Deeper Understanding, Faster Calculation: MLC, Fall 2011

Transcription

1 Deeper Understanding, Faster Calculation: MLC, Fall 211 Yufeng Guo June 8, 211

2 actuary88.com Chapter Guo MLC: Fall 211 c Yufeng Guo 2

3 Contents INTRODUCTION 9 The origin of this study guide Two sections each with a separate table of contents Wanting for more practice problems MLC syllabus change in Spring About Yufeng Guo ACTUARIAL MATHEMATICS: CHAPTER 3 SURVIVAL DISTRIBU- TIONS AND LIFE TABLES The Survival Function Time-until-Death for a Person Age x Curtate-Future-Lifetime Force of Mortality Life Tables Recursion Formulas Assumptions for Fractional Ages Some Analytical Laws of Mortality Modified DeMoivre s Law Select and Ultimate Tables Conclusion CHAPTER 3 Formula Summary Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS exams Solutions to Chapter ACTUARIAL MATHEMATICS CHAPTER 4 LIFE INSURANCE Insurances Payable at the Moment of Death 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 the End of the Year of Death CHAPTER 4 Formula Summary

4 actuary88.com Chapter Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS exams Solutions ACTUARIAL MATHEMATICS: CHAPTER 5 LIFE ANNUITIES Continuous Life Annuities The most important equation so far(!) Discrete Life Annuities Life Annuities with m-thly Payments CHAPTER 5 Formula Summary Continuous Annuities: Discrete annuities: Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS exams Solutions to Pre-2 Problems: Chapter ACTUARIAL MATHEMATICS: CHAPTER 6 BENEFIT PREMIUMS Fully Continuous Premiums Fully Discrete Premiums True m-thly Payment Premiums CHAPTER 6 Formula Summary Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS exams Solutions to Pre-2 Exam Questions: Chapter ACTUARIAL MATHEMATICS: CHAPTER 7 BENEFIT RESERVES Fully Continuous Benefit Reserves Other Methods for Calculating the Benefit Reserve ) Prospective Formula ) Retrospective Formula ) Premium Difference Formula ) Paid-Up Insurance Formula ) Other Reserve Formulas Fully Discrete Reserves Benefit Reserves on a Semi-Continuous Basis Benefit Reserves Based on True m-thly Benefit Premiums CHAPTER 7 Formula Summary Continuous Reserve Formulas: Discrete Reserves: Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS exams Solutions to Pre-2 Questions: Chapter Guo MLC: Fall 211 c Yufeng Guo 4

5 actuary88.com Chapter 6 ACTUARIAL MATHEMATICS: CHAPTER 8 ANALYSIS OF BENEFIT RESERVES Benefit Reserves for General Insurances Recursion Relations for Fully Discrete Benefit Reserves Benefit Reserves at Fractional Durations The Hattendorf Theorem CHAPTER 8 Formula Summary Chapter 8 More Formulas Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS exams Solutions to Pre-2 Problems: Chapter ACTUARIAL MATHEMATICS: CHAPTER 9 MULTIPLE LIFE FUNC- TIONS Joint Distributions of Future Lifetimes Joint Life Status 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 CHAPTER 9 Formula Summary Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS exams Solutions to Pre-2 Exam Questions: Chapter ACTUARIAL MATHEMATICS: CHAPTER 1 MULTIPLE DECREMENT MODELS Two Random Variables Probability density functions: Random Survivorship Group Deterministic Survivorship Group Associated Single Decrement Tables Basic Relationships Uniform Distribution Assumption for Multiple Decrements Construction of a Multiple Decrement Table CASE I : Two decrements that are uniformly distributed in the associated single decrement table CASE II : Three decrements that are uniformly distributed in the associated single decrement table Guo MLC: Fall 211 c Yufeng Guo 5

6 actuary88.com Chapter CASE III : Multiple decrements some are uniformly distributed in the associated single decrement table and some are not CHAPTER 1 Formula Summary Past SOA/CAS Exam Questions: Problems from Pre-2 SOA-CAS Exams Solutions to Pre-2 Problems: Chapter ACTUARIAL MATHEMATICS: CHAPTER 11 APPLICATIONS OF MUL- TIPLE DECREMENT THEORY Actuarial Present Values and Their Numerical Estimation Benefit Premiums and Reserves CHAPTER 11 Formula Summary ARCH Sample Exam Problem Solution: Past SOA/CAS Exam Questions: ACTUARIAL MATHEMATICS: CHAPTER 15 INSURANCE MODELS INCLUDING EXPENSES Expense Augmented Models More Expenses Asset Shares CHAPTER 15 Formula Summary Asset Shares Past SOA/CAS Exam Questions: SOLUTIONS to Past SOA-CAS Exam Problems: DANIEL CHAPTER 1 - MULTI-STATE TRANSITION MODELS FOR ACTUARIAL APPLICATIONS Introduction Non-homogeneous Markov Chains CHAPTER 2 CASH FLOWS AND THEIR ACTUARIAL PRESENT VALUES359 Section 2.1 Introduction Cash Flows while in states Cash Flows upon transitions Actuarial Present Values ARCH Warm-up Questions: Solutions Past SOA/CAS Exam Questions: DANIEL STUDY NOTE ON POISSON PROCESS The Poisson Process Counting Processes 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 Guo MLC: Fall 211 c Yufeng Guo 6

7 actuary88.com Chapter Nonhomogeneous Poisson Process Compound Poisson Process Conditional or Mixed Poisson Processes: Gamma-Poisson Model 398 CHAPTER 5 Formula Summary ARCH Warm-up Problems: Solutions: Past SOA/CAS Exam Questions: Practice Exam 417 Answer Key for Practice Exam SOLUTION TO MAY 27 MLC 449 Guo MLC: Fall 211 c Yufeng Guo 7

