THE GENETICS OF BREAST AND OVARIAN CANCER V: APPLICATION TO INCOME PROTECTION INSURANCE. By Baopeng Lu, Angus Macdonald and Howard Waters.

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1 1 THE GENETICS OF BREAST AND OVARIAN CANCER V: APPLICATION TO INCOME PROTECTION INSURANCE By Baopeng Lu, Angus Macdonald and Howard Waters abstract In Part IV we presented a comprehensive model of a life history of a woman at risk of breast cancer (BC), in which relevant events such as diagnosis, treatment, recovery and recurrence, were represented explicitly, and corresponding transition intensities were estimated. In this part, we study some applications to income protection insurance (IPI) business. We calculate premiums based either on genetic test results or more practically on a family history of BC. We then extend the model into an IPI market model by incorporating rates of insurance-buying behaviour, in order to estimate the possible costs of adverse selection, in terms of increased premiums, under various moratoria on the use of genetic information. keywords Breast Cancer; BRCA1 Gene; BRCA2 Gene; Family History; Income Protection Insurance contact address Angus Macdonald, Department of Actuarial Mathematics and Statistics and the Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, U.K. Tel: 44(0) ; Fax: 44(0) ; A.S.Macdonald@ma.hw.ac.uk 1. Introduction In its Report No.12, the Continuous Mortality Investigation Bureau (CMIB) presented a multiple state stochastic model for the analysis of IPI (previously known as Permanent Health Insurance (PHI)), as shown in Figure 1. The transition intensities σ x, ρ x,z and ν x,z were estimated and graduated by mathematical formulae, using data from UK insurers, in respect of individual IPI policies for males between 1975 and Our purpose is to estimate IPI policy values for a woman with a familial risk of breast cancer (BC). This leads us to distinguish BC from other illnesses causing IPI claims and to model the life history of a woman at risk of BC. In Part IV we presented such a model, incorporating risk factors usually taken into account in IPI underwriting, as shown in Figure 2. In Part IV we described and interpreted the model in Figure 2 and estimated the following transition intensities: (a) Tumour detection rates for women with BC at various stages (Part IV, Section 6). (b) Rates of recovery based on treatment options (Part IV, Section 9). (c) Recurrence rates of BC depending on the type of recurrence, primary treatment and the patient s genotype (Part IV, Section 11) In Part V, we will estimate the other transition intensities, which fall into two groups:

2 Application to Income Protection Insurance 2 σ State 0 x State 2 Healthy Sick ρ x,z µ x ν x,z State 3 Dead Figure 1: The CMIB s three-state semi-markov model for sickness. (a) Onset rates of sicknesses other than, and rates of recovery from these sicknesses (see Section 2). (b) Rates of mortality (see Section 3). In Section 4, we divide the population into five subpopulations depending on the risk that a woman and/or her family carry a mutation in either the BRCA1 or BRCA2 genes, which confer high risks of BC. We apply the model to IPI premium rating in Section 5, assuming the insurer may take account of known BRCA1/2 mutations or a family history of BC. In Section 6, we extend the model to represent the IPI market by considering insurancebuying behaviour, and estimate the costs of adverse selection under a moratorium on the use of genetic information. Our conclusions are given in Section 7. All references were given in Part I. 2. Intensities for Sicknesses other than Breast Cancer In Figure 2, we depict a model of the life history for a woman with BC, and we estimated relevant transition intensities concerning the detection, recovery and recurrence, respectively, in Part I. In addition, we assume that a woman at risk of BC may develop other sicknesses independently, and we parameterise these additional transition intensities as follows. 2.1 The Onset Rate of Other Sicknesses The basic IPI model was reported in CMIR 12 (1991), which showed the graduated sickness inception intensities for four deferred periods (DPs):

3 i µ 1,0 xt,z State i0 State i1 Other Sickness i µ 0,3 xt Normal i µ 0,1 i µ 0,4 xt xt i µ 0,2 xt State i2 ➀ Non-invasive BC Diagnosed i µ 0,5 xt State i3 ➀ Early BC Diagnosed State i4 ➁ LABC Diagnosed State i5 ➃ Metastatic BC Diagnosed i µ 2,6 z i µ 3,6 z i µ 4,8 z State i6 i µ 6,7 xt State i7 Able to Other Work i Sickness µ 7,6 xt,z i µ 6,10 z State i10 ➁ Non-distant i Recurrence µ 6,11 z State i11 ➃ Distant Recurrence State i8 i µ 8,9 xt State i9 Able to Other Work i Sickness µ 9,8 xt,z i µ 8,12 z State i12 ➂ Non-distant i Recurrence µ 8,13 z State i13 ➃ Distant Recurrence i µ 10,14 z i µ 14,15 xt i µ 12,16 z i µ 16,17 xt State i14 Able to Work State i15 Other Sickness State i16 Able to Work State i17 Other Sickness i µ 15,14 xt,z i µ 17,16 xt,z i µ 14,18 z i µ 16,19 z State i18 ➃ Distant Recurrence State i20 Dead State i19 ➃ Distant Recurrence Figure 2: A semi-markov model of the life history of a woman with genotype ith. ➀: treatment of BCS with radiotherapy; ➁: combined treatment of mastectomy (bilateral mastectomy for BRCA1/2 mutation carriers), radiotherapy and chemotherapy; ➂: chemotherapy for ipsilateral recurrence or combined treatment for contralateral recurrence; and ➃: systematic treatment. Intensities are functions of current age, x t, and duration in current state, z. Application to Income Protection Insurance 3

