# Calculating Survival Probabilities Accepted for Publication in Journal of Legal Economics, 2009, Vol. 16(1), pp

Save this PDF as:

Size: px
Start display at page:

Download "Calculating Survival Probabilities Accepted for Publication in Journal of Legal Economics, 2009, Vol. 16(1), pp. 111-126."

## Transcription

1 Calculating Survival Probabilities Accepted for Publication in Journal of Legal Economics, 2009, Vol. 16(1), pp David G. Tucek Value Economics, LLC Vinson Court St. Louis, MO Tel: 314/ Cell: 314/ I Introduction A recent discussion on AAEFE-L, an list server for members of the American Academy of Economic and Financial Experts, revealed that a concise explanation of how to calculate the survival probabilities used in the Life/Participation/Employment (LPE) method is absent from the forensic economics literature. 1 This note remedies this situation by presenting an overview of the columns found in a life table, followed examples of survival probability calculations for both whole and fractional ages. II Life Table Overview 2 Table 1 presents an abbreviated life table for all females, regardless of race. This abbreviated table is taken from Table 3 of United States Life Tables, 2004, published by the National Center for Health Statistics (NCHS). The first column in Table 1 of this paper specifies the age ranges for each row; these are one-year ranges except for the last, which corresponds to ages of 100 or more. This life table is a period life table: Rather than being based on the experience of a cohort of individuals born in the same year, a period table presents what would happen to a synthetic cohort through time if it experienced the death rates specified in column (2). These death rates are written as q(x), and equal the probability of dying between ages x and x+1, for integer values of x from 0 to 99. The probabilities are based on death registrations in the U.S. for ages 0 to 84; for ages 85 to 99, the 1

2 probabilities are based on Medicare death rates. In the last row of the table, q(100) equals the probability of dying for ages greater than 100. This probability is set equal to one. The q(x) and an initial synthetic cohort of 100,000 persons at age 0 determine the rest of the table. 3 Column (3) shows l(x), the number of persons surviving from the initial cohort to age x, and column (4) shows d(x), the number of persons dying in each age range. For x 1, l(x) equals l(x-1) d(x-1). The number of deaths is specified by the q(x) and l(x): d(x) = q(x) l(x). The number of person-years lived between ages x and x+1 is shown in column (5) and is written as L(x). For 1 x 99, L(x) = l(x) d(x)/2. For x=0, L(x) reflects an adjustment which accounts for the number of deaths in a year occurring for infants born in the previous year. L(100) is the result of an iterative process that continues until the number of person-years lived is essentially zero; this occurs somewhere between ages 110 and 120. Note that in the table, L(100) is the number of person-years lived from age 100 to infinity. Column (6) is the total number of person-years lived above age x, and is written as T(x). It is the sum of L(x) in column (5), starting at x and continuing through the end of the table. The last column of the table, column (7), is the remaining life expectancy at age x. It is represented by e(x) and equals T(x)/ l(x). III Calculating Survival Probability Whole Ages If the dates of the loss periods correspond to exact ages, the required survival probabilities are easily calculated from the l(x). For example, for a female, if the loss starts exactly at age 20, the probability of surviving one additional year is This probability is calculated in Table 2, and equals l(21)/l(20) or 98,899 divided by 98, This is the probability that would be used to reduce the first year of loss in a wrongful death case. 5 The fourth year of loss would be reduced by multiplying by , or l(24) divided by l(20). 2

3 Most economic damage reports divide losses between past and future periods, with the scheduled trial date serving as the point of demarcation between the past and the future. This presents a choice to the forensic economist in personal injury cases: Whether to calculate all survival probabilities as of the date of the injury or as of the trial date, setting the probability of survival equal to one for the past period. The first option will understate the loss estimates, since losses are reduced for the probability of an event that, if the case goes forward, did not happen. The second option recognizes the known, or assumed, circumstances as of the date of the trial. If the loss estimates are calculated using the LPE method, and if the probabilities of labor force participation and employment are set equal to their expected value as of the date of the injury, the second option will neither understate or overstate the loss estimates. 6 Under the second approach in a personal injury case, it is assumed that the plaintiff survives up until the start of the future loss period. Accordingly, the survival probability for past losses should be set equal to one, and the survival probabilities corresponding to each future loss should be based on the probability of surviving from the start of the future loss period. For example, suppose the plaintiff was a female injured on her 20 th birthday, but that because of trial delays the future loss period doesn t start until her 26 th birthday. The survival probability associated with the eighth future loss is , or l(34) divided by l(26). IV Calculating Survival Probability Fractional Ages 7 Of course, it would be an extreme coincidence if the start of the past and future loss periods corresponded exactly to a plaintiff s or decedent s birthday. 8 More often than not, survival probabilities for a fractional age will need to be calculated. These calculations must be based on some assumption about the functional form the survival curve determined by l(x) takes between integer values of x. It is common to assume either (1) a linear, (2) an exponential, or (3) a hyperbolic function. Each of these assumptions results in an equation for l(x+t), or for some function of l(x+t), that is a weighted average of 3

