

382

0.15

837

0.27

044

0.42

650

0.58

685

0.71

868

0.81

844

0.89

405

0.95

291

300.

2176

00.

3282

20.

4704

50.

6135

60.

7310

90.

8187

30.

8832

40.

9318

70.

9697

240

100.

0701

70.

1213

70.

2175

50.

3671

80.

5365

40.

6827

50.

7950

40.

8803

80.

9468

320

0.19

887

0.30

467

0.44

583

0.59

270

0.71

576

0.80

825

0.87

646

0.92

792

0.96

796

5010

0.14

344

0.23

076

0.36

279

0.51

852

0.65

992

0.76

980

0.85

160

0.91

341

0.96

151

2010

0.00

946

0.01

620

0.02

896

0.05

262

0.09

327

0.15

354

0.22

721

0.30

270

0.37

165

200.

0351

00.

0585

00.

0986

60.

1606

10.

2409

70.

3270

20.

4065

70.

4743

80.

5304

030

0.10

006

0.15

676

0.23

696

0.33

150

0.42

425

0.50

431

0.56

923

0.62

092

0.66

225

400.

2200

80.

3133

80.

4170

00.

5131

30.

5920

90.

6534

50.

7005

90.

7371

70.

7660

630

100.

0260

80.

0438

70.

0754

10.

1269

20.

1989

80.

2820

40.

3629

90.

4341

30.

4938

020

0.09

209

0.14

531

0.22

214

0.31

496

0.40

796

0.48

936

0.55

587

0.60

903

0.65

161

300.

2139

10.

3059

40.

4092

00.

5058

20.

5856

20.

6478

20.

6956

80.

7328

40.

7622

040

100.

0688

50.

1109

90.

1758

90.

2610

00.

3530

90.

4381

10.

5096

90.

5677

70.

6146

120

0.19

545

0.28

336

0.38

514

0.48

300

0.56

531

0.63

011

0.68

020

0.71

919

0.75

003

5010

0.14

087

0.21

314

0.30

612

0.40

496

0.49

421

0.56

740

0.62

515

0.67

054

0.70

662

Model 3 Model 4


TABLE 15

THRESHOLD STANDARD DEVIATIONS s*R AT WHICH ADVERSE SELECTION APPEARS FOR THE (–2,0) GENOTYPE,FOR WEALTH W = £100,000. FIGURES IN BOLD INDICATE VALUES OF s*R LOWER THAN THE ESTIMATE sR OF

ANTONIOU ET AL. (2002). MISSING VALUES INDICATE VALUES OF s*R TOO LARGE TO COMPUTE, IN PRACTICAL

TERMS MEANING THAT ADVERSE SELECTION WILL NEVER APPEAR.

Loss to Wealth RatioModel Age Term

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 20 10 0.266 0.592 0.914 1.184 1.405 1.595 1.768 1.938 2.12420 0.072 0.184 0.315 0.467 0.639 0.824 1.019 1.225 1.45730 0.057 0.143 0.243 0.360 0.497 0.655 0.835 1.041 1.29040 0.069 0.168 0.284 0.421 0.581 0.764 0.968 1.199 1.489

30 10 0.054 0.146 0.253 0.378 0.522 0.685 0.866 1.067 1.30020 0.052 0.134 0.228 0.338 0.466 0.616 0.789 0.990 1.23630 0.068 0.164 0.277 0.411 0.568 0.747 0.949 1.178 1.465

40 10 0.052 0.131 0.222 0.328 0.452 0.597 0.765 0.961 1.20020 0.070 0.168 0.284 0.420 0.579 0.760 0.962 1.188 1.468

50 10 0.086 0.205 0.344 0.507 0.690 0.888 1.094 1.315 1.5762 20 10 0.568 1.113 1.473 1.733 1.944 2.132 2.313 2.507 2.752

20 0.176 0.422 0.700 0.976 1.229 1.463 1.689 1.932 2.25130 0.136 0.323 0.544 0.788 1.037 1.285 1.539 1.835 2.28540 0.159 0.375 0.629 0.902 1.176 1.455 1.772 2.199 3.086

30 10 0.139 0.340 0.575 0.826 1.072 1.307 1.537 1.780 2.08620 0.127 0.303 0.511 0.744 0.987 1.232 1.484 1.772 2.20130 0.156 0.367 0.616 0.885 1.157 1.433 1.745 2.162 2.997

40 10 0.125 0.295 0.497 0.723 0.961 1.200 1.444 1.716 2.09720 0.160 0.375 0.629 0.900 1.170 1.441 1.743 2.140 2.876

50 10 0.195 0.455 0.749 1.038 1.306 1.563 1.837 2.186 2.7453 20 10 2.256 2.921 3.687 4.845

