THE EFFECT OF TURKISH MORTALITY 

IMPROVEMENTS ON THE COST OF ANNUITIES 

USING ENTROPY MEASURE 

 

 

TÜRKİYE’DEKİ ÖLÜMLÜLÜK DEĞİŞİMLERİNİN 

ANÜİTE FİYATLARI ÜZERİNDEKİ ETKİSİNİN 

ENTROPİ ÖLÇÜSÜ  

 

 

BÜKRE YILDIRIM 

 

 

ASSIST. PROF. DR. MURAT BÜYÜKYAZICI 

Supervisor 

 

 

Submitted to Institute of Graduate School of Science and Engineering of 

Hacettepe University as a Partial Fullfillment  to the Requirements  

for the Award of the Degree of Master of Sciences  

in Actuarial Science 

 

2014



i 
 

This work named “The Effect of Turkish Mortality Improvements on the Cost 

of Annuities Using Entropy Measure” by BÜKRE YILDIRIM has been approved 

as a thesis for the Degree of MASTER OF SCIENCE IN ACTUARIAL SCIENCE 

by the below mentioned Examining Committee Members. 

 

 

Assoc. Prof. Dr. A. Sevtap KESTEL   ………………………………… 

Head 

 

 

Assist. Prof. Dr. Murat BÜYÜKYAZICI   ………………………………… 

Supervisor 

 

 

Assist. Prof. Dr. Uğur KARABEY    ….…………………………….. 

Member 

 

This thesis has been approved as a thesis for the Degree of MASTER OF 

SCIENCE IN ACTUARIAL SCIENCE by Board of Directors of the Institute of 

Graduate School of Science and Engineering. 

 

Prof. Dr. Fatma SEVİN DÜZ 

Director of the Institute of 

Graduate Schools in Science 

and Engineering 



ii 
 

ETHICS 

In this thesis study, prepared in accordance with the spelling rules of Institute of 

Graduate School of Science and Engineering of Hacettepe University, 

I declare that 

 all the information and documents have been obtained in the base of the 

academic rules 

 all audio-visual and written information and results have been presented 

according to the rules of scientific ethics 

 in case of using other Works, related studies have been cited in accordance 

with the scientific standards 

 all cited studies have been fully referenced 

 I did not do any distortion in the data set 

 and any part of this thesis has not been presented as another thesis study 

at this or any other university. 

 

 

 /     / 2014 

 

 

BÜKRE YILDIRIM 

 

 

 

 

 

 

 

 



iii 
 

ABSTRACT 

 

 

THE EFFECT OF TURKISH MORTALITY IMPROVEMENTS 
ON THE COST OF ANNUITIES USING ENTROPY MEASURE 

 

 

Bükre YILDIRIM 

Master of Science, Department of Actuarial Science 

Supervisor: Assist. Prof. Dr. Murat BÜYÜKYAZICI 

June 2014, 60 pages 

 

 

Over the twentieth century there have been unprecedented and unexpected 

gradual decline in mortality rates due to considerable process of eliminating 

hazards to survival, which have increased the life expectancy at birth remarkably. 

Decrease in mortality rates often been underestimated by annuity and pension 

providers. Especially, mortality improvements for post-retirement ages have 

significant financial effect on the annuity and pension plans.  

To correctly measure the effect of mortality rate improvements on the life 

expectancy or annuity costs many demographers use the idea of entropy. Entropy 

has also been used as a measure of the rectangularization of the survival function, 

variability of the age-at-death and the convexity of the survival curve. 

In this thesis work, the Keyfitz definition of entropy is used to measure the effect of 

any changes in the force of mortality on the cost of a life annuity. Entropy 

parameters investigated to see how they affect the entropy measure. Firstly, 

Turkish period life tables are used to calculate the entropy measure for the cost of 

annuities. Then a mathematical model for the force of mortality is adopted to 

obtain a more theoretically satisfactory conclusion. 



iv 
 

This master thesis is comprised of 5 chapters, mainly as; introduction, entropy in 

actuarial science, entropy values using Turkish life tables, entropy values and its 

behavior using a mathematical model and finally results and conclusions. 

In Chapter 2; the formula for the entropy measure for life expectancy at birth is 

explained and then the corresponding formula for the life annuity is explained. In 

Chapter 3; the value of entropy measure H  is calculated using the Turkish Life 

Mortality Tables over the period 1931-2015. In Chapter 4; the value of entropy 

measure is calculated with an assumption that mortality is following a Gompertz 

mortality model. Lastly, Chapter 5 discusses the overall conclusions and makes 

some recommendations. 

 

 

Keywords: Entropy, force of mortality, annuity cost, rectangularization. 



v 
 

ÖZET 

 

 

TÜRKİYE’DEKİ ÖLÜMLÜLÜK DEĞİŞİMLERİNİN ANNUİTE 
FİYATLARI ÜZERİNDEKİ ETKİSİNİN ENTROPİ ÖLÇÜSÜ 

 

 

Bükre YILDIRIM 

Yüksek Lisans, Aktüerya Bilimleri Bölümü 

Tez Danışmanı: Yrd. Doç. Dr. Murat BÜYÜKYAZICI 

Haziran 2014, 60 sayfa 

 

 

20. yüzyıl boyunca beklenen yaşam süresini etkileyen tehlikelerin azaltılması 

sonucunda mortalite hızlarında, daha önce görülmemiş ve beklenmedik kademeli 

düşüşler olmuştur. Bu nedenle doğumda beklenen yaşam süresi önemli bir 

biçimde artmıştır. Günümüzde mortalite gelişmelerinin yavaşlamakta olduğuna 

dair ya da insan ömrünün belirli bir teorik üst limiti olduğuna dair kesin bir kanıt 

bulunmamaktadır. 

Tarihsel süreç boyunca, gözlemlenen, mortalite hızlarındaki azalmaların arkasında 

son derece önemli gelişmeler bulunmaktadır. Bunlardan bazıları; daha besleyici 

gıdaların üretilmesi, temiz su kaynakları, atık imha alanındaki gelişmeler, yüksek 

yaşam standartları, daha iyi barınma ve çalışma alanları, motorlu araç 

güvenliğindeki gelişmeler, temel sağlık hizmetlerine erişimin kolaylaşması, 

aşılamanın yaygınlaşması ve tıbbi gelişmelerdir. 

Türkiye’de toplam doğurganlık hızı Türkiye İstatistik Kurumu’nun hesaplamalarına 

göre 2012 yılı için 2.08’dir. Bu nüfus yenilenme hızı olarak kabul edilen 2.1 

değerinin de altındadır. Nüfusun yaşlanmakta olduğu ve nüfustaki genç insan 

sayısının, yani çalışma gücüne katkı sağlayan kişi sayısının azalmakta olduğuna 



vi 
 

şüphe yoktur. Uzun yaşama riski, azalan doğurganlık ve mortalite hızlarının 

etkisiyle daha da önemli bir hale gelmiştir.  

Mortalite hızındaki süregelen bu düşüş beklenenden daha fazla olduğu için 

genellikle dikkate alınamamıştır. Dolayısıyla primler eksik tahmin edilmektedir. 

Özellikle emeklilik sonrası yaşlar için ortaya çıkan mortalite iyileşmelerinin 

emeklilik planlarını düzenleyen kuruluşlar ve anüite garantisi veren şirketler 

üzerinde önemli finansal etkileri olmuştur. Sosyal güvenlik sisteminin yapısı 

nedeniyle mevcut olan eksiklikler de mortalitedeki gelişmelerin etkisi ile birlikte 

kötüleşmektedir. 

Entropi ilk olarak Alman fizikçi Rudolf Clasius tarafından 1865’de, Sadi Carnot ve 

Lord Kelvin’in önceki çalışmalarını da temel alarak tanıtılmıştır. Clasius sistemler 

arasındaki mükemel ısı transferlerinde bile kullanılabilir enerjinin bir kısmının 

kaybolmasının kaçınılmaz olduğunu bulmuştur ve bu kaybolmayı entropideki artış 

olarak tanımlamıştır. Entropi; termodinamiğin doğadaki en temel kanunlarından biri 

olan ikinci yasasıdır. Bu şu anlama gelir; tüm süreçler kaçınılmaz enerji kaybı 

(entropi artışı) nedeniyle %100 ’den daha az verimle çalışmaktadır. Entropi 

kavramı modern bir bilim dalı olan bilgi kuramında da önemli bir role sahiptir ve bir 

mesajın içerdiği bilginin beklenen değerini ifade eder. Bilgi kuramının 1948’de 

Amerikalı matematikçi Claude E. Shannon’ın öncü çalışması “A Mathematical 

Theory of Communication” da bulunduğu kabul edilmektedir.  

Geçtiğimiz yıllarda birçok demograf entropi ölçütünü toplumların mortalite 

karşılaştırması amacıyla kullanmıştır. İlk olarak Nathan Keyfitz entropi ölçütünü 

mortalite hızındaki herhangi bir değişimin beklenen yaşam süresi üzerindeki 

etkisini ölçmek için bir indeks olarak kullanmıştır. Yüksek entropi değeri düşük 

entropi değerine göre, beklenen yaşam süresinin mortalitedeki değişimlere daha 

duyarlı olduğunu göstermektedir.  

Entropi kavramı birçok araştırmacı tarafından yaşam fonksiyonu için 

dikdörtgenselleşmenin bir ölçütü olarak da kullanılmıştır. Dikdörtgenselleşme 

geçtiğimiz yüzyılda yüksek mortaliteden düşük mortaliteye geçişin neredeyse 

tamamlandığı birçok gelişmiş ülkede ortaya çıkmıştır. Dikdörtgenselleşme ya da 

başka bir deyişle, yaşam eğrisinin artan yaşama olasılıkları ve beklenen yaşam 

süresine bağlı olarak düzleşmesi, yaşam fonksiyonunun gittikçe dikdörtgen bir 



vii 
 

şekil almasıdır. Yaşama olasılığının ileri yaşlara kadar bire yakın kalıp kısa bir 

zaman aralığı içinde sıfıra düşmesi sonucu dikdörtgen bir şekil ortaya çıkar. 

Mortalite hızlarındaki gelişim insanların daha ileri yaşlarda ölmesine neden 

olmuştur. Bu, ölüm yaşındaki değişkenliğin azalması ve ölümlerin yaşamın daha 

ileri yıllarında birikim göstermesi anlamına gelir.  

Bu tez çalışmasında Keyfitz’in entropi tanımı, mortalitedeki değişimlerin anüite 

değerleri üzerindeki etkisinin bulunması için kullanılmıştır. Daha sonra entropide 

yaş, cinsiyet, mortalite değişimi ve faiz oranı gibi parametrelerinin etkileri 

incelenmiştir. Öncelikle  1931- 2015 yılları arasındaki Türkiye periyot hayat 

tabloları kullanılmış ve entropi anüite değerleri için hesaplanmıştır. Daha sonra, 

teorik olarak daha tatmin edici bir sonuca ulaşılabilmesi için mortalite hızı için 

Gompertz modeli kullanılmış, yeni hayat tablolarıyla işlemler tekrar edilmiştir. 

Hesaplamalar entropinin cinsiyet değişkenine çok duyarlı olmadığını ancak yaş, 

yıl, faiz oranı ve mortalite değişimlerine karşı duyarlı olduğunu göstermiştir. Anüite 

ve entropi değerlerinin kullanılan faiz ile ters, yaş ile doğru orantılı olduğu 

görülmüştür. 

Bu yüksek lisans tezi beş kısımdan oluşmaktadır. Bunlar başlıca: Giriş, Aktüerya 

biliminde entropi, Türkiye hayat tabloları kullanılarak entropi değerleri, 

Matematiksel model kullanılarak entropi değerleri ve son olarak Sonuç ve 

öneriler’dir. Bölüm 2’de; ilk olarak doğumda beklenen yaşam süresi için entropi 

ölçütü formülü açıklanmış daha sonra bu formül annüite değerleri için kullanılabilir 

hale getirilmiştir. 3. Bölüm’de entropi ölçütünün değeri, 1931-2015 Türkiye 

mortalite tabloları kullanılarak hesaplanmıştır. Bölüm 4’de entropi değerleri 

mortalitenin matematiksel bir modele uyduğu varsayımı altında hesaplanmış ve bu 

modelin parametrelerine göre davranışı incelenmiştir. son olarak Bölüm 5 elde 

edilen genel sonuçlar hakkında bilgi vermekte ve ileride yapılabilecek çalışmalar 

için önerilerde bulunmaktadır. 