8 actuary88.com Chapter -1 Guo MLC: Fall 211 c Yufeng Guo 8

9 Chapter INTRODUCTION The origin of this study guide Most of the study guide is from the original Arch manual written by by two gifted actuaries, Nathan Hardiman and Robin Cunningham. My great thanks to them for passing the Arch manual to me. In 26 I inherited the Arch manual when Nathan and Robin, who, each having a busy day job, no longer had time to maintain or expand the Arch manual. From 26 to the Fall 21, I had two study guides on MLC: the Arch and my Deeper Understanding manual. Managing two separate manuals by the same author on the same exam can be confusing to readers, who have difficulty deciding which one to buy. To overcome this confusion, I decided to discontinue the Arch manual starting from Spring 211 and transfer the Arch manual content to my Deeper Understanding manual for MLC. Thanks to the Arch manual I finally learned LaTeX. Many of my Deeper Understanding manuals 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 maintain. Wanting to break away from Word+MathType, I tried several times to learn LaTeX but always went nowhere. 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 and became proficient in LaTeX. Two sections each with a separate table of contents There are two sections that don t appear in the overall table of contents. These two sections have 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 toward the end of this book (so scroll toward the end of this document to read these two sections). Wanting for more practice problems The strength of this study guide is that it explains the core concepts well. It gets you up and running quickly toward learning the fundamentals of the core MLC knowledge. 9

10 actuary88.com Chapter The weakness of this study guide is that it doesn t have a great number of practice problems. If you use this book to learn the core concepts but need additional practice problems, you can download the past course 3 and Exam M problems from the SOA website http: // and the CAS website MLC syllabus change in Spring 212 SOA already announced that the MLC syllabus is to be changed in Spring 212. SOA stated: Changes are coming to the learning objectives and required readings for Exam MLC effective with the Spring 212 exam administration. What is changing and when? The learning objectives for Exam MLC are being revised to reflect current theory and practice. A draft of the revised learning objectives has been prepared to give advance notice for this change. Information regarding textbooks and study materials for the exam will be released by the end of the second quarter of 211. The first Exam MLC administration using the new learning objectives will be held in the Spring of 212. Why Spring 212? Currently, the life contingencies topic is taught as a twosemester fall/spring course sequence at most universities. As such the end of the spring semester is an ideal time to offer the first exam administration with the revised syllabus. A textbook covering all the learning objectives is not currently available. However, it is anticipated that a suitable textbook will be available in early 211. About Yufeng Guo Yufeng Guo was born in central China. After receiving his Bachelor s degree in physics at Zhengzhou University, he attended Beijing Law School and received his Masters of law. He was an attorney and law school lecturer in China before immigrating to the United States. He received his Masters of accounting at Indiana University. He has pursued a life actuarial career and passed exams 1, 2, 3, 4, 5, 6, and 7 in rapid succession after discovering a successful study strategy. Mr. Guo s exam records are as follows: Fall 22 Passed Course 1 Spring 23 Passed Course 2,3 Fall 23 Passed Course 4 Spring 24 Passed Course 6 Fall 24 Passed Course 5 Spring 25 Passed Course 7 Study guides by Mr. Guo: Guo MLC: Fall 211 c Yufeng Guo 1

11 actuary88.com Chapter Deeper Understanding, Faster Calc: P Deeper Understanding, Faster Calc: FM Deeper Understanding, Faster Calc: MLC Deeper Understanding, Faster Calc: MFE Deeper Understanding, Faster Calc: C Guo s Solution to Derivatives Markets: Exam FM Guo s Solution to Derivatives Markets: Exam MFE In addition, Mr. Guo teaches online classes for Exam P, FM, MFE, and MLC. For details see If you have questions, you can Mr. Guo at yufeng guo@msn.com. All materials contained herein are copyrighted by Yufeng Guo. This PDF study manual is for individual use for the sole purpose of taking Exam MLC. Reselling this manual is prohibited. Redistribution of this manual in any form is prohibited. Guo MLC: Fall 211 c Yufeng Guo 11

12 actuary88.com Chapter Guo MLC: Fall 211 c Yufeng Guo 12

13 Chapter 1 ACTUARIAL MATHEMATICS: CHAPTER 3 SURVIVAL DISTRIBUTIONS AND LIFE TABLES Option A reference: Actuarial Mathematics Chapter 3 Option B reference: Models for Quantifying Risk Chapter 5,6 This text forms the heart and soul of the exam syllabus. The basic principles of life insurance (and annuities) are explained throughout the book. You need to have a solid understanding of this material in order to pass the exam. However, you do not need to understand the majority of the underlying theory in this text. The key points that a student must learn from this text are: KEYPOINTS: 1. Notation much of this notation is new. While it can be confusing at first, there is some logic to it. It will help you to remember and understand the many symbols if you regularly translate the notation into words as you read. 2. Basic ideas for example, chapter four introduces a variety of types of insurance. You will want to make sure you have an understanding of these different products and their benefit designs. Another key point is that there are many parallels. Again in Chapter 4, the first part of the chapter considers products which pay a benefit immediately upon death. The second part of the chapter considers the same products except that the benefit is paid at the end of the year in which death occurs. It is helpful to realize that you are really learning only one set of products, with a couple of benefit options, rather than two sets of products. These parallels run throughout the text (e.g., continuous vs curtate functions). 13

14 3. Learn key formulas there is no substitute for being able to recall the formula for, say, a net level premium reserve for term insurance. If you can do this for most of the formulas, you will be ready to answer questions quickly. This manual has tools to help you learn these formulas, so don t feel overwhelmed! To the text!!! Guo MLC: Fall 211 c Yufeng Guo 14

