In our start-up market, we model the progression of the market from introduction of the product. In such a case there will be lives of different ages buying the product and influencing the adverse selection costs. Some lives at each age have already developed some form of cognitive or functional disability so to model forward from the current time, it is necessary to calculate the distribution of lives in each state in our model for each sex, genotype and age group as at 1st January, 2013. To do this, we will
Table 3.5: Distribution of APOE genotypes. Source: Farrer et al. (1997). Genotype Probability, PG(g) ε2ε2 0.135 ε2ε4 0.026 ε3ε3 0.609 ε3ε4 0.213 ε4ε4 0.018
first estimate a mix of lives at age 60. From this mix at age 60, we will calculate the occupancy probabilities for each state for the elderly population which we will then use to form our insurance market, split into 5-year age groups in terms of age last birthday: 60–64, 65–69,70–74, 75–79, 80–84 and 85–89.
Denote the probability that a life has genotypeg ∈ G at age 60, byPG(g). Further denote the probability that a life of sexς, has functional disability typei∈ {0,1,2,3}
at age 60 byPADL,ς(i).
Since we are concerned with dementias of old-age, we assume that no lives have any signs of decrease of cognitive function at age 60 and the distribution of genotypes is that used by Macdonald and Pritchard (2000), and derived by Farrer et al. (1997) (shown in Table 3.5). Farrer et al. (1997) performed a meta-analysis of 40 studies of APOE, and provided the genotype frequencies among both lives with AD and the controls split by ethnicity (the chosen results being Caucasian controls).
The remaining part of the mix of lives at age 60 exactly, is the prevalence of func- tional ability. We find the prevalences of functional ability at age 60 for males and females, which when projected forward following the transition intensities (for sim- plicity, we exclude mortality improvements) in our model of health (see Figure 2.7), best fits the prevalences observed in the first phase of the CFAS study, as reported by Akodu (2007) (see Tables 3.6 and 3.7). They list these at age groups in terms of age last birthday, 65–69,70–74,75–79,80–84,85–89,90+, which we represent by the set of midpoints (and 92.5 for 90+), denotedX0, whereX0 ={67.5,72.5,77.5,82.5,87.5,92.5}. By varying the proportions originally in each state at age 60, we use a weighted least squares method, with weights equal to the number of lives in each age group, as estimated by Office for National Statistics (2011) for 2010 (shown in Table 3.8), to minimise the difference between the observed and our calculated prevalences, thereby extrapolating Tables 3.6 and 3.7 to age 60. The function to minimise is,
X x∈X0 X j∈{0,1,2,3} Obs j x− P g∈G,i∈{0,1,2,3},k∈{0,4,8,12} x−60pi,j60,ς,g+kPG(g)PADL,ς(i) P g∈G,i∈{0,1,2,3},k∈S\{16} x−60pi,k60,ς,gPG(g)PADL,ς(i) 2 PX,ς(x), (3.1)
Table 3.6: Distribution of functional ability in males by age group in terms of age last birthday. Source: Akodu (2007).
Functional Age group
Ability 65–69 70–74 75–79 80–84 85–89 90+ None 0.8406 0.8112 0.6791 0.5382 0.3545 0.1757 IADL 0.0534 0.0602 0.0756 0.1191 0.1231 0.1081 1-ADL 0.0737 0.0860 0.1556 0.2075 0.3134 0.2568
>1ADL 0.0323 0.0426 0.0898 0.1352 0.2090 0.4595
Table 3.7: Distribution of functional ability in females by age group in terms of age last birthday. Source: Akodu (2007).
Functional Age group
Ability 65-69 70-74 75-79 80-84 85-89 90+ No ADLs 0.7759 0.6854 0.5486 0.3483 0.1813 0.0847 IADL 0.1013 0.1194 0.1382 0.1749 0.1827 0.1229 1-ADL 0.0851 0.1515 0.2272 0.302 0.3626 0.2542 >1ADL 0.0376 0.0436 0.0860 0.1749 0.2734 0.5381 where Obsj
x is the observed prevalence of functional ability j, at age x, subject to
the constraints that P i∈{0,1,2,3}
PADL,ς(i) = 1, and PADL,ς(i) ∈ [0,1]. The occupancy
probabilities,x−60pi,j60,ς,g+k, are found by solving the Kolmogorov forward equations, d dttp i,j x,ς,g = X k6=j tpi,kx,ς,gµ k,j x,t,ς,g−tpi,jx,ς,gµ j,k x,t,ς,g . (3.2)
This was done using a 4th order Runge Kutta algorithm with a step size of 2−12 and boundary conditions 0pi,jx,ς,g =δij, where δij is the Kronecker delta. The result of the
minimising Equation (3.1) is shown in Table 3.9, with no lives having any ADLs at age 60.
Using the assumptions and calculated distributions of lives at age 60, we can find the distribution of lives who are alive at age x, of sex ς, have genotype g and are in state j at 1st January 2013, pjx,ς,g, by calculating,
pjx,ς,g =
3
P i=0
PADL,ς(i)PG(g)tpi,j60,ς,gPSex,x(ς) P
ς∈{M,F},g∈G,s∈S
3
P i=0
PADL,ς(i) PG(g) tpi,s60,ς,gPSex,x(ς)
, (3.3)
Table 3.8: U.K. population by sex and age group in terms of age last birthday in 1,000s. Source: Office for National Statistics (2011).
Age Male Female 60–64 1840.08 1923.52 65–69 1412.11 1519.56 70–74 1160.31 1307.44 75–79 893.91 1107.84 80–84 607.09 885.55 85–89 326.08 608.47 90+ 131.89 331.54
Table 3.9: Proportion of lives of each sex in each functional disability type, at age 60 exactly.
No ADLs IADL 1-ADL >1 ADL
Male 1 0 0 0
Female 1 0 0 0
This will not necessarily produce an accurate depiction of the mix of lives of the U.K. population, however the purpose of such a distribution is to provide an approximate baseline from which we can illustrate how adverse selection may impact costs. The aim of our model is not to accurately estimate or project future demand for services and our results should not be used in this way.