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B 3 MOD Bfit.
ATE OF MORTALITY!
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JONES.
1 —
LIBRARY
OF THE -"^^
UNIVERSITY OF CALIFORNIA.
i
Chus
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A
SERIES OF TABLES
OF
ANNUITIES AND ASSURANCES
CALCULATED FROM
NEW RATE OF MOETALITT
T
AMONGST
ASSURED LIVES:
WITH
EXAMPLES
ILLUSTRATIVE OF THEIR CONSTRUCTION AND APPLICATION,
&c. &c. &c.
BY
JENKIN JONES,
ACTUARY TO THE NATIONAL MERCANTILE LIFE ASSURANCE SOCIETY
LONDON
PUBLISHED BY LONGMAN, BROWN, GREEN k. LONGMANS;
AND JONES & CAUSTON, 47, EASTCHEAP.
EDINBURGH: A. & C. BLACK.
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PRINTED BT JONES AND CAUSTOW, 47, EASTCHEAP, LONDON .
PHEFACE.
The object of the present publication, and an explanation of the data, from which the Tables have been computed, are set forth in the " Introduction/'
It was originally the Author's intention simply to publish a few Tables, with practical examples, illus- trative of their application ; but, in working out the examples, it occurred to him that it would not be unacceptable to those who take an interest in the subject, but who are not familiar with the theory of Annuities and Assurances, if he were also to explain, without using any Algebraic symbols, the principles upon which the Tables were constructed. This he has endeavoured to accomplish.
To those persons, therefore, who are acquainted with decimal arithmetic, the author thinks that they would find in the '^ Examples'" an ^' Elementary Treatise" on Annuities and Assurances, which would be of considerable service to them by way of prepa- ration for the study of the larger and more compre- hensive treatises by Milne, Bailey, and D, Jones.
The whole of the computations made from the '' New Rate of Mortality," have been carefully cal- culated by two separate computers
In the construction of some of the Tables, the
11
Author is indebted to Mr. Joseph J. Cleghorn, the efficient Deputy to Mr. Griffith Davies, the Actuary of the Guardian Assurance Company^ who had pre- viously computed them for the use of his own office, and which, upon comparison, were found to agree in every respect with those computed by the Author.
The Author is also indebted to Mr. Griffith Davies^ step-son, Mr. Evan Owen Glynne of the Legal and General Life Office, whose services he was fortunate enough to obtain, and by whom the greater portion of the calculations were made in Duplicate with the Author.
The Legal Decisions were compiled by the Author's friend, Mr. Hugh Owen, of the Poor Law Commis- sion Office.
The Author had intended to print, by way of Appendix, a Popular Exposition of the Principles of Assurance, with observations upon the various '^ad- vantages'' held out by the several Life Offices, and a comparison of their rates of premium, &c., but it has been suggested to him that it would be desirable to make a separate, and a very cheap publication of it, which the author purposes doing at a future opportunity.
National Mercantile
Life Assurance Society,
December "11, 1843.
CONTENTS.
Paste PREFACE.
INTRODUCTION.
EXAMPLES ILLUSTRATIVE OF
Compound Interest,
Definition of, , , \
To find the Amount of Sums at, 2
Deferred Sums certain,
Definition of, 4
To find the Present Values of, 5
Annuities certain,
Amounts of, 5
Present Values of, 9
Immediate, 9
Perpetual, 11
Deferred, 12
New Rate of Mortality, ., 14
Probabilities of Life, , 15
To find the probability of a Life surviving any Age, 18
Do. Do. failing in any year of Age, 18
To determine the number and amount of claims that a Life
Office may expect in a year, 18
To find the probability of two Lives surviving a term of years, 20
Expectation of Life
Definition of, 21
Mode of constructing Table of, 21
Comparative Expectations of Life, 23
Life Annuities and Assurances:
Construction of D, N, M, &c. columns, 24
To determine the value of Annuities by the D and N columns, 27
Do. do. Premiums for Assurances by D and
M columns, 31
Page EXAMPLES ILLUSTRATIVE OF Life Annuities,
Single Lives— Ho determine the value of an Annuity by the
ordinary method, 32
Joint Lives, ditto, ditto, ditto, 34
Two Joint Lives, and the Survivor, ditto ditto, 37
Absolute Reversions :
Present Values, 38
Life Assurances,
Single Lives — Ordinary method of determining Premiums, fur 39 Joint Lives, Ditto ditto,' 43
Last Survivor, Ditto ditto, 45
Valuation of Policies :
Construction of Preparatory Tables for, 47
What is the value of a Policy 48
Temporary Annuities and Assurances :
Comparison of the D, N, &c. method and ordinary method of calculating values of, 51
TABLES :
Compound Interest — Amounts, !•
Deferred Sums certain — Present Values, II.
Annuities certain — Amounts, ' * HI-
Do. Present Values IV.
New Rate of Mortality, V.
Probabilities of Life, VI-
Expectation of Life VU*
Comparative Expectations of Life, VIIT.
D, N, S, M, and R, columns,
2i per Cent IX.
3 „ X.
3^ „ XL
Life Annuities — Single Lives, XII.
Do. Joint Lives, XIII.
Absolute Reversions — Present Values, XIV.
Life Assurances— Single Lives— Single and Annual Premiums,. . XV.
Do. Joint do. do. XVI.
Do. Last Survivor, do. XVII.
Valuation of Policies— Preparatory Tables :
Annuities 3 per Cent. — Interpolated for Months, XVIII.
Single Premiums, do. — Do. XIX.
LEGAL DECISIONS.
INTRODUCTION.
The institution of Life Assurance Societies is generally admitted to be one of the most important and benevolent features in modern civilization : and it must be gratifying to all who take an interest in the welfare of society and in the happiness of their species^ to observe the great increase which has taken place in the number of these institutions so far as that fact may be taken as an indication of the increase in the numbers who have availed themselves of their advantages. The great danger^ however, is that by over competition parties may be (as some have been) induced to charge premiums which are too low to cover the risks incurred, and thus be productive of the very mischief which it is ostensibly their object to prevent.
The great importance of the subject will be manifest when it is considered that thousands of persons are annually investing a large portion of their income to provide subsistence for their families, in the
c
IV. INTRODUCTION.
event of their own premature death, and that a very large portion of the property of the country is depen- dent upon the tenure of human life, so that the welfare and future happiness of a large part of the commu- nity are entirely dependent upon the solvency of these institutions. It is therefore of the first im- portance that the tables of rates should be calculated from the most recent and most extensive experience that can be obtained ; so that, on the one hand, they should not be exorbitant; yet, on the other, that they should he fully adequate to cover the risks, and to meet all the liabilities incurred.
To determine the premiums, single or periodical, which ought to be charged for any description of assurance, it is first necessary to construct a table of mortality — that is, a table exhibiting out of a certain number born or who complete a given age, say 1 00,000, the number who die in each year of age, from birth, or the given age to the extreme of life. It is by means of such a table, combined with the interest of money, that the premiums charged by Life Offices are determined.
The earlier societies, such as the Amicable, and Royal Exchange, which were established in the ] 7th century, it appears, charged a premium of £5 per cent., on all lives assured without reference to age\ but it is needless to add that they have since adopted proportionate rates for the risk of each age ; and in the absence of better materials the premiums charged by the Equitable Life Office were deduced
INTRODUCTION. V
from the probabilities of life in London during a period of 20 years^ which included the year 1740, when the mortality was considered to be almost equal to that of a plague. These premiums, how- ever, were not deemed by the then Attorney-General to be sufficiently high, and the Crown, in consequence of his recommendation, refused to issue a charter, which naturally retarded very materially the progress of the society.
In the year 1776, however, the premiums were reduced 1- 10th, and in 1 780, Dr. Price's Northampton Table was adopted as the basis upon which a fresh set of rates was calculated, to which 15 per cent, was added for further security. This, however, was taken off in 1785, and the premiums from that date to the present have remained unaltered.
The Northampton Table was formed by Dr. Price from bills of the mortality during the years 1735 to 1780, in the parish of All Saints, Northampton, which contained little more than half the popula- tion of that town, and on the supposition of a sta- tionary population, whereas the population was then increasing. It is manifest that a rate of mortality so obtained and deduced from the experi- ence of one parish could not be taken as an index of the mortality throughout the kingdom, which con- tains upwards of 12,000 parishes. This, however, is the table used by the majority of the old Life Offices, and by some of the new, although it has apparently been proved^ at least by the experience
VI
INTRODUCTION.
of the Equitable, to represent the mortality much too high, especially at the younger and middle ages. This will be seen by the following table extracted from Mr. Morgan's " View of the rise and progress of the Equitable Society/' which shews the num- ber who died in the 12 years preceding 1829, out of a certain number of assurances in force, and contrasts that number with the number that they had reason, according to the Northampton rate to expect to have died in that period.
|
Age. |
No. |
Died. |
Should have died. |
|
20 to 30 |
4720 |
29 |
68 |
|
30 n 40 |
15951 |
106 |
243 |
|
40 // 50 |
27072 |
201 |
506 |
|
60 t 60 |
23307 |
339 |
545 |
|
60 n 70 |
14705 |
426 |
502 |
|
70 // 80 |
5056 |
289 |
290 |
|
80 // 95 |
701 |
99 |
94 |
Various other tables of mortality have been con- structed since the Northampton : of those of most note the first is the Swedish, which was constructed from returns collected in the years from 1 755 to 1776, inclusively, and which contained the whole population of Sweden and Finland. This Table has been since corrected from more recent data. The next is a table by Mons. De Parcieux exhibiting the mortality among-st the nominees of the French Tontine. The more recent tables, and those now generally used are the Carlisle and Equitable rates of mor-
INTRODUCTION. Vll
tality. The Carlisle was framed by Mr. Milne, from observations made by Dr Heysham, of the mortality in that town during the years 1779-1787, upon a population of 8,000 persons. The " Equitable" was framed by Mr. Griffith Davies, from the decrements of life among the members at the Equitable, and subsequently by Mr. Morgan, from more complete data, so that Mr. Davies' table can now only be considered as a graduated Carlisle.
The Carlisle table agrees very closely with the Equitable, but independently of the objection to a table based upon so few observations, it will be found, notwithstanding its close agreement with the Equi- table experience, that for the want of a greater number of observations at each age, and the table not being graduated, but confined strictly to the data afforded at each age, the Carlisle is imprac- ticable as a basis for temporary assurances, for, on account of the irregularities in the probabilities of dying in one year at several of the ages, the pre- miums deduced therefrom would, in some instances, be greater for young lives than for old ones. For example — at 45 the premium to assure ^1000 for one year would he £\4 8s. Or/., and at 50 it would be .^'IS Os. Od. The irregularities in the probabilities would also affect survivorship assurances, as the probability of surviving one year is an important element in the calculation of those contingencies.
Mr. Milne states that the Carlisle table differs very little from the general law that obtains throughout
Vlll INTRODUCTION.
the country, taking town and country together. But supposing the Carlisle, or any other table, to repre- sent accurately the mortality of the united kingdom, such a rate ought only to be used in the absence of the actual experience of the mortality amongst assured lives, for offices do not take lives indiscrimi- nately, but have the power of selection. Now if an office is prudently conducted, all doubtful lives are rejected; andif it were possible to select all good lives such a table as the Carlisle would manifestly repre- sent a mortality higher than that which would pre- vail amongst the lives actually assured. As there is also greater laxity in the selection of lives in some offices than in others, and as it will happen, even with the utmost vigilance exercised, that some unsound lives will be passed as eligible, it is manifest that a rate of mortality, deduced from the combined experience of the various Life Offices, is the most consistent, and the safest basis upon which the rates of assurance ought to be determined.
Mr. Griffith Davies, the able and experienced Actuary of the Guardian Life Office, in his observa- tions upon the data afforded by the Equitable observes, — '"^ It must be allowed that however doubtful the limited experience of a new institution might be regarded, the proportions stated by Mr. Morgan, repeated and confirmed as they have been for a period exceeding half a century, afford more satis- factory data for determining the rate of mortality among assuredlives, than any registers hitherto made public.''
INTRODUCTION. IX
Mr. Babbage, in his " Comparative view of the various Institutions for the Assurance of Lives/' says^ in reference to the best data for constructing a rate of mortality, that ^' It is, therefore, to be expected that the law of mortality which exists amongst assurers, should approach more nearly to that which takes place amongst select classes of mankind, such as amongst annuitants, (where it is the interest of each proprietor to select a good life) than to more indiscriminate bodies of people. Although there exist good observations of this kind, I am not aware of their having been employed as the basis of any table of premiums for assurances.'^
^' Having now pointed out the defects of the tables in general use, it will naturally be inquired what others it is proposed to substitute. To this it may be answered, that the best substitution would be a table actually constructed from the deaths occurring amongst a large body of persons of this very class whose law of mortality we wish to ascertain. Mate- rials for such a table exist, and probably in the best and most authentic form. The Equitable Society has been established sixty years, and it must possess records of the death, and cause of death, of all those who have had claims on its funds. Another society of considerable extent, the Amicable, has existed above a century, a vast quantity of valuable mate- rials is, without doubt, contained in the records of these two societies, and if they were each to com- municate to the public the facts of which they
INTRODUCTION.
are in possession, it would form a most important addition to our knowledge, and supply the most accurate materials for tables of this class which have yet been produced/^
By the liberality of several of the Life Offices, and the disinterested zeal and services of a Committee of some of the most experienced and eminent of the Actuaries, we have now^ data for the construction of a rate of mortality, not simply of the experience of the Equitable and Amicable, but of the combined expe- rience of no less than 17^ Life Offices, embracing 83,905 policies, and a rate of mortality has been adjusted by one of the most eminent Mathematicians on the Committee, from the combined town and country experience, embracing 62,537 assurances.