 

 

Anahtar Kelimeler: Entropi, mortalite hızı, anüite değeri, dikdörtgenselleşme. 



viii 
 

ACKNOWLEDGEMENTS 

I would like to express my deepest gratitude to my supervisor, Assist. Prof. Dr. 

Murat Büyükyazıcı, whose understanding and patience added considerably to my 

graduate experience. 

I also would like to express my sincere thanks to Assoc. Prof. Dr. A. Sevtap Kestel 

and Assist. Prof. Dr. Uğur Karabey for their suggestions and comments for 

improving this study. 

Thanks to my friends and collaborators Rümeysa Karataş and Selin Değirmenci 

for their friendship, support and help. 

Most importantly, none of this would have been possible without love and patience 

of my family. I owe earnest thankfulness to my dear mother Filiz Yıldırım and my 

father S. Şahin Yıldırım. I deeply appreciate their belief in me. 

 

 

 

 

 

 

 

 

 

 

 

 



ix 
 

TABLE OF CONTENTS 

                Page 

ABSTRACT..................................................................................................................................... iii 

ÖZET ................................................................................................................................................ v 

ACKNOWLEDGEMENTS ....................................................................................................... viii 

TABLE OF CONTENTS............................................................................................................. ix 

LIST OF TABLES ........................................................................................................................ xi 

LIST OF FIGURES ..................................................................................................................... xii 

1. INTRODUCTION...................................................................................................................... 1 

2. ENTROPY IN ACTUARIAL SCIENCE .............................................................................. 7 

2.1. Notation ................................................................................................................................... 7 

2.2. Methodology ........................................................................................................................ 12 

2.2.1. Entropy and Life Expectancy ...................................................................................... 12 

2.2.2. Entropy and Cost of Annuity ....................................................................................... 18 

3. ENTROPY VALUES USING TURKISH LIFE TABLES .............................................. 21 

3.1. Entropy Values of Age 0 .................................................................................................. 22 

3.2. Entropy Values of Age 60 and 70 ................................................................................. 25 

4. ENTROPY VALUES AND ITS BEHAVIOR 

USING A MATHEMATICAL MODEL .................................................................................... 28 

4.1. Gompertz Law ..................................................................................................................... 28 

4.2. Entropy Values Using Gompertz Law .......................................................................... 29 

5. RESULTS AND CONCLUSIONS ..................................................................................... 40 

REFERENCES ............................................................................................................................ 42 



x 
 

TABLE OF CONTENTS 

                Page 

APPENDIX .................................................................................................................................... 44 

CURRICULUM VITAE ............................................................................................................... 60 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



xi 
 

LIST OF TABLES 

        Page 

Table 4.1. 2000 base table mortality rates and estimated c  values ........................ 32 

Table 4.2. 2015 base table mortality rates and estimated c  values ........................ 32 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



xii 
 

LIST OF FIGURES 

    Page 

Fig. 1.1. Mortality rates of Turkey for females and males 

respectively over the years 1931 to 2015 ............................................................................. 2 

Fig. 2.1. Turkish female and male survival curves  

for selected calendar year ........................................................................................................ 17 

Fig. 3.1. Mortality rates for females and males in log scale 

over the period 1931-2015 ....................................................................................................... 21 

Fig. 3.2. Entropy values for Turkish life tables 

at different rates of interest for females at age 0 .............................................................. 23 

Fig. 3.3. Entropy values for Turkish life tables 

at different rates of interest for males at age 0 .................................................................. 24 

Fig. 3.4. Entropy values for Turkish life tables 

at different rates of interest for females at age 60 ............................................................ 25 

Fig. 3.5. Entropy values for Turkish life tables 

at different rates of interest for males at age 60................................................................ 26 

Fig. 3.6. Entropy values for Turkish life tables 

at different rates of interest for females at age 70 ............................................................ 27 

Fig. 3.7. Entropy values for Turkish life tables 

at different rates of interest for males at age 70................................................................ 27 

Fig. 4.1. Data and Gompertz model comparisons 

of mortality rates of 2000, in log scale at ages 0, 60 and 70 ......................................... 33 

Fig. 4.2. Data and Gompertz model comparisons 

of mortality rates of 2015, in log scale at ages 0, 60 and 70 ......................................... 34 

Fig. 4.3. Calculated values of ( , , , )xH b c    in year 2000 

for different values of   and   for ages 60 and 70 ....................................................... 36 



xiii 
 

LIST OF FIGURES 

    Page 

Fig. 4.4. Calculated values of ( , , , )xH b c    in year 2015 

for different values of   and   for ages 60 and 70 ....................................................... 36 

Fig. 4.5. H  values for Females and Males at ages 60 and 70  

in base year 2000 with different c  values and 0%   .................................................. 38 

Fig. 4.6. H  values for Females and Males at ages 60 and 70  

in base year 2015 with different c  values and 0%   .................................................. 38 

Fig. 4.7. H  values for Females and Males at ages 60 and 70 

in base year 2000 with different c  values and 4%   .................................................. 39 

Fig. 4.8. H  values for Females and Males at ages 60 and 70 

in base year 2015 with different c  values and 4%   .................................................. 39



1 
 

1. INTRODUCTION 

Over the twentieth century there have been unprecedented and unexpected 

gradual decline in mortality rates due to considerable process of eliminating 

hazards to survival, which have increased the life expectancy at birth remarkably.  

Aging is still an uncertain process and there is much debate as to whether there 

are upper limits to human longevity. However, it is clear that there is no conclusive 

evidence to suggest that current mortality improvements are tending to slow down 

and no fixed theoretical limit to human longevity is apparent today [1].  

Going back in history, there exist a number of extremely important developments 

behind the observed reductions in mortality rates. These include; more nutritious 

foods, clean water supply, improvements in waste removal, improvements in 

sanitary conditions, higher standards of living, better housing and working 

conditions, improvements in motor vehicle safety, access to primary medical care 

for the general population, availability of immunizations, medical advances, etc. 

Throughout the years each of these developments will make a smaller impact to 

mortality improvement. 

Further reductions in mortality will depend upon factors such as; development of 

new diagnostic and surgical techniques, emergence of new forms of disease, 

presence of environmental pollutants, cigarette smoking, improvements in 

exercise and nutrition, incidence of violence, ability and willingness to pay for the 

development of new treatments and technologies. 

Over the years, this evolution has led death causes to change from infectious 

diseases, such as tuberculosis, diphtheria and cholera, to degenerative and 

chronic diseases, such as cancer and cardiovascular diseases which effect mostly 

older ages [2]. In other words, mortality from exogenous causes is mostly 

eliminated, and the remaining variability in the age at death is generally caused by 

genetic factors [3].  

Figure 1.1. demonstrates the improvements in mortality rates of Turkey for 

females and males respectively over the years 1931 to 2015. It can be seen that 

mortality rates have fallen considerably. There are significant mortality 

improvements in some decades and almost no improvements in other decades. 



2 
 

 

 

Fig. 1.1.  Mortality rates of Turkey for females and males 

respectively over the years 1931 to 2015. 



3 
 

Life expectancy at birth in the early times among humans was likely 20 to 30 years 

[4] proven by tombstones inscription, skeletal remains, and genealogical records 

[5]. When the first population data in history began being collected in 1750 in the 

Nordic countries, life expectancy was around 35-40 years. Then rapid 

improvements began at the beginning of 20th century and life expectancy reached 

60-65 years by the middle of the 20th century. Most of the life expectancy increase 

occurred in the last two centuries. The 20th century in particular has made a huge 

increase in life span compared to all other centuries. Since then life expectancy 

gains mostly have been due to medical factors which have reduced mortality 

among older persons.  By the beginning of the 21st century, life expectancy at birth 

has reached about 70 years even 80 years in some countries, such as Japan [6]. 

In Europe, Eurostat states that the total fertility rate has dropped to 1.5 in 2012, 

which is below the rate of 2.1, considered to be the replacement rate of 

populations. In Turkey the total fertility rate for year 2012 is 2.08 according to 

calculations of Turkish Statistical Institute (TÜİK) [7]. Due to these reasons, 

without arguing that the world’s population is getting older and younger people in 

population who constitutes the work force is declining with the contribution of 

declining fertility rates. The effect of longevity risk is even more important with the 

consequences of falling fertility rates and improving mortality. It has agitated the 

plan providers, especially for products offering annuity guarantees. 

This persistent decrease in mortality rates was unexpected and as a result they 

have often been underestimated by annuity and pension providers. Especially, 

mortality improvements for post-retirement ages have a significant financial effect 

on the annuity and pension plans. Because of the tax-paid structure, the 

deficiencies of public finance are aggravated by mortality improvements. Also due 

to low fertility rates, the ratio between old people and young people decreased 

over time [2]. 

Which takes us to entropy: The term entropy in physical sciences evolves around 

the definitions of the thermodynamics [8]. In thermodynamics the second law 

states that in a system or process, entropy is the disorder or randomness. 

Thermodynamic entropy is a measure of how disordered a system is. A system 

which in an imbalanced state ,for example a hot region next to a cold region, will 



4 
 

always tend to even out, with the heat flowing from hot to cold region until the 

systems even out. 

Entropy was first introduced when the steam engine became science 

(thermodynamics) by the German physicist Rudolf Clasius in 1865, based on 

earlier work by Sadi Carnot and Lord Kelvin. Clasius found that even for perfect 

exchanges of heat energy between systems, a loss of useful energy is inevitable. 

He called this loss an increase in entropy. The increase in entropy is the amount of 

heat transfer divided by the absolute temperature. This is one of the basic laws of 

nature, known as the second law of thermodynamics. This means that all 

processes must operate at less than 100% efficiency due to the inevitable loss of 

heat energy, entropy rise [8]. 

The concept of entropy also plays an important role in the modern discipline of 

information theory. Information theory is generally considered to have been 

founded in 1948 by the American mathematician Claude E. Shannon in his 

pioneer work “A Mathematical Theory of Communication” [9]. In information 

theory, entropy quantifies the expected value of the information contained in a 

message. 

To correctly measure the effect of mortality rate improvements on the life 

expectancy or annuity costs many demographers use the idea of entropy. During 

the past decades, the concept of entropy has been adopted by several 

demographers. 

For example, Pollard [10] focused on the function of mortality rate and its effects 

on life expectancy. He used discrete differences in life expectancy at two moments 

in time, and analyzed the effect of difference between two mortality rates. The 

second approach by Keyfitz [11] introduced the concept of entropy into the field of 

demography as a measure of the elasticity of the life expectancy for proportional 

changes in mortality rates. 

As stated by Keyfitz, entropy of a survival model is the time derivative of the life 

expectancy. He used the entropy measure as an index to measure the effect of 

any change in the force of mortality on life expectancy. A high value of entropy 

measure indicates that the life expectancy has a greater sensitivity to a change in 



5 
 

the force of mortality than a lower value of entropy measure. So a high value of 

entropy measure means that future reductions in mortality rates would have an 

impact on life expectancy. On the other hand, a low value of entropy measure 

means that life expectancy is relatively ‘immune’ to future reductions in mortality 

rates. The particular advantage of the concept of entropy is that it allows the 

sensitivity of the life expectancy to changes in mortality rates to be summarized in 

a single figure index. 

The concept of entropy has also been used as a measure of the rectangularization 

of the survival function by a number of researchers (e.g. Goldman and Lord [12], 

Nagnur [13], Nusselder and Mackenbach [3]). Rectangularization showed up over 

the past century in most developed countries of the world in which the 

transformation from high mortality to low mortality is nearly complete [13]. 

Rectangularization or the flattening of the survival curve is a trend toward a more 

rectangular shape of the survival function due to increased survival rate and 

expectation of life. A rectangular curve emerges when the probability of survival 

stays close to one up to old ages and then drops to zero over a short age range 

(sharp down slope) at very old ages, due to demographic transition from high to 

low mortality levels. Improvements of the mortality rates have caused people to die 

at older ages. This means that; the variability in the age at death declines and 

deaths are concentrated to the upper years of life. Also, a way of analyzing the 

rectangularization is to observe the temporal variations in the values of entropy. 

Entropy is also a measure of the variability of the age-at-death and a measure of 

the convexity of the survival curve [5]. If the survival curve is completely 

rectangular, entropy is zero and everybody lives to a certain age and then dies at 

that age hence the variability in the age-at-death is zero. If the survival curve 

declines in a linear form with age and the number of deaths are equal at all ages, 

entropy is 0.5. If mortality is independent of age in other words the force of 

mortality is equal at all ages, entropy equals the unity. 