15 Chapter 3 is all about notation, definitions, and a few basic ideas that are essential to life contingencies. If you can make yourself comfortable with the symbols and methods of Chapter 3, the rest of Actuarial Mathematics will be easier to absorb The Survival Function Option A reference: Actuarial Mathematics Chapter Option B reference: Models for Quantifying Risk Chapter 5.1 Consider a newborn (i.e. a person whose attained age = ). Definitions X = newborn s age at death You can also think of X as the future lifetime of a newborn. Define F (x) = Pr (X x), where x. Read as the probability that death will occur prior to (or at) age x. In statistics, F (x) is the cumulative distribution function for the future lifetime of a newborn. If y > x, it is always true that F (y) > F (x). This makes sense. For a newborn, F (98), the probability of dying before age 98, is greater than F (94), the probability of dying before age 94. Define s(x) = 1 F (X) = 1 Pr(X x). The function s(x) is a survival function. Read it as the probability that death does not occur by age x or the probability of attaining (surviving to) age x. Pr(x < X z) = probability that a newborn dies between ages x and z = F (z) F (x) = [1 s(z)] [1 s(x)] = s(x) s(z) F(x) s(x) the pdf y=f(x) O x z The figure shows the probability distribution function f(x) for death at age x. For any value of x, F (x) is equal to the area under the curve y = f(x) and to the left of x. Similarly s(x) is equal to the area under the curve and to the right of x. By the way, you may have noticed that in our discussion, we dropped the subscript X in F X (x)... you can ignore it. I don t know if the authors realize it but they are being a little Guo MLC: Fall 211 c Yufeng Guo 15

16 inconsistent in their treatment of F and s! If two different random variables, say X and Y, referred to the future lifetimes of two different newborns, then you would need to keep the F and s straight for each kid. That s all the subscript is indicating Time-until-Death for a Person Age x Option A reference: Actuarial Mathematics Chapter Option B reference: Models for Quantifying Risk Chapter 5.3 Newborns are great, but if our pension and insurance companies are going to make money we need to be able to deal with people who are older than. So... Consider a person with attained age = x. The simple F (x) and s(x) functions no longer work, since we are now dealing with a person who has already survived to age x. We are facing a conditional probability situation. Pr(x < X z X > x) = probability that person living at age x will die between ages x and z = the probability that an x-year-old will die before turning z = = [F (z) F (x)] [1 F (x)] [s(x) s(z)] [s(x)] Why is this a conditional probability? Because it is the probability that a newborn will die before age z given that the newborn survives to age x. EXAMPLE: 1. Write two expressions (one with F only and one with s only) for the probability that a newborn dies between 17 and 4, assuming the newborn dies between 1 and Interpret the following expression in English (or the language of your choice!). S(2) S(35) 1 S(8) SOLUTION: 1. F (4) F (17) F (4) F (1) or s(17) s(4) s(1) s(4) Guo MLC: Fall 211 c Yufeng Guo 16

17 2. The probability of death between ages 2 and 35, given that the newborn will not attain age 8. Now, let the symbol (x) represent a person age x and let T (x) be the future lifetime of a person age x. (So T (25) is the future lifetime of (25), a twenty-five-year-old.) Two basic probability functions exist regarding T (x): tq x = probability that person age x will die within t years = Pr[T (x) t] where t tp x = probability that person age x will survive at least t years = Pr[T (x) > t] where t q p t x t x x x+t Age In the figure, t q x is the probability that (x) s death will occur in the age-interval (x, x+t), and tp x is the probability that (x) s death will occur in the age interval (x + t, ω). (ω represents the oldest possible age to which a person may survive.) Useful notes: t p is just s(t). If t = 1, the convention is to drop the symbol 1, leaving us with either p x or q x. Remember, these are the two basic functions. The formulas that follow are simply take-offs on t p x or t q x which you will learn with practice. The symbol t uq x represents the probability that (x) (that is, a person age x) survives at least t more years, but dies before reaching age x + t + u. This is equal to each of the following expressions, each of which you want to be able to put into words: Pr[t < T (x) t + u] t+uq x t q x tp x t+u p x (As with q x and p x, if u = 1, we drop it, leaving t q x, the probability that (x) will survive t years but not t + 1 years.) Guo MLC: Fall 211 c Yufeng Guo 17

18 Useful formulas: tp x = x+t p s(x + t) = xp s(x) tq x = 1 s(x + t) s(x) t uq x = s(x + t) s(x + t + u) s(x) s(x + t) s(x + t) s(x + t + u) = s(x) s(x + t) = t p x u q x+t This last equation makes sense. It says The probability of (x) dying between t and t + u years from now ( t u q x ) is equal to the probability that (x) will first survive t years ( t p x ) and then die within u years ( u q x+t ). If you don t remember anything else from the above, remember the following! tp x = s(x + t) s(x) CONCEPT REVIEW: 1. Write the symbol for the probability that (52) lives to at least age Write the symbol for the probability that a person age 74 dies before age Write the symbol for probability that (33) dies before age Write the symbol for probability that a person age 43 lives to age 5, but doesn t survive to age Write 5 6 q x in terms of F and then in terms of p. SOLUTIONS: p q q q q x = s(x + 5) s(x + 11) s(x) = F (x + 11) F (x + 5) 1 F (x) = 5 p x (1 6 p x+5 ) or = 5 p x 11 p x. To help memorize symbols, practice translating symbols into words and express words in symbols. You can also make flash cards and quiz yourself. Guo MLC: Fall 211 c Yufeng Guo 18

19 3.2.3 Curtate-Future-Lifetime Option A reference: Actuarial Mathematics Chapter Option B reference: Models for Quantifying Risk Chapter Suppose a person born on Jan 1, 19 died on Sept 3, 199. How old was he at death? The true age was about 9.75 years old. The curtate age was 9. To find the curtate age, first find the true age. Next, throw away all the decimals and keep the integer. If there s no decimal, then the curtate age is equal to the continuous (true) age. For example, if T (x) = 9, then K(x) = 9. (This book and others use Curtate and Discrete interchangeably.) Previously, we defined T (x) to be the future lifetime of (x). This is a continuous function. Now we define K(x) = curtate future lifetime of (x) = greatest integer in T (x) = number of future years completed by (x) prior to death = number of future birthdays (x) will have the opportunity to celebrate A couple of formulas apply: Pr(K(x) = k) = Pr(k T (x) < k + 1) = Pr(k < T (x) k + 1) = k p x k+1 p x = k p x q x+k = k q x (Remember, the 1 in front of q has been dropped.) EXAMPLE: If s(x) = 1 x 1 for every x, what is the probability that K = 19 for (18)? SOLUTION: Pr(K(18) = 19) = 19 q 18 = s(37) s(38) s(18) = = Guo MLC: Fall 211 c Yufeng Guo 19

