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 .. 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.

2 582 ACTUARIAL MATHEMATICS CHAPTER 4 LIFE Insurances Payable at the Moment of Death ..62 TYPES OF INSURANCE .. Level Benefit Insurance .. Endowment Insurance .. Deferred Insurance .. 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 .. 112 CHAPTER 5 Formula Summary .. 116 Continuous Annuities: .. 116 Discrete annuities: .. 117 Past SOA/CAS Exam Questions.

3 118 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 .. 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.

4 Benefit Reserves Based on Truem-thly Benefit Premiums .. 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 .. 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.

5 Joint Life Status .. 243 The following is important! .. Last Survivor Status .. More Probabilities and Expectations .. Dependent Lifetime Models .. Common Shock (Non-Theoretical Version) .. 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 .. Associated Single Decrement Tables .. Basic Relationships .. Uniform Distribution Assumption for Multiple Decrements.

6 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.. 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.

7 327 Solution: .. 327 Past SOA/CAS Exam Questions: .. 32910 ACTUARIAL MATHEMATICS: CHAPTER 15 INSURANCE MODELSINCLUDING Expense Augmented Models .. More Expenses .. Asset Shares .. 341 CHAPTER 15 Formula Summary .. 343 Asset Shares .. 343 Past SOA/CAS Exam Questions: .. 343 SOLUTIONS to Past SOA-CAS Exam Problems: .. 34711 DANIEL CHAPTER 1 - MULTI-STATE TRANSITION MODELS FORACTUARIAL Introduction .. Non-homogeneous Markov Chains .. 354 CHAPTER 2 CASH FLOWS AND THEIR ACTUARIAL PRESENT VALUES359 Section Introduction .. 359 Cash Flows while in states .. 359 Cash Flows upon transitions .. 364 Actuarial Present Values .. 364 ARCH Warm-up Questions: .. 369 Solutions .. 372 Past SOA/CAS Exam Questions: .. 37512 DANIEL STUDY NOTE ON POISSON The Poisson Process .. Counting Processes.

8 Definition of the Poisson Process .. Interarrival and Waiting Time Distributions .. Further Properties of Poisson Processes .. Conditional Distribution of the Arrival Times .. Generalizations of the Poisson Process .. 393 Guo MLC, Spring 2011c Yufeng Guo66 of Nonhomogeneous Poisson Process .. Compound Poisson Process .. Conditional or Mixed Poisson Processes: Gamma-Poisson Model 398 CHAPTER 5 Formula Summary .. 401 ARCH Warm-up Problems: .. 402 Solutions: .. 403 Past SOA/CAS Exam Questions: .. 40413 Practice Exam417 Answer Key for Practice Exam .. 43014 SOLUTION TO MAY 2007 MLC449 Guo MLC, Spring 2011c Yufeng Guo77 of 674 Chapter 0 INTRODUCTIONThe origin of this study guideMost of the study guide is from the original Arch manual written by by two gifted actuaries,Nathan Hardiman and Robin Cunningham.

9 My great thanks to them for passing the Archmanual to 2006 I inherited the Arch manual when Nathan and Robin, who, each having a busyday job, no longer had time to maintain or expand the Arch 2006 to the Fall 2010, I had two study guides on MLC: the Arch and my DeeperUnderstanding manual. Managing two separate manuals by the same author on the sameexam can be confusing to readers, who have difficulty deciding which one to buy. To overcomethis confusion, I decided to discontinue the Arch manual starting from Spring 2011 andtransfer the Arch manual content to my Deeper understanding manual for to the Arch manual I finally learned LaTeX. Many of my Deeper Understandingmanuals were written in Microsoft Word with the MathType plugin. Though Word+MathType gets one started fast and easy, documents produced this way are a nightmare to to break away from Word+MathType, I tried several times to learn LaTeX butalways went nowhere.

10 After taking over the Arch manual, I had no choice but to learn LaTeX(because the Arch was written in LaTeX). After struggling several days, finally I got it andbecame proficient in sections each with a separate table of contentsThere are two sections that don t appear in the overall table of contents. These two sectionshave separate tables of contents. One section is about Actuarial Mathematics Chapter 9 Multiple Life Functions; the other about the Poisson process. These two sections are towardthe end of this book (so scroll toward the end of this document to read these two sections).Wanting for more practice problemsThe strength of this study guide is that it explains the core concepts well. It gets you up andrunning quickly toward learning the fundamentals of the core MLC of 0 The weakness of this study guide is that it doesn t have a great number of practiceproblems.