It is a very common practice with some of the offices to announce their premiums as having been computed by an able Mathematician from the most recent and most extensive experience, without usually stating what such experience is, or giving the name of the able Mathematician, who is thus alleged to have constructed their tables. As, however, we have now very recent and extensive experience of the
* It may not be amiss here to observe that 13, out of the 17, contribu- ting offices are proprietary companies, who would thus appear to be animated by motives equally as disinterested as those of the " Equitable" and " Amicable," who, as Mr. Babbage observes, " have no private interests to oppose their publishing for the advancement of science, the results of that experience which it alone, by securing their stability, has enabled them to acquire, thus supplying the solid materials of further improvements, which must inevitably reflect back the greatest advan- ages on those most largely engaged in such transactions."
INTRODUCTION.
XI
mortality amongst assured lives^ such as ought to form the basis upon which all rates shall in future be calculated, it may be useful to explain the origin of the Committee, and the course adopted by them in their collection and employment of the data contributed by the several offices.
The Committee was formed at a Meeting of Actuaries, and others connected with Life Assu- rance Offices in London, held at the London Coffee- House, Ludgate Hill, on Monday the 19th March, 1 838, at which it was resolved unanimously :
" That in the opinion of the meeting, it is desirable that the different Assurance Offices, should from their records contribute the requisite data to the common fund, to afford the means of deter- mining the Law of Mortality which prevails among Assured Lives.
^* That such a Law of Mortality, truly determined, would prove generally useful, especially to the Life Offices themselves, and the numerous class of persons availing themselves of those Institutions.
*' That persons professionally engaged in similar investigations, are most likely to draw correct conclusions from existing data, and to classify the same into forms, showing the true rate of mortality among Assured Lives."
The following particulars were obtained from the offices that engaged to contribute their experience:—
|
For use of Office. |
Current Age at Entry. |
Year of |
If by Death, D. |
Sex, if Female F. |
Distinc- tion into Town, T. |
Cause of |
Special risks and Remarks. |
|
|
Entry. |
Exit. |
CouutrvC. Death. Irish i. |
||||||
D
Xll INTRODUCTION.
The following circular^ which was transmitted with a supply of forms to each of the contributing offices, will explain the particulars that were obtained from them : —
1, King Willimn Street, City,
2-5tk September, 1838. Sir,
** The Committee of Actuaries desire me, in forwarding the accompanying forms, which they have prepared for collecting the data, on which to found the experience of Assured Lives generally, to submit the following explanation of the nine columns into which the forms are divided.
'^ Column 1. — Headed * For use of Office,' is intended for the number of the policy, or any other distinguishing mark, by which the person employed to make the extract from the Policy Register, may note how far he has proceeded, and be enabled to resume the operation without difficulty.
<* Column 2. — Headed 'Current Age at Entry' is intended to contain the age next birth day of the party Assured, at the time the Assurance was effected.
" Column 3. — Headed * Year of Entry' is for the Year in which the Assurance was effected. The Committee require neither the month, nor the day of the month. The same observation applies to column 4, headed ' Year of Exit.' No distinguishing mark is required to show whether a Policy has become extinct by forfeiture, purchase, or expiration of term; but when extinguished by death, a D must be inserted in the next column, No. 5. The column marked ^ Exit' will be left blank, opposite all those Policies which were in force on the 31st December, 1837, to which date it is requested that the list be made up, if convenient.
" The next column is for distinguishing the sex, in which is to be put an F opposite all Policies on the lives of Females ; the blanks will indicate Males. Such Offices as have Agents are requested to insert a T opposite those Policies effected in Town ;
INTRODUCTION. Xlii
a C opposite to those Policies effected in the Country, and an I opposite those effected in Ireland, in the column marked * Dis- tinction into Town, Country, and Irish.'
'^ The cause of death is to be inserted in the next column, in a line with those Policies extinguished by death.
" The last column is intended for a notice of special or foreign risks, and for the insertion of any observation that may be considei'ed useful.
** The question of founding the experience from returns of Policies issued, or on Lives Assured, was fully discussed by the Committee, — to confine the returns to a list of the Lives Assured in each Office might at first appear desirable, as a means of avoiding the insertion of the same Life more than once, in cases where more than one Policy has been granted thereon ; but when it was considered that in combinino; the returns of several Offices, it would be impossible to prevent the repetition of the same life, as many are assured in several Offices, and that, in combining large numbers where Lives represented by duplicate Policies, are subject to the same ratio of mortality as those represented by single Policies, the result cannot be sensibly affected by the duplication, it was determined by the Committee to confine the lists to a record of Policies issued on single Lives."
I have &c.
Robert Christie, Hon. Sec.
From the returns received from the several offices in the prescribed form^ and which were blended together as they came in, ^^ so as to prevent any use being made of the returns separately,"' various tables have been prepared, and great care appears to have been exercised in the classification of the data, upon which the results in the tables have been obtained.
XIV INTRODUCTION.
The following is a list of the several tables,^ pre- pared by the committee.
Table A (3-6) — Shewing out of the number of Assurances effected in each current year of age, the respective numbers in each year of duration, cancelled by discontinuance and by death? and existing at the termination of the observations. (Separate tables for Male and Female lives, Town, Country, and Irish respectively.)
Table B (1-6) — Being an enumeration of entries, existences, discontinuances, and deaths, in each year of age, deduced from the foregoing tables, A (1-6) (separate tables for Town, Country, and Irish Male and Female lives respectively.)
Table C. — Shewing the number exposed to the risk of mor- tality, the actual number of deaths for Assurances on the lives of Males and Females, separately and collectively, and for Town, Country, and Irish Assurances separately, deduced from Tables B and the computed number of deaths, according to the Nor- thampton, Carlisle, and Mr. Davies's Equitable Tables of mortality, in decennial periods of age, calculated to the nearest whole number.
Table D. (1-5) — Shewing the number exposed to the risk of mortality, and the deaths in each year, with the probability of surviving one year, and the expectation or average duration of life; deduced from Tables B (1-6) (for Town, Country, and Irish Male and Female Lives separately, and for Town, Country, and Irish experience separately.)
Table E. — Shewing four times the number exposed to the risk of mortality, and four times the number of deaths in each year, with the probability of surviving one year, and the ex- pectation or average duration of life, deduced from Tables B (1) B (4) and other Town experience, which together comprise 48,702 Assurances.
Table F. — Shewing four times the number exposed to the risk
* These Tables are not published, and are only in the possession of the several Life Offices who subscribed for copies.
IlNTRODUCTION. XV
of mortality, and four times the number of deaths in each year with the probability of surviving one year, and the expectation or average duration of life; deduced from the total experience, which comprises 83,905 Assurances.
Table G.^ — Adjusted law of mortality, according to the com- bined Town and Country experience, deduced from Tables D, (4) and E, which comprise 62,537 assurances.
Equitable experience for separate classes.
Table H (1) — Shewing results on 7,259 lives admitted between the ages of 25 and 35 years.
Table H (2) — Shewing results on 6,270 lives admitted between the ages of 35 and 45 years.
Table H (3) — Shewing results on 3,436 lives admitted between the ages of 45 and 55 years.
Table H (4)— Shewing results on 1,317 lives admitted between the ages of 55 and Q6 years.
Table I (l)t — Shewing the expectation or average duration of life ; deduced from eight original Tables, and compared with the Northampton and Carlisle Tables.
Table I (2) — Shewing the expectation or average duration of life, for persons admitted at particular ages in the Equitable Society, and compared with Mr. Morgan's and Mr Davies's Tables of that Society's total experience.
Table K. — Shewing the mortality per cent, in each year of age; deduced from twelve original Tables.
Table L, — Shewing the annual number of deaths in quin- quennial periods of age, out of 10,000 persons living at each age according to various Tables of mortality.
It appears to have been originally the intention of the Committee ''to put the various offices, and those who might be interested in carrying out such inves- tigations^ in possession of what appeared to be the most useful and valuable classifications of the bare
* See Tables 5, 6, and 7. t See Table 8.
XVI INTRODUCTION.
facts comprised in the different returns^ without the introduction of any arbitrary or theoretical adjust- ments. However^ as some persons might be desirous to see an adjusted table of mortality, one has been deduced from the combined Town united with the Country Assurances^, which comprise the whole of the male and female lives that admit of being separated from the Irish/^
It would have been interesting to have had a classification of the causes of death amongst assured lives^ but it appears that '' the returns of the causes of death were deficient in so many of the lists that it was not considered desirable to make any classi- fication of them. ^'
The Author has examined the whole of the Tables with great care and with much interest^ but prefers set- ting forth the peculiar features exhibited by them in the language of the Committee in whose praise too much cannot be said for the valuable time and trouble which they have gratuitously given to this important and interesting subject.
The Committee state that the most striking features exhibited in these Tables^ are the great mortality that prevails among Irish lives, and the marked difference in the rate of mortality between males and females. The near agreement with each other of the Tables for " Town '' and " Country " Assurances is also very remarkable, considering that no adjustment has been employed.
On comparing the results given in tables C and
INTRODUCTION. XVU
L, the mortality annually, taking all ages together, is shown to be least amongst '' Town " Assurances, rather more amongst ^^ Country/' and greatest amongst ^' Irish'" Assurances. The mortality amongst assured females, taking all ages together, is also greater than amongst assured males ; and both these classes exhibit a greater rate of mortality than either " Town " or '' Country '' Assurances, which arises from the Irish Assurances being in- cluded amongst the males and females.
The mortality represented in table C, is con- siderably greater for females than males, between the ages of 20 and 50, from 50 to 70 years of age it is less, and after the latter age it is at some periods rather greater, but the numbers are too small to be of any import at these advanced periods of life. The ^^ Irish " Assurances are subject to rather less mortality under 60 years of age than is represented by the Northampton Table ; but after that age the mortality amongst them is greater : and taking all ages together, the deaths are more than 95 per cent, of what might be expected by that table.
On making a comparison of the different classes according to the expectations of life, as shewn in Table I, it will be seen that the average duration of male lives, under 36 years of age, is greater than that of females, and from 36 to 61 years of age, the average duration of the lives of females is greater than that of males, but after the age of 61, the ex- pectation is greater for males than females, which
Xviii INTRODUCTION.
may arise from the paucity of numbers at the ad- vanced periods of life. The expectation of life for the class designated '' Town '' (deduced from the facts contained in Table A), will be found to agree very nearly with Mr. Morgan's Equitable Table E^ being a little more^ but scarcely differing one with another a quarter of a year from 22 to 63 years^ after the latter age the expectation of life is sometimes a little more and sometimes less than by Mr. Morgan's Table, but on the whole exhibiting a close agreement. The '' Irish '' class gives a considerably less expecta- tion of life than Mr. Morgan's Table at all ages ; and after the age of 44, the expectation is even less than by the Northampton Table. The class designated '' Combined Town" in which the " Equitable " and ^^ Amicable" total experiences are combined with the other " Town " Assurances, will be found to give the expectation of life rather less than the latter, arising doubtlessly from the assurances in the two offices just named being of longer duration than those in most of the other offices. The expectation of life, deduced from the whole of the materials put together, it will be seen differs very little from the '' Combined Town," The four classes " Town," " Country," '' Combined Town," and " General," will be found to agree very closely with the ex- pectations of life deduced from Mr. Milne's Carlisle Table of Mortality, although generally giving a lower expectation than that Table."
In reference to the materials from which the
INTRODUCTION. xix
whole of the Tables have been formed^ the Com- mittee state that they represent a lower rate of mortality than can be expected to prevail in a longer period of time than that over which the present observations extend ; for the average duration of Policies embraced in nearly] one-half of the ex- perience is under 5^ years ; and taking the whole of the experience together, which includes that of the '^ Equitable " and " Amicable, '' the two oldest offices existing, the average duration of all the Policies is not 8|- years. This is readily ac- counted for when it is seen that more than half the Policies effected were .existing at the termination of the observations, and nearly a 'third had been discontinued during the life time of the parties assured. The circumstance of recent selection should not be lost sight of by such persons as may use these Tables either for the sake of comparison or as the basis of other tables for granting as- surances. The difference in the rate of mortality between recently selected lives and those of longer continuance in the society is clearly shewn by Mr. Galloway in the tables of mortality deduced by him from the experience of the '^^ Amicable Society,'^ and which that society, like the " Equitable,^' has recently so disinterestedly printed for the use of its members. ^^
It has been thought right to enter thus fully into the origin of the publication of the Tables, prepared under the superintendence of the Committee o^
E
XX INTRODUCTION.
Actuaries, and to set forth their opinion of the results obtained by them, as it is of the utmost importance that the public should be made acquainted with the fact that such a committee has been formed, and have availed themselves of the most extensive and special experience that could be obtained to determine the lawof mortality which prevails amongst assured lives, and have thus enabled every existing office to test the adequacy of its rate of premiums, and future offices to provide a rate for themselves on a secure basis.