4 Application to Income Protection Insurance 4 Table 1: The modifying factor M inception of claim inception rates based on the standard female experience for Source: CMIR 20 (2001). DP 1 week DP 4 weeks DP 13 weeks DP 26 weeks σ R12 x = exp( x x x 3 ) for DP 1 week exp( x x x 3 ) for DP 4 weeks exp( x x x 3 ) for DP 13 weeks exp( x x 2 ) for DP 26 weeks. (1) Here, σx R12 is the rate of onset of all sicknesses causing IPI claims at age x, based on CMIR 12. In CMIR 20, these were adjusted and extended to females, by applying a multiplicative factor, M sick, which was estimated from new data collected between 1995 and Let σx R20 represent these onset rates. In our model, we distinguish BC from other sick other sicknesses and represent the rate of other sickness, µ x by deducting the onset rate of BC from the adjusted rates for females: other sick µ x = σ R20 x µ BC x = M sick σ R12 x µ BC x. (2) However, CMIR 20 did not show M sick, but factors modifying claim inception rates, M inception, as in Table 1. Let d be the deferred period (DP) in years, and w = 52 approximate the number of weeks in a year. Let σ x denote the claim inception rate at age x, and π y,d (or d p SS y,0, equivalently) be the probability that an individual who falls sick at age y, will remain sick for at least the deferred period d of the policy. Let r d (y) be the proportion of potential claims actually reported at age y for policies with deferred period d. Denote quantities based on CMIR 12 and CMIR 20 by superscripts R12 and R20, respectively, for example rd R12 (y) and rd R20 (y). Since policyholders will have to stay sick over the DP to claim an IPI benefit, we have: that is: σ R20 yd σ R12 yd = σr20 y σy R12 πy,d R20 rr20 d (y d) πy,d R12 rr12 d (y d) M inception = M sick πr20 y,d rr20 d (y d) πy,d R12 rr12 d (y d) (3) (4)

5 Application to Income Protection Insurance 5 Transition intensity of sickness other than BC for females DP 1 week DP 4 weeks DP 13 weeks DP 26 weeks Age Figure 3: Onset rates of sickness other than BC for different DPs, based on the standard female experience for and hence: M sick = M inception πr12 y,d rr12 d (y d) We can compute this as follows. First, we have: ( π y,d = exp d 0 πy,d R20 rr20 d (y d). (5) ) (ρ ys,s ν ys,s ) ds. (6) Next, the proportion r d (y) was introduced in CMIR 13 (1993) (there denoted r d (y)) to make some allowance when comparing actual claim inceptions with those expected according to the model in CMIR 12. In CMIR 12, it was observed that for DPs 4, 13 and 26 weeks, the recovery rates during the four weeks immediately following the end of the DP were considerably lower than those for DP 1 week at the same durations. This was explained by the possibility that some policy holders whose sickness lasts only a little beyond the end of the DP do not bother to claim benefit, called non-reported claims. For DP d, define pss d (y,z 1,z 2 ) to be the probability that a life who became sick at age y and is still sick at duration z 1 remains sick until duration z 2, which can be computed in a similar way to Equation (6). Then: r d (y) = pss 1(y, d, d 4/w ) pss d (y, d, d 4/w ). (7)

6 Application to Income Protection Insurance 6 Table 2: The modifying factor M recovery of claim recovery rates based on the standard female experience for Source: CMIR 20 (2001). DP 1 DP 4 DP 13 DP 26 DP/w z < (DP 4)/w Otherwise Note as in Part IV, Section 9.2, women diagnosed with BC usually have sickness periods much longer than typical DPs and it is reasonable to assume that they will claim at the end of the DP. Therefore, in respect of them we would assume π y,d 1 and r d (y) = 1. Figure 3 shows the resulting intensities of other sickness for different DPs, based on the standard female experience for The Rates of Claim Recovery from Other Sicknesses The claim recovery rates depend on current age xt and sickness duration z. CMIR 12 introduced graduations of recovery rates based on data for males, and CMIR 20 suggested ways to modify these to obtain intensities appropriate for: (a) more recent data ( ) and; (b) females as well as males. The resulting recovery rates for females, , given deferred period DP weeks are: ρ R20 yz,z = M recovery ρ R12 yz,z = M recovery { ( max(4 wz, 0))} r Z (Y 50) e Z (8) where: y = the exact age at the time of falling sick; z = the sickness duration in years, wz = the sickness duration in weeks; { y z 5 Y = y z 5 otherwise; z z 1 Z = (z 1) 1 < z otherwise; { min( (wz DP), 1) DP = 4, 13, 26 weeks only r = 1 otherwise. From CMIR 20, the factors M recovery are as given in Table 2 for females in Mortality In Figure 2, each state has a transition into the Dead state. There are three categories of mortality depending on the states occupied.

