<<

's Table by Walter J. Mays, A.S.A. Abstract

The great antiquity of Ulpian•s Table, which dates from about 220 A.D., gives it a very prominent place in the history of mortality tables, but it has received rather scanty treatment in actuarial literature. There has been some uncertainty whether it was intended to represent what we know as the expectation of life or the present value of a life annuity under an implicit interest assumption. Greater uncertainty has been ex?ressed concerning the extent to which the values might have been based upon observation of an actual life group. The present paper ex.~bits a relationship, which does not seem to have been noticed before, indicating that Ulpian 1 s Table is a refinement of another table described by Ulpian's contemporary, Aemilius Macer, and that to this extent, at least, the values are not arbitrary. Arguments are presented in support of the belief that the values originally represented the expectation of life. Some exposition of the background of is given with quotations from the sources, including the text of Macer's statement of Ulpian 1 s Table together with a translation and notes. These appear in Appendix A. Mortality rates are derived from age 25 upward on the assumption that

Makeham 1 s Law holds. The mathematical theory used to develop such mortality rates is given in Appendix B.

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I. Introduction

The table attributed to the Roman praetorian prefect Domitius Ulpianus (composed legal treatises 211-222 A.D.; died 228 A.D.) antedates the early annuity tables of De Witt and Halley by more than 14 centuries and it was in actual use for valuing life annuities as late as the early 1800's, Although noticed in nearly all treatises on the history of mortality tables, it has attracted less investigation than its antiquity would seem to warrant. The references commonly cited are the papers presented to the Institute of 1 2 over a century ago by Hendriks and Hodge These papers, however, discuss Ulpian's Table only incidentally. It is included by Hendriks in a survey of many records from ancient and medieval times that relate in some way to the concept of , but the whole survey was preliminary to the main theme of his paper, the restoration of Johann De Witt's treatise on annuities. The subject of Hodge's paper was the history of interest rates, Hendriks and Hodge were both of the opinion that Ulpian's Table represented the expectation of life rather than the present values of a life annuity under some implicit interest assumption. Hodge, however, was not inclined to believe that the values were based on any accurate observation of lives.

1 Frederick Hendriks: Contributions to the and of the Theory of Life Contingencies. Assurance Magazine Vol. II, pages 224-5 2 William Berwick Hodge: On the Rates of Interest for the Use of Money in Ancient and Modern Times. Assurance Magazine, Vol. Vl, pages 313-4 -96- - 3 -

II. Relationship bet·•een Ulpian 1 s Table and Macer 1 s Table

Our ultimate authority for Ulpian 1 s Table is the praetorian prefect Aemilius Macer, another prominent Roman jurist and contemporary of Ulpian. Extracts from the writings of both Ulpian and Macer are contained in the Digesta, or of the opinions of eminent jurists collected by the order of the Emperor Justinian at Constantinople about the year 530 A.D. Although nearly one-third of this large compilation is taken from the writings of Ulpian, only a single paragraph attributed to Macer (Digest, Lib. XXXV, Tit. II, lxviii) contains the statement of Ulpian•s Table. (See text, translation and notes in Appendix A of the present paper.) In the same paragraph Macer gives an alternative method which he says was the one commonly used in computing the value of aliments and usufructs (forms of life income which are discussed in Section III of this paper). This was to use a constant or 30 years for ages under 30 and to decrease such factor by one year for each year or age over 29. For convenience, we shall refer to this as Macer's Table. It may be exhibited together ~ith

Ulpian 1 s Table in the following table, in which the values are designated as expectations or life for reasons that will be given later in this section.

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Table I.

Expectation of Life

~ Macer Ulpian 19 years and under 30 years 30 years 20 - 24 30 28 25 - 29 30 25 30- 34 29 - 25* 22 35 - 39 24 - 20<·' 20 40 19 19 41 18 18 42 17 17 43 16 16 44 15 15 45 14 14 46 13 13 47 12 12 48 11 11 49 10 10

50 - 54 9 - 5* 9 55 - 59 4 - 0* 7 60 and over 5 *Expectation of life by Macer's Table decreases uniformly by one year for each year increase in age in the interval,

The crudeness of Macer's Table is obvious. From age 29 upward, it would represent the curtate expectation of life on the completely unrealistic assumption that all those living at age 29 survived to age 59 and then died in the year before reaching age 60. Nevertheless, it ma.y be the "best" approximation to the expectation of life for some actual group, subject to the restriction that only two linear functions of age can be used for the approximation over the whole ~·-+:-r+-~-+~~+-rl·~+~~~+-~~-+~~~~~~~+-~i~'-+'~i 1 I ! i i ...

I I I ' i I r·· "' i i I f--.1 I e; ' ' • ~ I . j I ! I I! r-- I , •• I I ! '&,- I ' I i i I I

1 l-+-~-r~l-+~~-+~l-+_l~i-~~~-41-~i~~-+~~+·-~~~~~~~, - HI I : ·.~! ~· I I I I 1-- ! I I I i ! ' I ->f. : i'~~~~-r1-+-~-+~~~~-+-r+'~l-r+-~l ~1-+1~4-~11_~!, ~+-~~~~~ 1...1...

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range of ages.

~hen Ulpian's Table is represented graphically together with Macer's Table (Fig. l), the relationship strongly suggests that

Ulpian's Table is a more refined approxi~ation made by introducing ' additional linear functions as corrections to Macer's Table in the 1 t~o age ranges where the latter was likely to have been most seriously in error, that is, in the neighborhood of age 30 where the expectation of life had been overstated and of age 60 where no survivors had been assumed. Granted that without specific information on the source of the data we can only conjecture that both tables were intended to represent some real life group, the graphical comparison adds plausibility to the conjecture. Refinement of Macer's Table would have been pointless had the intention not been to bring the values into closer agreement with reality of some kind. There is, then, a justification for attempting to analyze the underlying mortality pattern, which justification, of course, could not exist if the values in Ulpian's Table were purely arbitrary. This justification is accentuated by the extreme paucity of the mortality data on ancient that have come down to us. Any uncertainty as to whether Ulpian's Table was originally intended to represent the expectation of life or the pres~nt values of a unit life annuity under an implicit interest assumption would seem to be resolved in favor of the expectation of life. If any appreciable interest rate were involved, the values at younger ages would exceed the values under comparatively modern tables, and the average level of mortality rates indicated would be far lower than could be accepted in view of what is known concerning -100- - 6 - mortality rates in antiquity. The following table compares the values from Ulpian's Table at mid-range ages 32 and below with life annuity values from some fairly recent tables (issued since 1910).