A rate of mortality having been determined, the next important point for consideration is the rate of interest which must be assumed, as that which the funds invested by a Life Office will realize. Those offices which have started at considerably ^^ lower premiums than any other office,^^ justify the reduc- tion in their rates on the ground of the mortality not being so great as that represented by the tables of mortality used generally by the offices, and also that they can realize a larger per centage on the monies invested, than that on which the rates are generally based. The mortality deduced from the combined experience of the various Life Offices will set all speculations at rest as to the rate of mortality which may be expected to prevail amongst assured lives. With respect to ^^ Interest, "" it will be admitted, at least, that it is liable to great fluctua- tion, and that money has been for a series of years gradually lessening in value. Mr. De Morgan
INTRODUCTION. Xxi
observes, in reference to this point, advanced by the advocates for low premiums. '^The rate of interest has been halved within the memory of man, and a heavy war might double it again. That same war with all its casualties, direct and indirect included, would not alter the mortality of the country by any serious amount. I consider it then as next to certain, that the insurance offices have more to look for, whether as matter of hope or fear, from the fluc- tuations of the rate of interest, than from those of mortality/' # # # ^ # #
'^ We are already in a very different position as to the rate of interest which has been gradually fall- ing since the war. # # # Assuming the neces- sity of calculating upon a rate of interest something less than that which can actually be attained, I should think that no office would be justified in sup- posing more than 3 per cent., ivilh tables ivhich are sirfficiently high to come any ways near to the actual experience of mortality. With regard to one point, and that of fundamental importance, namely, the possibility of a still further fall in the rate of interest, it may even be doubted whether, ivith such tables, a still lower rate of interest should not be allowed."'
But it is urged by the cheap offices, '' Oh, but we have a large protecting capital,'" which protecting capital, as Mr. De Morgan justly re- marks, would, '*^if the premiums were really too low, be an injury and not a benefit, for since this capital is really paid for in whole or in part out of
XXll INTRODUCTION.
premiums^ it would not preserve the office from insolvency, but would rather accelerate its progress towards bankruptcy/'
It is needless to observe that proprietors of Life Offices do not embark their capital to make up an anticipated deficiency, but like other in- vestments, their capital is sunk with the view of legitimate profit, and as a sqfeguai'd against any unexpected or sudden increase in the mortality, and in the fluctuation of interest. If they act prudently for their own interest, as well as for the safety of the assured, they will take care to charge such a rate of premium as will, in the opinion of an experienced and qualified Actuary, meet every probable risk, and cover the expenses of management, and will, in addition to the interest to be obtained by ordinary investment, also yield them a fair equivalent for the money which they have risked for the protection of the assured.
The Author is not contending for high or excessive rates ; all that is desired is, that the rates should be sufficient and fully adequate to meet the risks and expenses incurred. On this point Mr Griffith Davies makes the following excellent observations. '' The evil of charging excessive premiums cannot, however, long remain in a country where capital is allowed to flow freely from one channel to another, as the na- tural effects of competition must necessarily reduce the profits on Life Assurance to the level of that de- rived from other species of investments ; on the
INTRODUCTION. XXlll
contrary, the peculiar nature of the subject renders it extremely dangerous lest the rates for Life Assur- ance should be so far reduced as to diminish the security of those who may select this mode of ac- cumulating their savings for the benefit of their families ; for if the premiums charged by societies established for these purposes should, by excessive competition, be rendered inadequate to the pay- ments of the claims which, sooner or later, must come upon them, whatever honour, wealth, or pro- bity, the present managers of them may possess — whatever capitals they may boast of— or however prosperous they may appear to go on, even for a considerable time, the result must ultimately termi- nate in litigation, disappointment and ruin, and in- stead of a national benefit. Life Assurance in such a case would inevitably become a national calamity.'" The Equitable Life Office, whose great success is generally appealed to in justification of reduced pre- miums, it must be remembered not only enjoyed a monopoly, but, as has already been stated, the rate of premiums originally charged was enormously high, and, in addition to this, they were enabled to invest their funds in the purchase of government stock at very low prices, for, as observed by the late Mr. Morgan, ^^ during the long series of years in w^hich this society has existed, the nation, for a considerable part of the time, has been engaged in foreign wars. These, by depressing public credit, have afforded the opportunity of investing money in
XXIV INTRODUCTION.
the funds to great advantage^ and have thus contri- buted in no inconsiderable degree to create the surplus of the society. From the year 1777 to 1786, the average price of stock in the 3 per cents, was about 60 per cent., and from the year 1796 to 1816, the average price of the same stock was below 60 per cent., or 24 per cent, lower than its present price. But no reliance ought to be placed on advantages of this kind. Another war may reduce the value of stock in the funds to half its present value, or still lower, if some of our modern statesmen should succeed in breaking the public faith by destroying the sinking fund. It would be madness, therefore, to found any measure on a property so fluctuating. The addition to the surplus arising from the im- proved state of public credit is an accidental cir- cumstance, affording no proof of the excellence, any more than a deficiency in the capital arising from its depreciated state would have afforded proof of any defects in the construction of the society, and is mentioned merely as one of the causes which have produced its present opulence/^
And in 1828, when the pamphlet from which the above observations have been quoted was written, and when the price of consols varied from 82^ to 88|, he proceeds to observe — ^^That all the causes hitherto noticed as having conduced to promote the welfare of the society, no longer exist to enrich it. The premiums have been re- duced in some instances nearly one-half. The
INTRODUCTION. XXV
policies are seldom or ever forfeited ; and the pur- chases made in the public funds at their present price are more likely to be disadvantageous than beneficial to the society/^
From 1829 to the present year the average price of consols has been about 90, and the price at present is 96^^ so that it will appear that at the present time circumstances are peculiarly unfavourable, so far as the interest of money is concerned, for the success of any new undertaking which does not take the precau- tion of adding a considerable per centage to the net premiums to cover any extraordinary mortality, the expenses of management, and the fluctuation in interest.
By reference to Table 8, it will appear that the expectation of an Irish life at 20, is 34.95; at 30, 29.71; at 40, 23.36; at 50, 17.76; so that, as compared with the combined English experience, an office may calculate upon receiving upon an Irish life of 20, only thirty-five premiums, instead of forty-one; at 30, only thirty instead of thirty- five ; at 40 only twenty-four premiums instead of twenty-eight ; and at 50 only eighteen premiums instead of twenty-one. Notwithstanding this fact, in addition to the risk already incurred of charging too low a rate of premium even for the English lives; if report speaks true, some of the cheap offices do a very extensive Irish business, so that an extensive business, and the announcements which
* December 21, 1843.
XXvi INTRODUCTION.
are frequently seen among the advertisements of the day, to the effect that in addition to a large sub- scribed capital, the policy holders have the additional security of £ per annum for premiums, are not always to be taken as indicative of extensive security; for where much Irish business is transacted the ad- vertisement, strictly speaking, should run — ^^in addi- tion to a large subscribed capital the policy holders have the additional security of £ per annum annual income for premiums, £ of which are
from Irish Assurances, from which the society has reason to expect they will receive several premiums less than they ought, and than which they expect to receive on an English Assurance"
The offices generally are getting very cautious of Irish lives, and the circumstance is only mentioned here to point out an additional risk that the cheap offices incur.
These remarks have been extended to a much greater length than was intended, and the Author would, in conclusion, merely express a hope that the example of liberality set by the various private companies in contributing their experience, and of disinterested zeal displayed by the Actuaries who superintended the compilation of the materials, and deduced therefrom a rate of mortality amongst assured lives, will be followed by the government, and by their Actuary Mr. Finlaison, in supplying the materials which, it is presumed, they possess in abundance in several of the government depart-
INTRODUCTION. XXvii
ments relative to sickness and mortality, which might be worked out by Mr. Finlaison, or under his superintendence. In the mean time, it would not, perhaps, be considered too liberal on the part of the government, if they were to print, for the benefit of the public, the various tables on Life Contin- gencies, which their actuary has made from govern- ment records, and at the national expence, and, in reference to which, the following petition was printed, and signed by upwards of 40 gentlemen connected with Life Assurance Offices in the year 1837, but which was never presented, probably in the expectation that the agitation of the matter would be sufficient to induce their publication.
TO THE HONOURABLE THE COMMONS OF THE UNITED KINGDOM OF GREAT BRITAIN AND IRELAND, IN PARLIAMENT ASSEM- BLED.
The humble Petition of the undersigned Actuaries of Life Assurance Offices, in London, and of others connected therewith.
Sheweth,
That a very large portion of the property of this country is held upon tenures depending upon the duration of human life, and that the business of Life Assurance has of late ex- tended so as to affect the interests and future happiness of large numbers of all classes in the community.
XXVm INTRODUCTION,
That one of the principal elements in all calculations of the value of property depending on human life, and of the value of the risks of Life Assurances, is the average duration of human existence, as determined by observations : and the means by which such calculations are made or facilitated, are tables of the value of Life Annuities, deduced therefrom.
That as the accuracy of Annuity, and other tables, founded on the rate of mortality, depends upon the extent of the observations from which they are derived, every addition to them is of national importance.
That to adjust equitably the value of church property, and other life interests,— 'to measure truly Life Assurance risks, and to afford the means of satisfying the public of the just application of correct principles in such valuations, it is highly necessary that every authentic information bearing upon the subject, should be made generally accessible.
That very extensive tables, have been calculated at the national expense, from data, furnished by Government Records, which were printed by order of your Honourable House, in 1829 : and that on these tables the Government now grant Annuities on lives, and it has recently been pro- posed in your Honourable House, that the value of church property, should be estimated by the same standard.
That of these tables a very limited portion only has hitherto been made available to the public.
Your Petitioners, therefore, humbly pray that your Honourable House will be pleased to order the pub- lication of all tables founded upon the same data as those upon which the Government now grant Annuities on Lives. These tables will comprehend Annuities on Single Lives for males and females
INTRODUCTION. xxix
separately, and on every combination of two or more joint lives, at every rate of interest at wh ich they have been respectively computed. London, June^ 1837.
Joshua Milne, Sun Life Office
Arthur Alorgan, Equitable Assurance Office
George Kirkpatrick, Law Life Assurance Office
Charles Ansell, Atlas Assurance Office
Griffith Davies, Guardian Office
J. D. Bayley, Royal Exchange Assurance Office
Benjamin Gompertz, Alliance Office
W. S. Lewis, Rock Life Assurance Office
James J. Downes, Economic Office
Samuel Ingall, Imperial Life Office
Robert Christie, Universal Life Office Thomas Lewis, Union Assurance Office
J. M. Rainbow, Crown Assurance Office
Thomas Galloway, Begistrar, Amicable Society
E. Charlton, Albion Insurance Office
"W. Bury, Hope Assurance Office
H. P. Smith, Eagle Assurance Office
M. Saward, Promoter Life Office
Robert John Bunyon, ]S"orwich Union Life Assurance Office
M. Tate, Pelican Insurance Office
Edward Hulley, Globe Office
Henry Marshall, Metropolitan Office
J. Tullock, Minerva Life Assurance Office
Charles Jellicoe, Protector Life Office
John Robertson, Argus Life Office
Ebenezer Femie, British Commercial Life Office
J. M. Terry, Hand-in-Hand Life Office
John Laurence, London Assurance
G. H. Heppel, Standard of England Office
J. T. Clement, Licensed Victualler's and General Fire and Life
Assurance Office. Joseph Marsh, Xational Provident Institution C. B. Smith, National Life Assurance Society Edwin James Farren, Asylum Life Office B. A. M. Boyd, Resident Director^ North British Company
XXX INTRODUCTION.
J. C. C. Boyd, Secretary J United Kingdom Life Assurance
J. T. Barber Beaumont, Managing Director, Provident Life
Office Charles M. Willich, Secretary §• Actuary, University Life
Society F. G. Smith, for Scottish Union Assurance Company Charles Lewis, West of England Insurance Office David Foggo, Secretary, European Life Insurance Office T. R. Edmonds, Actuary of Legal and General Life Assurance
Society T. Pinckard, of the Clerical, Medical and General Office
As the above petition lias never been presented^ it has been thought desirable to print it in this Intro- duction, as it is important that the public should know that there are some valuable and very exten- sive tables in the hands of the government calculator, which "have been computed at the national ex- pence/' and which it is submitted ought to be printed for the public information.
.bvi%>-
<v
ur:rv>:RsiTY *
or
PRACTICAL EXAMPLES
ILLUSTRATIVE OF
THE CONSTRUCTION AND APPLICATION
OF
THE TABLES.
COMPOUND INTEREST,
TABLE I.
Interest is a remuneration allowed by a party bor- rowing money to the party lending it^ and is payable at periods agreed upon at a certain annual rate for every £ 1 00. Where it is so paid^ the Interest is called '' Simple Interest," but where it is not so paid and is added to the sum lent whereby the sum due from the borrower is increased by that amount upon which (instead of upon the original sum) he will have to pay interest— such interest is called ^^ Compound Interest."
EXAMPLE 1.
What will £450 amount to in 12 years at 4 per cent. Compound Interest ?