7 Application to Income Protection Insurance 7 Death Ratio Because of BC Figure 4: Crude and graduated proportions of total female deaths that are due to BC. Age 3.1 Baseline Mortality Our baseline mortality is that of English Life Tables No. 15 (Females) (ELT15F) for women in normal health. Women in the Able to Work states are just recovering from BC treatment and any extra mortality because of BC should be accounted for by later BC events. Therefore, we assume baseline mortality for them, but removing deaths caused by BC. We graduated the ratio, denoted φ x, of the number of deaths from BC, to the total number of deaths, using UK population data in (OPCS 1991, 1993(a), (b); ONS, 1999) as follows: 6.58 φ x = Γ(15.37) ( e 0.32x x ) x 54 (9) x x 2 x > 65 with linear interpolation between ages 54 and 65. The results are shown in Figure Mortality Since Diagnosis of Breast Cancer After onset of BC, patients suffer additional risk of death depending on the duration since diagnosis and the stage of BC. Let t p cancer x,z be the t-year survival probability for a BC patient at age x with duration z since diagnosis and let µ cancer x,z be the corresponding force of mortality; define t p standard x and µ standard x analogously for the standard population. Here, we assume: where µ extra z µ cancer x,z = µ standard x µ extra z is additional mortality for women diagnosed with BC, which depends only

8 Application to Income Protection Insurance 8 Relative survival rate SEER Rates 95% CI Fitted Functions Time since diagnosis Figure 5: Relative survival rate for women who develop early breast cancer. on sickness duration, z. Then we define relative survival rates (RSR) for women suffering BC as follows: RSR = z p cancer x,0 zp standard x = exp ( z 0 (µstandard xs µ extra s )ds ) exp ( z 0 µstandard xs ds ) = exp ( z 0 ) µ extra s ds. (10) Using SEER Stat version and based on the SEER 9 Regs Public-use ( ) database, we produced RSR for women suffering each type of BC, and fitted them using smooth functions of the duration since diagnosis, z, as shown in Figures 5 7. Here, the standard population is the US Standard Population in 2000 and we assume similar mortality trends in respect of women at risk of BC in the US and in the UK. Therefore, the extra mortality, µ extra z, can be deduced from the fitted RSR functions for each stage of BC: µ extra z = z z z z z 5 for Early BC z z z z z 5 for LABC z z z z z 5 for Metastatic BC. (11)

9 Application to Income Protection Insurance 9 Relative survival rate SEER Rates 95% CI Fitted Functions Time since diagnosis Figure 6: Relative survival rate for women who develop locally advanced breast cancer. Relative survival rate SEER Rates 95% CI Fitted Functions Time since diagnosis Figure 7: Relative survival rate for women who develop metastatic breast cancer.

10 Application to Income Protection Insurance 10 Table 3: The modifying factor M sick mort of mortality rates after onset of sickness other than BC based on the standard female experience for Source: CMIR 20 (2001). DP 1 week DP 4 weeks DP 13 weeks DP 26 weeks Mortality after Onset of Other Sickness The mortality rate while ill was reported in CMIR 12 and adjusted for females based on updated data in CMIR 20. We reproduce the results in Equation (12): ν R20 yz,z = M sick mort νyz,z R12 { = M sick mort ( Y Y 2 ) exp ( /(Z } ) ) (Y Z) e (Z ) (12) where y, z and Y are as in Equation (8), Z = min(z, 5) and M sick mort is given in Table Risk Subpopulations and The Onset of Family History 4.1 Risk Subpopulations Given our estimates in the previous sections and in Part IV, we can straightforwardly calculate premiums based on genotype. However, in most countries, insurers are prevented from knowing or using genetic test results, except perhaps for very large policies. Therefore, family history is probably the most important item of information that may be used in underwriting and insurers need to estimate the probabilities: P[ Applicant carries a causative mutation Family history ]. (13) Gui et al. (2006) introduced an approach to estimating these by: (a) partitioning the population of insurance applicants based on knowing whether or not a BRCA1 or BRCA2 mutation is present in the family and in the applicant; and (b) calculating the probability of a family history developing in each such subpopulation. Then Bayes Theorem gives the probabilities (13). Following their approach, we divided the general population into five subpopulations labeled with index i: (a) Family with no BRCA1 or the BRCA2 mutation carriers (i=0); (b) BRCA1 mutation carrier family but applicant is not a carrier (i=1); (c) BRCA1 mutation carrier family and applicant is a carrier (i=2); (d) BRCA2 mutation carrier family but applicant is not a carrier (i=3); (e) BRCA2 mutation carrier family and applicant is a carrier (i=4); where a mutation carrier family was defined as a nuclear family (parents and siblings) in which one parent carries a mutation. Table 4 shows the resulting proportions of the five

11 Application to Income Protection Insurance 11 Table 4: Proportions of the five subpopulations in the whole population. Source: Gui et al. (2006). i Subpopulation Proportion 0 Non BRCA1/2 carrier family BRCA1 carrier family, applicant non-brca1 carrier BRCA1 carrier family, applicant BRCA1 carrier BRCA2 carrier family, applicant non-brca2 carrier BRCA2 carrier family, applicant BRCA2 carrier Table 5: Adjusted proportions of the five subpopulations in the whole population, where BRCA1/2 mutation carriers who undergo bilateral prophylactic mastectomy are included in subpopulation 0. i Subpopulation Proportion 0 Non BRCA1/2 carrier family BRCA1 carrier family, applicant non-brca1 carrier BRCA1 carrier family, applicant BRCA1 carrier BRCA2 carrier family, applicant non-brca2 carrier BRCA2 carrier family, applicant BRCA2 carrier subpopulations in the whole population. See Gui et al. (2006) for further details. The life history of a member of subpopulation i is represented by the model in Figure 2, with states labelled ij and transition intensities labelled i µ jk x and i µ jk x,z. However, in Part IV, Section 8.2 we have shown that unaffected BRCA1/2 mutation carriers who undergo bilateral prophylactic mastectomy (BPM) are exposed to the population risk of BC, and hence would be included in subpopulation 0. From Part IV, we assume the uptake rate of BPM for unaffected BRCA1/2 mutation carriers is 20% and show the adjusted results in Table The Development of a Family History of BC Gui et al. (2006) defined a dynamic model of family history in respect of BC and ovarian cancer (OC), which is based on underwriting practice and uses first-degree relatives only (FDRs, meaning siblings and parents). In their model, a family history is defined as two FDRs who develop BC or OC before age 50. They also simplify the family structure by assuming that the applicant s siblings are all the same age as her, and their mother is 30 years older. Then they estimate the rate of onset of a family history in each subpopulation. See Gui et al. (2006) for further details. We follow Gui et al. (2006) but ignore OC and define a family history as two FDRs with BC before age 50. Figure 8 shows rates of onset of family history of BC for each subpopulation. The discontinuity in these rates is caused by the NHS Breast Screening Programme.