4 l(x) and l(x+1), or of expressions involving l(x) and l(x+t), with the weights being determined by the value of t (for 0 t 1). The linear function assumption is equivalent to assuming that deaths between age x and x+1 are uniformly distributed, so that, for 0 t 1, l(x+t) decreases linearly until reaching l(x+1). The slope of the line connecting l(x) and l(x+1) equals l(x+1) l(x), so that l(x+t) = l(x) + t [ l(x+1) l(x)] =t l(x+1) + (1-t) l(x) [1] The exponential function assumption is also known as the constant force of mortality assumption, and assumes that l(x+t) = ab t. Under this assumption, l(x) = ab 0 and l(x+1) = ab 1. From this it follows that a = l(x) and b = l(x+1)/l(x), and that l(x+t) = l(x) [ l(x+1)/l(x)] t ln(l(x+t)) = ln(l(x)) + t [ln(l(x+1) ln(l(x)] ln(l(x+t)) = t ln(l(x+1)) + (l t) ln(l(x)) [2] The hyperbolic function assumption (also known as Balducci s assumption) assumes l(x+t) = 1/[a + b t] or, equivalently, 1/l(x+t) = a + b t. Setting t equal to 0 and 1 in the latter expression produces 1/l(x) = a and 1/l(x+1) = a + b. From this it follows that b=1/l(x+1) 1/l(x) and that 1/l(x+t) = 1/l(x) + t [1/l(x+1) 1/l(x)] 1/l(x+t) = t/l(x+1) + (1 t)/l(x) [3] Equations [1], [2] and [3] all express l(x+t) (or some form of l(x+t)) as a weighted average of some form of l(x) and l(x+1), with the weights determined by the value of t. Table 3 shows the 4

5 calculation of the survival probabilities that would be used in calculating the economic losses of a female plaintiff born on July 4, 1960 and injured on November 1, 2008, using each of these equations. In Table 3, it is assumed that the loss period starts on the day of the plaintiff s injury and that the future loss period starts on February 1, 2010, the date of the trial. Accordingly, the past loss period runs from November 1, 2008 through January 31, In keeping with the second option discussed above, the survival probabilities for this period are assumed to equal one. If losses in future years are presented as ending at the end of each calendar year, then the required survival probabilities correspond to the probability of surviving from January 31, 2010 to December 31, 2010; to December 31, 2011; to December 31, 2012; and so on. The first ten columns in Table 3 are common to all three assumptions, with the first column showing the dates the injured plaintiff survives to. The date in the first row is the end of the past loss period, and survival probabilities are calculated from this date to the date found in each of the subsequent rows. The second column shows the age of the plaintiff as of the date in the first column. Column (3), Date for l(x), is the birthday prior to the date in the first column the first row corresponds to the plaintiff s 49 th birthday, the second row corresponds to her 50 th birthday, and so on. Column (4), l(x), is the number of persons in the synthetic cohort living to the exact age 49, age 50, and so on. Similarly, column (5) corresponds to the plaintiff s birthday immediately following the date in the first column. These birthdays correspond to age 50, age 51, age 52, and so on. The entries in column (6) correspond to the number of persons in the synthetic cohort living to the exact age 50, age 51, age 52, and so on. The Days1 column contains the number of days from Date for l(x) until Date. Similarly, the Days2 column contains the number of days from Date until Date for l(x+1). That is, Days1 corresponds to the interval starting with the birthday prior to Date and ending with Date, while Days2 corresponds to the interval from Date until the next birthday. Their sum is shown in column (9) and equals 365 except when the period includes February 29 th in a leap year. These values determine 5

6 the value of t in the equations above: t equals Days1 divided by the total number of days and is shown in column (10). Columns (11) and (12) show, respectively, the interpolated values of l(x) corresponding to Date, and the probabilities of surviving from the end of the past loss period until Date for the uniform distribution assumption. For example, consider the probability of surviving from January 31, 2010 until year-end The first step is to calculate l(49.6), the value of l(x) corresponding to January 31, The birthdays immediately prior and after this point in time are July 4, 2009 and July 4, These are the dates for l(49) and l(50), respectively. The value for l(49) is taken directly from the NCHS life table for all females and equals 95,733. Similarly, l(50) equals 95,445. There are 211 days from July 4, 2009 until January 31, 2010, and 154 days until the plaintiff s next birthday. The total number of days equals 365, and t equals (211 divided by 365). Assuming that deaths between integer ages are uniformly distributed, the interpolated value of l(49.6) equals ,445 + ( ) 95,733, or 95,567. The second step is to calculate l(53.5), the value of l(x) for December 31, The birthdays immediately before and after year-end 2013 are July 4, 2013 and July 4, These are the dates for l(53) and l(54), respectively. The value for l(53) equals 94,462, and l(54) equals 94,085; these values come directly from the NCHS life table for all females. There are 180 days from July 4, 2013 until yearend 2013, and 185 days from year-end 2013 until the plaintiff s next birthday. The total number of days equals 365, and t equals (180 divided by 365). One minus t equals Assuming that deaths between integer ages are uniformly distributed, the interpolated value of l(53.5) equals 94,276 ( , ,462). The probability of surviving from January 31, 2010 until December 31, 2013 is (94,276 divided by 95,567). 6

