North American Actuarial Journal
ISSN: 1092-0277 (Print) 2325-0453 (Online) Journal homepage: http://www.tandfonline.com/loi/uaaj20
Pricing Critical Illness Insurance from Prevalence
Rates: Gompertz versus Weibull
Fabio Baione & Susanna Levantesi
To cite this article: Fabio Baione & Susanna Levantesi (2018) Pricing Critical Illness Insurance
from Prevalence Rates: Gompertz versus Weibull, North American Actuarial Journal, 22:2,
270-288, DOI: 10.1080/10920277.2017.1397524
To link to this article: https://doi.org/10.1080/10920277.2017.1397524
Published online: 28 Mar 2018.
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North American Actuarial Journal, 22(2), 270–288, 2018
Copyright C 2018 Society of Actuaries
ISSN: 1092-0277 print / 2325-0453 online
DOI: 10.1080/10920277.2017.1397524
Pricing Critical Illness Insurance from Prevalence Rates:
Gompertz versus Weibull
Fabio Baione 1 and Susanna Levantesi 2
1Department of Economics and Management, University of Florence, Florence, Italy
2Department of Statistics, Sapienza University of Rome, Rome, Italy
The pricing of critical illness insurance requires specific and detailed insurance data on healthy and ill lives. However, where the
critical illness insurance market is small or national commercial insurance data needed for premium estimates are unavailable, national
health statistics can be a viable starting point for insurance ratemaking purposes, even if such statistics cover the general population,
are aggregate, and are reported at irregular intervals. To develop a critical illness insurance pricing model structured on a multiple state
continuous and time-inhomogeneous Markov chain and based on national statistics, we do three things: First, assuming that the mortality
intensity of healthy and ill lives is modeled by two parametrically different Weibull hazard functions, we provide closed formulas for
transition probabilities involved in the multiple state model we propose. Second, we use a dataset that allows us to assess the accuracy
of our multiple state model as a good estimator of incidence rates under the Weibull assumption applied to mortality rates. Third, the
Weibull results are compared to corresponding results obtained by substituting two parametrically different Gompertz models for the
Weibull models of mortality rates, as proposed previously. This enables us to assess which of the two parametric models is the superior
tool for accurately calculating the multiple state model transition probabilities and assessing the comparative efficiency of Weibull and
Gompertz as methods for pricing critical illness insurance.
1. INTRODUCTION
Medical data are available from public health and social security databases, but these do not always provide sufficiently detailed
information. The publicly available illness data usually refer to prevalence rates (the proportion of people in the population who
have a particular disease or disability at a specific point in time) because prevalence statistics are easier to collect than incidence
rates, which provide the annual number of people who have a new illness or disability. This is the case in Italy, where national
health data (referring to the general population) are scarce and available at irregular intervals and additionally limited in the sense
that only aggregate prevalence data are provided. The Italian health insurance market is undersized (Swiss Re 2012; Baione et al.
2016) and consequently unable to provide sufficient data as a basis for reliable incidence rate estimates. However, Italian national
prevalence statistics can be considered as a viable starting point for the calculation of national incidence rates and, consequently,
as a starting point for assessment of critical illness (CI) insurance premiums. CI insurance is still underdeveloped in Italy, but this
has not limited the local development of this insurance product: currently, Italian CI policies cover more than 50 diseases, although
the most common ones are cancer, stroke, and myocardial infarction. The CI coverage can be added as a rider to other insurance
plans and can enable complete or partial acceleration of payment of the face amount. The most popular form of CI in Italy is CI
added as an accelerated benefit rider to term insurance.
Some CI insurance pricing methods are based on multiple state models that deal with the estimation of transition intensities or
probabilities operating under Markov Chain discipline. Success of these methods depends on the consistency of reported prevalence
data from one period to the next. The estimation of transition intensities (probabilities) in the time-continuous (time-discrete)
Markov models for health insurance has been addressed by some authors, such as Dash and Grimshaw (1993), Cordeiro (2002),
Czado and Rudolph (2002), and Helms et al. (2005). The problem of the estimation of illness incidence rates, given an extensive
dataset, has been addressed by Ozkok et al. (2014a), Ozkok et al. (2014b), and Dodd et al. (2015). In these three papers, statistical
models are developed for estimation and graduation of CI incidence rates and insurance premiums. The numerical illustrations
provided are based on U.K. data for the period 1999–2005, published by the Continuous Mortality Investigation (CMI) of the
Address correspondence to Fabio Baione, DiSEI, Via delle Pandette, 9-50127, Florence, Italy. E-mail: [email protected]
270
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 271
Institute and Faculty of Actuaries. There is no comparable statistical CI database in Italy because available data are too sparse. In
our previous work (Baione and Levantesi 2014) we proposed a parametric model to price health insurance products when only
aggregate information on morbidity is available and the only illness data consist of prevalence rates, while incidence statistics are
unavailable. Our proposal amounted to using prevalence data and parametric mortality models as a basis for inferring CI incidence
data as a balancing feature of the model.
We repeat this work substituting a Gompertz model with two sets of parameters for the Weibull model. We then assess whether
the Weibull model or the Gompertz model more accurately replicates the Italian prevalence data. Many hazard functions have been
used to describe mortality, the Gompertz and Weibull functions being widely used. The Gompertz (1825) and Weibull (1951) laws
are useful for building mortality models, yet they have some limitations and drawbacks. However, they are suitable for a restricted
age range as is the case with CI insurance where entry ages range between 18 and 55 years, the age at end of coverage is typically
65 years, and policy terms are not over 10 years (see Munich Re 2001; Eppert 2014). Restriction of coverage at older ages is
motivated by the steep increase in incidence rates with age progression. It is felt that the needs of seniors are better addressed
through long-term care insurance.
The choice of the Weibull function to represent the mortality of healthy lives as well as the mortality of sick lives due to CI is
motivated as follows. First, the Weibull hazard function is well known and frequently used in actuarial science and statistics (e.g.,
Carriere 1992; Juckett and Rosenberg 1993; Biffis 2005). Second, the Gompertz and Weibull functions imply different biological
causes of demographic aging because they differ in the way in which mortality increases with age: the terms explaining the age
dependence of mortality are multiplicative in the Gompertz model and additive in the Weibull model. The interesting study by
Juckett and Rosenberg states that the Gompertz function is a better descriptor for all causes of deaths and combined disease
categories, while the Weibull function is preferable for single causes of death. These authors compare the Gompertz and Weibull
functions with respect to goodness of fit to human mortality experience, using mortality and incidence data from both the United
States and Japan. The causes of death they consider are several, cancer (total and specific) included.
