Deeper Understanding, Faster Calculation: MLC, …

1 Deeper understanding , Faster calculation : MLC, Spring 2011 yufeng GuoNovember 28, 20101 of 674 Contents0 INTRODUCTION9 The origin of this study guide ..9 Two sections each with a separate table of contents ..9 Wanting for more practice problems ..9 MLC syllabus change in Spring 2012 ..10 About yufeng Guo ..101 ACTUARIAL MATHEMATICS: CHAPTER 3 SURVIVAL DISTRIBU-TIONS AND LIFE The Survival Function .. Time-until-Death for a Person Agex.. Curtate-Future-Lifetime .. Force of Mortality .. Life Tables.

2 Recursion Formulas .. Assumptions for Fractional Ages .. Some Analytical Laws of Mortality ..33 Modified DeMoivre s Law .. Select and Ultimate Tables ..35 Conclusion ..38 CHAPTER 3 Formula Summary ..39 Past SOA/CAS Exam Questions: ..43 Problems from Pre-2000 SOA-CAS exams ..56 Solutions to Chapter 3 ..582 ACTUARIAL MATHEMATICS CHAPTER 4 LIFE Insurances Payable at the Moment of Death ..62 TYPES OF INSURANCE .. Level Benefit Insurance .. Endowment Insurance .. Deferred Insurance.

3 Varying Benefit Insurance .. Insurances Payable at the End of the Year of Death .. Relationships between Insurances Payable at the Moment of death and theEnd of the Year of Death ..79 CHAPTER 4 Formula Summary ..8133 of 0 Past SOA/CAS Exam Questions: ..83 Problems from Pre-2000 SOA-CAS exams ..93 Solutions ..953 ACTUARIAL MATHEMATICS: CHAPTER 5 LIFE Continuous Life Annuities ..97 The most important equation so far(!) .. Discrete Life Annuities .. Life Annuities withm-thly Payments.

4 112 CHAPTER 5 Formula Summary .. 116 Continuous Annuities: .. 116 Discrete annuities: .. 117 Past SOA/CAS Exam Questions: .. 118 Problems from Pre-2000 SOA-CAS exams .. 131 Solutions to Pre-2000 Problems: Chapter 5 .. 1334 ACTUARIAL MATHEMATICS: CHAPTER 6 BENEFIT PREMIUMS Fully Continuous Premiums .. Fully Discrete Premiums .. Truem-thly Payment Premiums .. 147 CHAPTER 6 Formula Summary .. 151 Past SOA/CAS Exam Questions: .. 152 Problems from Pre-2000 SOA-CAS exams.

5 168 Solutions to Pre-2000 Exam Questions: Chapter 6 .. 1705 ACTUARIAL MATHEMATICS: CHAPTER 7 BENEFIT RESERVES Fully Continuous Benefit Reserves .. Other Methods for Calculating the Benefit Reserve .. 1771) Prospective Formula .. 1772) Retrospective Formula .. 1773) Premium Difference Formula .. 1804) Paid-Up Insurance Formula .. 1805) Other Reserve Formulas .. Fully Discrete Reserves .. Benefit Reserves on a Semi-Continuous Basis .. Benefit Reserves Based on Truem-thly Benefit Premiums.

6 187 CHAPTER 7 Formula Summary .. 188 Continuous Reserve Formulas: .. 188 Discrete Reserves: .. 188 Past SOA/CAS Exam Questions: .. 190 Problems from Pre-2000 SOA-CAS exams .. 194 Solutions to Pre-2000 Questions: Chapter 7 .. 196 Guo MLC, Spring 2011c yufeng Guo44 of 06 ACTUARIAL MATHEMATICS: CHAPTER 8 ANALYSIS OF Benefit Reserves for General Insurances .. Recursion Relations for Fully Discrete Benefit Reserves .. Benefit Reserves at Fractional Durations .. The Hattendorf Theorem.

7 208 CHAPTER 8 Formula Summary .. 212 Chapter 8 More Formulas .. 213 Past SOA/CAS Exam Questions: .. 214 Problems from Pre-2000 SOA-CAS exams .. 234 Solutions to Pre-2000 Problems: Chapter 8 .. 2377 ACTUARIAL MATHEMATICS: CHAPTER 9 MULTIPLE LIFE Joint Distributions of Future Lifetimes .. Joint Life Status .. 243 The following is important! .. Last Survivor Status .. More Probabilities and Expectations .. Dependent Lifetime Models .. Common Shock (Non-Theoretical Version).

8 Insurance and Annuity Benefits .. Survival Statuses .. Special Two-Life Annuities .. Reversionary Annuities .. Simple Contingent Functions .. 265 CHAPTER 9 Formula Summary .. 268 Past SOA/CAS Exam Questions: .. 270 Problems from Pre-2000 SOA-CAS exams .. 283 Solutions to Pre-2000 Exam Questions: Chapter 9 .. 2858 ACTUARIAL MATHEMATICS: CHAPTER 10 MULTIPLE Two Random Variables .. 287 Probability density functions: .. Random Survivorship Group .. Deterministic Survivorship Group.

9 Associated Single Decrement Tables .. Basic Relationships .. Uniform Distribution Assumption for Multiple Decrements .. Construction of a Multiple Decrement Table .. 300 CASE I : Two decrements that are uniformly distributed in the asso-ciated single decrement table .. 300 CASE II : Three decrements that are uniformly distributed in the as-sociated single decrement table .. 300 Guo MLC, Spring 2011c yufeng Guo55 of 0 CASE III : Multiple decrements some are uniformly distributed inthe associated single decrement table and some are not.

10 300 CHAPTER 10 Formula Summary .. 305 Past SOA/CAS Exam Questions: .. 307 Problems from Pre-2000 SOA-CAS Exams .. 317 Solutions to Pre-2000 Problems: Chapter 10 .. 3209 ACTUARIAL MATHEMATICS: CHAPTER 11 APPLICATIONS OF MUL-TIPLE DECREMENT Actuarial Present Values and Their Numerical Estimation .. Benefit Premiums and Reserves .. 324 CHAPTER 11 Formula Summary .. 327 ARCH Sample Exam Problem .. 327 Solution: .. 327 Past SOA/CAS Exam Questions: .. 32910 ACTUARIAL MATHEMATICS: CHAPTER 15 INSURANCE MODELSINCLUDING Expense Augmented Models.