20 3.2.4 Force of Mortality Option A reference: Actuarial Mathematics Chapter Option B reference: Models for Quantifying Risk Chapter The force of mortality can be thought of as the probability of death at a particular instant given survival up to that time. This is an instantaneous measure, rather than an interval measure. There is good bit of theory in this section, but the most important items are the following formulas and the table of relationships. µ(x) = f(x) 1 F (x) = s (x) s(x) (3.2.13) It is very important to know the relationships and requirements given in Table These will probably be tested on the exam. Below is a summary of the useful information in this table. Each row shows 4 ways to express the function in the left column. F (x) s(x) f(x) µ(x) F (x) F (x) 1 s(x) x f(u) du 1 x e µ(t) dt s(x) 1 F (x) s(x) x f(u) du e x µ(t) dt f(x) F (x) s (x) f(x) µ(x) e x µ(t) dt µ(x) F (x) 1 F (x) s (x) s(x) f(x) s(x) µ(x) EXAMPLE: Constant Force of Mortality If the force of mortality is a constant µ for every age x, show that 1. s(x) = e µx 2. t p x = e µt SOLUTION: 1. s(x) = e x µ dt = e µx. 2. tp x = s(x + t) s(x) = e µt. Guo MLC: Fall 211 c Yufeng Guo 2

21 Life Tables Option A reference: Actuarial Mathematics Chapter Option B reference: Models for Quantifying Risk Chapter 6 Life Table is widely used actuarial practice. Even today, Life Tables are often loaded into systems for calculating reserves, premium rates, and the surrender cash value of an insurance policy. Learning Life Tables will not only help you pass Exam MLC, it also helps you when you become an actuary. Definitions: l = number of people in cohort at age, also called the radix l i = number of people in cohort at age i (those remaining from the original l ) ω = limiting age at which probability of survival = (s(x) = for all x ω) nd x = number alive at age x who die by age x + n Relationships: l x = l s(x) q x = l x l x+1 l x nq x = l x l x+n l x np x = l x+n l x nd x = l x l x+n Illustrative Life Table: Basic Functions Age l x d x 1, q x 1,. 2, , , , , , , , Guo MLC: Fall 211 c Yufeng Guo 21

22 EXAMPLE: Life Table Mortality Above is an excerpt from the Illustrative Life Table in the book. The following questions are all based on this excerpt. 1. Find s(42). 2. Find 4 d Find 38 q Find 2 q 4. SOLUTION: 1. s(42) = 92, , = d 2 = l 2 l 42 = q 3 = 1 38 p 3 = 1 l 41 l 3 = 1 92, ,76.6 = q 4 = 2 p 4 q 42 = 92, ,131.6 (.32) = Concepts which follow from the Life Table: Based on Equation (3.2.13) on an earlier page, we can determine that the probability density function f(t) for T (x) is given by f(t) = t p x µ(x + t). This says that the probability that (x) will die at age x + t, symbolized by f(t), is equal to the probability that (x) will survive t years and then be hit at that instant by the force of mortality. Among other things, this tells us that tp x µ(x + t)dt = f(t)dt = 1. The complete-expectation-of-life is the expected value of T (x) (or E[T (x)] for fans of Statistics) and is denoted e x. If you remember how to find the expected value of a continuous random variable, you can figure out that e x = E[T (x)] = tp x dt Var[T (x)] = 2 t tp x dt e 2 x (3.5.4) The book shows how to figure both of these formulas out with integration by parts in Section I suggest that you memorize these two expressions. The median future lifetime of (x) is denoted m(x) and simply represents the number m such that m p x = m q x. In other words, it is the number of years that (x) is equally likely to survive or not survive. It can be found by solving any of the following: Pr[T (x) > m(x)] = 1 2 Guo MLC: Fall 211 c Yufeng Guo 22

23 or or s[x + m(x)] s(x) mp x = 1 2. = 1 2 The curtate-expectation-of-life is E[K(x)] and is denoted e x (no circle). To remember the difference between e x and e x, remember life is a continuous circle. So a circle means continuous. e x = E[T (x)] and e x = E[K(x)]. Here are the formulas, note the Continuous/Curtate parallel: e x = E[K(x)] = kp x 1 Var[K(x)] = (2k 1) kp x e 2 x 1 EXAMPLE: Constant Force of Mortality Find e and e 5 if the force of mortality is a constant µ. SOLUTION: [ ] 1 e = tp dt = e µt dt = µ e µt = 1 µ [ ] 1 e 5 = tp 5 dt = e µt dt = µ e µt = 1 µ If the force of mortality is constant, your future expected lifetime is the same whether you are (a newborn) or 5. EXAMPLE: DeMoivre s Law for Mortality (We ll learn DeMoivre later in this chapter.) If s(x) = for all x between and 5, find e and e 45. { 5 x 5 < x < 5 Otherwise SOLUTION: e = 5 1 tp = t 5 = t Guo MLC: Fall 211 c Yufeng Guo 23

24 = 5 1 (5)(51) = e 45 = 5 tp 45 = s(45 + t) s(45) = = t 5 = 2. More Life Functions: The expression L x denotes the total expected number of years, full or fractional, lived between ages x and x + 1 by survivors of the initial group of l lives. Those who survive to x + 1 will live one year between x and x + 1, contributing one full year to L x. Those who die during the year will contribute a fraction of a year to L x. L x = 1 l x+t dt The expression m x is the central death rate over the interval x to x + 1. Make sure not to confuse m x with m(x), the median future lifetime! m x = (l x l x+1 ) L x L x and m x can be extended to time periods longer than a year: nl x = n l x+t dt nm x = l x l x+n nl x The remaining of Section has obscure symbols T x and α(x). They rarely show up in the exam. Don t spend too much time on them. Let T x be the total number of years lived beyond age x by the survivorship group with l initial members (i.e. the l x people still alive at age x). Be careful with notation. This is not T (x), the future lifetime of (x). T x = Note from the definitions that you can think of T x as L x. l x+t dt (3.5.16) The final symbol is α(x). It s the expected death time given x dies next year. Guo MLC: Fall 211 c Yufeng Guo 24