Table II Value at Age Name of Table ~ 22 27 32 Ulpian's 30,0 28.0 25.0 22,0

Life Annuity a" English Life No. 8, Male, 3~ 22.8 20.7 19.6 18.4 u.s. Life 1939-41, White Males, 2i% 29.2 26.0 24.5 22.8

1941 cso 2~ 28.6 25.5 24.0 22.3 1937 Std. Annuity, Basic Male, 2~ 30,0 27.0 25.6 23,9 The values in Ulpian's Table and Macer's Table coincide from age 39 through age 50. The unrealistic decrease of one year in value for each year's increase in age in this range is tantamount to ignoring interest and deferring all deaths among those living at age 39 until after age 50. If the Romans had no clear notion of mortality decrements, the same cannot be said for the element of interest. They had great familiarity with its operation in everyday financial transactions. Had the intention been to assume an interest rate, the error in using one-year decrements in annuity values over any extended period of years would very likely have been recognized,

Moreover, there is no hint of interest in the two fo~ms of life income mentioned in the Latin text of Ulpian's Table, the aliment and the usufruct, The fact that Ulpian's Table was used many -101- - 7 -

centuries later as if it were a table of present values of a life annuity with some implicit interest assumption may be attributed to the improvement in mortality rates that took place in the intervening centuries. The improvement in mortality rates was roughly compensated by assuming an interest rate, and the values remained applicable. The coincidence of the values of Ulpian's Table and Macer's Table between ages 39 and 50 seems to have escaped notice on account of misinterpretation of the Latin text. (See the text in Appendix A, lines 16-18.) The part pertaining to Macer's Table, " ••••• ab annis vera triginta, tot annorum computationem inire, 1 quat ad annum sexagesimum deesse videntur", has been read as if it said that from 30 years of age onward, as many years enter the computation as appear to be wanting from the person's age to age 60. The correct interpretation takes the preposition ad to mean

"to, but not including", as opposed to~~ used in other parts of the text to mean "to and including". The quoted passage is then to be translated, "·····from exact age 30 years upward, for as many years to enter the computation as appear to be wanting to (but not including) the 60th year of life". This interpretation makes the values coincide between ages 39 and 50, whereas the former interpretation would make the values by Macer's Table one year in excess of those by Ulpian's Table throughout such age range. It seems likely that the last survivors under Macer's Table were assumed to die in the 60th year of life, and hence the values in both tables represent what we call the curtate expectation of life~ •so by Dr. Sprague, article "Annuities", Encycl. Britannica, 9th Ed. -102- - 8 -

III. The Falcidian Law

Ulpian's Table was originally used for valuing bequests in the form of life income to determine compliance with the Falcidian Law, This law, dating from 40 S.C., provided that a testator had to

leave a clear' fourth (~uarta Falcidia) of his estate to his heir (heres) or heirs free of legacies (legati) to third parties (legatarii). (See Apoendix A for a quotation from Justinian's Instltutiones concerning the Falcidian Law,) Under Roman law the heirs not only succeeded to the testator's property at his death but also were charged with the responsibilities of settling the claims of creditors and carrying out the various bequests to third parties. Wealthy Romans had a predilection for benefitting persons other than the heirs through legacies. Such bequests were made in consideration of friendship to patrons and advocates, or for reasons of political expediency to the Emperor and r~s favorites, or, strangely enou~~. for no clear reason to important personages with whom the testator had no close relationship.

Slaves were given their freedom throu~~ legacies. Macer mentions (Appendix A) legacies left to the State for support of the public exhibitions. Not infrequently, the heirs had littl~ or nothing left for themselves after settling the various claims and legacies, and numerous contentions arose over the distribution of estates, The Falcidian Law was thus directed toward securing a just share of the estate for the heirs, and it provided that excessive legacies should be reduced to whatever extent necessary, in order that each heir should get at least a clear fourth of what his

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share would have been in the absence of legacies. In Macer's statement of Ulpian's Table, there is specific mention of two forms of life income to which the table applied. These were aliments (alimental and usufructs (ususfructus). An aliment provided support. In the singular the word alimentum meant food or nutriment; in the plural aliments was used by the Roman jurists to include not only food but all the necessaries of life. An early meaning of the word was the support or reward that children owed their parents in return for their rearing. A vestige of this ancient alimentary obligation is found today in the justification sometimes cited for the of a parent in the life of his child, the loss of support in old age in event of the child's untimely death. In Scottish law the obligation of support applies reciprocally between parent and child, husband and wife, and grandparent and grandchild. Funds intended for such support are designated as alimentary funds and are exempt from claims of creditors. In bankruptcy cases the funds set aside for support of the bankrupt person are known as alimentary funds. The usufruct meant the ri~~t to enjoy the profits or fruits of property belonging to another without impairing the substance. An example would be the right to receive the rent from a piece of property for life under the obligation to maintain the property in its original condition and without acquiring title to it. Essentially, the aliment and usufruct to which Ulpian 1 s Table applied convey the notion of life income without any suggestion of of idle funds.