As an alternative to the Weibull model, the mortality due to other causes apart from CI could be represented by the family of
frailty models (Butt and Haberman 2004) such as the Perks laws that constitute the result of applying an individual Gompertz
law to persons belonging to a heterogeneous population. Representing heterogeneity inside a cohort is an interesting exercise but
beyond the scope of this article. Our first purpose is to investigate whether the Weibull model is preferable to the Gompertz model
in studying the impact of a single cause of death, such as cancer or myocardial infarction. Our second purpose is to investigate
whether the statement in Juckett and Rosenberg on Weibull preferability over Gompertz is supported by our dataset and whether
it is relevant to the pricing of CI insurance. Our third purpose is to compare critical illness net single premium rates calculated
using the Weibull model with those calculated using the Gompertz model as measures for assessing the comparative merits of
the two models. The article is organized as follows. Section 2 describes the Gompertz and the Weibull hazard functions under
different parameterizations. Section 3 first describes the actuarial model for a critical illness insurance, illustrating the multiple
state model and calculation of premium rates, and, second, presents the use of the theoretical model for a CI insurance to estimate
transition probabilities under the Weibull model. A numerical application with Italian cancer statistics, both for all cancers and
single categories thereof, is presented in Section 4 where we make a comparison of the models proposed to test the goodness of fit
and to compute insurance premiums. Section 5 concludes the article.
2. GOMPERTZ VERSUS WEIBULL MODELS OF MORTALITY
The Gompertz and Weibull models can be considered the most popular ones describing mortality. The Gompertz law was first
published in 1825 and has been shown to apply over limited age ranges. The Weibull law was first fully described in 1951 and has
been used to successfully model reliability risk in mechanical systems. Biological systems failures can be likened to mechanical
systems failures, and, since models of failure of mechanical systems can be described in the form of configurations of components
arranged in parallel and serial form, the extension of the distribution formulas from mechanical systems to biological systems
such as human beings makes sense. The two models are, unfortunately, not suitable for modeling population mortality over a wide
range of ages. For example, they do not reproduce the “accident hump” (mortality due to accidents) that is a feature of almost all
population mortality experience in the age group 15 to 30 years and is more pronounced in the case of male lives. The Gompertz
model is widely used to represent senescent mortality. Gigliarano et al. (2017) have drawn attention to a limitation of the Weibull
model: Its hazard function “must be monotonic, whatever the values of its parameters. This may be inappropriate in some settings,
for example when the course of the disease is such that mortality reaches a peak after some finite period, and then slowly declines.”
The Gompertz and Weibull models mainly differ in the way age-independent and age-dependent components of mortality
operate within the model formulas. The Gompertz function develops exponential mortality rate increases with advancing age.
The Weibull survival function exhibits mortality rates that increase as a power function of advancing age. Both models include
a variable representing an initial force of mortality (see Ricklefs and Scheuerlein 2002): This initial force of mortality must be
a positive number in the Gompertz formula, where the force of mortality increases exponentially as a multiple of the force of
mortality at age zero. In the Weibull model the initial force of mortality is zero.
272 F. BAIONE AND S. LEVANTESI
TABLE 1
Formulations of the Gompertz and Two-Parameter Weibull Function
Form Gompertz Weibull
Hazard rate μx = A · eBx μx = a · xb
Survival function S(x) = e? A
B ·(eBx?1) S(x) = e? a
b+1 ·xb+1
Linearized hazard rate log[μx] = log A + Bx log[μx] = log a + b · log x
Linearized survival
function
log[ log[S(x)]] = log A
B + log[eBx 1] log[ log[S(x)]]
= log a log(b + 1) + (b + 1) · log x
Three parameter versions are available for the Weibull distribution: three-parameter (scale parameter, shape parameter, and
location parameter), two-parameter (obtained by setting the location parameter equal to zero), and one-parameter (obtained by
assuming that the shape parameter is known a priori from past experience with identical or similar risks.
In Table 1 we define the two-parameter Weibull and the Gompertz function in the form of hazard rate μx, survival function
S(x), and their linearized forms. Note that in the Weibull model the hazard rate and the survival function are presented in log-linear
form.
In the Gompertz model, the initial force of mortality is A. B is a multiplicative factor applied to age and determines the rate
at which the force of mortality increases with advancing age. Note: In this article we use the terms “hazard rate” and “force of
mortality” interchangeably. The parameters of the Gompertz model incorporate two different mortality components: the initial
mortality intensity A affecting all ages, and parameter B representing the slope of the increase with age. In the Weibull function,
a > 0 is the scale parameter, and b > 0 is the shape (or slope) parameter. The most common parameterization of the Weibull model
is obtained by setting b = k 1 and a = k
λk . The hazard function then becomes
μx = kλxλk1, (2.1)
where λ > 0 is the scale parameter and k > 0 is the shape parameter. Otherwise, another parameterization could be obtained by
setting b = β ? 1 and a = α · β, and the Weibull hazard function can be rewritten as follows:
μx = α · β · xβ1
. (2.2)
In the above parameterization, α > 0 is the scale parameter and β > 0 is the shape parameter. The Weibull model is versatile, and
its distribution can take on the characteristics of other types of distributions, based on the value of the shape parameter.
3. CRITICAL ILLNESS INSURANCE
CI insurance provides a lump sum when the insured individual is diagnosed with a serious illness included within a set of
diseases specified by the policy conditions. The most common diseases are heart attack, coronary artery disease requiring surgery,
cancer, and stroke. CI policies are available in a number of differing designs, as described in detail by Gatzert and Maegebier
(2015). In the following section, we will consider two main types of coverage: the acceleration rider and the standalone policy. CI
insurance combined with a term life insurance as an acceleration rider provides a proportion of the basic policy’s death benefit on
the diagnosis of a specified illness and maintains life insurance for the remainder of the death benefit. A policy providing term life
with an acceleration rider is cheaper than buying two separate policies, one term insurance and the other standalone CI, because
the face amount is paid only once.
3.1. Multiple-State Model and Net Single-Premium Rates
To model the above-defined CI policies, we introduce a multiple state model with state space S = {1 = healthy, 2 = ill, 3 =
dead due to CI, 4 = dead due to other causes} and a set of transitions depicted in Figure 1.