In this thesis work, the Keyfitz definition of entropy is used to measure the effect of 

any changes in the force of mortality on the cost of a life annuity. Then we 

investigate some parameters of the entropy to see how they affect the entropy 

measure -for example; the effects of the age and gender of the pensioner at 



6 
 

inception and the rate of interest. Firstly we use period life tables to calculate 

entropy measure from annuity values and life expectations. Then we consider a 

mathematical model for the force of mortality to obtain a more theoretically 

satisfactory conclusion. 

This master thesis is comprised of 5 chapters, mainly as; introduction, entropy in 

actuarial science, entropy values using Turkish life tables, entropy values and its 

behavior using a mathematical model and finally results and conclusions. 

In chapter 2; the formula for the entropy measure for life expectancy at birth is 

explained and then the corresponding formula for the life annuity is explained. In 

chapter 3; the value of entropy measure H is calculated using the Turkish Life 

Mortality Tables over the period 1931-2015. In chapter 4; the value of entropy 

measure is calculated with an assumption that mortality is following a Gompertz 

mortality model. Lastly, chapter 5 discusses the overall conclusions and makes 

some recommendations. 

 

 

 

 

 

 

 

 

 

 

 

 



7 
 

2. ENTROPY IN ACTUARIAL SCIENCE 

In this chapter we take into account the formula of the entropy measure for life 

expectancy at birth and then introduce the corresponding formula for the life 

annuity. After brief explanations of some basic actuarial terms in Section 2.1, 

Section 2.2 presents the theory of entropy in life expectancy from Keyfitz [11] in 

2.2.1., and the entropy measure for annuity costs in 2.2.2. 

2.1. Notation 

Some important terms which frequently used in the following sections will be 

defined basically here. 

A life table shows fundamental parameters of a population for each age or age 

group, such as; the number of survivors, the number of deaths, the probability that 

they die or live to their next birthday and the life expectancy. It describes the 

mortality and survival pattern of a population. From this point forward, a number of 

fundamental statistics can be derived. 

There are two types of life tables: 

I. Period (static) life tables show the current probability of death (for people 

with different ages, in the current year). It is based on the mortality 

experience of an entire population. 

II. Cohort, or generation, life tables are based on mortality experience over the 

lifetime of people from a given cohort (which usually covers one year) over 

the course of their lifetime.  

Mortality data and life tables, originate from observations concerning a whole 

national population or a specific part of a population (e.g. retired workers, disabled 

people, etc.) or an insurer’s portfolio, and so on. 

The past life table data does not assure future outcome. Hence in order to price 

insurance products, and obtain the solvency of insurance companies, actuaries 

must develop projections of future insured events. To do this actuaries develop 

mathematical models for mortality, next life tables are generated. The life tables 



8 
 

are called mortality tables if they show death rates and morbidity tables if they 

show various types of sickness or disability rates. 

The standard notations required for a life table includes: 

Number of Survivors 

Radix is the number of people alive at the age of 0. It is shown as 0l  and this is the 

starting point for xl . xl  is the number of survivors at age x . It is the number of 

those original radix who are still alive on their thx  birthday. As age increases the 

number of people alive decreases to a finite limit of 0. Also,   is the limiting age of 

the mortality table and nl  is zero for all n  . 

Also, 0l  can be calculated as; 

0 1

0

w

x

x

l d


  

Number of Deaths 

Number of people who die between age x  and x n  is defined as n xd . It can be 

calculated using the formula; 

1 1...n x x x x nd d d d      .   

     x x nl l    

Probability of Living and Dying 

Probability of dying between age x  and x n  is defined as n xq . 

n x
n x

x

d
q

l
  

Also, when mortality is following a mathematical model 0x q  can be calculated as; 



9 
 

0

0

1 exp[ ]

x

x aq da    

Probability of surviving between age x  and x n  is defined as n xp . 

x n
n x

x

l
p

l

  

Also, when mortality is following a mathematical model 0x p  can be calculated as; 

0

0

exp[ ]

x

x ap da   

Since the only possible alternatives from one age x  to the next x n  are living and 

dying, the relationship between these two probabilities; 

1n x n xq p  . 

Life Expectancy 

Another statistic that can be obtained from a life table is life expectancy. Life 

expectancy is the average number of years of life expected to remain for a person 

at age x . It is defined as xe . 

1

x t x

t

e p




   

Also, when mortality is following a mathematical model, xe  can be calculated as; 

0

x t xe p dt



   

The expected lifetime is often used to compare mortality in various populations. 

The definition of xe  is based on the probability distribution of the lifetime 

conditional on being alive at age x . Thus when 0x   the mortality at all ages 

contributes to the value of xe . 



10 
 

Within populations, differences in life expectancy exist with regard to gender. 

Females tend to outlive males in all populations, and have lower mortality rates at 

all ages, starting from infancy. 

Force of Mortality 

Force of mortality refers to an instantaneous rate of mortality at a certain age 

measured on an annualized basis. In a life table, we consider the probability of a 

person dying between age x  and age x t ; as t xq . In the continuous case, we 

could also consider the conditional probability that a person who has attained age 

x  will die between age x  and age x t . This is called the force of mortality and 

defined as follows: 

0 0

[ ]
lim limx t x

x
t t

T t q

t t


 

 
   

Where xT  is the remaining lifetime of a person at age x . And hence it represents 

the instantaneous rate of mortality at a given age x .  

From 

0 0 0 0

0

( ) ( ) ( ) ( )
[ ] ( )

1 ( ) ( )
x x

F x t F x F x t F x
T t F t

F x S x

   
    


 

we obtain 

0 0 0

0

( )
( ) ( ) ( )

lim ln ( )
( ) ( ) ( )

x
t

d
S x

F x t F x f x ddx S x
tS x S x S x dx





 

      

Hence, once the survival function ( )S x  has been assigned, the force of mortality 

can be derived. 

Life tables can be constructed using projections of future mortality rates, but more 

often they are snapshots of age-specific mortality rates in the recent past. For 

these reasons, the older ages in the life table may have greater chance of not 

being representative of what lives at these ages may experience in future, as it is 



11 
 

predicated on current advances in medicine, public health, and safety standards 

that did not exist in the early years of this cohort. 

Annuity Cost  

People generally feel a necessity to make savings to help them when they retire. 

An annuity specifically provides that. Everyone at the same age group makes 

equal payments to a life insurance company to be a beneficiary of the 

accumulated payments in the future under the condition of reaching a certain age. 

In this manner the survivors from this group gets more than they pay initially. No 

one knows when they will die and the payment is conditional on staying alive 

therefore, life annuities are conditional annuities. Making regular savings, rental 

payments, coupon and bond interest payments, life insurance premiums are 

examples of annuity problems. If not recognized, the changing mortality rates can 

and will cause companies to fail to meet their liabilities. 

A life annuity consists of a series of payments which are made while the annuitant 

(of initial age x ) lives. To ensure the future payments insured needs to make 

contributions to the fund, this quantity is called net premium. The net single 

premium of a life annuity is its expected present value, xa . It is defined as 

0

exp[ ]x t xa p t dt


   

Where   is the force of interest rate. 

Survival Function 

Survival function gives the proportion of individuals reaching the age x . Assume 

that the function ( )S t , called the survival function and defined for 0t   as 

follows: 

0( ) [ ]S t T t    

0T  denotes the random lifetime for a newborn. Where w  is the limiting age it is 

usual to assume that the possible outcomes of xT  lie in (0, )w  and the probability 

measure outside this interval is zero. 



12 
 

The survival curve moves towards a rectangular shape with time due to 

improvements in mortality rates and increases in life expectancy, and hence the 

term rectangularization is used to describe this feature.  

Rectangularization of survival curve is also associated with a reduction in the 

variability of age at death. As deaths become concentrated in a narrow age range, 

the slope of the survival curve becomes steeper, and the curve itself begins to 

appear more rectangular.  

The point of maximum downwards slope of the survival curve progressively moves 

towards the very old ages; this feature is called the expansion of the survival 

function. 

2.2. Methodology 

2.2.1. Entropy and Life Expectancy 

A measure of uncertainty associated with a random variable comes from the field 

of information theory and is called entropy. In actuarial science entropy is a single 

figure index that quantifies the effect of proportional change in the force of 

mortality over the whole range of age on life expectancy. It gives the percentage 

change in life expectancy produced by a reduction of one percent in the force of 

mortality if equal progress is achieved against mortality at all ages.  

Entropy determines the expected uncertainty on the result of an experiment and 

presents information about the foreseeability of the result of a random variable. 

The larger the entropy the less compressed the distribution and thus an 

observation about random variable provides less information. Measure of entropy 

may be regarded as a descriptive quantity of the corresponding probability density 

function as it belongs to the class of measures of variability like the median, mode, 

variance or the standard deviation [14]. 

The most known measure of entropy, Shannon’s information entropy equation, is 

defined by; 

( ) ( ) ln ( )
x

H X p x p x    

or  



13 
 

( ) ( ) ln ( )H X f x f x dx   

Where, ( )p x  is the probability mass function of X  and ( )f x  is the probability 

density function of X  depending on whether the random variable X  is discrete or 

continuous, respectively [9]. 

The Shannon’s entropy, measures the information content of a random variable 

and represents a lower bound on how much that random variable can be 

compressed without losing its content. Also, Shannon’s entropy measure initiated 

the mathematical study of information theory, and his results are the bases of 

communication over networks today. 

The formula for the entropy measure for actuarial work frame differs from that 

associated with information theory. 

The expectation of life at age x  
0

xe , calculated from life tables is a commonly 

used summary measure for population’s level of mortality. Variations in life 

expectancies also can be used to compare between countries. The role of single-

figure indices such as 
0

xe , summarizing the lifetime probability distribution, should 

not be underestimated. It does suffer from some of the usual disadvantages of 

single-figure indices but at the same time there are good reasons for adopting 

such a summary measure. Apart from the fact that a single index such as 
0

xe  

condense the information of a full mortality schedule considerably, life 

expectancies also have the advantage of easing the interpretation.  

Assume that the overall force of mortality at age x  is x , and the force of 

mortality changes by 100  percent where   is a constant change in the force of 

mortality at all ages, and could be positive or negative depending on whether the 

mortality rates are increasing or decreasing respectively. Then, the new force of 

mortality at age x  is, 
* (1 )x x    . 

This new force of mortality leads to a new survivorship curve, and the new 

probability of surviving from age 0  to x  becomes 



14 
 

* *

0

0

exp[ ]

x

x ap da        

      
0

exp[ (1 ) ]

x

a da     

      1

0

exp[ ]

x

ada     

      
1

0( )x p                   (2.1) 

The new expectation of life is 

* 1

0 0

0

( )

w

o

ae p da  ,                (2.2) 

where w  is the maximum limit age of a population in the life table. 

Keyfitz [11] has shown how the logarithm of the survivorship curve, [ln ]n xp , contains 

useful information about the effects of changes in force of mortality on life 

expectancies. So as to determine the effect of changes in force of mortality, x , on 

the expectation of life, 
0

xe , we consider the derivative of equation (2.2) with 

respect to  ; 

*
10

0 0

0

ln( )( )

wo

a a

de
p p da

d





                (2.3) 

In the neighborhood of 0   based on a Taylor expansion, we have the following 

approximation  

0 00

0 0

0

0

0

0

[ln ]

w

a a

w

a

p p da
e

e
p da



 
 

  
 
 
 




               (2.4) 

To make H  a positive quantity we define it as minus the expression in brackets. 



15 
 

0

0

0

0

e
H

e



                   (2.5) 

Equation (2.5) means that the proportional change in life expectancy can be 

approximated in simple terms as H  times  , where   means that a small 

proportional change in the force of mortality if equal progress is achieved against 

mortality at all ages.  

The quantity H  as the entropy measure is shown as 

0 0

0

0

0

[ln ]

w

a a

w

a

p p da

H

p da

 



               (2.6) 

Entropy is a positive quantity and the ratio of the integrals is always negative 

because 0a p  is always between zero and one and 0ln( )a p is always negative. 

Eventually entropy is a weighted average of values of 0ln a p  as age ranges from 

0  to w . As t xp  declines from one to zero over the age span ln t xp  increases 

from zero to infinitely large values. Therefore, there is no apparent mathematical 

explanation of why ( )xH   should be bounded from above [12]. Hence, it has no 

upper bound and life tables may have entropy values above one. Those life tables 

exhibit extremely high death rates in the young ages.  