12 Application to Income Protection Insurance 12 Rate of Onset of Family History For subpopulation 0 For subpopulations 1 and 2 For subpopulations 3 and Age Figure 8: Rates of onset of family history of BC in risk subpopulations, following the pproach of Gui et al. (2006). 5. Application of the Life History Model to IPI Business 5.1 Numerical Procedures Define H to be the set of all healthy states, including Normal and Able to Work states; S to be the set of Other Sickness states; and B to be the set of BC states (including recurrences). Suppose an applicant in subpopulation i, insured at age x, starting in state i0, is in state ij at age x t and has been in that state for duration z. A continuous premium at annual rate a ij xt would be paid if ij H or ij S B with z d; and a continuous benefit at annual rate b ij xt will be claimed if ij S B with z > d. Let i z,t Vx j be the prospective policy value for a life in state ij at age xt with duration z since entry into the state. If any transition out of a state depends on duration, we call that state duration-dependent. To realize a semi-markov model numerically, Gutiérrez & Macdonald (2004) used the fact that the policy liability can be extinguished at any time by payment of the prospective policy value (we imagine the payment is a lump-sum premium to a reinsurer). If such a reinsurance payment is carried out immediately upon entering any duration-dependent state ij, only the policy values i 0,t Vx j upon entry need be calculated. Thus z is eliminated and the model is restored to a Markov framework, and expected present values (EPVs) can be found numerically by Thiele s Equations. All states in S B are duration dependent, so our approach is to start with terminal (dead) states and work backwards, tabulating values of i 0,t Vx j for any duration-dependent state we meet. Since all i 0,t Vx j s are calculated by numerical integration, we call this the backward integration method.

13 Application to Income Protection Insurance 13 For each state ij, let I ij be the set of states into which there is a transition from state ij. These are called inferior states of ij; and ij is called a superior state to states in I ij. Let d be the DP in years and TE the time to policy expiry. We define some useful probabilities. Let i t p jj x,z be the probability that a person in state ij at age x with duration z will remain in state ij for time t, at which time the duration must be z t. We can directly compute: ( ) t i p jj x,z = exp t 0 ik I ij i µ j,k xs,zs ds. (14) Also, let i w,t p jk x,z be the probability that a person in state ij at age x with duration z will be in state ik at age x t, with duration w. If w z t it is irrelevant and we just write i w,t p jj x,z = i t p jj x,z. We can compuet i t p jj x,z and i w,t p jk x,z using the the following differential equations: t ti p jj x,z w w,ti p jk x,z = = t i p jj x,z ik I ij t w i p jj x,z t i µ j,k xt,zt s i p jj x,z 0 ik I ij S ( i µ j,k xt w wi p kk xt w,0) w < t ik I ij 0 w t ( i µ j,k xs,zs i t s p kk xs,0 iµ k,j xt,t s) ds (15). (16) Using these, policy values with force of interest δ can be calculated as follows: (a) If ij B, TE 0,t i Vx j = e δs si p jj xt,0 b ij xts ds d TE ik I ij 0 d 0 e δs si p jj xt,0 a ij xts ds e δs si p jj xt,0 iµ j,k xts,s i 0,ts Vx k ds. (17) (b) If ij H, TE 0,t i Vx j = e δs 0 TE 0 TE 0 ik I ij B ik I ij S ( s i p jk xt,0 d,s i p jk xt,0) b ik xts r d (x t s) ds e δs si p jj xt,0 a ij xts ds e δs ik I ij S TE 0 ( d,s i p jk xt,0 a ik xts) ds e δs si p jj xt,0 iµ j,k xts,s i 0,ts Vx k ds (18)

