Scand. Actuarial J. 2003; 1: 28–50 ORIGINAL ARTICLE

The Genetics of Breast and Ovarian Cancer II: A Modelof Critical Illness Insurance

ANGUS S. MACDONALD, HOWARD R. WATERS andCHESSMAN T. WEKWETE

Macdonald AS, Waters HR, Wekwete CT. The genetics of breast andovarian cancer II: a model of critical illness insurance. Scand. Actuarial J.2003; 1: 28–50.

We present a model of breast cancer (BC) and ovarian cancer (OC) andother events that would lead to a claim under a Critical Illness (CI)insurance policy, and estimate its transition intensities, mainly usingUnited Kingdom population data. We use this to estimate the costs of CIinsurance in the presence of a family history of BC or OC, using theprobabilities from Part I of carrying a BRCA1 or BRCA2 mutation,given the family history. In practice, the family history may not includeall relevant facts; we look at the range of costs depending on what isknown. We show the effect of lower penetrance than is observed inhigh-risk families. Finally, we consider what the cost of adverse selectionmight be, were insurers unable to use genetic test or family historyinformation. Key words: Ad�erse selection, breast cancer, BRCA1 gene,BRCA2 gene, critical illness insurance, o�arian cancer, underwriting.

1. INTRODUCTION

1.1. Part I

Recent advances in human genetics, in this case the discovery of the BRCA1 andBRCA2 genes that predispose to breast cancer (BC) and ovarian cancer (OC), havebegun to offer quantitative insights into inherited disease. In Part I, we proposed amodel of BC and OC with separate states representing onsets at ages �50, 50–65and �65 (typical of those used in the underwriting of family history). We used themodel to represent the life histories of relatives of a woman applying for insurance,and hence family histories of BC and OC based on the genotypes of the familymembers.

The following notation and conventions from Part I will also be used here.

(a) Let M be the family size, including the applicant, her mother, her sisters andaunts. Let X= (x1, x2, . . . , xM) be the ages of the family members when theapplicant is born. We use the following conventions: the applicant is the firstfamily member (so x1=0); her mother is the second; and the applicant’smother and aunts are all exactly 30 years older than the applicant and her

© 2003 Taylor & Francis. ISSN 0346-1238 DOI: 03461230110106318

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 29

sisters (so x2=30). Let G= (g1, g2, . . . , gM) be the genotypes of the familymembers (defined in detail later). Finally, let C(t)= (c1(t), c2(t), . . . , cM(t)) bethe family history when the applicant is age t ; that is, ci(t) is the state in themodel occupied by the ith family member at age xi+ t.

(b) We combine heterozygous and homozygous BRCA1 and BRCA2 genotypes, sothere are four genotypes: (0, 0) means no mutations, (1, 0) means a BRCA1mutation, (0, 1) means a BRCA2 mutation and (1, 1) means mutations in bothgenes.

(c) Table 1 in Part I gave estimates of the allelic frequencies; in this work we havetaken the estimates (b) and (c) of Parmigiani et al. (1998) as low and highestimates, respectively, of the allelic frequencies.

From the relatives’ model, �ia Bayes’ Theorem, we found probabilities of theform:

P[Applicant’s genotype � Family history]

in which the family history could be incomplete; in particular, the underwritermight not know M, or the history of relatives without BC or OC.

1.2. Outline

In Section 2, we describe the main features of Critical Illness (CI) insurance, and wepropose a model with BC and OC as distinct causes of claims. This is called ‘theapplicant’s model’. Section 3 describes the underwriting of a family history of BCand OC. Some of the intensities in the model have already been estimated in Part

Table 1. Typical underwriting of a family history of breast or o�arian cancer for CIinsurance. Source: Swiss Re Life & Health

PremiumNumber ofType ofApplicant’s Age atage diagnosis ratingcancer affected

relatives

Any +100Breast 1 �50Any DeclineBreast �2 �50

Standard50–651BreastAnyAny +50 upBreast �2 50–65Any Breast Any �65 Standard

+150�501Ovarian�40�40 Ovarian 1 50–65 +50

Standard�40 Ovarian 1 �65�40 Ovarian �2 �50 Decline�40 Ovarian �2 �50 +15041–50 Ovarian 1 �50 +10041–50 �50 Standard1Ovarian41–50 Ovarian �2 �50 Decline

�50�2Ovarian41–50 Standard�50 Ovarian Any Any Standard

A. S. Macdonald et al. Scand. Actuarial J. 130

I; we estimate the remainder in Section 4. Using the model, we find genotype-spe-cific costs of CI insurance by solving Thiele’s equations and, in Section 5, we applythe probabilities calculated in Part I to these costs to determine the impact ofknown or unknown genetic information on underwriting. In Section 7 we discussthe potential for adverse selection. Our conclusions are in Section 8.

2. A MODEL FOR CRITICAL ILLNESS INSURANCE

CI insurance policies pay an assurance benefit on the occurrence of a serious butnot immediately fatal event, such as the diagnosis of a malignant cancer orprogressive illness (for example, multiple sclerosis), heart attack or coronary bypasssurgery. Most are sold as ‘accelerated benefits’ riders with life insurance policies.Usually, in the U.K., the claim is paid only if the insured survives for 28 days afterthe diagnosis of the critical illness. This is a negligible condition for most cancers,but not for other CI events. In this study, therefore, a claim may be paid if one ofthe following ‘CI events’ occurs:

(a) breast cancer;(b) ovarian cancer; or(c) another insured illness, followed by survival for for 28 days.

These lead to the continuous-time, discrete-state model in Fig. 1, with transitionintensities g�x

0j (j=1, 2, 3, 4), where ‘g ’ indicates genotype. We call this ‘thisapplicant’s model’.

Fig. 1. A model for an applicantwith genotype g, buying Critical Ill-ness insurance.