It should be noted that the transitions considered herein are exhaustive for our purposes, but the same consideration does not
hold for a population study developing a model for estimating the incidence rates. In this latter case, both inward and outward
migration, as well as the ill-to-healthy transition, should be taken into account, while they can be deemed irrelevant in an insurance
study. In this article, this choice is also motivated by the features of the dataset used in the numerical application that does not take
into account cases where the cancer sufferer is cured and maintains the observed life in its impaired state. Let [0, T] be a fixed
finite time horizon, x(x ≥ 0) be the entry age, and S(t) the state of the policyholder at time t, with S(0) = 1. The process describing
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 273
FIGURE 1. Set of States and Set of Transitions for CI Benefits.
the development of a single policy in continuous time, {S(t)}t∈[0,T], is Markovian, and the transition intensity from state i to j is
μi j
x = lim
t→0t pi jxt
t ∈ [0, T], i, j ∈ S, i = j, (3.1)
where
t pi j
x = P{S(x + t) = j |S(x) = i} t ∈ [0, T], i, j ∈ S, i = j. (3.2)
t pii
x = P{S(x + z) = i for all z ∈ [0, T], S(x) = i} (3.3)
are, respectively, the transition probability of a policyholder being in state j at age x + t given that the policyholder is in state i
at age x, and the probability of a policyholder being in state i at age x remaining in the same state up to age x + t. The transition
probabilities involved in the multiple state model considered herein are solutions of the Kolmogorov forward differential equations
(see, e.g., Haberman and Pitacco 1999):
t p11
x+u + μ14
x+udu
, (3.4)
t p12
x =t
x · μ12
x+u ·t?u p22
x+u
du, (3.5)
t p223
x+u + μ24
x+u du
. (3.6)
Let v(0,t) = exp(δt) be the value at time 0 of a monetary unit to be paid at time t, where δ is the force of interest. For simplicity
purposes, we do not consider the waiting period (i.e., the period of time specified in the policy between the date of policy issuance
and the date coverage begins) and the elimination period usually included in the design of a CI policy to reduce the risk of adverse
selection. Therefore, the net single premium rates for the two types of CI coverages here considered are defined as follows:
Standalone CI with an policy term N, where the sum insured is payable on occurrence of one of the diseases specified by
the policy conditions:
(1)A
(CI)
x:N =Nt p11
x · μ12
x+t · v(0,t) dt. (3.7)
CI full acceleration rider of a term life insurance with policy term N:
(2)A
(CI)
x+t + μ14
x+t
v(0,t)dt. (3.8)
Note that Equation (3.8) can be rewritten as
(2)A
(CI)
x:N = (1)A
(CI)
t p11
x · μ14
x+t · v(0,t) dt. (3.9)
274 F. BAIONE AND S. LEVANTESI
3.2. A Model to Estimate Transition Probabilities for CI Insurance: The Weibull Assumption
In Baione and Levantesi (2014) we provide transition probabilities estimation starting from the prevalence rates of sickness,
rather than from incidence rates, as would be preferable. This approach can be considered particularly useful in countries where
national health statistics are sparse and noncontinuous and only aggregated information on mortality and morbidity is available. In
fact, when incidence rates of sickness are available, the transition intensity from healthy to ill, μ12
x , can be directly estimated from
data by, for example, parametric methods. However, if only prevalence rates are available, a method based on the relationships
between prevalence rates and transition probabilities should be implemented to provide estimates of μ12
x (see Olivieri 1996; Haberman
and Pitacco 1999). Following Olivieri (1996), we suppose that transition intensity from healthy to ill can be described by a
piecewise constant function, given a certain number of prevalence rates available from statistical data. Concerning the mortality
intensities, we suppose that μ14
x and μ23
x are described by two independent Weibull hazard functions. The lack of statistics on
the deaths due to causes other than CI of ill lives justifies the use of simplifying assumptions that allow the estimation of μ24
x . In
actuarial practice, the mortality of individuals in poor health is usually expressed in relation to the standard mortality, appropriately
adjusting the standard probabilities of death. The adjustment can be made according to an additive model, a multiplicative model,
or a combination of these two models. Because of medical progress in the treatment of chronic diseases, the evolution of some
illnesses, such as certain types of cancer, has a short recovery time; in this case, a decreasing extra-mortality model should be
prioritized. In the field of CI insurance, Dash and Grimshaw (1993) describe three different approaches to the mortality of CI sufferers
depending on the available statistics. Among these, we assume the multiplicative approach based on the comparison of the
mortality of CI sufferers from causes other than CI with the mortality of healthy lives. We suppose that μ24
x exceeds the mortality
of healthy lives, μ14
x , by an extra mortality of γ . It is important to note that the extra mortality of ill lives may assume different
values depending on the specific disease and the γ parameter should be estimated accordingly. However, if the CI insurance covers
a set of different diseases, such as heart attack, cancer, or stroke, the extra mortality parameter should be representative of all the
diseases included within the coverage. This article extends our previous work assuming that mortality intensity of both healthy and
ill lives are modeled by two parametrically different Weibull functions instead of Gompertz models. Other assumptions underlying
the model remain valid. To compare the mortality intensity assumptions from the Weibull and Gompertz models, we summarize
the functions involved in the multiple model (see Figure 1) in Table 2. Note that in order to simplify calculations, we use the
parametrization expressed in Equation (2.1) for the Weibull model.
In this article we provide the transition probabilities under the Weibull assumptions, while for the Gompertz model the reader
may refer to Baione and Levantesi (2014). The probability of remaining in state 1 until time t defined in Equation (3.4) has the
following solution:
t p11
x = exp σk+1 · t bh1bh
2 + 1(x + t)bh2+1 xbh2+1
for k = 0, 1,..., n 2, xk < x ≤ xk+1, and t ≤ xk+1 x (3.10)
and for k = n ? 1, x > xn?1, and t,
and n is the number of prevalence rates.
TABLE 2
Summary of Models Used to Estimate Transition Intensities for CI Benefits
Transition Model Hazard rate Parameters
1 → 4 Gompertz μ14
x = β˙h
1 · exp(βh
2 x) β˙h
1 , βh
2 >0
Weibull μ14
x = bh
1 · xbh
2 bh
1, bh
2 > 0
2 → 3 Gompertz μ23
x = β˙ci
1 · exp(βci
2 x) β˙ci
1 , βci
2 >0
Weibull μ23
x = bci
1 · xbci
2 bci
1 , bci
2 > 0
2 → 4 Extra μ24
x = μ14
x · (1 + γ ) γ > 0
mortality
1 → 2 Piecewise
constant
function
μ12
x =
0 x ≤ x0
σk+1 xk < x ≤ xk+1
σn xn?1 < x
σk+1 > 0
k = 0, 1,..., n ? 2
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 275
Using Equation (3.6), we derive the probability of remaining in state 2 until time t, for all x,t, as
t p22
x = exp? bci1bci2 + 1(x + t)bci
2 +1 xbci
2 +1× exp(1 + γ )bh
1bh2 + 1
(x + t)
bh
2+1 ? xbh
2+1
. (3.11)
From Equation (3.5), under the Weibull assumptions on mortality intensities and the assumption that μ12
x is piecewise constant
(see Table 2), we obtain
t p12
du (3.12)
for k = 0, 1,..., n 2, xk < x ≤ xk+1, and t ≤ xk+1 x
and for k = n 1, x > xn1, and t.