7 The next two columns, numbered (13) and (14), show the interpolated values of l(x) and the corresponding survival probabilities for the constant force of mortality assumption. Consider again the probability of surviving from January 31, 2010 until year-end The interpolated value of l(49.6) is calculated using equation [2] above: ln(l(49.6)) = ln(95,445) + ( ) ln(95,733) = This gives e or 95,566 for l(49.6). The value of l(53.5) is given by ln(l(53.5)) = ln(94,085) ln(94,462) = This gives e or 94,276 for l(53.5). The probability of surviving from January 31, 2010 until December 31, 2013 is (94,276 divided by 95,566). The interpolated values of l(x) and survival probabilities corresponding to the Balducci assumption are shown in columns (15) and (16). The interpolated value of l(x) corresponding to January 31, 2010 is calculated using equation [3] above: 1/[l(49.6)] = /95,445 + ( )/95,733 = The interpolated value for l(49.6) equals the inverse of or 95,566, the same value as obtained under the constant force of mortality assumption. The value of l(x) corresponding to yearend 2013 is given by 1/[l(53.5)] = /94, /94,462 =

8 The interpolated value for l(53.5) equals the inverse of or 94,276. Again, this is the same value as was obtained under the constant force of mortality assumption. The probability of surviving from January 31, 2010 until December 31, 2013 is (94,276 divided by 95,566). The rightmost column of Table 3 shows the maximum difference between the three interpolated survival probabilities for each row of the table. The largest difference in this example is and the average of the maximum differences in In order to investigate the magnitude of the differences among the survival probabilities calculated under each of the three assumptions discussed above, the probability of surviving from age 1 through age 99 was calculated by month, where a month was interpreted an equal to one-twelfth of a year. For females, the largest difference equaled and the average of the largest differences (for the interpolated ages only) equaled The comparable results for male survival probabilities are and The resulting interpolated values of the l(x) allow one to calculate the survival probability between any two of the ages shown. This was done for all integer starting ages from 1 to 75, through all ending monthly ages to age 99. The resulting maximum differences among the three assumptions for each starting age, along with the means of the maximum difference for only the interpolated ages, are shown in Table 4. For both females and males, the maximum difference is less than and the average of the maximum difference for the interpolated ages is less than These are not differences that are likely to significantly impact calculations of economic losses. V Rounding Values of l(x) All of the calculations presented in this paper rounded the l(x) before the survival probabilities were calculated in order that the results presented could be duplicated with a hand calculator. Since the NCHS makes Excel versions of its life tables available containing non-integer values for l(x), and for 8

9 d(x), L(x) and T(x) as well, it is natural to ask if rounding such values makes a significant difference. To investigate this question, the life tables for all females and all males were duplicated, using rounded values of l(x), d(x), L(x) and T(x) in each row. For females, the difference in the resulting l(x) was no more than 2 persons, corresponding to ages where the number of survivors in the synthetic cohort exceeded 82,000 persons. For males, the difference in the resulting l(x) was no more than 4 persons. These differences corresponded to ages where the number of cohort survivors exceeded 94,000 persons. VI Conclusion The examples presented above show that calculating survival probabilities for whole or fractional ages is relatively straightforward. Moreover, there does not seem to be a significant difference among the results obtained for interpolated ages using the three common assumptions concerning the shape of the survivor curve between integer ages. In this regard, it is worth noting that the assumption of uniformly distributed deaths between exact ages x and x+1 is equivalent to the assumption made in Arias (2007) that l(x) declines linearly between x and x+1 for ages 1 to 99. This serves as a basis for a small preference for the uniformly distributed assumption over the other two. Finally, using rounded values of l(x), d(x), L(x) and T(x) will result in some minor differences in the derived values of l(x) from the published tables. Given that Excel versions with the unrounded values are readily available, loss calculations should probably not utilize rounded values. However, doing so is not likely to result in materially different results. 9

10 Table 1: Abbreviated Life Table for All Females, United States, 2004 (1) (2) (3) (4) (5) (6) (7) Probability Number Person-years Total of dying dying lived number of between Number between between person-years Expectation ages x surviving to ages x ages x lived above of life to x+1 age x to x+1 to x+1 age x at age x Age q(x) l(x) d(x) L(x) T(x) e(x) , ,465 8,038, , ,368 7,938, , ,332 7,839, , ,309 7,740, : : : : : : : : : : : : : : ,194 1,505 5,442 18, ,689 1,230 4,075 13, or over ,460 3,460 8,935 8,

11 Table 2: Survival Probability for All Females by Start of Loss Period Age That Loss Occurs l(x) = Number Surviving to Starting Age Age x , , Equals 98,899 divided by 98,944 or l(21) divided by l(20) , , Equals 98,169 divided by 98,661 or l(34) divided by l(26) , , , , , , , , , , , , , ,

12 Table 3: Calculation of Survival Probability from 1/31/2010 to year-end through age 99.5 Date of Birth is: 7/4/1960 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Date for Date for Total Date Age l(x) l(x) l(x+1) l(x+1) Days1 Days2 Days t 1/31/ /4/ ,733 7/4/ , /31/ /4/ ,445 7/4/ , /31/ /4/ ,139 7/4/ , /31/ /4/ ,813 7/4/ , /31/ /4/ ,462 7/4/ , : : : : : : : : : : : : : : : : : : : : 12/31/ /4/2057 7,988 7/4/2058 6, /31/ /4/2058 6,194 7/4/2059 4, /31/ /4/2059 4,689 7/4/2060 3,