20 1.612 2.465 3.50930 1.435 2.521 4.55140 1.600 3.324

30 10 1.461 2.285 3.22720 1.383 2.436 4.36630 1.578 3.248

40 10 1.352 2.328 3.91720 1.587 3.153

50 10 1.712 3.0064 20 10 2.227 2.831 3.368 4.137 4.892 7.375

20 1.575 2.337 3.109 4.17830 1.392 2.334 3.61540 1.545 2.878

30 10 1.424 2.171 2.863 3.67320 1.340 2.257 3.40030 1.524 2.816 4.658

40 10 1.312 2.168 3.082 4.75620 1.535 2.759 4.571

50 10 1.665 2.746 4.493


However, as in the discrete polygenotype model, we assume that the insureradjusts the premium to reflect the composition of the risk pool. If everyone withgenotype ( p,0) with p < p* has declined to insure and has left the risk pool, theactuarial premium for those who remain is:

Pc( p*) =,

, , , , , ,.

p dp

p p dp p p dp p p dpP P P

1 0

0 0 1 1 2 2

,*

, , ,*

P Mp

P M P M P Mp

-

+ +

3

3

3

3

33

-

--

f

f f f

#

###

] ^

^ ] ^ ^ ] ^ ^ ] ^

g h

h g h h g h h g h

where the notation makes explicit the dependence of the actuarial premiumon p*. Following the same general idea as before, we find the threshold valuesof the polygenotype p*, such that those with genotypes ( p,0) (for p < p*) donot insure at premium Pc( p*), and everyone else does. These thresholds aregiven in Table 16 for utility Model 1 and in Table 17 for utility Model 2. Theyare markedly higher than those found previously. Indeed there are instanceswhere nearly the entire BRCA0 subpopulation would decline to buy insurance.This could easily ‘spill over’ into the BRCA1 and BRCA2 subpopulations,although for the purposes of demonstration we have assumed it does not. Suchbehaviour would be disastrous for CI business.

However, we find that adverse selection does not usually occur under utilityModels 3 and 4. It is so infrequent that instead of showing tables we list theexceptions here (all of which occur when L/W = 0.1).

(a) In Model 3 a policy with entry age 40 and term 10 years has p* = –3.74representing 0.1% of the population.

(b) In Model 4 a policy with entry age 30 and term 20 years has p* = –2.86representing 1% of the population, and a policy with entry age 40 andterm 10 years has p* = 2.74 representing 98.5% of the population.

4. CONCLUSIONS

4.1. Background

It is clear that a major development of genetics in future will concern the inter-actions of multiple genes contributing to common disorders. One of the very fewepidemiological studies so far to estimate onset rates associated with a complexdisorder is that of Antoniou et al. (2002). They modelled a polygene influencingBC risk, in addition to the known BRCA1/2 major genes. Risky polygenes arecommon enough that adverse selection becomes an option for a larger pro-portion of the population. Furthermore, the relative risks attributed to the poly-gene range from 0.04 to 23.62, many times more extreme than the assumptionsin Macdonald, Pritchard & Tapadar (2006) for example. Our aim has been toapply this model to the study of adverse selection in a CI insurance market, bothalong the lines of previous studies and from more of an economic viewpoint.

(8)

398 A.S. MACDONALD AND K.R. MCIVORT

AB

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9

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3.25

3.21

3.15

3.09

–3–3

–3–3

–3(9

9.4%

)(9

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)(9

9.2%

)(0

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)(0

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3.63

3.60

3.57

3.53

3.49

3.43

3.34

3.22

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3.40

3.38

3.35

3.32

3.27

3.22

3.14

3.00

2.68

(99.

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(99.

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(99.

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(99.

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(99.

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(99.

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(99.

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(99.

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(98.

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403.

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203.

173.

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86–3

(99.

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(99.

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(99.

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(99.

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(99.

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(99.

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(99.

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(98.

8%)

(0.0

%)

3010

3.68

3.65

3.62

3.58

3.54

3.48

3.40

3.29

3.06

(99.

6%)

(99.

6%)

(99.

6%)

(99.

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(99.

6%)

(99.

5%)

(99.

5%)

(99.

4%)

(99.

1%)

203.

413.

393.

363.

333.

283.

233.

153.

022.

72(9

9.5%

)(9

9.5%

)(9

9.5%

)(9

9.4%

)(9

9.4%

)(9

9.3%

)(9

9.3%

)(9

9.1%

)(9

8.4%

)30

3.25

3.23

3.21

3.18

3.14

3.08

3.01

2.87

–3(9

9.4%

)(9

9.3%

)(9

9.3%

)(9

9.3%

)(9

9.2%

)(9

9.2%

)(9

9.1%

)(9

8.8%

)(0

.0%

)

4010

3.42

3.40

3.37

3.33

3.29

3.23

3.15

3.02

2.73

(99.