To solve Equation (3.12) we assume the following approximations according to a Taylor series expansion:
(x + u)
k =x +t
2k+u t2; (3.13)
hence, Equation (3.12) becomes
t p12
x ～= σk+1 · e bh
1bh2+1(x+t)
bh2+1xbh2+1
+γ bh
(x+ t2 )bh
2 (x t2 ·bh2 )(x+t)
bh2+1× ebci
1bci2 +1(x+ t2 )
bci2 (x t2 ·bci2 )(x+t)
bci2 +1× eσk+1+γ ·bh
1 (x+ t2 )bh
2+bci
1 (x+ t2 )
σk+1 + γ · bh
. (3.14)
Therefore, considering the prevalence rates of sickness as the probability of being ill at age x for a policyholder with initial age
x0 and according to Equations (3.10) and (3.14), we can estimate the unknown parameters σk+1 for all k = 0, 1,..., n 1 via an
iterative approach starting from the initial age group (x0, x1 ) (see Baione and Levantesi [2014] for further details). As a consequence
of the assumption of μ12
x the standalone CI net single-premium rate on the covered age period (x, x + N) is
(1)A
(CI)
x:N = n1
k=0yk+1
yk
t p11
x μ12
x+tv(0,t) dt (3.15)
with yk+1 = xk+1 x and set y0 = 0, yn = N. The generic integral in Equation (3.15) defined on the subinterval (yk, yk+1] has the
following solution:
yk+1
yk
t p11
x μ12
x+tv(0,t)dt ～= σk+1 × e? bh
1bh2+1x+ yk+yk+1
2bh2+1?x
bh
2+1× e
yk+yk+1
2 Dk× e(σk+1+δ+Dk )yk e（σk+1+δ+Dk )yk+1
σk+1 + δ + Dk
, (3.16)
276 F. BAIONE AND S. LEVANTESI
where Dk = bh
1bh2+1 (bh2 + 1)(x + yk+yk+12 )bh2 . The premium rate for a CI full acceleration rider of a term life insurance is obtained by
splitting the integral in Equation (3.9) into the sum of subintegrals yk+1
yk t p11
x μ14
x+tv(0,t) dt. Using the approximated formula (3.13),
the generic subintegral on (yk, yk+1] has the following solution:
2 + 1, (σk+1 + δ + Dk ) (x + yk )
(bh2 + 1, (σk+1 + δ + Dk ) (x + yk+1 ), (3.17)
where indicates the Gamma function (a) = ∞
0 et
t
a1dt (with a > 0). It is worth noting that the piecewise constant function
assumption for the transition intensity μ12
x allows us to obtain estimates of the incidence that are fully consistent with the observed
prevalence rates (see the analysis of the growth rate of prevalence rates shown in the following section). The choice of a piecewise
constant function has the advantage of no constraints on the shape of the transition intensity and thus is able to depict the natural
trend of the phenomenon. An alternative to the piecewise constant function could be a parametric function capable of solving
Equations (3.4)–(3.6). For example, a GM(0,2) (see Baione and Levantesi 2014) or a Weibull distribution (see Appendix C for
further details) can be a reasonable choice if supported by empirical data. In general, different functions can be used for modeling
μ12
x , but only some of them allow closed form solutions for transition probabilities and premium rates.
4. NUMERICAL APPLICATION
CI insurance can cover well more than 30 different illnesses, but the four most common ones are cancer, heart attack, stroke,
and coronary artery bypass surgery. These four cases have been classified as the “basic four” critical illnesses (see, e.g., Gatzert
and Maegebier 2015). Hereinafter, due to the availability of a specific dataset, we shall consider only cancer, which is the second
cause of death after heart disease in Italy in terms of deaths, although the mortality rate of cancer is higher than heart disease. The
product we model is not typical CI insurance inasmuch as it pays out on cancer incidence only. When, in the rest of this article,
we use the term “CI” we are referencing insurance with the incidence event limited to cancer diagnosis. Other CI events are not
insured or considered. In the numerical application we use cancer data for Italy publicly available for the period 1970–2015 and
provide the estimates of prevalence, mortality, and incidence. Therefore, in addition to the prevalence rate, these data also provide
incidence rates allowing us to check whether the model produces reliable estimates of incidence rates.
4.1. Data Set
Data are downloaded from the website www.tumori.net (Istituto Superiore di Sanitá 2015), which publishes the results of
epidemiological research in oncology, on both an Italian and international level. It arises from a research project of the IRCCS
Foundation of the National Cancer Institute of Milan, in collaboration with the National Institute of Health (Istituto Superiore di
Sanitá [ISS]). Published information concerns the main types of cancer, but the website also provides information on rare cancers
and childhood and adolescent cancers. Data are divided into five-year age groups (from 0 to 99 years). As stated in the introduction,
our approach is suitable to situations in which health statistics are sparse, noncontinuous, and aggregated by age groups. By way
of example, for the analysis, we have selected year 2009, and we provide all the results for all the main cancers available in the
dataset, hereafter labeled as “Total cancers.” Similar results are also obtained for the other years. The dataset used in the numerical
application contain the following tables, which are presented in Appendix A:
a. People reporting chronic conditions by type of cancer, gender, and age group, Italy, year 2009
b. Mortality rates by age group, gender, type of cancer, and year of death, Italy, year 2009
c. Mortality table by age and gender, Italy, year 2009 (downloadable from www.tumori.net).
The prevalence rates of sickness (data of type [a]) and the mortality rates by type of cancer (data of type [b]) are given in
Table A.1 and Table A.3 for males and Table A.2 and Table A.4 for females, respectively (see Appendix A). The main categories
of cancer collected by ISS are stomach, colorectal, lung, malignant melanoma, breast, cervix uteri, and prostate. We restrict our
analysis to the age range 20–69, consistent with the typical CI insurance age limits.
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 277
TABLE 3
Parameters and Accuracy Measures of Gompertz and Weibull μ14
x Mortality Models, Total Cancers, Year 2009
Gompertz β˙h
1 SE p value βh
2 SE p value
Sex
Male 0.000074 0.232681 1.57E-11 0.071027 0.004878 1.47E-07
Female 0.000018 0.214348 2.11E-12 0.084008 0.004494 1.64E-08
Weibull bh
1 SE p value bh
2 SE p value
Sex
Male 6.224008E-08 1.354396 6.47E-07 2.751176 0.360524 3.24E-05
Female 3.615433E-09 1.402257 2.23E-07 3.284108 0.373264 1.03E-05
TABLE 4
Parameters and Accuracy Measures of Gompertz and Weibull μ23
x Mortality Models, Total Cancers, Year 2009
Gompertz β˙ci
1 SE p value βci
2 SE p value
Sex
Male 0.013404 0.177314 1.61E-09 0.029435 0.003718 2.40E-05
Female 0.008430 0.107555 7.45E-12 0.020625 0.002255 7.48E-06
Weibull bci
1 SE p value bci
2 SE p value
Sex
Male 0.000524 0.488468 8.64E-08 1.222286 0.130024 5.97E-06
Female 0.000866 0.273219 9.48E-10 0.857448 0.072728 8.95E-07
4.2. Parameter Estimation and Accuracy of the Estimates of Mortality Models
To estimate transition intensities μ14
x and μ23
x we should consider the mortality rates of both healthy and ill lives by age and
gender. However, since the mortality rates of ill lives collected by ISS belong to five-year age groups, we first construct an abridged
multiple state life table. Considering the paucity of information about the relationship between the mortality rate of healthy lives
and mortality rates of ill lives from causes other than CI, we set the extra mortality parameter, γ , to 0, therefore μ24
x = μ14
x .
