Comparison of prices of life insurances using different mortality rates models

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School of Education, Culture and Communication

Division of Applied Mathematics

Comparison of prices of life insurances using different mortality rates

models

by

Belinda Straß

MASTER THESIS IN MATHEMATICS/ APPLIED MATHEMATICS

DIVISION OF APPLIED MATHEMATICS

MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN

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School of Education, Culture and Communication

Division of Applied Mathematics

Master thesis in mathematics / applied mathematics

Date:

2018-06-01

Project name:

Comparison of prices of life insurances using different mortality rates models

Author:

Belinda Straß

Supervisor(s):

Milica Ranˇci´c and Karl Lundengård

Reviewer: Anatoliy Malyarenko Examiner: Ying Ni Comprising: 30 ECTS credits

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I dedicate this thesis to my parents, Regina and Karl, through whose encouragement and love I could achieve my dreams.

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Acknowledgements

I want to thank my supervisor Milica Ranˇci´c for her amazing support and guidance throughout this journey, as well as Karl Lundengård.

I want to express my thanks to Professor Anatoliy Malyarenko for taking his time reviewing my thesis and giving great feedback. Additionally, I want to thank my examiner Ying Ni. Especially, I want to thank Andromachi and Pablo for all the laughs, tears, despairs and achievements we shared throughout the last two years.

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Abstract

Capturing mortality became a crucial modelling problem throughout the years due to the rais-ing demand of life insurances and annuities. Fittrais-ing three models, namely, logistic, Heligman– Pollard HP4 and power–exponential model, to real life data shows that latter two models represent the actual data quite well. Pricing a term life insurance and a whole life annuity, implemented using the MATLAB software, based on these models ends in the result that the Heligmann–Pollard HP4 model is the less preferable model, in perspective of an insured, than the logistic or power–exponential ones.

Keywords: Pricing life insurances, modelling mortality rate, logistic model, Heligmann– Pollard HP4, power–exponential model.

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6 Summary of reflection of objectives in the thesis 39 6.1 Objective 1: Knowledge and understanding . . . 39 6.2 Objective 2: Methodological knowledge . . . 39 6.3 Objective 3: Critically and Systematically Integrate Knowledge . . . 40 6.4 Objective 4: Independently and Creatively Identify and Carry out Advanced

Tasks . . . 40 6.5 Objective 5: Present and Discuss Conclusions and Knowledge . . . 40 6.6 Objective 6: Scientific, Social and Ethical Aspects . . . 40

Bibliography 40

A MATLAB Code 43

A.1 Term Life Insurance . . . 43 A.2 Whole Life Annuity . . . 47

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List of Figures

2.1 Example of the logarithm of the logistic mortality rate . . . 13 2.2 Development of the logarithm of the logistic mortality rate from 1950 to 2010. 14 2.3 Example of the logarithm of the Heligman–Pollard HP4 mortality rate. . . 17 2.4 Development of the logarithm of the Heligman–Pollard HP4 mortality rate

from 1950 to 2010. . . 18 2.5 Example of the logarithm of the power–exponential mortality rate. . . 20 2.6 Development of the logarithm of the power–exponential mortality rate from

1950 to 2010. . . 21 2.7 Mortality of 1950 and 2010 with three mortality models: logistic, Heligman–

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List of Tables

1.1 Mortality rates’ models. . . 9

4.1 Term life insurance for Sweden. . . 32

4.2 Term life insurance for the USA. . . 33

4.3 Term life insurance for Germany. . . 34

4.4 Whole life Annuity with quarterly payments for the logistic, Heligman–Pollard HP4 and power–exponential mortality rate. . . 35

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Chapter 1 Introduction

In the early eighteenth century the first version of life insurances came alive among companies and their employees. The liabilities were calculated by the deaths occurred within the insured lives in that year. From that day forward the interest of modelling mortality increased rapidly [Dickson et al., 2013]. Hence, the question arises how to capture, or rather, model mortality in general which is needed to define and price life insurances. The answer for mortality rates was given by de Moivre in 1725 when he transformed John Graunts life tables into simple mathematical laws. This set the start for many developments and suggestions on capturing mortality. The current pattern of mortality consists on three parts; the first part describes the mortality of infants, the second period includes the way into adulthood and the third period describes a slow geometric increase of mortality with age [Forfare, 2006, Hannerz, 2001]. However, the second part is quite tricky. It describes the so–called “accident hump” which refers to a higher mortality among adolescents due to accidents by cars or motorbikes as well as maternal mortality among women [Hannerz, 2001].

Mortality laws are mathematically parameterised functions of age that specify the age pattern in functional form beforehand. We will focus on parameterised functions, but it is noticeable that there exists other models like multi-exponential models or principal component models [Booth and Tickle, 2008].

In Table 1.1 a list of parametrised models from the literature on mortality rates can be found. We start with a very famous mortality law known as the Gompertz–Makeham model. After Gompertz introduced 1825 the first law of mortality which concluded the geometric increase of the force of mortality with age, Makeham extended 1867 the aforementioned law by one term which is autonomous of age and therefore, senescent deaths, i.e. accidents [Forfare, 2006]. The Gompertz–Makeham model is a 3-parameter model and Pham [Pham, 2008] states that the model shows improvement at younger ages in comparison to Gompertz law but it also over estimates the old ages.

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each term represents one of the three periods mentioned above, whereas the last term de-scribes Gompertz law for senescent. The Modified Perks model is a 4-parameter model, using a gamma frailty distribution which makes it preferable compared to the Gompertz–inverse Gaussian model, due to its better fit and logistic form in older ages. The Double Geometric mortality law is a 5-parameter model [Forfare, 2006, Butt and Haberman, 2004].

Another modification of the Gompertz law is the Gompertz–inverse Gaussian model and it is said that it is not very good in predicting mortality for older ages and it is less receptive to data variations. [Butt and Haberman, 2004].

In 1951 Weibull introduced his rate of mortality. It is a 2-parameter formula, where these parameters describe the scale and the shape, respectively, of the distribution [Forfare, 2006, Pham, 2008].

Another well known model is the Heligman–Pollard HP model which has four different ver-sions. The Heligman–Pollard HP1 model has three terms where each term represents again the infant period, young adulthood and older ages. It is a 8-parameter formula and all pa-rameters are estimated in one step [Booth and Tickle, 2008]. The second Heligman–Pollard HP2 model is similar to the first one except the term for older ages differs. Furthermore, the Heligman–Pollard HP3 and Heligman–Pollard HP4 are 9-parameter functions and cover the entire lifespan. The only difference, within all the versions, is the last term which represents senescent mortality [Forfare, 2006].

In 1999, Hannerz first proposed a 5-parameter law of mortality, only for Swedish females and got excellent fitting results. This model would not be adequate for the male population due to its “accident hump” throughout males’ way into manhood. That is why Hannerz pro-posed a proper mortality rate for males and therefore, uses different models for each gender [Hannerz, 2001].

A logistic shifting model from Bongaarts ended in a one-factor parametrization for humans over 25 [Booth and Tickle, 2008]. The log-logistic model should be used if there exists a liability of adolescence. Bruderl and Schussler say that it will perform better than a Makeham model because it allows a monotonically fall and a hazard function of an inverted U-shape [Bruderl and Schussler, 1990].

Lundengård et al. propose a single-hump model, named power-exponential function, repre-senting the high mortality in young adults, and a multi-hump model describing all humps during the lifespan [Lundengård et al., 2017].

It is important to discuss these models because they are necessary to evaluate the payments for policy holder. These payments are dependent on the death or survival of the insured person. Because the death date of a person is uncertain, the amount of the payments is uncertain and therefore, it can be modelled as a random variable. That means that the expected present value (EPV) of these claims is a function of time and death.

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The traditional life insurances are whole life, term life and endowment insurances. More specific, a whole life insurance covers the whole life and it will pay the benefits when the person actually dies and it will then become a claim.

A term life insurance covers a certain period of time, e.g. ten years, and the payments can only be received if the person dies within this fixed term.

Pure endowment insurances are special cases of endowment policies. It depends on the sur-vival of the policyholder and will only pay the benefits to the insured life if it is still alive after the maturity date. This version is mostly used in combination with a term insurance. It is then called endowment insurance, and the payment is made, either if the person dies within a fixed period of, say n, years or it is paid at the end of year n, if the individual survives these n years. Except the pure endowment, these insurance payouts can occur continuously — immediately after death, annually — at the end of the year of death or mth-ly — at the end of one of the mth term in the year of death, for example quarterly or monthly. Therefore, m would be either four or twelve.

Furthermore, the insurance policy can be deferred which means that the insured person can negotiate the contract now but it will not start before a predetermined period of time has passed.

Another life contingent claim is called annuity. It is defined as a series of compensations to or from the insured person for as long as the person still lives on the payment date. The present value is dependent on the future lifetime, and therefore, it represents a random variable. We distinguish annuities again in whole life, term life and endowment annuities. The payments can be made as well annually, mth-ly or continuously. Except for the continuous case all other variations can be deferred or be immediate. The annuity–certain is the equivalent to the pure endowment and it states that it has no uncertainty. [Slud, 2001, Dickson et al., 2013]

The aim of this work is to compare different mortality models with respect to pricing life insurances. That means that we use three different mortality laws, namely, the logistic, the Heligman–Pollard HP4 and the power-exponential hazard rate by Lundengård et al. We fit these to actual mortality data, create expected present value tables and calculate a term life insurance with annually payments, and a whole life annuity with quarterly payments. Finally, we discuss our results.