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(99.

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(99.

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(99.

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(99.

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(99.

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(99.

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(98.

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203.

263.

243.

213.

183.

143.

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012.

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(99.

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(99.

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(99.

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(99.

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(99.

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(99.

2%)

(99.

1%)

(98.

8%)

(0.0

%)

5010

3.32

3.30

3.27

3.23

3.18

3.12

3.03

2.87

–3(9

9.4%

)(9

9.4%

)(9

9.4%

)(9

9.3%

)(9

9.3%

)(9

9.2%

)(9

9.1%

)(9

8.8%

)(0

.0%

)

THE BREAST CANCER POLYGENE 399T

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)(9

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)(0

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)(0

.0%

)(0

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)(0

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)(0

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)(0

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)(0

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3.61

3.55

3.47

3.37

3.23

–3–3

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(99.

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(99.

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(99.

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(99.

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(99.

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(0.0

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(0.0

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(0.0

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303.

383.

333.

273.

183.

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(99.

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(99.

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(99.

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(99.

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(99.

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(98.

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(0.0

%)

(0.0

%)

(0.0

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403.

233.

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)(0

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)(0

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)(0

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)(0

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)

3010

3.66

3.60

3.52

3.43

3.30

3.08

–3–3

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)(9

9.6%

)(9

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)(9

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)(9

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)(9

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)(0

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)(0

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)(0

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3.39

3.34

3.28

3.19

3.06

2.83

–3–3

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3.23

3.19

3.13

3.06

2.94

–3–3

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(99.

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(99.

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(99.

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(99.

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(98.

9%)

(0.0

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(0.0

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(0.0

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(0.0

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4010

3.40

3.35

3.28

3.19

3.05

2.81

–3–3

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)(0

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3.24

3.20

3.14

3.06

2.94

–3–3

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(99.

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(99.

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(99.

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(99.

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(98.

9%)

(0.0

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(0.0

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(0.0

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(0.0

%)

5010

3.30

3.24

3.17

3.08

2.92

–3–3

–3–3

(99.

4%)

(99.

4%)

(99.

3%)

(99.

2%)

(98.

9%)

(0.0

%)

(0.0

%)

(0.0

%)

(0.0

%)

4.2. Multi-State Market Models

The most striking conclusion is the magnitude of the possible adverse selectioncosts associated with the polygene. In the worst case (Scenario (a) small insurancemarket, high level of genetic testing) the necessary premium increase approaches17%. We must bear in mind that this cost arises solely from the contributionof BC/OC, and that there are several other genetic disorders which must have apolygenic component when major genes have been allowed for. This contrastswith the findings of most studies of major genes alone, where the rarity of muta-tions keeps adverse selection costs very small.

The inflated costs in the small market are perhaps a cause for concern.Should polygene testing become available, youthful insurance markets may beexposed to high costs from a moratorium on genetic testing (even at modestlevels of testing).

The contrast between Tables 4 and 6 is perhaps the most revealing. In Table 4we assumed all genetic tests to follow the development of a relevant family his-tory. The premium increases, even under severe adverse selection, were of theorder of 0.001%. In Table 6 we assumed that testing for polygenes was freedfrom this regime. Even under more moderate adverse selection, the premiumincreases were of the order of 1%, a thousand-fold increase.

Nevertheless, these are worst case outcomes. They assume the existence ofaccurate, freely available genetic tests, no treatments to modify the known risks,active and well-informed market participants (who can all afford insurance) andso on. And, women may not be all that likely to curtail their cover because ofa beneficial test result, because they would lose cover against all other illnessesas well as BC. Reality is likely to fall well short of the worst outcome even ifconditions would let it exist.

4.3. An Economic Framework for Adverse Selection

Introducing the utility framework, with informed purchasers versus risk-pooling insurers, characterises adverse selection in another way, by mappingout when it ought or ought not to appear. This approach too has many assump-tions and simplifications, including single premiums, perfectly rational agents,universal knowledge (on the part of individuals) of genotype, fixed insurancecoverage and a known utility function. We suspect, nevertheless, that blurringall these in more realistic ways is liable to reduce rather than to increase adverseselection.

Our chief observation is the strong dependence on the utility functionassumed. It is comforting that those that were parameterised using empiricaldata (Models 3 and 4) showed adverse selection to be very limited. In mostcases, it would not appear at all under the assumed conditions. However, thoseutilities chosen just to illustrate lower risk aversion (Models 1 and 2) led toadverse selection in most cases, including sometimes for polygenic risks lessextreme than those of the empirical BC model.