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IV. The Mortality Rates Underlying Ulpian's Table

Let us consider first the age range below age 20, for which the expectation of life is stated to be 30 years. If this value is the average for individual ages ranging back into the infant years and if the mortality rates follow a characteristic pattern, high infant mortality would reduce the expectation of life at the earliest ages to such extent that it would rise to a maximum or more than 30 years at some intermediate age, It is reasonable to suppose that the range of ages below age 20 in Ulpian's Table did not extend entirely back to birth, but we cannot be sure what was the lowest age included (see note to line 2 of the Latin text, Appendix A). Nevertheless, on the assumption that there is a maximum value of the complete expectation of life or roughly 33 years somewhere in the range, the corresponding force or mortality would be the reciprocal of 33, or .030. (In Equation (1) of Appendix B, set e~ = 33 and~: 0.) For ages 25 and over, we assume that the mortality rates follow Makeham's Law. To determine the constants A, B, and c, we choose the values of the complete expectation of life (values from Ulpian's

Table increased by~) at ages 27, 45, and 57 as the three anchor values. The choice of age 45 is dictated by the consideration that it lies in the middle of the age range where Ulpian's Table and Macer's Table coincide, and if the former is indeed a correction of the latter, then in the range where no correction was deemed necessary. We recognize, of course, that there are errors in this range on account of the unrealistic decrease of one year in the expectation of life for each year of increase in age, but note that if the value at age 45 is correct, the maximum error in the range is held within the lowest bounds, Ages 27 and 57 are chosen -105- - 11 -

for anchor values, because they lie close to the ages at which Macer's Table was most seriously in error, and presumably the corrected values would have been considered most carefully at such ages. The Makeham constants derived in Appendix B are. as follows: A = .01613025; B = .0000213685; c : 1.14536554. The resulting mortality rates for quinquennial ages are set forth in the following

table, together with mortality rates from other tables for compa~ison.

Table III

Age 1000 q,., Ulpian as % trip{ an All India Halley 1958 CSO U.S. Total of U.S. Total Male ( Breslau Male Population Population II X 1901- 1687- 1959-61 1959-61 1911 1691)

I' 25 16.67 20.3 12.35 1.93 1.26 1323 % 30 17.32 23.5 15.07 2.13 1.43 1211 35 18.60 27.8 18.37 2.51 1.94 959 II 40 21.12 32.3 20,22 3.53 3.00 704 45 26.07 37.2 25.19 5.35 4.76 548 1,1II' 50 35.74 42.8 31.79 8.32 7.74 462 55 54.54 49.9 34.25 13.00 11.61 470 60 90.52 59.8 41.32 20.34 17.61 514 \I' 65 157.47 75.5 52.08 31.75 26.22 601 70 275.37 101.7 77.46 49.79 38.66 712 75 461.64 148.0 113.64 73.37 57.99 796

It may be that the mortality rates for Ulpian's Table shown above are fairly representative of some cross section of the ancient Roman population, at least up to about age 55. If so, one could conclude that mortality in e.ncient Rome in the middle age range was about the same as that which prevailed in European cities a century or so before the beginning of the Industrial Revolution, and better than the -106-

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mortality in some areas of the world, like India, as late as the beginning of the present century. However, the clear purpose in Ulpian's Table was to satisfy the requirements of the Falcidian Law in favor of the heirs. Judicial conservatism might well have increased the values of the expectation of life in order to produce a safe table for such purpose, and the higher values would be consonant with mortality rates lower than those actually prevailing. Beyond age 55 the mortality rates in Ulpian's Table increase very rapidly until Rt the oldest ages they far exceed those in any other table shown in Table III. This pattern is associate~ with the high value of the Makeham constant c (in conjunction with a hi~~ value of B). In most Makehamized tables c is close to 1.09, but the value 1.14536554 used 1 here is almost the same as the value 1.1448 used by Ackland in graduating the All India Male Table (1901-11). His constant B was different and he employed Makeham's Seco~d Modification(~= A+Hx+Bc~). If the very high rates of mortality at the older ages under Ulpian's Table actually prevailed, they would indicate a very primitive state of society in which malnutrition and disease took excessively hi~~ toll among the losing competitors _in the struggle for a share of the necessities of life. On the other hand, judicial conservatism, operating in a different direction than cited above, may have lowered the values of the expeJtation of life beyond age 50 below the true values as a compromise with Macer's termination of the life span in the year before attaining age 60, with the result that mortality rates in the range are too high.

Thomas G. Ackland: On the Estimated Age Distribution of the IndiRn Population as Recorded at the Census of 1911 and the Estimated Rates of Mortality Deduced from a Comparison of t~e Census Returns from 1901 to 1911. JIA, Vol. XLVII, p. 315 -107- - 13 -

Appendix A

The excerpt from Aemilius Macer on Ulpian 1 s Table is contained in Justinian's Digests, Lib. XXXV, Tit. II, lxviii. The text quoted below is from the edition of the Digests in Volum.e I of Corpus Iuris Civilis by Georgius Christianus Gebaver (Gebauer) and Georgius Augustus Spangenberg, Gottingen, MDCCLXXVI, a copy of which is preserved in the Cincinnati Public Library.

1 Aemilius Macer libro II ad legem vicesimam hereditatium. Computation! 2 in alimentis faciendae hanc formam esse Ulpianus scribit ut a prima 3 aetate usque ad annum vicesimum quantitas alimentorum triginta annorum 4 computetur, eiusque quantitatis Falcidia praestetur: ab annis vero 5 viginti usque ad annum vicesimumquintum, annorum vigintiocto: ab 6 annis vigintiquinque usque ad annos triginta, annorum vigintiquinque: 7 ab annis triginta usque ad annos trigintaquinque, annorum vigintiduo; 6 ab ennis trigintaquinque usque ad annos quadraginta, annorum viginti: 9 ab annis quadraginta usque ad annos quinquaginta, tot annorum 10 computatio fit, quot aetati eius ad annum sexagesimum deerit, remisso 11 uno anno: ab anno vero quinquagesimo usque ad annum quinquagesimum 12 quintum, annorum novem: ab annis quinquagintaquinque usque ad annum 13 sexagesimum, annorum septem: ab annis sexaginta cuiuscumque aetatis 14 fit, annorum quinque: eoque nos iure uti Ulpianus sit, et circa 15 computationem ususfructus faciendam. Solitum est taman a prima aetate 16 usque ad annum trigesimum computationem annorum triginta fieri: ab 17 annis vero triginta, tot arJlorum computationem inire, quot ad annum 16 sexagesimum deesse videntur: nunquam ergo amplius quam triginta 19 annorum computatio initur. Sic denique, et si Reipublicae ususfructus 20 legetur, sive simpliciter, sive ad ludos, triginta annorum computatio 21 fit. -108- - 14 -

