Exam MLC Preparation Notes

Exam MLC Preparation Notes Yeng M. Chang 213 All Rights Reserved

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Contents Preface 7 1 Continuous Survival Models 9 1.1 CDFs, Survival Functions, µ x+t............................... 9 1.2 Parametric Survival Models................................. 15 1.2.1 Uniform Distribution (de Moivre s Law)..................... 15 1.2.2 Exponential Distribution............................... 15 1.2.3 Gompertz Distribution................................ 16 1.2.4 Makeham Distribution................................ 16 1.3 kth Moments (Exam P/1 Review)............................. 16 1.4 International Actuarial Notation: T x............................ 18 1.5 More Formulas to Know................................... 21 1.6 SOA Released Questions................................... 22 1.7 Formula Summary Sheet................................... 23 2 Discrete Survival Models 25 2.1 Commonly Used Symbols................................... 25 2.2 K x, K x, R x........................................... 26 2.3 Probabilities and Expectation of K x............................ 27 2.4 Term Expectation of Life................................... 29 2.5 Temporary Curtate Expectation of Life.......................... 3 2.6 SOA Released Questions................................... 31 2.7 Formula Summary Sheet................................... 34 3 Life Tables 35 3.1 Calculating Probabilities................................... 35 3.2 Fractional Age Assumptions................................. 38 3.2.1 Uniform Distribution of Deaths (UDD) Assumption.............. 38 3.2.2 Constant Force of Mortality (CF) Assumption.................. 4 3.3 Select and Ultimate Life Tables............................... 41 3.4 SOA Released Questions................................... 42 3.5 Formula Summary Sheet................................... 43 3

4 CONTENTS 4 Valuation of Insurances 45 4.1 Introduction........................................... 45 4.2 Types of Insurances...................................... 46 4.3 Variances and Moments.................................... 49 4.3.1 Discrete Whole Life Insurance............................ 49 4.3.2 Continuous Whole Life Insurance.......................... 49 4.3.3 Term Insurances.................................... 5 4.3.4 Deferred Insurances.................................. 5 4.3.5 Pure Endowment Insurance............................. 51 4.3.6 Endowment Insurances................................ 51 4.4 Mortality Assumptions on the Entire Domain of T x................... 52 4.4.1 Continuous Insurances................................ 52 4.4.1.1 Fundamental Equations.......................... 52 4.4.1.2 Whole Life Insurance (Āx)........................ 53 4.4.1.3 Term Insurance (Ā1 ).......................... 53 x n 4.4.1.4 Deferred Whole Life Insurance ( n Ā x )................. 54 4.4.1.5 Deferred Term Insurance ( u Ā 1 ).................... x n 54 4.4.1.6 Endowment Insurance (Āx n )...................... 54 4.4.1.7 Calculation of Higher Moments..................... 55 4.4.2 Discrete Insurances.................................. 55 4.4.2.1 Fundamental Equations.......................... 55 4.4.2.2 Whole Life Insurance (A x )........................ 57 4.4.2.3 Term Insurance (A 1 ) x n.......................... 57 4.4.2.4 Deferred Whole Life Insurance ( n A x )................. 58 4.4.2.5 Deferred Term Insurance ( u A 1 ).................... x n 58 4.4.2.6 Endowment Insurance (A x n )...................... 58 4.4.2.7 Calculation of Higher Moments..................... 58 4.5 UDD Assumption: Continuous-Discrete Relationships.................. 59 4.6 mthly Insurances........................................ 62 4.7 UDD Assumption: Discrete-mthly Relationships..................... 65 4.8 Insurances with Variable Benefits.............................. 65 4.8.1 General Cases..................................... 65 4.8.2 Arithmetically Changing Benefits.......................... 66 4.9 Recursive Relationships.................................... 68 4.1 Actuarial Accumulated Value................................ 69 4.11 SOA Released Questions................................... 7 4.12 Formula Summary Sheet................................... 71 4.13 Appendix: Fully-Variable Insurances............................ 74 5 Valuation of Annuities 77 5.1 Review from FM/2....................................... 77 5.1.1 Discrete Annuities................................... 77 5.1.2 Continuous Annuities................................. 79 5.1.3 mthly annuities.................................... 8 5.2 Introduction........................................... 8 5.3 Whole-Life Annuities..................................... 81

5.4 Whole Life Annuity-Insurance Relationships and Moments............... 82 5.5 Non-Insurance Annuity APV Formulas........................... 83 5.6 kth Moments of Annuities.................................. 85

6 CONTENTS

Preface This set of notes is meant to prepare you to take exam MLC by the Society of Actuaries. Numbers surrounded by [ ] indicate a specific source cited in the bibliography, presented at the end of this set of notes. I base this set of notes on Models for Quantifying Risk by Cunningham, et al., (4th edition) which I will abbreviate as MQR throughout this text (see [1]); Actuarial Mathematics for Life Contingent Risks by Dickson, et al., which I will abbreviate as AMLCR (see [3]); their respective supplementary notes (see [2], [4]); concepts which are in the SOA sample questions and released exams for exam MLC; and my experiences from taking Math 45 and Math 46 at the University of Wisconsin - Eau Claire with Prof. Herschel Day. I will rely immensely on exam P and FM material as I go through this text. Information in footnotes and appendices consist of technical information that is needed to add coherence to the discussion, but is unnecessary for the the purpose of studying for the exam. This set of notes would not exist without the help and support of many people. First of all, I thank the professors whose courses I took as an undergraduate at the University of Wisconsin - Eau Claire. Specifically, I would like to thank Dr. Carolyn Otto, Dr. Chris Ahrendt, and Dr. Dandrielle Lewis for teaching me how to write mathematics and for pushing me to learn the L A TEX typesetting language. I would also like to thank Dr. Vicki Whitledge, my advisor and Probability professor during my undergraduate years, as this text would not exist had I not taken her Probability course. Also, I would like to thank Prof. Herschel Day for teaching me the course MLC material in Math 45 and Math 46. I would also like to thank Jim Daniel, Abraham Weishaus, and Steve Paris for correcting errors, helping out with proofs, and/or giving suggestions as I was creating these notes. In addition, I would like to thank the various people who gave me feedback on my notes through the online Actuarial Outpost. Last, but definitely not least, I would like to thank my girlfriend Katie Miller for her support as I took on this ambitious project during my undergraduate years. I wish you the best of luck as you prepare for this exam! If you find any typos, mistakes, or have any suggestions for this set of notes, please contact me at (e-mail address omitted). Yeng Chang July 18, 213 7