13 Table 3 (Continued) (11) (12) (13) (14) (15) (16) (17) Constant ** Uniform Distribution ** *** Force of Mortality *** ** Balducci Assumption ** Absolute Difference in Survival Probability Probability of Probability Probability Interpolated Interpolated Interpolated Surviving of Surviving of Surviving l(x) l(x) l(x) from from from for "Date" 1/31/2010 for "Date" 1/31/2010 for "Date" 1/31/ , , , , , , , , , , , , , , , : : : : : : : : : : : : : : 7, , , , , , , , , Largest Difference: Mean Difference:

14 Table 4: Differences in Interpolated Results for Survival Probabilities Through Age 99 Starting Age Largest Difference Females Mean of Difference Largest Difference Males Mean of Difference

15 Table 4: Differences in Interpolated Results for Survival Probabilities Through Age 99 (continued) Starting Age Largest Difference Females Mean of Difference Largest Difference Males Mean of Difference

16 References Arcones, Miguel A., Manual for SOA Exam MLC, Winstead, CT: ACTEX Publications. Arias, Elizabeth, United States Life Tables, 2004, National Vital Statistics Report, Volume 56, Number 9, National Center for Health Statistics. Bowers, Newton L. Jr., Gerber, Hans U., Hickman, James C., Jones, Donald A. and Nesbit, Cicil J. Actuarial Mathematics, 1986, The Society of Actuaries, New York. Townsend, Jules A Date of Injury or Date of Trial: A Comment on Work Life Expectancy Calculations, Litigation Economics Digest, Vol. II, No. 2: In addition to the LPE method, survival probabilities can be used by forensic economists to calculate the expected present value of a pension, or to reduce the expected loss to survivors in a death case to reflect their own mortality. 2 The discussion in this section is based on Arias (2007). 3 For the first and last age ranges, additional information is required for the number of person-years lived shown in column (5). See Arias (2007) and the discussion below. 4 The columns after l(x) in Table 2 correspond to the age the loss period starts; the rows correspond to the age in the year each subsequent loss occurs. There is nothing special about the ending age of 26 for the columns or of 35 for the rows. The table was stopped at these ages only due to size considerations. Note that l(x) in the Excel spreadsheet from the NCHS underlying Table 1 are not integer values even though they are displayed that way. The l(x) in Table 2 and all subsequent tables have been rounded to the nearest whole number in order that the calculations can be duplicated using a calculator. 5 Alternatively, the probability of surviving only six additional months might be used to reduce the first year loss. This approach would be consistent with the logic underlying mid-period discounting for example, though the difference in the overall results would be negligible. Under this approach, the subsequent losses would be reduced using the probabilities of surviving 18 months, 30 months, 42 months, and so on. 6 Townsend (1997) has addressed this issue in the context of work life expectancy (WLE). He maintains that the pre-injury WLE expectancy should be used, since use of a post-injury WLE may enrich the plaintiff by extending the total loss period. While this is may be true, use of the pre-injury WLE implicitly assumes that the probability of surviving during the past loss period was less than one as of the trial date, contrary to the assumed or known circumstances. 7 The calculations discussed in this section are based on Bowers, et. al. (1986) and on Arcone (2008). 8 Even if they did, survival probabilities for fractional ages might be needed, for example, if annual loss estimates corresponded to a calendar year. 16

### A Comparison of Period and Cohort Life Tables Accepted for Publication in Journal of Legal Economics, 2011, Vol. 17(2), pp. 99-112.

A Comparison of Period and Cohort Life Tables Accepted for Publication in Journal of Legal Economics, 2011, Vol. 17(2), pp. 99-112. David G. Tucek Value Economics, LLC 13024 Vinson Court St. Louis, MO

### The Impact of the SSA Cohort Tables on Work Life Expectancy: An Application of the Skoog-Ciecka-Krueger Transition Probabilities

2014 The Impact of the SSA Cohort Tables on Work Life Expectancy: An Application of the Skoog-Ciecka-Krueger Transition Probabilities David G. Tucek Value Economics, LLC 13024 Vinson Court St. Louis, MO

### Contents. 3 Survival Distributions: Force of Mortality 37 Exercises Solutions... 51

Contents 1 Probability Review 1 1.1 Functions and moments...................................... 1 1.2 Probability distributions...................................... 2 1.2.1 Bernoulli distribution...................................

### I. CENSUS SURVIVAL METHOD

I. CENSUS SURVIVAL METHOD Census survival methods are the oldest and most widely applicable methods of estimating adult mortality. These methods assume that mortality levels can be estimated from the survival

### Calculating Changes in Worklife Expectancies and Lost Earnings in Personal Injury Cases

Calculating Changes in Worklife Expectancies and Lost Earnings in Personal Injury Cases Michael Nieswiadomy Eugene Silberberg Abstract This paper utilizes the Bureau of Labor Statistics (BLS) new worklife

### Using Life Expectancy to Inform the Estimate of Tail Factors for Workers Compensation Liabilities

Using Life Expectancy to Inform the Estimate of Tail Factors for Workers Compensation Liabilities Michael Shane, FCAS, MAAA and Dawn Morelli, ACAS, MAAA Abstract Traditional accident year paid and/or reported

### APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS

CHAPTER 8 Current Monetary Balances 395 APPENDIX Interest Concepts of Future and Present Value TIME VALUE OF MONEY In general business terms, interest is defined as the cost of using money over time. Economists

### Discount Rates and Life Tables: A Review

Discount Rates and Life Tables: A Review Hugh Sarjeant is a consulting actuary and Paul Thomson is an actuarial analyst at Cumpston Sarjeant Pty Ltd Phone: 03 9642 2242 Email: Hugh_Sarjeant@cumsar.com.au

### The Relation between Two Present Value Formulae

James Ciecka, Gary Skoog, and Gerald Martin. 009. The Relation between Two Present Value Formulae. Journal of Legal Economics 15(): pp. 61-74. The Relation between Two Present Value Formulae James E. Ciecka,

### Manual for SOA Exam MLC.

Chapter 5. Life annuities. Section 5.7. Computing present values from a life table Extract from: Arcones Manual for the SOA Exam MLC. Spring 2010 Edition. available at http://www.actexmadriver.com/ 1/30

### Manual for SOA Exam MLC.

Chapter 4. Life Insurance. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam MLC. Fall 29 Edition. available at http://www.actexmadriver.com/ c 29. Miguel A. Arcones.

### Research Note: Assessing Household Service Losses with Joint Survival Probabilities

Research Note: Assessing Household Service Losses with Joint Survival Probabilities Victor A. Matheson and Robert A. Baade December 2006 COLLEGE OF THE HOLY CROSS, DEPARTMENT OF ECONOMICS FACULTY RESEARCH

### COMMUTATION INSTRUCTIONS

COMMUTATION INSTRUCTIONS The following examples illustrate various methods of commuting permanent disability and life pension benefits. Examples A, B and C apply to permanent disability and utilize Table

### Exponentials. 1. Motivating Example

1 Exponentials 1. Motivating Example Suppose the population of monkeys on an island increases by 6% annually. Beginning with 455 animals in 2008, estimate the population for 2011 and for 2025. Each year

### Practice Exam 1. x l x d x 50 1000 20 51 52 35 53 37

Practice Eam. You are given: (i) The following life table. (ii) 2q 52.758. l d 5 2 5 52 35 53 37 Determine d 5. (A) 2 (B) 2 (C) 22 (D) 24 (E) 26 2. For a Continuing Care Retirement Community, you are given

### Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

### The Impact of the IRS Retirement Option Relative Value

University of Connecticut DigitalCommons@UConn Honors Scholar Theses Honors Scholar Program May 2005 The Impact of the IRS Retirement Option Relative Value Robert Folan University of Connecticut Follow

### Measurement of Incurred but Unreported Deaths in Life Settlements Donald F. Behan

Measurement of Incurred but Unreported Deaths in Life Settlements Donald F. Behan ABSTRACT The traditional application of incurred but unreported insurance claims adds unreported claims to the reported

### THE MATHEMATICS OF LIFE INSURANCE THE NET SINGLE PREMIUM. - Investigate the component parts of the life insurance premium.

THE MATHEMATICS OF LIFE INSURANCE THE NET SINGLE PREMIUM CHAPTER OBJECTIVES - Discuss the importance of life insurance mathematics. - Investigate the component parts of the life insurance premium. - Demonstrate

### Introduction to survival analysis

Chapter Introduction to survival analysis. Introduction In this chapter we examine another method used in the analysis of intervention trials, cohort studies and data routinely collected by hospital and

### Memorandum. To: Pension Experience Subcommittee From: Bob Howard Date: July 31, 2013 Subject: CPM-RPP14 Mortality Tables Document 213060

Memorandum To: Pension Experience Subcommittee From: Bob Howard Date: July 31, 2013 Subject: CPM-RPP14 Mortality Tables Document 213060 1 BACKGROUND This memorandum sets out the method that I used for

### 10/7/2010. I. Population Demography A. Inputs and Outflows B. Describing a population C. Summary II. Population Growth and Regulation

/7/ Population Ecology I. Population Demography A. Inputs and Outflows B. Describing a population C. Summary II. Population Growth and Regulation A. Exponential Growth B. Logistic Growth C. What limits

### Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

### Since the early 1980s, economic experts have been using computerized spreadsheets to make their

Calculating Lost Earnings: Algebraic vs. Spreadsheet Method Frank D. Tinari* Abstract Since the early 1980s, economic experts have been using computerized spreadsheets to make their calculations of lost

### 1. Introduction. 1.1 Objective

Second International Comparative Study of Mortality Tables for Pension Fund Retirees T.Z.Sithole (Kingston University London), S.Haberman (Cass Business School, City University London) and R.J.Verrall

### INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 INSTITUTE AND FACULTY OF ACTUARIES EXAMINATION 8 October 2015 (pm) Subject CT5 Contingencies Core Technical

### Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page

Errata for ASM Exam C/4 Study Manual (Sixteenth Edition) Sorted by Page 1 Errata and updates for ASM Exam C/Exam 4 Manual (Sixteenth Edition) sorted by page Practice exam 1:9, 1:22, 1:29, 9:5, and 10:8

### Mathematics 96 (3581) CA (Class Addendum) 3: Inverse Properties Mt. San Jacinto College Menifee Valley Campus Spring 2013