If £100 were lent for one year at 4 per cent, its
B
amount at the end of the year would be £100 + 4 = £104, and this divided by 100 would give the amount of £l at the same rate at the end of the year, or £1.04 from which we may easily determine the amount of any other sum in one or more years ; for if 1 : 1.04:: 1.04: (1.04 x 1.04) = 1.04^ =1.0816 the amount of £l at 4 per cent, at the end of two years, and in like manner; if I : 1.04:: 1.04' : (1.04x 1.04") = 1 .04^ = 1.1 24864 — the amount of £ 1 at 4 per cent, in three years; and so on for any number of years; the amount of £l obtained for any given number of years at the given rate of interest multiplied by the amount of £l at the same rate for one year will give the amount for the succeeding year, and in this manner Table I. has been constructed, on reference to which, under the head of 4 per cent., against 12 years, we find £1,601032 which multiplied by 450 will give £720.4644^ = £720 9s. 3Jd., the amount of £450 in 12 years, at 4 per cent, as required; and so with any other amount at the same or any other rate per cent. The Rule being — Find the amount of £l in the Table under the given rate per cent, against the given number of years, and multiply it by the sum of which the amount at the same rate and for the same period is required.
If the interest is payable half yearly the rule is —
* To persons unacquainted with Decimals, it would be useless to give a rule for the conversion of shillings, pence, and farthings into decimals, and vice versa, such persons, therefore, are referred to works on Arithmetic. To those who are acquainted with Decimals it is unnecessary to do so.
Take one half of the annual interest and double the number of years, and proceed as iV* the case where interest is paid annually. For example^ if in the above case the interest were payable half yearly, the amount would be obtained thus — Under column 2 per cent., in Table 1, and against 24 years, we find £1.608437 — the amount of £l at 2 per cent, per annum in 24 years, or, which is the same thing, the amount of £l at 2 per cent, per half year in 24 half years, which, multiplied by 450, gives £723.79665 =£723 15s. lid.— Answer. And if interest were payable quarterly the rule would be — Take one fourth of the annual interest and multiply the number of years by 4, and proceed as in the case where interest is paid annually. If, in the above example, the interest were paid quarterly, we should refer to column headed 1 per cent., in Table 1, and against 48 years, we should find £1.612227 the amount of £l at 1 per cent, per annum in 48 years; or, which is the same thing, the amount of £l at 1 per cent, per quarter, for 48 quarters of a year, which, multiplied by 450, would giw^ £725.50215 = £725 10s. 0|d.— Answer.
It evidently matters not whether the ^' rate" be called the rate per annum, or per half year, or per quarter, as the amount of any sum at a given rate of interest manifestly depends upon the number of conversions of interest into principal.
EXAMPLE 2.
The amount of £450 in 12 years at 4 per cent.,
Compound Interest, payable
annually, - - being £720 9s. 3^d.
n payable half yearly // £723 15s. lid.
// // quarterly // £725 lOs OW.
it is required to find the total amount of interest realized. This will evidently be the difference be- tween the sum lent and its amount at the end of the time, and will be respectively, £720 9s. 3|d.— £450 = £270 9s. 3|d., Amount of
interest realised upon £450 in 12 years, at 4 per cent, interest, payable yearly.
£723 i5s. 11 d.— £450 = £273 15s. lid., do. do. pay-
able half-yearly.
£725 10s. 0|d.— £450 = £275 10s. 0|d, do. do. pay-
able quarterly^
DEFERRED SUMS CERTAIN.
TABLE II.
The present value of a sum of money to be received at the end of any number of years, is that which, laid out at a given rate per cent, will amount at that rate, to the sum to be received at the expiration of the given period.
EXAMPLE.
In exmaple J, of Compound Interest £720 9s. 3Jd. = £720.4644 is stated to be the amount of £450 at 4 per cent, in 12 years. £450, therefore, ought to be shewn by Table 2, to be the present value of £720 9s. 3^d. to be received at the expiration of 12 years, supposing interest to be 4 per cent.
On referring to Table 2, under the head 4 per
cent., and against 12 years, will be found £".624597, the present value of £'1 to be received at the expi- ration of 12 years, which multiplied by 720.4644 will give £450, the present value required. This sum might have been obtained by dividing£720.4644 by 1.601032 the amount of £l in 12 vears. For if 1.04 ; 1 :: 1 : r^j the present value of £l at 4 per cent. Compound Interest to be received at the ex- piration of one year; and similarly, — if 1.04 : 1 :: i^ : i:^ — present value of £l at the same rate to be received at the expiration of two years : and so on for any number of years. In this manner Table 2 has been formed — ujiity being divided by the amount against each age at the several rates per cent, in Table 1 ; and it is manifest that when the present value of £l for any number of years at a given rate is found, that the Rule for finding the present value of any other sum at any rate per cent, will be Mul- tiply the present value of £\ at the given rate and the given number of years by any other amount of which at that rate and for that teryn the present value is re- quired.
ANNUITIES CERTAIN— AMOUNTS.
TABLE III.
An Annuity Certain, is a sum of money pavable at fixed periods without being subject to anv contin- gency.
6
EXAMPLE.
What will an Annuity of ^20 per annum amount to in five years, at 6 per cent. Compound Interest ?
On reference to Table 3_, under 6 per cent, against 5 years will be found 5.637093, the amount of an Annuity of .^'l at that rate and for that term, (or, as it is usually called, the number of years purchase,) which multiplied by 20 gives £'112,74 1 86 =.£^112 14s. lOd. — Answer.
The results contained in the Table were obtained thus : — The last payment of an Annuity of £l, at 6
per cent, and upon which no Interest is
received is £1.000000
The last payment but one, and upon which
one year's Interest accrued 1.060000
Their Sum — Amount of Annuity in 2 years 2.060000 The last payment but two, with 2 years'
interest 1.123600
Their Sum — Amount of Annuity in 3 years 3, 183600 The last payment but three, with 3 years'
Interest 1.191016
Their Sum — Amount of Annuity in 4years 4.374616 The last payment but four, with 4 years'
Interest 1.262477
Their Sum £5.637093
amount of Annuity of £l forborne 5 years (or the number of years purchase) and agrees with the
amount given above as taken from the Table ; and by proceeding in this manner the Amount of an An- nuity for any rate and for any period may be obtained. The Rule for the construction of the Table being To £l .00000, add the amount of £l at the expiration of one year, at the ^iven rate of interest obtained from Table 1, which will give the amount of an anjiuity at that rate forborne two years, to this sum add the amount of £l in two years, which will give the amount of the aimuity for three years, and so on (as in the above example) to the end of the period required. The Table being formed, the rule for finding the amount of any other sum annually will he,Obtainfrom Table 3 the Amount of an Annuity of£\ at the^iven rate per cent, and for the given tenn, ajid multiply it by the annuity, whose amount, at the same rate and for the same period is required.
If the annuity is payable half yearly, Take the quantity from the Table under half of the rate per cent, opposite twice the number of years, and multiply it by one- half of the annuity.
If payable quarterly, Take the quantity opposite one-fourth the rate per cent, and opposite four times the number of years, and multiply it by one-fourth of the annuity.
Or the amount of an Annuity might be found by the following Rule : —
Obtain from Table 1 the amount of £ I at the given rate of Literest and against the given nmnber of years ; subtract unity from it and divide the remainder by the Interest of £{ for one year at the same rate.
8
which will give the mnourit of an Annuity of £\ at that rate and for that term, and multiply the quotient by the Annuity ivhose amount is required Table 3 might also have been formed in this manner though not so readily.
The reason of this rule is manifest, for when unity is deducted from the amount of ^£"1 at the given rate and for the given term obtained from Table \, the remainder must be the total amount of interest realised, and this amount accrued by putting by the interest due each year, upon which also interest was obtained, therefore the diiference between the amount of ^1, at any rate and for any term, and £\, the sum originally laid out^ is equal to the amount of an annuity of the interest of £l B,t the same rate and for the same term.
The above example might therefore have been obtained thus. From the amount of ^£"1 at 6 per cent, in five years, which, by Table 1, is ^1.338226, take £\, the original sum laid out, and the difierence <£0.338226 is the total interest realised, or the amount of an annuity of £.06 at 6 per cent, in five years ; then, by the common rule of proportion : — If ^.06 : £0 338226 ::£'l : .56371 -the quantity, given above as obtained from Table 3, to the nearest 4th place of decimals, which, multiplied by 20, gives ^112.742=^112 14s.l0d. as before.
Table 3, has been constructed upon the suppo- sition that the annuity is payable at the end of the year ; if it were payable at the beginning of the year each of the amounts in that Table ought to be in-
9
creased by one year's interest ; the amount of the last payment, therefore, reckoning interest at 6 per cent, upon which one year's interest accrued
would be =£^1.060000
The last but one upon which two years
interest had been received 1.123600
Their sum f 2.183600
the amount of an annuity payable at the beginning of the year, laid by for two years, which is equal to the amount of an annuity payable at the end of the year for three years less unity ; so that where the annuity is payable at the hegirming of the year, the rule is Subtract unity from the amount of an annuity payable at the end oj the year — in Table 3 — at the given rale of interest opposite one year more than the time.
ANNUITIES CERTAIN— PRESENT VALUES.
TABLE IV.
1st. Immediate Annuities. — The present value
of an Annuity to be entered upon immediately and
to continue for a term of years, is that sum
which paid down now and invested at a given rate
of Interest will, at the expiration of the term,
c
10
amount to the same sum as will the Annuity itself invested in like manner.
EXAMPLE 1.
What is the present value of an Annuity of .^'SO per annum to continue 4 years^ reckoning Interest at 4 per cent.?
On referring to Table 4^ under the head of 4 per cent, and opposite to 4 years will be found .^'S. 629895 the present value of an Annuity of £} at that rate and for that term, which multiplied by 30, gives £'108.89685=^108 18s.— Answer.
Proof. — By Table 1, under the head of 4 per cent, and against 4 years we find ,£'1.169859^ the amount of c^'l in 4 years at 4 per cent.^ which multiplied by 1 08.89685 =<£'127. 3938 =£127 7s. lOd., the sum to which £108.89685 the present value of an An- nuity of £30 at 4 per cent, will amount to in 4 years^ and
By Table 3, under 4 per cent, and against 4 years will be found £4.246464, the amount of an Annuity of £l in 4 years at 4 per cent.^ which multiplied by 30 =£127.3939 = £127 7s. lOd. the amount of an Annuity of £30 at the same rate and for the same term^ thus proving the accuracy of the present value as determined from Table 4.
The total present value of an Annuity for a term of years is manifestly equal to the sum of the present values of each yearns payment^ and by the continued addition of these at the several rates of Interest Table 4 has been formed. For example — by Table 2.
11
.£0.961538 is given as the present value of .^l to be received at the expiration of 1 year at 4 per cent. Interest.
£0.924556 ditto ditto at the expiration of 2 years.
£1.886094 Sum of the above^ or present value of an Annuity of £\ for 2 years.
£0.888996 present value of£l to be received at the expiration of 3 years.
£2.775090 Sum of the above, or present value of an Annuity of £l for 3 years.
£0.854804 present value of £l to be received at the expiration of 4 years
£3^629894 Sum of the above^ or present value of an Annuity of £l for 4 years^ &c. &c.
2nd. Perpetual Annuities. — The present value of a Perpetual Annuity is that sum which paid now and invested at a given rate of Interest will per- petually produce the same amount as will the An- nuity itself invested in like manner.
It is manifest that if £100 were sunk at 5 per cent, that it would be the present value of a Perpe- tual Annuity of £5^ and consequently that £20 would be the present value of a Perpetual Annuity of £l^ for— If £ 5 : 100 :: 1 : 20 or
If £.05 : £1 :: 1 : 1^= 20— and in a similar manner the present value of a perpetuity at any other rate of Interest might be found^ there-
12
fore The present value of a perpetuity of £\ may he found by dividing £l by the Interest of £\ at the given rate for one year, and the quotient multiplied hy any other perpetuity will give the present value of such perpetuity,
EXAMPLE 2.
What is the present value of a Freehold Estate producing £l50,per annum^ reckoning Interest at 4 per cent.?
At the end of Table 4^ under column headed 4 per cent, will be found 25 =;^^ which multiplied by £150 = £3750. — Answer.
Now at 4 per cent. £3,750 sunk will yield .£150 per annum, therefore £3,750 invested at 4 per cent, and never withdrawn, is equal to a Perpetual An- nuity of £150 invested in like manner, it producing annually exactly that sum.
3rd. Deferred Annuities. — The present value of an Annuity not to be entered upon until the expi- ration of a given period, is that sum which paid down now and invested at a given rate of Interest will, at the end of the period during which the Annuity is deferred, amount to the sum which will then^ at the same rate of Interest, purchase the Annuity in ques- tion to be entered upon immediately,
EXAMPLE 3.
What is the present value of an Annuity of £30, to be entered upon at the expiration of 4 years and then to continue 10 years, reckoning Interest at 4 per cent.?
13
By the exemplification of the construction of Table 4, it has been shewn that the total value of an annuity for any given term is equal to the total ot the present values of each year's payment through- out the term, consequently if the present value of the firsts or any number of year's annuity, is deducted from the present value of the annuity for the whole term, the difference will be the present value of the annuity for the remainder of the term.