Translation

Aemilius Macer writes in Book II with reference to the Twentieth

Law of irL~eritances: -- Ulpian writes that the formula for making a computation in case of an aliment is this: that from the earliest age to the 20th year of life inclusive the amount of the aliment should be computed by using 30 years, and the Falcidian Fourth of this amount should be maintained: but from exact age 20 years to the 25th year of life inclusive, 28 years: from exact age 25 years to exact age 30 years, 25 years: from exact age 30 years to exact age 35 years, 22 years: from exact age 35 years to exact age 40 years, 20 years: from exact age 40 years to exact age 50 years, the computation is made by using as many years as are wanting from the person's age to the 60th year of life, with one year left off: but from exact age 50 years to the 55th year of life inclusive, 9 years: from exact age 55 years to the 60th year of life Inclusive, 7 years: from exact age 60 years upward, of whatever age the person may be, 5 years: and Ulpian says that we may with justness use the above also in making the computation of a usufruct. It has been customary, nevertheless, for the computation to be made from the earliest age to the 30th year of life inclusive by using 30 years: but from exact age 30 years upward, for as many years to enter the computation as appear to be wanting to the 60th year of life: therefore the computation is never entered with more than 30 years. Thus, finally, even if a usufruct is bequeathed as a legacy to the State, whether simply or for the exhibitions, the computation is made by using 30 years. -109- - 15 -

Notes on the Text and Translation l. 2 This could mean "from birth" but probably excludes the period of infancy in which the death rate was excessively high. "From the earliest age" seems to be a more appropriate translation.

11 l. 3 usque ad ~ vicesimum Usque ad means to and including". Annum vicesimum means the 20th year of life, that is, the year commencing on the 19th birthday and extending through the day just before the 20th birthday. The entire phrase means the interval of age ending just before exact age 20

years, ?li thout usque, the phrase ad ~ vicesimum would mean the interval of age ending just before exact age 19 years, that is, excluding the 20th year of life. This distinction is very important for consistent interpretation of the entire text. l. 3 al1mentorum The plural of alimentum is used by the Roman jurists to mean "a life support", provision for food and all other necessaries to support someone for life. See Section III of the main part of this paper. l. 4 Falcidia Understand Quarta before Falcidia, meaning the Falcidian Fourth of the estate which was required by the Falcidian Law to be left for the heirs free of legacies to third parties, See Section III. l. 4 ab annis ~ viginti The plural of annus followed by a cardinal number, when used in regard to age, is to be interpreted as the exact age on the birthday. This means of expressing the limit of the age interval is to be -110- - 16 -

contrasted with the singular of~ followed by the

ordinal number as explained in the note to li~e 3 above. 1. 10 quot aetati eius Supply annorum, Genitive of the Whole, after quot. Literally, the whole passage beginning with tot reads: 11 The computation is made with as many years as the number of the years of his age lacks to the 60th year".

1. 10 rem1sso ~ ~ -- This ablative absolute, meaning "one year left off", is used in the sense of an added explanation to make clear that the 60th year of life is excluded -- that the proper difference is the number of years between the person's age and age 59, which fact is already indicated by

the preposition~ in the phrase ad~ sexagesi~.

1. 11 ab ~ ~ quinquagesimo -- This must be interpreted to mean from the end of the 50th year of life. Perhaps the

text should be emended here to read ab annis ~ quinquaginta to agree with similar phrases before and after·it.

1. 13 ab annis sexaginta cuiuscumque aetati fit -- Supply usque ad~ before cuiuscumque and aetas before fit: literally, "from 60 years to the year of whatever age the age happens to be" • 1, 15 ususfructus -- usufruct, a form of life income -- see Section III of the main part of this paper. 1. 19 Reipublicae -- The State would be presumed to have perpetual existence and a legacy in the form of annual income to the State would amount to a perpetuity, but even in this case no more than 30 years' purchase would be used to determine the value in establishing compliance of the testator with the Falcidian Law,

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11 11 l. 20 simpliciter simply , that is, without conditions attached 1. 20 ludos "spectacles" or "exhibitions" given !l.s free entertainment to the public at the Roman Circus

The following excerpt on the Falcidian Law is taken from Justinian's Institutiones II, Tit. xxii, Holland's Second Edition, , 1881.

Novissime lata est lex Falcidia, qua cavetur, ne plus legare liceat quam dodrantem totorum bonorum, id est ut, sive unus heres institutus esset sive plures, apud eum eosve pars quarta remaneret.

Translation Most recently, there was brought forth the Falcidian Law, whereby one is cautioned that it is not lawful to bequeath in legacies more than three-fourths of all one's property, that is, whether one heir II' or several have been appointed, a fourth part should remain to such heir or heirs.

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I I IIIII - 18 - Appendix B

The problem of determining the Makeham constants A, B, and c from

0 given values of the complete expectation of life e~ at three different ages is an interesting one that presents not a few complexities. No particular claim for originality is made for the solution that follows. 1 2 Makeham himself exhibited the role of the Gamma Function and McClintock derived the expansion in infinite series for a continuous life annuity in terms of the Makeham constants. Nevertheless, it seems desirable to develop here the mathematical theory from first principles.

1. The Differential Equation Connecting ex and fL~ The texts on life contingencies. establish the following general relationship:

( 1)

It is readily derived from the definition of e~ in terms of an 0 f. .. infinits integral, e" = o -t 1>,.. alL •

W, M. Makeham: On the Integral of Gompertz•s Function for Expressing the Values of Sums depending upon the contingency of life. The Assurance Magazine, Vol. XVII, p. 305 2 Emory McClintock: On the Computation of Annuities on Mr. Makeham's Hypothesis. The Assurance Magazine, Vol, XVIII, p. 242.

