STOCHASTIC MODELLING OF SYSTEMATIC MORTALITY RISK UNDER COLLATERAL DATA AND ITS APPLICATIONS

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Abstract

Many actuaries worldwide use Systematic Mortality Risk (SMR) to value actuarial products such as annuities and assurances sold to policyholders. Data availability plays an essential role in ascertaining the SMR models’ accuracy, and it varies from one country to another. Incorrect stochastic modeling of SMR models due to paucity of data has been a problem for many Sub-Saharan African countries such as Kenya, thus prompting modifications of the classical SMR models used in those countries with limited data availability. This study aimed at modelling SMR stochastically under the collateral data environment such as Sub-Saharan African countries like Kenya and then apply it in the current actuarial valuations. This thesis has formulated novel stochastic mortality risk models under the collateral data setup. Kenya population data is preferably integrated into the commonly applied stochastic mortality risk models under a 3-factor unitary framework of age-time- cohort. After testing SMR models on the Kenyan data to assess their behaviours, we incorporate the Bühlmann Credibility Approach with random coefficients in modeling. The randomness of the classical SMR models was modeled as NIG distribution instead of Normal distribution due to data paucity in Kenya (use of collateral data environment). The Deep Neural Network (DNN) technique solved data paucity during the SMR model fitting and forecasting. The forecasting performances of the SMR models were done un- der DNN and, compared with those from conventional models, show powerful empirical illustrations in their precision levels. Numerical results showed that SMR models become more accurate under collateral data after incorporating the BCA with NIG assumptions. The Actuarial valuation of annuities and assurances using the new SMR offered much more accurate valuations when compared to those under classical models. The study’s findings should help regulators such as IRA and RBA make policy documents that protect all stakeholders in Kenya’s insurance, social protection firms, and pension sectors. For areas for further research, one can use the BCA approach for Sub-Saharan African countries with similar demographic characteristics and Hierarchical BCA in SMR modeling.
 






Table of Contents

Declaration i
Acknowledgement ii
Abstract iii
Table of Contents v
List of Tables viii
List of Figures x
Abbreviations and Acronyms xi
List of Publications xiii

Chapter One: Introduction
1.1 Motivation and Background 1
1.2 Actuarial Notations, Definitions, and Terminologies 4
1.3 Statement of the Problem 5
1.4 Objectives of the study 5
1.5 Significance of the Study 6
1.6 The Structure of the Thesis 6

Chapter Two: Literature Review
3 Systematic Mortality Risk Modeling Under Three-Factor Structure 13
3.1 Stochastic Mortality Risk Models 13
3.2 Force of Systematic Mortality 15
3.3 The Age-Time-Cohort Modeling Framework of SMR 17
3.4 Mortality Model Fitting 20
3.5 Systematic Mortality Risk Projection 29
3.6 Analysis and Results 34

Chapter Four: Bühlmann Credibility Approach Incorporation into Systematic Mortality Risk Modeling
4.1 Bühlmann Credibility Model Description 36
4.2 Heavy Tailed Distribution 40
4.3 Incorporating the BCA into the Mortality Models 49
4.4 Fitting and forecasting of Models 59
4.5 Analysis and Results 61

Chapter Five: Systematic Mortality Risk Forecasting Under Deep Learning Technique
5.1 Deep Learning Integration 62
5.2 SMR Modeling Under Deep Learning 64
5.3 Mathematical Application and Results 69
5.4 Results 73

Chapter Six: Actuarial Valuation of Life Products in Kenya
6.1 Life Annuities and Assurance Life Products without BCA 74
6.2 Life Annuities and Assurance Life Products Under BCA 79
6.3 A Comparison of Assurances and Annuities in Kenya and The UK Data .  81 6.4 Results 84

Chapter Seven: Conclusions and Recommendations 85
7.1 Conclusions 85
7.2 Recommendations 86
References 88
Appendices 93
 