In the present case 4 + 10 = 14 — the period during which the annuity is deferred, added to the period it is to be continued when entered upon, and on re- ference to Table 4, under 4 per cent., and against 14 years, will be found 10.563123, the present value of an annuity of ^1, to be entered upon immediately, and to continue 14 years, and in the same column opposite 4 years will be found £"3.629895, the present value of an annuity of fl, to be entered upon imme- diately, and to continue four years, therefore ^10.563123— £3. 629895 =f6.933228,present value of an annuity of £\, to be entered upon at the ex- piration of four years, and then to continue ten years, which, multiplied by30 = £207,99684 = £207 1 9s. 1 1 d. the present value of an annuity of £"30 deferred for the like period and to be continued for the same term.
Proof. — On reference to Table 1, under the head of 4 per cent., and against four years, will be found £"1.169859, the amount of £l in four years, at 4 per cent. which,multiplied by 207. 99684 gives£243. 3268, which will be found to be the present value of an annuity of £30, to be entered upon immediately, and
14
to continue ten years ; for, by Table 4, under 4 per cent, and against ten years, we find £^. 1 10(396^ the present value of an annuity of.^"!^ to be entered upon immediately, and to continue ten years, which, multi- plied by 30, will give ^243.3268, as before.
If the annuity were a Deferred Perpetuity, the present value would be found in a similar manner ; the general rule being, From the present value of the annuity for the whole of the term, at the given rate of interest, subtract the present value of the ajinuity at the same rate for the term during ivhich it is to he defeiTed. And, consequently, the value of a deferred annuity subtracted from the value of the whole term annuity, will leave the value of the temporary an- nuity, i. e, of the annuity for the term deferred.
NEW RATE OF MORTALITY.
TABLE V.
The numbers in column 2, of Table 5, against each age in column 1 are the numbers which have com- pleted or survived those ages out of the 100,000 who completed their 10th year of age, and from which, by the simple rule of Proportion, the number who might be expected to survive any given age or die within the term, out of any other number, at any age, &;C. may be ascertained.
15
EXAMPLE 1.
Out of 3,500 persons living at the age of 20, how many may be expected to survive the age of 40 ?
On reference to Table 5, it will be found that there are at 20 years of age 93/268 persons living, of whom 78.653 survive the age of 40; then
As 93.268 : 78,653:: 3500 : 2952 /zmr/j/, tlie num- ber out of 3500 at the age of 20 who may be ex- pected to survive the age of 40.
EXAMPLE 2,
It is required to determine the number of deaths that may be expected out of 3500 persons alive at the age of 20 during the next 20 years ?
By Table 5, it appears that the number living at the age of 20 is 93,268 and the number livino- at the age of 40 is 78,653, therefore 93.268 — 78.653 = 14.615 the number who died during the interval, hence 93,268 : 14.615 :: 3500 : 549 the number who may be expected to die in 20 years or before attain- ing 40 years of age, out of 3500 alive at 20 years of age.
PROBABILITIES OF LIFE.
TABLE VI. EXAMPLE 1.
Required, the probability of a person aged 30, dying within and surviving one year ?
16
On reference to column 2^ in Table 6, and against 30 years of age, will be found .0084248, the proba- bility of a person aged 30 dying in one year ; and on reference to column 3, in the same Table and against the same age, will be found .9915752, the probability of a person aged 30 surviving one year ; and the two added together will give unity or cer- tainty, for it is manifestly certain that a person at any age will either survive a given period or die within it, from which it follows that if we know the probability of a person at any age dying within any given period, and subtract it from unity, the dif- ference or remainder will be the contrary proba- bility, or the probability of surviving the given period ; and, on the other hand, if we subtract the probability of surviving from unity , the remainder will give the probability of not surviving, or of dying within the given period.
The probabilities of dying within one year are obtained by dividing the number of deaths against each age by the number living at the same age, and the quotient subtracted from unity gives the pro- bability of surviving one year. Or, the probability of surviving one year may be obtained by dividing the number living one year older than the given age by the number living at the given age, and the quotient subtracted from unity gives the probability of dying within one year. And in this manner Table 6 was constructed.
17
EXAMPLE 2.
Required the probability of a person aged 16, sur- viving the age of 20 ?
This will evidently be the number living at the age of 20, divided by the number living at the age of 16, or by Table 5, |?|- = .97190
EXAMPLE 3.
Required the probability of a person aged 16, dying in the 21st year of his age.
The number who die in the 21st year of age, being the decrement set against age 20, according to Table 6, is 680, and this divided by 95965, the number living at 16, will evidently give the probability of one of that number dying in the 21st year of age, or J-^; this probability might have been obtained by subtracting the probability of the life surviving the 21st year of age, from the proba- bility of its entering upon that age, or the probability of its surviving the 20th year of age, thus : —
.Q3268 92588 680 ' . . .
— . — — — = as beiore, and
95965 95965 95965 '
this will be manifest upon inspection, as the first numerator is the number living at 20, and the second, the number living at 21, and the difference is the number of deaths which occur within the 21st year, and the denominator the number living at 16, is common to each of the three fractions.
From the above it will appear that, the rule for
D
18
determining the probability of a life surviving any age is.
Divide the number living at the advanced age by the number living at the present age. And of its failing in any year of age, Divide the number of deaths tvhich occur in that year"^ by the Jiumher living at the present age; or sub- tract the probability of the life surviving the given year from the probability of its entering upon that year.
EXAMPLE 4.
Suppose a Life Assurance Office to have 2000 policies in force, averaging £1000 each policy, viz., 200 at 25 years of age ; 300 at 30 ; 400 at 35 ; 500 at 40 ; 300 at 45 ; 200 at 50 ; 50 at 55 ; and 50 at 60 ; it is required to determine the number and the amount of claims by deaths that may be expected to be made within one year.
The probabilities of surviving and of dying in Table 6, being the probabilities of one person at the given ages dying within, or surviving one year, it is manifest that the probabilities of any other number dying within, or surviving that period, will be obtained by multiplying such probabilities by the number in question. Hence,
Probability of one Number Probable
Age. Person dying in of number of
one year. Persons. Deaths.
25 .0077700 x 200 = 1.55400 30 .0084248 x 300 = 2.52744
* The number of deaths which occur in any year is represented by the decre- ment set opposite the next younger age.
19
|
Age. |
Probability of one Person dying in one year. |
Number of Persons. |
Probable number of Deaths. |
|
35 |
.0092877 X |
400 = |
3.71508 |
|
40 |
.0103619 X |
500 = |
5.18095 |
|
45 |
.0122120 X |
300 = |
3.66360 |
|
50 |
.0159386 X |
200 = |
3:18772 |
|
55 |
.0216643 X |
50 = |
1.08321 |
|
60 |
.0303362 X 50 = I number of Deaths that may |
1.51681 |
|
|
Tota] |
|||
|
be |
expected. |
. |
22.42881=22^ |
nearly, which multiplied by £1000, the amount of each policy, gives ^£^22,429, the whole amount of claims that may be expected. This number, and the amount being determined from the policies in force at the beginning of the year, only indicates the probable number and amount of claims that may be expected to arise out of that number only, and upon the supposition that all the policies continue in force, except those which become claims. But as an addition will be made during the year, by the introduction of new business, and as some policies may lapse, or be surrendered, they must be taken into account before a comparison can be made of the number of deaths that might be expected, with the number that actually occurred. Of the new policies, and those surrendered, it may be assumed that taking one with another, they were each in force one half- year, or, which is the same thing, that one-half of them were in force the whole of the year. In making tlie comparison at the end of the year, there-
20
fore, one-half of tlie number of neiv policies at each age, should be added to the number in force at each age at the beginning of the year, and one-half of those lapsed or surrendered at each age should be deducted from the number in force at each age, the numbers being thus corrected, the number of deaths expected according to the Table may be obtained as above. An office may, therefore, with very little difficulty, ascertain whether the amount of claims during the year is more or less than they had reason to expect.
EXAMPLE 5.
Required the probability of two lives aged 16 and 21, both surviving 5 years ?
The probability of a life aged 16, surviving 5 years,, by Table 5 is ^; and of a life aged 21, surviving 5 years, is ^^; and these two quantities multiplied to- gether will give the probability in question. For unity or certainty bears the same ratio to either of the probabilities as the remaining probability does to that required, viz.,
^g . 92588 .. 89137 . 92588 ^ 89137 * 95965 " 92588 * 95965 92588
= ^^^^^ = .92885 Answer. 95965
Then to find the probability of any two given lives, both surviving a given period, the rule is simply,
21
3Iulliphj the separate probabilities together, and the product will be the probability/ required ; and the same rule applies to the probability of any two lives, both failing in, or within any given period, and in a similar manner the probabilities of three or more lives surviving, or failing within a given period, may be obtained.
EXPECTATION OF LIFE.
TABLE VII.
By Expectation of Life is meant the average number of years that a person, at any age, may yet expect to live, taking one life with another. For example, a person aged 30, (see Table 7, 30 years of age,) according to the experience amongst assured lives many expect to live 34J years nearly, or, in other words, he may expect to attain the age of 64|- years nearly.
The total-existence enjoyed in any one year by the number of persons alive at any age at the expiration of one year, will manifestly be as many years as there are persons who survive the year, added to the existence enjoyed by those who die within the year. And of those who die within the year, it is probable that as many die at equal intervals during the first half year, as die at the same intervals during the last half of the year, or, in other words, that of
22
those who die in any one year, taking one life with another, it may fairly be assumed that, upon an average, they each enjoy one-half year's existence — therefore, the total existence enjoyed at the expira- tion of a year, by those alive at any given age at the beginning of the year, is equal in years to the number who survive the year, plus one-half of those who died within the year.
EXAMPLE.
Required the number of years that a person aged 90, may expect to live.
|
On reference to Table 5, i( that, of 13] 9 persons alive at of 90. |
appears the age |
Who enjoyed between them in each year as many years as there are persons, or the under- mentioned number of years. |
To which we must add one- half of the num- ber who died in each year or |
Which gives. |
|
892 survived 1 |
year. |
892 |
2134 |
I105i |
|
570 „ 2 |
^» |
570 |
161 |
731 |
|
339 „ 3 |
>) |
339 |
115i |
454* |
|
184 „ 4 |
?y |
184 |
774 |
2614 |
|
89 „ 5 |
}? |
89 |
474 |
136* |
|
37 „ 6 |
yy |
37 |
26 |
63 |
|
13 „ 7 |
yy |
13 |
12 |
25 |
|
4 „ 8 1 „ 9 0 „ 10 |
yy yy |
4 1 0 |
H 1 2 |
n 1 2 |
Sum =
^-2788*
2129 + 659| And this divided by 1319, gives 2.11, or % years expectation of life to a person aged 90, and agrees with the expectation as given in Table 7, opposite to 90 years of age.
The 659*, the sum of all the halves of the number
23
of deaths in each year, is manifestly one-half of the number who were alive at the age of 90 ; the expec- tation might, therefore, have been obtained by dividing the sum of all who survived that age 2129, by the number alive at that age 1139, and adding to the quotient ^ for
27S8i _ 2129 + 6591- _ 2129 ^ i
1319 1319 1319
+ i = 2.11
so that a Table of the Expectations of Life may easily be formed, by first obtaining the successive sums of the numbers surviving each age. and then dividing them by the number living at each age, and adding ^ to the quotient, and in this manner Table 7 was constructed.
COMPARATIVE EXPECTATIONS OF LIFE.
TABLE VIII.
This Table speaks for itself, and sets forth the Expectations of Life as deduced from various rates of mortality, and also amongst the different descrip- tions of assured lives, and will be found not only very interesting, but very important, particularly as from the Irish experience, it appears that, of that class of assurances, at some of the younger ages, the Expectation of Life is as much as 6 years less than that obtained from the combined English town and
24
country experience. — (See observations on the Irish experience, in ^' Introduction/'')
LIFE ANNUITIES AND ASSURANCES. TABLES IX. X. AND XI,
ANNUITIES.
Required the Value of an Annuity of £l per annum, on a life aged 97, reckoning interest at 3 per cent ?
If this were an annuity certain, its value would be equal to the sum of the present values of ^1, to be received at the expiration of 1 and 2 years, but as the payment of the annuity is contingent upon the existence of the life the value of each year's pay- ment of the Life Annuity will be less than that of an annuity certain^ in the ratio oi unity or certainty to the probability of the life surviving each year.
By Table 2, under the head of 3 per cent., we find.
.970874 = present value of £\y to be received at the expiration of one year.
.942596 = ditto ditto, two years, and, by Table 5 we find the number living at the ages 98 and 99 to be respectively 4 and 1, and these, divided by 13, the number living at 97 will give -^ and ^, the probability of a life aged 97 surviving 98
25
and 99 years of age; the latter — the oldest age which can be survived according to the Table. The present value of the first year's payment^ therefore, on a life aged 97, will be
As 1 : -i :: .970874 = i-^-^^^
And of the second,
As 1 : 7^ :: .942596 = —
And the total value will be
(4 X 970874) + 1 x (.942596) __ 4.826092
13 "" 13 =^•^'^1
as given in Table 12, in column headed 3 per Cent., opposite to 97 years of age.
Now the value of a fraction is not altered in any degree by multiplying its numerator by any quantity provided Y^e also multiply its denominator by the same quantity. For example, if we multiply the numera- tor and denominator of the fraction ^, by 2 and by 30, we get \, and ^-g, each of which is still equivalent to ^, for if the fraction in question be of 60 shillings, ^ of it is 30s., and ^ of it being 15s., ^ths. is necessarily 30s., and, in like manner^ ^th. of 60s. being Is., the |J ths. must be 30s. and so with any other fraction. If, for example, we say, the probability of a person living 1 year is |^, of another ?, and of a third g^ths, their probabilities are each equal to |^, this being premised, what follows will appear clear.