Setting γ = 0 leads to higher premium rates than the assumption γ > 0 and includes a profit margin for the insurer. However, it
is important to keep in mind that cancer death experience is not homogeneous at all. Even after cure, the risk of remission varies
greatly across types of cancers; for example, the extra mortality of cured breast cancer cases tends to zero at approximately five
years after successful treatment, whereas the extra mortality of stabilized leukemia or lung cancer patients declines much more
slowly. Therefore, when data are available, the γ parameter should be estimated on the set of diseases considered, as we have done
for the mortality parameters. We assume that the force of mortality remains constant over each age group (x, x + n). According to
the assumptions made in Table 2 we graduate μ14
x and μ23
x first with two Gompertz functions and then with two Weibull functions.
In the following section, we show the results for parameters’ estimation and model fitting calculated from the dataset on total
cancers. In both mortality models here proposed the relationship between the variables is not linear and the linear model should
be applied only after having transformed the dependent and/or the explanatory variable; the linearized form after a logarithmic
transformation has been reported in Table 1. In the Gompertz model we have to make a logarithmic transformation of the dependent
variable only, while the Weibull model requires the logarithmic transformation of both dependent and explanatory variable (age).
The values of parameters are obtained by using the minimum mean square error (MMSE) estimation. The statistical estimation of
the parameters is based on the traditional linear regression model with constant variance (homoscedasticity assumption) without
any assumption on the distribution of μ12
x . In general, in the basic linear regression model both MMSE and maximum likelihood
estimators are equivalent in the case of a linear Gaussian system. In our case the Gaussian assumption is made on the logarithm of
transition intensities (see the linearized form of the Gompertz and Weibull models given in Table 1). The estimates of the regression
coefficients, the standard error (SE) of each parameter estimate and its p value are reported in Table 3 for the mortality model of
healthy lives (μ14
x ) and in Table 4 for the mortality model of ill lives (μ23
x ). The values of both SE and p value indicate that the
278 F. BAIONE AND S. LEVANTESI
TABLE 5
Accuracy Measures of Gompertz and Weibull Mortality Models, Total Cancers, Year 2009
Mortality model μ14
x μ23
x
Gompertz R2 s R ? 2 s?
Sex
Male 95.95% 25.50% 87.45% 19.49%
Female 97.53 23.37 90.29 11.83
Weibull R2 s R 2 s
Sex
Male 86.65% 46.31% 90.76% 16.73%
Female 89.71 47.69 93.92 9.36
TABLE 6
R2 of Gompertz and Weibull μ14
x Mortality Models, Year 2009
Colorectal Lung Stomach
Gender Gompertz Weibull Gompertz Weibull Gompertz Weibull
Male 96.16% 87.06% 96.18% 87.05% 96.17% 87.08%
Female 98.88 93.06 98.86 92.98 98.87 93.08
parameters are significant, and we can conclude that the functional forms assumed for the transition rates are consistent with the
observed data. Tables 3 and 4 also show the standard error of each estimate of the regression coefficients and the corresponding p
value. Note that parameters β˙h
1 and β˙ci
1 are the exponential of the linear regression estimators obtained by using MMSE estimation.
Therefore, to calculate the p values it is necessary to apply the logarithmic transformation to β˙h
1 and β˙ci
1 .
The accuracy of the estimation procedure is addressed by computing the (R2) value1 that measures how successful the fit is in
explaining the variation of the data. R2 is also known as the coefficient of determination, and it is a commonly used statistic to
evaluate the model goodness of fit. The R2 values calculated on ISS dataset on total cancers are given in Table 5 and show that the
Gompertz model fits better than the Weibull model only for μ14
x , while the Weibull model has higher levels of accuracy for μ23
x . In
the same table we also show the standard errors2 s(μ14
x ) and s(μ23
x ) for the dependent variable estimate of the regression on μ14
x and
μ23
x , respectively. The standard error estimates the standard deviation of the observed values of the transition intensities from the
regression line, and, thus, it can be considered a measure of the accuracy of predictions: the lower the values of the standard error,
the better the accuracy of the prediction. The standard error values will be used to define the confidence intervals of the transition
intensities’ estimates.
To test the efficacy of the Weibull model in describing a single cause of death as stated by Juckett and Rosenberg (1993), we
have tested the model over some categories of cancer having a high incidence according to data collected in the cancer registry of
Italy: colorectal, lung, and stomach. Results are presented in Table 6 for the mortality model of healthy lives (μ14
x ) and in Table 7
for the mortality model of ill lives (μ23
x ).
The value of the coefficient of determination R2 does not reliably identify a good fit of the mortality models we consider.
Nevertheless, the Weibull model turns out to be preferable to the Gompertz model only in one instance (the colorectal mortality
rates of sick lives). Thus our results are not consistent with those obtained by Juckett and Rosenberg (1993). Finally, we calculate
the confidence intervals of the mortality intensities μ14
x and μ23
x according to a p-percentile level of confidence. In the linear
regression models, under the well-known assumption that the standardized residuals (as well as the standardized residuals with
1In a linear regression model, given a dependent variable y and its estimate ?y, the coefficient of determination is calculated by the ratio between the deviance
of the estimate and the deviance of the variable: R2 = dev( y)
dev(y) = 1 dev(e)
dev(y) , where e = y y is the vector of the residuals.
2In a linear regression model, the standard error for the y estimate is defined as is = √(i e2in?2 ), where ei = yi yi is the ith residual and n is the number of
observations. In other words, s is the standard error of the residuals.