List of Tables

3.1 Summary of Popular Stochastic Mortality Models 15
3.2 The Log Likelihood and BIC, AIC(c), and AIC values (order of ranking within brackets) of the SMR models for males 27
3.3 The Log Likelihood and BIC, AIC(c), and AIC values (order of ranking within brackets) of the SMR models for females. 27
3.4 LR test statistics for General models (H0) within Specific models (H1) for Males 28
3.5 LR test statistics for General models (H0) within Specific models (H1) for Females 28
3.6 ARIMA(p,d,q) models for the time index k(i) , i = 1, 2, 3 of males in SMR models. 30
3.7 ARIMA(p,d,q) models for the time index k(i) , i = 1, 2, 3 of females in
SMR models 30
3.8 ARIMA(p,d,q) models for the cohort index Wt−c of female and male
SMR models 31
3.9 Expected values (grading order in brackets) of MAPE and MAE of the predicting period 2010–2020 using fitted jump-off rates for Kenyans 31
4.1 JB Normality test for Model A of Males and Females Respectively 46
4.2 Variance and Mean Matrices Estimates for Model B 47
4.3 DH Normality test for Model B of Males and Females Respectively 48
4.4 A Multivariate Shapiro-Wilk Test for Normality of Males and Females Respectively 48
4.5 AD Test for Model C of Males and Females Respectively 49
4.6 Estimations for the values of θ ,υ and b 55
4.7 Model A With and Without BC for Males and Females(brackets) 60
4.8 Model B with and without BC for Males and Females(brackets) 60
4.9 Model C with and without BC for Males and Females(brackets) 60
5.1 Kenyan Supervised Deep Learning Dataset 70
5.2 Testing set years as per Nation 71
5.3 Best ARIMA by Nation and Gender 71
5.4 LSTM & ARIMA Performances in the testing set for every Nation 72
6.1 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted WLA 76
6.2 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted WLA 76
6.3 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted WLA 76
6.4 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted WLA 77
6.5 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted WLA 77
6.6 EPV of MAE and MAPE measures for Males and Females (in brackets)
for a whole life annuity 78
6.7 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted temporary life annuity 78
6.8 EPV of MAE and MAPE measures for Males and Females (in brackets)
for WLA 79
6.9 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted temporary ELA 79
6.10 MAE and MAPE EPV measures for Males and Females (in brackets) for
10 year predicted Pure ELA 79
6.11 MAE and MAPE EPV measures for Males and Females (in brackets) for
10 year predicted Endowment life Assurance 80
6.12 EPV of MAE and MAPE measures with (ranking order in brackets) for
10 year predicted Deffered ELA 80
6.13 EPV of MAE and MAPE measures for Males and Females (in brackets)
for a Whole life annuity 80
6.14 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted temporary life annuity 81
6.15 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted WLA 81
6.16 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted Temporary ELA 82
6.17 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted Pure ELA 82
6.18 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted ELA 82
6.19 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted Deffered ELA 83
6.20 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted Whole life annuity 83
6.21 EPV of MAE and MAPE measures for Males and Females (in brackets)
for 10 year predicted temporary life annuity 83
7.1 An Extract of the new life Table for Males in Kenya in 2030 94
7.2 An Extract of the new life Table for Females in Kenya in 2030 95
7.3 An Extract of the new life Table for Males in Kenya in 2040 96
7.4 An Extract of the new life Table for Females in Kenya in 2040 97
 