14 Application to Income Protection Insurance 14 We have the following comments: (a) i µ j,k x,z can be written i µ j,k x or i µ j,k z if it depends only on current age x or duration z in the current state, respectively. (b) Equations (17) and (18) are recursive and for each state ij, i 0,t Vx j can be calculated once i 0,ts Vx k has been obtained for all its inferior states ages over x t. (c) Equations (15) and (16) were solved by a fourth-order Runge-Kutta method with step-size years and initial conditions i 0 p jj x,z = 1 and i 0,t p jk x,z = 0, for all t. We solved Equations (17) and (18) by numerical integration using Simpson s rule with step-size 1/156 year (1/3 week) and force of interest δ = 0.05; (d) In Equation (18), for each state ij H, by defining the probabilities i t p jj x,z and i w,t p jk x,z, we can discount all prospective cashflows in every state ik I ij S back to the time of entering state ij. In this way, healthy states and their corresponding Other Sickness states are treated as a unit and hence the periodic transitions between them are eliminated. 5.2 Premium Ratings for the General Population The EPV of a unit benefit or unit premium can be obtained by setting the other to zero in Equations (17) and (18). In Table 6, we present the EPVs of IPI cover of 1 per annum for various DPs and various scenarios. These can be interpreted as pure single premiums payable at outset. Their respective quotients give us the net rates of premium payable continuously, as shown in Table 7. We observe the following features: (a) For a given starting age and policy term, EPVs of benefit and level net premiums decrease as DPs increase. The reason is that shorter DPs imply earlier and longer benefit claims. (b) Level net premiums increase with entry age and policy term. Some are very large. For example, with DP 1 week, an applicant age 50 will pay per annum for a benefit of 1 per annum. If, in the model in Figure 2, we modify i µ 0,1 xt to include the onset rate of BC, and set the intensities i µ 0,2 xt, i µ 0,3 xt, i µ 0,4 xt and i µ 0,5 xt to zero, we should reproduce, approximately, the basic three-state model of CMIR 12 as modified in CMIR 20. Tables 8 and 9 show the resulting EPVs of the unit benefit and the level net premium. We can see that these values are close to, although a little bit lower than, their counterparts in Tables 6 and 7, which could be considered to support our estimates of the intensities in Figure 2.

15 Table 6: Expected present values of the unit benefit of IPI cover of 1 per annum payable continuously while sick with duration exceeding the DP, according to the standard female experience for and based on the model in Figure 2. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs DP DP DP DP Table 7: Level net premium for the unit claim annuity of IPI cover of 1 per annum, according to the standard female experience for and based on the model in Figure 2. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs DP DP DP DP Application to Income Protection Insurance 15

16 Table 8: Expected present values of the unit benefit of IPI cover of 1 per annum payable continuously while sick with duration exceeding the DP, according to the standard female experience for and based on the basic model in CMIR 12 and CMIR 20. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs DP DP DP DP Table 9: Level net premium for the unit claim annuity of IPI cover of 1 per annum, according to the standard female experience for and based on the basic model in CMIR 12 and CMIR 20. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs DP DP DP DP Application to Income Protection Insurance 16

17 Application to Income Protection Insurance Premium Ratings for Mutation Carriers In Table 10 we show level premium rates for women with a known BRCA1/2 mutation, as a percentage of the standard rates, which we take to be those for non-mutation-carriers. We make the following comments: (a) Generally, the relative premium increases for IPI are much smaller than those for life insurance or CI insurance reported by Gui et al. (2006). (b) Results for BRCA1 mutation carriers vary with the DP. All cases with DP 1 or 4 weeks are well within the typical underwriting limit (in the UK) of 250% of the standard rate. For longer DPs, some cases are outside this limit. (c) A BRCA2 mutation is much less severe than a BRCA1 mutation as we should expect from the rates of onset. Extra premiums for all BRCA2 cases are less than 50%, so all cases for BRCA2 mutation carriers would be insurable. (d) All cases with DP 1 week have premium increases less than 40%. In particular, for BRCA2 mutation carriers seeking DP 1 week policies, premium increases are less than or around 10%. In practice, this might mean no extra premium would be charged. 5.4 Premium Ratings for Women with a Family History Given family history only, a level premium is calculated satisfying the equivalence principle, where the EPVs of the benefit and the premium are weighted averages of the EPVs in respect of each subpopulation (see Section 4), the weights being the probabilities that an applicant age x with a family history is in each subpopulation. (see Section 4.2). (In practice, these weights are most easily computed using the market model which we will introduce in Section 6.2, simply summing the occupancy probabilities in all non-dead states subsequent to the onset of a family history.) Table 11 shows the premiums for applicants with a family history of BC as a percentage of those for standard risks. Our main observations are as follows: (a) Premiums follow a similar pattern to those in Table 10, but are smaller. Most cases with shorter DPs have extra premiums below 10%. Even for longer deferred periods, the highest extra premiums would not exceed 30%. Therefore, all women with a family history of BC are insurable and applicants with DP 1 week might often be offered ordinary rates. (b) For policies that expire after age 50, extra premiums fall as the age at entry and policy term increase. This is because surviving free of symptoms decreases the probability that the applicant is a mutation carrier. For example, the extra premium would not exceed 6% for policies with DP 1 week which start at age 30 and expire after age 50. (c) In most cases, extra premiums are higher for applicants with a family history than for BRCA2 mutation carriers. The reason is that the penetrance of BRCA2 mutations is much lower than that of BRCA1 mutations before age 40, so a family history that includes at least one affected sister makes it much more likely that a BRCA1 mutation is present.

