Chapter 1
This theory 1

Ⅰ.
Basic Theory 2
1.
Unit price function · 2
2.
Simple and Compound Interest · 5
3.
Current price and discount · 8
4.
Nominal Interest Rate and Nominal Discount Rate · 13
5.
History and Discount Power · 17
6.
Fixed pension (payment period = payment period per installment) · 21
7.
Fixed pension (payment period > each payment period) · 29
8.
Basic Variable Pension (Term = Payment Period) · 36
Practice Problem 1.1 · 44
Ⅱ.
General Theory 51
1.
Fixed pension (phone period 2.
Continuously Fixed Pension · 53
3.
General Variable Annuity · 58
4.
Continuously Variable Annuity · 62
5.
When the period and interest rate are unknown · 64
6.
Measuring the Rate of Return (Rate of Return) · 69
7.
Installment repayment (equal principal and interest repayment) and sinking fund · 75
8.
Bonds and Stocks · 87
Practice Problem 1.2 · 94

Chapter 2
Survival Distributions and Life Tables 101

Ⅰ.
Basic Theory 102
1.
The Concept of Probability · 102
2.
Life Table · 106
3.
Chance of survival · 113
4.
Life expectancy · 116
5.
Selection Table · 119
Practice Problem 2.1 · 122
Ⅱ.
General Theory 125
1.
Probability Theory · 125
2.
Survival Function · 136
3.
Future Survival Period of () · 139
4.
() Future Estimated Survival Period · 142
5.
Death · 144
6.
Life Table · 152
7.
Functions on Life Tables · 154
8.
Approximate life expectancy and life expectancy · 166
9.
Assumptions about the singular part · 171
10.
The Law of Death · 177
11.
Special Survival Distribution · 179
Practice Problem 2.2 · 183

Chapter 3
Life Insurance 193

Ⅰ.
Basic Theory 194
1.
Basics of Insurance Premium Calculation · 194
2.
Survival Insurance · 195
3.
Life Insurance · 197
4.
Life and Death Insurance · 207
5.
Accumulated insurance costs · 208
6.
Calculator and General Formulas · 210
Practice Problem 3.1 · 212
Ⅱ.
General Theory 215
1.
Year-end insurance premiums · 215
2.
Year-end insurance premiums · 228
3.
Insurance benefits paid immediately upon death · 232
4.
The Relationship Between Immediate Death Benefits and Year-End Benefits · 244
5.
Recursive (ignition) · 251
6.
Special Survival Distributions and Life Insurance · 252
Practice Problem 3.2 · 256

Chapter 4
Life Annuity 267

Ⅰ.
Basic Theory 268
1.
Life annuity · 268
2.
Organic Life Pension · 271
3.
Deferred Annuity · 272
4.
Guaranteed Life Annuity · 274
5.
Forborne Annuity · 275
6.
Variable annuity · 281
7.
Calculator and General Formulas · 286
8.
Life annuity in case of annual payment · 287
9.
The Relationship Between Life Insurance and Life Annuity Lump-Sum Premiums · 295
Practice Problem 4.1 · 298
Ⅱ.
General Theory 303
1.
Life annuity paid once a year · 303
2.
Life annuity paid by annual membership · 317
3.
Continuous Life Annuity · 328
4.
Calculator · 340
5.
Recursive (ignition) · 347
6.
Single-Term Payment Pension (Full Pension) and Single-Term Return Pension · 349
7.
Traditional Approximation · 359
8.
Special Survival Distributions and Life Annuities · 365
Practice Problem 4.2 · 369

Chapter 5
Net premium 379

Ⅰ.
Basic Theory 380
1.
Annual premium · 380
2.
Annual installment payment insurance premium · 389
3.
Insurance with a return of premiums · 395
Practice Problem 5.1 · 397
Ⅱ.
General Theory 403
1.
Principles of Insurance Premium Calculation · 403
2.
Annual premium · 404
3.
Annual installment payment of insurance premium (in case of equal payment) · 416
4.
Continuous Payment Insurance Premium · 419
5.
Installment payment insurance premiums · 428
6.
Unused Premium Return Insurance · 430
7.
Premiums for term life insurance and life-and-death mixed insurance with a premium refund policy · 436
8.
Special Survival Distributions and Average Premiums · 445
Practice Problem 5.2 · 449