25 α (x) = E [T T < 1] = 1 tf(t)dt 1 f(t)dt = 1 ttpx µ(x+t)dt 1 = tpxµ(x+t)dt 1 lx+t t lx µ(x+t)dt 1 l x+t = lx µ(x+t)dt 1 t l x+t µ(x+t)dt 1 l x+t µ(x+t)dt If UDD, then f(t) = q x = c is a constant. Then α(x) = 1 t cdt 1 cdt = 1 tdt 1 dt = 1 tdt = 1 2. EXAMPLE: Constant Force of Mortality If l = 1 and the force of mortality is a constant µ =.1, find (A) L 5 (B) m 5 (C) T 5 Guo MLC: Fall 211 c Yufeng Guo 25

26 SOLUTION: (A) Since we have L 5 = 1 l 5+t dt = 1 t p 5 l 5 dt. l 5 = l e µ 5 = 1e.5 = 66.5, 1 L 5 = 66.5 e.1t dt = 66.5 [ 1e.1t] 1 [ ] = (1 e.1 ) = (B) (C) m 5 = l 5 l = =.1 L This approximates the rate at which people were dying between the 5th and 6th years. T 5 = 1e.1(5+t) dt = 66.5 e.1t dt = 665 So if we add up all of the time lived by each of the people alive at t = 5, we expect to get a total of 665 years, or 1 years per person. Relationship: T x l x = e x This relationship makes sense. It says that the average number of years lived, e x, by the members of l x is equal to the total number of years lived by this group divided by l x. We can determine the average number of years lived between x and x + n by the l x survivors at age x as: nl x l x = n t p x dt nl x l x = n-year temporary complete life expectancy of (x) = e x:n (p.71) Guo MLC: Fall 211 c Yufeng Guo 26

27 3.5.2 Recursion Formulas Option A reference: Actuarial Mathematics Chapter Option B reference: Models for Quantifying Risk Chapter 6 These are basically ways to avoid working integrals. They are based on the Trapezoid Rule for integration maybe you remember the trapezoid rule from calculus. Backward: u(x) = c(x) + d(x) u(x + 1) Forward: u(x + 1) = u(x) c(x) d(x) Note that the Forward Method is simply an algebraic recombination of the Backward Method. Note also that this Forward formula is different from the book work out the formulas yourself to convince yourself of their equivalence. Then, learn whichever form you find more straightforward. The text shows how to use these formulas to compute e x and e x starting with e ω and e ω and working backward. For e x, using the recursion once will produce e ω 1, the second iteration will produce e ω 2, etc. until you get all the way back to e, when you will have produced a list of e x for every x between and ω. The formulas are: for e x, u(x) = e x c(x) = p x d(x) = p x Starting Value = e ω = u(ω) = So to start, set x + 1 = ω and the recursion will produce u(x) = u(ω 1). For e x, u(x) = e x c(x) = 1 s p x ds d(x) = p x Starting Value = e ω = u(ω) = Guo MLC: Fall 211 c Yufeng Guo 27

28 3.6 Assumptions for Fractional Ages Option A reference: Actuarial Mathematics Chapter 3.6 Option B reference: Models for Quantifying Risk Chapter 6.5 (OK, you can start paying attention again...) The random variable T is a continuous measure of remaining lifetime. The life table has been developed as an approximation of T, using a curtate variable K. As we ve discussed, K is only defined at integers. So, we need some way to measure between two integer ages. Three popular methods were developed. For all of the methods that follow, let x be an integer and let t 1. Suppose that we know the value of s(x) for the two integers x and x + 1 and we want to approximate s at values between x and x+1. In other words, we want to approximate s(x+t) where t 1. Method 1: Linear Interpolation: s(x + t) = (1 t)s(x) + t s(x + 1) This method is also known as Uniform Distribution of Deaths, or UDD. Under UDD, s(x + t) and t p x are both straight lines between t = to t = 1. This method assumes that the deaths occurring between ages x and x + 1 are evenly spread out between the two ages. As you might imagine, this is usually not quite correct, but is a pretty good approximation. (Please note: the linearity of s(x + t) and t p x is only assumed to hold up to t = 1!) One key formula for UDD you might want to memorize is: f(t) = q x To see why, please note that the number of deaths from time zero to time t is a fraction of the total deaths in a year s(x) s(x + t) = t[s(x) s(x + 1)] Here for convenience we interpret s(x + t) as the number of people alive at age x + t. For example, if s(x +.5) =.9, we say that for each unit of people at age x, we have.9 unit of people at age x +.5, with one unit being one billion, one million, or any other positive constant. tq x = s(x) s(x + t) s(x) s(x) s(x + 1) = t = tq x s(x) f(t) = d dt t q x = q x Guo MLC: Fall 211 c Yufeng Guo 28

29 You can also come up with f(t) = q x by intuitive thinking. Under UDD, death occurs at a constant speed. If 12 people died in one year, then one person died each month. So f(t) must be a constant. Then: q x = 1 1 f(t)dt = f(t) dt = f(t) Method 2: Exponential Interpolation: Forget about the complex formula: log s(x + t) = (1 t) log s(x) + t log s(x + 1) All you need to know is that under the constant force of mortality, µ(x + t) = µ for t 1. Method 3: Harmonic Interpolation: This method is more commonly called the Balducci assumption or the Hyperbolic assumption. Forget about the complex formula 1 s(x + t) = 1 t s(x) + t s(x + 1) All you need to know about Balducci assumption is this: 1 tq x+t = (1 t)q x The above formula says that if you are x + t years old (where t 1), then your chance of dying in the remainder of the year is a fraction of your chance of dying in the whole year. You can derive all the other formulas using 1 t q x+t = (1 t)q x. Later I ll show you how to derive other formulas in Balducci assumptions. Uniform Constant Function Distribution Force Hyperbolic tq x tq x 1 p t x tq x 1 (1 t)q x µ(x + t) q x 1 tq x log p x q x 1 (1 t)q x 1 tq x+t (1 t)q x 1 tq x 1 p 1 t x (1 t)q x yq x+t yq x 1 tq x 1 p y x yq x 1 (1 y t)q x tp x 1 tq x p t x p x 1 (1 t)q x tp x µ(x + t) q x p t x log p x q xp x [1 (1 t)q x] 2 Guo MLC: Fall 211 c Yufeng Guo 29