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 279
TABLE 7
R2 of Gompertz and Weibull μ23
x Mortality Models, Year 2009
Colorectal Lung Stomach
Gender Gompertz Weibull Gompertz Weibull Gompertz Weibull
Male 66.91% 73.70% 12.53% 4.16% 53.48% 37.43%
Female 69.57 76.83 93.77 87.25 16.96 15.78
zero mean) are distributed as a Student’s t with n 2 degrees of freedom, e?E(e)
S ～ T(n2), the confidence interval for a generic
dependent variable y is
Pr( y tp · s ≤ y ≤ y + tp · s) = 1 p,
where s is the standard error and tp is the p-percentile of a Student’s t distribution with n 2 degrees of freedom, so that Pr(T ≤
tp) = Pr(T ≥ tp) = p
2 . Therefore, the estimate of the confidence interval in the Gompertz as well as in the Weibull model for
μ14
x (Eq. [4.1]) and μ23
x (Eq. [4.2]) is determined according to a fixed confidence level p and to the standard errors of the estimates,
s(μ14
x ) and s(μ23
x ), as
4.3. Transition Probabilities from Healthy to Ill
The ISS dataset used in the numerical application provides information on both prevalence and incidence rates. Therefore, we
can compare ISS incidence rates with the incidence rates estimated by our model. However, ISS affirms (see www.tumori.net) that
cancer incidence is not observable at a national level and is obtained by applying the estimation methodology implemented in the
Mortality Incidence Analysis MODel (MIAMOD) program. According to this methodology, incidence is modeled by ISS with ageperiod-cohort
models and is back calculated by a regression on observed cancer deaths (see AIRTUM [2014] for further details).
Therefore the incidence rates published by the ISS are not observed, but estimated from observed data. In contrast to ISS, our work
is based on pairs of Gompertz and Weibull mortality models, each pair addressed by parameters appropriate for healthy lives and
lives impaired by cancer. Our work is also based on cancer morbidity prevalence rates that are incorporated in our models. We have
no input data relating to incidence rates; these are assumed to be piecewise constant function. Instead, the MIAMOD works with
detailed time series and not with age groups, and does not make distributional assumptions about the shape of mortality of ill lives
that is based on cancer registry data; on the other hand, the incidence is modeled by a logistic function depending on age, period,
and cohort for each gender and type of cancer (see Verdecchia et al. 1989). In fact, the MIAMOD generally assumes that the logit
of incidence is well approximated by a second degree polynomial. Although a real comparison between our estimates and the
values provided by ISS is not fully appropriate for the aforementioned reasons, instead of considering data points, we compare the
ISS incidence rates to the range within which the incidence rates estimated by our model should fall according to the confidence
intervals of the mortality intensities (μ14
x and μ23
x ) at a 95% confidence level. The range of transition probabilities p12
x obtained by
our calculations compared to the ISS point estimates is plotted in Figure 2 for males and Figure 3 for females: black points depict
the ISS estimation of incidence rates, white points show our central estimate, and the gray area defines the low and high bounds
of our estimate.
To measure the estimation accuracy, we show in Table 8 the following goodness-of-fit measures:
Nash-Sutcliffe Efficiency (NSE): a normalized statistic that determines the relative magnitude of the residual variance
compared to the measured data variance. Given a dependent variable y and its estimate y, NSE can be defined as
NSE = 1
(yy)
2
(yE(y))2 . The NSE ranges from ∞ to 1: if NSE = 1, there is a perfect match of modeled to the observed
data; if NSE = 0, the model predictions are as accurate as the mean of the observed data, if ∞ < NSE < 0, the observed
mean is a better predictor than the model.
Root Mean Square Error (RMSE) is the square root of the mean square error (MSE). The values range from 0 to +∞. If
RMSE = 0, there is a perfect match of modeled to the observed data.
280 F. BAIONE AND S. LEVANTESI
FIGURE 2. Transition Probabilities p12
x and Confidence Intervals, Total Cancers, Males, Year 2009.
FIGURE 3. Transition Probabilities p12
x and Confidence Intervals, Total Cancers, Females, Year 2009.
Normalized Root Mean Square Error: the ratio between the RMSE and the standard deviation of the observations. It is
also called RSR and ranges from 0 to +∞. It is particularly useful in the comparison between datasets or models with
different scales.
The GoF measures show high levels, especially for males. The values in Table 8 are consistent with Figures 5 and 6. It can
be observed that the values of transition probability p12
x are depicted on different y axis scales for males and females graphically
emphasizing and penalizing females’ estimates compared to what can be observed from the GoF measures.
Note that, especially for females, our estimates of transition probabilities are not strictly nondecreasing, because of certain
features of the data (prevalence rates and mortality rates of healthy and ill lives) used in the calculation. As already described in
TABLE 8
Goodness-of-Fit Measures for Estimates to ISS Estimate of Incidence Rates, Total cancers, Year 2009
Males Females
GoF measure Gompertz Weibull Gompertz Weibull
NSE (?∞,1] 0.958351 0.975032 0.876537 0.858839
RMSE [0,+∞) 0.000702 0.000544 0.000652 0.000697
RSR [0,+∞) 0.193608 0.149903 0.333341 0.356434
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 281
FIGURE 4. Growth Rate of Prevalence Rates, Total Cancers, Year 2009.
FIGURE 5. Transition Probabilities p14
x and p23
x by Gender and Model, Total Cancers, Year 2009.
Baione and Levantesi (2014), such behavior could be ascribed to the different size of the increases observed in prevalence rates
between the age groups. To better understand the prevalence rates increment, we calculate the growth rate of prevalence rates as
gr =
xr+1?x1 for r = 1, 2,..., n. The gr values shown in Figure 4 clearly indicate a change in the trend between the age
groups 50–55 and 65–69 for females. The trend change is in the age groups 55–59 for males, although it is not as visible as for
females.
FIGURE 6. Transition Probabilities p12
x by Gender and Model, Total Cancers, Year 2009.
282 F. BAIONE AND S. LEVANTESI
This result is not surprising because the incidence of certain cancers moderates with age, based on ideas about the timing
of “gene expression” or overall decline in susceptibility (for example, see the incidence of Hodgkin’s lymphoma and thyroid in
the United States in 1999–2013, available at https://nccd.cdc.gov/uscs [United States Cancer Statistics]). However, when dealing
with aging-associated diseases (e.g., atherosclerosis, cardiovascular disease, some types of cancer, arthritis, osteoporosis, and
hypertension), whose incidence increases rapidly with age, it may be appropriate to graduate the prevalence rates with a strictly
increasing function to guarantee the morbidity rates’ growth. In this regard, the assumption of a piecewise constant function to
describe the incidence allows one to bring out the “real” behavior of the phenomenon, without assuming a predefined shape. In
Appendix B, we show the figures of the transition probability from healthy to ill, p12
x , compared to the observed incidence rates
published by ISS and the relative range according to the confidence level p of the mortality intensities, for each analyzed category
of cancer: colorectal, lung, and stomach. The estimation accuracy is measured by the above-described goodness-of-fit measures
(RMSE, RSR, and NSE). Even for those categories of disease for which the quality of the mortality models fit is poor, the estimate
of p12
x does not differ much from the ISS estimates. The estimates of transition probabilities p14
x and p23
x are depicted in Figure 5
and the transition probabilities p12
x in Figure 6.