Mathematics 96 (358) CA (Class Addendum) 3: Inverse Properties Mt. San Jacinto College Menifee Valley Campus Spring 203 Name This class handout is worth a maimum of five (5) points. It is due no later

### Size of a study. Chapter 15

Size of a study 15.1 Introduction It is important to ensure at the design stage that the proposed number of subjects to be recruited into any study will be appropriate to answer the main objective(s) of

VALUING YOUR PENSION BENEFITS AND THE ASSET ALLOCATION IMPLICATIONS By William W. Jennings and William Reichenstein Asset allocation should be based on the family s extended portfolio that includes the

### The terms and conditions of the plan, analysis and assumptions should be understood before the negotiations begin.

HOW TENABLE IS YOUR PENSION VALUATION? The following is provided courtesy of Contributing Editor Garrick G. Zielinski, CFP, CDFA, CDS. Mr. Zielinski is President of Divorce Financial Solutions, LLC specializing

### To give it a definition, an implicit function of x and y is simply any relationship that takes the form:

2 Implicit function theorems and applications 21 Implicit functions The implicit function theorem is one of the most useful single tools you ll meet this year After a while, it will be second nature to

### Math 419B Actuarial Mathematics II Winter 2013 Bouillon 101 M. W. F. 11:00 11:50

Math 419B Actuarial Mathematics II Winter 2013 Bouillon 101 M. W. F. 11:00 11:50 1 Instructor: Professor Yvonne Chueh Office: Bouillon 107G (Tel: 963-2124) e-mail: chueh@cwu.edu Office hours: M-Th 1-1:50

### Question 2: How do you solve a matrix equation using the matrix inverse?

Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients

### SOA EXAM MLC & CAS EXAM 3L STUDY SUPPLEMENT

SOA EXAM MLC & CAS EXAM 3L STUDY SUPPLEMENT by Paul H. Johnson, Jr., PhD. Last Modified: October 2012 A document prepared by the author as study materials for the Midwestern Actuarial Forum s Exam Preparation

### Canadian Insured Payout Mortality Table 2014 (CIP2014)

Mortality Table Canadian Insured Payout Mortality Table 2014 (CIP2014) Annuitant Experience Subcommittee Research Committee February 2015 Document 215006 Ce document est disponible en français 2015 Canadian

### Math 370/408, Spring 2008 Prof. A.J. Hildebrand. Actuarial Exam Practice Problem Set 2 Solutions

Math 70/408, Spring 2008 Prof. A.J. Hildebrand Actuarial Exam Practice Problem Set 2 Solutions About this problem set: These are problems from Course /P actuarial exams that I have collected over the years,

### Linear Programming Notes VII Sensitivity Analysis

Linear Programming Notes VII Sensitivity Analysis 1 Introduction When you use a mathematical model to describe reality you must make approximations. The world is more complicated than the kinds of optimization

### 7: Compounding Frequency

7.1 Compounding Frequency Nominal and Effective Interest 1 7: Compounding Frequency The factors developed in the preceding chapters all use the interest rate per compounding period as a parameter. This

### A power series about x = a is the series of the form

POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to

### Joint Probability Distributions and Random Samples (Devore Chapter Five)

Joint Probability Distributions and Random Samples (Devore Chapter Five) 1016-345-01 Probability and Statistics for Engineers Winter 2010-2011 Contents 1 Joint Probability Distributions 1 1.1 Two Discrete

### PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be

Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our

### 2.4 Applications (Exponential Growth/Decay)

26 CHAPTER 2. METHODS FOR SOLVING FIRST ORDER ODES 2.4 Applications (Exponential Growth/Decay) Many things grow (decay) as fast as exponential functions. In general, if a quantity grows or decays at a

### 9.1 Observations about the exponential function. In a previous chapter we made an observation about a special property of the function.

Chapter 9 Exponential Growth and Decay: Differential Equations 9.1 Observations about the exponential function In a previous chapter we made an observation about a special property of the function namely,

### David G. Tucek Value Economics, LLC 13024 Vinson Court St. Louis, MO 63043 Tel: 314/434 8633 Cell: 314/440 4925 David.Tucek@valueeconomics.

1 Economic Controversy in Personal Injury Cases An Example of a Bad Example Submitted for Publication in the Journal of the Missouri Bar, (Draft 2/14/2012) David G. Tucek Value Economics, LLC 13024 Vinson

### SOCIETY OF ACTUARIES. EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE QUESTIONS

SOCIETY OF ACTUARIES EXAM MLC Models for Life Contingencies EXAM MLC SAMPLE QUESTIONS The following questions or solutions have been modified since this document was prepared to use with the syllabus effective

### By: Garrick G. Zielinski, CFP, CDFA, CDS Divorce Financial Solutions, LLC Milwaukee, Wisconsin

ANATOMY OF A COLLABORATIVE PENSION VALUATION By: Garrick G. Zielinski, CFP, CDFA, CDS Divorce Financial Solutions, LLC Milwaukee, Wisconsin The terms and conditions of the plan, analysis and assumptions

### ICASL - Business School Programme

ICASL - Business School Programme Quantitative Techniques for Business (Module 3) Financial Mathematics TUTORIAL 2A This chapter deals with problems related to investing money or capital in a business