Chapter 6
Contractor's savings amount 461

Ⅰ.
Basic Theory 462
1.
Calculation (evaluation) method of liability reserve · 462
2. IFRS 4-based liability reserves and IFRS 17-based liability reserves · 465
3. IFRS 4-based surrender value and IFRS 17-based surrender value · 469
4.
Net cost method liability reserve and contractor's reserve · 472
5.
Calculating the Future-Law Contractor's Accumulated Amount for Fixed-Rate Products · 472
6.
Interest-linked insurance policyholder savings amount · 491
Practice Problem 6.1 · 492
Ⅱ.
General Theory 496
1.
Definition of a Random Variable · 496
2.
Policyholder's accumulated amount (insurance benefit paid immediately upon death, continuous premium payment) · 498
3.
Contractor's savings (year-end insurance payment, annual premium) · 509
4.
Policyholder's accumulated amount (insurance benefit upon death, annual premium) · 517
5.
Decomposition and Recursive Formula of Net Premium · 518
6.
Contractor's accumulated amount in case of single-term contract · 525
7.
Contractor Accumulation and Differential Equations · 531
8.
Special Survival Distribution and Contractor Accumulation Amount · 533
Practice Problem 6.2 · 536

Chapter 7
Sales insurance premium 547

Ⅰ.
Basic Theory 548
1.
Calculating Sales Insurance Premiums (Ⅰ) · 548
2.
Calculating Sales Insurance Premiums (II) · 551
3.
Cash Flow Method Premium Calculation · 556
4.
Korea's Insurance Premium Calculation System · 557
Practice Problem 7.1 · 563
Ⅱ.
General Theory 565
1.
Calculating Business Insurance Premiums · 565
2.
Premium Calculation Method and Additional Risk Calculation Method · 569
3.
Diversification of Future Losses · 574
4.
Probability of Future Loss · 582
5.
Percentile Insurance Premium · 591
6.
Portfolio Percentile Premium Calculation Principle · 597
7.
Sales Insurance Premium Liability Reserve · 602
Practice Problem 7.2 · 607

Chapter 8
Yeonseong Model 617

Ⅰ.
Basic Theory 618
1.
Life Probability of United Life · 618
2.
The History of Union Life · 631
3.
Conditional probability of life · 636
4.
Simultaneous Survivors' Annuities and Life Insurance · 644
5.
Final Survivor Annuity and Annuity Insurance · 650
6.
Conditional Life Insurance · 660
7.
Survivor's Pension · 666
Practice Problem 8.1 · 674
Ⅱ.
General Theory 679
1.
Overview of Probabilistic Models · 679
2.
Joint distribution of future survival periods · 680
3.
The Joint-Life Status · 685
4.
The Last-Survivor Status · 693
5.
Expected Value of Dependent Variables · 702
6.
Common Shock Model · 708
7.
Actuarial Present Value of Life Insurance and Life Annuities · 712
8.
Actuarial Suspension Based on Mortality Assumptions · 718
9.
Conditional Life Insurance · 725
10.
Reversionary Annuities · 731
11.
Special survival distribution · 734
Practice Problem 8.2 · 736

Chapter 9
Multiple withdrawal model 745

Ⅰ.
Basic Theory 746
1.
Multiple Withdrawal Remaining Table (Multiple Withdrawal Table) · 746
2.
Multiple withdrawal probability · 747
3.
Calculating Insurance Premiums Using Multiple Withdrawal Tables · 755
4.
Multiple Attrition Models Used in Practice · 758
Practice Problem 9.1 · 768
Ⅱ.
General Theory 775
1.
Multiple Attrition Model · 775
2.
Multiple withdrawal votes · 786
3.
Basic Relationships Related to Multiple Withdrawal Rates · 789
4.
Calculating the Absolute Withdrawal Rate · 791
5.
Calculating the Multiple Withdrawal Rate · 798
6.
APV (EPV) of multiple withdrawal benefits · 804
7.
Multiple Withdrawal Model and Surrender Benefits · 808
8.
Change of Contract · 817
9.
Retirement Pension · 820
Practice Problem 9.2 · 828