The effect of a mortality change on the life expectancy depends directly on the 

concavity of the survival curve (same as t xp , or x , which t xp was derived) [15]. 

As mortality improve over time, in other words the curve of t xp  turns down more 

sharply which is the same as saying that x  turns up more sharply, more people 

die in a narrow range of old ages before w . This leads H  closer to zero. Hence, 

a change in x  has almost no effect on life expectancy. The lower H , the greater 

the tendency for all of us to die at about the same age, a tendency that apparently 

accompanies mortality decline. Indeed, H  would be 0  if all mortality were 

concentrated at a single age (everyone dies at the same age). Hence, a change in 



16 
 

x  is reflected by an equal proportionate change in 0e . Also, in the case of 

complete rectangularization of the survival curve, entropy is zero regardless of the 

age at which all members of the population die. 

Entropy is roughly the percentage change in life expectancy originated from a 

reduction of 1%  in the force of mortality at all ages. Such as if entropy is 1, a 

decrease in mortality rates by 1%  results in an increase in life expectancy at birth 

by 1% . Similarly, if entropy is 0.2, a uniform decrease in the force mortality by 1%  

results in an increase in life expectancy at birth by 0.2% .  

Analyses point out that there is an inverse relationship between the period 

expectation of life and the entropy measure H . This decline in H  is, in large 

measure, an outcome of the progress of reducing deaths in infancy- the age at 

which the highest number of years of life expectancy are lost. Another result of the 

progress in reducing deaths in infancy is a shift in the ages where further 

improvements of mortality would be most effective in increasing life expectancy. 

About a century ago, most of the potential for saving years of life was 

concentrated in the first years of childhood; today, in developed countries most is 

in old age. In the course of mortality declines and life expectancy increases, there 

is an upward shift in the ages where further declines in mortality would be most 

effective in increasing life expectancy. This gives rise to life expectancy being less 

sensitive to further declines in the force of mortality, and so we would expect the 

entropy measure to decrease over time as mortality improves. As a consequence, 

future movement towards rectangularization and a reduction in the force of 

mortality by one percent at all ages would produce a much lower increase in life 

expectancy today than would have been the case fifty years or a century ago [16]. 

Rectangularization occurs when the age-at-death distribution shows an increased 

building up around a certain old age, as the survival function takes more and more 

a rectangular shape. It is often being seen as a sign that the life expectancy is 

approaching to its biological limit. Approaching the average life span (biological 

limit of the life expectancy) implies that increases in total life expectancy must 

decelerate [16]. 



17 
 

Great developments were made in the 20th century towards minimizing hazards to 

survival at the young ages. Survival rates at the older ages are also continuing to 

improve. As a result of these developments; Figure 2.1 presents the 

rectangularization of the Turkish female and male survival curves respectively for 

selected calendar years 1931, 1950, 1970, 1990 and 2015.  

 

 

Fig. 2.1. Turkish female and male survival curves for selected calendar years. 



18 
 

Rectangularization is gradually evolving over the last 8 decades. The differences 

between men and women is more pronounced at the older ages in 2015 than in 

1931. Women made greater gains than man and remarkable improvements have 

been registered in the probabilities of survival to older age. 

Therefore, entropy measure can be seen as an indicator of rectangularization and 

it can also be interpreted as a measure of the heterogeneity of the population with 

respect to mortality at different ages [11], [16]. 

The theory presented in this section has considered the concept of entropy in 

relation with the expectation of life at birth. We now investigate how to extend this 

approach in order to find the corresponding entropy value for annuity costs. 

2.2.2. Entropy and Cost of Annuity 

This section provides a method to find the effect of a uniform proportional change 

in the force of mortality over the whole age range on the cost of an annuity.  

Using the same assumption as considered in Section 2.1: Assume that the overall 

force of mortality at age x  , x , changes by 100  percent where   is a constant 

change in the force of mortality at all ages, and could be positive or negative 

depending on whether the mortality rates are increasing or decreasing 

respectively. This new force of mortality leads to a new value of a life annuity. 

The new value of a life annuity at age x  becomes 

*
1

0

( ) exp[ ]x t xa p t dt 


                (2.7) 

To determine the effect of changes in force of mortality,  , on the value of a life 

annuity, xa , we consider the derivative of equation (2.7) with respect to  . 

*
1

0

(ln )( ) exp[ ]x t x t x

d
a p p t dt

d

 




               (2.8) 

And using a Taylor expansion for the neighborhood of 0  , it concludes that, 



19 
 

0

0

(ln )( )exp[ ]

exp[ ]

t x t x
x

x

t x

p p t dt
a

a
p t dt











 
 

  
 

 
 




                  (2.9) 

( )xH   can be thought of as minus the weighted average value of ln t xp , weighted 

by exp[ ]t xp t . Hence; 

( )
x

x
x

a
H

a
 


                (2.10) 

Where ( )xH   is the entropy measure of an annuity value for a person at age x

with the force of interest,  . 

0

0

(ln )( )exp[ ]

( )

exp[ ]

t x t x

x

t x

p p t dt

H

p t dt











 








           (2.11) 

Here, ( )xH   is a positive quantity because the numerator of the integrals in (2.11) 

is always negative. 

When mortality rates improve over time, we would expect a larger percent of 

deaths to occur at older ages because of the decrease in volatility of age-at-death 

distribution. Also, the value of ( )xH   decreases and becomes closer to 0 . As 

stated previously, the pace at which this happens depends on the concavity of the 

survivorship curve. Additionally, as the interest rates increases, ( )xH   is expected 

to decrease, because any change of the force of mortality is expected to have a 

lower impact on the cost of a life annuity at higher interest rates. 

Equation (2.11) can be written as weighted average of x sa   with using integration 

by parts; 



20 
 

0( )

s
x sx s s x

x
x

p e a ds

H
a













            (2.12) 

Hereby,  ( )xH   can be seen as a measure of heterogeneity easily, as s x x sp    is the 

probability density function for the time of death of a person aged x . 

To compute the values of entropy measure, we first need to arrange the ratio of 

integrals in equation (2.11), by using a numerical approach designed by Pollard 

[10]. 

Let  

0

ln ln

t

x t
t x x u t x

x

l
Q du p

l
 



 
     

 
 , 

and  

exp( )t x t xE p t  . 

Using these two equations and the mean value theorem for integrals; the integrals 

in (2.11) are replaced by sums of the one-year integrals. As a result of this, it leads 

the entropy measure to the following approximation: 

1 1

0 2 2

1

0 2

( )

w

x x
t t

t

x w

x
t

t

Q E

H

E


 











             (2.13) 

Where w is the maximum limit age in the mortality tables. Hence, equation (2.13) 

can be used directly to calculate the values of entropy measure for different values 

of x  and  . 

 

 

 

 



21 
 

3. ENTROPY VALUES USING TURKISH LIFE TABLES 

İn this section of the thesis, entropy measure ( )xH   is calculated using the Turkish 

Life Mortality Tables over the time interval between 1931 and 2015. These 

mortality tables for each year was derived from the computed regression levels of 

the Construction of Turkish Life and Annuity Tables Project [17]. The project 

assumes that Turkish Life Tables and the Cole-Demeny West Model Life Tables 

[18] are alike. And the project data covering all age groups of the Turkish 

population had originally been derived from the Turkish population registers. 

 

 

Fig. 3.1. Mortality rates for females and males in log scale 

over the period 1931-2015. 



22 
 

Firstly the plots of mortality rates were considered to determine the properties of 

population data. From the following Figure 3.1, it can be seen that mortality rates 

have been decreasing throughout the years. To facilitate understanding these 

rates have been plotted on a logarithmic scale. These life tables will allow us to 

investigate the entropy measure changes over an almost 85 year period. 

The entropy measure is calculated separately for males and females to see and 

examine the importance of gender in mortality. Furthermore, it is calculated at 

different rates of interest (e.g. 0% , 2% , 4% , 10% , 8% , 10% ). 

Currently, until the year of 2036 the retirement age in Turkey is 60 for males and 

58 for females as stated in the law number 4447 [19]. Beginning from 2048 the 

age of retiring from Social Security Institution is expected to be 65 for both females 

and males [20]. As the primary area of interest for the application of the concept of 

entropy for annuities is for people who retired, the age ranges 60-100 and 70-100 

are considered, as well as the whole age range 0-100 to have a general outlook. 

Hence in the next subtitles 0 ( )H  , 60 ( )H   and 70 ( )H   are computed and investigated 

respectively. 

3.1. Entropy Values at Age 0 

Firstly, the values of the entropy measure for the whole age range, 0 ( )H  , is 

calculated. The change in the mortality rates at all ages shows its overall impact 

on 0 ( )H  . Also, different interest rates are taken into consideration when using the 

generated Turkish Life Tables between the years 1931-2015. The results are 

shown in Figures 3.2. and 3.3. for females and males respectively. Also, tables of 

these values are contained in the Appendix 2. 

From the figures it can be clearly seen that, with the decreased mortality rates 

over the years, the value of 0 ( )H   also decreased. This decrease in 0 ( )H   means 

that any improvement in the force of mortality in a population will have a much 

smaller increasing effect on the expectation of life and the cost of the life annuity. 

Also as expected, the decreasing of entropy measure over the years has lesser 

impact at higher rates of interest. Because, the value of 0 ( )H   decreases when 

interest rate increases.  



23 
 

As the number of survivors to older ages has increased and life expectancy at 

birth has improved over the years, the value of 0 ( )H   has decreased. 

Consequently, the effect of proportional improvement of the age specific mortality 

rates on the cost of life annuity has diminished [13]. For recent mortality 

experiences in developed countries, 0 ( )H   values appear to be below 0.2 [12]. 

As seen on the Figures 3.2. and 3.3., after the year of 1940 for all interest rate 

levels and each year, entropy measure is lower for females than males, due to the 

higher expectations of life for females than males in each year because females 

tend to outlive males and have lower mortality rates, at all ages. Also the female 

regression levels of the Construction of Turkish Life and Annuity Tables Project 

are lesser than men. 

 

Fig. 3.2. Entropy values for Turkish life tables 

at different rates of interest for females at age 0.  



24 
 

 

Fig. 3.3. Entropy values for Turkish life tables 

at different rates of interest for males at age 0. 

 It can also be seen from the tables 3.2. and 3.3. that the changes in the values of 

0 ( )H   fall into two distinct periods, for both males and females. Approximately 

before 1950s, when the levels of mortality were high, any change in the force of 

mortality has a stronger decreasing impact on the value of the entropy measure. 

However, this effect lessens with the increase of the interest rate. Hence, at high 

interest rates, the curve is more horizontal than the lower interest rates. The same 

applies in later years, when mortality is at a lower level. However in later years, 

any change in the mortality does not distinctively affects the entropy measure. 

It can be explained by approximately dividing the curve into two sections. One on 

the left is where the mortality level is high and one on the right is where the 

mortality level is low. In high mortality part, the steeper slope reflects the fact that 

the value of 0 ( )H   is more susceptible, and changes in the force of mortality will 

have a strong effect on the value of the entropy measure. However, for the low 

mortality part of the table, the more horizontal slope of the curve shows that the 

0 ( )H   is insusceptible and changes in the force of mortality do not have a great 

effect on the value of 0 ( )H  .  



25 
 

3.2. Entropy Values at Age 60 and 70 

The primary area of interest is the effect of any change of the force of mortality on 

the cost of a life annuity at retiree ages with different rates of interest. In this 

section the values for ( )xH   at ages 60 and 70 are computed for both genders. As 

in Section 3.1, the same data set and interest rates are used. Figures 3.4. and 3.5. 

show the values of the entropy measure at age 60, 60 ( )H  , for females and males 

respectively. Also, tables of these values are contained in the Appendix 3 and 4. 

 

Fig. 3.4. Entropy values for Turkish life tables 

at different rates of interest for females at age 60. 

Because the mortality improvements (e.g. medical advances) are mostly in the 

older ages, for females it can be seen from the Tables 3.2. and 3.4. that the values 

of 60 ( )H   are higher than the values of 0 ( )H  , in Section 3.1. This again shows the 

fact that greater effect of changes in the force of mortality on the expectation of life 

and the cost of a life annuity is in the higher age group (where mortality rates are 

much higher compared to younger ages).  

At any year, the higher the interest rate the lower is the value of 60 ( )H   and the 

effect of changes in the force of mortality on the cost of life annuity. As expected, 

at any interest rate the value of 60 ( )H   is decreasing while levels of mortality get 



26 
 

lower through 1931 to 2015. Hence, for the 2015 life table and 0.1   the values 

of 60 ( )H   are lowest. 