### Manual for SOA Exam MLC.

Chapter 4. Life Insurance. Extract from: Arcones Manual for the SOA Exam MLC. Fall 2009 Edition. available at http://www.actexmadriver.com/ 1/14 Level benefit insurance in the continuous case In this chapter,

### May 2012 Course MLC Examination, Problem No. 1 For a 2-year select and ultimate mortality model, you are given:

Solutions to the May 2012 Course MLC Examination by Krzysztof Ostaszewski, http://www.krzysio.net, krzysio@krzysio.net Copyright 2012 by Krzysztof Ostaszewski All rights reserved. No reproduction in any

### MAT Solving Linear Systems Using Matrices and Row Operations

MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 2. Cashflows. Section 2.4. Dollar weighted and time weighted rates of return. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for

### Number Skills for Public Health

Number Skills for Public Health Rounding, Percentages, Scientific Notation and Rates These notes are designed to help you understand and use some of the numerical skills that will be useful for your studies

### 6 Gaussian Elimination

G1BINM Introduction to Numerical Methods 6 1 6 Gaussian Elimination 61 Simultaneous linear equations Consider the system of linear equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 +

### Note on growth and growth accounting

CHAPTER 0 Note on growth and growth accounting 1. Growth and the growth rate In this section aspects of the mathematical concept of the rate of growth used in growth models and in the empirical analysis

### PERPETUITIES NARRATIVE SCRIPT 2004 SOUTH-WESTERN, A THOMSON BUSINESS

NARRATIVE SCRIPT 2004 SOUTH-WESTERN, A THOMSON BUSINESS NARRATIVE SCRIPT: SLIDE 2 A good understanding of the time value of money is crucial for anybody who wants to deal in financial markets. It does

### Manual for SOA Exam MLC.

Chapter 4. Life insurance. Extract from: Arcones Fall 2009 Edition, available at http://www.actexmadriver.com/ (#1, Exam M, Fall 2005) For a special whole life insurance on (x), you are given: (i) Z is

TABLE OF CONTENTS 1. Survival A. Time of Death for a Person Aged x 1 B. Force of Mortality 7 C. Life Tables and the Deterministic Survivorship Group 19 D. Life Table Characteristics: Expectation of Life

### Preliminaries. Chapter Compound Interest

Chapter 1 Preliminaries This chapter introduces interest rates and growth rates. The two topics are closely related, so we treat them together. The concepts discussed here are not in Barro, but they will

### Eigenvalues and eigenvectors of a matrix

Eigenvalues and eigenvectors of a matrix Definition: If A is an n n matrix and there exists a real number λ and a non-zero column vector V such that AV = λv then λ is called an eigenvalue of A and V is

### VALUING FLOATING RATE BONDS (FRBS)

VALUING FLOATING RATE BONDS (FRBS) A. V. Rajwade * Valuing Floating Rate Bonds (FRBs) 1. The principal features of floating rate bonds can be summarised simply: these are bonds day T-bill, refixed every

### ALASKA DEPARTMENT OF LABOR ALASKA WORKERS' COMPENSATION BOARD 2005 WEEKLY COMPENSATION RATE TABLES

ALASKA DEPARTMENT OF LABOR ALASKA WORKERS' COMPENSATION BOARD 2005 WEEKLY COMPENSATION RATE TABLES FOR INJURIES THAT OCCURRED ON OR AFTER JANUARY 1,2005 THROUGH DECEMBER 31,2005 THE MAXIMUM WEEKLY COMPENSATION

### The Advantage Testing Foundation Solutions

The Advantage Testing Foundation 013 Problem 1 The figure below shows two equilateral triangles each with area 1. The intersection of the two triangles is a regular hexagon. What is the area of the union

### Annuities. Lecture: Weeks 9-11. Lecture: Weeks 9-11 (STT 455) Annuities Fall 2014 - Valdez 1 / 43

Annuities Lecture: Weeks 9-11 Lecture: Weeks 9-11 (STT 455) Annuities Fall 2014 - Valdez 1 / 43 What are annuities? What are annuities? An annuity is a series of payments that could vary according to:

### Version 3A ACTUARIAL VALUATIONS. Remainder,Income, and Annuity Examples For One Life, Two Lives, and Terms Certain

ACTUARIAL VALUATIONS Remainder,Income, and Annuity Examples For One Life, Two Lives, and Terms Certain Version 3A For Use in Income, Estate, and Gift Tax Purposes, Including Valuation of Pooled Income

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### Contents. Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!)

Page Contents Introduction and Notes pages 2-3 (These are important and it s only 2 pages ~ please take the time to read them!) Systematic Search for a Change of Sign (Decimal Search) Method Explanation

### Chapter Two. THE TIME VALUE OF MONEY Conventions & Definitions

Chapter Two THE TIME VALUE OF MONEY Conventions & Definitions Introduction Now, we are going to learn one of the most important topics in finance, that is, the time value of money. Note that almost every

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 2. Cashflows. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall 29 Edition,

### Elasticity. I. What is Elasticity?

Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in

### Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

### Probability Models for Continuous Random Variables

Density Probability Models for Continuous Random Variables At right you see a histogram of female length of life. (Births and deaths are recorded to the nearest minute. The data are essentially continuous.)