List of Figures

3.4.1 α , β (1) and κ(1) parameters estimates for males (blue top panels) & females (red bottom panels) for personal ages fitted from 2010 to 2022 for Model A 21
3.4.2 α , β (1) and κ(1) parameters estimates for males (blue top panels) & females (red bottom panels) for personal ages fitted from 2010 to 2022 for Model B 22
3.4.3 α , β (1) and κ(1) parameters estimates for males (blue top panels) & females (bottom panels) for personal ages fitted from 2010 to 2022 for Model C 22
3.4.4 A: The Estimated parameters of α , k(1),β (1) for Males fitted aged 20–100 23
3.4.5 B: k(1) , β (1),k(2)and γ estimated parameters for Males, aged 60–100 . 24
3.4.6 C: k(1) , β (1)and γ estimated parameters for Males aged 60–100 . . . 24
3.4.7 A: Residuals of Deviance for males (top consoles) & females (bottom consoles) for duration 2010-2020 from ages 60–100 for Kenya 25
3.4.8 B: Residuals of Deviance for males (top consoles) & females (bottom consoles) for duration 2010-2020 from ages 60–100 for Kenya 26
3.4.9 C: Residuals of Deviance for males (top consoles) & females (bottom consoles) for duration 2010-2020 from ages 60–100 for Kenya 26
3.5.1 Long-term SMR prediction of A, and B models fitted from 2010 to 2020 and projections from 2020 to 2050 for ages 60 to 100 for both males & fe- males for confidence levels of 50%, 80% and 95% intervals of prediction
respectively 33
3.5.2 Long-term SMR prediction of C models fitted from 2010 to 2020 and projections from 2020 to 2050 for ages 60 to 100 for both males & fe- males for confidence levels of 50%, 80% and 95% intervals of prediction respectively 33
3.6.1 A(up) and B(down) for females(right) & males(left) 35
3.6.2 C model for females(right) & males(left) 35
4.2.1 NIG Probability Density plots 42
4.3.1 loge (m(x,t) ) & logit(q(x,t) ) against time for Kenyan Males 50
4.3.2 loge (m(x,t) ) & logit(q(x,t) ) against time for Kenyan Females 50
4.3.3 Qx,t against time for Kenyan Males and Females 51
5.1.1 A Normal Representation of feed-forward Artificial Neural Network (ANN) 63
5.2.1 A LSTM Block Structure with Its Internal Information Forward Flow Design 68
5.3.1 Mortality Prediction Under ARIMA vs LSTM 73



 
Abbreviations and Acronyms

AI: Artificial Intelligence
AIC: Akaike Information Criterion
AIC(c): Modified Akaike Information Criterion
AD: Anderson-Darling
ANN: Artificial Neural Network
AR: Autoregressive Model
ARIMA: Autoregressive Integrated Moving Average
ATC: Age-Time-Cohort
BIC: Bayesian Information Criterion BCA: Bühlmann Credibility Approach BC: Bühlmann Credibility
CBR: Central Birth Rates
CDF: Cumulative Distribution Function
CDR: Central Death Rates
CFM: Constant force of mortality
CMI: Continuous Mortality Investigation
CMR: Combined Mortality Rates
DB: Defined Benefit Pension Plan
DC: Defined Contribution Pension Plan
DH: Doornik-Hansen
DL: Deep Learning
DNN: Deep Neural Network e.g.: for the sake of example EPV: Expected Present Value
ELA: Endowment Life Assurance EW: Expanding the Window HMD: Human Motality Database i.e.: that is (to say)
i.i.d.: Independent and identically distributed
IRA: Insurance Regulatory Authority
IG: Inverse Gaussian
IMF: International Monetary Fund
JB: Jarque–Bera
LR: Likelihood Ratio
LRM: Linear Regression Mortality Model
LSTM: Long Short-Term Memory
M.L.: Maximum Likelihood
M.L.E.: Maximum Likelihood Estimation
MA: Moving Average Model
MAE: Mean Absolute Error
MAPE: Mean Absolute Percentage Error
MAPFE: Mean Absolute Percentage Forecast Error
ML: Machine Learning MLP: Multilayer Perceptron MW: Moving the Window
NIG: Normal Inverse Gaussian
OLS: Ordinary Least Squares
PDF: Probability Distribution Function RBA: Retirement Benefit Authority ReLU: Rectified Linear Unit
RMSE: Root Mean Square Error RNN: Recurrent Neural Networks RR: Reduction Ratio
SLR: Systematic Longevity Risk SMR: Systematic Mortality Risk SVD: Singular Value Decomposition UDB: Uniform Distribution of Births UDD: Uniform Distribution of Deaths VaR: Value at Risk
WLA: Whole Life Assurance
XOR: Exclusive OR
 








Chapter 1 
Introduction

In this chapter, we have discussed the background and motivation of the study, actuarial notations, definitions and terminologies used, the statement of the study’s problem and objectives, and both general and specific objectives. In addition, we have highlighted the significance as well as the overall structure of the thesis document.