In Chapter 2, we give a definition of a survival function, derive the survival function of the aforementioned mortality laws and discuss them in greater detail with suitable figures. Chap-ter 3 has the mathematical representation of the Chap-term life insurance and whole life annuity. In Chapter 4, we give a review of the non-linear least square fitting method used in MATLAB, the application to real data as well as a discussion based on the results of the term life insurance and annuity. Chapter 5 gives a short conclusion and further suggestions of research1.

1_{Part of the findings in this project will be reported in the full version of [Lundengård et al., 2018] and will be}

presented at the 5th International Conference on Stochastic Modeling Techniques and Data Analysis, SMTDA 2018, in June 2018.

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Table 1.1: Mortality rates’ models.

Model Mortality rate

Gompertz-Makeham [Forfare, 2006] µ (x) = a + becx

Thiele [Forfare, 2006] µ (x) = a1e−b1x+ a2e−b2

(x−c)2

2 + a₃eb3x

Modified Perks [Forfare, 2006] µ (x) = a

1 + eb−cx+ d Double Geometric [Forfare, 2006] µ (x) = a + b1bx2+ c1cx2 Gompertz-inverse Gaussian

[Butt and Haberman, 2004]

µ (x) = e a−bx √ 1 + e−c+bx Weibull [Forfare, 2006] µ (x) =a b x b a−1

Heligman – Pollard HP1 [Forfare, 2006] µ (x) = a(x+a₁ 2)a3+ b1e −b2ln x b3 2 + c1cx₂ Heligman – Pollard HP2 [Forfare, 2006] µ (x) = a(x+a₁ 2)a3+ b1e

−b2ln x b3 2 + c1c x 2 1 + c1cx₂ Heligman – Pollard HP3 [Forfare, 2006] µ (x) = a(x+a₁ 2)a3+ b1e

−b2ln x b3 2 + c1c x 2 1 + c3c1cx₂ Heligman – Pollard HP4 [Forfare, 2006] µ (x) = a(x+a₁ 2)a3+ b1e

−b2ln x b3 2 + c1c xc3 2 1 + c1cx₂c3 Hannerz [Hannerz, 2001] µ (x) =g(x)e

G(x) 1 + eG(x), g(x) = a1 x2+ a2x+ a3e cx_, G(x) = a0− a1 x + a2x2 2 + a3 c e cx Logistic [Booth and Tickle, 2008] µ (x) = ae

bx 1 +ac

b(e bx_{− 1)} Log-logistic [Bruderl and Schussler, 1990] µ (x) = abx

a−1 1 + bxa Power-Exponential [Lundengård et al., 2017] µ (x) = c1

xe−c2x+

∑

~a∈A a1 xe−a2x a3 , µ (x) = c1 xe−c2x+ a1 xe −a2xa3

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Chapter 2 Derivation of Survival Functions

A survival function as well as the force of mortality are models for capturing human mortality. They calculate the probability of survival or death at specific ages. They are both fundamental concepts of future lifetime modelling.

Survival functions can either be calculated from life tables, which use integer ages, or through mortality laws which are continuous functions of time [Dickson et al., 2013].

2.1 The Survival Function

Definition 1. Let Fx(t) = P {Tx≤ t} be the lifetime distribution, with x representing the current age of a life, and Tx be the continuous random variable of the future lifetime, then a survival function for a life aged x surviving t years is defined as:

S_x(t) = 1 − Fx(t) = P {Tx≥ t} .

Definition 2. The mortality rate µx at age x, with T0 is the future lifetime at birth, is defined as: µx= lim dx→0+ 1 dxP {T0≤ x + dx | T0> x} = limdx→0+ 1 dx(1 − Sx(dx)) .

Due to the topic of mortality laws, the survival function can be derived by Theorem 1 to which the later derivations will refer.

Theorem 1. Let Sx(t) be the survival function and µx the mortality rate. Then, the survival function described by the force of mortality is:

S_x(t) =S0(x + t) S₀(x) = exp − Z x+t x µrdr = exp − Z t 0 µ(x+s)ds .

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These following three conditions must be satisfied by Sx(t) to be called a survival function.

Condition 1 A life at age x surviving 0 years must have the probability of 1: Sx(0) = 1. Condition 2 All individuals die: lim

t→∞Sx(t) = 0.

Condition 3 The function of t must be non-increasing: S0_x(t) ≤ 0.

Note, that the third condition will not be proven in all three models because exp (−x) is defined as a decreasing function.

For detailed proof of Theorem 1, the reader may refer to [Dickson et al., 2013].

2.2 Logistic model

The logistic mortality model is a one-factor function with three parameters. It is basically the Gompertz law with a logistic shifting with whom the logistic model addresses the bias in older ages. Meaning, that at that age range the Gompertz mortality function overestimates the observed data. The parameter a represents the level of Gompertz mortality, b symbolizes the exponential increase of mortality with age and c stands for the age-invariant background mortality. If we plot the logistic force of mortality, it is pictured as a rise with increasing age and for older lifes it plateaus [Booth and Tickle, 2008, Wilson, 1994].

Theorem 2. The survival function with the logistic mortality law is

S_x(t) = exp ( −1 cln " ac(eb(x+t)− 1) + b ac(ebx_{− 1) + b} #) . (2.1) Proof. S_x(t) = exp ( − Z t 0 aeb(x+s) 1 +ac_b(eb(x+s)_{− 1)}ds ) .

Considering only the integral

Z t

0

aeb(x+s)

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and solve it by substituting the denominator by u and rearranging it for ds: u= 1 +ac b(e b(x+s)_{− 1),} du ds = ace b(x+s)_, ds = du aceb(x+s).

Then, solve the new integral

Z _aeb(x+s) u du aceb(x+s) = Z ₁ u du c = 1 c Z _du u = 1 cln u.

Our last step is to insert the adjusted upper and lower bounds and simplify the terms:

1 cln " ac(eb(x+s)− 1) b + 1 # 1+ac(eb(x+t)−1_b ) 1+ac(ebx−1_b ) = 1 c ln " ac(eb(x+t)− 1) b + 1 # − ln ac(e bx_{− 1)} b + 1 ! = 1 cln " ac(eb(x+t)− 1) b + 1 ! · ac(e bx_{− 1)} b + 1 −1# = 1 cln " ac(eb(x+t)− 1) + b ac(ebx_{− 1) + b} # .

This results in the survival function

S_x(t) = exp ( −1 cln " ac(eb(x+t)− 1) + b ac(ebx_{− 1) + b} #) .

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Condition 1 Sx(0) = 1. S_x(0) = exp ( −1 cln " ac(eb(x+0)− 1) + b ac(ebx_{− 1) + b} #) = exp −ln(1) c = exp(0) = 1. Condition 2 lim t→∞Sx(t) = 0. lim t→∞exp ( −1 cln " ac(eb(x+t)− 1) + b ac(ebx_{− 1) + b} #) = 0.

The Figure, Fig. 2.1, describes the logistic mortality model of Sweden male population. This illustration simplifies the understanding on the mortality rate. The x-axis represents the age of the individual in years, whereas the y-axis depicts the logarithm of the force of mortality.

Figure 2.1: Example of the logarithm of the logistic mortality rate

The logistic model has only three parameters, namely a, b and c. The parameter a shifts the whole line along the y-axis. Parameter b tilts the graph clockwise and c produces a steep curve for the old ages for values greater than 0.9.

It is clearly seen that the logistic model is an unsuitable function for presenting the mortality rate. It does not include the “accident hump” nor does it represent the infant period correctly.

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The function is an increasing straight line which can be interpreted as the mortality for the ageing body.

Nonetheless, we will compare the logistic mortality rate for several years. We chose to use the parameter of fitted data from 1950 to 2010 in 20 year steps.

Figure 2.2: Development of the logarithm of the logistic mortality rate from 1950 to 2010. We observe in Fig. 2.2 that the overall mortality rate decreases with the years throughout an individual’s life. Furthermore, the greatest drop of overall mortality is between the years 1970 and 2010.

The above mentioned plateau which the logistic model should provide according to the theory, shows in ages over 120. If we choose a smaller age range, the beginning of the steep curve would already be seen at age 80.

2.3 Heligman–Pollard HP4

The Heligman–Pollard HP4 mortality rate is a 9-parameter function divided into three terms. Every parameter holds a demographic interpretation; for instance, parameter a1 shows the level of infant mortality, b1 describes the severity of the “accident hump” and c1 is the level of adult mortality. Furthermore, each term expresses one part of a human life. Specifically, the early childhood, the “accident hump” of males’ way into manhood and women’s maternal mortality, as well as the senescent mortality. It is a continuous function and applicable for the entire lifespan. Additionally, the curve is quite adequate flexible and can be adapted to a broad range of mortality patterns. The fourth version of Heligman–Pollard is said to be the best in fit, especially in older ages [Forfare, 2006, Heligman and Pollard, 1980].