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 31

3. FAMILY HISTORY IN UNDERWRITING PRACTICE

Underwriters do take family history into account, but typically in less detail thanthe cross-checked pedigrees obtained by clinical geneticists. The key piece ofinformation is the number of first-degree relatives (sisters, mother and aunts) whohave had BC or OC in certain age bands; those we use here (�50, 50–65 and�65) are typical. Information often not taken into account includes:

(a) the number of first-degree relatives;(b) the family structure; namely the numbers of sisters, maternal aunts and

paternal aunts; and(c) which relatives have had BC or OC.

Although this information might well be obtained, it is not usually used. Note,also that even clinical geneticists with access to much more information thanunderwriters often find it difficult to establish an accurate history. Table 1 shows atypical example of family history underwriting.

Applicants whose extra premiums would exceed +200% to +250%, will usuallybe declined CI cover (roughly half of the upper limits used in life insuranceunderwriting).

4. ESTIMATION OF THE TRANSITION INTENSITIES

In Part I we estimated:

(a) incidence rates of BC and OC in the population, assumed to apply to womenwith no BRCA1 or BRCA2 mutations; and

(b) incidence rates of BC and OC for women with BRCA1 or BRCA2 mutations,or both.

These are used in the applicant’s model as well. In this section, we estimate:

(a) incidence rates of other CI events, specifically ‘other cancers’, strokes and heartattacks (all assumed to be independent of BRCA1 or BRCA2 genotype); and

(b) rates of mortality with the other transitions in Fig. 1 excluded.

Note that these intensities are needed only at ages at which CI policies are inforce. Although we have estimated intensities over a wide age range, the importantparts of the graduations are those over ages 20–65.

As in Part I, all intensities have a time unit of one year.

4.1. Cancers other than breast cancer and o�arian cancer

Following the definition used in CI policies, ‘other cancers’ means all cancers exceptBC, OC and skin cancers other than malignant melanoma. We used new cancerregistrations in 1990–92 (O.N.S., 1999), using mid-year population estimates as theexposed-to-risk. These were graduated by the following functions at lower andhigher ages:

A. S. Macdonald et al. Scand. Actuarial J. 132

�xother=exp(−10.3995+0.08235x) for x�40 (1)

�xother=0.00808−0.00019x+0.000016(x−35)2−0.000000144(x−35)3

for x�64 (2)

and between ages 40 and 64, the two functions were blended by linear interpolation.Fig. 2 shows the results.

4.2. Stroke

Stewart et al. (1999) studied stroke incidence using a prospective community strokeregister in a multi-ethnic South London community of almost 250 000 persons.Table 2 gives their incidence rates for first-ever stroke in women. We graduatedthese with the Gompertz function �x

stroke=exp(−11.45+0.085x) shown in Fig. 3.

Fig. 2. Crude and graduated annual incidence rates of all first cancers other than (a) breast and ovariancancer; and (b) skin cancers except malignant melanoma, for females.

Table 2. Annual incidence rates of first-e�er stroke among women. (Source: Stewartet al. (1999))

Age Age IncidenceAge IncidenceIncidenceraterate rate

�15 0.00000 35–44 0.00034 65–74 0.0044515–24 0.0089875–840.0007845–540.00005

0.01887�850.0013655–640.0000925–34

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 33

Fig. 3. Crude and graduated incidence rates of all first strokes, for females.

4.3. Heart attack

The Morbidity Statistics from General Practice Survey (McCormick et al. 1995)produced a CD-ROM with details of all first-ever cases of heart attacks (ICD 410and 414) between September 1991 and August 1992. From the same CD-ROM it isalso possible to calculate the exposed-to-risk exactly. The incidence rates are shownin Table 3, and were graduated with the Gamma function (Fig. 4):

�xheart=0.58

�0.1616.34 exp(−0.16x)x15.34

�(16.34)�

. (3)

4.4. The 28-day sur�i�al period

The 28-day survival period can be neglected for cancers, but not for strokes andheart attacks. We have to reduce the transition intensity into the ‘other CI events’

Table 3. Annual incidence rates of first-e�er heart attack (ICD 410 and 414) amongwomen. (Source: McCormick et al. (1995))

IncidenceAge Age IncidenceIncidence Agerate raterate

0.00001030–29 0.007905275–790.002156455–590.00781630.0001160 60–6430–44 0.0027862 80–84

45–49 0.008899685–890.004165765–690.00046680.006964190–940.004777270–740.001012550–54

A. S. Macdonald et al. Scand. Actuarial J. 134

Fig. 4. Crude and graduated annual incidence rates of all first heart attacks, for females.

state, and increase the intensity into the ‘dead’ state by the same amount. Thefollowing 28-day survival probabilities were based on information supplied to us bySwiss Re Life and Health:

pxheart=0.8983095−0.00235911x−0.00001359781x2 (4)

pxstroke=0.8718412+0.001566578x−0.00003711161x2. (5)

4.5. Total ‘other critical illness’ incidence rate

Table 4 shows the incidence rates of CI claims by cause, for females, given to us bythe authors of a study of 1991–97 data by the Faculty and Institute of ActuariesHealth Care Study Group (Dinani et al., 2000). BC and OC account for over 50%of cancers at ages 40–55. Together, bypass surgery, multiple sclerosis, total andpermanent disability and ‘other’ account for about 30% of claims up to age 30, andabout 15% at other ages. The total intensity, allowing approximately for these‘minor’ causes is:

g�x03=1.15(�x

other+pxstroke�x

stroke+pxheart�x

heart)+0.15(0�x01+0�x

02) (6)

for all genotypes g, where the prefix 0 in the last term indicates absence ofmutations.