1.1 Motivation and Background
At the start of this century, systematic mortality risk modeling has been of particular significance in actuarial valuations, making it a feature in many; British, European, and American actuarial journals, among many more. Many medical inventions in this century have made it possible for people to live longer than expected by a drop in their death rates (reduced SMR) at the older ages. For example, according to WHO reports, in 2018, the mean life expectancy worldwide increased to 72 years from 68 years in 2010. According to UN statistics in the Department of Economics and Social Affairs in 2018, a male adult in Kenya had a mortality rate of 5695 (per 100,000 male adults), which decreased from 9730 in 2004 Raalte (2021).

Many International Organizations such as World Economic Forum and IMF have also been interested in mortality trends since it affects many countries’ fiscal and monetary policies, both developed and developing countries Case and Deaton (2017). In SMR, the company has to correctly determine the number of years in which life is antici- pated/expected to survive or die to a given age before deciding the monthly pension money payable to the policyholder. When this is done incorrectly, the pension providers are likely to pay more, thus decreasing the chances of survival due to an increased probability of ruin or insolvency rates.

The current mortality tables used in the Kenyan market may not correctly predict the years an individual retiree may live after attaining a mandatory retirement age of 60. Today, most insurance companies in Kenya borrow tables from more developed countries like the United States of America, Sweden, the United Kingdom, and other countries. They apply scale factors during actuarial product pricing and valuation. Most policyholders in Kenya often outlive their expected death times since they are borrowed from those countries with different demographic characteristics reports Authority (2017). This effect is compounded by decreased systematic mortality risks that cut across all ages and gender.

Many pension and insurance firms, government, and annuity providers often earn periodically payable annuities in terms of pension money that their surplus process has bought, leading to the application of ruin theory methodology in actuarial life valuation defined by Blake and Hunt (2016). Systematic mortality risk, in many cases, is never easy to understand, transfer, or even try to manage; however, a life assurance company can apply financial derivatives as a financial risk hedging.

With most countries experiencing aging populations, mortality risk has been recognized as among the most common actuarial risks, especially in pension and life assurance mathematics, when formulating pension schemes and life products worldwide. Recently, governments, insurance companies, social security firms, and pension providers worldwide have adopted DC schemes to reduce the risks associated with DB schemes as defined by (Mitchell, 2020) to address reducing global SMR, which is essential in life products pricing.

Many actuaries have a long tradition of using collateral data when improving SMR estimates (Jewell, 1975a). Three main approaches used to accomplish this improvement include model life tables, mortality laws, and relational methods. During the confrontation with estimating SMR in small populations or populations where mortality data do not exist for all the age strata, collateral data offers a solution. In these situations, the pop- ulation data of the sample alone may not be enough to get reasonable estimates of more than one parameter Bozikas and Pitselis (2020). The condition means collateral data can be used when substituting for limited data. However, most SMR studies are concerned only for short periods, and most populations are always open for departures related to non-death or new group member entrance.

The SMR for people of different ages in life-insured cohorts often constitutes potential collateral data to all other ages within a similar cohort. To use this information, one must have an SMR model Party (2015). SMR Models for different populations exist in two forms, namely deterministic models and stochastic models. Each of these types has various advantages and disadvantages during modeling and projection. While deterministic models are simple to use in modeling SMR, they do not consider the market changes, hence the shortcomings as they do not represent the realities in the market (see literature review chapter). Most actuaries today use stochastic models since they present realities in the markets, thus leading to correct modeling and pricing of actuarial products beneficial for policyholders.

Incorporating data from a standard life table to compensate for the scanty deaths in the extreme ages is essential for direct data sets. Incorporating collateral data from a standard life table can solve the data paucity problem during the modeling of SMR Najafabadi (2010) and Kim and Jeon (2013). It means the higher the similarity degree between the study population (Kenya) and the standard (the U.K.), the higher the benefits of its use during modeling and actuarial products pricing.

Under collateral data, one can use Credibility theory and machine learning techniques as a solution to data paucity problems. Credibility theory is a topic in actuarial science that uses mathematical modeling to make decisions purely based on historical data Jewell (1975b). While actuaries have been using Credibility theory in their lines of duty, two types of credibility approaches are used when calculating expected risk Buhlmann and Gisler (2005).

On the other hand, the Bühlmann credibility approach looks at the different variances experienced across the population to help actuaries decide depending on the different levels of risks it assesses Ralevic´ (2020).