The following are the quantities given above, from
D
20.
which the value of an annuity, on a life aged 97, at
3 per cent, interest, was obtained .
(4 X .970874) + (1 x .942"596) _
J3 -0.371
which, expressed in words, is the number living at 98 years of age, multiplied by the present value of £l, to be received at the expiration of 1 year,^Zws the number living at the age of 99, multiplied by the present value of ^1, to be received at the expiration of 2 years, divided by the number living at the age of 97.
Now, if we multiply each of the quantities in the
numerator and denominator by .056858 the present
value of <£l, to be received at the expiration of 97
years, (the same as the age of the life,) we shall get
(4 X .055202) + (1 X .053594)
13 X .056858
i e. the number living at 98, multiplied by the pre- sent value of ^1, to be received at the expiration of 98 years, plus the number living at 99, multiplied by the present value of .f'l to be received at the expira- tion of 99 years, divided by the number living at 97, multiplied by the present value of ^1, to be received at the expiration of 97 years, which is equal to
:^^ = .37. as before,
and in a similar manner, the value of an annuity
at any other age may be obtained.
![_ But the D and N columns for the rates 2|^, 3, and
•27
3 J per cent, in Tables 9, 10, and II, contain the numerator and denominator that will obtain at each age ; the quantities in column D being the number living at each age, multiplied by the present value of £l, to be received at the expiration of as many years as the age, and the quantities in column N, opposite to each age, are respectively the sum of all the quantities in column D., at all the ages older than the given age; therefore,
T/ie quantity in column N, opposite to any age, divided by the quantity in column J), at the same age, will give the value of an annuity at that age.
And in this manner the values of the annuities at 2^ 3, and 3h per cent, in Table 12 were obtained.
For example, at 2|- per cent. (See Table 9.)
Nat 98 = 00^676 ^ .^44 the value D at 98 = 0.35573
of an annuity of £l per annum on a life aged 98, and agrees with the value given in Table 12.
N at 97 = 0^44249 ^ 3^3 ^^^ ^^^^^ D at97 = 1.18503 of an annuity of £\ per annum, on a life aged 97, as also given in Table 12-
Column S in Tables 9, 10, and 11, is the sum of the quantity at each age, and at all the ages older than the given age in column N, and is useful in find- ing the values of increasing and decreasing annuities.
ASSURANCES. The diiference between the value of an Annuity
28
and that of an Assurance is, that in the former, as has already been shewn, each yearns payment depends upon the probability of the life surviving each year of age, whereas, in the latter, the value depends upon the probability of the Mi^ failing in each year, and in the calculation of the premiums, the sum assured is, in all cases, assumed to be payable at the expiration of the year in which the life fails.
The present Value, therefore, or ^' Single Pre- mium '' for an assurance on a life at any age, is equal to the sum of the present values of <£l certain, to be received at the expiration of 1, 2,3, &c., &c. years to the end of life, multiplied respectively by the proba- bility of the life failing in each year.
EXAMPLE.
Required, the single premium to secure £\ on a life aged 97, reckoning interest at 3 per cent.
By Table 2,— .970874 = Present value of £l to be
received at the expira- tion of 1 year. „ .942596 = ditto ditto, 2 years
„ .915142 = ditto ditto, 3 „
And by Table 5, — — = Probability of a life aged 97
failing in or before comple- ting the 98th year of age. ^ = ditto 99th year.
1= ditto 100th ditto.
Then,
13
(9 X .970874) + (3 x .942596) + (1 x .915142) ^^
yn ^ = .96005
29
the Single Premium required ; but if^ as in the case of Annuities (see page 26) we multiply the numerator and denominator by .056858 the present value of ^'l to be received at the expiration of 97 years, (the same number of years as the age,) the value will not be altered, and we shall have (9 X .055202) + (3 X .053594) + (1 x .052033)
1 3 X. 056858 -.Jb005
as before, and in a similar manner the single pre- mium for an assurance at any other age may be found. But we have already got each of the denominators that would obtain at each age (the number living at each age multiplied by the present value of ^1, to be received at the expiration of the same number of years as the age) — in column D, and the quantity in column M, opposite to any age, is equal to the sum of the decrements opposite to that age, and all the ages older than the given age in Table 5, multiplied respectively by the present value oi £\. to be received at the expiration of one year more than the given age, as, for example :
Present value
Deere- ""^ ^^ ^"^,** ^Se- ment the end of one year more than
the age.
99 1 X .052033 = .052033 =M, opposite to 99 years
of age 98 3 X .053594 = .160782
Sum = .212815 = M, ditto, 98 97 9 X. 055202 = .4968 18
Sum = .709633 =M, at 97
and the last quantity, .709633, is the sum of the pro- ducts in the numerator above, and agrees with the
30
quantity in Table 10, in column M, opposite to 97 years of age^ and the quantity in column D^ opposite to 97 is .73915^ and corresponds with the pro- duct of the quantities in the above denominator.
Then —^ =.96005 as before, and agrees with the quantity in Table 15^ in column headed^ ^^ Single Premium^'^ opposite to 97 years of age, so that, where the columns D and M, are formed the rule to determine the single premium is. Divide the quantity in column M opposite to the age hy the quantity in column D, opposite to the same age. and, in this manner, the single premiums at each asre in Table 15 were obtained.
If the annual premium for an Assurance were £l per annum, its equivalent present value, or ^^ Single Premium," would manifestly be £l paid down,^ added to the present value of an annuity of £*!, to be paid during the life in question, or on a life aged 97 .27439 =N, opposite to 97 "^ .73915 = D, do.
,.,. ,, .73915 + .27439 1.01354
which IS equal to ^g^^^ = -j^gy^
then by the simple rule of proportion. If
1.01354 , .70962 = M, at97 .70962 .73915
.73915' .73915 =D, at 97 • .73915 1.01354
.70962 = M, at 97 „^^. .
= . . XT ^ n/? .70014
1.01354 =N, at 96 the annual premium for an assurance of jf'l on a life aged 97, and corresponds with the quantity given in
* The Annual Premium for an assurance is always paid at the beginning of the year.
31
Table 15, in column headed ^^ Annual Premium/' opposite to 97 years of age.
The rule^ therefore, to determine the annual pre- mium for an assurance of £^1 is,
D ivide I he quantity in cohuim M, opposite to the given age, hy tJie quantity in column iV, opposite to the age one year younger; and, in this manner, tlie annual premiums at each age, in Table 15, were obtained.
It is also manifest from the above that the annual premium might have been obtained by the following rule :
Divide the Single Premium by 1 plus the value of an annuity on the life at the given age.
Column R is the sum of the quantity at each age, and all the ages older than the given age in column M, and is useful in finding the values of increasing and decreasing assurances.
LIFE ANNUITIES.— SINGLE LIVES.
TABLE XII.
It has already been shewn, in page 27 that the rates 2J, 3, and 3^ per cent, in this Table, have been constructed from the D and N columns in Tables 9, 10, and 11, but as D and N columns have not been constructed for any other rates of interest, it was found to occupy less time to calculate the remaining rates by the ordinary method.
As the payment of an annuity depends upon the
32
party being alive when it becomes due, and as an
annuity is considered to be due at the end of each
year, it is manifest that the value of an annuity on a
life aged 99, the oldest age in the Table, is equal to
0 ; and on a life aged 98, the value, if the life were
certain to survive the year, would at the end of the
year be equal to £l, plus an annuity on a life aged
99, the present value of which reckoning interest
at 3 per cent, is manifestly.
1 +0 X .970874 = .970874; but as the life is not certain
to survive the year, this value must be diminished in
the ratio of certainty or unity to the probability of its
surviving the year, and will be
A . 1 c^n^onA .970874 ^., As 1 : |:: .970874 : — - — = .243
and corresponds with the value given in Table 12, under 3 per cent, and opposite to 98 years, and by proceeding in this manner from the oldest to the youngest age, the rates 2, 4, 4|^, 5, 6, 7, and 8, per cent, have been computed, and is the method adopted by Mr. Milne in his excellent treatise on annuities.
The rule being
Multiply U7iily added to the value of an annuity on a life one year older than the given life by the present value of £\, due at the end of 1 year, and by the pro- bability of the given life surviving 1 year, and the product ivill be the value of an annuity on the given
life.
The table being formed, the value of any other amount at any given age and rate of interest, may be readily obtained by the following rule :
33
Multiply the anmiity of £\ at the given age and rate per cent, by the annuity^ whose amount is required, and the product will be the value of such annuity.
EXAMPLE 1.
Required the value of an Annuity of ^150 per annum, on a life aged 54 reckoning interest at 3 per cent ?
By Table 12, opposite to 54 years of age, will be found 12.385. the present value of ^1 per annum on a life at that age, which, multiplied by 150 = £1857.75 =^1857. 15 the value required
If it were required to find what annuity should be granted in consideration of a sum to be paid down, the rule would manifestly be
Divide the sum to be paid down by the present value of an annuity of £\ on the given life at the given rate of iideresiy as for
EXAMPLE 2.
What Annuity ought to be granted on a life aged 54 in consideration of jf" 1857. 15 paid down, reckon- ing interest at 3 per cent ?
12.385 was shewn in the last example to be the value of an annuity at 3 per cent, on a life aged 54.
1857.75 ^,_ ^ J ,
then .^ oog = X 1 50 Answer, — and corresponds 12.385 ^ ^
with the annuity in example 1, whose present value was shewn to be £1857. 75 = f 1857.15.
E
34 LIFE ANNUITIES— JOINT LlV^ES.
TABLE XIII.
The same reasoning employed with respect to Annuities on Single Lives^ is applicable to Joint Lives, the rule to determine the value of an annuity on the latter being,
Multiply unily added to the value of an annuity on two Joint LiveSy respectively , one year older than the two given lives, by the present value of £\, due at the end of one year, and by the probability of the two given lives jointly surviving one year,
EXAMPLE.
Required, the value of an Annuity on two Joint Lives aged 89 and 84, reckoning interest at 3 per cent ?
The two lives one year older than these respectively, are aged 90 and 85, and, on reference to Table 13, in the column headed, '' Older," will be found 90, and in the column on the right, headed, " Younger," will be found 85, opposite to which, in the column headed, 3 per cent, will be found,
0.946 the value of an annuity on two joint lives, aged 90 and 85, And on reference to Table 6, it will be found that .7076180 is the probability of a life aged 89,
surviving one year .8103215 ditto 84 years, ditto
35
then .7070180 x .8103215 = .57340 the probability
of the lives jointly
and by Table 2 .970874
surviving one year present value of <£*! at 3 per cent, due at the end of one year, then 1,946 x .970874 x .57340 = 1.083 the value of an annuity on the two lives aged 89 and 84 as required^ and which corresponds with the value in Table 13, opposite to 89 and 84, in column head- ed 3 per cent., and in this manner by beginning at the ages
at the several rates of in- terest, all the joint lives, where the difference of age is 5 were obtained, but it was not thought necessary to print the values of any joint lives at an older age than 90.
|
99 |
& M\ |
|
|
len |
98 |
II 93 |
|
// |
97 |
// 92 |
|
if |
m |
// 91 |
|
if |
95 |
// 90 |
|
if |
94 |
// 89 |
|
If |
93 |
// 88 |
|
ff |
92 |
// 87 |
|
n |
91 |
// 86 |
|
a |
90 |
// 85 |
|
H |
89 |
// 84 |
|
&c. |
&c. |
And in a similar manner all the other quantities at the several rates of interest and differences of age in Table 13 were obtained; the value of the oldest of the two given lives at the given difference of age being first obtained, and then the values of the next two respectively, one year younger, &c.
36
The Table being formed, the value of an Annuity for any amount at any of the given ages, and rates of interest, may be obtained in the following manner.
Multiply the value of the annuity of £\ at the given ages and rate of interest by the annuity, ivhose value is required, and the product ivill be the value of such annuity.
EXAMPLE 1.
Required the value of an Annuity of £^0 per annum on two joint lives aged 71 & 51, reckoning interest at 3|- per cent ?
On reference to Table 13, in column '' 3 J per cent/' opposite to 71 & 51, will be found 5.487, which, multiplied by 30, gives 164,610 = ^164 12 2, the value required?
EXAMPLE 2.
Required the value of an Annuity of £50 per an- num on two joint lives aged 7 1 and 53, reckoning interest at ^^ per cent ?
It will be found, on reference to Table 13, that both these ages are not contained in the Table, but against 71, the older age (in finding the values of annuities on joint lives, the older age is the index of the two ages), we find opposite to the column headed ^* Younger,"' that age 53 falls between 5 1 and 56, and the value at 3^ per cent, on 71 & 51 is 5.487 and 71 // 56 // 5.240 Dilference 0,247
37
and this being the difference for 5 years, -|th or, 049 subtracted from 5.487 will give the
value on 71 & 52
^^« // 098 ditto ditto, on 71 & 53
-^ths // 147 ditto ditto, // 71 // 54
-> // 196 ditto ditto, // 71 // 55
then 5.487-098 = 5.389, which, multiplied by 50
= ^269,450 =£"260 9, the value of an annuity of
£50 per annum on two joint lives, aged 71 & 53, as
required.