In contrast with the differences observed in the intensity of mortality between the Gompertz and the Weibull model (Fig. 5),
the estimate of the transition probability p12
x (Fig. 6) is not affected by the underlying mortality assumptions (except after about
70 years of age). However, the differences on p14
x and p23
x are not negligible, and they will have a significant impact on the pricing
of CI products. Taking gender into account, it is worth noting that the p12
x male curve is lower than the female curve until age
60, that is, younger and adult male cancer incidence rates are lower than those of females. On the other hand, males show higher
probabilities of death than females due to causes other than cancer. It is important to note that the differences between our model and
the MIAMOD mortality assumptions (Gompertz/Weibull versus observed values) as well as cancer incidence (piecewise constant
function versus logit function) cannot be investigated in terms of premium rates because MIAMOD data cannot be directly used
in the calculation of pure premium rate (see Eqs. [3.15]–[3.17]). A fair comparison between the two models in terms of insurance
premium can be made only when closed solutions for transition probabilities based on MIAMOD underlying assumptions can be
obtained.
4.4. Premium Rates
From the CI insurer’s point of view it is very important to analyze the effects of the estimated transition probabilities according to
the Gompertz and Weibull models on premium rates. Therefore, we calculate the net single premium rates for both standalone and
full acceleration rider CI insurance with a 10-year duration and a force of interest δ = ln(1.02). The resulting net single premium
rates for total cancers according to the Gompertz and Weibull models are reported in Table 9 and Table 10 for standalone and full
acceleration rider CI insurance, respectively.
The choice of the mortality model (Gompertz or Weibull) to arrange a multiple life table has a relevant impact on premium
rates only in case of full acceleration rider. When considering the premium rate for a full acceleration rider in Table 10, (2)A
(CI)
x:N , the
relative percentage differences between the Gompertz and the Weibull models are in the order of (15.48%, 11.92%) for males
and (6.84%, 14.34%) for females, while the relative percentage differences between premium rates for the standalone CI (see
Table 9) are on the order of (0.97%, 3.66%) for males and (0.63%, 2.20%) for females. The standalone CI premium rates,
for both males and females, are very similar due to the values of μ12
x estimated by the two mortality models that are very close to
each other. On the contrary, in the case of full acceleration rider, the premium rates are consistent with the trend of μ14
x that shows
higher values between ages 25 and 55 for both the genders in the Weibull model.
TABLE 9
Net Single Premium Rates (per 1000 m.u.) of Standalone ((1)A
(CI)
x:N ) CI Insurance, Gompertz Model versus Weibull Model, Total
Cancers, Year 2009
Males Females
Age Gompertz Weibull % Difference Gompertz Weibull % Difference
20 0.75 0.74 1.35 2.13 2.12 0.47
30 2.05 2.07 ? 0.97 8.45 8.48 0.35
40 6.77 6.87 ? 1.46 19.02 19.14 0.63
50 26.38 26.26 0.46 26.67 26.71 0.15
60 84.46 81.48 3.66 35.29 34.53 2.20
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 283
TABLE 10
Net Single Premium Rates (per 1000 m.u.) of Full Acceleration Rider ((2)A
(CI)
x:N ) CI Insurance, Gompertz Model versus Weibull
Model, Total Cancers, Year 2009
Males Females
Age Gompertz Weibull % Difference Gompertz Weibull % Difference
20 4.76 4.75 0.21 3.47 3.45 0.58
30 10.21 12.08 15.48 11.58 12.43 6.84
40 23.36 26.73 12.61 26.24 28.05 6.45
50 60.07 60.65 0.96 43.11 43.66 1.26
60 151.71 135.55 11.92 72.5 63.41 14.34
5. CONCLUSIONS
The article deals with the pricing of CI insurance contracts when data are scarce and only mortality and prevalence rates are
available. In this context, assuming a Weibull function to describe mortality rates, we find closed-form solutions for the transition
probabilities of a multiple state model for a CI insurance and make a comparison with the solutions obtained by substituting a
Gompertz formula for mortality rates. The Weibull and the Gompertz models are well known and frequently used in actuarial
science and statistics because they have a simple analytic expression and are relatively easy to estimate. In addition, these models
imply different biological causes of aging and differ in the way they develop mortality increases with age; some authors (see, e.g.,
Juckett and Rosenberg 1993) state that the Weibull model is preferable to Gompertz when modeling single causes of death. We
have tested this assertion and have found the Gompertz model to be a better descriptor of mortality (of both healthy and ill lives)
than the Weibull model.
We have also addressed the relevance of the mortality model to CI premium calculation: We separately calculated the effect
of both the Weibull and the Gompertz models on certain net single premium rates for CI insurance. Finally, from our analysis we
conclude that the use of the Gompertz mortality model rather than the Weibull model does not produce meaningful differences in
estimates of transition probabilities from healthy to ill lives (p12
x ) within our multistate calculations.
This result should be expected because the illness transition rates, μ12
x , do not directly depend on the mortality model used.
Moreover, the probabilities of remaining in state1 or in state 2, respectively p11
x and p22
x , both needed for calculating p12
x (see
Eq. [3.5]) have very similar values across almost the entire age range for both the Gompertz and Weibull models. The greatest
differences are observed in the older age group and are ascribed to the different slopes of μ14
x and μ23
x . The slope of the mortality
transition intensity of healthy lives, μ14
x , has a considerable impact on premium calculation only in the case where a full acceleration
rider is provided.
ACKNOWLEDGMENTS
The authors thank two anonymous referees whose suggestions greatly improved the article.
ORCID
Fabio Baione http://orcid.org/0000-0002-9926-7869
Susanna Levantesi http://orcid.org/0000-0002-4644-4358
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APPENDIX A. DATABASE
In this appendix, we show the prevalence rates of cancer sickness and the mortality rates by type of cancer downloaded from
the website http://www.registri-tumori.it/cms/ (Istituto Superiore di Sanitá 2015) for year 2009 and for the age range 20–69 .
The cancer categories are classified according to the International Classification of Disease, version 10 (ICD-10), published by
the World Health Organization (WHO 2007): stomach (ICD-10: C16), colorectal (ICD-10: C18-21), lung (ICD-10: C33-C34),
malignant melanoma (ICD-10: C43), breast (ICD-10: C50), cervix uteri (ICD-10: C53), prostate (ICD-10: C61).
TABLE A.1
Prevalence Rates of Cancer by Age Group, Males, Italy, 2009 (Rates per 1000)
Age group C16 C18–C21 C33–C34 C43 C61 Total cancers
20–24 0.01 0.02 0.01 0.20 0.00 0.24
25–29 0.02 0.05 0.01 0.42 0.00 0.50
30–34 0.05 0.11 0.03 0.74 0.01 0.93
35–39 0.10 0.26 0.06 1.13 0.02 1.57
40–44 0.22 0.60 0.17 1.58 0.07 2.64
45–49 0.43 1.30 0.42 2.03 0.25 4.43
50–54 0.80 2.69 1.00 2.50 1.00 7.98
55–59 1.38 5.22 2.16 2.99 3.49 15.24
60–64 2.13 8.93 4.01 3.21 9.34 27.63
65–69 3.39 15.36 6.97 3.67 22.79 52.17
Source: National Institute of Health (ISS).