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 35

Table 4. Annual incidence rates (per 1 000) of CI claims by cause, for females in theU.K. in 1991–97. (Source: Dinani et al., personal communication)

Incidence rate per 1 000 at age

Cause �30 31–40 41–50 51–60 �617.7741.199Cancer 0.215 0.434 0.9620.3380.0740.0140.0040.000Heart attack

0.166 0.169Stroke 0.024 0.028 0.0720.037 0.000Bypass surgery 0.000 0.000 0.023

0.072 0.000Multiple sclerosis 0.049 1.5210.0430.1840.0990.0210.034 0.000Total permanent disability

0.0150.015Other 0.014 0.018 0.169

9.9711.6781.2550.5450.337Total

4.6. Rate of mortality excluding CI e�ents

The intensity g�x04 is that of English Life Tables No. 15 (Females) (ELT15F)

adjusted as follows:

(a) We removed deaths caused by conditions that would lead to a CI claim,namely cancer, heart attack, stroke, kidney failure, multiple sclerosis,Alzheimer’s disease, Parkinson’s disease and benign brain tumor. The ratio �x

of the number of deaths from these causes to the total number of deaths in1990–92 (O.P.C.S., 1991, 1993a,b; O.N.S., 1999) was graduated below age 30by:

�x= −0.02612913+0.1046405x−0.01181445x2+0.0004671351x3

−0.000005790098x4 (7)

and above age 35 by:

�x= −1.345136+0.08972161x−0.001199781x2+0.000004867845x3, (8)

which were blended linearly between these ages. The results are shown in Fig.5.

(b) Deaths within 28 days of heart attacks and strokes were added back: that is:

(1−pxstroke)�x

stroke+ (1−pxheart)�x

heart.

5. THE COSTS OF CRITICAL ILLNESS INSURANCE

5.1. Premiums by genotype and in aggregate

Given the applicant’s age and genotype, we can calculate any moments of thepresent value of:

(a) a benefit payable on transition into any of the states in Fig. 1; or(b) a premium payable continuously while in the ‘healthy’ state of Fig. 1

A. S. Macdonald et al. Scand. Actuarial J. 136

Fig. 5. Crude and graduated proportion of total deaths that are from causes that would give rise to aclaim under a Critical Illness insurance policy, for females.

(Norberg, 1995). Norberg’s equations for first moments reduce to Thiele’s equa-tions. In what follows, we solve these numerically, with a force of interest of�=0.05 per annum, using a Runge-Kutta procedure with step-size 0.0005 years.

Table 5 shows level net annual premiums (payable continuously) for a CI benefitof £1, for various terms and entry ages, for each genotype and in aggregate, basedon estimated mutation frequencies. The first four lines are the net premiums thatwould be charged if:

Table 5. Le�el net annual premium for Critical Illness co�er of £1, depending ongenotype, and based on low and high estimates of mutation frequencies

Age 30 at entry Age 40 at entry Age 50 at entryGenotype

Term Term TermTerm Term Term

10 Yrs20 Yrs10 Yrs30 Yrs20 Yrs10 Yrs

0.005428 0.0083650.003319(0, 0) 0.001373 0.0037650.0022600.035145 0.037363 0.060753 0.061216 0.0626990.025186(1, 0)

0.002883 0.010979(0, 1) 0.019463 0.0894150.0422860.0259710.026951(1, 1) 0.1449360.0920830.0819180.041403 0.046386

0.003812 0.005470 0.008423Low 0.001393 0.002287 0.0033460.003849High 0.005504 0.0084690.001409 0.002307 0.003367

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 37

(a) the results of a genetic test were known;(b) all the model assumptions were correct; and(c) the modelled intensities were appropriate for the particular applicant.

This may be an appropriate point, therefore, to remind the reader of the majorassumptions, namely: for each gene, all mutations confer the same risk, which is notdose related, and is consistent with observed risk in families with a known historyof BC or OC. It is likely that these overstate the risk in the general population (seeSection 6).

The last two lines in Table 5 are the net premiums assuming that the insuredpopulation is the same as the general population, using the low and high estimatesof mutation frequencies from Parmigiani et al. (1998) ((c) and (b) in Table 1 of PartI, respectively). They hardly differ from those in respect of the (0, 0) genotype.

Some of the net premiums in Table 5 are extraordinarily high. The most extreme(age 50, 10-year term, genotype (1, 1)) would result in a surviving policyholderhaving paid 145% of the benefit in premiums, surely beyond the bounds of reasonfor insured and insurer alike.

Table 6 shows the genotype-specific premium rates as a percentage of theaggregate rates, with high estimates of mutation frequencies.

5.2. Premium rating with complete knowledge of the family history and structure

In this section, we give some examples of how CI net premiums vary depending onknown family histories in known family structures. For brevity, we considerfamilies with cases of BC before age 50 only. In Section 5.3 we consider morerealistic information, in respect of both BC and OC family histories. As before, wecould show examples without limit, but we have picked just a few that illustrate themain features.

Once the applicant’s genotype probabilities have been estimated, the expectedpresent values (EPVs) of CI benefits and premiums are given simply by weightingthe EPVs of benefits and premiums (underlying Table 5, but not shown) and the

Table 6. Le�el net premium, depending on BRCA1 and BRCA2 genotype, based onhigh estimates of mutation frequencies, as a percentage of the aggregate premium

Age 30 at entryGenotype Age 50 at entryAge 40 at entry

TermTermTermTermTermTerm

20 Yrs 10 Yrs10 Yrs 20 Yrs 30 Yrs 10 Yrs

%%%%%%

(0, 0) 99 9997 98 99 98(1, 0) 1 788 1 523 1 110 1 578 1 112 740

1 056768675(0, 1) 578476205(1, 1) 1 913 1 795 1 378 2 128 1 673 1 711

A. S. Macdonald et al. Scand. Actuarial J. 138

pure net premiums allowing for the given information are the ratios of thesequantities. In the following, we express premiums as percentages of aggregate netpremiums for the whole population (the last two lines of Table 5).