18 Table 10: Level net premiums for IPI cover as a percentage of the premium for standard risks, for women with a known BRCA1 or BRCA2 mutation. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry Gene 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs Mutation % % % % % % % % % % BRCA1 DP DP DP DP BRCA2 DP DP DP DP Table 11: Level net premiums for IPI cover as a percentage of the premium for standard risks, for women with a family history of BC. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs % % % % % % % % % % DP DP DP DP Application to Income Protection Insurance 18

19 Application to Income Protection Insurance The Effect of Reduced Onset Rates Macdonald, Waters & Wekwete (2003b) allowed for ascertainment bias in an approximate fashion, by reducing by 50% or 75% the rate of onset for mutation carriers based on Ford et al. (1998), because the latter study used highly selected families. Antoniou et al. (2003), upon which this study is based, used less highly selected families but the possibility of ascertainment bias is still present, since selection was through an index (affected) patient. Therefore we allow approximately for possible bias by also considering onset rates 50% of those estimated directly from Antoniou et al. (2003). Tables 12 and 13 show level net premiums expressed as a percentage of the standard rates for mutation carriers and women with a family history, respectively, with incidence rates of BC for mutation carriers reduced by 50% of those estimated from Antoniou et al. (2003). We note that: (a) Apart from following the same pattern as the above tables, the effect of reducing the penetrance is striking, especially where the genetic risk is highest. With onset rates 50% of those observed, extra premiums drop by roughly half. (b) All premiums for BRCA1 mutation carriers fall within the typical underwriting limit of 250% of the standard rates, and hence all BRCA1 mutation carriers become insurable. (c) Most BRCA2 mutation carriers and women with a family history become quite moderate risks, especially for policies with shorter DPs, where extra premiums could even be negligible. 6. The Cost of Adverse Selection in an IPI Market 6.1 Adverse Selection Adverse selection can be a problem when there is asymmetric information between the seller of insurance and the applicants; in particular, insurance will often not be profitable when buyers have better information about their risk of claiming than has the seller. In the underwriting process, insurers pool accepted applicants with roughly equivalent levels of risk into underwriting classes and charge the same premium within each underwriting class. However, the existence of a moratorium could force very different risks into the same underwriting class. When there is adverse selection, people who know they have a higher risk of claiming than the average of the group may buy more insurance, while those who know they have a below-average risk may decide it is too expensive to be worth buying. In this case, premiums set according to the average risk in the underwriting class will not be sufficient to cover the claims that actually arise, because among the people who have bought insurance above-average risks will be over-represented. Putting up the premium may not solve this problem, for as the premium rises the insurance policy will become unattractive to more of the people who know they have a lower risk of claiming. If this cycle continues, it might, in theory, lead to the collapse of the market.

20 Table 12: Level net premiums for IPI cover as a percentage of the premium for standard risks, for persons with a known BRCA1 or BRCA2 mutation. Excess BC incidence rates are 50% of those observed. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry Gene 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs Mutation % % % % % % % % % % BRCA1 DP DP DP DP BRCA2 DP DP DP DP Table 13: Level net premiums for IPI cover as a percentage of the premium for standard risks, for persons with a family history of BC. Excess BC incidence rates are 50% of those observed. Age 20 at Entry Age 30 at Entry Age 40 at Entry Age 50 at Entry 10 Yrs 20 Yrs 30 Yrs 45 Yrs 10 Yrs 20 Yrs 35 Yrs 10 Yrs 25 Yrs 15 Yrs % % % % % % % % % % DP DP DP DP Application to Income Protection Insurance 20

21 Application to Income Protection Insurance 21 State i0 uninsured, not i µ 0,1 xt State i1 Insured, not State i6 Sick i µ 6,7 xt,z State i7 Dead i µ sickness xt i µ recovery xt,z i µ Dead xt tested, no FH i µ 0,2 xt State i2 uninsured, not tested, FH i µ 2,4 xt State i4 uninsured, i µ 2,3 xt i µ 4,5 xt tested, no FH State i3 Insured, not tested, FH State i5 Insured, tested, FH tested, FH Figure 9: A semi-markov model of family history, genetic testing and IPI purchase for a person in the i th risk subpopulation (FH = family history present). 6.2 The Life History Model of a Woman at Risk of Breast Cancer in an IPI Market The model in Figure 2 depicts the life history of a woman after she has bought insurance. Since adverse selection arises because of the decision to buy insurance, or not, we extend the model as shown schematically in Figure 2, so that buying insurance is an event represented by a transition between states. It represents a person s life history in an IPI market. The decision to buy insurance may be affected by the person s knowledge that she has a family history, or that she has had a genetic test and knows the result. Only women with a family history may be tested, which is realistic in countries such as the UK where access to testing is controlled by the public health service. After buying insurance (states i1, i3 and i5) a woman will follow the life history depicted in Figure 2. With this model, we can represent the following features of an IPI market: (a) Mutation frequencies are represented by the proportions in each subpopulation. (b) The development of family history and the incidence of genetic testing are represented explicitly by states and transitions between states, governed by transition intensities. (c) Underwriting classes are defined as sets of insured states whose composition may be determined by the presence of a moratorium on the use of genetic information. (d) The market size is represented by the rate of purchase of IPI insurance for persons without a family history. (e) Adverse selection is represented by: (i) the rate of insurance purchase after receiving an adverse test result and, if family history may not be used; (ii) the rate of insurance purchase given a family history before testing; and (iii) any tendency to buy larger amounts of insurance given any adverse information.