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In order to solve Equation (1), consider first the solution of the differential equation: de,.- _ e_,,--" u. .:. 0 ...... ••. (2) "-" Separating the variables, we have eLl,. -;;;; •••••••••••.•••••••••••••••.•••••••• (3) --;;- = fLx. .,e,. Integrating both terms of (3), we obtain

./Pj 1x. = fix <.6< + f~ C J ••••••••••••••••••••••••• (4) or ..1,}£ = c ~./ j-{y,dx. •••••••••••.••••.•••..••..•.••• ( 5)

In (5) e is the base of natural logarithms, 2.71828~, and C is a constant of integration in so far as the solution of Equation (2) is concerned. For the present , the limits under the integral sign are left off. Consider C, however, as a function of x to be determined, if possible, in such manner that Equation (5) shall be the solution to Equation (1). Differentiating the terms of Equation (5) with respect to x, we obtain d. e" d c .J.u._. e~x , r I,.,., cbt u_ = - -€. r-.. -1'"\...e • ,--x -;;;:; oL"

= .e/JJ¥rl."(:: +tL><) ...... (6)

0 ell,. Substituting the expressions for .e,. and ;;;r,. from (5) and (6) in (1), we obtain .efJ-L,k (:: +tL")- C e.f~.L~ ~ : - I (7) -114- - 20 -

which simplifies to

-I (R./~-'-"tk) iC :: ••••••••••••••••••••••••••••••••• (8) From (8) we obtain by integration -fJA-,_.1.> C = - .e. .J..J< K 1 + ••••••••••••••••••••••••• ( 9) where K is a constant of integration. The solution to Equation (1) results when the expression for c from (9) is substituted in Equation (5): • [1"-M A( f -/!-'-•""" ) ..€..,. = .e. - ~ _c(" + K •••••.•••••.•••• ( 10)

This is the general solution for any form of the function ~~ which is integrable over the range of values of x. Under llakeham's Law,fA"~A+Bc!', and the expression for;,. becomes

<> A>< + ~: (-J -A><-~ J. + I< ) ..e.,. = .L ~ . " ...... (11) where the constant K is now to be determined.

2. Determination of the Constant K

The limits of integration for the integral on the right-hand of (11) are-• andx.. In order that Equation (11) should represent the expectation of life over a portion of the range • of values of" which includes Jt.,.oo, where .e~~.: 0 , it is necessary that

•••••••••••••.•.••• (12)

To evaluate the infinite integral in (12), change the variable by making the substitution ~~: = z Noting that the -115- - 21 -

corresponding lim1 ts or integration for z are now o and oo , we obtain ~ 00 _t,. _ 1 -z. K : .J_ (.,!_ )0; { z. "::e £ J..~ ~c l'1e J, ••.••••••.•.••••• ( 13)

Now, the integral on the right-hand side or (13) is in the rorm or the well known Gamma Function, and we have

K = ~)::,e) ~c l (- ~.J ,...... ,( 14)

Since the Gamma Function is ordinarily tabulated for values of its argument between 1 and 2, and since in practice Q <. ~c <. I , let us make use of the relationship f'c.,tz.) ,. ~('j'"9f'l-,) and write ( 14) as follows: -..!... (~ \ ~'-. cu--:S~ K · t,e ~c.J (- ~ ..)(1- z;;<.) ••••••••••••.••.•••••••••• (15) . 3. Expression for ex in Infinite Series r -A1-~ Let the integral J e ~e""" be integrated by parts. Set Bell -"" ,£(." e- ~otc and d-11• e -"' Then /1.4"-" • "'" - jvoW...

••.•• ( 16)

Repeated integration by parts in this manner yields . s• ~ ~ -A• - ~c .,{,( : -..!.. e -A" - ~~ - .!!.-:, (. ( t..,.:.-A)'- '?~ e A A (£,3".,1.) 1 ad' .. ( :z. t.. c -A) X - ~ e . ~ • "'"1~ A (I- c -A)(2b<;c •A) ~ A' Be" -, (n c- .,--. -r 9~ ·~ q~

-···-~(A C"';c.-A./ .... ""-~l~c.-A' •J " Be. J:_~t.,._-A ) "-- """.,~.;, ... (t7) A (~c.-A)··· (,.l.,c.-A) -116- - 22 -

Substituting in (ll) the expressions obtained in (15) and (17) and dropping the term involving the integral in (17) which

becomes vanishingly small as ~ increases without limit, we have

e : _!. [ {I + -~-. (Be.") I . Be"') .2. .. A (1- ..d...) A A •) ( t..c ~7' + ( 1-- )(.2.-- 7.:"-,c + · .. " I '?< L?< I . (~).... } ~:.(!ft.")~'. T(:t-0J] + + ..... - e ~' 1 ,.. { 1- t.,J( 2-~J··· (n- ~) ~c - T.;c •••••••••••••••• (18) The series within the brace signs in (18) is convergent for all values of the age .!• provided that .£ is not equal to l and ~cis not equal to a positive integer, which conditions are satisfied by all values of the Makeham constants encountered in practice. The ratio of the general term in this series to the preceding :Be." term is~ , which approaches zero as n increases without n-...1!- - limit. The~llmit of this ratio is thus less than unity and the series is convergent by the well known criterion. For large B values of .!• especially whenL,"is relatively large, the series may converge so slowly that a great many terms may be required to obtain the desired degree of approximation. Fortunately, the convergence is rapid for the ages that we use in finding the mortality rates underlying Ulpian. 1 s Table. Except for difference in symbols, the expression for~ in ( 18) is essentially the same result that was obtained by McClintock nearly a century ago 1 for the case of a life annuity at a zero interest rate.

l McClintock, op. cit.