Deep learning offers experience rating systems new dimensions in SMR modeling by considering individual experiences regarding limited population samples collected Odhiambo, Weke, and Ngare (2021). Credibility theory approaches help use data to improve the estimation accuracy of the conventional models applied in SMR modeling by values, which are reasonable to the extent of using the historical data when forecasting the future SMR Hardy and Panjer (1998). These approaches help customize individual policyholder characteristics according to personal needs, thus increasing actuarial product satisfaction among sophisticated customers in the 21st century.

Recently, governments, insurance companies, social security firms, and pension providers, including Kenya, have adopted DC schemes to reduce the risks associated with DB schemes, as demonstrated by Opoku and Hsu (2019). In many cases, this has been established by legislation where both DC and DB plans are meant to offer employees adequate financial means, thus enabling them to retire and keep a particular standard of living lifestyle during retirement.

If actuaries can model SMR properly, it would be easy to make better projections that would enable the companies to save money, which is often lost from inaccurate estima tions when using models that do not consider the different realities of developing markets. Ultimately, this would greatly benefit these insurance firms, thus reducing financial losses that are often experienced today in the world’s economic recession, especially after the endemic Covid-19 pandemic, as discussed by Blake and Cairns (2021a).
 
1.2 Actuarial Notations, Definitions, and Terminologies

1.2.1 Actuarial Notations
lx+t: The total number of people who live at a particular age x + t, where x, t = 0, 1, 2.....
t px: The probability that an individual of age exactly x will survive to age of (x + t) for all x, t = 0, 1, 2.....
tqx: The probability that a person of age exactly x will die between age of x and age of
(x + t) for all x, t = 0, 1, 2.....
dx: The exact number of people who are aged exactly x will die between age of x and age of (x + 1) where x = 1, 2.....

1.2.2 Definitions and Terminologies

Systematic Mortality Risk: It is known as the risk of dying earlier than expected.

Force of Systematic Mortality: It is an instantaneous death rate.

Back-Testing: Back-testing is testing a predictive model on specific historical data.

Bootstrapping: Bootstrapping uses random sampling to replace accuracy measures (an error, variance, prediction error, and confidence intervals, among others) with the sample estimate.

Model Robustness: Robustness of a model means strength of a statistical model, pro- cedures, and tests, as per the specific statistical analysis conditions a study aspires to achieve.

A DB scheme: A scheme where benefits are dependant upon on the amount of money submitted to the pension firm, last salary, and years worked.

A DC scheme: It is where your employee and employer’s contributions are invested, and the returns are used to buy a pension plan with benefits at the age of retirement.

Collateral Data: Data from a standardized mortality table.

Artificial Intelligence: Artificial Intelligence is the simulation of statistical processes under human intelligence by using computer systems machines.

Machine Learning: This is the use of computer algorithms to enhance statistical experience automatically using data.

Deep Learning: This is a class of ML algorithms, which uses artificial neural networks with data representation learning and forecasting.
 
1.3 Statement of the Problem
Actuarial modeling and pricing of life products such as annuities and assurances sold in Sub-Saharan African countries like Kenya depend on the existing SMR. The accuracy of SMR depends on the models used and the availability of data during the valuation in the respective Sub-Saharan countries.

Correct modeling of SMR by life assurance companies, pension firms, and government agencies determines the prices of these life products sold to customers. During actuarial pricing of products, overestimating Systsmeatic Mortality Risk will lead to higher rates, resulting in higher costs, making them expensive and unappealing to Kenyans.

Conversely, a lower estimation of SMR will lead to underpricing of these products, making it unsustainable for the companies offering to pay benefits, leading to a higher probability of ruin and insolvency in the long run. This problem calls for correct modeling and forecasting of SMR by incorporating modern methods that will help both parties, namely the companies and policyholders, to have a win-win situation for the growth and satisfaction of the insurance and pension industry in the Sub- Saharan African countries like Kenya.

When correct estimations and valuations of life products are done using the Expected Present Value of assurances and annuities, Kenyans will buy correctly priced annuities and assurance products that serves their needs. The pricing and reserving of assurances and annuities products are done using old period-based assumptions; the liabilities under- estimation is expected because of reduced mortality levels.