30 Table Table summarizes UDD, constant force of mortality, and the Balducci assumption. Don t try to memorize the whole table. Learn basic formulas and derive the rest on the spot. Under UDD, for t 1: tq x = t f(t) = q x = constant f(t)dt = t t q x dt = q x dt = tq x tp x = 1 t q x = 1 tq x µ(x + t) = µ x (t) = f(t) tp x = q x 1 tq x yp x+t = tp x µ(x + t) = f(t) = q x s(x + t + y) = s(x) t+yp x = t+y p x = 1 (t + y)q x s(x + t) s(x) t p x tp x 1 tq x yq x+t = 1 y p x+t = yq x 1 tq x 1 tq x+t = (1 t)q x 1 tq x Under constant force of mortality, for t 1: µ x (t) = µ tp x = e t µdt = e µt p x = e µ µ = ln p x tp x = e µt = (e µ ) t = (p x ) t yp x+t = t+y p x tp x = (p x) t (p x ) (t+y) = (p x) y Finally, let s derive log s(x + t) = (1 t) log s(x) + t log s(x + 1) s(x + t) = s(x)e µt log s(x + t) = log s(x) µt s(x + 1) = s(x)e µ log s(x + 1) = s(x) µ (1 t) log s(x) + t log s(x + 1) = (1 t) log s(x) + t log s(x) µt = log s(x) µt log s(x + t) = (1 t) log s(x) + t log s(x + 1) Guo MLC: Fall 211 c Yufeng Guo 3

31 Under Balducci assumption: The starting point of Balducci assumption is 1 tq x+t = (1 t)q x You can derive all the other formulas from this starting point. For example, let s derive the formula for t p x. T (x) t 1 Age x x + t x + 1 Number of people alive s(x) = 1 s(x + t) = s(x + 1) = p x p x 1 (1 t)q x This is how to derive s(x + t) = p x 1 (1 t)q x. First, we set the starting population at s(x) = 1 for convenience. You can set it to any positive constant and get the same answer. After setting s(x) = 1, we ll have s(x + 1) = p x. This is because p x = s(x+1) s(x). Next, let s find s(x + t) using the formula 1 t q x+t = (1 t)q x. 1 tq x+t = 1 s(x + 1) s(x + t) = 1 p x s(x + t) = (1 t)q x s(x + t) = p x 1 (1 t)q x However, t p x = s(x+t) s(x) = s(x + t). This gives us: t p x = Next, let s derive the formula f(t) = t p x µ(x + t) = t p x µ x (t) p x f(t) = d dt t p x = d = dt 1 (1 t)q x p x 1 (1 t)q x. q x p x [1 (1 t)q x ] 2 Derive µ(x + t): µ(x + t) = f(t) tp x = q x 1 (1 t)q x Derive y q x+t : yq x+t = 1 p x 1 (1 t)q x Use the formula: s(x + t) = Replace t with t + y, assuming t + y 1: s(x + t + y) s(x + t) s(x + t + y) = p x 1 (1 t y)q x Guo MLC: Fall 211 c Yufeng Guo 31

32 yq x+t = 1 s(x + t + y) s(x + t) = 1 1 (1 t)q x 1 (1 t y)q x = p x 1 (1 t y)q x Finally, let s derive 1 s(x+t) = 1 t s(x) + t s(x+1) s(x + t) = p x 1 (1 t)q x 1 s(x + t) = 1 (1 t)q x = 1 (1 t)(1 p x) = (1 t)p x + t = (1 t) + t p x p x p x p x Since s(x) = 1 and s(x + 1) = p x, we have: 1 t s(x) + t s(x + 1) = 1 t + t p x 1 s(x + t) = 1 t s(x) + t s(x + 1) Now you see that you really don t need to memorize Table Just memorize the following: Under UDD, f(t) = q x is a constant. Under constant force of mortality, µ(x + t) = µ. Under Balducci, 1 t q x+t = (1 t)q x. A couple of time-saving formulas that are valid under UDD only!! e x = e x Var(T ) = Var(K) EXAMPLE: You are given that q x =.1. Find (A).5 q x, (B).5 q x+.5 under each of Uniform Density of Deaths (UDD) Exponentially distributed deaths (constant force) Harmonic Interpolation (Balducci, hyperbolic) SOLUTION: (A) UDD:.5q x = (.5)(.1) =.5 Guo MLC: Fall 211 c Yufeng Guo 32

33 (B) CF: Balducci: UDD: CF: Balducci:.5q x = 1 (.9).5 =.513.5q x = (.5)(.1) 1 (.5)(.1) = q x+.5 = (.5)(.1) 1 (.5)(.1) =.526.5q x+.5 = 1 (.9).5 =.513.5q x+.5 = (.5)(.1) = Some Analytical Laws of Mortality Option A reference: Actuarial Mathematics Chapter 3.7 Option B reference: Models for Quantifying Risk Chapter 5.2 Although computers have rendered Analytical Laws of Mortality less imperative to our profession, they are still important for understanding mortality and particularly for passing the exam. The book describes four basic analytical laws/formulas. Of these 4 laws, De Moivre s Law is frequently tested in the exam. The other 3 laws are rarely tested in the exam (I would skip these 3 laws). De Moivre s Law: µ(x) = (ω x) 1 and s(x) = 1 x ω, where x < ω Gompertz Law: µ(x) = Bc x and s(x) = exp[ m(c x 1)] where B >, c > 1, m = B log c, x Makeham s Law: µ(x) = A + Bc x and s(x) = exp[ Ax m(c x 1)] where B >, A B, c > 1, m = B log c, x Weibull s Law: µ(x) = kx n and s(x) = exp( ux n+1 ) Guo MLC: Fall 211 c Yufeng Guo 33