Chapter 10
Multistate Model 837

Ⅰ.
Basic Theory 838
1.
Stochastic Processes · 838
2.
Discrete-Time Markov Chains · 840
3.
Multistate Model Form · 854
4.
Applications of Markov Chains · 860
5.
Discrete-Time Markov Models and Insurance Product Design · 881
Practice Problem 10.1 · 908
Ⅱ.
General Theory 917
1.
Assumptions of the Multistate Model · 917
2.
Multistate Model 2 · 920
3.
Kolmogorov's forward equations (KFE) · 939
4.
Kolmogorov's forward equations and multistate models · 945
5.
Kolmogorov's forward equations and multistate models 3 · 950
6.
Insurance Premiums in Multistate Models · 955
7.
Contractor Accumulation Amount in the Multi-State Model · 978
8.
Multistate and Multi-Departure Models · 990
9.
Multistate and Linked Models · 998
10.
Continuous-Time Markov Models and Insurance Product Design · 1015
Practice Problem 10.2 · 1049

Chapter 11
Insurance Liabilities Market Value 1059

Ⅰ.
Basic Theory 1060
1.
Overview of Market Value of Insurance Liabilities · 1060
2. IFRS 17 Liability Reserves · 1061
3.
Components of the liability reserve fund · 1063
4. Comparison of Insurance Liabilities under IFRS 4 and IFRS 17 · 1066
5.
Monthly Multiple Withdrawal Rate Calculation Overview · 1067
6.
Multiple withdrawal rate for calculating premiums and accrued amounts · 1068
7.
Multiple withdrawal rate for debt assessment purposes · 1070
8. Discount Rate Calculation Methods Based on IFRS 17 and Supervisory Regulations · 1078
9.
Deterministic Discount Rate Scenario · 1079
10.
Probabilistic Discount Rate Scenario · 1083
Practice Problem 11.1 · 1091
Ⅱ.
General Theory 1096
1.
Insurance Liability Valuation Model Overview · 1096
2.
Future Cash Flow Model · 1098
3. BEL calculation formula · 1102
4.
Future Cash Composition and Future Cash Flows · 1105
5.
Overview of Interest-Linked UL Whole Life Insurance · 1109
6.
Future Cash Flow Calculation Assumptions · 1113
7.
Insurance Liability Market Value Assumptions and Analysis Products · 1116
8.
Market Value Assessment of Insurance Liabilities for Fixed-Rate Products · 1120
9.
Death benefits and bonus accumulation amounts for interest-linked products · 1132
10.
Market Value of Insurance Liabilities for Interest Rate-Linked Products Case 1 · 1139
11.
Summary of Market Value of Insurance Liabilities · 1153
Practice Problem 11.2 · 1155

Chapter 12
Analysis of Changes in Insurance Liabilities and Profit and Loss Recognition 1163

Ⅰ.
Basic Theory 1164
1.
Meaning of Participatory Characteristics · 1164
2.
Types of Accounting Models Based on Participation Characteristics · 1165
3.
Insurance Finance Costs of Insurance Liabilities by Accounting Model · 1167
4. IFRS 17 Profit and Loss Recognition · 1174
5. Comparison of Profit and Loss Recognition under IFRS 4 and IFRS 17 · 1180
6.
Best Estimated Liabilities (BEL) Change Analysis · 1182
7.
Risk-Adjusted (RA) Variance Analysis · 1186
8.
Analysis of Changes in Contractual Margin (CSM) · 1186
9.
Changes in Insurance Liabilities and Recognition of Profit and Loss · 1189
10.
Analysis of Insurance Liabilities and Structural Analysis of Profit and Loss Recognition · 1190
Practice Problem 12.1 · 1196
Ⅱ.
General Theory 1199
1.
Overview · 1199
2.
Analysis Assumptions and Premium Calculation for Fixed-Rate Products · 1200
3.
Market Value of Insurance Liabilities for Fixed-Rate Products (Time 0, Initial Measurement) · 1203
4.
Market Value of Insurance Liabilities for Fixed-Rate Products (K1, Subsequent Measurement) · 1217
5.
Analysis of BEL fluctuations in the first year of fixed-rate products · 1224
6.
Analysis of First-Year BEL Fluctuations for Fixed-Rate Products · 1230
7.
and · 1245
8.
Analysis of the First-Year Contractual Margin (CSM) Fluctuations in Fixed-Rate Products · 1248
9.
First-Year Financial Statements for Fixed-Rate Products · 1251
10.
Second-Year Fluctuation Analysis of Fixed-Rate Products · 1258
Practice Problem 12.2 · 1265