-117- - 23 - 4. Determination or the Makeham Constants from Given Values or eA at Three Different Ages

The Makeham constants A, B, and c are to be determined from We first seek a set or trial values or the constants, which will be tested by substitution in (18). First order corrections will then be derived and applied to obtain a more accurate set or values. It is most important that the trial values be as close to the true values as possible, because first order corrections are valid only in a.very narrow range around the true values. A satisfactory method or obtaining the trial values is the one now to be described, which involves the use or the generalized Makeham tables derived by H. P. Calderon1 • His Table No. I and Table No. II are appended to this section. Since Calderon's paper was presented some 70 years ago and is not readily apcessible to those who do not have copies of the Journal or the Institute or Actuaries dating back that rar, the following explanation or the tables may be helpful. Table No. I shows the force or mortality and Table No. II the corresponding

complete expectation or life in terms or the unit or time~ ror

which log~c = .005. The functions are tabulated for 10-unit intervals, n+t, ranging from n+t = 0 to n+t: !190, and for six equidistant values or A, ranging rrom A = 0 to A = .0011513. The assumed value of B used in constructing the tables is .000002213, but such value is not explicitly involved in the use

1 H. P. Calderon: Some Notes on Makeham's Formula ror the Force of Mortality. Vol. XXXV, JIA, page 157. -118- - 24 - of the tables, as will appear. Now, if we choose a new unit of

time such that log1.c' = • 005m, we can find the force of mortality in terms of the new unit by multiplying the values in Table No.I by .!!! and dividing n-+ t by .!!!• The corresponding values of the complete expectation of life can be found by dividing the values in Table No. II by.!!! and likewise dividing n-+ t by.!!! to get the intervals in terms of the new unit of time. The Makeham constants A and B are multiplied by.!!! to determine their values in terms of the new time interval. The above results of the change in the unit can be understood by analogy with a continuous annuity certain. Suppose that we have a tabulation of the values of such annuity based on a force of interest, log (1+ i), corresponding to an interest rate of i per annum, tabulated for the remaining integral number of years to run, and representing the present value of 1 per annum. We wish to obtain the present values of a continuous annuity of 1 payable over each two years at the same interest rate. The force of interest in terms of the new two-year period is 2log(l+ i), and the present values are half of those already tabulated. Fractional valaes of .!!! are admissible in using Calderon's tables and interpolation is necessary. Even though the Makeham constant

B' under the changed unit of time should not equal mB, if log, c 1 0 is equal to .005m and A' is equal to mA, the values in Table No. I multiplied by.!!! and those in Table No. II divided by.!!! will be those for the changed unit of time except for a constant difference to be added or subtracted in the new period -...;_"'

Reverting now to our given values e,.7 ::ZS.!r , e•s •11/-·S 1 and e.s., ~ 7-s , we find that no factor .!!! produces correspondence with the values -119- - 25 -

1n Table No. II. However, if a column for A~ .0013816 is extrapolated by first differences in Table No. I and by third differences in Table No. II, the factor m = 11.5 will produce very close correspondence. Thus, when we multiply the ages 27,

45, and 57 each by 11,5 and obtain for n+t the respective quantities

310.5, 517.5, and 655.5, the corresponding values of e~·~ in the extrapolated column for A are very close to the given expectations of life, 25.5, 14.5, and 7.5 each multiplied by 11.5, that is, 293.25, 166.75, and 86.25 respectively. Hence, we have immediately two of the constants for Ulpian 1 s Table:

log~c' : 11.5 X .005 .0575, or c' = 1.14156 A' = 11.5 X .0013816 = .0158884

To determine the trial value for the constant B1 , we proceed as follows, Using Table No. I, we extrapolate a value for the force of mortality for A= .0013816 and n+t = 310.5. Such extrapolated value is found to be .0014601. The corresponding value of (A 1 +B'b~) is 11.5 X .0014601, or .0167912. Using the values of A' and c' already found, we obtain B1 = .0000252983. Substituting the values of A I I B' I and c I in the expression for ex given in ( 18) I we find e:/.1 = 25.630, e-45·14.635, and es.,• 7.663, which are too high by .130, .135, and .163 respectively.

Using Equation ( 18) I we recalculate e,.? I e. ..5 I and eS'T I with A I increased by an increment of .00001, but with B' and c' unchanged. We make a similar recalculation with only B' increased by an increment of .0000001 and a third recalculation with only c' increased by an increment of .0001. The results may be summarized in the following form. -120- - 26 -

Increase in Decrease .produced in ~ ~ en A' , .00001 .00405 .00127 .00023 B' , ,0000001 .01698 .01860 .01518 c', .0001 .01989 .02359 .02115

Letting 1 represent the multiple of .00001 to be added to A',~ the multiple of .0000001 to be added to B', and~ the multiple of .0001 to be added to c' to produce the desired corrections in the trial values already found, we set up simultaneous linear equations as follows: .004051+ ,Ol698m+ .Ol989n .13084

.00127l+.Ol860m+ .02359n ~ .13533 • 000231+. Ol518m + .02115n .16315 These yield the solution: 1:; 24.185; m-=. -12.640; n: 38.0554. When corrections are made accordingly and the corrected values of A', B', and c' are substituted in Equation (18), the results, which will. not be given here, show need of further correction. However, the correction can be obtained by reworking Equation (18) for an incremental change in B' only. The final values for A, B, and c are obtained by adjusting the original trial values obtained from Calderon's Tables as follows: A= .015884-t- .00024185 :. • 01613025 B: .0000252983 - .0000039298::. .0000213685 c = 1.14156 -t- • 00380554 : 1.14536554 The numerical details of proving these values by substitution in Equation ( 18) atte exh1 bi ted on page 28 • The resulting values of the complete. expectation. of life are. as follows. e... ; 25.500; e.,..l4.498; e.r ; 7.497 1 -121- - 27 -

We regard the above values as final, despite the differences tn the

• third decimal place in e~S and es 7 They are near the limit of accuracy afforded by some of the tables used (for example, the logarithms of the Gamma Function tabulated by intervals of .001 in Glover's Tables). No comment is needed concerning the computation of the force of mortality, which is readily done by substituting the final values of the constants in the formula, f>< = A+Bc" • In order to determine the corresponding mortality rates, q~ , we derive the formula, as follows:

fo;~P" = -1~ f'-xH dt. .: _;:,'(A +Ec" .. ':.kc

::

--- ~"'+,.... 13 c·--" C:-1.. ., .. )

!3 " C-1 ) I b - 1/3'/2'/'14 (A+ c.. u.c. _,c.,,., /:C. - -. ~"