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Theorem 3. The survival function including Heligman–Pollard HP4 force of mortality is Sx(t) = exp ( 1 a₃ (−1) 1 a3 (ln (a1)) 1 a3 " γ 1 a₃, − ln (a1) (x + t + a2) a3 − γ 1 a₃, − ln (a1) (x + a2) a3 # − b1b2b₃ 2 " (x + t)1−2b2_{− x}1−2b2 1 − 2b2 # − Z t 0 c₁c(x+s)₂ c3 1 + c1c(x+s) c3 2 ds ) . (2.2) Proof. S_x(t) = exp ( − Z t 0 " a(x+s+a2)a3 1 + b1e −b2ln x+s b3 2 + c1c (x+s)c3 2 1 + c1c(x+s) c3 2 # ds ) .

By definition, we know that the sum rule for integrals allows us to separate it into three indi-vidual parts, therefore the proof falls naturally into three parts.

The first integral uses a substitution of

u= (x + s + a2)a3,

ds = du

a3(x + s + a2)a3−1 .

This gives the new integral with a different upper bound of u2= (x + t + a2)a3and lower bound of u1= (x + a2)a3

Z u₂

u1

au₁ du

a₃(x + s + a₂)a3−1.

Additionally, we express s in terms of u and transform the equation, we get

Z u₂ u1 au₁du a3u a3−1 a3 = 1 a₃ Z u₂ u1 exp (u ln(a1)) · u 1 a3−1_du.

We use a second substitution of

uln (a1) = −t, and

du = −du ln (a1)

,

with which we can transform the integral to the gamma-function: 1 a₃ Z t₂ t1 e−t t ln (a1) 1 a3−1 dt ln (a1) ,

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with the new bounds t₁= − ln (a₁)u₁, t₂= − ln (a1)u2. Thus, 1 a3 (−1) 1 a3 (ln (a1)) 1 a3 Z t₂ t1 e−t t 1 a3−1_{dt =} 1 a3 (−1) 1 a3 (ln (a1)) 1 a3 Z t₂ 0 e−tt 1 a3−1_{dt −} Z t₁ 0 e−tt 1 a3−1_dt .

These last two integrals are the expressions for the gamma-function [Abramowitz and Stegun, 1964] and after inserting the bounds and changing back the values we get

1 a₃ (−1) 1 a3 (ln (a1)) 1 a3 " γ 1 a₃, − ln (a1) (x + t + a2) a3 − γ 1 a₃, − ln (a1) (x + a2) a3 # .

The second part is quite straightforward. By reducing the term the remaining integral consists of b₁b2b2 3 Z t 0 (x + s)−2b2_ds.

The substitution of u = (x + s) with ds = du, gives the integral b₁b2b2 3 Z x+t x 1 u−2b2du.

Using the power rule and inserting the bounds we get u1−2b2 1 − 2b2 x+t x = b1b2b₃ 2 " (x + t)1−2b2_{− x}1−2b2 1 − 2b2 # .

under the restriction of b26=1₂.

The last term has no analytical solution and will therefore be solved numerically in MATLAB. Combining all three parts again, we receive the survival function as in Eq.(2.2) based on Heligman–Pollard (HP4).

The two conditions reassure the survival function:

Condition 1 Sx(0) = 1

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Condition 2 lim t→∞Sx(t) = 0 lim t→∞ exp ( 1 a3 (−1) 1 a3 (ln (a₁)) 1 a3 " γ 1 a3 , − ln (a1) (x + t + a2)a3 − γ 1 a3 , − ln (a1) (x + a2)a3 # − b1b2b₃ 2 " (x + t)1−2b2_{− x}1−2b2 1 − 2b2 # − Z t 0 c1c(x+s) c3 2 1 + c1c (x+s)c3 2 ds ) = 0.

Figure 2.3 shows the course of the mortality rate of Heligman–Pollard HP4.

Figure 2.3: Example of the logarithm of the Heligman–Pollard HP4 mortality rate. The black line represents the entire Heligman–Pollard HP4 mortality rate. Term 1, the green line, describes the infant mortality which decreases rapidly in the first few years of life. The red parabola shows the “accident hump” of males into manhood. The third term describes the continuously increase of mortality of the ageing body .

All nine parameters have reasonable values, but we still do not know how the parameters affect their own term.

Figure 2.3 shows also the parameters and their directional influence on their corresponding term. The three parameters a1, a2, a3of the first term influence the infant mortality by moving along the x-axis. Additionally, the first parameter a1is responsible for the slope. Parameter a2 can shift the green line only until the start of the “accident hump” which is usually around the age of 15. However, the last parameter, a3, of term 1 moves along the x-axis and therefore, the mortality of the ageing body.

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The second term which expresses the “accident hump”, has a great impact on the shape of the mortality rate. As we can see, the parameter b1as well as the parameter b3change the position of the hump. For example, the first parameter shifts the hump up or down, whereas the later one moves along the x-axis. Only b2varies the shape from wider to narrower.

The last observation focuses on the term for older ages. The third term also has three parame-ters, namely c1, c2and c3. The first parameter shifts the entire graph up and down. The second parameter defines the start of the third term, and therefore the beginning of the mortality rate. Parameter c3tilts the graph clockwise.

Another interesting aspect on this topic is how the mortality rate changed over time. The next figure, Fig. 2.4, demonstrates the development of the force of mortality through time. We chose the years 1950, 1970, 1990 and 2010.

Figure 2.4: Development of the logarithm of the Heligman–Pollard HP4 mortality rate from 1950 to 2010.

It is clearly observable that humans death rate is lower in 2010 than it was 60 years ago. Furthermore, more infants until the age of seven died in the 50s. It is interesting that the “accident hump” of young males had its peak in the 70s around the age of 20, whereas in the other decades the peak is at age 23.

2.4 Power-exponential function

Authors of [Lundengård et al., 2017] introduced a new model for modelling mortality. The power–exponential model is seen as a good alternative to other mortality models which have

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either more or less parameters. It performs as well as the Heligman–Pollard HP4 model for some countries and it is said that the fit of this model is quite good to observed data [Lundengård et al., 2017].

Theorem 4. The survival function with power–exponential mortality rate, single hump, is described by: S_x(t) = exp ( − c₁[Ei (c2(x + t)) − Ei (c2x)] − a1 (a2a3x)a3+1 [γ (a3+ 1, a2a3(x + t)) − γ (a3+ 1, a2a3x)] ) . (2.3) Proof. S_x(t) = exp − Z t 0 c₁ (x + s) e−c2(x+s) + a1 (x + s) e−a2(x+s)a3 ds .

We begin again the proof by dividing Eq.(2.3) into two parts, precisely two individual inte-grals.

The first integral is solved by substituting c2(x + s) with u:

c2(x + s) = u ⇒ ds = 1 c₂du. It follows, Z c₂(x+t) c2x c₁ u c2e −u du c₂ = c1 Z c₂(x+t) c2x eu udu,

whereas the last integral describes the Exponential integral [Abramowitz and Stegun, 1964]. Therefore,

c₁Ei (u)|c2(x+t)

c2x = c1[Ei (c2(x + t)) − Ei (c2x)] .

Considering the second part, we use the change in variables of a2a3(x + s) = u ⇒ ds =_a₂1_a₃du. Thus, Z a₂a₃(x+t) a2a3x a1 _ua3 (a₂a₃)a3e −u du a₂a₃ = a₁ (a₂a₃)a3+1 Z a₂a₃(x+t) a2a3x ua3_e−u_du = a1 (a₂a₃)a3+1γ (a3+ 1, u) a2a3(x+t) a2a3x = a1 (a2a3)a3+1 [γ (a3+ 1, a2a3(x + t)) − γ (a3+ 1, a2a3x)] .

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Combining both parts, we have proven that a survival function with power-exponential func-tions exists, it is Eq.(2.3).

Lastly, we confirm the Theorem 4 through the two conditions:

Condition 1 Sx(0) = 1 Sx(0) = exp {0} = 1. Condition 2 lim t→∞Sx(t) = 0 Sx(t) = lim t→∞exp ( − c1[Ei (c2(x + t)) − Ei (c2x)] − a1 (a2a3)a3+1 [γ (a3+ 1, a2a3(x + t)) − γ (a3+ 1, a2a3x)] ) = 0.

Again, we take a closer look at the graphs of the power–exponential mortality law: The Fig-ure 2.5. It shows that the first term influences the death with age. The second term is respon-sible for the “ accident hump”.

Figure 2.5: Example of the logarithm of the power–exponential mortality rate.

If we change the parameters of the first term, we observe that c1 shifts the entire rate up for smaller values and down for bigger ones. Despite that, the second parameter in term 1 tilts the graph counter clockwise. It affects both ages, the infant mortality and the senescent mortality.

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The second term changes the position and shape of the “accident hump”. For instance, a1 moves the “accident hump” according to the y-axis. The second parameter a2 shifts it to the left or right and the last parameter a3shapes the “accident hump”; wider for higher values and it becomes narrower for smaller values.

Finally, we compare again the development of mortality through time. We chose the same decades as in the aforementioned models.