Table 7 shows the net premiums for families of size M=4 or M=6, with exactlyone or at least two relatives with BC before age 50, using high estimates ofmutation frequencies (Parmigiani et al. (1998) (b) in Table 1 of Part I). Table 8shows net premiums with exactly two relatives with BC before age 50, using highand low estimates of mutation frequencies. Comparing these with Table 1, we findboth similarities and differences:

(a) Even with the higher estimates of mutation frequencies, extra premiums givenone case of BC before age 50 are well below the +100% of Table 1.

(b) The extra premiums suggested for women with two, or two or more relativeswith BC before age 50 vary quite widely. For an applicant age 30, longer termcontracts have lower extra premiums, some well below +100%. Table 1suggests declinature.

(c) Applicants age 50 have lower extra premiums than applicants age 30. Table 1suggests extra premiums independent of the applicant’s age and policy term(for BC).

(d) Lower mutation frequencies do reduce the loadings, but not by much. How-ever, we noted in Part I that high estimates of mutation frequencies mayimplicitly allow for the effect of other, as yet unknown, genes implicated infamilial BC.

Table 7. Le�el net premium as a percentage of the aggregate premium, gi�en onerelati�e, or at least two relati�es, with BC before age 50, for M=4 or 6. Highestimates of mutation frequencies

Premium as percentage of aggregate premiumNumberNumberof auntsof sisters

2 or more cases of BC�501 case of BC�50

Age 30 Age 50 Age 30 Age 30 Age 30 Age 50Age 30 Age 30

30 Yrs20 Yrs10 Yrs10 Yrs30 Yrs 10 Yrs20 Yrs10 Yrs

% % % % % % % %

2 0 129 122 115 106 440 357 268 192108 105 2921 247 197 1711 116 112

2 111 108 1060 104 235 203 168 1491214 0 128 165265354435103114

13 156199251297102108116 11210510811122 102 238 205 169 146104 102 206 181 153 1371 3 107 1061030 102 186 165 143 1304 105 104

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 39

Table 8. Le�el net premium as a percentage of the aggregate premium, gi�en tworelati�es with BC before age 50, for M=4 or 6. High and low estimates of mutationfrequencies

Premium as percentage of aggregate premiumNumberNumberof auntsof sisters

High mutation frequencies Low mutation frequencies

Age 30 Age 50 Age 30 Age 30 Age 30 Age 50Age 30 Age 30

10 Yrs 20 Yrs 30 Yrs 10 Yrs 10 Yrs 20 Yrs 30 Yrs 10 Yrs

% % % % % %% %

263 185 331 273431 211351 15502194 167 220 1911 160 1421 286 242163 147 179 160 139196 1290 2252

3454 260 150 325 268 209 1310 4242443 195 144 221 192 160 1271 288

163 137 180 160 140196 123226221471 131 158 144 129 1193 193 171136 125 144 133 122 1151540 4 171

5.3. Premium rating with incomplete knowledge of the family history and structure

Now, we suppose the family structure is unknown. Whereas in previous sections weshowed only a few illustrative examples, here we show a more complete set,allowing for low and high mutation frequencies, and both BC and OC familyhistories.

(a) Table 9 shows net premiums, as a proportion of the aggregate net premium,assuming the family structure (M, X) is unknown, using high and low estimatesof mutation frequencies. As shown in Part I, Section 7.2, limiting the totalfamily size M� to 9 is a reasonable compromise. The same features as in Section5.2 are apparent, in particular the dependence on policy term and applicant’sage, quite low extra premiums with just one affected relative, and feasible extrapremiums with either two or at least two affected relatives. With three or moreaffected relatives, however, extra premiums rise steeply; the rarity of thesefamilies is reflected in the very small difference between having two, or at leasttwo, affected relatives. With low mutation frequencies the extra premiums arenot much smaller, and in some cases (5 affected relatives) are slightly higher.We comment on this below.

(b) Table 10 shows the effect of relatives with BC at ages 50–65. The extrapremiums are much reduced, but sometimes the dependence on age and policyterm is reversed; when there are many affected relatives, longer term policies,and older applicants, have larger extra premiums. The reason is that BRCA2may be more strongly indicated than BRCA1 by these family histories and thatlonger term policies include more of the ages of high BRCA2 risk.

A. S. Macdonald et al. Scand. Actuarial J. 140

Table 9. The effect of the family history (BC before age 50) on le�el net premiumsas a percentage of the aggregate premium, unknown (M, X). Maximum family sizeM� =9. Uses high and low estimates of mutation frequencies

Number of relatives Premium as percentage of aggregate premiumwith BC before 50

High mutation frequencies Low mutation frequencies

Age 30 Age 30 Age 30 Age 50 Age 30 Age 30 Age 30 Age 50

20 Yrs10 Yrs 20 Yrs 30 Yrs 10 Yrs 10 Yrs 30 Yrs 10 Yrs

% %%% %%%%

1 116 113 108 105 109 107 105 1032 263 224 182 155 207 181 153 1353 618 501 366 284 543 440 323 2524 806 652 468 355 793 639 458 347

3705 860 695 498 371 863 696 497269 184�2 137154183210158228

Table 10. The effect of the family history (BC between ages 50–65) on le�el netpremiums as a percentage of the aggregate premium, unknown (M, X)

Number of relatives Premium as percentage of aggregate premiumwith BC between50–65 High mutation frequencies Low mutation frequencies

Age 30 Age 30 Age 30 Age 50 Age 30 Age 30 Age 30 Age 50

10 Yrs 20 Yrs 30 Yrs 10 Yrs 10 Yrs 20 Yrs 30 Yrs 10 Yrs

%%%% %%% %

1 102 103 102 102 101 101 101 1012 111 115 114 112 107 108 108 1073 129 152 154 153 119 133 134 1324 148 215 231 260 139 192 203 2155 153 254 282 377 150 245 271 344

108 107�2 111 115 114 113 107 109

(c) Table 11 shows extra premiums for family histories of OC before age 50. Withone affected relative, these are much lower than those in Table 1. The same istrue of the older applicant with two or more affected relatives. However, wehave chosen applicants age 30 and 50, while the age ranges in Table 1 are �40,41–50 and �50, and within these age ranges extra premiums will vary.