 

Fig. 3.5. Entropy values for Turkish life tables 

at different rates of interest for males at age 60. 

Similar characteristics are valid in the entropy values for males. Expectedly, the 

values of 60 ( )H   for males are higher than the values of 0 ( )H   for males, and after 

the year 1931 they are also higher than the values of 60 ( )H   for females. Also, the 

higher the interest rates the lower is the effect of changes in the force of mortality 

on the cost of life annuity, thus the lower are the values of 60 ( )H  . We expect the 

value of 60 ( )H   to decrease when levels of mortality are lower so that, through 1931 

to 2015 the values of 60 ( )H   are on the decrease at all rates of interest.  

Figures 3.6. and 3.7 show the values of 70 ( )H   for both females and males 

respectively. As expected, the values of 70 ( )H   are higher than the values of 60 ( )H   

and 0 ( )H   at all interest rates.  Again after the year 1931, for the all interest rate 

levels and each year, the value of entropy measure is lower for females than 

males due to higher expectations of life for females than males. 



27 
 

 

Fig. 3.6. Entropy values for Turkish life tables 

at different rates of interest for females at age 70. 

 

Fig. 3.7. Entropy values for Turkish life tables 

at different rates of interest for males at age 70. 

 

 

 



28 
 

4. ENTROPY VALUES AND ITS BEHAVIOR 

USING A MATHEMATICAL MODEL 

If the progress against mortality will continue, what will happen to entropy measure 

and annuity costs? Some insight into these questions can be gained by a simple 

mathematical mortality model. 

In this section of the thesis, the value of entropy measure, ( )xH  , is calculated with 

an assumption that mortality is following a chosen mathematical model 

considering the data from Construction of Turkish Life and Annuity Tables Project 

as the base mortality tables for 2000 and 2015. In this manner, mortality does not 

depend completely on observed data and hereby a cohort outlook is adopted 

rather than a period life table outlook. 

By using the mathematical model assumption, it allows for a better examination of 

the different characteristics of entropy and how the value changes with different 

factors of a mathematical model. 

Section 4.2 presents the entropy values under the assumption that force of 

mortality is following the Gompertz model. Also in this model, improvements in 

mortality rates over the years are allowed with a reduction factor. For the ease of 

calculation, the reduction factor is assumed to depend only on time. 

4.1. Gompertz Law 

Gompertz [21] suggested that a geometric progression pervades in mortality after 

a certain age. The mortality rates, particularly for ages over 30, suitably fits the 

Gompertz model. The model he proposed is often used to explain the distribution 

of adult lifespans by demographers and actuaries.  

There are other and newer models for modelling particular features of mortality 

rates, but the Gompertz model has advantages over them for users, such as 

implementation simplicities and familiarity with formula. 

The model suggests that 

exp( )x b cx    



29 
 

Where b  denotes the level of the force of mortality at age 0 (location parameter), 

c is the rate of aging (shape parameter). 

Considering the decreasing rates of mortality, a simple exponential reduction 

factor for the force of mortality, dependent only on the time past after the base 

year, ( ( ) exp[ ]RF t t  ) is used, so that 

( )t o

x u x uRF t                     (4.1) 

Where 
o

x u   is the mortality rate in the chosen base year at age x u  and 
t

x u   

is the mortality rate in the tth year after the chosen base year at age x u . 

For these reasons we use the Gompertz model to demonstrate the application of 

entropy measure to cost of annuities. Therefore, annuity costs for different ages 

and time can be studied. 

The behavior of the expectation of life, 
0 ( )xe t , and life annuity values as functions of 

time can be explored. As mentioned in Vaupel [16], it is straightforward to 

demonstrate that 

0 0( ) 1 ( )t

x x xe t e t
t c





   

               (4.2) 

So that, in the long run, 

0 ( )xe t
t c





 

and is roughly constant. Thus, the rate of mortality change over time in the 

expectation of life at the end reaches a stable level (the same applies to annuity 

values). This is one of the driving forces behind the changes that have been 

demonstrated in ( )xH   from the historical perspective of the Turkish life table 

calculations depicted in Figures 3.4., 3.5., 3.6. and 3.7. 

4.2. Entropy Values Using Gompertz Law  

The Gompertz law [21] with parameters b  and c  with a familiar exponential 

function in the form, 



30 
 

exp( )x b cx    

is used. 

Hereby, the force of mortality of the base table (time 0) at age x u  can be 

expressed as 

0 exp[ ( )]x u b c x u     . 

As in section 3.3, a reduction factor is used which depends only on time t for 

simplicity, in the form exp[ ]t , 
0 exp[ ]t

x u x u t     . Hence, the probability 

of a person at age x  lives to be x t  is 

*

0

exp
x

t

u

t x up du 

 
  

 
  

      0

0

exp ( )

t

x uRF u du 

 
  

 
  

      
0

exp exp[ ( )]exp[ ]

t

b c x u u du
 

     
 
  

      
0

exp exp[ ( ) ]

t

b c x u u du
 

     
 
  

      
( )

0 1
exp

c t

x

e

c






  
   

  

                       (4.3) 

Here,   is a bounded to be less than c , so the probability of survival in Equation 

(4.3) will be less than 1. 

Entropy is now calculated on a cohort basis with the use of a reduction factor and 

it is a function of the parameters b  and c ,   and  . Cohort life tables are 

preferable to period life tables at projecting a population into the future when 

mortality rate is expected to change over time, and for analyzing the general 

trends in mortality. 

Hereafter, for the notation of the entropy measure, ( , , , )xH b c    is used. 



31 
 

* *

0

*

0

ln( )

( , , , )

t

t x t x

x

t

t x

p p e dt

H b c

p e dt





 
















 

        

 

 

0
( ) ( )

0

0

0
( )

0

exp 1

1

exp 1

t

c t c tx

x

t

c t tx

e e dt
c

c
e e dt

c

  

 





 



  

 

  
   

   
   

   
  





                  (4.4) 

One beneficial side of using a mathematical model for the force of mortality is that 

it enables to see the effects of its parameters in order to test their effects on

( , , , )xH b c   . For example; the effects of the base table used in computation ( c ), the 

assumed level of mortality improvement ( ) and the interest rate ( ), can be 

seen on ( , , , )xH b c   . Hence, it helps to understand the behavior of the entropy 

measure. 

Integrals in Equation (4.4.) can be written as the sums of one-year integrals using 

the same numerical approach and the mean value theorem for integrals in Chapter 

3. 

( , , , )xH b c    value has been calculated using data from the Construction of Turkish 

Life and Annuity Tables Project [17] for females and males at ages 0, 60 and 70. 

The base mortality tables used in calculations are the 2000 and 2015 mortality 

tables. 

 The values of 
0

x , c ,   and   are needed to calculate ( , , , )xH b c    values from 

the Equation (4.4). The values for 
0

0 , 
0

60  and 
0

70  are taken directly from the 

2000 and 2015 base mortality tables. After that, Gompertz mortality model has 

been fitted to these data using MATLAB (R2013a) in order to obtain estimated c  

values by a simple regression approach. Note that assuming a lower c  means 

that mortality level of the base table is at lower level. Table 4.1 and 4.2 shows the 

mortality rates and estimated c  values for ages 0, 60 and 70, for the base tables 

of 2000 and 2015 respectively.  

 



32 
 

Table 4.1. 2000 base table mortality rates and estimated c  values. 

 Female Male 

Age 
0

x  c  
0

x  c  

0 0,014789 0,0741 0,026659 0,0664 

60 0,011753 0,1037 0,020131 0,0924 

70 0,037185 0,0996 0,052360 0,0910 

 

Table 4.2. 2015 base table mortality rates and estimated c  values. 

 Female Male 

Age 
0

x  c  
0

x  c  

0 0,006029 0,0869 0,017054 0.0722 

60 0,007890 0,1112 0,017071 0,0953 

70 0,028844 0,1050 0,046953 0,0932 

 

Mortality rates of the base year 2000 and Gompertz model’s mortality rates are 

compared in Figure 4.1. After that mortality rates of the base year 2015 and 

Gompertz model’s mortality rates are compared in Figure 4.2, which gives the 

similar results as in mortality rates of the year 2000. As explained in Section 4.1., 

Gompertz model is not a good fit for ages smaller than 30. This can be seen on 

the Figure 4.1 and 4.2 that Gompertz model is not a good fit for age 0, but for ages 

60 and 70 mortality rate estimations are consistent with the data. For this reason 

age 0 is not taken into consideration in next sections.  

 

 



33 
 

 

Fig. 4.1. Data and Gompertz model comparisons 

of mortality rates of 2000, in log scale at ages 0, 60 and 70. 



34 
 

 

Fig. 4.2. Data and Gompertz model comparisons 

of mortality rates of 2015, in log scale at ages 0, 60 and 70. 



35 
 

 

Afterwards, the sensitivity of ( , , , )xH b c    to changes in the values of   and   are 

tested. Different levels of mortality improvement are considered by allowing   to 

vary between -0.1 and 0.1. Note that, positive values of   stands for improving 

mortality rates and negative values of   stands for deteriorating mortality rates 

over time. Also, different rates of interest are considered from 0%  to 10% .  

As can be seen from Figures 4.3 and 4.4 the value of entropy, ( , , , )xH b c   , 

decreases in all four cases as the interest rate increases. This effect arises from 

the decreasing effect of mortality improvement on the cost of annuities when the 

interest rate is higher. As Figures 4.3 and 4.4 show when the interest rate is lower 

the value of entropy is at its highest and directly proportional with the mortality rate 

improvement level. 2.19%  increases with the   in direct proportion up until a level 

(breaking point of H ) which thereafter, anymore improvement in mortality rate 

decreases H . Hence mortality rate improvements start to lose their effect on the 

cost of a life annuity. Despite that, when the rate of interest is high, ( , , , )xH b c    

decreases almost continuously and there is no peak point as mortality improves. 

H  values and the breaking points are consistently smaller for women than man. 

This reflects the relative difference in favor of women in the survival probabilities, 

annuity costs and life expectations at various ages. Within each gender, with the 

age increase from 60 to 70, H  values and the breaking point for H  increase too. 

As a summary with 0.05   and 0.04   changes in the annuity values are found 

with the Equation (2.10) by simply multiplying minus the entropy measure, H , with 

the  . For the base year 2000 the change in the cost of annuity for males; at age 

60 is 1.68%  and at age 70 is 2.59% . For the base year 2015 these values change 

to 1.54%  at age 60 and 2.46%  at age 70. For the base year 2000 the change in 

the cost of annuity for females; at age 60 is 1.26%  and at age 70 is 2.19% . For the 

base year 2015 these values change to 1.01%  at age 60 and 1.93%  at age 70. 



36 
 

 

Fig. 4.3. Calculated values of ( , , , )xH b c    in year 2000 

for different values of   and   for ages 60 and 70. 

 

Fig. 4.4. Calculated values of ( , , , )xH b c    in year 2015 

for different values of   and   for ages 60 and 70. 



37 
 

Additionally, effect of the base mortality table on ( , , , )xH b c    can be tested by 

different values of c . To exemplify, the values of c  between 0.09 and 0.115 is 

taken into consideration. Figures 4.5 and 4.6 shows results for 0%   and Figure 

4.7 and 4.8 shows results for 4%  . 

Figures 4.5 and 4.6 shows entropy values for the base life tables 2000 and 2015 

with the interest rate equal to zero. ( , , , )xH b c   , as a function of  , is peaked at one 

point (breaking point of H ). When the reduction factor is greater than one (when 

  is negative), the values of ( , , , )xH b c    are higher in lower mortality level ( 0.09c  ) 

than the ones in higher mortality levels ( 0.115c  ). On the contrary, when the 

reduction factor is less than one (when   is positive), this relationship is reversed 

with improving mortality rates. 

Initially this verifies the conclusion that annuity cost with a high c  value (high 

mortality group) is less sensitive to changes in the mortality rate than with a low c  

value, when the mortality rate is high. At low levels of mortality this is reversed that 

a high c  value is more sensitive to changes in the mortality rate than a low c  

value. 

As expected, when the interest rate increased to 4% , shown in Figure 4.7 for base 

year 2000 and Figure 4.8 for base year 2015, the effect of changes in the force of 

mortality on cost of life annuity is less distinctive and it results to lower values of 

( , , , )xH b c   . This is based on the inverse proportion between interest rate and 

entropy measure, as in previous figures. 