34 where k >, n >, u = k (n + 1), x Notes: Gompertz is simply Makeham with A =. If c = 1 in Gompertz or Makeham, the exponential/constant force of mortality results. In Makeham s law, A is the accident hazard while Bc x is the hazard of aging. We will be seeing more of these laws later in the book. To be ready for the exam, you should become intimately familiar with De Moivre s Law. DeMoivre s Law says that at age x, you are equally likely to die in any year between x and ω. Here are some key life-functions for DeMoivre s Law in the form of an example. If you don t read the proofs, still make an effort to understand the formulas and what they mean. EXAMPLE: Under DeMoivre s law, show that tp x = ω x t ω x tq x = t ω x e = ω 2 e = ω 1 2 e x = ω x 2 e x = ω x 1 2 Guo MLC: Fall 211 c Yufeng Guo 34

35 SOLUTION: 1. tp x = s(x + t) s(x) = ω x t ω x tq x = 1 t p x = t ω x [ ] ω ω t e = ω dt = (ω t)2 = ω 2ω 2 4. e = ω kp = 1 ω 1 ω k ω = ω 1 ω ω 1 k = ω ω x ω x t e x = ω x dt = ω x 2 6. Since mortality is uniform over all years under DeMoivre s law, it is uniform over each individual year, so UDD applies. Therefore e x = e x 1 2 = ω x 1 2 We could have done (4) this way also. Modified DeMoivre s Law Often on the exam, a modified version of DeMoivre s Law arises. This occurs, for example, when µ(x) = c ω x where c is a positive constant. This gives rise to a set of formulas for each of the quantities found for standard DeMoivre s Law (c = 1) in the example above. All of the formulas for Modified DeMoivre s Law are listed in the formula summary at the end of this chapter. 3.8 Select and Ultimate Tables Option A reference: Actuarial Mathematics Chapter 3.8 Option B reference: Models for Quantifying Risk Chapter 6.6 Guo MLC: Fall 211 c Yufeng Guo 35

36 Suppose you are trying to issue life insurance policies and two 45 year-old women apply for policies. You want to make sure you charge appropriate premiums for each one to cover the cost of insuring them over time. One of the women is simply picked from the population at large. The second women was picked from a group of women who recently passed a comprehensive physical exam with flying colors significantly healthier than the general population. Is it equitable to charge the same premium to the two women? No because you have additional information about the second woman that would cause you to expect her to have better mortality experience than the general population. Thus, to her premiums, you might apply a select mortality table that reflects better the mortality experience of very healthy 45 year olds. However, after 15 years, research might show that being very healthy at 45 does not indicate much of anything about health at age 6. So, you might want to go back to using standard mortality rates at age 6 regardless of status at age 45. After all, 15 years is plenty of time to take up smoking, eat lots of fried foods, etc. This simple scenario illustrates the idea behind select and ultimate tables. For some period of time, you expect mortality to be different than that for the general population the select period. However, at some point, you re just not sure of this special status anymore, so those folks fall back into the pack at some point the ultimate table. Consider table The symbol [x] signifies an x-yr old with select status. Note that for the first two years (columns 1,2), select mortality applies with q [x] and q [x]+1. However, at duration 3 (column 3), it s back to standard mortality, q x+2. This table assumes the selection effect wears off in just 2 years. Excerpt from the AF8 Select-and-Ultimate Table in Bowers, et al. (1) (2) (3) (4) (5) (6) (7) [x] 1, q [x] 1, q [x]+1 1, q x+2 l [x] l [x]+1 l x+2 x , , , , ,9.58 9, , , , , , , , , , Table Here are a few useful formulas and relationships. In general, q [x] < q [x 1]+1 < q x Why would this hold? The expression q x represents a pick from the general population. The expression q [x] indicates special knowledge about the situation for example, recently passing a physical exam (This formula assumes we are trying to select out healthy people, of course). The expression q [x 1]+1 indicates special knowledge about the situation as before for example, recently passing a physical exam, but this time the applicant has had a year Guo MLC: Fall 211 c Yufeng Guo 36

37 for health to deteriorate since she was examined at age x 1 (one year ago) rather than at age x. EXAMPLE: Select and Ultimate Life Table Using the select and ultimate life table shown above find the value of ) 1 (3q 32 3 q [32]. Guo MLC: Fall 211 c Yufeng Guo 37

38 SOLUTION: 3q 32 deals only with the ultimate table so I am only interested in the values of l x + 2 in Column (6). 3q 32 = 1 3 p 32 = 1 l 35 l 32 = 1 9,888 9,91 =.1313 For 3 q [32], we need l [32] and l [32]+3 which is just l 35 since the select period is only 2 years. So 3q [32] = 1 l 35 = 1 9,888 l [32] 9,899 =.1111 So the answer is.22. Two important points regarding this example: The probability that [32] will die in the next 3 years is lower if [32] is taken from a select group. People you are sure are healthy should be less likely to die than someone drawn from the general population. To follow the people alive from the selected at age 32, you first follow the numbers to the right until you hit the ultimate column and then proceed down the ultimate column. This is useful! You can quickly evaluate that 5p [31] = by counting off 5 years 2 to the right and then 3 down. Conclusion Chap. 3 introduces a lot of new concepts and notation. Make sure you understand the notation in Table this is the foundation for the rest of the text. Chapter 3 Suggested Problems: 1(do first row last), 5, 6, 7, 9, 12, 18abc, 2, 28, 3, 36, 39 There are lots for this chapter, some chapters in this book will have very few. (Solutions are available at archactuarial.com on the Download Samples page.) Guo MLC: Fall 211 c Yufeng Guo 38

