PENSION MATHEMATICS with Numerical Illustrations

PENSION MATHEMATICS with Numerical Illustrations Second Edition Howard E. Winklevoss, Ph.D., MAAA, EA President Winklevoss Consultants, Inc. Published by Pension Research Council Wharton School of the University of Pennsylvania and University of Pennsylvania Press Philadelphia © Copyright 1977 (first edition) and 1993 (second edition) by the Pension Research Council of the Wharton School of the University of Pennsyl vania All rights reserved Library of Congress Cataloging-in-Publication Data Winklevoss, Howard E. Pension mathematics with numerical illustrations / Howard E. Winklevoss. -2nd ed. p. em. Includes bibliographical references and index. ISBN 0-8122-3196-1 I. Pensions-Mathematics. 2. Pensions-Costs-Mathematics. 3. Pension trusts-Accounting. I. Title. HD7105.W55 1993 331.25'2-dc20 92-44652 CIP Printed in the United States ofAmerica Chapter 3 Basic Actuarial Functions The purpose of this chapter is to introduce several actuarial functions used in the development of pension mathematics throughout the remainder of the book. The discussion begins with the composite survival function and interest function, per haps the two most basic concepts in pension mathematics. Pen sion plan benefit functions are then presented, followed by a dis cussions of annuities, the latter representing a combination of interest and survival functions. COMPOSITE SURVIVAL FUNCTION The composite survival function represents the probability that an active plan participant survives in service for a given pe riod, based on all of the decrement rates to which the employee is exposed. Whereas the probability of surviving one year in a sin gle-decrement environment is equal to the complement of the rate of decrement, the probability of surviving one year in a mul tiple-decrement environment is equal to the product of such complements for each applicable rate of decrement. The proba bility of an active participant aged x surviving one year is p~T} = (1 - q~m») (1 - qi'») (1 - qid ») (1 - qir») , (3.1a) or equivalently, m d r PY) = Pl ) pi') pl ) pl ). (3.lb) This same probability can be expressed in terms of multiple decrement probabilities: m d PY) = 1 - (qi ) + qi') + qi ) + ql'»). (3.2a) 31 32 Pension Mathematics A common approximation for multiple-decrement probabilities is illustrated for the mortality probability as follows: qlm) '" q;m) (1 _ ~q;t»)(l _ ~q~d») (1 _ ~q~r»). (3.2b) The probability of surviving in active service for n years is equal to the product of successive one-year composite survival proba bilities: n-I (T) - II p(T) nP x - x+t· (3.2c) 1=0 Table 3-1 shows the probability of surviving to age 65 from each attained age, based on nine different entry ages, under the various rates of decrement set out in Chapter 2. Since the termi nation decrement includes a 5-year select period, probabilities are given for each of the first five years subsequent to the nine entry ages. The chance of reaching retirement, at least under the illustrative assumptions, is quite low even up through attained age 50, where the probability is 0.62. Since retirement-related pension cost estimates are in direct proportion to the survival probability (Le., the lower the probability the lower are such es timates), it is clear that discounting for the various decrements reduces pension costs significantly. Obviously, the cost reduc tion is mitigated to the extent that a benefit is provided with each type of decrement (e.g., a vested termination benefit, a death ben efit, or a disability benefit). The calculation of survival probabilities in accordance with the above definitions can be quite arduous. The probabilities, however, are readily obtainable from a service table. This table shows the hypothetical number of employees, from an arbitrary initial number, who survive to each future age. The initial num ber, or radix, is generally taken to be some large value, such as 100,000 or 1,000,000, and IY) is the notation for the survivors at age x. 1 The total number of employees leaving active service dur ing the year is denoted by d;T) and defined as d;T) = IY)qY). (3.3a) 1Although not indicated by the symbol Ix(T), it is understood to represent the number of survivors at age x who entered the plan at age y. At a later point the more general symbollx\9 will be used, which makes the entry age variable explicit. 