Again, H  values and the breaking points are consistently smaller for women than 

man. This reflects the relative difference in favor of women in the survival 

probabilities, annuity costs and life expectations at various ages. Within each 

gender, with the age increase from 60 to 70, H  values and the breaking point for 

H  increase too. 



38 
 

 

Fig. 4.5. H  values for Females and Males at ages 60 and 70  

in base year 2000 with different c  values and 0%  . 

 

Fig. 4.6. H  values for Females and Males at ages 60 and 70  

in base year 2015 with different c  values and 0%  . 



39 
 

 

Fig. 4.7. H  values for Females and Males at ages 60 and 70 

in base year 2000 with different c  values and 4%  . 

 

Fig. 4.8. H  values for Females and Males at ages 60 and 70 

in base year 2015 with different c  values and 4%  . 



40 
 

5. RESULTS AND CONCLUSIONS 

This persistent decrease in mortality rates was unexpected and it is producing a 

number of new issues and challenges at multiple levels. As a result of this 

unexpected decrease they have often been underestimated by annuity and 

pension providers. Mortality improvements and rising longevity for post-retirement 

ages have a significant financial effect on the annuity and pension plans. Because 

of the tax-paid structure, the deficiencies of public finance are aggravated by 

mortality improvements.  

In several countries, defined benefit pension plans have continued to be replaced 

with defined contribution plans, which will most likely lead to the same result. 

Furthermore a high number of governments including Turkey are planning to 

increase retirement ages by at least 2 to 5 years to be able to take into account 

the changing dynamics of mortality improvements, and the impacts of ageing 

populations upon the financing of pensions. 

In this paper, the entropy measure has been examined in different scenarios to 

measure the effect of any changes in the force of mortality on the cost of life 

annuity. The entropy measure has been calculated with different mortality and 

interest rate assumptions. Also, it has been calculated over the whole age range 

(from age 0 to 100) and at older ages (from ages 60 and 70 to 100). 

Computation results show that entropy is almost insensitive to gender of the 

person but it is highly sensitive to age, base year, interest rate and mortality 

changes. We see that cost of annuity and value of entropy measure have inverse 

proportion with interest rate and direct proportion with age. 

There is a special case with the use of entropy measure to measure the effect of 

any changes in the force of mortality on the cost of life annuity: If the survival curve 

is already very rectangular shaped (when entropy is very close to zero), an 

expansion of the survival curve will have a great impact on the cost of life annuities 

but this would not be caught by the entropy measure. 

For all examined cases, the lower the force of interest, the higher is the value of 

entropy. This means that there is a higher effect of mortality risk on the present 



41 
 

value of annuity payments. This reflects the importance of mortality risk in the 

context of life annuity portfolios in particular with a low interest environment. 

At very high and low levels of mortality, the numerical results suggest that the 

effect of mortality changes on the cost of life annuity is very small. This reflects the 

fact that if the mortality is already very high or very low, any positive or negative 

change in the force of mortality is less likely to have a marked effect on the cost of 

life annuity. This implies that even if mortality continues to improve in the future, it 

will reach a level beyond which any more improvements would not markedly affect 

the annuity costs. 

For the future work, it would be useful to investigate the entropy measure with 

different models of mortality such as Lee Carter mortality forecasting model and 

Cairns-Blake-Dowd (CBD) stochastic mortality model. In addition, mortality model 

could be tested with distributional assumptions for its parameters. Also a 

stochastic model for the interest rate could be incorporated to see its effect on cost 

of life annuity. 

 

 

 

 

 

 

 

 

 

 

 

 



42 
 

REFERENCES 

[1] Khalaf-Allah, M., Haberman, S., Verrall, R., Measuring the effect of mortality 

improvements on the cost of annuities, Insurance: Mathematics and 

Economics, 39, 231–249, 2006. 

[2] Fiig Jarner, S., Masotti Kryger, E., Dengsøe, C., The evolution of death 

rates and life expectancy in Denmark, Scandinavian Actuarial Journal, 2-3, 

147-173, 2008. 

[3] Nusselder, W. J., Mackenbach, J. P., Rectangularization of the Survival 

Curve in the Netherlands, 1950-1992, The Cerontologist, Vol. 36, No. 6, 

773-782, 1996. 

[4] Preston, S. H., Elo, I. T., Are Educational Differentials In Mortality 

Increasing in the United States?, PARC Working Paper Series, WPS 95-01, 

1995. 

[5] Demetrius, L., Demographic Parameters and Natural Selection, Proc. Nat. 

Acad. Sci. USA, Vol. 71, No. 12, 4645-4647, 1974. 

[6] Comparing demographic, economic and social trends among the G-20 

members, Eurostat News release, 166/2012 - 28 November 2012. 

[7] TÜIK, Basic fertility and mortality indicators, 

http://www.tuik.gov.tr/UstMenu.do?metod=temelist (December 2013). 

[8] Definitions of the thermodynamics 

 www.Idolphin.org/entropynotes.html (March 2014). 

[9] Shannon, C. E., A mathematical theory of communication, Bell System 

Technical Journal, 27 (3), 379–423, 1948. 

[10] Pollard, J. H., The expectation of life and its relationship to mortality, JIA, 

109, 225-240, 1982. 

[11] Keyfitz, N., Applied Mathematical Demography, 2nd Edition, Springer 

Verlag Inc, New York, 1985. 

[12] Goldman, N., Lord, G., A New Look at Entropy and the Life Table, 

Demography, Vol. 23, No. 2, 275-282, 1986. 

[13] Nagnur, D., Rectangularization of the survival curve and entropy: The 

Canadian Experience, 1921-1981, Canadian Studies in Population, Vol.13 

(1), 83-102, 1986. 



43 
 

[14] Sachlas, A., Papaioannou, T., Residual and Past Entropy in Actuarial 

Science and Survival Models, Methodol Comput Appl Probab, DOI 

Methodology and Computing in Applied Probability, Volume 16, Issue 1, pp 

79-99, 2014. 

[15] Demetrius, L., Relations between demographic parameters, Demography, 

Vol.16 (2), 329-338, 1979. 

[16] Vaupel, J. W., How change in age-specific mortality affects life expectancy, 

Population studies 40, 147-157, 1986. 

[17] Construction of Turkish Life and Annuity Tables Project, 2010, 

Sigorta Bilgi ve Gözetim Merkezi, 

http://www.sbm.org.tr/?p=mortaliteIstatistik, Kasım, 2013. 

[18] Coale, A. J., Demeny, P., Regional Model Life Tables and Stable 

Populations, 1966, New York: Academic Press. 

[19] Sosyal Güvenlik Kurumu 

http://www.sgk.gen.tr/ssk/sgk-emeklilik-sartlari.html (June 2014). 

[20] 5510 Sayılı Sosyal Sigortalar ve Genel Sağlık Sigortası Kanunu, Resmi 

Gazete, 26200, 31.05.2006.  

[21] Gompertz, B., On the nature of the function expressive of the law of human 

mortality, and on a new mode of determining the value of life contingencies, 

Philosophical Transactions of the Royal Society 115: 513–585, 1825. 

 

 

 

 

 

 

 

 

 



44 
 

APPENDIX 

Appendix 1: Construction of Turkish Life and Annuity Tables Project’s female and 

male regression levels. 

Appendix 2: Entropy values for Turkish life tables at different rates of interest for 

females and males at age 0. 

Appendix 3: Entropy values for Turkish life tables at different rates of interest for 

females and males at age 60. 

Appendix 4: Entropy values for Turkish life tables at different rates of interest for 

females and males at age 70. 

Appendix 5: Entropy values for different mortality levels and rate of interests for 

the base years 2000 and 2015 at ages 60 and 70 for females and males. 

 

 

  



45 
 

Appendix 1: Construction of Turkish Life and Annuity Tables Project’s female and 

male regression levels. 

Life 

Table 

Female 

Regression 

Level 

Male 

Regression 

Level 

Life 

Table 

Female 

Regression 

Level 

Male 

Regression 

Level 

Life 

Table 

Female 

Regression 

Level 

Male 

Regression 

Level 

1930 0,00 0,00 1969 19,00 19,60 2008 23,97 23,12 

1931 5,57 8,20 1970 19,16 19,72 2009 24,07 23,19 

1932 7,03 9,67 1971 19,32 19,84 2010 24,17 23,26 

1933 8,05 10,65 1972 19,48 19,95 2011 24,27 23,33 

1934 8,86 11,40 1973 19,63 20,06 2012 24,37 23,39 

1935 9,55 12,02 1974 19,79 20,17 2013 24,47 23,46 

1936 10,15 12,56 1975 19,93 20,28 2014 24,57 23,53 

1937 10,69 13,03 1976 20,08 20,39 2015 24,67 23,59 

1938 11,18 13,45 1977 20,23 20,49 

1939 11,63 13,83 1978 20,37 20,59 

1940 12,05 14,18 1979 20,51 20,70 

1941 12,44 14,51 1980 20,65 20,80 

1942 12,81 14,81 1981 20,79 20,89 

1943 13,15 15,09 1982 20,92 20,99 

1944 13,48 15,36 1983 21,06 21,09 

1945 13,80 15,62 1984 21,19 21,18 

1946 14,10 15,86 1985 21,32 21,27 

1947 14,39 16,09 1986 21,45 21,36 

1948 14,67 16,31 1987 21,58 21,45 

1949 14,94 16,52 1988 21,70 21,54 

1950 15,19 16,72 1989 21,83 21,63 

1951 15,44 16,92 1990 21,95 21,72 

1952 15,69 17,11 1991 22,07 21,80 

1953 15,92 17,29 1992 22,19 21,89 

1954 16,15 17,46 1993 22,31 21,97 

1955 16,37 17,63 1994 22,43 22,05 

1956 16,59 17,80 1995 22,55 22,14 

1957 16,80 17,96 1996 22,66 22,22 

1958 17,01 18,12 1997 22,78 22,30 

1959 17,21 18,27 1998 22,89 22,37 

1960 17,40 18,42 1999 23,00 22,45 

1961 17,60 18,56 2000 23,11 22,53 

1962 17,78 18,70 2001 23,22 22,61 

1963 17,97 18,84 2002 23,33 22,68 

1964 18,15 18,97 2003 23,44 22,76 

1965 18,33 19,10 2004 23,55 22,83 

1966 18,50 19,23 2005 23,65 22,90 

1967 18,67 19,36 2006 23,76 22,97 

1968 18,84 19,48 2007 23,86 23,05 



46 
 

Appendix 2: Entropy values for different Turkish life tables at different rates of 

interest for females and males at age 0. 