And in a similar manner the value of an annuity
at any other ages not found in the Table may be
obtained.
TWO JOINT LIVES AND THE SURVIVOR.
An Annuity on the Last Survivor of two lives signifies an Annuity to be paid until the expiration of both lives.
It is manifest that an annuity during the joint conti- nuance of two lives added to an annuity on the last survivor, are together equal to the sum of similar annuities on each of the lives, for in the case of the JointLives, the annuity would cease at the first death, and in the other on the death of the last survivor, consequently the value of the annuity on the last sur- vivor may be obtained by subtracting the value of an annuity on the Joint Lives from the sum of the annuities on the two single lives.
EXAMPLE.
Required the value of an Annuity of £30 per
38
annum, on the last survivor of two lives aged 51 and 36, reckoning interest at 3^ per cent?
On reference to Table 12, in column headed 3 J per cent, will be found opposite to ages 5 1 and 36
12.795 = Value of annuity of jfi'l on a life aged 51,
17.037= ditto ditto 36^
29.832 = Sum
And on reference to Table 1 3, in column headed 3j per cent, will be found 11.260, the value of an annuity on the two joint lives; then 29.832 — 11.260 = 18.572, which multiplied by 30, gives 55,7160 = £65 14 4 — Answer. And in a similar manner the value of an annuity of any other amount may be obtained, the rule being.
From the sum of the valuea of an annuity of £\ on the separate lives at the given late, deduct the value of a similar annuity at the same rate on the Joint Lives and midtiply the difference by the annuity whose value is required.
ABSOLUTE REVERSIONS— PRESENT
VALUES.
TABLE XIV.
The mode of constructing this Table is explained in page 42.
EXAMPLE.
What is the present value of the Reversion to ^£"5000, or which is the same thing, the Single Pre- mium for an assurance of £5000 to be received at
39
tlie end of the year, in wliicli a life aged (JO may fail, reckoning interest at 4 per cent. ?
By column 4 per cent, in Table 14, opposite to 60 years of age will be found .59943, the present value of the reversion of ^1 on the failure of the life in question ; then
.59943 X 5000 =^2997.15 =f 2997 3 0 the value required.
LIFE ASSURANCES— SINGLE LIVES. TABLE XV. EXAMPLE I,
What Single Premium should be charged for an assurance of f 2500 on a life aged 55, reckoning interest at 3 per cent. ?
By column headed ^^ Single Premium," in Table 15, and opposite to 55 years of age will be found .62075 the Single Premium to assure £l on the given life; then ,62075 X 2500 =^155L875 =^1551 17 6, the Single Premium required.
EXAMPLE 2.
What Annual Premium should be charged for an assurance of £4000 on a life aged 65^ reckoning in- terest at 3 per cent. ?
By column headed Annual Premium in Table 15, and opposite to 65 years of age, will be found. 07745, the Annual premium for an assurance of ^1 on the given life, then
.07745x4000 =^309.8 =.£309 16, the Annual Premium required.
40
The quantities in Table 15 were obtained by means of the D. N, and M, columns in Table 10, as explained in pages 28—31. The mode of obtaining the same results by the ordinary method will be illustrated in the following
EXAMPLE. 3.
Required the Single Premium for an assurance of ^1 on a life aged 97, reckoning interest at 3 per
cent.
By Table 2 .970874= Present value of £], to
be received at the ex- piration of 1 year. // .942596= ditto ditto, 2 years.
// .915142= dittoditto, 3 //
and by Table 5 ^^~^ = Probability of a life aged
97 failing in or before completing the 98th year of age.
^~~^ = ditto ditto, in 99th ditto
13
// ^= ditto ditto, in 100th ditto
Then (see page 28)
(1^ X .970874) + (^ X .942596) +
f_i_x .915142) =.96005 Single Premium re-
^ 13 quired as contained in Table 15, in column headed '' Single Premium/' opposite to 97 years of age.
41
Let us, however, separate the positive from the negative quantities, and we shall have ( if x .970874) + (^ X 942596) + (i§ X .915142)=Positive quantities. If we divide each of these by .970874, the present value of ^1, to be received at the expiration of one year, and multiply them again by that quantity, their value will still be the same, and we shall have
.970874[j|+(^x 970874) + (j^x942596)|
But the sum of the two last quantities, as was shewn in page 26, is equal to jf 0.37 1— the value of an annu- ity on a life aged 97, if, therefore, we substitute this value we shall have
^ + 0.371 )> = 970874 + (.970874 x 0.371)
Let us now bring down the negative quantities from the original expression which are,
( ^ X 970874) + (^ X 942596)
But these have just been shewn to be equal to .0.371 the value of an annuity on a life aged 97, this quan- tity, therefore, must be subtracted from the above expression, which will give
,970874 + (.970874 X .0.371) -0.371. Now the middle quantity is the present value of £0.371 to be received at the expiration of one year, (for the present value oi £i due at the end of any number of years, multiplied by any other sum, gives the present value of that sum for the same period), and if we subtract it from the last quantity we shall have .01082 or the discount for one year of the value of
42
the annuity;* then 970874— .01 082 = .96005, as before.
The rule, therefore, for finding the Single pre- mium for an assurance by the ordinary method is
From the prese7it value of £\ at the given rate of
interest due at the end of one year subtract the discount
for one year of the value of an annuity of £\ on the
given life at the same rate of interest. And by this
rule the quantities in Table 14 were obtained.
The Rule to determine the annual premium as shewn in page 31, is
Divide the single premium by 1 plus the value of an annuity on the life.
And in a similar manner it might be shewn, that the Rule to determine the Single Premium for an assu- rance on two Joint Lives is
From the present value of £\ at the given rate of interest due at the end of one year^ subtract the discount for one year of the value of an annuity of £\ on the Joint Lives at the same rate of interest.
And for the Annual Premiums
Divide the single premium by 1 plus the value of an annuity on the Joint Lives.
And in this manner Table 16 was formed.
And similarly —
To find the Single Premium for an Assurance on the Last Survivor of Two Lives.
* The discount of any sum is manifestly the difference between that sum and its present value, and may be obtained by multiplying the discount of £1 by any other sum, whose discount is required.
43
From the present value of £\ at the given rate of interest due at the end of one year, subtract the discount for one year of the value of an annuity of £\ on the last survivor, of the two lives at the same rate of interest.
And for the Annual Premium —
Divide the single premium by 1 plus the value of an annuity on the last survivor.
And in this manner Table 17 was formed.
LIFE ASSURANCES.— JOINT LIVES.
TABLE XVI.
The quantities in this Table were constructed by the following rules (see page 42.)
To find the Single Premium.
From the present value of £\ at the given rate of interest, due at the end of one year, subtract the discount for one year of the value of an annuity of <£] oji the Joint Lives at the same rate of interest.
To find the Annual Premium :
Divide the Single Premium by £\ plus the value of an annuity on the Joint Lives,
EXAMPLE 1.
Required the single and annual premium for an assurance of £\ on two lives aged 53 and 18^ rec- koning interest at 3 per cent,?
By column 3 per cent, in Table 2^ and opposite to one year^ will be found
44
,970874 the present value of ^1 at 3 per cent.
due at tlie end of one year.
And 1-. 970874 = .029126 = discount of ^1 at the
same rate for one year.
By column 3 per cent in Table 13^ opposite to 53
and 18, will be found,
11.776, the value of an annuity of £l on the
two Joint Lives.
And .029126 x 1 1.776 = . 34297 = the discount of the
annuity for one year.
Then .970874 -.34297 = .62790 the single premium
required, and corresponds with the quantity in column
^^ Single Premium," in Table 16, opposite to ages
53 and 18.
^ , .62790 .62790 ^.^.. .i a
1 -1-11 776 ""^ 12 77(3 = .04915 the Annual
Premium required, and corresponds with the quan- tity in column *^ Annual Premium," in Table 16, opposite to ages 53 and 18.
And in a similar manner, the premiums at all the other ages in the Table were calculated, from which the Premiums, for assurances of any other amount may be readily obtained as shewn in the following examples.
EXAMPLE 2.
Required the Single Premium that would be charged according to Table 16, to effect an assu- rance of ^2000 on two lives, aged 54 and 29 ?
On reference to the Table in column, headed " Single Premium," and opposite to ages 54 and 29,
45
will be found .64306, the Single Premium for an assurance of £"1 on the two lives, which, multiplied by 2000 gives ^1286.12 =£1286 2 5, the Single Premium required.
EXAMPLE 3.
What Annual Premium should be charged for the above assurance ?
On reference to Table 16 in column, Annual Pre- mium per <£l, and opposite to ages 54 and 29, will be found .05247 which multiplied by 2000 = 104.94 = ^104 18 10, the Annual Premium required.
LIFE ASSURANCES.— LAST SURVIVOR.
TABLE XVII.
The quantities in this Table were constructed by the following rules, (see page 42.)
To find the Single Premium :
From the present value of £i at the given i^ate oj
interest clue at the end of one year, subtract the discount
for one year, of the value of an Annuity of £l on the
last Survivor of the two lives at the same rate of interest.
To find the Annual Premium :
Divide the Single Premium by 1 plus the value of an annuity on the last survivor,
EXAMPLE 1.
What Single and Annual Premium should be charged for an assurance of <£l on the last survivor
46
of two lives aged 46 and 41, reckoning interest at 3 per cent.?
By Table 12, in column 3 per cent, the value of an annuity of ^1, on a life aged 46 years is 15.204
Ditto, ditto, 41 ditto .16.821
Sum =32.025
By Table 13, in column 3 per cent. the value of an annuity of ^1 on two joint lives aged 4 6 and 4 lis. . 12.488
Difference. • 19.537 = Value of
an annuity of ^1 on the last survivor, (see page 37).
By Table 2, the present value of £l at 3 per cent, due at the end of 1 year = .970874 and 1— .970874 = .029126 the discount of ^1 at 3 per cent, for one year.
Then .029126 x 19.537= .56902 the discount for one year of the annuity on the last survivor. And .97087— .56902 = .40185 = the Single Premium required, and corresponds with the quantity in column *^^ Single Premium"^ in Table 17, opposite to ages 46 and 41.
The Annual Premium, therefore, is equal to
.40185 .40185 ^^^^^ , ,
1 + 19.537 = 20:537= -^^^^^^ ^^^ corresponds
with the quantity in column " Annual Premium per £{," in Table 17, opposite to the ages 46 and 41.
And in a similar manner the Premiums at the other ages in the Table were found, from which the value of an assurance of any other amount may readily be obtained as shewn in the following examples.
47
EXAMPLE 2.
What Single Premium should be charged for an assurance of £5000 on the last survivor of two lives aged 60 and 50^ reckoning interest at 3 per cent.?
By Table 17_, opposite to ages 60 and 50^ in column *^ Single Premium per £\," will be found »51671, the single premium for the assurance of £l, on the survivor of the two lives^ which_, multiplied by 5000, gives £2583.55=^2583 11 0 the single premium required,
EXAMPLE 3.
What Annual Premium should be charged for the above assurance ?
Bj Table 17^ opposite to ages 60 and 50^ in column headed *^ Annual Premium per£j_,''^ will be found .03114^ the annual premium for the assurance of £l^ on the last survivor of the two lives, which^ multiplied by 5000 gives £155.70 = £155 14, the annual premium required.
VALUATION OF POLICIES— SINGLE LIVES.
TABLES XVIII & XIX.
Let it be assumed that the Annual Premium upon an assurance is ^1.
Then the value of all the future Premiums_, where the Annual Premium has just been paid, is evidently equal to the value of an annuity of ^1 on the given life.
48
And where the premium is just due, but not paid, the value is evidently greater by that amount, and is equal to ^1 plus the value of an annuity of £l on the given life.
The value of the future premiums, when estimated at any intermediate period between two successive payments, may, therefore, be obtained by deducting the value of ^1 on the age of the assured, at the date of the last payment, from the value increased by unity of a similar annuity on the age at the next payment, and adding to the former a part of the difference, proportional to the time elapsed since the last pay- ment became due ; and the several values thus ob- tained are given for each year and month in Table 18,
And the value of the future payments of any other Annual Premium may be obtained by multiplying the quantities in the Table by such Annual Premium.
The quantities in Table 19, show the Single Pre- mium required for an assurance of <£ 1 on each age, from 1 0 to 70 with interpolated values for months in each year.
And the value for any other amount may be obtained by multiplying the quantities in the Table by such amount.
The Value of a policy at any time is manifestly the difference between the *^ Single Premium,^'' for the sum assured on the age of the party, at the time the policy is proposed to be valued, and the then value of all the future premiums, expected to be received on such policy.
49
EXAMPLE 1.
Required the value of a policy of £4000, effected at an annual premium of ^100 13 4 = £100.667 on a life aged 39, but now aged 57 years and four months ?
By Table 19, in column, headed 4 months, and opposite to 57 years, will be found .64561, the single premium for an assurance of £l on a life aged 57 years and 4 months.
Then .64561 x 4000 = 2582.4= Single Premium for
an assurance of
£4000 on a life
aged 57 years and
4 months.
And by Table 18, in column, headed 4 months, and
opposite to 57 years, will be found 11.501, the value
of the future premiums of f 1 per annum, on a life
aged 57 years and 4 months.