Figure 2.6: Development of the logarithm of the power–exponential mortality rate from 1950 to 2010.

The assumptions that we made for the Heligman–Pollard HP4 and the logistic model agree with the power–exponential model. The amount of people dying in total decreased with the decades. Furthermore, the “accident hump” is shifted more to the left, towards younger people, in the 70s.

Additionally, it is very clear that the power–exponential function has a much smoother transi-tion from high to low mortality in the infant period, compared to the Heligman–Pollard HP4 model.

The three models were explained and studied individually. However, we still do not know which model reflects the reality the best. In the next Figure, Fig. 2.7 we have the mortality data of 1950 and 2010, as well as the three mentioned models.

The real mortality, represented in blue and retrieved from the Human Mortality Database (HMD), is captured very well by Heligmann–Pollards HP4 model in both years. The power– exponential model fits the infant mortality and the “accident hump” very well too, but the mortality for older humans differs quite a bit from the actual data. The logistic model is not

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adequate to make profound assumptions on the course of the mortality.

In 2010, the mortality for children was lower and less consistent than in 1950. The logis-tic model shows again a poor reproduction of the mortality. Furthermore, we observe that the mortality model of Heligmann–Pollard HP4, as well as the power–exponential function, describe the “accident hump” and the senescent perfectly. The only difference exists in the in-fant mortality. Heligmann–Pollard HP4 models this period better than the power-exponential mortality rate. We described earlier that the power–exponential function has a smoother tran-sition from younger to older ages, which seems to be a disadvantage for capturing the infant mortality.

(a) Mortality of 1950 (b) Mortality of 2010

Figure 2.7: Mortality of 1950 and 2010 with three mortality models: logistic, Heligman– Pollard HP4 and power–exponential.

Based on the Figures in Fig. 2.7, we can say that the Heligman–Pollards fourth version is the most preferable model for representing actual data. Making statements based on this mortality rate can be assumed as quite accurate. Nonetheless, the power-exponential is a good alternative with the advantage of having less parameters to estimate.

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Chapter 3 Life insurance and Annuity

Life insurances are contracts that pay a face amount when that life ends. They are mostly based on the date the policy holder is no longer alive and therefore, the payment belongs to the beneficiary. We distinguish between whole life and term life insurances. The payments can be either received continuously, right after death of the insured person, annually, at the end of the year, or e.g. at the end of the month which is denoted by m = 12. Another form is an endowment insurance which is a combination of a pure endowment, the policyholder survives a specific time period, and a term life insurance.

Annuities are periodic payments for life, either continuously, annually or at an _m1 interval per year. If the first payment occurs at time 0, we call it due, otherwise it is an annuity-immediate.

These aforenmentioned contracts are basic insurances. By combining these in various ways as well as adding special restrictions, e.g. the cause of death, they become more complex [Mitchell et al., 1999, Slud, 2001, Dickson et al., 2013].

This chapter explains the term life insurance and the whole life annuity in detail and focuses on the payments a policyholder would receive from an insurance company which is represented by the expected present value.

3.1 Expected Present Value (EPV) tables

To determine the amount a policyholder would receive according to its contract is shown in EPV tables for each life insurance or annuity. That means, we create tables which show the EPV of insurances with an insured sum of 1$. Doing that, we need to introduce the following equations. [Dickson et al., 2013].

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3.2 Term life insurance

Once again, a term life insurance is a contract that pays the insured sum if and only if the policyholder dies within a prespecified period of time. The annual case indicates that the death benefit is paid at the end of the year of death [Dickson et al., 2013].

Definition 3. A term life insurance with an annual payment is

A1_x:n = n−1

∑

k=0 νk+1_k|qx, (3.1) where • _k|q_x= S_x(k) − S_x(k + 1) • Z = νk+1 • ν = e−δ • δ = log (1 + i) • annual interest rate i

• time period within the person must die n.

The random variable [Wackerly et al., 2008] of the present value Z is restricted to:

Z= ( νk+1, Kx≤ n − 1 0, K_x≥ n, where Kx de f

= [Tx] is a random variable. This represents the whole years an individual lived in the future. By definition, we know that_k|q_x= Sx(k) − Sx(k + 1). Note,k|qx means that an individual with age x survives k years and will die within the following year. k|1qxis the long version of the aforementioned, but because of the subsequent year the 1 can be dropped. A is the actuarial notatiom for life insurances, the superscript 1 in A1_x:n states that we use the annual payment and n is the period in which the individual must die. Furthermore, we define, for simplicity, that our interest rate i is constant through time [Dickson et al., 2013].

Below follow versions of Eq.(3.1) when logistic, Heligman–Pollard HP4 and power–exponential models are applied.

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Logistic model A1_x:n = n−1

∑

k=0 νk+1 " exp ( −1 cln " ac(eb(x+k)− 1) + b ac(ebx_{− 1) + b} #) # − exp ( −1 cln " ac(eb(x+k+1)− 1) + b ac(ebx_{− 1) + b} #) . Heligman-Pollard HP4 A1_x:n = n−1

∑

k=0 νk+1 " exp ( 1 a₃ (−1) 1 a3 (ln (a1)) 1 a3 " γ 1 a₃, − ln (a1) (x + k + a2) a3 − γ 1 a₃, − ln (a1) (x + a2) a3 # − b1b2b₃ 2 " (x + k)1−2b2_{− x}1−2b2 1 − 2b2 # − Z k 0 c1c(x+s) c3 2 1 + c1c (x+s)c3 2 ds ) − exp ( 1 a₃ (−1) 1 a3 (ln (a1)) 1 a3 " γ 1 a₃, − ln (a1) (x + k + 1 + a2) a3 − γ 1 a₃, − ln (a1) (x + a2) a3 # − b1b2b₃ 2 " (x + k + 1)1−2b2_{− x}1−2b2 1 − 2b2 # − Z k+1 0 c₁c(x+s)₂ c3 1 + c1c (x+s)c3 2 ds )# . Power-exponential function A1_x:n = n−1

∑

k=0 νk+1 " exp ( − c₁[Ei (c2(x + k)) − Ei (c2x)] − a1 (a2a3x)a3+1 [γ (a3+ 1, a2a3(x + k)) − γ (a3+ 1, a2a3)] ) − exp ( − c1[Ei (cs(x + k + 1)) − Ei (c2x)] − a1 (a2a3)a3+1 [γ (a3+ 1, a2a3(x + k + 1)) − γ (a3+ 1, a2a3x)] )# .

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3.3 Whole Life Annuity

An Annuity is defined as a series of payments made to an individual as long as the person is alive on the date of payment. Several different payment variations are defined for annuites, for instance continously, annualy, monthly or weekly. We choose to use annuities payable mthly. That means, that a yearly sum will be paid m times per year, for instance quarterly, this means m= 4. Hence, the quarterly payments to the policyholder will be at the same amount. The actuarial notation for annuities is ¨a[Dickson et al., 2013].

Definition 4. A whole life annuity with m-thly payments is

¨ a(m)x = ∞

∑

r=0 1 mν r m r mpx, (3.2) where

• frequency of payments per year m • discount factor νmr

• r mpx

de f

= S_x _mr

Below follow versions of Eq.(3.2) when logistic, Heligman–Pollard HP4 and power–exponential models are applied.

Logistic model ¨ a(m)_x = ∞

∑

r=0 1 mν r m _exp ( −1 cln " ac(eb(x+mr)− 1) + b ac(ebx_{− 1) + b} #) . Heligman-Pollard HP4 ¨ a(m)x = ∞

∑

r=0 1 mν r mexp ( 1 a₃ (−1) 1 a3 (ln (a1)) 1 a3 " γ 1 a₃, − ln (a1) x+ r m+ a2 a3 − γ 1 a₃, − ln (a1) (x + a2) a3 # − b1b2b₃ 2 " x+_mr1−2b2_{− x}1−2b2 1 − 2b2 # − Z _mr 0 c₁c(x+s)₂ c3 1 + c1c (x+s)c3 2 ds ) .

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Power-exponential function ¨ a(m)x = ∞

∑

r=0 1 mν r mexp ( − c1 h Eic2 x+ r m − Ei (c2x) i − a1 (a2a3)a3+1 h γ a₃+ 1, a2a3 x+ r m − γ (a3+ 1, a2a3x) i ) .

In the following chapter, we will display the expected present values of the term life insurance and the whole life annuity. For calculating these tables, we use the aforementioned equations.

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Chapter 4 Application and Comparison

There are several different mortality laws available but we still do not know which one is the most preferable for the insured live. With the help of the insurance methods we want to discover which mortality rate will provide the best profit for the insured. Meaning, that the highest value represents the model which should be chosen.

For that purpose, it is necessary to fit the models to actual data. We will use the built–in least square method of MATLAB to estimate the parameters. By using the programming language MATLAB we create the data for the EPV tables.

4.1 Least Square Method

The least square method has the objective to either determine if the data fits the theory or to estimate parameters to fit the model to the data. If we talk about the best fitted model we want to minimize the distance between the actual data and the results of the used model, e.g. minimize the error introduced by the model. [Engström et al., 2016]

4.1.1 Linear Least Square Problem

The least square data fitting problem starts with the assumption that we have m data points, e.g. (t1, y1), (t2, y2), ..., (tm, ym) with the relation of

yi= Γ(ti) + ei, i= 1, 2, ..., m.