supplement

1269

Appendix 1: Frequently Used Mathematical Formulas · 1270
Appendix 2 A.
Critical survival function and singular part assumption · 1279B.
Differentiation of Actuarial Functions · 1285
Appendix 3 Table 1287
Appendix 4: Practice Problem Solutions · 1319
Appendix 5 Insurance Claims Symbol Index · 1324

After nearly 15 years of intense debate following the introduction of Phase 1 International Financial Reporting Standards (IFRS), IFRS 17 finally took effect in 2023. The changes in actuarial science brought about by the introduction of IFRS 17 represent a fundamental paradigm shift, unlike any seen over the past hundreds of years.
The theory related to the valuation of insurance liabilities and recognition of profit and loss presents completely different content from the actuarial theory we have studied so far.
Because such a major change is already underway, there was a pressing need to revise this book, which had previously introduced the latest domestic and international theories, to reflect the new content of the new era. Now, that need has been met.

Over the past decade, the author has served as the Chairman of the Financial Supervisory Service's IFRS 17 Introduction Committee and, in his sole individual capacity, as a member of the Financial Services Commission's Insurance Capital Soundness Advancement Promotion Team. This has allowed him to directly and deeply engage in the key contents and changes of IFRS 17, the supplementation and revision of the domestic actuarial system in accordance with the introduction of IFRS 17, and the proposal and regulation of the new solvency system. This has given him the opportunity to research, hold meetings, and discuss matters related to IFRS 17 and the new solvency system from beginning to end.
The author has published a paper proposing the introduction of CFP in Korea (2005) and also published a paper proposing the introduction of a new solvency system (2015), which is currently being implemented.
Over the past 15 years, the author has published more than 90% of domestic papers related to IFRS 17, either alone, as a co-author, or with graduate students from Hanyang University.
These presentations and numerous discussions and regulatory work in the committee formed the most important foundation for publishing this book containing the contents of IFRS 17.

Since IFRS 17 is a system that will be first implemented in 2023, there are no domestic or foreign textbooks explaining the market value of insurance liabilities and profit and loss recognition under IFRS 17.
Moreover, because the US has not adopted IFRS 17, even the US textbooks that have been the primary references do not explain IFRS 17. The first publication on IFRS 17 liability valuation and profit and loss recognition requires a truly innovative approach, as it differs from existing actuarial theory and paradigms and requires a new theoretical framework. Because no other IFRS 17-related papers or textbooks exist outside of the author's published papers, the author had to creatively define the terms, formulas, and symbols necessary to implement the theoretical framework, from beginning to end.
Implementing this theoretical framework is a challenging task, requiring creativity and innovation from beginning to end. A systematic understanding of the IFRS 17 standard, Korean SAP, PAP (solvency provision accounting), and past and current domestic actuarial regulations is essential to consistently and clearly define new terms and symbols. Therefore, writing IFRS 17-related textbooks demands a significant level of knowledge, creativity, and courage. Furthermore, beyond a high level of knowledge and courage, the task of systematically explaining the mark-to-market valuation of insurance liabilities requires understanding numerous new product-related content and revised regulations, and developing examples in an easily understandable and logical manner. This, too, is a challenging and creative endeavor.
The new symbols and examples appearing in this book are the result of the author's creativity and extensive trial and error.
It is hoped that the newly defined terms, symbols, and examples for implementing this new theoretical system will evolve in a progressive manner in the future.
From this perspective, this book is the world's first textbook to theoretically introduce and explain IFRS 17, and we hope it will serve as a useful guide for studying IFRS 17-related topics in the future.
The various definitions, formulas, symbols, etc. used by the authors to implement the new theoretical system are the authors' own creations without any external references, and therefore any errors in the content are the sole responsibility of the authors.