Suppose actuaries can improve their estimation methods for systematic mortality risk. In that case, they can hedge unpredictable financial losses due to poor product development in Kenyan markets while increasing the number of Kenyans purchasing these products.

1.4 Objectives of the study

1.4.1 General Objective
The study’s general objective is to model Systematic Mortality Risk Stochastically Under Collateral Data and its Applications in the Valuation of Actuarial products sold in the Kenyan market.

1.4.2 Specific Objectives
The Specific objectives of the study are to:
 
1. Model Systematic Mortality Risk Stochastically under the three-factor of Age- Time-Cohort Structure for Kenyan population.

2. Determine Systematic Mortality Risk by Incorporation of Bühlmann Credibility Approach into Stochastic Mortality Models for Kenyan population.

3. Forecast Systematic Mortality Risk Under Deep Learning Technique.

4. Determine the Expected Present Value of Assurance and Annuities under the Inte- grated Bühlmann Credibility Approach for Kenyan population.

1.5 Significance of the Study
By modeling SMR Stochastically under the Age-Time-Cohort Structure for the Kenyan population, insurance companies can adjust their valuations to deal with the actuarial (life) products sold within the Kenyan market. In addition, insurance and pension players develop reasonably priced pension and life products such as annuities and assurances that work for the Kenyan population.

Incorporating the Bühlmann Credibility Approach helps model SMR for the Kenyan pop- ulation, and policyholders can adjust policies that protect Kenyans from the insurance companies that take advantage of the uninformed citizens. In addition, to ensure high levels of efficiency, the use of Deep Learning techniques when Forecasting SMR is sig- nificant to levels of precision, thus helping insurance companies keep reserves. Regulators such as IRA and RBA can check the reserves held by insurance companies to reduce the firm’s ultimate probability of ruin for higher survival chances in a competitive market.

Using correct SMR is essential for the insurance companies when they price the EPV of assurance and annuities under the Integrated Bühlmann Credibility Approach for the Kenyan population. This phenomenon will make the specific insurance company prices competitive, especially in Kenya, with over 50 companies competing in a market with less than ten percent insurance coverage.

1.6 The Structure of the Thesis
This doctoral thesis is organized into seven chapters. Chapter 1 introduces the SMR concept and credibility theory approaches and the benefits of applying the concept to mortality modeling. In addition, it outlined the importance of proper estimation and man- agement in today’s financial and actuarial world. It enabled us to look at the general and specific objectives of the research study.
 
Chapter 2 reviews the actuarial literature on SMR modeling, ranging from some primitive deterministic models and stochastic models used for single age and cohort specifications. We also look at the various gaps identified by the previous researchers and how they were fulfilled.

Chapter 3 explores the SMR Modeling Under the Age-Time-Cohort structure (3-factor systematic mortality risk framework) while considering the Kenyan population set-up. In addition, it looked at the backtesting assumptions on how these mortality models can behave on a population with data paucity using popular SMR models in actuarial research areas.

In Chapter 4, we first introduce the mathematical concept of Buhlmann’s credibility to model mortality risk and compared it to the model considering the randomness assumptions. It also looked at the Credibility-based approaches to modeling risk while assuming that randomness does not follow a normal distribution but a NIG (Normal Inverse Gaussian) statistical distribution, a heavy-tailed distribution. The concept of heavy-tailed distribution took into consideration such as shocks in the mortality upsurge. We compared the method with the standard model from the calculated MAPE and RMSE values.

In chapter 5, we forecasted SMR determined Under Deep Learning. We used a simple CBD Model to Forecast SMR Under DL technique, which is essential in calculating life assurance products sold within the Kenyan market. We note the difference between our novel model when compared to the classical models.

In chapter 6, we conducted the actuarial valuation of the products from the projected mortality rates, including life annuities and life assurance products, while comparing how different the values compared to conventionally projected mortality rates. Besides, a comparison of the classical models is determined in the Kenyan set up with the UK data to show the levels of accuracy under the CBA incorporation under the different mortality models in both countries.

In chapter 7, we finally gave general and specific conclusions and recommendations from the study while expanding room for further research for actuaries, academics, and actuarial science studies researchers who wish to continue with studies of SMR modeling.
We have references and projected complete life tables for Kenyans in the Appendix section.
 

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