### Basic Formulas in Excel. Why use cell names in formulas instead of actual numbers?

Understanding formulas Basic Formulas in Excel Formulas are placed into cells whenever you want Excel to add, subtract, multiply, divide or do other mathematical calculations. The formula should be placed

### CHAPTER 17, DEFINED BENEFIT ACCRUALS

CHAPTER 17, DEFINED BENEFIT ACCRUALS by Avaneesh Bhagat, (Western) Ann Trichilo, (Rulings and Agreements) INTERNAL REVENUE SERVICE TAX EXEMPT AND GOVERNMENT ENTITIES TABLE OF CONTENTS INTRODUCTION:...

### Memorandum. I means the improvement rate in mortality for persons aged x nearest birthday at the start of. To: From:

Memorandum To: From: Pension Experience Subcommittee Bob Howard Date: February 13, 2014 Subject: 1. BACKGROUND CPM 2014 Mortality Tables Document 214014 This memorandum sets out the method that I used

### Methodology For Calculating Damages in the Victim Compensation Fund

Methodology For Calculating Damages in the Victim Compensation Fund The calculation of presumed economic loss will generally follow the guidelines established in the original VCF. The Special Master will

### Why the Gamboa-Gibson Disability Work- Life Expectancy Tables Are Without Merit

Thomas Ireland. 2009. Why the Gamboa-Gibson Disability Work-Life Expectancy Tables Are Without Merit. Journal of Legal Economics 15(2): pp. 105-109. Why the Gamboa-Gibson Disability Work- Life Expectancy

### Calculating Railroad Retirement Employee Annuities - Benefit Information

Calculating Railroad Retirement Employee Annuities - Benefit Information Many railroad workers and annuitants want to know how their railroad retirement benefits are calculated. The following describes

### ACTUARIAL NOTATION. i k 1 i k. , (ii) i k 1 d k

ACTUARIAL NOTATION 1) v s, t discount function - this is a function that takes an amount payable at time t and re-expresses it in terms of its implied value at time s. Why would its implied value be different?

### ECONOMICS #1: CALCULATION OF GDP

ECONOMICS #1: CALCULATION OF GDP Dr. Paul Kutasovic Mathematics by Dr. Sharon Petrushka Introduction Gross Domestic Product or GDP is the broadest measure of the health of the US economy. Real GDP is defined

### NPV Versus IRR. W.L. Silber -1000 0 0 +300 +600 +900. We know that if the cost of capital is 18 percent we reject the project because the NPV

NPV Versus IRR W.L. Silber I. Our favorite project A has the following cash flows: -1 + +6 +9 1 2 We know that if the cost of capital is 18 percent we reject the project because the net present value is

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 3. Annuities. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall 2009 Edition,

### WEEK 6: FORCE, MASS, AND ACCELERATION

Name Date Partners WEEK 6: FORCE, MASS, AND ACCELERATION OBJECTIVES To develop a definition of mass in terms of an object s acceleration under the influence of a force. To find a mathematical relationship

### A, we round the female and male entry ages and

Pricing Practices for Joint Last Survivor Insurance Heekyung Youn Arkady Shemyakin University of St. Thomas unded by the Society of Actuaries Abstract In pricing joint last survivor insurance policies,

### ECONOMIC DAMAGES CHECKLIST PERSONAL INJURY

ECONOMIC DAMAGES CHECKLIST PERSONAL INJURY 1. Official name of case as filed: Person for whom loss is to be calculated: 2. Attorney s name: Firm name: Attorney s email address: Attorney s phone number:

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Estimating economic loss - recent developments. Richard Cumpston

Estimating economic loss - recent developments Richard Cumpston This paper has been written for presentation at the Victorian State Conference of the Australian Plaintiff Lawyers Association, 23 to 25

### Matrix Operations on a TI-83 Graphing Calculator

Matrix Operations on a TI-83 Graphing Calculator Christopher Carl Heckman Department of Mathematics and Statistics, Arizona State University checkman@math.asu.edu (This paper is based on a talk given in

### Econ-UA 323 Development Economics. Outline of Answers to Problem Set 7

Professor Debraj Ray 19 West 4th Street, Room 608 Office Hours: Mondays 2.30 5.00pm email: debraj.ray@nyu.edu, homepage: http://www.econ.nyu.edu/user/debraj/ Webpage for course: click on Teaching, and

### COMPTROLLER GENERAL OF THE UNITED STATES WAslllNtTON O.C. 2OW@

1 COMPTROLLER GENERAL OF THE UNITED STATES WAslllNtTON O.C. 2OW@ B-213348 125452 SEf'TEMBER27,1384 The Honorable Nancy L. Johnson House of Representatives RELEASED Subject: Comparison of Estimates of Effects

### SUPPLEMENT TO CHAPTER

SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved

### Developing Statistical Based Earnings Estimates: Median versus Mean Earnings

Lawrence M. Spizman. 2013. Developing Statistical Based Earnings Estimates: Median versus Mean Earnings. Journal of Legal Economics 19(2): pp. 77 82. Developing Statistical Based Earnings Estimates: Median

### The Simplex Solution Method

Module A The Simplex Solution Method A-1 A-2 Module A The Simplex Solution Method The simplex method is a general mathematical solution technique for solving linear programming problems. In the simplex