The major changes in this revised edition can be divided into revisions to existing content and additions of new content.
The most important thing about the revision of the existing contents is that the net reserve liability reserve introduced in Chapter 6 is no longer a Korean reserve liability reserve in the IFRS 17 era, so the term “net reserve liability reserve” was completely deleted from the entire textbook and the related contents were revised.
Currently, in Korea, the term "sunburden reserve" has been replaced with the term "contractor's reserve," which is the basic data for calculating surrender value, and is used only when calculating surrender value.
In the past, net insurance reserves were a core component of actuarial theory. However, now, their importance has significantly diminished, as they are used solely for calculating surrender value. (In countries where they aren't even used for this purpose, their use has become even more limited.) In the IFRS 17 era, the key component is not the policyholder reserve (net insurance reserves), but the market value of insurance liabilities.

The new additions in this revised edition are as follows:
First, many parts were revised by changing the term ‘sunbo-sik liability reserve fund’ to ‘contractor’s reserve amount.’
Second, the authors systematized the multiple withdrawal theory used domestically and introduced it in Chapter 7.
We believe this will be of great help to readers who have had difficulty understanding the pricing of domestic insurance products.
Moreover, since this multiple withdrawal theory is used in the market value valuation of insurance liabilities, it is necessary to explain the IFRS 17 insurance liability valuation.
Third, a new chapter, Chapter 11, entitled Market Value of Insurance Liabilities, was created.
In Chapter 11, formulas for calculating monthly retainers, payers, deceased, and cancelers using multiple withdrawal theory were presented, and a structure for calculating monthly discount rate scenarios using financial engineering theory was explained.
Based on this, a model and new symbols for market valuation of insurance liabilities of fixed-rate products and interest-linked products were systematically presented, and the process of market valuation of insurance liabilities was explained in detail through examples.
These are the parts that required the most creativity from the authors.
Fourth, a new chapter, Chapter 12, on analysis of changes in insurance liabilities and recognition of profit and loss, was created.
In Chapter 12, a three-year term life and death hybrid insurance policy was used to illustrate all future cash flows for the market valuation of insurance liabilities. Furthermore, IFRS 17 insurance liabilities and financial statements were explained by linking the changes in insurance liabilities and profit and loss recognition through market valuation and fluctuation analysis of liabilities.
In explaining Chapter 12, the authors created many new terms, formulas, and symbols that implement the theoretical system.

In this revised edition, the term "soon-in-kind liability reserve" has been removed and Chapters 11 and 12, which introduce the market value of insurance liabilities, have been added. However, it is expected that there will be much more to supplement and add in the future.
While reading a collection of essays by poet Jeong Ho-seung, I thought that the essay titled "Don't miss the scent of wine while drinking makgeolli" well expressed the current situation related to actuarial science textbooks and actuarial science.
The content of the prose is that if you miss the past (the scent of wine) more than today, you will eventually lose today. However, if you look at the 4th edition of the latest insurance mathematics, the times have completely changed to IFRS 17, but in many places in the textbook, the term for the net insurance liability reserve has been changed to the policyholder's reserve, but it remains the same.
I believe that many experts, including the author, are unable to abandon the policyholder reserve fund (the policyholder reserve fund) because it was so important and familiar in the past and continues to be attached to the policyholder reserve fund (like the wine scent of the past).
I think the current 4th edition of the Actuarial Science textbook is in that state.
Even in the latest revised edition of the insurance law textbook, the policyholder's accumulated amount, which has no use other than the simple calculation of the surrender refund, should be introduced at an appropriate level in Chapter 6 and boldly discarded in the remaining chapters.
The policyholder's reserve (net insurance reserve) is no longer the basis of actuarial science.
In the lectures and exams of insurance law, it is now necessary to boldly abandon the use of policyholder savings, leaving only the weight for simple calculation of surrender value.
The empty space left behind must be filled with information related to the market value assessment of insurance liabilities.
The theory of market valuation of insurance liabilities, the related profit and loss recognition, profitability analysis based on this, risk management, and the new solvency system should be the core contents of modern actuarial science.
In the next revised edition, we plan to boldly delete the policyholder's accumulated amount, which has lost its intended use after Chapter 7, and instead add new chapters related to the market value evaluation of insurance liabilities.

