辅导ISSN留学生、讲解Java/Python编程语言、辅导Python/c++程序
讲解数据库SQL|辅导留学生Prolog
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.
Submit your article to this journal
Article views: 86
View Crossmark dataNorth 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: fabio.baione@studioacra.it
270PRICING 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 describingPRICING 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 ? 2PRICING 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). Accordi