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Chapter 1 Continuous Survival Models References: AMLCR, Chapters 2.2-2.5, 2.6.2, 2.7; MQR, Chapters 5.1-5.3.5. 1.1 CDFs, Survival Functions, µ x+t In this chapter, assume that all random variables are continuous. Recall, from exam P/1, the cumulative distribution function (CDF) of a random variable X: F X (x) = P (X x) (1.1) where P is the probability, hence P (X x) is read as the probability that X is less than or equal to the value x. You may also see Pr(X x) instead of the above. Suppose that X is a nonnegative continuous random variable, i.e., X. The CDF of X must satisfy the following properties: 1. F X is continuous and non-decreasing for all x. 2. F X () = 3. lim x F X (x) = 1. Define the survival function to be the complement of the CDF, denoted S X. For all x, we write S X (x) = 1 F X (x). Then the survival function has the following properties: 1. S X is continuous and non-increasing for all x. 2. S X () = 1 F X () = 1 3. lim x S X (x) = lim x [1 F X (x)] =. The reason why the complement of the CDF is known as the survival function will be clear very shortly. For exam MLC, the main focus is on random variables and the properties and descriptive quantities of random variables. Hence, much of the basis for what is in MLC can be derived by using principles from probabilty. Consider the time-to-failure random variable of someone age x, denoted T x. This random variable should be viewed as the amount of years that someone age x will fail (or die) - hence, its domain is in the interval (, ). This particular random variable will have CDF F Tx (t) and survival function S Tx (t). To make our lives easier, we will remove the T in the subscript throughout the course; hence, the CDF and survival functions of T x will be F x (t) and S x (t) respectively. 9

1 CHAPTER 1. CONTINUOUS SURVIVAL MODELS Notice that F x (t) = P (T x t). This can be interpreted as the probability that someone currently age x will die within t years. The survival function, S x (t) = P (T x > t). This is the probability that someone age x will die at some point after t years, or in other words, survive at least t years. Hence this is the reason why the complement of the CDF is known as the survival function. Let us now consider the intution behind the properties of the survival function using the random variable T x : 1. S x () = 1: the probability of someone age x surviving to time is 1. 2. S x (t) is continuous and non-increasing for all t: the probability of someone age x surviving t years decreases as t increases. 3. lim t S x (t) = : the probability of someone age x surviving years is. Eventually, all lives age x must die. Of course, as you should remember from probability, the probability density function (PDF) of a continuous random variable X, denoted f X, is the derivative of the CDF of X. Similarly, for the random variable T x, d dt [F T x (t)] = f Tx (t) = f x (t) (1.2) which is the PDF of the time-to-failure of someone age x. Again, we drop the T in the subscript for the PDF. We can also write the PDF in terms of the survival function: d dt [S x(t)] = d dt [1 F x(t)] = f x (t) d dt [S x(t)] = f x (t). (1.3) Hence, the negative derivative of the survival function of T x is the PDF of T x. This relationship holds for any continuous random variable in general. We can also integrate the PDF to get the CDF of the survival function: t d du [F x(u)] du = F x (t) F x () = F x (t) = t t t f x (u) du f x (u) du We can find a similar relationship with the survival function: 1 F x (t) = 1 S x (t) = 1 t t f x (u) du (1.4) f x (u) du f x (u) du.

1.1. CDFS, SURVIVAL FUNCTIONS, µ x+t 11 Because T x takes on values in the interval (, ), f x (u) du = 1. Hence, for any value t such that < t <, t f x (u) du + t t f x (u) du = 1 f x (u) du = 1 t f x (u) du = S x (t) Hence, we can write S x (t) = t f x (u) du. 1 (1.5) For the rest of this course, we will use the notation (x) to refer to a person or thing whose age is x. For exam MLC, one particular value of interest for x is the case when x =. Our goal is to analyze how we can relate our general formulas mentioned so far in this chapter to those of when x =. Consider T, the time-to-failure random variable of someone age, or that of a newborn. Given T > x, T is defined in the equation T = x + T x. To understand the intuition behind this formula, suppose that () is known to fail at age d. Then of course, T = d. After one year, () is now (1). Thus, T 1 = d 1 = T 1. After another year, (1) is now (2) and T 2 = (T 1) 1 = T 2. Repeat this recursion for any positive integral (or integer) n and you get the relationship T n = T n, given T > n. It may be more intuitive to understand this relationship when written in the form T x = T x T > x. Let x be constant and t be variable. We assume that F x (t) = P (T x t) = P (T x + t T > x). (1.6) This assumption is definitely reasonable. In words, what (1.6) states is that the probability of (x) surviving t years is the probability that a newborn, (), dies within x + t years, given survival to x years. By use of conditional probability (from exam P/1), F x (t) = P (T x t) = P (T x + t T > x) = P (T x + t T > x) P (T > x) = P (x < T x + t). (1.7) P (T > x) Using CDFs, we can write F x (t) = P (T x t) = P (x < T x + t) P (T > x) = F (x + t) F (x). (1.8) 1 F (x) 1 In addition, we assume that S x (t) is differentiable for all t >. This guarantees the existence of the force of mortality, which we will mention later in this chapter.

12 CHAPTER 1. CONTINUOUS SURVIVAL MODELS Using survival functions, we can write Taking the complement of this, we have F x (t) = P (T x t) = F (x + t) F (x) 1 F (x) = [1 S (x + t)] [1 S (x)] S (x) = S (x) S (x + t) S (x) = 1 S (x + t) S (x). (1.9) S x (t) = 1 [1 S (x + t) S (x) ] = S (x + t) S (x). (1.1) Equation (1.1) has a very intuitive interpretation: the probability that (x) survives t years is the probability that () survives x + t years, given () survives x years. This is essentially the use of conditional probability in survival functions. We now focus on a quantity called the force of mortality for someone currently age x, denoted µ x. 2 We define P (T x + dx T > x) µ x = lim = lim dx + dx dx + 1 = lim dx + 1 = lim dx + = S (x + dx) S (x) dx S (x) S (x + dx) S (x) dx S (x) [S (x) S (x + dx) ] dx [S (x) S (x + dx) ] dx + dx [S (x + dx) S (x) ] dx + dx 1 S (x) lim = 1 S (x) lim = 1 S (x) lim [S (x + dx) S (x) ] dx + (x + dx) x = 1 d S (x) dx [S (x)]. (1.11) 2 This is an unfortunate source of notational ambiguity with µ X, which in exam P/1, is often used to signify the expected value of the random variable X. For the rest of this course, assume µ x is used to represent the force of mortality for someone currently age x.