(d) Finally, Table 12 shows the effect of a family history of OC between ages50–65.

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 41

Table 11. The effect of the family history (OC before age 50) on le�el net premiumsas a percentage of the aggregate premium, unknown (M, X)

Premium as percentage of aggregate premiumNumber of relativeswith OC before 50

Low mutation rrequenciesHigh mutation frequencies

Age 50 Age 30 Age 30 Age 30Age 30 Age 50Age 30Age 30

10 Yrs 10 Yrs 20 Yrs 30 Yrs 10 Yrs30 Yrs10 Yrs 20 Yrs

% % % % % %% %

122 114 108 117 113 108 1051 129204 316 262 204249 1743294022336 730 586 420 3243 764 615 441366 856 689 491492 3666888534

710 507 374 885 713 507 3755 880205 317 263 205250 175330�2 404

Table 12. The effect of the family history (OC between ages 50–65) on le�el netpremiums as a percentage of the aggregate premium, unknown (M, X)

Premium as percentage of aggregate premiumNumber of relativeswith OC between

High mutation frequencies Low mutation frequencies50–65

Age 30 Age 30 Age 50 Age 30 Age 30 Age 30 Age 50Age 30

10 Yrs 10 Yrs 20 Yrs 30 Yrs 10 Yrs30 Yrs10 Yrs 20 Yrs

% % % % % %% %

102 105 104 103 1011 109 107 105115 137 128 119132 1091471612153 251 217 179 1343 320 272 216226 479 397 300340 1934545484

5 417 296 664 544 400 272688 566115 137 129 119132 109148161�2

It is surprising, at first sight, that lower mutation frequencies can leadto higher extra premiums, but it is easily seen that this must sometimeshappen. If the mutation frequencies were 0% or 100%, the extra premium wouldbe nil (in both cases there would be only one genotype). Therefore as themutation frequencies increase from 0% to 100%, the extra premium will rise andthen fall.

A. S. Macdonald et al. Scand. Actuarial J. 142

6. THE EFFECT OF LOWER PENTRANCE OF BRCA1 AND BRCA2MUTATIONS

In Part I, Section 8, we reduced the excess incidence rates of BC and OC to 50%and 25% of those observed, because the observations were in respect of veryhigh-risk families (25% corresponds roughly to the penetrance suggested by Hopperet al. (1999)). The effect is twofold:

(a) the probabilities that the applicant has a mutation fall substantially; and(b) the insurance costs in respect of mutation carriers will also fall.

Table 13 shows level net premiums expressed as a percentage of the aggregatepremiums with high mutation frequencies (compare with Table 6). (The aggregatepremiums assuming high and low mutation frequencies were almost identical.) Theeffect is substantial, especially where the additional genetic risk is highest. Forexample, the premium in respect of a woman age 50 with genotype (1, 1), for policyterm 10 years, falls from 1 711% to 518% of the aggregate. This is still much higherthan current underwriting limits; only BRCA2 mutations at younger ages fallwithin these. These are pure risk premiums; the premium for (say) an endowmentassurance with accelerated benefit would be a much smaller percentage of thestandard rate.

Table 13 assumes that the genotype is known; the reduction in premiums givenonly a family history will be even more significant. Table 14 shows the effect whenthe family structure is unknown, and only the number of affected relatives (in thiscase, with BC before age 50) is known. If the excess incidence of BC and OC is only25% of that observed in the highest risk families, a woman with two affectedrelatives is at negligible extra risk, and only when four or five relatives are affected

Table 13. Le�el net premium, depending on genotype, based on high estimates ofmutation frequencies, as a percentage of the aggregate premium. Excess BC and OCincidence rates 50% or 25% of the le�els obser�ed among high-risk families

Excess BC/OC risk Age 50 at entryAge 40 at entryAge 30 at entryGenotypeas % of observed

Term Term Term Term Term Term

10 Yrs20 Yrs10 Yrs30 Yrs20 Yrs10 Yrs

% % % % % %

50% (0, 0) 99 99 99 99 99 9950% (1, 0) 968 857 642 852 608 41950% (0, 1) 153 297 380 396 471 59550% (1, 1) 1 035 1 033 856 1 151 954 92525% 10010099999999(0, 0)25% (1, 0) 540 491 381 479 355 259

35229625025225% 201127(0, 1)(1, 1) 51856663551825% 592575

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 43

Table 14. Le�el net premium as a percentage of the aggregate premium, gi�en ahistory of BC before age 50, unknown (M, X). High estimates of mutation frequen-cies. Excess BC and OC incidence rates 50% and 25% of the le�els obser�ed amonghigh-risk families

Relatives with Premium as percentage of aggregate premiumBC before 50

Excess 50% of observed Excess 25% of observed

Age 50 Age 30 Age 30 Age 30 Age 50Age 30 Age 30 Age 30

10 Yrs 10 Yrs 20 Yrs 30 Yrs20 Yrs 10 Yrs10 Yrs 30 Yrs

% % % % %%%%

101 101 101 1011 100103 103 102106 105 105 103115 1022 121126

1673 130 120 117 112 106218 1982564 174 157 149 135 118373 327

201 215 200 170311 1374064665122 115 107 105 105 103 102�2 126

would substantial extra premiums be considered. This appears to change theunderwriting outlook completely.

However, we ought to consider how likely it is that all the cases shown in Table14 will occur in practice. It is possible that the incidence of BC and OC observedin high risk families is, in fact a fair estimate in the presence of a strong familyhistory. Then, Table 5 might be appropriate for women whose mutations werediscovered because of their family history, whereas Table 13 might be appropriateif mutations were discovered for other reasons, for example screening. It is thendifficult to judge what parts of Table 14 might be appropriate in cases of a modest,but not extreme, family history. Possibly the extra information contributed by someknowledge of the family structure would be helpful.