39 CHAPTER 3 Formula Summary s(x) = 1 F (X) = 1 Pr(X x) Pr(x < X z) = F (z) F (x) = s(x) s(z) Pr(x < X z X > x) = [F (z) F (x)] [1 F (x)] = [s(x) s(z)] [s(x)] [ ] t tp x = e µ(x+s)ds tp x = s(x + t) s(x) tq x = 1 t p x t uq x = t p x u q x+t t u q x = t+u q x t q x t u q x = t p x t+u p x Pr(K(x) = k) = k p x k+1 p x = k p x q x+k = k q x Life Tables: l x = l s(x) nd x = l x l x+n q x = l x l x+1 l x p x = l x+1 l x nq x = l x l x+n l x np x = l x+n l x Constant Force of Mortality If the force of mortality is a constant µ for every age x, s(x) = e µx e x = 1 µ tp x = e µt Var[T ] = 1 µ 2 Guo MLC: Fall 211 c Yufeng Guo 39

40 Expected Future Lifetime e x = E[T (x)] = tp x dt Var[T (x)] = 2 t tp x dt e 2 x e x = E[K(x)] = kp x Var[K(x)] = 1 (2k 1) kp x e 2 x 1 Under UDD only, e x = e x Var(T ) = Var(K) Median future lifetime Pr[T (x) > m(x)] = 1 2 s[x + m(x)] s(x) = 1 2 mp x = 1 2 Make sure not to confuse m x with m(x), the median future lifetime! L x = 1 l x+t dt nl x = n l x+t dt m x = (l x l x+1 ) L x nm x = l x l x+n nl x T x = l x+t dt T x = e nl n x x = l x l x t p x dt = e x:n a(x) is the average number of years lived between age x and age x + 1 by those of the survivorship group who die between age x and age x + 1. With the assumption of uniform distribution of deaths over the interval (x, x + 1), Without this assumption, a(x) = 1/2. a(x) = 1 t l x+t µ(x + t)dt 1 l x+t µ(x + t)dt = 1 t tp x µ(x + t)dt 1 tp x µ(x + t)dt By the way, a(x) is a minor concept in MLC. I ll be surprised if SOA tests this obscure concept. I would skip it. Guo MLC: Fall 211 c Yufeng Guo 4

41 De Moivre s Law: µ(x) = (ω x) 1 and s(x) = 1 x ω, where x < ω Gompertz Law: µ(x) = Bc x and s(x) = exp[ m(c x 1)] where B >, c > 1, m = B log c, x Makeham s Law: µ(x) = A + Bc x and s(x) = exp[ Ax m(c x 1)] where B >, A B, c > 1, m = B log c, x Weibull s Law: µ(x) = kx n and s(x) = exp( ux n+1 ) where k >, n >, u = k (n + 1), x DeMoivre s Law and Modified DeMoivre s Law: If x is subject to DeMoivre s Law with maximum age ω, then all of the relations on the left below are true. The relations on the right are for Modified DeMoivre s Law with c >. DeMoivre Modified DeMoivre µ(x) = 1 ω x µ(x) = c ω x s(x) = ω x ω l x = l ω x ω e x = E[T ] = ω x 2 ) c ω ) c s(x) = ( ω x l ( ω x ω e x = ω x c+1 Var[T ] = (ω x)2 12 Var[T ] = (ω x)2 c tp x = ω x t ω x tp x = (c+1) 2 (c+2) ( ) c ω x t ω x µ x (t) = 1 ω x t µ x (t) = c ω x t Be careful on the exam - Modified DeMoivre problems are often disguised in a question that starts something like You are given s(x) = ( 1 x 8 ) 2 In cases like this, you have to recognize that the question is just a modified DeMoivre written in a different algebraic form. Guo MLC: Fall 211 c Yufeng Guo 41

42 Note that in this table, the function listed at left are given in terms of the functions across the top row! F (x) s(x) f(x) µ(x) F (x) F (x) 1 s(x) x f(u) du 1 x e µ(t) dt s(x) 1 F (x) s(x) x f(u) du e x µ(t) dt f(x) F (x) s (x) f(x) µ(x) e x µ(t) dt µ(x) F (x) 1 F (x) s (x) s(x) f(x) s(x) µ(x) Assumptions for fractional ages. Uniform Constant Function Distribution Force Hyperbolic tq x tq x 1 p t x tq x 1 (1 t)q x µ(x + t) q x 1 tq x log p x q x 1 (1 t)q x 1 tq x+t (1 t)q x 1 tq x 1 p 1 t x (1 t)q x yq x+t yq x 1 tq x 1 p y x yq x 1 (1 y t)q x tp x 1 tq x p t x p x 1 (1 t)q x tp x µ(x + t) q x p t x log p x q xp x [1 (1 t)q x] 2 Guo MLC: Fall 211 c Yufeng Guo 42

43 Past SOA/CAS Exam Questions: All of these questions have appeared on SOA/CAS exams between the years 2 and 25. You will find that they often involve some clever thinking in addition to knowledge of actuarial math. These questions are used with permission. 1. Given: (i) e = 25 (ii) l x = ω x, x ω (iii) T (x) is the future lifetime random variable. Calculate Var[T (1)]. (A) 65 (B)93 (C) 133 (D) 178 (E) 333 Solution: Key: C ω ( e = 1 t ) dt = ω ω2 ω 2ω = ω 2 e 1 = 4 = 25 ω = 5 ( 1 t ) dt = (2)(4) = 2 [ t 2 4 ( Var [T (x)] = 2 t 1 t ) dt (2) 2 = t3 3 4 ] 4 (2) 2 = For a certain mortality table, you are given: (i) µ(8.5) =.22 (ii) µ(81.5) =.48 (iii) µ(82.5) =.619 (iv) Deaths are uniformly distributed between integral ages. Calculate the probability that a person age 8.5 will die within two years. (A).782 (B).785 (C).79 (D).796 (E).8 Guo MLC: Fall 211 c Yufeng Guo 43