1.1. CDFS, SURVIVAL FUNCTIONS, µ x+t 13 We can also write µ x = 1 d S (x) dx [S (x)] d = dx [S (x)] S (x) = f (x) S (x) (1.12) using equation (1.3). Notice that f (x) = F (x). µ x can be interpreted as the instantaneous rate of failure at age x, given survival to age x. Similarly, we can define a force of mortaility for (x) t years after, denoted µ x+t. Let x be fixed and t be variable. µ x+t is defined as: 1 d µ x+t = S (x + t) d(x + t) [S (x + t)] 1 d = S (x + t) dt [S (x + t)] 1 d = S (x + t) dt [S (x)s x (t)] by equation (1.1) = S (x) d S (x + t) dt [S x(t)] = 1 d S x (t) dt [S x(t)] by equation (1.1) (1.13) = f x(t) by equation (1.3) (1.14) S x (t) Some references may also use µ x (t) to emphasize that the force of mortality depends on t. On exam MLC, both of these notations will be used. 3 Furthermore, notice that we can also write µ x+t = d dt [S x(t)] S x (t) = d dt [ln S x(t)]. (1.15) Thus, t µ x+s ds = t d ds [ln S x(s)] ds = [ln S x (t) ln S x ()] = [ln S x (t)] since S x () = 1 ln S x () =. 3 See Notation and Terminology used on Exam MLC, found in http://www.soa.org/files/edu/edu-213-springmlc-notation.pdf for the spring 213 sitting. For the most current document, check the most current syllabus.

14 CHAPTER 1. CONTINUOUS SURVIVAL MODELS Solving for S x (t), we get [ln S x (t)] = ln S x (t) = t S x (t) = exp µ x+s ds t µ x+s ds t µ x+s ds. (1.16) where exp(x) = e x. Be sure to practice these concepts using the SOA sample question list in section 1.6 after reading the rest of this chapter.

1.2. PARAMETRIC SURVIVAL MODELS 15 1.2 Parametric Survival Models While we did work with quite a few probability concepts in section 1.1, we did not apply them to any specific probability distributions. This section outlines possible distributions we could use for the random variable T x. 1.2.1 Uniform Distribution (de Moivre s Law) Recall from exam P/1 that a random variable following the uniform distribution has domain in a closed interval [a, b]. In the case of survival models, suppose that T (, ω]. 4 In other words, suppose that ω is the limiting age, or the highest age at which a person can live. For (x), if T x follows a uniform distribution, then f x (t) is constant in the interval (, ω x] and 1 f x (t) = ω x t F x (t) = ω x S x (t) = ω x t ω x 1 µ x+t = ω x t E[T x ] = ω x 2 (ω x)2 V [T x ] = 12 (1.17) (1.18) (1.19) (1.2) (1.21) (1.22) where E[T x ] and V [T x ] are the expected value and variance of T x respectively, 5 and t < ω x. In exam MLC, the uniform distribution can referred to as de Moivre s Law. 1.2.2 Exponential Distribution Recall from exam P/1 that a random variable following the exponential distribution has domain in the interval (, ). If T x follows an exponential distribution, then f x (t) = µe µt (1.23) F x (t) = 1 e µt (1.24) S x (t) = e µt (1.25) E[T x ] = 1 µ (1.26) V [T x ] = 1 µ 2. (1.27) There is a valid reason why µ is used as the parameter for the exponential distribution. Notice that µ x+t = µe µt = µ. Thus, the force of mortality is constant for the exponential distribution. Hence, e µt the exponential distribution is sometimes referred to as the constant force distribution. 4 The symbol is used to mean in or is a member of. 5 The expected value and variance of T x will be further discussed in the next section.

16 CHAPTER 1. CONTINUOUS SURVIVAL MODELS 1.2.3 Gompertz Distribution This distribution is defined by its force of mortality: µ x+t = Bc x+t (1.28) where B (, 1) and c > 1. The survival function is given by S x (t) = exp [ Bcx (c t 1) ]. (1.29) ln c The proof is left to the reader. You can use equation (1.14) to derive the PDF. 1.2.4 Makeham Distribution This distribution is also defined by its force of mortality - it adds an additional constant A to the force of mortality of the Gompertz distribution: µ x+t = A + Bc x+t (1.3) where B >, c > 1, and A > B. The survival function is given by S x (t) = exp [ Bcx ln c (1 ct ) At]. (1.31) The proof is left to the reader. You can use equation (1.14) to derive the PDF. 1.3 kth Moments (Exam P/1 Review) Recall that the kth moment of a continuous random variable X, for k, is given by E[X k ] = x k f X (x) dx. (1.32) In particular, the mean of X is the first moment, i.e., the case when k = 1. The variance (or the second central moment) of X, denoted V [X] or Var[X], is defined by V [X] = E[(X E[X]) 2 ] = E[X 2 2XE[X] + E[X] 2 ] = E[X 2 ] 2E[X]E[X] + (E[X]) 2 = E[X 2 ] (E[X]) 2. (1.33)

1.3. kth MOMENTS (EXAM P/1 REVIEW) 17 Assume that X takes on nonnegative values. If the mean of X exists and is finite, an alternate formula for the mean of X, where X (a, ) is E[X] = a + Now consider T x. The kth moment of T x is given by E[(T x ) k ] = a S x (x) dx. 6 (1.37) t k f x (t) dt. (1.38) 6 Here is how this formula is proven: suppose X is a nonnegative random variable taking values in (a, b). Then, by definition, E[X] = a b xf X (x) dx. Using integration by parts with u = x and dv = f X (x)dx, we have du = dx and v = F X (x) + C, where C is an arbitrary constant. Since it does not matter what value C takes, we set C = 1. Then E[X] = a b xf X (x) dx = b[f X (b) 1] a[f X (a) 1] Using the fact that F X (b) = 1 and F X (a) =, we have In particular, if X (, b), then E[X] = a( 1) E[X] = = a + b a b a b a b [F X (x) 1] dx [F X (x) 1] dx. [1 F X (x)] dx. (1.35) [1 F X (x)] dx = b S X (x) dx. (1.36) Equations (1.35) and (1.36) also apply for the case in which b, provided that E[X] exists and is finite. Even more generally, we can show that for integral k >, if X (a, b), where a, E[X k ] = a b x k f X (x) dx = b k [F X (b) 1] a k [F X (a) 1] = a k + k provided that the moments exist and are finite. a b a b [F X (x) 1]kx k 1 dx x k 1 [1 F x (x)] dx (1.37)