We must emphasise that we are only able to show a few examples, and that thosewe have shown may tend to include some of the higher risks.

7. THE POTENTIAL FOR ADVERSE SELECTION

To model the potential for adverse selection we must introduce the rate at which CIinsurance is purchased, or equivalently the size of the CI insurance market. Thelarger the market, the less impact adverse selection can have. Following Macdonald(1997, 1999), and Subramanian et al. (1999) we extend the multiple state model sothat having a genetic test and buying insurance are represented by transitionsbetween states (Fig. 6). A person starts at age x with no CI insurance, not havinghad a genetic test, and may then be tested and/or buy CI insurance, expiring at agex+T. Each genotype is represented by such a model with different intensities, sothere are 24 states in total; a life with genotype gi moves from state ij to state ik

A. S. Macdonald et al. Scand. Actuarial J. 144

Fig. 6. A Markov model of the insurance purchase and CI insurance events for a person with genotypegi.

according to the intensity �x+ tijk . On such a transition, a payment bx+ t

ijk may bemade. While in state ij, a premium may be payable, continuously, at rate bx+ t

ij . LetVx+ t

ij be the statewise prospective reserve (the EPV of the future loss if the life, atage x+ t, is in state ij ). It satisfies Thiele’s equations:

dVx+ tij

dt=�Vx+ t

ij +bx+ tij − �

k� j

�x+ tijk (bx+ t

ijk +Vx+ tik −Vx+ t

ij ), (9)

which can be solved backwards from the terminal conditions Vx+Tij =0. We used a

force of interest �=0.05 per annum, and a Runge-Kutta method with step size0.0005 years.

Here, premiums are payable while in an insured state, and a benefit on movementinto the ‘CI Event’ state (representing stand-alone CI). A problem arises indetermining the premium rate bx+ t

ij . It can depend on the current age, as shown,but in the Markov framework it cannot depend on age at entry. This is easilysolved in theory by taking bx+ t

ij to be the population average intensity of a CI eventat age x+ t (times the sum assured), following the common practice of unit-linkedbusiness of charging for risk as it arises. We proceed by assuming the mutationfrequencies of Table 1 of Part I at age 30, solving the Kolmogorov forwardequations for the occupancy probabilities in Fig. 6, then using these to weight theintensities of CI events at age x+ t to obtain bx+ t

ij . We also use these occupancyprobabilities to find genotype frequencies for applicants at ages x�30.

Adverse selection is modelled by choosing intensities and benefits so that:

(a) mutation carriers may be more likely to have a genetic test;(b) those who test positive may be more likely to buy insurance;(c) those who test positive may tend to buy larger amounts of insurance; and(d) the insurer charges everyone a premium based on the incidence of CI claims in

the whole population.

Note that (d) above is a very strong assumption, likely to overstate the extent ofadverse selection, because it ignores any extra premiums that may be charged on

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 45

the basis of family history. It would be possible to subdivide each genotype further,into those with a strong family history and those without, but we lack any basis fordoing so at present.

We consider the reserves in the starting states, rather than in the insured states.This allows for insurance-buying behaviour; the more likely someone is to buyinsurance, the higher the necessary reserve at outset. The size of the market in theabsence of genetic tests is then represented by the intensities �x+ t

i01 which we supposeall to be the same. According to Dinani et al. (2000) sales of individual CIinsurance are consistently rising, to almost 700 000 policies in 1998. About2 400 000 policies were estimated to be in force at the end of 1998, with another50 000 lives covered by group policies, amounting to 9% of the working population.Most policies are accelerated benefit riders written with term or endowmentassurances. The market is expected to expand, especially as sales of traditionalsavings contracts decline.

We choose intensities �x+ ti01 =0.001, 0.01 and 0.05 to represent low, medium and

high levels of market growth. With �x+ ti01 =0.001, a person has about a 1% chance

of buying a policy within 10 years, or a 3% chance within 30 years. With�x+ t

i01 =0.05, these chances are about 40% and 80%, respectively.Table 15 shows the EPVs of the benefit cost under a level £1 CI contract, for

various ages and ‘terms’, for a woman who is untested and uninsured at outset,with no adverse selection, assuming high estimates of mutation frequencies andexcess BC and OC risk 100% of that observed. (Note that ‘term’ here does notmean policy term; it means the period during which uninsured women may buyinsurance expiring at the end of the ‘term’. For example, ‘age 30, term 10 years’means that women uninsured at age 30 buy insurance during the next 10 years, allpolicies expiring at age 40.) The results for lower frequencies and penetrances arevery similar. The EPVs of the losses are zero (to 5 decimal places).

To measure the costs of adverse selection, we choose transition intensities torepresent the levels of genetic testing and insurance purchase, and find the EPV ofthe loss in respect of a woman picked at random from the population. This will

Table 15. Expected Present Value (EPV) of benefit under a CI insurance of £1, fora woman untested and uninsured at outset, with no ad�erse selection. High estimatesof mutation frequencies, and excess BC and OC incidence rates 100% of thoseobser�ed. �x+t

i01 represents the normal rate at which CI insurance is purchased

�x+ti01 Age 30 at entry Age 40 at entry Age 50 at entry

Term Term Term Term Term Term

10 Yrs10 Yrs 20 Yrs 30 Yrs 10 Yrs 20 Yrs

0.0003180.001 0.000060 0.000328 0.000870 0.000157 0.0007060.0030910.01 0.000583 0.003082 0.007930 0.001521 0.0066440.0136090.0259410.0066720.05 0.0274970.0119070.002554

A. S. Macdonald et al. Scand. Actuarial J. 146

increase because of the assumed behaviour of those with gene mutations. The costof adverse selection is then:

EPV of loss with adverse selection−EPV of loss without adverse selectionEPV of benefit without adverse selection

. (10)

Expressed as a percentage, this is exactly the increase in premiums that should becharged to everyone to meet the cost of the adverse selection.