The noise free data is described by the pure–data function Γ(ti) and e1, e2, ..., emrepresent the data error.

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The goal is to receive an approximation to Γ(t), preferably in the interval [t1,tm], whereas this approximation is determined by the fitted model M(x,t) and the n–parameter vector x = (x1, x2, ..., xn)> which characterizes the model and is specified by the given noisy data.

The fitting model has the form of

M(x,t) = n

∑

j=1

x_j f_j(t),

with fj(t) representing the model basis functions which are dependent on the exact goal of the data fitting, and n is the order of the fit.

For determining the least square fit the introduction of the residual riof the data is necessary: ri= yi− M(x,ti), i= 1, 2, ..., m.

Note, the residuals are a function of the vector x.

Therefore, the least square fit is the minimization of the sum of squares of residuals using the parameter vector with respect to x:

min x m

∑

i=1 r_i(x)2= min x m

∑

i=1 (yi− M (x,ti))2

The resulting parameter of vector x specifies used model hence, it fits given data in least square sense[Hansen et al., 2013].

4.1.2 Nonlinear Least Square Problem

A parameter α of a function f is called nonlinear if its derivation with respect to α is a func-tion of α.

In a parametrized fitting model, M(x,t), non-linearity appears if one of the parameters in x is nonlinear [Hansen et al., 2013].

In that case the nonlinear least square problem is defined as min x f(x) ≡ minx 1 2kr(x)k 2 2= min x 1 2 m

∑

i=1 r_i(x)2,

with x ∈ Rn and the vector- valued function r(x) = (r1(x), r2(x), ..., rm(x))> of the residuals of all data points, including yirepresenting the observed data corresponding to ti:

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4.2 Application in MATLAB

Determining which of the three aforementioned mortality rates models is the most preferable can be achieved by pricing life insurances. The model with the highest value after pricing will be our choice.

We retrieved the actual mortality data from the Human Mortality Database, imported these to MATLAB and used the least-square method to estimate our parameters for each model. The following shows the for-loops which creates the expected present value tables in MAT-LAB. In the case of the term life insurance, the values are the discounted present value of a total insured sum of $1. The annuity tables show the sum of the quarterly rewards of a $1_m payment:

Term life insurance a g e s = 1 : 1 0 0 ; t e r m = [ 1 0 , 1 5 , 2 0 ] ; i n t = 0 . 0 5 ; d e l t a = l o g ( 1 + i n t ) ; v= exp (− d e l t a ) ; c o l = 3 ; row = 6 1 ; a g e s t a r t = 2 0 ; a g e P = a g e s ( a g e s t a r t ) : a g e s ( a g e s t a r t +row ) ; r e s u l t 0 = z e r o s ( row , c o l ) ; f o r j = 1 : row f o r p = 1 : c o l f o r k = 0 : ( t e r m ( p ) −1) r e s u l t ( j , p ) = ( v ^ ( k + 1 ) ) ∗ S u r v i v a l _ f u n c t i o n ; r e s u l t 0 ( j , p ) = r e s u l t 0 ( j , p ) + r e s u l t ( j , p ) ; end end end

The rows express ages, the columns represent the three periods of the term life insurances within which the insured must die. Survival f unction represents the derived survival function of the used mortality rate at age ageP( j)

Annuity

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m= 4 ; i n t = 0 . 0 5 ; d e l t a = l o g ( 1 + i n t ) ; v= exp (− d e l t a ) ; c o l = 1 ; row = 4 ; a g e s t a r t = 2 0 ; a g e P = [ 2 0 40 60 8 0 ] ; r e s u l t a n n = z e r o s ( row , c o l ) ; f o r j = 1 : row f o r r = 0 : ( ( 1 0 0 − a g e P ( j ) ) ∗ 4 ) r e s u l t ( j ) = 1 /m∗ ( v ^ ( r /m) ) ∗ S u r v i v a l _ f u n c t i o n ; r e s u l t a n n ( j ) = r e s u l t a n n ( j ) + r e s u l t ( j ) ; end end

The rows again express the ages, this time until the age of 100 because the insured receives payments until the insured dies. We can assume that the actual life expectancy is not higher than this age. The value m = 4 represents the quarterly payments.

The Appendix contains the code for the term life insurance and for the whole life annuity for these three models.

4.3 Term Life Insurance

We create tables for the three mortality rate models, logistic (Log), Heligman–pollard HP4 (HP4) and power–exponential (PE) and choose the ages x between 20 and 80 for males’ pop-ulation because these ages seem realistic for a person to have a life insurance. The period within the insured must die is 10, 15 and 20 years.

The countries we take a closer look at are Sweden, the USA and Germany. Before the year of 1990 we will have two separate values for, respectively, West Germany, written in blue, and East Germany, written in red.

We choose the years 1964, 1990 and 2010. The first two years are of greater interest because 1964 is part of the “baby boomer” generation, known as the year with the highest birthrate. Therefore, the year 1990 is said to be the year in which these “baby boomers” started families. For Sweden we see in Table 4.1 that the values are changing as the theory claims. That means, that the benefits for the term life insurance increase with the age of the insured person

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Table 4.1: Term life insurance for Sweden. 1964 A1 x:10 A 1 x:15 A 1 x:20

x Log HP4 PE Log HP4 PE Log HP4 PE

20 0.0144 0.0029 0.0029 0.0144 0.0039 0.0049 0.0223 0.0046 0.0068 40 0.0430 0.0092 0.0224 0.0670 0.0132 0.0324 0.0922 0.0170 0.0416 60 0.1731 0.0519 0.1191 0.2538 0.0741 0.1647 0.3263 0.0917 0.1988 80 0.5417 0.2528 0.4871 0.6484 0.2744 0.5411 0.6851 0.2772 0.5508 1990 A1 x:10 A 1 x:15 A 1 x:20

20 0.0065 0.0077 0.0002 0.0113 0.0080 0.0013 0.0113 0.0082 0.0024 40 0.0302 0.0033 0.0150 0.0502 0.0049 0.0221 0.0689 0.0064 0.0290 60 0.1446 0.0218 0.0942 0.2191 0.0316 0.1344 0.2873 0.0398 0.1669 80 0.5116 0.1150 0.4599 0.6259 0.1304 0.5200 0.6684 0.1337 0.5316 2010 A1 x:10 A 1 x:15 A 1 x:20

20 0.0083 0.0021 0 0.0083 0.0021 0 0.0083 0.0022 0 40 0.0223 0.0008 0.0066 0.0324 0.0013 0.0101 0.0479 0.0017 0.0136 60 0.0991 0.0069 0.0520 0.1554 0.0103 0.0779 0.2131 0.0135 0.1020 80 0.4313 0.0448 0.3470 0.5602 0.0524 0.4242 0.6198 0.0543 0.4492

[Dickson et al., 2013]. However, the Heligman–Pollard HP4 model does not follow this rule in 1990 and 2010 for the ages from 20 to 60.

In the year of 1964 we see that the logistic model has the highest benefits throughout all ages and the different terms. This phenomenon seems to be the case for the other two years as well, followed by the power–exponential model and lastly, the Heligman–Pollard HP4 sur-vival function. One exception is in 2010, for an age of 20 the power–exponential is the least preferable model.

It is interesting that the logistic model has the same values for the term insurance in 1990 for 15 and 20 years, for an insured of 20 years. In 2010 the logistic model, as well as the power–exponential have the same benefits for a person age 20 for all three terms.

For Sweden the best payouts has the logistic model, and therefore we can say that this model should be chosen for calculating life insurances.

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Table 4.2: Term life insurance for the USA. 1964 A1 x:10 A 1 x:15 A 1 x:20

20 0.0075 0.0013 0.0046 0.0130 0.0018 0.0065 0.0175 0.0022 0.0081 40 0.0309 0.0053 0.0195 0.0561 0.0076 0.0279 0.0777 0.0098 0.0356 60 0.1377 0.0285 0.0974 0.2094 0.0405 0.1354 0.2702 0.0504 0.1652 80 0.4569 0.1364 0.4099 0.5728 0.1595 0.4755 0.6277 0.1662 0.4939 1990 A1 x:10 A 1 x:15 A 1 x:20

20 0 0.0007 0.0022 0.0059 0.0009 0.0034 0.0102 0.0012 0.0044 40 0.0229 0.0028 0.0126 0.0386 0.0041 0.0183 0.0539 0.0054 0.0236 60 0.1052 0.0162 0.0694 0.1565 0.0236 0.0989 0.2091 0.0302 0.1239 80 0.3747 0.0904 0.3391 0.4947 0.1137 0.4116 0.5641 0.1239 0.4384 2010 A1 x:10 A 1 x:15 A 1 x:20

20 0.0087 0.0961 0.0008 0.0135 0.1065 0.0013 0.0135 0.1110 0.0019 40 0.0234 0.0544 0.0090 0.0337 0.0632 0.0134 0.0490 0.0678 0.0176 60 0.0850 0.0475 0.0568 0.1339 0.0588 0.0825 0.1800 0.0665 0.1053 80 0.3487 0.1154 0.3132 0.4695 0.1391 0.3880 0.5444 0.1474 0.4180

From the Table 4.2 we can clearly see that in 1964 for all three term life insurances, the logistic function produces the highest benefits throughout all ages, followed by the power–exponential survival function and lastly, the Heligman–Pollard HP4 model.