18 CHAPTER 1. CONTINUOUS SURVIVAL MODELS In particular, since T x (, ), we can write, by equation (1.36), E[T x ] = S x (t) dt (1.39) provided that E[T x ] exists and is finite. Equation (1.39) will be very important throughout this course. Although not mentioned in any of the sources for exam MLC, by equation (1.37), it can be shown that E[(T x ) k ] = k t k 1 S x (t) dt (1.4) provided that these moments exist and are finite. Equation (1.4) can be very useful in calculating the variance and the moments of T x. 1.4 International Actuarial Notation: T x In 1844, David Jones, a British actuary, wrote a book titled The Value of Annuities and Revisionary Payments, from which a new system called International Actuarial Notation was created. 7 Exam MLC uses International Actuarial Notation in its questions, so it will be necessary to be able to read and interpret this system of notation. This section will briefly introduce this system of notation; we will learn more on this system of notation as we go throughout the course. For the material mentioned so far in this text, we can write for (x): t p x : the probability that (x) survives t years. Notice that this is equivalent to S x (t). tp x Age x x + t Recall that t p x = S x (t) = P (T x > t). The arrow in the timeline above indicates where failure or death of (x) would occur in the computation of the probability t p x. t q x : the probability that (x) dies or fails within t years. Notice that this is equivalent to F x (t). tq x Age x x + t Recall that t q x = F x (t) = P (T x t). The arrow in the timeline above indicates where failure or death of (x) would occur in the computation of the probability t q x. 7 For further details, see the article The International Actuarial Notation by Frank P. Di Paolo in the newsletter The Actuary, March 1976 - Vol. 1, No. 3. See also International Actuarial Notation by F.S. Perryman from Proceedings of the Casualty Actuarial Society, Vol. XXXVI, pp. 123-131, for an actuarial symbol reference.

1.4. INTERNATIONAL ACTUARIAL NOTATION: T x 19 u t q x : the probability that (x) survives u years and dies or fails within the t years following the u years of survival. In other words, this is the probability that (x) survives u years and dies or fails within u + t years. This is known as a deferred mortaility probability, i.e., it is the probability that death occurs in some interval following a deferred period. We define u tq x = P (u < T x u + t) = S x (u) S x (u + t) = F x (u + t) F x (u). Consider the timeline below. u tq x Age x x + u x + u + t The arrow in the timeline above indicates where failure or death would occur for (x) in computing the probability u t q x. Now consider the following timeline: S x (u) S x (u + t) Age x x + u x + u + t The arrows in the timeline indicate where failure or death would occur in the computations of the probabilities S x (u + t) = P (T x > u + t) and S x (u) = P (T x > u). Think of S x (u) S x (u + t) as subtracting intervals of failure, indicated by the arrows. If you subtract the interval of failure described by S x (u) from the interval of failure described by S x (u + t), you get the interval of failure for u t q x. By removing the dashed lines in the timeline below, you get the timeline of the interval of failure for u t q x. Hence, u t q x = S x (u) S x (u + t). S x (u) S x (u + t) Age x x + u x + u + t As a notational convention, if t = 1, t p x, t q x, and u t q x are written as p x, q x, and u q x respectively.

2 CHAPTER 1. CONTINUOUS SURVIVAL MODELS Using these new notations, we can write: tp x = 1 t q x (1.41) u tq x = u p x u+t p x (1.42) d µ x+t = dt [ tp x ] = d tp x dt [ln tp x ] by equations (1.13), (1.15) (1.43) f x (t) = t p x µ x+t by equation (1.14) (1.44) tp x = exp tq x = tp x = t t t µ x+s ds by equation (1.16) (1.45) tp x µ x+t dt by equations (1.4), (1.44) (1.46) tp x µ x+t dt by equations (1.5), (1.44) (1.47) tp x = x+t p xp by equation (1.1). (1.48) It is a strongly suggested exercise for the reader to verify these relationships using their survival function definitions. These relationships will be important throughout the course. Another notational convention that is used is to write e x = E[T x ]. e x is known as the complete expectation of life. 8 By the discussion in section 1.3, assuming that E[T x ] exists and is finite, e x = = = = tf x (t) dt t t p x µ x+t dt by equation (1.44) (1.49) S x (t) dt by equation (1.39) tp x dt. (1.5) Intuitively, equation (1.5) makes sense. If you sum up all of the infinitesimally small intervals that (x) can survive, you should get the expected lifetime of (x). Using equation (1.4), the kth moment of T x is given by, assuming each moment exists and is finite, E[(T x ) k ] = k t k 1 tp x dt. (1.51) 8 The adjective complete is used to make the distinction between the complete and curtate expectations of life. Discussion on the curtate expectation of life will be in the next chapter.

1.5. MORE FORMULAS TO KNOW 21 Needless to say, the amount of new notation is massive in this section. In section 1.7, we provide a formula summary sheet. 1.5 More Formulas to Know Using equation (1.1), we have for u, t >, S x (t + u) = S (x + t + u) S (x) = [ S (x + t) S (x) ] [S (x + t + u) S (x + t) ] = S x (t)s x+t (u) by equation (1.1). (1.52) Using the actuarial notation in section 1.4, equation (1.52) can be written as: t+up x = t p x u p x+t. (1.53) Intuitively, equation (1.53) makes sense. The probability that (x) survives t + u years is the probability that x survives t years and, given T x > t, the probability that (x + t) survives u years. (1.53) can be looked at as an application of the general multiplication rule for two events. Recall from probability that this rule states that for two events E and F, P (E)P (F E) = P (E F ). If we let E be the event of (x) surviving to x + t and F be the event of surviving to x + t + u, given (x) survives to x + t, then we have P ((x) survives to x + t) P ((x) survives to x + t + u (x) survives to x + t) = P ((x) survives to x + t (x) survives to x + t + u) = P ((x) survives to x + t + u) Equation (1.53) will prove useful in future chapters. However, in general, failure probabilities do not follow the general multplication rule. That is, for u, t >, t+uq x t q x u q x+t. When trying to find probabilities of failure given probabilities over disjoint periods, it is usually best to use equation (1.41) to switch to survival probabilities, and then use equation (1.53) as necessary. Using equations (1.42) and (1.53), we can write: u tq x = u p x u+t p x = u p x u p x t p x+u = u p x (1 t p x+u ) = u p x t q x+u. (1.54) Intuitively, (1.54) makes sense. The probability that (x) fails within the age interval (x+u, x+u+t] 9 is the probability that (x) survives to age x + u and fails within the following t years. 9 See the discussion on page 17. For T x, it does not matter whether we use (), [], (], or [), since T x is continuous. However, for the discrete time-to-failure random variable, the meaning of these intervals does differ. A discussion on the discrete time-to-failure random variable is in the next chapter.