Female Age=0 Interest Rate 

Life Table %0 %2 %4 %6 %8 %10 

1931 0,218407 0,146166 0,11769 0,112047 0,115556 0,122012 

1932 0,204658 0,132944 0,104239 0,097862 0,100369 0,105737 

1933 0,200603 0,125872 0,096036 0,088963 0,090793 0,095473 

1934 0,192701 0,119239 0,089613 0,082281 0,083669 0,087856 

1935 0,187732 0,114317 0,084573 0,076965 0,077987 0,081781 

1936 0,183656 0,110185 0,080312 0,072465 0,073176 0,076638 

1937 0,179878 0,106582 0,076666 0,068631 0,069082 0,072263 

1938 0,176623 0,10341 0,073435 0,065229 0,065449 0,068383 

1939 0,173916 0,100654 0,070591 0,062226 0,062241 0,064955 

1940 0,17137 0,098099 0,067965 0,059456 0,059283 0,061796 

1941 0,168911 0,095772 0,065614 0,056981 0,056638 0,05897 

1942 0,166588 0,093581 0,063402 0,054654 0,054152 0,056313 

1943 0,164482 0,091568 0,061338 0,052456 0,051784 0,053772 

1944 0,162593 0,089728 0,05941 0,05037 0,049514 0,05132 

1945 0,16079 0,087974 0,057574 0,048382 0,047353 0,048987 

1946 0,159104 0,086327 0,055858 0,046537 0,045357 0,04684 

1947 0,157562 0,084807 0,054289 0,044874 0,043578 0,044943 

1948 0,156077 0,083345 0,052781 0,043276 0,041869 0,04312 

1949 0,154644 0,081936 0,051329 0,041737 0,040224 0,041366 

1950 0,153225 0,080604 0,049985 0,040324 0,038719 0,039763 

1951 0,151841 0,079325 0,048703 0,038981 0,037289 0,038242 

1952 0,150499 0,078086 0,047462 0,03768 0,035906 0,03677 

1953 0,149196 0,076884 0,046258 0,03642 0,034565 0,035344 

1954 0,148042 0,075776 0,045139 0,035249 0,033321 0,034021 

1955 0,146974 0,074728 0,044076 0,034137 0,032141 0,032767 

1956 0,145931 0,073707 0,043041 0,033054 0,030992 0,031547 

1957 0,144911 0,072712 0,042033 0,032 0,029873 0,030359 

1958 0,143912 0,07174 0,041051 0,030973 0,028785 0,029203 

1959 0,142883 0,070809 0,040138 0,03003 0,027788 0,028146 

1960 0,141876 0,069899 0,039246 0,029109 0,026815 0,027114 

1961 0,14089 0,069009 0,038374 0,028208 0,025864 0,026106 

1962 0,139923 0,068137 0,03752 0,027327 0,024933 0,025119 

1963 0,138975 0,067283 0,036685 0,026465 0,024022 0,024153 

1964 0,138084 0,066471 0,035896 0,025657 0,023174 0,023256 

1965 0,137218 0,06568 0,035128 0,024873 0,022351 0,022386 

1966 0,136365 0,064902 0,034374 0,024103 0,021544 0,021533 

1967 0,135527 0,064139 0,033635 0,023349 0,020752 0,020696 

1968 0,134702 0,063389 0,032909 0,022608 0,019975 0,019874 

1969 0,133891 0,062652 0,032196 0,021881 0,019212 0,019068 

1970 0,133162 0,061992 0,031551 0,021215 0,018509 0,018323 



47 
 

1971 0,132443 0,061343 0,030916 0,020561 0,017818 0,017591 

1972 0,131733 0,060703 0,030292 0,019917 0,017139 0,016871 

1973 0,131033 0,060073 0,029677 0,019283 0,016471 0,016163 

1974 0,130343 0,059452 0,029072 0,01866 0,015813 0,015466 

1975 0,129661 0,05884 0,028476 0,018046 0,015165 0,01478 

1976 0,128986 0,058245 0,027907 0,017469 0,014562 0,014143 

1977 0,128317 0,057664 0,027362 0,016922 0,013994 0,013547 

1978 0,127657 0,05709 0,026824 0,016384 0,013435 0,012959 

1979 0,127004 0,056524 0,026293 0,015853 0,012883 0,012379 

1980 0,126358 0,055966 0,02577 0,015329 0,012339 0,011808 

1981 0,12572 0,055414 0,025254 0,014812 0,011803 0,011244 

1982 0,125089 0,054869 0,024744 0,014302 0,011273 0,010688 

1983 0,124439 0,054336 0,024267 0,013838 0,0108 0,010194 

1984 0,123761 0,053813 0,023829 0,013433 0,010397 0,00978 

1985 0,123087 0,053296 0,023397 0,013033 0,009999 0,00937 

1986 0,122418 0,052784 0,02297 0,012638 0,009606 0,008966 

1987 0,121754 0,052277 0,022547 0,012248 0,009218 0,008566 

1988 0,121094 0,051775 0,022129 0,011861 0,008834 0,008171 

1989 0,12044 0,051277 0,021715 0,01148 0,008455 0,007781 

1990 0,11979 0,050784 0,021306 0,011102 0,00808 0,007395 

1991 0,119142 0,05031 0,020927 0,010762 0,007746 0,007053 

1992 0,118496 0,049848 0,020568 0,010447 0,007441 0,006742 

1993 0,117853 0,04939 0,020213 0,010135 0,007138 0,006433 

1994 0,117213 0,048935 0,019861 0,009826 0,00684 0,006129 

1995 0,116575 0,048483 0,019512 0,009521 0,006544 0,005827 

1996 0,11594 0,048035 0,019167 0,009219 0,006251 0,005529 

1997 0,115308 0,047589 0,018824 0,008919 0,005961 0,005233 

1998 0,114678 0,047147 0,018485 0,008623 0,005674 0,004941 

1999 0,114051 0,046708 0,018149 0,00833 0,005391 0,004652 

2000 0,113422 0,046295 0,017857 0,008094 0,005173 0,004434 

2001 0,112793 0,045883 0,017568 0,00786 0,004956 0,004217 

2002 0,112164 0,045473 0,017281 0,007628 0,004741 0,004003 

2003 0,111536 0,045064 0,016996 0,007398 0,004529 0,00379 

2004 0,110908 0,044658 0,016713 0,00717 0,004318 0,00358 

2005 0,11028 0,044253 0,016432 0,006944 0,004109 0,003371 

2006 0,109653 0,04385 0,016152 0,00672 0,003902 0,003164 

2007 0,109027 0,043448 0,015875 0,006498 0,003697 0,002959 

2008 0,108402 0,043048 0,015599 0,006277 0,003494 0,002756 

2009 0,107775 0,042667 0,015354 0,006094 0,003332 0,002598 

2010 0,107146 0,042294 0,015125 0,00593 0,003192 0,002463 

2011 0,106513 0,041921 0,014896 0,005768 0,003053 0,002329 

2012 0,105879 0,041549 0,014669 0,005606 0,002916 0,002196 

2013 0,105242 0,041177 0,014442 0,005446 0,002779 0,002064 

2014 0,104604 0,040805 0,014217 0,005286 0,002643 0,001933 

2015 0,103963 0,040433 0,013992 0,005128 0,002508 0,001803 

 



48 
 

Male Age=0 Interest Rate 

Life Table %0 %2 %4 %6 %8 %10 

1931 0,210125 0,133419 0,101218 0,092709 0,094034 0,098783 

1932 0,202081 0,1232 0,09016 0,080959 0,081464 0,085313 

1933 0,19517 0,116381 0,083248 0,073723 0,073749 0,077056 

1934 0,189706 0,111291 0,078184 0,068448 0,068133 0,071048 

1935 0,186084 0,107427 0,074179 0,064235 0,063637 0,066239 

1936 0,182547 0,10396 0,070626 0,060468 0,059587 0,061884 

1937 0,179403 0,100915 0,067517 0,057176 0,056048 0,058081 

1938 0,176356 0,098318 0,064972 0,054503 0,05318 0,055001 

1939 0,173579 0,095957 0,06266 0,052076 0,050577 0,052206 

1940 0,171577 0,094036 0,060705 0,050011 0,048363 0,049832 

1941 0,170198 0,092472 0,059041 0,048241 0,046467 0,047802 

1942 0,168897 0,09101 0,057489 0,046591 0,044699 0,045909 

1943 0,167539 0,089611 0,056042 0,045064 0,043067 0,044164 

1944 0,166011 0,088242 0,054696 0,043664 0,041579 0,042574 

1945 0,164563 0,086947 0,053422 0,04234 0,04017 0,04107 

1946 0,163186 0,085715 0,052212 0,041081 0,038832 0,039642 

1947 0,161925 0,084567 0,051079 0,039903 0,03758 0,038307 

1948 0,160799 0,083508 0,050025 0,03881 0,03642 0,037072 

1949 0,159719 0,082493 0,049016 0,037762 0,035309 0,035888 

1950 0,15868 0,081517 0,048047 0,036756 0,034243 0,034751 

1951 0,157678 0,080578 0,047114 0,035788 0,033216 0,033658 

1952 0,156739 0,079692 0,046236 0,034879 0,032253 0,032634 

1953 0,155851 0,078852 0,045403 0,034019 0,031344 0,031667 

1954 0,154991 0,078038 0,044598 0,033187 0,030465 0,030732 

1955 0,154156 0,07725 0,043818 0,032381 0,029613 0,029826 

1956 0,153345 0,076485 0,043062 0,0316 0,028787 0,028948 

1957 0,152556 0,075742 0,042327 0,030842 0,027986 0,028097 

1958 0,151793 0,075049 0,041645 0,030135 0,027238 0,0273 

1959 0,151051 0,074383 0,040992 0,029459 0,02652 0,026537 

1960 0,150327 0,073734 0,040356 0,028799 0,025822 0,025793 

1961 0,149621 0,073102 0,039735 0,028157 0,025141 0,025068 

1962 0,14893 0,072484 0,03913 0,02753 0,024477 0,024361 

1963 0,148255 0,07188 0,038539 0,026919 0,023828 0,023672 

1964 0,147594 0,07129 0,037962 0,026321 0,023195 0,022998 

1965 0,14694 0,0707 0,037396 0,025747 0,022595 0,022362 

1966 0,146298 0,070118 0,036842 0,025189 0,022012 0,021747 

1967 0,145668 0,069549 0,036299 0,024642 0,021442 0,021144 

1968 0,14505 0,06899 0,035768 0,024107 0,020883 0,020554 

1969 0,144444 0,068443 0,035247 0,023583 0,020336 0,019976 

1970 0,143848 0,067906 0,034736 0,023068 0,0198 0,01941 

1971 0,143263 0,067379 0,034235 0,022564 0,019273 0,018854 

1972 0,142688 0,066861 0,033744 0,022069 0,018757 0,018308 

1973 0,142157 0,066382 0,033287 0,021608 0,018276 0,0178 

1974 0,141661 0,065935 0,032859 0,021175 0,017823 0,017322 



49 
 

1975 0,141172 0,065495 0,032439 0,02075 0,017379 0,016851 

1976 0,140691 0,065062 0,032025 0,020332 0,016942 0,016389 

1977 0,140217 0,064636 0,031619 0,019921 0,016512 0,015935 

1978 0,13975 0,064216 0,031219 0,019516 0,016089 0,015488 

1979 0,139289 0,063804 0,030825 0,019118 0,015673 0,015048 

1980 0,138835 0,063397 0,030437 0,018726 0,015263 0,014615 

1981 0,138387 0,062996 0,030055 0,018341 0,01486 0,014189 

1982 0,137945 0,062601 0,029679 0,01796 0,014463 0,013769 

1983 0,137431 0,06219 0,029322 0,017621 0,01412 0,013413 

1984 0,136913 0,061781 0,028971 0,01729 0,013787 0,013068 

1985 0,136402 0,061378 0,028625 0,016964 0,013459 0,012728 

1986 0,135897 0,060981 0,028284 0,016642 0,013136 0,012392 

1987 0,135397 0,060588 0,027947 0,016325 0,012816 0,012061 

1988 0,134902 0,060199 0,027614 0,016012 0,012502 0,011735 

1989 0,134413 0,059816 0,027286 0,015703 0,012191 0,011413 

1990 0,13393 0,059437 0,026962 0,015398 0,011884 0,011095 

1991 0,133451 0,059063 0,026642 0,015097 0,011581 0,010781 

1992 0,132977 0,058692 0,026326 0,014799 0,011283 0,010471 

1993 0,132508 0,058326 0,026013 0,014505 0,010987 0,010165 

1994 0,132004 0,057956 0,025713 0,014232 0,010717 0,009888 

1995 0,131484 0,057584 0,02542 0,013971 0,010462 0,009627 

1996 0,130968 0,057217 0,025131 0,013713 0,01021 0,009369 

1997 0,130456 0,056852 0,024844 0,013458 0,009961 0,009114 

1998 0,129948 0,056491 0,024561 0,013206 0,009714 0,008862 

1999 0,129444 0,056134 0,02428 0,012956 0,009471 0,008613 

2000 0,128943 0,055779 0,024003 0,01271 0,00923 0,008366 

2001 0,128446 0,055428 0,023728 0,012466 0,008991 0,008123 

2002 0,127953 0,05508 0,023456 0,012224 0,008756 0,007881 

2003 0,127463 0,054735 0,023187 0,011985 0,008522 0,007643 

2004 0,126977 0,054393 0,02292 0,011748 0,008292 0,007407 

2005 0,126495 0,054054 0,022656 0,011514 0,008063 0,007173 

2006 0,126016 0,053717 0,022394 0,011282 0,007837 0,006942 

2007 0,125597 0,053412 0,022156 0,011074 0,007637 0,006739 

2008 0,125212 0,053124 0,021931 0,01088 0,007453 0,006554 

2009 0,124828 0,052838 0,021708 0,010688 0,00727 0,00637 

2010 0,124446 0,052554 0,021487 0,010498 0,007089 0,006187 

2011 0,124065 0,052272 0,021268 0,010309 0,00691 0,006007 

2012 0,123686 0,051991 0,021051 0,010123 0,006732 0,005828 

2013 0,123307 0,051713 0,020835 0,009937 0,006556 0,005651 

2014 0,122931 0,051436 0,020621 0,009754 0,006382 0,005475 

2015 0,122555 0,051162 0,020409 0,009572 0,006209 0,005301 

  



50 
 

Appendix 3: Entropy values for different Turkish life tables at different rates of 

interest for females and males at age 60. 