In 1990, the power–exponential model is better than the logistic model, only in the 10 years insurance. Otherwise, the models behave as described before. In the last year considered, the logistic model is again better than the power–exponential model and the Heligman–Pollard HP4.

This means that for the USA, the preferable mortality rate model for pricing term life in-surances would be the logistic model.

Lastly, we take a closer look at Germany. The Table 4.3 shows that in 1964 we consider both parts of Germany, East and West. It is surprising, that the values for East Germany are slightly higher compared to West Germany. This is the case for the three periods and also for the three considered survival functions. The observations which we made above, agree with the study

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Table 4.3: Term life insurance for Germany. 1964 A1 x:10 A 1 x:15 A 1 x:20

20 0.0071 0.0013 0.004 0.0124 0.0018 0.0057 0.0161 0.0023 0.0071 0.0072 0.0013 0.0042 0.0125 0.0018 0.0059 0.0162 0.0022 0.0075 40 0.0291 0.0052 0.0181 0.0496 0.0076 0.0262 0.0718 0.0098 0.0337 0.0349 0.0053 0.0195 0.0581 0.0078 0.0283 0.0787 0.0102 0.0365 60 0.1449 0.0313 0.0971 0.2162 0.0452 0.1361 0.2852 0.0569 0.1668 0.1507 0.0340 0.1075 0.2286 0.0492 0.1502 0.2989 0.0618 0.1831 80 0.4884 0.1664 0.4275 0.6032 0.1852 0.4916 0.6511 0.1883 0.5076 0.5187 0.1791 0.4654 0.6304 0.1958 0.5236 0.6717 0.1981 0.5353 1990 A1 x:10 A 1 x:15 A 1 x:20

20 0.0096 0.0006 0.0020 0.0148 0.0008 0.0030 0.0148 0.0010 0.0040 40 0.0221 0.0024 0.0119 0.0327 0.0036 0.0175 0.0486 0.0047 0.0229 60 0.1072 0.0160 0.0730 0.1710 0.0237 0.1051 0.2267 0.0308 0.1325 80 0.4201 0.1006 0.3799 0.5458 0.1170 0.4512 0.6071 0.1209 0.4725 2010 A1 x:10 A 1 x:15 A 1 x:20

20 0.0071 0.0003 0.0010 0.0071 0.0005 0.0015 0.0071 0.0006 0.0020 40 0.0079 0.0016 0.0070 0.0187 0.0024 0.0104 0.0305 0.0032 0.0139 60 0.0719 0.0117 0.0490 0.1135 0.0178 0.0726 0.1619 0.0237 0.0944 80 0.3391 0.0869 0.3076 0.4657 0.1074 0.3848 0.5436 0.1144 0.4154

of Germany. The logistic model is the most preferable model for pricing life insurances, no matter in which year we are or which time period we choose.

4.4 Whole Life Annuity

For the whole life annuity we choose the same countries and years as in the section before. Additionally, the payments to the insured are quarterly, and only if the insured is still alive at maturity.

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age of 100. For instance, a person age 20, lives another 80 years. Thus, he/she will receive 60 · 4 = 240 payments of $1₄.

Table 4.4: Whole life Annuity with quarterly payments for the logistic, Heligman–Pollard HP4 and power–exponential mortality rate.

Sweden

1964 1990 2010

20 18.38 18.71 18.72 18.77 19.03 19.04 19.08 19.50 19.40 40 15.75 16.12 16.15 16.29 16.64 16.66 17.01 17.58 17.35 60 11.34 11.38 11.43 11.89 12.00 12.04 12.98 13.52 13.08 80 6.04 5.40 5.44 6.35 5.69 5.73 7.15 7.02 6.50 USA 1964 1990 2010

20 18.65 6.50 18.85 19.08 6.13 19.19 19.11 3.34 19.38 40 16.28 5.71 16.55 16.95 5.58 17.19 17.22 4.17 17.50 60 12.18 4.18 12.20 13.16 4.4 13.16 13.64 3.84 13.59 80 6.94 2.10 6.33 7.83 2.5 7.18 8.11 2.3 7.50 Germany 1964 1990 2010

20 18.68 6.52 18.11 18.93 4.65 19.17 19.33 5.11 19.44 18.62 6.58 18.05 40 16.28 5.74 15.04 16.91 4.17 17.10 17.60 4.65 17.67 16.09 5.76 14.91 60 11.96 4.14 9.79 12.83 3.12 12.86 13.89 3.62 13.81 11.73 4.09 9.54 80 6.61 1.89 4.08 7.29 1.51 6.66 8.18 1.90 7.54 6.27 1.79 3.83

The payments decrease with higher ages due to the fact that older people have less years to live, and therefore receive less payments.

In Sweden, all annuities calculated by the three different survival functions throughout the ages and years, have approximately the same value. Interesting is that the power–exponential function gives the highest values in 1964 and 1990 for the ages of 20, 40 and 60. Nonetheless, in 2010 Heligman–Pollard’s mortality model calculates the highest benefits for a life aged 20, 40 or 60. For age 80, the logistic model is to be preferred.

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highest values except for the age of 80, and in 2010 for the age of 60. The logistic model is better for these years. Furthermore, it is surprising that the difference between the two mentioned models and the Heligman–Pollard HP4 model is relatively big.

Due to the separation of Germany in 1964, we calculate two values. West Germany has higher payouts, as well as the logistic model seems to have the best results in all considered ages. After reunification, it turns out that the power–exponential function is better than the logistic and the Heligman–Pollard HP4 model. However, at age 80 the logistics model is again preferable.

4.5 Discussion of the Results

Even though we assumed in chapter 2 that the Heligman–Pollard HP4 model would be the best, it turns out that this model is the weakest of all three. A possible explanation could be the implementation to MATLAB. For example, in Germany, some of the fitted parameters are extremely high, therefore parts of the equation become awfully high or small and MATLAB returns not–a–number. That is why we needed to calculate not only the third part of Heligman– Pollard HP4 numerically but also, in some cases, the second part.

Furthermore, we argued that the logistic model is not adequate for displaying the mortality, we get results which are higher in the term life insurance, but the power–exponential model is mainly preferred for the annuities.

We should also mention that the power–exponential model prepared us some difficulties while programming it to MATLAB. For instance, in Germany 1964 the model has the problem of overfitting. The real data shows that there is not a remarkable “accident hump” present. That is why, the parameter a3 becomes extremely high and due to that we are not obtaining any reasonable results. Therefore, we needed to include the condition that a3would become zero, if the parameter exceeds 100. Hence, we ignore the “accident hump”. This might explain the bad results for the term life insurance.

If we need to choose one model for calculating the price of an annuity, we would suggest the power–exponential model. It can be fitted extremely well to real data, it represents the “acci-dent hump” as well as calculates reasonable good prices for the annuity.

For the term life insurance, the logistic model does give higher values than the Heligman– Pollard HP4 and the power–exponential model, and from the perspective of the insured this is favourable because it means a higher benefit. However, these values are based on the model that cannot represent the reality, and should not be used by the insurance companies as the risk of failing to predict is higher. To be on the save side, not taking advantage of the in-sured, but also not go into ruin, a middle ground should be found. Overall, it seems that the power–exponential model is a “fair” option for all.

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Chapter 5 Conclusion and Future Work

5.1 Project summary

The first life insurances were available at the beginning of the 18th century. But to determine the prices and payouts for those, we need to derive the survival functions which are based on the human mortality rate. Several different mortality models were introduced since then. The most important models to mention are Gompertz–Makeham, Modified Perks, Hannerz mor-tality rate and Heligman–Pollards four different versions.

In greater detail the logistic model, the Heligman–Pollard HP4 and a newly introduced model, called power–exponential, are discussed. The assumptions, drawn from this discussion, say that the Heligman–Pollard model captures the actual mortality very well and has the best fit. The empirical study of males’ population of Sweden, the USA and Germany shows that the calculated prices for term life insurances are the highest with the logistic model, whereas the power–exponential is mostly preferred for annuities.

5.2 Future Work

In this thesis, we only take three mortality models into perspective. Therefore, we suggest to compare the power–exponential mortality model to already introduced models which are different from the ones used in this project, and try to improve the representation of the infant and senescent mortality.

Additionally, we suggest to try to get to the bottom of the rather bad results of the Heligman– Pollard HP4 model in relation to pricing insurances.

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Another perspective could be on the programming and implementation difficulty, specifically, the numerical instability of the power–exponential model.

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Chapter 6 Summary of reflection of objectives in the

thesis

6.1 Objective 1: Knowledge and understanding

This thesis clearly shows the knowledge of the topic of actuarial mathematics. The theoretical knowledge was gained through reading various scientific papers and focusing in deeper under-standing of three models. It also represents both, the mathematical underunder-standing– derivation of the survival function– and the application to the financial world. The student showed further knowledge in current research by focusing on the new developed mortality law, called power– exponential model, by Lundengård et al. Furthermore, the programming language MATLAB as well as Latex is now very familiar to the student.