22 CHAPTER 1. CONTINUOUS SURVIVAL MODELS 1.6 SOA Released Questions Be sure to download the SOA 39 sample questions. 1 I will refer to these as the SOA 39. In addition, be sure to have a copy of the tables provided with exam MLC. 11 I will refer to the Illustrative Life Table in the tables provided for MLC as the ILT. At the time of this writing, the May 212, November 212, and May 213 MLC exams have been released by the SOA. Relevant Questions: SOA 39: #13, 22, 32, 59, 98, 116, 155, 171, 188, 2. A distribution present in some of the SOA s sample questions is known as the Modified de Moivre s Law, where for a constant α >, S x (t) = ( ω x t α ω x ) (1.55) α µ x+t = (1.56) ω x t E[T x ] = ω x α + 1. (1.57) 1 At the time of this writing, they can be found at http://www.soa.org/files/edu/edu-213-spring-mlc-que.pdf. 11 At the time of this writing, the ILT can be found at http://www.soa.org/files/edu/edu-213-mlc-tables.pdf.

1.7. FORMULA SUMMARY SHEET 23 1.7 Formula Summary Sheet T x = T x T > x Uniform Distribution (de Moivre s Law): 1 f x (t) = ω x t F x (t) = ω x S x (t) = ω x t ω x 1 µ x+t = ω x t E[T x ] = ω x 2 (ω x)2 V [T x ] = 12 Exponential Distribution (constant force of mortality µ): Gompertz Distribution: where B (, 1) and c > 1. Makeham Distribution: f x (t) = µe µt F x (t) = 1 e µt S x (t) = e µt E[T x ] = 1 µ V [T x ] = 1 µ 2. µ x+t = Bc x+t S x (t) = exp [ Bcx (c t 1) ] ln c where B >, c > 1, and A > B. µ x+t = A + Bc x+t S x (t) = exp [ Bcx ln c (1 ct ) At]

24 CHAPTER 1. CONTINUOUS SURVIVAL MODELS Relationships among formulas: tp x = S x (t) = 1 t q x t+up x = t p x u p x+t tq x = F x (t) = 1 t p x u tq x = u p x u+t p x = u p x t q x+u d µ x+t = dt [ tp x ] = d tp x dt [ln tp x ] f x (t) = t p x µ x+t tp x = exp tq x = tp x = t t tp x = x+t p xp e x = E[T x ] = = E[(T x ) k ] = k t tp x µ x+t dt tp x µ x+t dt t t p x µ x+t dt tp x dt µ x+s ds t k 1 tp x dt

Chapter 2 Discrete Survival Models References: AMLCR, Chapter 2.6, Supplementary Notes 1 ; MQR, Chapters 5.3.6., 6.3.3., 6.3.4. Before we discuss discrete survival models, we discuss symbols that will be used throughout the rest of this text. 2 2.1 Commonly Used Symbols In order to abbreviate statements throughout this text, we define the following sets: 1. The set of integers is denoted Z, which is the set {, 3, 2, 1,, 1, 2, 3, }. Z is the set of all positive and negative whole numbers. We say that x is an integer by writing x Z. 2. The set of natural numbers is denoted N, which is the set {1, 2, 3, }. 3 N is the set of all whole numbers greater than or equal to 1. We say that x is an natural number by writing x N. 3. The set of rational numbers is denoted Q, which is the set of numbers that can be written in the form a, where a, b Z and b. In other words, the rational numbers is the set of b all ratios of integers with a nonzero denominator. We say that x is an rational number by writing x Q. 4. The set of irrational numbers is denoted R Q, 4 which is the set of all numbers that are not rational numbers. We say that x is an irrational number by writing x R Q. 5. The set of real numbers, denoted R, is the set R = Q (R Q). We say that x is an real number by writing x R. 1 These are provided in the current syllabus. For the spring 213 sitting, see http://www.soa.org/files/edu/edu- 212-spring-mlc-studynotes.pdf. 2 Readers familiar with undergraduate real analysis may skip section 2.1 without loss of continuity. For the reader familiar with undergraduate linear algebra or abstract algebra, it is suggested that he or she goes through the definitions of the floor and ceiling functions. 3 Some textbooks use as the lowest number of N, rather than 1. 4 The symbol is used to denote difference among sets. For example, if X, Y are sets, X Y is the set consisting of elements that are in X but not Y. Some texts may also use to denote difference among sets. 25

26 CHAPTER 2. DISCRETE SURVIVAL MODELS Let x R and R Z be defined by x = max{k Z k x}. 5 is known as the floor function, and x is known as the floor of x. One can think of the floor of a number as the rounded down value of the number to the nearest integer, or the integer part of the number. Examples. 1. 5.4 = 5 2. 6.97 = 6 3. 7 = 7. Let x R and R Z be defined by x = min{k Z k x}. is known as the ceiling function, and x is known as the ceiling of x. One can think of the floor function as the rounded up value of the number to the nearest integer. Examples. 1. 5.4 = 6 2. 6.97 = 7 3. 7 = 7. 2.2 K x, K x, R x We define three new random variables in this section. First, K x = T x. (2.1) K x6 is known as the curtate time-to-failure random variable. It is the integer part of the time-to-failure random variable T x defined in chapter 1. 7 We define another random variable as the fractional part of T x, denoted R x, such that T x = K x + R x. (2.2) R x is a continuous random variable where R x [, 1). We define another random variable K x = K x + 1. 8 (2.3) K x is known as the time interval of failure random variable. For example, suppose that (85) fails at age 87.5. Below is a timeline outlining the relationships between K 85, K 85, and T 85. 5 max{k Z k x} is the largest integer k such that k is less than or equal to x. 6 K(x) may also be used on exam MLC. 7 Notice that K x is a discrete random variable. 8 Notice that K x is a discrete random variable.