Female Age=60 Interest Rate 

Life Table %0 %2 %4 %6 %8 %10 

1931 0,717799 0,646583 0,587351 0,538665 0,498954 0,466693 

1932 0,678189 0,607655 0,549094 0,501074 0,462021 0,4304 

1933 0,673751 0,59599 0,532906 0,482219 0,441706 0,409376 

1934 0,649908 0,574644 0,513293 0,463809 0,424147 0,392439 

1935 0,636652 0,561249 0,499887 0,450487 0,410971 0,379441 

1936 0,626087 0,550354 0,488843 0,439423 0,399973 0,36856 

1937 0,615894 0,540364 0,479038 0,429796 0,39052 0,359275 

1938 0,607316 0,531809 0,470543 0,421393 0,38223 0,35111 

1939 0,600429 0,524701 0,463327 0,414155 0,375028 0,343979 

1940 0,593931 0,51807 0,456642 0,407478 0,368401 0,337429 

1941 0,587435 0,511745 0,450464 0,401432 0,362478 0,33162 

1942 0,581333 0,505809 0,444669 0,395763 0,356925 0,326176 

1943 0,57601 0,500652 0,439649 0,390861 0,35213 0,321476 

1944 0,571484 0,496296 0,435425 0,386748 0,348111 0,317541 

1945 0,567178 0,492151 0,431409 0,382837 0,344291 0,3138 

1946 0,563216 0,488307 0,427662 0,379175 0,340705 0,310283 

1947 0,559702 0,484837 0,424238 0,375801 0,337384 0,307016 

1948 0,556325 0,481504 0,420951 0,372563 0,334198 0,303882 

1949 0,553072 0,478296 0,417789 0,36945 0,331134 0,300869 

1950 0,549629 0,474987 0,414585 0,366333 0,328093 0,297893 

1951 0,546208 0,471723 0,411443 0,363289 0,325128 0,294996 

1952 0,5429 0,468568 0,408406 0,360345 0,322263 0,292197 

1953 0,539696 0,465512 0,405464 0,357495 0,319488 0,289487 

1954 0,53687 0,462717 0,402704 0,354774 0,316811 0,286855 

1955 0,534259 0,460081 0,400066 0,352152 0,314217 0,284296 

1956 0,531712 0,457514 0,397499 0,349602 0,311695 0,281808 

1957 0,529225 0,455012 0,394999 0,347119 0,30924 0,279388 

1958 0,526784 0,452561 0,392555 0,344696 0,306846 0,277029 

1959 0,524062 0,449942 0,390018 0,342227 0,304438 0,274675 

1960 0,521406 0,447385 0,387541 0,339819 0,302089 0,272379 

1961 0,518809 0,444886 0,385122 0,337466 0,299795 0,270137 

1962 0,516271 0,442444 0,382758 0,335168 0,297553 0,267946 

1963 0,513787 0,440055 0,380445 0,332919 0,295361 0,265804 

1964 0,51175 0,43808 0,378519 0,331036 0,293515 0,263989 

1965 0,509832 0,436217 0,376701 0,329257 0,291768 0,26227 

1966 0,507948 0,434388 0,374917 0,327511 0,290055 0,260584 

1967 0,506095 0,432591 0,373166 0,325797 0,288374 0,25893 

1968 0,504273 0,430825 0,371445 0,324114 0,286723 0,257307 

1969 0,502473 0,429081 0,369746 0,322453 0,285095 0,255706 

1970 0,500081 0,426726 0,367431 0,320184 0,282875 0,253538 



51 
 

1971 0,49773 0,424411 0,365157 0,317955 0,280694 0,251409 

1972 0,495418 0,422137 0,362923 0,315765 0,278553 0,249317 

1973 0,493143 0,4199 0,360726 0,313612 0,276447 0,247261 

1974 0,490905 0,417699 0,358566 0,311495 0,274378 0,24524 

1975 0,488702 0,415534 0,35644 0,309413 0,272342 0,243252 

1976 0,486648 0,413526 0,354475 0,307491 0,270463 0,241417 

1977 0,484713 0,411643 0,352639 0,305699 0,268712 0,239705 

1978 0,482805 0,409788 0,35083 0,303933 0,266988 0,23802 

1979 0,480924 0,407959 0,349047 0,302193 0,265289 0,236359 

1980 0,479067 0,406154 0,347289 0,300478 0,263614 0,234723 

1981 0,477236 0,404375 0,345556 0,298787 0,261964 0,23311 

1982 0,475427 0,402619 0,343846 0,29712 0,260336 0,23152 

1983 0,473324 0,400581 0,341866 0,295194 0,25846 0,229692 

1984 0,470826 0,398167 0,339525 0,292922 0,256253 0,227546 

1985 0,468356 0,395782 0,337215 0,29068 0,254076 0,225429 

1986 0,465915 0,393425 0,334933 0,288466 0,251927 0,22334 

1987 0,463501 0,391097 0,332679 0,286281 0,249805 0,221278 

1988 0,461114 0,388795 0,330452 0,284123 0,247711 0,219243 

1989 0,458754 0,38652 0,328252 0,281991 0,245643 0,217233 

1990 0,456419 0,384271 0,326078 0,279885 0,2436 0,215249 

1991 0,453912 0,381857 0,323745 0,277627 0,241411 0,213124 

1992 0,451295 0,379339 0,321313 0,275274 0,239132 0,210914 

1993 0,448703 0,376846 0,318907 0,272947 0,236879 0,20873 

1994 0,446136 0,374379 0,316527 0,270647 0,234652 0,206572 

1995 0,443592 0,371936 0,314172 0,268371 0,23245 0,204438 

1996 0,441073 0,369518 0,311842 0,266121 0,230273 0,202329 

1997 0,438577 0,367124 0,309536 0,263894 0,22812 0,200243 

1998 0,436103 0,364753 0,307253 0,261691 0,22599 0,19818 

1999 0,433646 0,362398 0,304987 0,259506 0,223878 0,196135 

2000 0,430825 0,359702 0,302399 0,257015 0,221478 0,193818 

2001 0,428026 0,357028 0,299834 0,254549 0,219102 0,191526 

2002 0,425248 0,354377 0,297293 0,252106 0,216751 0,189258 

2003 0,422491 0,351748 0,294774 0,249687 0,214422 0,187012 

2004 0,419755 0,349141 0,292278 0,247291 0,212117 0,18479 

2005 0,417039 0,346555 0,289805 0,244917 0,209834 0,182589 

2006 0,414344 0,343991 0,287353 0,242565 0,207572 0,18041 

2007 0,41167 0,341448 0,284922 0,240235 0,205333 0,178253 

2008 0,409016 0,338926 0,282513 0,237926 0,203114 0,176116 

2009 0,40609 0,336157 0,279878 0,235409 0,200705 0,173804 

2010 0,403037 0,333274 0,277139 0,232799 0,198211 0,171415 

2011 0,400001 0,330409 0,274422 0,230211 0,195738 0,169047 

2012 0,396982 0,327564 0,271724 0,227643 0,193287 0,166701 

2013 0,393981 0,324738 0,269046 0,225096 0,190857 0,164376 

2014 0,390998 0,321931 0,266389 0,222569 0,188447 0,16207 

2015 0,388033 0,319143 0,26375 0,220062 0,186057 0,159785 

 



52 
 

Male Age=60 Interest Rate 

Life Table %0 %2 %4 %6 %8 %10 

1931 0,712237 0,641729 0,583085 0,534913 0,495672 0,463848 

1932 0,697509 0,620888 0,558463 0,508108 0,467723 0,435399 

1933 0,679877 0,602909 0,54034 0,489995 0,449723 0,417575 

1934 0,665508 0,588884 0,526613 0,476536 0,43651 0,404589 

1935 0,657177 0,579704 0,516925 0,466586 0,426463 0,394548 

1936 0,648432 0,57098 0,508245 0,457977 0,417945 0,386131 

1937 0,64069 0,563322 0,500673 0,450496 0,410561 0,378848 

1938 0,632982 0,556477 0,49443 0,444668 0,405021 0,373513 

1939 0,62597 0,55026 0,488767 0,439386 0,400004 0,368682 

1940 0,621402 0,545801 0,484399 0,435101 0,395794 0,364542 

1941 0,618776 0,542744 0,48107 0,431618 0,392237 0,360962 

1942 0,616269 0,539861 0,477951 0,428365 0,388922 0,357629 

1943 0,613323 0,53677 0,474776 0,425155 0,385708 0,354433 

1944 0,609479 0,533153 0,471327 0,421831 0,382481 0,351284 

1945 0,605848 0,529736 0,468068 0,418691 0,379434 0,34831 

1946 0,602404 0,526496 0,464978 0,415714 0,376544 0,34549 

1947 0,599263 0,52349 0,462076 0,412894 0,373791 0,342795 

1948 0,596471 0,52074 0,459365 0,410223 0,371161 0,340204 

1949 0,593795 0,518106 0,45677 0,407667 0,368644 0,337726 

1950 0,591227 0,515579 0,454281 0,405215 0,36623 0,33535 

1951 0,588756 0,513148 0,451888 0,402859 0,363911 0,333067 

1952 0,586413 0,510821 0,449582 0,40058 0,361663 0,330852 

1953 0,584179 0,508586 0,447357 0,398375 0,359484 0,328702 

1954 0,582017 0,506425 0,445207 0,396244 0,357378 0,326626 

1955 0,579921 0,504332 0,443125 0,394182 0,355342 0,324618 

1956 0,577888 0,502303 0,441108 0,392185 0,353369 0,322673 

1957 0,575914 0,500333 0,439151 0,390247 0,351456 0,320787 

1958 0,573651 0,498126 0,43698 0,388099 0,349321 0,318656 

1959 0,571336 0,495882 0,43478 0,385922 0,347152 0,316485 

1960 0,569083 0,493698 0,432639 0,383804 0,345042 0,314373 

1961 0,566888 0,491571 0,430554 0,381742 0,342988 0,312316 

1962 0,564748 0,489498 0,428522 0,379732 0,340987 0,310313 

1963 0,562661 0,487475 0,426539 0,377771 0,339035 0,308359 

1964 0,560623 0,485501 0,424604 0,375857 0,337129 0,306452 

1965 0,559049 0,483958 0,423097 0,374391 0,335708 0,305079 

1966 0,557616 0,482547 0,421721 0,373061 0,334432 0,303862 

1967 0,556211 0,481166 0,420373 0,371758 0,333182 0,30267 

1968 0,554834 0,479811 0,419053 0,370481 0,331957 0,301503 

1969 0,553482 0,478483 0,417758 0,36923 0,330757 0,300358 

1970 0,552155 0,477179 0,416488 0,368002 0,329579 0,299236 

1971 0,550852 0,4759 0,415241 0,366798 0,328424 0,298135 

1972 0,549572 0,474643 0,414018 0,365616 0,32729 0,297054 

1973 0,548207 0,473295 0,412692 0,364319 0,326027 0,295826 

1974 0,546783 0,471883 0,411296 0,362942 0,324671 0,294492 



53 
 

1975 0,545384 0,470496 0,409924 0,361589 0,323339 0,293183 

1976 0,544007 0,469132 0,408576 0,36026 0,322031 0,291897 

1977 0,542652 0,467791 0,407251 0,358953 0,320746 0,290634 

1978 0,541319 0,466472 0,405948 0,357669 0,319482 0,289392 

1979 0,540007 0,465173 0,404665 0,356405 0,318239 0,28817 

1980 0,538715 0,463895 0,403404 0,355162 0,317017 0,286969 

1981 0,537441 0,462637 0,402162 0,353939 0,315813 0,285786 

1982 0,536187 0,461397 0,400938 0,352734 0,314629 0,284622 

1983 0,534381 0,459647 0,399236 0,351075 0,31301 0,28304 

1984 0,53254 0,457867 0,397507 0,349392 0,311368 0,281436 

1985 0,530726 0,456113 0,395803 0,347733 0,30975 0,279857 

1986 0,528937 0,454383 0,394124 0,346099 0,308156 0,278301 

1987 0,527172 0,452678 0,392469 0,344488 0,306586 0,276767 

1988 0,525432 0,450996 0,390837 0,3429 0,305038 0,275256 

1989 0,523714 0,449337 0,389227 0,341334 0,303511 0,273766 

1990 0,522019 0,4477 0,387639 0,339789 0,302006 0,