22 Application to Income Protection Insurance Parameterisation In the model in Figure 9, i µ sickness xt, i µ recovery xt,z and i µ 6,7 xt,z correspond to intensities in the basic IPI model in CMIR 12. However, we should note that these events occur before the individual becomes insured. Since, to our knowledge, direct estimates have not been reported by now, we use the estimates described in Section 2 with DP 1 week, where no significant difference would be expected. The rate of onset of a family history of BC was estimated following the approach used in Gui et al. (2006) (see Section 4.2). For the other parameters, we follow the choices of Gui et al. (2006) for a life insurance market, summarised as follows: (a) The market size: We take i µ 0,1 xt = 0.05 to represent a large insurance market and i µ 0,1 xt = 0.01 to represent a smaller market. (b) The purchase rate given a family history: People with a family history, who are offered a higher than standard premium, may buy less insurance. We assume i µ 2,3 xt = 0.05, or 0.0 per annum to represent their rate of insurance purchase in the large market; and i µ 2,3 xt = 0.0 in the smaller market. (c) The level of adverse selection: We set i µ 4,5 xt = 0.25 to represent severe adverse selection for BRCA1/2 applicants; and moderate adverse selection is represented by a rate of purchase of i µ 4,5 xt = 2 i µ 0,1 xt; that is, twice the normal rate. (d) The rate of genetic testing: We choose i µ 2,4 xt = 0.02 per annum, which implies that about 18% of people at risk will have a test in 10 years. However, this estimate can be low in respect of severe disorders, especially if treatment is non-existent, traumatic or has a low rate of success. We will use double these rates in sensitivity analysis. In practice, genetic testing is only offered to women with a (strong) family history, and this often means an extensive pedigree collected by the clinician, so there is not an exact match between this and onset of family history in our model, which is closer to insurance practice. Nevertheless, a rate of 0.02 per annum seems consistent with levels of genetic testing generally. 6.4 Numerical Procedures We model an IPI market operating between ages 20 and 65, assuming that all women are uninsured at age 20, and all policies expire at age 65. Since a woman may have already developed a family history by age 20, she may be in one of the states i0 or i2 at outset, but we assume that there is no genetic testing before age 20. In the model in Figure 9, an underwriting class C is represented by a set of insured states. Within each underwriting class, the same premium rate ρ C xt should be charged to cover the average risk using the Principle of Equivalence. Here, we achieve this by setting ρ C xt to the weighted average of the premium rate for each policy in C, with the weights being the probabilities of being in C. In Figure 9, let i tp jj x be the probability that a person in state ij at age x will be in the same state at age x t; and i tp jj x be the probability that a person in state ij at age x will remain in state ij for time t. We define i P j x:n as the annual net rate of premium for unit benefit for policyholders in state ij who bought insurance at age x, with policy term n. Then:

23 Application to Income Protection Insurance 23 ρ C 20t = k=0,2,4 ( ij C k=0,2,4 t i 0 sp kk 20 ( ij C iµ k,j 20s t s i p jj 20s ip j 20s :45 s ds ) t 0 si p kk 20 iµ k,j 20s t si p jj 20s ds ) (19) where i tp jj x and i tp kk x are computed using equations analagous to Equations (14) and (15), respectively, and i P j x : n is calculated using the model in Figure 2. With these rates of premium, policy values conditional on being in any state are obtained using the backward integration method as in Equations (17) and (18). Then the expected loss in the entire market is the weighted average of these policy values in each starting state, the weights being the corresponding proportions at outset. If insurance is purchased exactly as assumed when calculating ρ C xt, there is no adverse selection and the expected loss is zero. However, when there exists a moratorium, adverse selection may occur, and more insurance might be purchased than assumed. If we keep charging the premium rate ρ C xt, a loss emerges, which is the cost of adverse selection. To absorb it, insurers would have to increase all premiums by: EPV of loss with adverse selection EPV of loss without adverse selection. EPV of premiums payable with adverse selection 6.5 Underwriting Classes Based on Various Moratoria A moratorium may restrict or forbid the use of genetic information. Some moratoria may allow insurers to use negative test results that would allow the applicant to be granted the standard premium rate. The effect of a moratorium is to re-arrange the states that constitute current underwriting classes into a set of new underwriting classes. We investigate three forms of moratorium, as shown in Table Moratoria on Genetic Test Results Alone Suppose underwriters cannot use any genetic test information and the only permissible information is family history. Then there will be two underwriting classes defined by the presence or absence of the onset of family history. A different moratorium may allow someone who discloses a clear genetic test result to be offered the standard premium rate. An extra premium will be charged to an individual with a family history who cannot produce a clear genetic test. However, insurers would still be allowed to take only the family history into account. Tables show the premium increases required to recoup the cost of adverse selection, under moratoria on all genetic test results and adverse test results, respectively, for IPI markets with different DPs. The values are very small, even negligible, but there are some features to note: (a) Under the same circumstances, longer DPs would produce more losses. (b) As we expect, the impact on the small market is larger than that on the large market.