2.3. PROBABILITIES AND EXPECTATION OF K x 27 R 85 =.5 Age 85 86 87 87.5 88 T 85 = 2.5 K 85 = K 85 = 1 K 85 = 2 K85 = 1 K85 = 2 K85 = 3 The reason why Kx is known as the time interval of failure of (x) is because it is the (annual) interval at which (x) fails. Using the example in the timeline above, there are three age intervals: 85 to 86, 86 to 87, and 87 to 88. Since (85) fails at 87.5, (85) fails in the third age interval, i.e., 87 to 88, and thus, K85 = 3. Also, if 1 T 85 < 2, i.e., if failure were to occur between ages 86 and 87, then K85 = 2, which is the second age interval. Similarly, if T 85 < 1, i.e., if failure were to occur between ages 85 and 86, then K85 = 1, which is the first age interval. Since (85) fails at 87.5, note that K 85 = 2.5 = 2. 9 The fractional part of 87.5 is.5 = R 85. Of course, if T 85 = 2.5, K 85 + R 85 = 2 +.5 = 2.5 by the definition provided in equation (2.2). 2.3 Probabilities and Expectation of K x The concepts in this section can easily be extended to Kx equation (2.3). by using the relationship given by Let k N {}. 1 Notice that K x = k if and only if (x) fails at age x + k or between ages x + k and x + k + 1. 11 Thus, to compute probabilities on K x, we note that P (K x = k) = P (k T x < k + 1) = k q x by the definition on p. 17 12 (2.4) = k p x q x+k by equation (1.54). (2.5) Here is another piece of International Actuarial Notation: we denote e x is known as the curtate expectation of life. 13 e x = E[K x ]. (2.6) 9 This is the rounded down value of 2.5 to the nearest integer. 1 N {} is the set of natural numbers in addition to the number. 11 Not including age x + k + 1, since the floor of k + 1 is k + 1 and not k. 12 Since T x is a continuous random variable, P (k T x < k + 1) = P (k < T x k + 1), which is what is stated in the definition on p. 17. 13 Note the differences between the curtate and complete expectations of life.

28 CHAPTER 2. DISCRETE SURVIVAL MODELS Provided that E[K x ] exists and is finite, by definition, e x = k P (K x = k) k= = k k q x by equation (2.4) (2.7) = k= k= k( k p x k+1 p x ) by equation (1.42) = + (p x 2 p x ) + 2( 2 p x 3 p x ) + 3( 3 p x 4 p x ) + = p x + 2 p x + 3 p x + = k=1 kp x. (2.8) Equation (2.8) follows the same line of intuition as equation (1.5) does. The only difference between these two equations is that (2.8) applies to the discrete case, whereas (1.5) applies to the continuous case. Let r N. Provided each moment of K x exists and is finite, the rth moment of K x is given by E[(K x ) r ] = k= k r ( k p x k+1 p x ) = + (p x 2 p x ) + 2 r ( 2 p x 3 p x ) + 3 r ( 3 p x 4 p x ) + = p x (1 r r ) + 2 p x (2 r 1 r ) + 3 p x (3 r 2 r ) + = k=1 kp x [k r (k 1) r ]. (2.9) Formula (2.9) is not as easy to use as formula (1.51), so we present the formula for the second moment. By (2.9), provided that the second moment exists and is finite, E[(K x ) 2 ] = kp x [k 2 (k 1) 2 ] k=1 = kp x [k 2 (k 2 2k + 1)] k=1 = kp x [2k 1] k=1 = 2 k k p x k=1 k=1) kp x = 2 k k p x e x. (2.1) k=1

2.4. TERM EXPECTATION OF LIFE 29 2.4 Term Expectation of Life T x, T x n Consider the random variable min(t x, n) =, where n R is such that n >. We n, T x > n call this random variable W 1. 14 The expected value of this random variable is given by, using the double expectation formula from exam P/1, 15 Given T x n, the PDF of T x is given by: E[W 1 ] = E[T x T x n]p (T x n) + E[n T x > n]p (T x > n). f Tx T x n(t) = P (T x = t T x n) = P (T x = t T x n) P (T x n) = P (T x = t) P (T x n) = t p x µ x+t nq x. Thus, E[W 1 ] = n n = = n = t ( t p x µ x+t ) dt n q x + n n p x nq x t t p x µ x+t dt + n n p x t ( d dt [ tp x ]) dt + n n p x by equation (1.43) Using integration by parts with u = t and dv = d dt [ tp x ] dt, n t ( d dt [ tp x ]) dt + n n p x. (2.11) E[W 1 ] = n n p x = n n p x + = n n n tp x dt + n n p x tp x dt + n n p x tp x dt. (2.12) The standard symbol for equation (2.12) is e x n, which is known as the term expectation of life. 16 The term expectation of life is the expected future lifetime for (x) over the next n years only. 14 Note that there is no symbol or standard letter given to this variable in the texts for exam MLC. We are calling this W 1 for convenience. 15 The double expectation formula for two random variables X and Y is E[X] = E Y [E X [X Y ]]. 16 It is also known as the partial expectation of life.

3 CHAPTER 2. DISCRETE SURVIVAL MODELS 2.5 Temporary Curtate Expectation of Life Let n N. There is an analogous concept in the discrete case of what is in section 2.4. Consider the K x, K x n random variable min(k x, n) = n, K x > n. We call this random variable W 2 for convenience. As W 2 is a function of K x, we write Notice that E[W 2 ] = n k= k ( k p x k+1 p x ) + k=n+1 n ( k p x k+1 p x ) by equations (2.4) and (1.42). ( k p x k+1 p x ) = n+1 p x n+2 p x + n+2 p x n+3 p x + n+3 p x + k=n+1 = n+1 p x. Thus, n ( k p x k+1 p x ) = n n+1 p x. (2.13) k=n+1 Now, n k= Therefore, k ( k p x k+1 p x ) = + (p x 2 p x ) + 2( 2 p x 3 p x ) + 3( 3 p x 4 p x ) + + n( n p x n+1 p x ) = p x + 2 p x + 3 p x + + n p x n n+1 p x = n k=1 kp x n n+1 p x. (2.14) n E[W 2 ] = kp x n n+1 p x + n n+1 p x = k=1 n k=1 kp x. (2.15) The standard symbol for equation (2.15) is e x n, which is known as the temporary curtate expectation of life. The temporary curtate expectation of life is the expected number of whole years that (x) lives over the age interval (x, x + n].