-122- Final Computation e45 • and e57 /rom Etp.~ation (18) {10) (11) (12-) ('3) (14) (IS) Prngre.ss~ _,_ (jo) >t(?) /Jrotlucr..s o.f (!oj;<{F) ~:::::"4 n +(') {it) clownwarol (f 3) clown ..,arJ. J. OOODDOOO 1· OOL'O 000 J.oooo"" 6 o (1) A • . 016 t.3o.z> I. 134 ¥155.] · 0069U2olll .oo/,91~21 .OioZ,'I&t8o _.,g,:Z.,96 'I/09148I09 .l/tJ9t48 II ->.11 S8f,,f .oo.Jz,?:Jb,'li -oooo.2279 ,o3759,.ZilLij .oo.Jolfjl -I!JI'~~.J/ ·078'1/JOI/ t;/J B = . oooo2136i!i" 2. ·.1'17ofU lio · Do21.J.J1J"V'T . 02>/S¥9/S.t. 7 .OODtJ74/ ./25 13/ 2.7.2 . ••911119.1 ·•9269o!i!l~/ . 04D5JII"f..t (3} .».3 · 01/1223!)321. .oo 0 00 •4- . oo•D r.?.J~ ··::to481.959· ·0731,~>ol'1 IJ.J ~~e. c = .1.3572164 4 .obl3otJt2o . o••oo413 s .J7DOJ.4'-59 .oS!l!/927-'1' c.ooo••L~ (S"J A_ > . 1181146 ltJ6') ·145"J£4.,f'1 1"/frc ' (6} 1 -A_ : . 8ffii5'.389.JI I-,, e. (7) Be•• •. oo6 14,23163' 7.iif (8) Jlc f.l" = • CJ7072.98S~9' L.J,c (9) Be"' ,.J,oS:t-2.4So9

I .I·!J·t! 1\) f-' 0> w "'I ./ Be-"7 1 ( 7) -lo.f,. --r,- = -2.. 2 II .3 9 I 04- ";7•"- {IKJ .loj,.e • -4.142944919 {!9) z;..-..Be" . /oq,.(/' e = {7) >< ( tJ) =. oo2r.t. 'J:l-1-451 7 (2o) ) • (5)x(l7) =--zr.zg,s t-,,c.ft.. ·(lt,q(/,. J¥.,.,. • 1, (21} /.,!/•• {1-/;;c) "' lo; .• {1.) =-. oS4943R5

(2z) /.,J·· r(2-4;J _ . otJ'l!J 2./.5'.5 (23) (1 ')) -1 ~0) - (21) +(22-) 9· 71Soo9f.5-to • 13907083 {.g_4) ~.. (2-3) . S9S,7S4 1-377434-

&n frota!. f,..~ {'.J, t12), tt4J, Q'J f. Ot>{, 'l'lioS /. 0 sJJ ti.J ft., 1. 49S3St24 . I:Z092.2U l!' J ~!i") - PI> -411322~5 . !283 85672 e_._,. 7.497 e-~s . 14--+93 Sonae 1Yole8 on .JUakeAan/s Fornauln fur the Force uf .llurt.:lity.

TAm.E No. I. TADJ.E No. J. -toulin:~ed. Force of Jlortalitg for iulc1'Vnl, for wkicf, 1ogc=·005. Force of lllorlnlitg for inlcn·nl, jOr wlzic/1 log c= ·005. p.'n+t=A+Bcn+t p.'n+t=A+Dcn+t "

--~·------· A- A- A- A- A- A- A• A- A- A- A- A~ A- A• n+l l I•000•3008 ·()()()D:lJO •OOQO()(h.l •0002303 ·000-1605 •0006908 •0009210 •0011513 ·oor361'1 ••• ilOOOOOO •0002303 ·000-1G05 '0011513 •001311, ------0 •000002:? •0002325 il001627 •0006930 •0009232 •0011535 600 •0006698 il009301 ·oou6o3 •0013908 'OU1620S •0018511 2328 4630 6933 9:?35 11538 610 7852 •0010165 12457 H7t.i0 17062 19365 oo.ZI6'8 10 25 .oo:Z2,1lll 20 28 2331 4633 6936 9238 11541 620 8810 llll3 13416 15il8 18020 20323 141!)1 30 31 2334 4636 6939 9241 1154,t, I 630 9886 12189 16i04 19096 21309 40 35 2338 4640 6943 9245 115-JS 540 •001109 13393 16695 1if*98 20300 22613 50 •0000039 •00023-1-2 ·OOO

------~------~----- .Votes - Some . on 11/aA·elurm's Formula /o1· l!te Force of .llorlality.