6.2 Objective 2: Methodological knowledge

A great deal of mathematical methodology and description was gained. The challenge of describing the mathematical derivations shortly but understandable was mastered quite well. Furthermore, the mathematical expressions written into Latex and MATLAB is succeeded well by the student.

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6.3 Objective 3: Critically and Systematically Integrate

Knowl-edge

The given literature references from the supervisor team were a first insight to the topic and is expanded by the student herself. The different sources agree with each other on the theory but are different in their scientific findings. The student tries to summarize the most important differences in the theory.

6.4 Objective 4: Independently and Creatively Identify and

Carry out Advanced Tasks

The student figured out on her own a way to implement the pricing calculations into MATLAB and used the help of her supervisors for problems due to numerical calculations.

6.5 Objective 5: Present and Discuss Conclusions and

Knowl-edge

The theoretical knowledge is supported by the mathematical derivations and are further dis-cussed by using several figures. The application results are shown in tables in a fair amount of values, so that the reader is not overwhelmed with numbers and cannot follow the drawn conclusion. A first assumption was made after representing the theory which is contradicted by the results of the application. The student gives suggestions of solutions and probable problems which might be the reason for this contradiction.

6.6 Objective 6: Scientific, Social and Ethical Aspects

This thesis is part of a bigger research that my supervisors, Milica Ranˇci´c and Karl Lun-dengård, conduct and parts of the results will be presented at the ? conference. Furthermore, the MATLAB code for fitting the models to actual data is provided by Karl Lundengård and can be used for all countries who have a mortality database. Nonetheless, the code for pricing is provided by the student.

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Bibliography

[Abramowitz and Stegun, 1964] Abramowitz, Milton and Stegun, Irene A. Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. Diver, New York, 1964.

[Booth and Tickle, 2008] Booth, Heather and Tickle, Leonie. Mortality modelling and fore-casting: A review of methods. Annals of Actuarial Science, 3:3-43, 2008.

[Bruderl and Schussler, 1990] Bruderl, Josef and Schussler, Rudolf. Organizational Mor-tality: The Liabilities of Newness and Adolescence. Administrative Science Quaterly, 35:530-547, 1990.

[Butt and Haberman, 2004] Butt, Zoltan and Haberman, Steven. Application of frailty-based mortality models using generalized linear models.Astin Bulletin, 34:175-197, 2004. [Dickson et al., 2013] Dickson, David C.M, Hardy, Mary R. and Waters, Howard R. Actuarial

Mathematics for Life Contingent Claims.Cambridge University Press, 2013

[Engström et al., 2016] Engström, Christoph, Lundengård, Karl and Silvestrov, Sergei. Course Compendium, Applied Matrix Analysis.2016.

[Forfare, 2006] Forfare, David O. Mortality Laws. Mathematical Formulae, 1-6. Encyclope-dia of Actuarial Science, 2006.

[Hannerz, 2001] Hannerz, Harald. Manhood trials and the law of mortality. Demographic Research, 4:185-202, 2001.

[Hannerz, 2001] Hannerz, Harald. An extension of relational methods in mortality estimation. Demographic Research, 4:337-368, 2001.

[Hansen et al., 2013] Hansen, Per Christian, Pereyra, Víctor and Scherer, Godela. Least Squares Data Fitting with Applications. The Johns Hopkins University Press, 2013. [Heligman and Pollard, 1980] Heligman, Larry and Pollard, J. H. The age pattern of

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[HMD, 2017] Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany). Available at

http://www.mortality.orgor

http://www.humanmortality.de(2017-06-14)

[Lundengård et al., 2017] Lundengård, Karl, Ranˇci´c, Milica and Silvestrov, Sergei. Mod-elling mortality rates using power exponential functions. Mälardalen University, 1-6, 2017.

[Lundengård et al., 2018] Lundengård, Karl, Strass, Belinda, Boulougari, Andromachi, Ranˇci´c, Milica and Silvestrov, Sergei. Application of a power–exponential function based model to mortality forecasting. Book of Abstracts of 5th Annual Stochastic Model-ing Techniques and Data Analysis International Conference. June 12-15, 2018, Chania, Crete, Greece.

[Mitchell et al., 1999] Mitchell, Olivia S., Poterba, James M., Warshawsky, Mark J., Brown, Jeffrey R. New Evidence on the Money’s Worth of Individual Annuities. The American Economic Review, 89:1299-1318, 1999.

[Pham, 2008] Pham, Hoang. Recent Advances in Reliability and Quality in Design. Springer, 2008.

[Slud, 2001] Slud, Eric V. Actuarial Mathematics and Life-Table Statistics. CRC Press Inc, 2001.

[Wackerly et al., 2008] Wackerly, Dennis D., Mendenhall, William and Schaeffer, Richard L. Mathematical statistics with applications. Thomson Learning, 2008.

[Wilson, 1994] Wilson, David L. The analysis of survival (mortality) data: Fitting Gompertz, Weibull and logistic fucntions. Mechanism of Ageing and Development, 74:15-33, 1994. [Yashin et al., 2000] Yashin, Anatoli I., Iachine, Ivan A. and Begun, Alexander S. Mortality

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Appendix A

MATLAB Code

The Appendix include the code constructed by the author.

The codes for the parameter fitting as well as the code for retrieving the data is provided by Karl Lundengård and can be obtained by request.

A.1 Term Life Insurance

This section contains the code for calculating the term life insurance, using the formulas from Section 3.2, with the three mortality rate models discussed in Chapter 2 using the sur-vival functions of the logistic, Eq.(2.1), the Heligman–Pollard HP4, Eq.(2.2) and the power– exponential, Eq.(2.3), model.

Logisic This code calculates the EPV for a term life insurance of 10, 15 and 20 years with the survival function of the logistic mortality rate model.

%Term L i f e I n s u r a n c e w i t h L o g i s t i c M o r t a l i t y r a t e c l e a r a l l %F i t t e d P a r a m e t e r s o f Germany 2010 a = 0 . 0 0 0 0 3 2 5 8 4 6 3 1 1 5 4 ; b = 0 . 0 8 7 8 3 6 0 0 2 8 6 1 3 5 2 ; c = 0 . 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 ; a g e s = 1 : 1 0 0 ; t e r m = [ 1 0 , 1 5 , 2 0 ] ; i n t = 0 . 0 5 ; %I n t e r e s t r a t e d e l t a = l o g ( 1 + i n t ) ;

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v= exp (− d e l t a ) ; %D i s c o u n t f a c t o r

c o l = 3 ; %Each column r e p r e s e n t s one t e r m

row = 6 1 ; %Ages f r o m 20 t o 80 a g e s t a r t = 2 0 ; a g e P = a g e s ( a g e s t a r t ) : a g e s ( a g e s t a r t +row ) ; %EPV m a t r i x r e s u l t 0 = z e r o s ( row , c o l ) ; f o r j = 1 : row f o r p = 1 : c o l f o r k = 0 : ( t e r m ( p ) −1) r e s u l t ( j , p ) = ( v ^ ( k + 1 ) ) ∗ ( exp ( ( − 1 / c ) ∗ . . . l o g ( ( a ∗ c ∗ ( exp ( b ∗ ( a g e P ( j ) + k ) ) − 1 ) + b ) / . . . a ∗ c ∗ ( exp ( b ∗ a g e P ( j ) ) − 1 ) + b ) ) − . . . exp ( ( − 1 / c ) ∗ . . . l o g ( ( a ∗ c ∗ ( exp ( b ∗ ( a g e P ( j ) + k + 1) ) −1 )+ b ) / . . . a ∗ c ∗ ( exp ( b ∗ a g e P ( j ) ) − 1 ) + b ) ) ) ; r e s u l t 0 ( j , p ) = r e s u l t 0 ( j , p ) + r e s u l t ( j , p ) ; end end

Heligman–Pollard HP4 This code calculates the EPV for a term life insurance of 10, 15 and 20 years with the survival function of the Heligman–Pollard HP4 mortality rate model.