3. Basic Actuarial Functions 33 TABLE 3-1 Probability of Surviving in Service to Age 65, ..-xp!} Select Ages Entry Attained Age y y+l y+2 y+3 y+4 x Age 20 0.02 0.03 0.04 0.05 om 0.08 25 0.10 26 0.11 27 0.13 28 0.15 29 25 om 0.09 0.11 0.13 0.15 0.17 30 0.19 31 0.21 32 0.23 33 0.25 34 30 0.14 0.17 0.20 0.23 0.25 0.28 35 0.30 36 032 37 034 38 0.36 39 35 0.24 0.28 0.31 0.33 0.36 0.38 40 0.40 41 0.43 42 0.45 43 0.47 44 40 0.35 0.38 0.41 0.44 0.47 0.49 45 052 46 054 47 057 48 0.60 49 45 0.47 0.51 0.54 0.57 0.60 0.62 50 0.65 51 0.69 52 0.72 53 0.75 54 50 0.51 0.55 0.58 0.61 0.64 0.79 55 0.80 56 0.81 57 0.82 58 0.84 59 55 0.59 0.63 0.67 0.70 0.74 0.85 60 0.87 61 Q.89 62 0.92 63 0.96 64 60 0.74 0.79 0.84 0.89 0.94 1.00 65 34 Pension Mathematics Total decrements from the active population equal the sum of each separate decrement: diT) = d;m) + di /) + did) + dir) (3.3b) m d r = IY) (qi ) + qi') + qi ) + qi ». (3.3c) Table 3-2 illustrates the concept of a service table for 1,000,000 entrants at age 20, based on the decrement assumptions specified in Chapter 2. 2 The probability of an age-20 entrant surviving in active service to age 65 is easily found from this table: (T) _ Ig) _ 24,448 6S-20P20 -/(T) - 000000 = 0.0244. (3.4a) 20 1, , Similarly, the probability of an employee age 40 surviving to age 65 would be l (T) = Ig = 24,448 = 0.3745. (3.4b) 2SP4Q (T) 65276 140 ' Although service tables are an important source of computa tional efficiency, even when working with high speed computers, they are used only occasionally in presenting the theory of pen sion mathematics in this book. INTEREST FUNCTION The interest function is used to discount a future payment to the present time. It plays a crucial role in determining pension costs and, like the survival function of the previous section, it re duces such values. If i, is the interest rate assumed for the tth year, the present value of one dollar due in n years is given by 1 (3.5a) (1 + il) (1 + i2) ... (1 + in) and, ifh =iz =... =in ,we have 1 (3.5b) (1 + if 2To the extent that any rates are entry-age dependent. such as termination rates, a separate service table is applicable to each entry age. 3. Basic Actuarial Functions 3S TABLE 3-2 Service Table x I}T) d~m) dY) d~d) dY) dY) 20 1,000,000 442 243,002 263 0 243,708 21 756,292 350 169,718 201 0 170,270 22 586,023 286 . 121,314 158 0 121,757 23 464,265 238 88,543 126 0 88,907 24 375,358 202 65,921 103 0 66,226 25 309,132 176 49,933 85 0 50,194 26 258,938 156 38,460 72 0 38,688 27 220,251 140 3O,OW 62 0 30,251 28 189,999 129 23,814 53 0 23,996 29 166,004 119 19,113 47 0 19,280 30 146,724 112 15,529 56 0 15,697 31 131,027 107 12,754 50 0 12,911 32 118,116 103 10,576 45 0 10,725 33 107,392 101 8$15 41 0 9,017 34 98,375 99 7,510 38 0 7,647 35 90,727 98 6,419 35 0 6,552 36 84,176 98 5,534 41 0 5,673 37 78,503 99 4,816 46 0 4,960 38 73,543 100 4,224 50 0 4,374 39 69,169 102 3,738 54 0 3,893 40 65,276 104 3,338 57 0 3,499 41 61,777 108 3,004 60 0 3,172 42 58,605 114 2,7Z7 69 0 2,910 43 55,695 123 2,491 76 0 2,690 44 53,006 133 2,290 83 0 2,506 45 50,499 144 2,121 89 0 2,354 46 48,145 156 1,969 94 0 2,219 47 45,926 169 1,841 99 0 2,108 48 43,818 181 1,721 107 0 2,009 49 41,808 194 1,616 115 0 1,925 50 39,884 206 1,517 121 0 1,845 51 38,039 219 1,424 127 0 1,769 52 36,270 230 1,335 135 0 1,700 53 34,570 241 1,244 142 0 1,628 54 32,942 252 1,159 148 0 1,559 55 31,383 267 0 156 0 423 56 30,960 286 0 166 0 452 57 30,508 305 0 182 0 487 58 30,020 326 0 203 0 529 59 29,491 350 0 235 0 585 60 28,907 377 0 281 0 659 61 28,248 405 0 348 0 753 62 27,495 433 0 436 0 869 63 26,626 459 0 549 0 1,008 64 25,618 485 0 685 0 1,170 65 24,448 0 0 0 24,448 24,448 36 Pension Mathematics The following simplifying definition is used in connection with the present value function: v = _1_.

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