'rAnr,Jt: No. 11. TAtll.t; No. f].. r:cmfhw-:J. ('!(, ·1plete ll.'t1Jt'Cffllio11s of Lifr i11 tr.r.Jis of tmif of thm· ,:; J-;_,'}Jrt'laliuJls o.f Life i11 lrnttll n.f tmif c;i time r, for tohic/, logc=·005. p.'n+t=A+lJc'Ht 1vhirh logc=·OOG. I'· n+t=A+Br"+t A~ T-A~ --~~=-~=--~A- : A= h --- ~--- h- I A-_----;_-1. .\- A- I •< I I 1/J/ ·OOOOOOtJ ·(1002303 ·000!60S 11!106DOB 11009210 ,-00115131 '001381' ------ooooooo ·0002303 ·0001605 '000691~ ·OOV0210 . ·0011513 ·DOI3Bih· (I 693·307 639·582 591•428 M8·1B8 509'207 474·2-IG , 1500 210•781 20-1•928 l99-311 193•{)16 ~~183-752 ---- 10 i g~g 202•311 196'880 191·GGO 1M·610 181·813 I 17i·167 11P--'90 20 ~~~--:~f ~~~:~~g ;~~:~:~ :~~:~g~ ~g~:~h~ ~~~:~:~l i 193·958 188•927 181·fJ87 179··126 174"939 170'51-l ,,,.439 30 GG3·3G1 613•962 56D·493 52D·3D7 4D3·1B8 460··J26 5:10 185·729 181"079 176·600 172'281 168-119 16l"l02 40 653"383 605·385 562•117 623·0-19 4B7·721 455"714 510 177·630 173·341 169·206 165·213 161·360 157·G-1S 50 643-407 596·7D2 564•710 516•660 482·206 450•950 169·669 550 160·722 161•912 158·230 151•6i1 151·:?:?!.1 60 633·433 588·1B1 647·270 510·228 476•642 4-16•133 560 161•853 158'229 154'727 151•338 14S·OGO 14-1·885 70 623·463 579·552 539•'i99 603"754 471•02B 441"262 570 154o189 150·B70 147·658 14-l·5·17 141•533 BO 570•!.107 632•296 497•237 465•365 146·684 13B·612 613-495 436·337 5BO 143·651 140'713 137•861 135·101 132"420 90 603·531 562·2·J.5 524•761 490•67B 459'652 431•358 139•345 590 136•581 133·900 131'298 128•772 126"3li 517•195 100 593•571 553•567 4B!-077 453'889 426'323 GOO 132•179 129'667 127·22B 124•857 122•553 110 54J.·872 509•597 47Nai 4-1B·077 120•311 5B3·614 421"234 610 125·194 122"916 120•703 ll8·54S 116 453 114·412 120 573 663 536•161 501·96B 470·748 4·J.2•214 416"089 620 UB·397 116•335 114-332 112"380 110·-!SO 491•3()(1 108•627 130 563716 527"436 46.J.·021 436•300 410•888 630 111•785 109'931 108•124 106·361 104·642 140 518•695 486·619 457•251 428·337 105•377 102·965 553·774 405·631 6-10 103•712 102-085 10!N9i 9B·948 97•431 600·9-JO 47B·B!l!) 450·439 42·!·323 4()()-318 150 613•839 650 99•174 97'681 96"222 9!·796 93"403 92·0!2 9o.7/2 160 509•170 471·14D 443"586 41B·25B 660 533·910 394·919 93•179 91'B47 90'543 B9·266 BB·01B 86·796 85-597 170 523•988 492·388 463•370 436·691 412·1·H 389•522 670 B7'399 86"212 85"050 B3•910 82•796 81"703 1BO 483•592 455•562 429"756 405"980 384·040 6BO 514'074 81•836 8Q-782 79·749 78·736 77·7H 76·7iO 190 50-1•169 471-"785 4·J.7•726 422·780 399•765 37B·500 690 76'493 75'561 74·646 73•7-18 72·868 72'003 I 465·966 439•862 415·763 393'501 372•90! 700 200 49-1·273 71•371 70•551 6D·743 68·950 68·171 67'406 >-' 431·972 387•188 710 N 210 4M·3SS 457·137 40B·70B 367·252 66•476 65•754 65·014 64•846 63•660 720 62·984 220 474•514 4-t8·299 424·056 401'613 380•836 361•5.J3 61•805 61'17< 60-551 59•938 59·336 58·7-&2 I 439·452 416•114 39.J.•480 374•4:?5 730 "' 230 46-t-·652 355'77B 57'360 56'809 56·265 55•729 55·:?02 5-1"681 430•597 408·14B 387·310 367·957 740 240 454"805 349·957 63•138 52'660 52•186 51•719 51"260 50"805 421•737 400•160 750 250 44-t-·972 380"103 861"451 344·0B1 49•139 48·725 48"314 471109 47·510 47'115 260 435·155 412·B72 392•149 372·B60 354•899 338·151 760 45·357 45•003 4i·6t8 4-.J.·2D8 43"!153 43•6()(1 270 425•357 404•003 3B4·11B 365'58-J. 348·302 332"166 770 41•801 41•491 41·1B6 40•BS3 40·586 -10•289 7BO 2BO 415'577 3!.15•133 376·068 358•274 M1•661 326·12R 38•449 3B·1B5 37·923 37'661 37·407 37•153 290 405•819 386·262 36B·OOO 350"932 334•976 320·038 790 35'307 35•082 34•B5B 3·1"635 34·-n6 3-&·198 BOO 300 396'0B·l 377'393 359·917 313•560 328•249 313·B97 32'366 32·173 31•9B2 31·793 31•607 31·4:?3 .294.149 B10 310 3B6·374 368•528 351•819 336•160 321•482 307·705 29•622 29·458 29•295 29·13S 28·979 28·8:!2 289-030 B20 320 376·691 359·6BO 3-13·710 328•733 314•676 301'466 27-o6o 26'924 26·786 2G·651 26·518 26•3!_)() 330 367·03B 350•819 335•591 321•2B1 307'833 295'178 B30 24•691 24•570 24•452 24·336 24·:?22 2-1·110 lHO 357'417 341•979 327'46·1 313·807 300•966 288·846 B40 22•490 22·389 22•289 22'191 22•09-J. 21•998 350 3<7'832 333·153 319•33-1 306·313 291•045 282·470 B50 20'457 2(1•372 20•288 20'205 20•122 20·0-!1 860 360 338"28-J. 324·343 311·201 298•802 2B7·101 276·053 18·579 18•511 18·441- 18•378 18•314 18•251 B70 370 32B•777 315·553 303·070 291·276 280'136 269'50B 16·B50 16·792 16·735 16"679 16"625 16·572 B80 SBO 319•316 306·7R5 294·9-l2 2B3•739 273'143 263·106 15·265 15•217 15•170 10·1:?3 15·0i9 15•031 390 309·900 298·015 2B6·B23 276•194 266•12B 256·582 BOO 13·81J 13·771 13·732 13'694 13·6a7 13•(i:?l 400 300"538 289·33•l 27B·716 268•615 259•005 250·027 1100 12·480 12--146 12·413 12·3S2 12·35:? 12•323 P10 410 291•232 280•658 270•625 261·005 252•018 243•446 11•266 11·238 11"211 11·1~ n·wo 11•137 920 420 2BI·9B6 272·021 262•553 253·549 244-991 236·8J2 10·158 10·13-J. 10·lll lO•(){l(l 10-070 10'0.i1 430 272·BOI 263"427 254'506 246·012 237•926 230·220 930 9'151 9·131 9•112 9·005 9·0i9 9·064 940 4-10 263·6!.12 254•882 246··189 23B·