In publishing this revised edition, I received much assistance from Hanyang University graduate school professors and alumni.
First of all, I would like to express my gratitude to Professors Choi Yang-ho, Shim Hyeon-woo, and Jang Cheol of the Graduate School for their great help as fellow professors.
I would also like to thank the authors and Professor Cho Seok-hee, Director Lee Yun-gu, Director Park Jong-hyeon, Director Oh Su-yeon, Director Oh Chang-yeong, Dr. Lee Seong-ho, Dr. Lee Chang-wook, Director Moon Seong-cheol, Director Park Su-won, Director Jeon Yong-seok, Director Kim Gyu-yong, Director Lee Jae-deuk, Director Kang Myeong-su, Director Kang Beom-su, Director Lee Seong-yong, Director Lee Haeng-geun, Dr. Byeon Jae-woong, Dr. Kim Se-jung, Dr. Jeong Hee-moon, and Dr. Park Young-jun who published related papers in the early stages of IFRS adoption.
I would also like to thank Manager Seo Yu-nam, Dr. Park Gyu-seo, Manager Kang Won-jae, Team Leader Park Jong-gak, Manager Baek Jin-wook, Deputy Manager Lim Hyeon-su, Deputy Manager Kim Seong-su, Managing Director Eun Jae-gyeong, Deputy Manager Lee Hyeon-jeong, Manager Kim Su-eun, Managing Director Jeong Jong-guk, Dr. Song Sang-wook, Dr. Choi Pan-gyun, Dr. Kim Dong-guk, Deputy Manager Kim Hyeong-jo, Manager Lee Yong-seung, and Manager Park Ho-gyun, who presented related papers during the mid-to-late stage of IFRS introduction.
I would like to express my gratitude to these individuals, as their knowledge gained through co-authoring related papers over the past 15 years has been a crucial foundation for the publication of this book.


I would also like to express my gratitude to Vice President Oh Hong-seok, CEO Kim Jeong-nam, Manager Kim Hyeong-jo, Dr. Yoo In-hyeon, Consultant Kim Ji-woon, Supervisor Lee Eun-joong, Manager Koo Bon-seung, Manager Kwak Jae-gu, Manager Kim Yong-kwon, Managing Director Lee Hong-yong, Manager Eom Yun-gyeong, and Manager Kim Ja-gyeong, who provided invaluable assistance in my research at Hanyang University Graduate School and in the industry.
I would also like to express my sincere gratitude to the many former senior actuaries, including Chairman Byun In-cheol of the Life Insurance Senior Actuaries Association, who have co-hosted numerous seminars on the new system twice a year with Hanyang University Graduate School for over 15 years.
I would also like to thank Chairman Kang Young-gu, Director Oh Seung-cheol, Managing Director Lee Jong-hoon, Vice President Lee Jun-seop, CEO Yoo Hyeok, Auditor Ahn Hyeong-jun, Auditor Kim Jeong-cheol, Committee Member Park Sang-rae, Managing Director Chae Hee-seong, Director Mok Jin-yeong, Director Jeong Geun-hwan, Senior Director Park Chan-jae, CEO Park Gyu-hong, Chairman Kim Yeon-su, Dr. Yoo Hong-rim, Dr. Shin Dong-wook, and Committee Member Lee Sang-il for providing a lot of materials and helping with proofreading and encouraging the writing process related to the revised edition.
I also extend my deepest gratitude to the authors who proposed the K-ICS as a paper in 2015, and to the Financial Supervisory Service's Risk System Office staff, including Director Park Jin-hae and Director Lee Tae-gi, who held seminars together and engaged in meetings, deliberations, and discussions over a long period of time (2015-2022) to develop supervisory regulations.
Lastly, I would like to thank Directors Jo Seong-ho and Kim Seon-min of Park Young-sa for their hard work in helping me successfully complete this challenging manuscript.

2023.
8. 30.

author