Table 16 shows the effect of extremely high levels of genetic testing (�x+ ti02 =1.0)

and adverse selection (�x+ ti23 =1.0 if a mutation is present) but no tendency for

‘adverse selectors’ to take out more than the average sum assured. Such a high levelof genetic testing is not far short of mass screening of the population over a fewyears, and this level of insurance purchase means that a woman with a mutation istwenty times more likely to insure than a woman without a mutation, even in thelarge market, and 1 000 times more likely to insure in the small market. Theseassumptions ought to be worse than is ever likely in practice.

Table 16 is based on the higher estimates of mutation frequencies. The lowerestimates result in costs of adverse selection consistently just over half of thoseshown, and we omit them.

Clearly, the size of the market has an overwhelming influence on the cost ofadverse selection, which could be very large indeed. In the most favourablecircumstances (low penetrance and mutation frequencies and a large market) thecost amounts to less than 10% of premiums, and less than 1% in many cases. Thisshould not be regarded as negligible, since we are considering just one rare geneticcondition, but on the other hand it is based on extreme assumptions.

Table 16. Percentage CI premium increases arising from high le�els of genetic testing(�x+t

i01 =1.0) and high ad�erse selection (�x+ti23 =1.0 if a mutation is present). Ad�erse

selectors take out the a�erage CI sum assured. High estimates of mutation frequen-cies. �x+t

i01 represents the normal rate at which CI insurance is purchased

�x+ti01 Age 50 at entryExcess BC/OC risk Age 40 at entryAge 30 at entry

as % of observedTerm TermTerm Term Term Term

10 Yrs20 Yrs10 Yrs30 Yrs20 Yrs10 Yrs

% % % % % %

9125579167 964000.001100%100% 0.01 39 16 8 25 99

5 2 20.05100% 7 3 10.001 217 9850% 50 163 63 76

50% 0.01 21 9 5 16 6 70.0550% 113124

48379328531120.00125%93511 40.0125% 5

125% 0.05 2 1 0 2 1

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 47

In the U.K., the size of the market is already such that the worst cases in Table16 are unlikely to occur, but there are significant implications for countries in whichCI insurance may, in future, build up from a small base, such as the U.S.A.

Table 17 shows the costs of adverse selection, assuming much lower rates ofgenetic testing (�x+ t

i02 =0.1) and adverse selection (�x+ ti23 =0.1 if a mutation is

present); however, these are still quite high rates. The cost falls quite dramatically.Table 18 shows the effect of adverse selectors buying two or four times the

average CI sum assured. For brevity, only the lower penetrances are shown, and theother assumptions are as in Table 17. This aspect of adverse selection presents avery significant risk, again bearing in mind that we are considering just one raredisorder. It supports the general conclusion reached by other studies, that anyrestrictions on the use of genetic information in underwriting should should beconsidered carefully in relation to high sums assured.

Overall, we conclude that adverse selection is only likely to be significant if:

(a) the CI insurance market is very small (smaller than its current size in the U.K.);(b) very high sums assured can be obtained without disclosing genetic test results

or family history; or(c) the higher penetrances observed in the highest risk families apply more widely.

The exclusion of family history underwriting may mean that we have overstatedthe costs of adverse selection; on the other hand, BC/OC is just one single-genedisorder, and it may be unwise to extrapolate these conclusions to the wholecollection of such disorders.

Table 17. Percentage CI premium increases arising from moderate le�els of genetictesting (�x+t

i02 =0.1) and moderate ad�erse selection (�x+ti23 =0.1) if a mutation is

present). Ad�erse selectors take out the a�erage CI sum assured. High estimates ofmutation frequencies. �x+t

i01 represents the normal rate at which CI insurance ispurchased

�x+ti01 Age 50 at entryExcess BC/OC risk Age 40 at entryAge 30 at entry

as % of observedTermTerm Term Term Term Term

10 Yrs20 Yrs10 Yrs30 Yrs20 Yrs10 Yrs

% % % % % %

21313050 10570.001100%100% 0.01 5 4 3 23 1

0 00.05100% 1 0 0 00.001 31 3150% 20 20 15 8

50% 0.01 3 3 2 2 1 10.0550% 0 00000

610121217160.00125%1112 010.0125%

25% 0.05 0 0 0 0 0 0

A. S. Macdonald et al. Scand. Actuarial J. 148

Table 18. Percentage CI premium increases arising from moderate le�els of genetictesting (�x+t

i02 =0.1) and moderate ad�erse selection (�x+ti23 =0.1) if a mutation is

present). Ad�erse selectors take out the two or four times the a�erage CI sum assured.High mutation frequences and excess BC and OC incidence 25% of that obser�ed.�x+t

i01 represents the normal rate at which CI insurance is purchased

Age 40 at entrySum assured of Age 50 at entry�x+ti01 Age 30 at entry

‘adverse selectors’Term Term Term TermTerm Term

30 Yrs 10 Yrs 20 Yrs 10 Yrs20 Yrs10 Yrs

% % % %%%

24 23 19 112×average 0.001 32 352 2 2 132×average 30.01

12×average 0 0 0 00.05 0704×average 49 47 39 230.001 64

5 5 4 2760.014×average1 1 1 1 1 00.054×average

8. CONCLUSIONS

8.1. Genotype-specific critical illness insurance premiums

We calculated level CI insurance premiums assuming a BRCA1 or BRCA2mutation was known to be present, based on incidence rates consistent with studiesof families with a strong history of BC or OC. Level extra premiums were veryhigh, some exceeding +1 000%, and depended on age and term. Only one example(applicant age 30, term 10 years, BRCA2 mutation) would have been insurable atall, given current practice.