%Term L i f e i n s u r a n c e w i t h HP4 m o r t a l i t y r a t e c l e a r a l l % f i t t e d P a r a m e t e r s o f West Germany 1964 a1 = 0 . 0 1 2 4 2 1 5 7 9 ; a2 = 2 . 3 6 E−14; a3 = 0 . 4 3 7 5 8 5 5 2 8 ; b1 = 0 . 0 0 0 1 0 1 4 1 ; b2 = 1 1 0 9 5 . 4 3 0 1 9 ; b3 = 2 0 . 1 1 3 2 7 2 3 6 ; c1 = 0 . 0 0 0 2 6 7 7 9 5 ; c2 = 1 . 0 0 4 5 1 1 2 4 8 ; c3 = 1 . 6 3 4 3 4 7 0 4 8 ; a g e s = 1 : 1 0 0 ; t e r m = [ 1 0 , 1 5 , 2 0 ] ; i n t = 0 . 1 ; %I n t e r e s t r a t e d e l t a = l o g ( 1 + i n t ) ; v= exp (− d e l t a ) ; %D i s c o u n t f a c t o r

c o l = 3 ; %Each column r e p r e s e n t s one t e r m

row = 6 1 ; %Ages f r o m 20 t o 80

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a g e P = a g e s ( a g e s t a r t ) : a g e s ( a g e s t a r t +row − 1 ) ; %EPV m a t r i x r e s u l t 0 = z e r o s ( row , c o l ) ; f o r j = 1 : row f o r p = 1 : c o l f o r k = 0 : ( t e r m ( p ) −1) r e s u l t ( j , p ) = ( v ^ ( k + 1 ) ) ∗ . . . exp ( ( 1 / a3 ) ∗ ( ( − 1 ) ^ ( 1 / a3 ) / ( l o g ( a1 ) ^ ( 1 / a3 ) ) ) ∗ . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . . ( a g e P ( j ) + k+ a2 ) ^ ( a3 ) , 1 / a3 ) ) − . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . . ( ( a g e P ( j ) + a2 ) ^ ( a3 ) ) , 1 / a3 ) ) − . . . b1 ∗ b3 ^ ( 2 ∗ b2 ) ∗ ( ( ( a g e P ( j ) + k )^(1 −2∗ b2 ) − . . . a g e P ( j )^(1 −2∗ b2 ) ) / ( 1 − 2 ∗ b2 ) ) − . . . i n t e g r a l (@( s ) t h i r d _ t e r m _ H P ( a g e P ( j ) + s , . . . c1 , c2 , c3 ) , 0 , k ) ) − . . . exp ( ( 1 / a3 ) ∗ ( ( − 1 ) ^ ( 1 / a3 ) / ( l o g ( a1 ) ^ ( 1 / a3 ) ) ) ∗ . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . . ( a g e P ( j ) + k +1+ a2 ) ^ ( a3 ) , 1 / a3 ) ) − . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . . ( ( a g e P ( j ) + a2 ) ^ ( a3 ) ) , 1 / a3 ) ) − . . . b1 ∗ b3 ^ ( 2 ∗ b2 ) ∗ ( ( ( a g e P ( j ) + k +1)^(1 −2∗ b2 ) . . . −ageP ( j )^(1 −2∗ b2 ) ) / ( 1 − 2 ∗ b2 ) ) − . . . i n t e g r a l (@( s ) t h i r d _ t e r m _ H P ( a g e P ( j ) + s , . . . c1 , c2 , c3 ) , 0 , k + 1 ) ) ) ; i f i s n a n ( r e s u l t ( j , p ) ) d i s p ( ’ NaN r e s u l t , c o m p u t i n g n u m e r i c a l l y ’ ) r e s u l t ( j , p ) = ( v ^ ( k + 1 ) ) ∗ . . . ( ( exp ( ( 1 / a3 ) ∗ ( ( − 1 ) ^ ( 1 / a3 ) / ( l o g ( a1 ) ^ ( 1 / a3 ) ) ) ∗ . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . . ( a g e P ( j ) + k+ a2 ) ^ ( a3 ) , 1 / a3 ) ) − . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . . ( ( a g e P ( j ) + a2 ) ^ ( a3 ) ) , 1 / a3 ) ) − . . . i n t e g r a l (@( t ) s e c o n d _ t e r m _ h p ( a g e P ( j ) + t , . . . b1 , b2 , b3 ) , 0 , k ) − . . . i n t e g r a l (@( x ) t h i r d _ t e r m _ H P ( a g e P ( j ) + x , . . . c1 , c2 , c3 ) , 0 , k ) ) ) − . . . ( exp ( ( 1 / a3 ) ∗ ( ( − 1 ) ^ ( 1 / a3 ) / ( l o g ( a1 ) ^ ( 1 / a3 ) ) ) ∗ . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . .

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( a g e P ( j ) + k +1+ a2 ) ^ ( a3 ) , 1 / a3 ) ) − . . . ( gamma ( 1 / a3 ) ∗ gammainc (− l o g ( a1 ) ∗ . . . ( ( a g e P ( j ) + a2 ) ^ ( a3 ) ) , 1 / a3 ) ) − . . . i n t e g r a l (@( t ) s e c o n d _ t e r m _ h p ( a g e P ( j ) + t , . . . b1 , b2 , b3 ) , 0 , k + 1 ) − . . . i n t e g r a l (@( x ) t h i r d _ t e r m _ H P ( a g e P ( j ) + x , . . . c1 , c2 , c3 ) , 0 , k + 1 ) ) ) ) ; end r e s u l t 0 ( j , p ) = r e s u l t 0 ( j , p ) + r e s u l t ( j , p ) ; end end end

where we need the following functions for calculating the integrals numerically:

%S e c o n d Term o f t h e HP4 m o r t a l i t y r a t e f u n c t i o n y= s e c o n d _ t e r m _ h p ( ageP , b1 , b2 , b3 ) y=b1 . ∗ exp (−b2 ∗ l o g ( a g e P / b3 ) . ^ 2 ) ; end % T h i r d t e r m o f HP4 m o r t a l i t y r a t e f u n c t i o n y= t h i r d _ t e r m _ H P ( ageP , c1 , c2 , c3 ) y= c1 ∗ c2 . ^ ( a g e P . ^ c3 ) . / ( 1 + c1 ∗ c2 . ^ ( a g e P . ^ c3 ) ) ; end

Power–exponential This code calculates the EPV for a term life insurance of 10, 15 and 20 years with the survival function of the power–exponential mortality rate model.

%p o w e r _ e x p o n e n t i a l t e r m l i f e i n s u r a n c e c l e a r a l l %F i t t e d P a r a m e t e r o f USA 2010 c1 = 0 . 0 0 0 4 7 6 5 1 2 ; c2 = 0 . 1 1 1 0 6 3 0 0 5 ; a1 = 8 . 4 1 8 9 5 4 8 4 1 ; a2 = 0 . 0 3 5 2 9 2 5 3 9 ; a3 = 1 5 . 3 4 2 7 8 6 7 6 ; a1 = ( a2 ^ a3 ) ∗ exp ( a3−a1 ) ; a g e s = 1 : 1 0 0 ; t e r m = [ 1 0 , 1 5 , 2 0 ] ; i n t = 0 . 1 ; %I n t e r e s t r a t e d e l t a = l o g ( 1 + i n t ) ; v= exp (− d e l t a ) ; %D i s c o u n t f a c t o r

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row = 6 1 ; %Ages f r o m 20 t o 80 a g e s t a r t = 2 0 ; a g e P = a g e s ( a g e s t a r t ) : a g e s ( a g e s t a r t +row − 1 ) ; e x p i n t i = @( x ) − e x p i n t (−x)−1 i ∗ p i ; %EPV m a t r i x r e s u l t 0 = z e r o s ( row , c o l ) ; f o r j = 1 : row f o r p = 1 : c o l f o r k = 0 : ( t e r m ( p ) −1) % I f a3 i s t o b i g i g n o r e hump t e r m i f a3 > 100 S1 = exp ( −( c1 ∗ ( e x p i n t i ( c2 ∗ ( a g e P ( j ) + k ) ) − . . . e x p i n t i ( c2 ∗ a g e P ( j ) ) ) ) ) ; S2 = exp ( −( c1 ∗ ( e x p i n t i ( c2 ∗ ( a g e P ( j ) + k + 1 ) ) − . . . e x p i n t i ( c2 ∗ a g e P ( j ) ) ) ) ) ; e l s e S1 = exp ( −( c1 ∗ ( e x p i n t i ( c2 ∗ ( a g e P ( j ) + k ) ) − . . . e x p i n t i ( c2 ∗ a g e P ( j ) ) ) . . . −a1 / ( ( a2 ∗ a3 ) ^ ( a3 + 1 ) ) ∗ ( gamma ( a3 + 1 ) ∗ . . . gammainc ( a2 ∗ a3 ∗ ( a g e P ( j ) + k ) , a3 + 1 ) . . . −gamma ( a3 + 1 ) ∗ . . . gammainc ( a2 ∗ a3 ∗ a g e P ( j ) , a3 + 1 ) ) ) ) ; S2 = exp ( −( c1 ∗ ( e x p i n t i ( c2 ∗ ( a g e P ( j ) + k + 1 ) ) − . . . e x p i n t i ( c2 ∗ a g e P ( j ) ) ) . . . −a1 / ( ( a2 ∗ a3 ) ^ ( a3 + 1 ) ) ∗ ( gamma ( a3 + 1 ) ∗ . . . gammainc ( a2 ∗ a3 ∗ ( a g e P ( j ) + k + 1 ) , a3 + 1 ) . . . −gamma ( a3 + 1 ) ∗ . . . gammainc ( a2 ∗ a3 ∗ a g e P ( j ) , a3 + 1 ) ) ) ) ; end r e s u l t 0 ( j , p ) = r e s u l t 0 ( j , p ) + r e s u l t ( j , p ) ; end end end

A.2 Whole Life Annuity

This section displays the codes for calculating the prices for the whole life annuity with quar-terly payments using the formulas of Section 3.3. Including the three derived survival func-tions of the logistic, Eq.(2.1), the Heligman–Pollard HP4, Eq.(2.2) and the power–exponential, Eq.(2.3), mortality rate model.

Comparison of prices of life insurances using different mortality rates models

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