Unlike the multi-state market model, the economic model does not let usestimate what the costs of adverse selection would be. However, under whatmight be the most realistic assumptions, of Section 3.6, we could estimate theproportion of the population who would leave the CI market under equilibrium.Under Models 1 and 2 this proportion was generally very high, in many casesthe market effectively disappearing. Under Models 3 and 4 this proportionwas nearly always zero, adverse selection being absent. Clearly, relatively modestshifts in preferences as expressed by utilities can move the market from oneextreme to the other.

ACKNOWLEDGEMENTS

This work was carried out at the Genetics and Insurance Research Centre atHeriot-Watt University. We would like to thank the sponsors for funding, andmembers of the Steering Committee for helpful comments at various stages.KM was funded by the Engineering and Physical Sciences Research Council.

REFERENCES

ANTONIOU, A.C., PHAROAH, P.D.P., MCMULLAN, G., DAY, N.E., STRATTON, M.R., PETO, J., PON-DER, B.J. and EASTON, D.F. (2002) A comprehensive model for familial breast cancer incor-porating BRCA1, BRCA2 and other genes. British Journal of Cancer, 86, 76-83.

EISENHAUER, J.G. and VENTURA, L. (2003) Survey measures of risk aversion and prudence.Applied Economics, 35(13), 1477-1484.

GUI, E.H., LU, B., MACDONALD, A.S., WATERS, H.R. and WEKWETE, C.T. (2006) The genetics ofbreast and ovarian cancer III: A new model of family history with insurance applications.Scandinavian Actuarial Journal, 2006, 338-367.

GUTIERREZ, M.C. and MACDONALD A.S. (2007) Adult polycystic kidney disease and insurance:A case study in genetic heterogeneity. North American Actuarial Journal, 11(1), 90-118.

HOY, M. and WITT, J. (2007) Welfare effects of banning genetic information in the life insurancemarket: The case of BRCA1/2 genes. Journal of Risk and Insurance, 74(3), 523-546.

LANGE, K. (1997) An approximate model of polygenic inheritance. Genetics, 147, 1423-1430.LEMAIRE, J., SUBRAMANIAN, K., ARMSTRONG, K. and ASCH, D.A. (2000) Pricing term insurance in

the presence of a family history of breast cancer. North American Actuarial Journal, 4, 75-87.MACDONALD, A.S. (1999) Modeling the impact of genetics on insurance. North American Actu-

arial Journal, 3(1), 83-101.MACDONALD, A.S. and MCIVOR, K.R. (2006) Application of a polygenic model of breast and

ovarian cancer to critical illness insurance. Annals of Actuarial Science, 1, 319-343.MACDONALD, A.S., PRITCHARD, D.J. and TAPADAR, P. (2006) The impact of multifactorial genetic

disorders on critical illness insurance: A simulation study based on UK Biobank. ASTIN Bul-letin, 36, 311-346.

MACDONALD, A.S. and TAPADAR, P. (2009) Multifactorial genetic disorders and adverse selection:Epidemiology meets economics. To appear in Journal of Risk and Insurance.

MACDONALD, A.S., WATERS, H.R. and WEKWETE, C.T. (2003a) The genetics of breast and ovar-ian cancer I: A model of family history. Scandinavian Actuarial Journal, 2003, 1-27.

MACDONALD, A.S., WATERS, H.R. and WEKWETE, C.T. (2003b) The genetics of breast and ovariancancer II: A model of critical illness insurance. Scandinavian Actuarial Journal, 2003, 28-50.

ROPKA, M.E., WENZEL, J., PHILLIPS, E.K., SIADATY, M. and PHILBRICK, J.T. (2006) Uptake ratesfor breast cancer genetic testing: A systematic review. Cancer Epidemiology Biomarkers andPrevention, 15(5), 840-855.


STRACHAN, T. and Read, A.P. (2004) Human Molecular Genetics 3. Garland Publishing.SUBRAMANIAN, K., LEMAIRE, J., HERSHEY, J.C., PAULY, M.V., ARMSTRONG, K. and ASCH, D.A.

(1999) Estimating adverse selection costs from genetic testing for breast and ovarian cancer:The case of life insurance. Journal of Risk and Insurance, 66, 531-550.

ANGUS MACDONALD

Department of Actuarial Mathematics and Statisticsand the Maxwell Institute for Mathematical SciencesHeriot-Watt UniversityEdinburgh EH14 4AS, U.K.Tel: +44(0)131-451-3209Fax: +44(0)131-451-3249E-mail: [email protected]


Documents

MODELLING ADVERSE SELECTION IN THE PRESENCE OF A