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 285
TABLE A.2
Prevalence Rates of Cancer by Age Group, Females, Italy, 2009 (Rates per 1000)
Age group C16 C18–C21 C33–C34 C43 C50 C53 Total cancers
20–24 0.01 0.02 0.01 0.48 0.10 0.00 0.62
25–29 0.02 0.05 0.02 0.82 0.38 0.02 1.32
30–34 0.04 0.12 0.04 1.27 1.22 0.11 2.81
35–39 0.09 0.28 0.10 1.72 3.24 0.35 5.77
40–44 0.17 0.61 0.20 2.16 7.31 0.78 11.23
45–49 0.29 1.22 0.34 2.51 13.20 1.18 18.74
50–54 0.48 2.30 0.55 2.84 20.83 1.40 28.40
55–59 0.75 4.04 0.88 3.16 28.92 1.37 39.12
60–64 1.07 6.27 1.29 3.22 34.85 1.21 47.92
65–69 1.62 9.76 1.80 3.56 42.11 1.06 59.90
Source: National Institute of Health (ISS).
TABLE A.3
Mortality Rates by Age Group and Type of Cancer, Males, Italy, 2009
Age group C16 C18–C21 C33–C34 C43 C61 Total cancers
20–24 11.76% 3.13% 36.36% 0.65% 0.00% 2.43%
25–29 10.53 4.88 33.33 0.85 0.00 2.45
30–34 10.78 4.76 32.73 0.91 0.00 2.78
35–39 9.80 4.68 31.76 0.95 2.38 3.40
40–44 9.48 4.60 31.31 0.94 1.90 4.49
45–49 10.50 5.10 32.44 1.18 1.95 6.25
50–54 10.98 5.30 34.13 1.26 1.87 7.77
55–59 12.49 5.32 34.28 1.33 1.32 8.37
60–64 13.37 5.48 35.00 1.52 1.23 8.47
65–69 14.05 5.68 34.84 1.67 1.31 7.93
Source: Authors’ processing of data from National Institute of Health (ISS).
Note: The values in the column “Total cancers” are obtained as a weighted average of the number of deaths for each type of cancer with
weight being the proportion of sickness of a specific cancer over cancer’s sickened population.
TABLE A.4
Mortality Rates by Age Group and Type of Cancer, Females, Italy, 2009
Age group C16 C18–C21 C33–C34 C43 C50 C53 Total cancers
20–24 11.11% 5.88% 15.38% 0.28% 1.36% 0.00% 0.98%
25–29 17.86 5.62 18.18 0.43 1.72 2.38 1.53
30–34 15.29 5.45 17.20 0.42 1.65 2.59 1.74
35–39 14.78 5.22 17.94 0.47 1.66 2.19 1.97
40–44 13.24 5.02 17.61 0.51 1.65 1.95 2.09
45–49 11.91 4.87 22.22 0.56 1.58 2.35 2.24
50–54 10.94 4.61 24.87 0.61 1.40 2.56 2.26
55–59 11.41 4.24 25.29 0.66 1.40 2.95 2.42
60–64 11.64 4.20 27.23 0.83 1.47 3.55 2.76
65–69 12.23 4.51 30.14 0.90 1.51 4.44 3.16
Source: Authors’ processing of data from National Institute of Health (ISS).
Note: The values in the column “Total cancers” are obtained as a weighted average of the number of deaths for each type of cancer with
weight being the proportion of sickness of a specific cancer over cancer’s sickened population.
286 F. BAIONE AND S. LEVANTESI
APPENDIX B. TRANSITION PROBABILITIES FROM HEALTHY TO ILL FOR SINGLE CATEGORIES OF CANCER
FIGURE B.1. Transition Probabilities p12
x and Confidence Intervals, Lung Cancer, Males, Year 2009.
FIGURE B.2. Transition Probabilities p12
x and Confidence Intervals, Lung Cancer, Females, Year 2009.
FIGURE B.3. Transition Probabilities p12
x and Confidence Intervals, Stomach Cancer, Males, Year 2009.
PRICING CRITICAL ILLNESS INSURANCE FROM PREVALENCE RATES 287
FIGURE B.4. Transition Probabilities p12
x and Confidence Intervals, Stomach Cancer, Females, Year 2009.
FIGURE B.5. Transition Probabilities p12
x and Confidence Intervals, Colorectal Cancer, Males, Year 2009.
FIGURE B.6. Transition Probabilities p12
x and Confidence Intervals, Colorectal Cancer, Females, Year 2009.
288 F. BAIONE AND S. LEVANTESI
TABLE B.1
Goodness-of-Fit Measures for our Estimates to the ISS Estimate of Incidence Rates, Lung Cancer, Year 2009
Males Females
GoF measures Gompertz Weibull Gompertz Weibull
RMSE [0,+∞) 0.000175 0.000153 0.000031 0.000044
RSR [0,+∞) 0.173213 0.151728 0.128754 0.180129
NSE (∞,1] 0.966664 0.974421 0.981581 0.963948
TABLE B.2
Goodness-of-Fit Measures for our Estimates to the ISS Estimate of Incidence Rates, Stomach Cancer, Year 2009
Males Females
GoF measures Gompertz Weibull Gompertz Weibull
RMSE [0,+∞) 0.000055 0.000045 0.000026 0.000024
RSR [0,+∞) 0.229971 0.186011 0.255736 0.240267
NSE (∞,1] 0.941237 0.961555 0.927332 0.935857
TABLE B.3
Goodness-of-Fit Measures for our Estimates to the ISS Estimate of Incidence Rates, Colorectal Cancer, Year 2009
Males Females
GoF measures Gompertz Weibull Gompertz Weibull
RMSE [0,+∞) 0.000138 0.000132 0.000072 0.000070
RSR [0,+∞) 0.156957 0.149770 0.163255 0.157050
NSE (∞,1] 0.972627 0.975076 0.970386 0.972595
APPENDIX C. FULL WEIBULL MODEL
We assume that transition intensity from a healthy to ill state is described by a Weibull function:
μ12
x = b1
hci · xbhci
2 , (C.1)
where bhci
1 > 0 and bhci
2 are the Weibull parameters. The solution to Equations (3.4) and (3.6) becomes, respectively:
t p11(C.3)
while the solution to Equation (3.5) under the full Weibull model is
2 +1 ·eDx[(bhci
2 +1,D(x+t))(bhci
2 +1,Dx)]
, (C.4)http://www.6daixie.com/contents/3/1992.html
where D = [?bhci
1 (x + t2 )bhci2 + γ bh
1(x + t2 )bh2 + bci1 (x + t2 )bci2 ].
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