2.6. SOA RELEASED QUESTIONS 31 2.6 SOA Released Questions Relevant Questions: SOA 39: #21, 28, 65, 12, 161, 171, 189, 27; November 212: #3; April 213: #2. There are a few concepts that are covered in the sample questions that are not covered in AMLCR or MQR. 17 For questions in which µ x+t is a piecewise function, the following will be useful. Consider for (x), e x n. Suppose we wanted to write this in terms of e x m, where m < n. Noting that e x n = n tp x dt, we have e x n = Set u = t m. 18 Then du = dt and n tp x dt = m = e x m + tp x dt + m n m n tp x dt. tp x dt e x m + m n tp x dt = e x m + n m m m n m = e x m + = e x m + n m = e x m + m p x m+up x du m+up x du mp x u p x+m du n m up x+m du = e x m + m p x e x+m n m. (2.16) (2.16) should be memorized. It comes up in quite a few of the sample questions as an extremely convenient shortcut. We can also derive a similar formula for the discrete case. Consider for (x), e x n. Then for integral m < n, n e x n = kp x = k=1 m k=1 kp x + = e x m + n k=m+1 n k=m+1 kp x kp x 17 I thank Jim Daniel for suggesting that I add this section. 18 Credit goes to Steve Paris for giving me this step.

32 CHAPTER 2. DISCRETE SURVIVAL MODELS Set u = k m. Then e x m + n k=m+1 kp x = e x m + = e x m + n m m+up x u+m=m+1 n m m+up x u=1 n m = e x m + mp x u p x+m u=1 n m = e x m + m p x up x+m u=1 = e x m + m p x e x+m n m. (2.17) Notice that (2.17) and (2.16) are almost identical, with the exception of the on top of the e symbol for (2.16). (2.16) and (2.17) should be considered first priority for this exam, as they do come up in the sample questions. The following equation has not come up in the released questions, but may be useful. Consider now, for (x), e x. For n <, we have Set u = t n. Then du = dt and e x = e x n + lim w n w tp x dt = n = e x n + tp x dt + n = e x n + lim w tp x dt = e x n + lim = e x n + lim w n tp x dt n w w n tp x dt tp x dt. w u+np x du n n w n = e x n + n p x lim w = e x n + n p x np x u p x+n du w n up x+n du up x+n du = e x n + n p x e x+n. (2.18)

2.6. SOA RELEASED QUESTIONS 33 Similarly, in the discrete case, Set u = k n. Then e x = e x n + lim k=1 w w k=n+1 kp x = n k=1 kp x + = e x n + lim k=n+1 w w k=n+1 kp x = e x n + lim kp x kp x. w n w n+up x n+u=n+1 w n = e x n + lim w np x u p x+n u=1 = e x n + n p x e x+n. (2.19) Again, notice how (2.18) and (2.19) are nearly identical. The case in which n = 1 of (2.19) does come up in the sample questions: note that this states that e x = e x 1 + p x e x+1 = p x + p x e x+1 = p x (1 + e x+1 ). (2.2)

34 CHAPTER 2. DISCRETE SURVIVAL MODELS 2.7 Formula Summary Sheet K x = T x T x = K x + R x K x = K x + 1 P (K x = k) = P (k T x < k + 1) = k q x E[K x ] = e x = k p x q x+k = k P (K x = k) k= = kp x k=1 E[(K x ) r ] = kp x [k r (k 1) r ] e x n = E[min(T x, n)] = e x n = E[min(K x, n)] = k=1 n tp x dt = e x m + m p x e x+m n m, for m < n n k=1 kp x = e x m + m p x e x+m n m, for m < n e x = e x n + n p x e x+n e x = e x n + n p x e x+n = p x (1 + e x+1 )

Chapter 3 Life Tables References: AMLCR, Chapters 3.2-3.3, 3.6-3.9 1 ; MQR, Chapters 5.4, 6.1-6.4, 6.6.1-6.6.2, 6.7. 3.1 Calculating Probabilities What is a life table? Let x N {}. Below is an example of a life table. x l x 5 5 51 49 52 47 53 43 54 37 55 29 As usual, x represents the age of a person or thing. l x represents the expected number of lives of a population who are alive of age x. 2 A life table is a table which has (at the very least) the following information: 1. n ages x, x 1,, x n 1 where each x i N {} i N 3 with a starting age x () = min{x, x 1,, x n 1 } 4 and limiting age ω = max{x, x 1,, x n 1 }, so that x () x i ω x i ; 2. n values l x, l x1,, l xn 1 corresponding to each respective age x i. The number of lives with the starting age, i.e., l x(), is known as the radix. Of course, for these definitions to make sense, we assume that each x i and l xi x i. 5 For the life table illustrated above, x () = 5 is the starting age, l 5 = 5 is the radix, and ω = 55 is the limiting age. 1 The supplementary notes to AMLCR provide background information for the sections from AMLCR that we have omitted in this chapter (3.4 through 3.5). We will not focus too much on these sections, as they only provide background information, and nothing really with assistance in solving exam MLC problems, judging by the released exams. We will use the background from these chapters to give a context to the concepts in this chapter, but we will not dedicate a whole section to them, as AMLCR does. 2 MQR uses l x instead of l x. MQR s notation will not be used on exam MLC. 3 is read as for all or for each. 4 Note that AMLCR uses x for the starting age. We use x () to indicate that it is an order statistic, i.e., the smallest value. 5 A more rigorous approach to the life table can be found in [3, pp. 41-42]. 35