24 Application to Income Protection Insurance 24 Table 14: Possible underwriting classes with five sub-populations: i = 1: no family history risk; i = 2: BRCA1 family history but not BRCA1 mutation carrier; i = 3: BRCA1 family history and BRCA1 mutation carrier; i = 4: BRCA2 family history but not BRCA2 mutation carrier; i = 5: BRCA2 family history and BRCA2 mutation carrier. (T) denotes persons who have had a genetic test and (NT) denotes that persons who have not. (F) denotes persons who have developed a family history and (NF) denotes persons who have not. (ALL) denotes all of the insured states. Factors Allowed in Underwriting Composition of Underwriting Classes Negative Positive Rated for Rated for Family Test Test OR Family Genetic History Results Results Class History Test Yes No No i = 1, 2, 3, i = 1, 2, 3, 4, 5 (NF) 4, 5 (F) Yes Yes No i = 1, 2, 3, i = 1, 3, 5 (F) and 4, 5 (NF) and i = 2, 4 (F,NT) i = 2, 4 (F,T) Yes Yes Yes i = 1, 2, 3, i = 1, 3, 5 (F) and i = 3 (F,T) and 4, 5 (NF) and i = 2, 4 (F,NT) i = 5 (F,T) i = 2, 4 (F,T) No No No i = 1, 2, 3, 4, 5 (ALL) (c) The costs are slightly higher under a moratorium on all genetic test results because, under a moratorium on adverse test results alone, the cost is distributed over a larger standard premium class which includes those offering clear test results. (d) It is natural that higher rates of adverse selection will lead to higher costs; however, its effect is rather small, compared with the effect of market size. This is because we consider just one rare disease. (e) The impact on an IPI market is much smaller than on a life or CI insurance market (see Gui et al., 2006). 6.7 A Moratorium on Family History and Genetic Testing Results A stricter moratorium will also forbid the use of family history, and all applicants will be charged the same premium rate. Its effect is divided into two parts: (a) A new underwriting class is created: Everyone has access to insurance on the same terms. Policyholders who previously were charged higher premiums will now be offered standard premiums. Even if we assume no adverse selection based on undisclosed genetic risk, riskier individuals who now buy insurance normally cause the standard premium to increase. (b) Adverse selection appears: Those at risk might also buy insurance at a higher rate than normal, which is adverse selection, which will further increase premiums. Here, people at risk include those with a family history but who are untested, and those who have had an adverse test result. We show both severe and moderate rates of adverse selection.

25 Application to Income Protection Insurance 25 Table 15: Percentage increases in premium rates for IPI cover with DP 1 week, under a moratorium on all genetic test results and adverse results respectively, for a market operating between ages 20 and 65. Deferred Adverse Market Rate of Purchase with Moratorium on using Period Selection Size A Family History All test results Adverse results % % Same as normal DP 1 Severe Large Half as normal Nil Small Nil Same as normal DP 1 Moderate Large Half as normal Nil Small Nil Table 16: Percentage increases in premium rates for IPI cover with DP 4 weeks, under a moratorium on all genetic test results and adverse results respectively, for a market operating between ages 20 and 65. Deferred Adverse Market Rate of Purchase with Moratorium on using Period Selection Size A Family History All test results Adverse results % % Same as normal DP 4 Severe Large Half as normal Nil Small Nil Same as normal DP 4 Moderate Large Half as normal Nil Small Nil

26 Application to Income Protection Insurance 26 Table 17: Percentage increases in premium rates for IPI cover with DP 13 weeks, under a moratorium on all genetic test results and adverse results respectively, for a market operating between ages 20 and 65. Deferred Adverse Market Rate of Purchase with Moratorium on using Period Selection Size A Family History All test results Adverse results % % Same as normal DP 13 Severe Large Half as normal Nil Small Nil Same as normal DP 13 Moderate Large Half as normal Nil Small Nil Table 18: Percentage increases in premium rates for IPI cover with DP 26 weeks, under a moratorium on all genetic test results and adverse results respectively, for a market operating between ages 20 and 65. Deferred Adverse Market Rate of Purchase with Moratorium on using Period Selection Size A Family History All test results Adverse results % % Same as normal DP 26 Severe Large Half as normal Nil Small Nil Same as normal DP 26 Moderate Large Half as normal Nil Small Nil

27 Application to Income Protection Insurance 27 Table 19: Percentage increases in premium rates for IPI cover with DP for 26 weeks, under a moratorium on all genetic test results and family history, for an IPI market with different DPs operating between ages 20 and 65. Cost From Cost From Cost From Deferred Market New Underwriting Severe Adverse Moderate Adverse Period Size Classes Selection Selection % % % DP 1 Large Small DP 4 Large Small DP 13 Large Small DP 26 Large Small Table 19 shows the premium increases for different DPs. We see that the appearance of a new underwriting class has an effect much larger than that of additional adverse selection, which is comparable to the effect under other moratoria. However, market size has a much greater effect now; for example, under severe adverse selection the increase in the standard premium in the small market is now double that in the large market. 7. Conclusions We have developed a continuous-time semi-markov model for the life history of a woman at risk of BC, and extended it to a model of an IPI market. Major events such as diagnosis, treatment, recovery and recurrence are represented explicitly in the model and corresponding transition intensities are estimated based on current medical studies. We calculated level IPI premiums assuming: (a) a BRCA1 or BRCA2 mutation was known to be present; or (b) a family history of BC has developed. Our main conclusions are summarised as follows: (a) The premium increases for IPI are much small than those for life or CI insurance. (b) For a given age at entry and policy term, extra premiums increase as DPs increase. (c) Most BRCA1 mutation carriers are insurable given typical underwriting limits. (d) All BRCA2 mutation carriers are insurable given typical undrwriting limits. (e) All women with a family history (as defined here) are insurable. Extra premiums for applicants with DP 1 week might even be negligible. (f) The effect of reduced onset rates, allowing for likely ascertainment bias, is substantial. Extra premiums could decrease by a half or more in most cases. All cases for BRCA1 mutation carriers become insurable. (g) Using reduced onset rates, most cases for BRCA2 mutation carriers or women with a family history of BC have extra premiums of the order of 10%. In practice, these applicants could probably be offered ordinary rates.