8.2. Family history underwriting

CI premiums are considerably lower if there is a family history, but no genetic testresult is known. In all the examples we considered, having one relative with BC orOC before age 50 implied an extra premium of no more than about +25%; havingtwo such relatives implied a substantial extra premium, and in a few cases (of OC)was higher than the usual underwriting limits.

The extra premiums varied considerably with age, policy term, and the level ofknowledge of the family structure, as well as the family history.

8.3. Dependence on penetrance and mutation frequencies

We considered excess incidence rates of BC and OC of 50% or 25% of thoseobserved, since the observed rates are based on families selected because of theirextreme histories.

(a) Extra premiums in the presence of a mutation fell significantly, but the onlycases that fell within the typical underwriting limits were BRCA2 mutations,for younger applicants.

Scand. Actuarial J. 1 The genetics of breast and o�arian cancer II 49

(b) Extra premiums based on family histories also fell, to the extent that withexcess incidence rates 25% of those observed, only women with at least three orfour affected relatives might be charged significant extra premiums.

If these mutations are heterogeneous in their effects then a genetic test result willhave different implications, depending on the family history or the reasons forhaving had the test. For example, a woman who was tested because of her familyhistory might be at much greater risk than a woman who discovered she had amutation through screening. It may also be difficult to make reliable risk estimatesbased only on family histories, where these are not extreme. In this way, BC andOC are quite different from conditions like Huntington’s disease, which only rarelyoccur sporadically. If genetic tests for BRCA1 and BRCA2 mutations are everconsidered for insurance use, they perhaps should only be used in conjunction withother information.

The frequencies of BRCA1 and BRCA2 mutations in the population are uncer-tain; we took high and low estimates from a recent study (Parmigiani et al., 1998).Higher estimates gave higher extra premiums in almost all cases, so would appearto be conservative. Also, higher frequencies could be regarded as implicit allowancefor the possible existence of other BRCA genes in which mutations have similareffects to mutations in BRCA1 and BRCA2. However, higher frequencies do notnecessarily result in higher extra premiums, and the fact that they usually did hereis only an empirical observation.

8.4. Ad�erse selection

We estimated the costs of adverse selection in terms of the percentage increases inordinary premiums needed to recoup losses from this source. Adverseselection appeared to be significant in small CI insurance markets, if pene-trance was as high as that observed in high-risk families, or if higher than averagesums assured could be obtained. Our results may overstate the costs, because weassumed high rates of genetic testing and adverse selection, and a form of premiumpayment that leads to higher costs than would level premiums; on the other handwe are only considering one, rare, genetic disorder. The potential impact on smallor emerging markets may be of some practical significance.

ACKNOWLEDGEMENTS

This work was funded by Swiss Re Life and Health, to whom we are grateful for financial support andfor many discussions with their actuarial, medical and underwriting staff and consultants. In particular,we wish to thank Douglas Keir, Dr. David Muiry and Dr. Hanspeter Wurmli.

REFERENCES

Dinani, A., Grimshaw, D., Robjohns, N., Somerville, A. S., Spry, A. & Staffurth, J. (2000). A criticalre�iew: report of the critical illness healthcare study group. Presented to the Staple Inn ActuarialSociety, London, on 14 March 2000.

A. S. Macdonald et al. Scand. Actuarial J. 150

Hopper, J. L., Southey, M. C., Dite, G. S., Jolley, D. J., Giles, G. G., McCredie, M. R. E., Easton,D. F., Venter, D. J. & The Australian Breast Cancer Family Study (1999). Population-based estimatesof the a�erage age-specific cumulati�e risk of breast cancer for a defined set of protein-truncatingmutations in BRCA1 and BRCA2. Preprint, University of Melbourne Centre for Genetic Epidemiol-ogy.

Macdonald, A.S. (1997). How will improved forecasts of individual lifetimes affect underwriting?Philosophical Transactions of the Royal Society B, 352, 1067–1075, and (with discussion) BritishActuarial Journal 3, 1009–1025 and 1044–1058.

Macdonald, A. S. (1999). Modeling the impact of genetic on insurance. North American ActuarialJournal 3 (1), 83–101.

McCormick, A., Fleming, D. & Charlton, J. (1995). Morbidity statistics from general practice. Fourthnational study 1991–1992. Series MB5 No. 3. OPCS, Government Statistical Service.

Norberg, R. (1995). Differential equations for moments of present values in life insurance. Insurance:Mathematics & Economics 17, 171–180.

O.N.S. (1999). Cancer 1971–1997 CD-ROM. Office for National Statistics, London.O.P.C.S. (1991). 1990 mortality statistics: Cause. Series DH 12 No. 17. Office of Population Censuses

and Surveys.O.P.C.S. (1993a). 1991 mortality statistics: Cause. Series DH 12 No. 18. Office of Population Censuses

and Surveys.O.P.C.S. (1993b). 1992 mortality statistics: Cause. Series DH 12 No. 19. Office of Population Censuses

and Surveys.Parmigiani, G., Berry, D. & Aguilar, O. (1998). Determining carrier probabilities for breast cancer

susceptibility genes BRCA1 and BRCA2. American Journal of Human Genetics 62, 145–158.Stewart, J., Dundas, R., Howard, R., Rudd, A. & Wolfe, C. (1999). Ethnic differences in incidence of

stroke: prospective study with stroke register. British Medical Journal 318, 967–971.Subramanian, K., Lemaire, J., Hershey, J. C., Pauly, M. V., Armstrong, K. & Asch, D. A. (1999).

Estimating adverse selection costs from genetic testing for breast and ovarian cancer: The case of lifeinsurance. Journal of Risk and Insurance 66, 531–550.

Address for correspondence:Angus S. Macdonald, Howard R. Waters and Chessman T. WekweteDepartment of Actuarial Mathematics and StatisticsHeriot-Watt UniversityEdinburgh EH14 4AS, UKE-mail: A.S.Macdonald@ma.hw.ac.uk