36 CHAPTER 3. LIFE TABLES Let s try calculating probabilities with this table. Examples. Assume that the following questions are based on a population whose life table is given above. 1. What is the probability that someone age 5 survives to age 51? Solution. There are currently 5 lives age 5. If this particular person survives to age 51, then he or she must be one of the 49 people who survived to age 51. This probability is p 5 = l 51 = 49 l 5 5 =.98. 2. What is the probability that someone age 5 dies before reaching age 51? Solution. We want to find q 5. Notice that q 5 = 1 p 5 = 1.98 =.2. When working with life tables, the relationships we have developed still apply. Alternatively, there are currently 5 lives age 5. 49 of these people survive to age 51, which implies that 1 have died before reaching age 51. Thus, q 5 = 1 5 =.2. 3. What is the probability that someone age 53 survives to age 55? Solution. We want to find 2 p 53, i.e., the probability that someone age 53 survives two years. There are 43 people who are age 53. 29 of them survive to age 55, so therefore, 2p 53 = 29 43.674. Alternatively, by equation (1.53), we know that 2p 53 = p 53 p 54. The probability of someone age 53 surviving one year is l 54 = 37. The probability of someone l 53 43 age 54 surviving one year is l 55 = 29 l 54 37. Thus, 2p 53 = p 53 p 54 = ( 37 43 ) (29 37 ) = 29 43.674.6 4. What is the probability that someone age 5 survives to age 52 but dies before reaching age 53? Solution. This probability is given by 2 q 5 = 2 p 5 q 52 by equation (1.54). Now notice that 2p 5 = l 52 = 47 l 5 5 =.94 and q 52 = 4.85. Using the exact values for these two quantities, 47 we multiply them to get.8. Make sure you try the examples above on your own until you are comfortable calculating probabilities with life tables. We define for t ω x (), l x() +t = l x() t p x(). (3.1) This makes sense, as it says that the expected number of lives who are alive with age x () + t is the number who are alive at x () multiplied by the probability that they survive t years. 6 In this case, using (1.53) was not necessary, but depending on the information you are given in a question, you may have to use (1.53). It will be especially useful when estimating probabilities on life tables for non-integral ages of failure.

3.1. CALCULATING PROBABILITIES 37 Thus, if x is such that x () x x + t ω, l x+t = l x() +x+t x () = l x() x+t x() p x() by (3.1) = l x() x x() p x() t p x() +x x () by (1.53) = l x() x x() p x() t p x = l x() +x x () t p x by (3.1) = l x t p x. (3.2) Therefore, tp x = l x+t l x. (3.3) We used this equation in the examples in this section without mathematically formalizing it. See examples 1 and 3. By equation (1.41), we have that tq x = 1 l x+t l x = l x l x+t l x. (3.4) Consider the numerator l x l x+t. What is this actually describing? If we look at the example life table in this section, for example, l 5 l 52 = 3. What does this quantity describe? It is the number of deaths that have occured in the age interval (5, 52). The number of deaths over this period is described by the symbol 2 d 5, i.e., from the ages of 5 to 52. In general, the number of deaths in the interval (x, x + t) is t d x = l x l x+t. 7 Thus, tq x = t d x l x. (3.5) We used this equation in example 2 without mathematically formalizing it. We could also present a formula for u t q x, but as you see from example 4, it is best just to use equation (1.54) with (3.3) and (3.4). 7 AMLCR does not mention this notation, nor is it in International Actuarial Notation by Perryman. However, both of these sources do mention d x, the number of deaths that occur in the age interval (x, x + 1). The Notation and Terminology used on Exam MLC document for spring 213 does not provide any comment on whether or not this notation will be used.

38 CHAPTER 3. LIFE TABLES As an exercise, complete the following life table. The values that have been filled in are the ones stated in the example, in addition to d 55. Round final answers to three decimal places at most. x l x p x q x d x 5 5.98.2 51 49 52 47 53 43 54 37 55 29 1 The solution is given below. Due to rounding, p 51 + q 51 does not equal 1. x l x p x q x d x 5 5.98.2 1 51 49.96.41 2 52 47.915.85 4 53 43.86.14 6 54 37.784.216 8 55 29.655.345 1 Here are some observations to notice about life tables that we can get from this particular life table: 1. l x is non-increasing as the age, x, increases. We do not introduce additional lives into l x at any age. This would cause our p x and q x values to fall outside the interval (, 1). 2. As l x is non-increasing as x increases, p x is non-increasing as x increases. 3. As p x is non-increasing as x increases, q x and d x are non-decreasing as x increases. 3.2 Fractional Age Assumptions If you look at any of the life tables in section 3.1, you would notice they all are based on integral ages. But what if we want to calculate probabilties for non-integral ages? For example, what is the probability that (52) survives to age 53.25? If we are just given a life table like the ones in section 3.1, we would have to make assumptions about the behavior between integral ages. 8 3.2.1 Uniform Distribution of Deaths (UDD) Assumption Suppose that s (, 1) is fixed. If, for some x N {}, we are given l x and l x+1, where x = x + s, and wanted to find l x+s, the simplest way to estimate l x+s would be to use linear interpolation. 8 [1, pp. 12-13] covers the hyperbolic/balducci assumption; however, this has since been dropped from the MLC syllabus and will not be covered in this text.

3.2. FRACTIONAL AGE ASSUMPTIONS 39 Let x be the independent variable and l x be the dependent variable. From Calculus, the slope, m, of the secant line connecting the points (x, l x ) and (x + 1, l x+1 ) is m = l x+1 l x (x + 1) x = l x+1 l x. 9 (3.6) We assume this slope m is constant throughout the interval (x, x + 1). This would imply that l x+s = (l x+1 l x )(x + s) + b for some b R. 1 Of course, l x+ = l x, so that l x = (l x+1 l x )x + b. Thus, we can write l x+s = (l x+1 l x )(x + s) + b = (l x+1 l x )x + b + (l x+1 l x )s = l x + sl x+1 sl x = (1 s)l x + sl x+1. (3.7) Now here is where things get very interesting: define x and s as stated earlier in this section. Then sq x = l x l x+s l x = l x (1 s)l x sl x+1 l x by equation (3.7) = sl x sl x+1 l x = s ( l x l x+1 l x ) = sq x by equation (3.4). (3.8) Equation (3.8) is known as the uniform distribution of deaths (UDD) assumption. 11 The reason why this is known as the uniform distribution of deaths assumption is the following: recall that T x = K x + R x by equation (2.2). It can be shown that (3.8) is equivalent to stating that, for x N {}, R x follows a uniform distribution in (, 1), and K x and R x are independent. 12 Of course, you can use previously developed relationships among formulas to gather many new ways to rewrite formulas under the UDD assumption, but we will focus on only two of these in particular. Notice that, for variable s (, 1), by equation (3.8), But also remember that by definition, s q x = P (T x s) = F x (s). Thus, d ds [ sq x ] = q x. (3.9) d ds [ sq x ] = q x = d ds [F x(s)] = f x (s) 9 Of course, m, as l x is non-increasing as x increases. 1 This is simply the familiar linear equation y = mx + b in a different form. 11 AMLCR calls this the UDD1 assumption. See [3, p. 44]. 12 AMLCR calls this statement the UDD2 assumption. For the purpose of studying for exam MLC, proving this statement is not particularly useful. A proof of this statement can be found in [3, pp. 45-46].