Then 11.501 x 100.667 = 1157.8 = Value of future
Premiums. And 2582.4—1157.8 = 1424.6 = f 1424 12 the value of the policy as required.
EXAMPLE 2.
Required the value of a policy of £"3000, effected at an Annual Premium of ^68 8 0, =68,4 on a life aged 36, but now aged 60, upon which the premium is just due, but not paid.
In this case the premium being just due, but not paid, the value of the future premiums will be 11.188,
G
50
the quantity in Table 1 8, opposite to 59 years and 12 months, {i.e. unity added to 10.188, the quantity opposite to 60 years of age,) multiplied by 68.4, which gives 765.25.
And by Table 19, the Single Premium for an assurance of ^''l, on a life aged 60, is .67414, which, multiplied by 3000, is equal to 2022.42. Then, 2022.42— 765.25 =£^1257. 17 =£1257 3 5 = the present value required.
If the premium in this case had been just paid, the value of the future premiums would be equal to 10.188, the quantity opposite to 60 years of age mul- tiplied by 68.4=696.85.
And 2022.42-696.85 = 1325.57=^1325 11 5 = the value required; which, it will be observed, is equal to the above value, plus £6S Ss., so that the value of a policy, when the premium has just been paid, is equal to the value of the policy upon which the premium is due and not paid, plus the payment then made.
If one or more bonuses have been added to a policy, find the value at the present age of the sum assured by the policy, plus the amount of such bonuses, and proceed as before.
The value of a policy which had been effected by the payment of a single premium is manifestly equal to the single premium that would be required for an assurance of the same amount at the present age, and may be obtained from Table 19.
51
TEMPORARY ANNUITIES AND ASSURANCES.
Comparative Advantages of the Z), iY, and M Method, and the Ordinary Method of Calculating the Values of Annuities and Assurances. The D and N system was first employed by Mr.
Griffith Davies^ the Actuary of the Guardian Assu- rance Company, and the Formulae used by him are somewhat analagous to those originally pointed out by the late Mr. Barrett.
The following examples will serve to show the superiority of the new method.
EXAMPLE 1.
Required the value of an Annuity of £'20 per annum on a life aged 36^ to continue 10 years, reck- oning interest at 3 per cent.
Rule by the D. and N. columns.
From the quantity in column N at the present as^e, subtract the quantity in the same column at the advanced a<^e, and divide the difference by the quantity in column D at the present age.
In Table 10, 515312.329 = the quantity in column
N, opposite to 36 the
present age.
,, 287000.704 ditto, opposite to 46,
■ the advanced age.
22831 1.625 = difference.
,, 28228.483 =: the quantity in column
D. opposite to 36 the
present age.
52
228311.625 ^ ^^^ ,^ then "oqooq'Tqq ="*^^"^ ^^^ value required.
Rule, by the common method — From the value of an annuity on the life at the pre- sent age, subtract the value of an annuity on the life at the advanced age, multiplied hy the present value of£l at the given rate of interest due at the end of the term for which the annuity is to continue^ and by the proba- bility of the life at the present age, surviving that term. By column, headed 3 per
cent, in Table 12 18.255 = present value of
an annuity of £^1
on a life aged 36.
Do. do. 15.204=do. do. 46.
By Table 2, in column 3
per cent, opposite 10 years, .744094 = Present value
of £\ at 3 per cent, due at the endof 10 years.
By Table 5 73526 = Probability of a
81814 life aged 36,
living 10 years.
73526 Then 15.204 x ,744094 x ^YgY^ = 10. i 67
And 18.255—10.167 =8.088 as before.
The rule to find the value of a DEFERRED ANNUITY, by the D and N columns is.
Divide the quantity in column N, at the age the Annuity is to be entered upon by the quantity in column D at the present age.
53
EXAMPLE 2.
Required the Single Premium for an assurance of £3000 on a life aged 40 for the term of 7 years, reckoning interest at 3 per cent.? Rule by the D and M columns. From the quantity in column M at the present age subtract the quantity in the same column at the advanced age, and divide the difference hy the quantity in column D at the present age.
In Table 10, 11384. 144 = the quantity in column
M, opposite to 40, the present age. u 9732.454 Ditto opposite to 47 the
advanced age,
1651.690 = difference u 241 11,6 15= the quantity in column
D, opposite to 40, the present age.
^, 1651.690 ^^^^ - . , I,. ,. , ,
Ihen 04111 ^ig = '^Q^^^ which, multiplied by 3000
gives f'205.5=.£'205 10 0, the single premium re- quired.
Rule by the common method.
From the value of an annuity on the life, at the present age, subtract the value of an annuity on the life at the advanced age y multiplied by the present value of £\. due at the end of the term for which the assur- ance is to continue, and by the probability of the life surviving that term ; and multiply the difference thus
54
obtained by the discount of £ I, for one year; then sub- tract this product frofn the present value of £\, due at the end of one year, multiplied by unity minus the product of the probability of the life surviving the term, and the present value of £l, due at the end of the term. In column 3 per cent
of Table 12 17. 123 = the present value of
an annuity of <£l at three per cent, on a life aged 40. M 14.864= do. do 47
In ditto of Table 2, .813092 = the present value of
fl^ at 3 per cent, due
at the end of 7 years
Ditto .970874 = do. do. at the end
of 1 year. And 1—970874 = 029 126= discount of£l at 3
per cent, for one year. From Table 5 we obtain 4lil, the probability of a life aged 40 surviving 7 years.
From which we obtain, according to the rule
.970874[l— ^mi X .813092 |-.029126J^17.I23—
-ggx. 813092x14.864] =
.24240— .17388 = .06852. And .06852 x 3000- £205.5 = £205 10 as before.
The rule to find the value of a DEFERRED ASSURANCE by the D and M columns is,
55
Divide the quantity in column M, at the advanced age, by the quantity in column D at the present age.
The above examples in Temporary Annuities^ and Assurances, without exhibiting the length of the operations of the multiplications and divisions, are sufficiently illustrative of the superiority of the D and N method. Other examples, much more striking, might be given, but the subject will be found fully illustrated in the treatise on Annuities and Assu- rances, by D. Jones, published by the Society for the Diffusion of Useful Knowledge, in which will also be found a very extensive collection of formulae for all cases involving one and two lives.*
*This Formulae is contained in No. 7, of the work, price sixpence, which may probably be obtained separately, and as it is printed in octavo, it might with advantage be bound up with the present work.
TAB LES
COMPOUND INTEREST,
Showing the Amount of £1 improved at Compound Interest, for any number of years not exceeding 100.
Years.
1 2 3 4 5
G 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
36
37 38 39 40
41 42 43 44 45
46 47 48 49 50
1 ^ Cent.
li ^ Cent. 2 W Cent
2i W Cent.
1.010000 1.020100 1.030301 1.040604 1.051010
1.061520 1.072135 1.082856 1.093685 1.104622
1.115668 1.126825 1.138093 1.149474 1.160969
1.172579 1.184305 1.196148 1.208109 1.220190
1.232392 1.244716 1.257163 1 269735
1.282432
1.295256 1.308209 1.321291 1.334504 1.347849
1.361327 1.374940 1.388689 1.402576 1.416602
1.430768 1.445076 1.459527 1.474122
1.488863
1.503752 1.518790 1.533978 1.549318 1.. 5648 11
1.580459 1.596264 1.612227 1.628349 1.644632
1.015000 1.030225 1.045678 1.061363 1.077284
1.093444 1.109845 1.126492 1.143389 1.160540
1.177948 1.195616 1.213550 1.231754 1.250231
1.268984 I
1.288019
1.307339
1.326948
1.346851
1.367055 1.387562 1.408376 1.429502 1.450945
1.472709 1.494800 1.517222 1.539980 1.563080
1.586527 1.610324 1.634479 1.658997 1.683882
1.709141 1.734777 1.760799 1.787211 1.814019
1.841229 1.868847 1.896879 1.925333 1.954212
1.983525 2.013277 2.043477 2.074129 2.105240
1.020000 1.040400 1.061208 1.082432 1.104081
1.126162 1.148686 1.171659 1.195093 1.218994
1.243374 1.268242 1.293607 1.319479 1.345868
1.372786 1.400241 1.428246 1.456811 1.485947
1.515666 1.545980 1.576899 1.608437 1.640606
1.673418 1.706886 1.741024 1.775845 1.811362
1.847589 1.884541 1.922231 1.960076 1.999890
2.039887 2.080685 2.122299 2.164745 2.208040
2.252200 2.297244 2.343189 2.390053 2.437854
2.486611 2.536344
2.587070 2.638812 2.691588
3#'Cent. Sic^Cent.
1.025000 1.050625 1.076891 1.103813 1.131408
1.159693 1.188686 1.218403 1.248863 1.280085
1.312087 1.344889 1.378511 1.412974 1.448298
1.484506 1.521618 1.559659 1.598650 1.638616
1.679582 1.721571 1.764611 1.808726 1.853944
1.900293 1.947800 1.996495 2.046407 2.097568
2.150007 2.203757 2.258851 2.315322 2.373205
2.432535 2.493349 2.555682 2.619574 2.685064
2.752190 2.820995 2.891520 2.963808 3.037903
3.113851 3.191697 3.271490 3.353277 3.437109
1.030000 1.060900 1.092727 1.125509 1.159274
1.194052 1.229874 1.266770 1.304773 1.343916
1.384234 1.425761 1.468534 1.512590 1.557967
1.604706 1.652848 1.702433 1.753506 1.806111
1.860295 1.916103 1.973587 2.032794 2.093778
2.156591 2.221289 2.287928 2.356566 2.427262
2.500080 2.575083 2.652335 2.731905 2.813862
2.898278 2.985227 3.074783 3.167027 3.262038
3.359899 3.460696 3.564517 3.671452 3.781596
3.895044 4.011895 4.132252 4.256219 4.383906
1.035000 1.071225 1.108718 1.147523 1.187686
1.229255 1.272279 1.316809 1.362897 1.410599
1.459970 1.511069 1.563956 1.618695 1.675349
1.733986 1.794676 1.857489 1.922501 1.989789
2.059431 2.131512 2.206114 2.283328 2.363245
2.445959 2.531567 2.620172 2.711878 2.806794
2.905031 3.006708 3.111942 3.220860 3.333590
3.450266 3.571025 3.696011 3.825372 3.959260
4.097834 4.241258 4.389702 4.543342 4.702359
4.866941 5.037284 5.213589 5.396065 5.584927
TABXiS X.
COMPOUND INTEREST,
Showing the Aiiiount of£l im})roved at Compoun<l Interest, for any number of vears not exceedinnr 100.
|
Years |
. 4 #" Cent. |
4^ #• Cent |
. 5#'Cent. |
6 f Cent. |
7 4f Cent. |
8 ^ Cent. |
|
1 |
1.040000 |
1.045000 |
1.050000 |
1.060000 |
1.070000 |
1.080000 |
|
o |
1.081000 |
1.092025 |
1.102500 |
1.123600 |
1.144900 |
1.166400 |
|
3 |
1.124864 |
1.141166 |
1.157625 |
1.191016 |
1.225043 |
1.259712 |
|
4 |
1.169859 |
1.192519 |
1.215506 |
1.262477 |
1.310796 |
1.360489 |
|
5 |
1.216653 |
1.246182 |
1.276282 |
1.338226 |
1.402552 |
1.469328 |
|
6 |
1.265319 |
1.302260 |
1.340096 |
1.418519 |
1.500730 |
1.586874 |
|
7 |
1.315932 |
1.360862 |
1.407100 |
1.503630 |
1.605781 |
1.713824 |
|
8 |
1.368569 |
1.422101 |
1.477455 |
1.593848 |
1.718186 |
1.850930 |
|
9 |
1.423312 |
1.486095 |
1.551328 |
1.689479 |
1.8384.59 |
1.999005 |
|
10 |
1.480244 |
1.552969 |
1.628895 |
1.790848 |
1.967151 |
2.158925 |
|
11 |
1.539454 |
1.622853 |
1.710339 |
1.898299 |
2.104852 |
2.331639 |
|
12 |
1.601032 |
1.695881 |
1.795856 |
2.012196 |
2.252192 |
2.518170 |
|
13 |
1 .665074 |
1-772196 |
1.885649 |
2.132928 |
2.409845 |
2.719624 |
|
14 |
1.731676 |
1.851945 |
1.979932 |
2.260904 |
2.578534 |
2.937194 |
|
15 |
1.800944 |
1.935282 |
2.078928 |
2.396558 |
2.759032 |
3.172169 |
|
16 |
1.872981 |
2.022370 |
2-182875 |
2.540352 |
2.952164 |
3.425943 |
|
17 |
1-947901 |
2.113377 |
2.292018 |
2.692773 |
3.158815 |
3.700018 |
|
18 |
2.025817 |
2.208479 |
2-406619 |
2.854339 |
3.379932 |
3.996020 |
|
19 |
2.106849 |
2.307860 |
2.526950 |
3.025600 |
3.616528 |
4.315701 |
|
20 |
2.191123 |
2,411714 |
2.653298 |
3.207135 |
3.869684 |
4.660957 |
|
21 |
2-278768 |
2.520241 |
2-785963 |
3.399564 |
4.140562 |
5.0.338.34 |
|
22 |
2.369919 |
2.633652 |
2.925261 |
3.603537 |
4.430402 |
5.436540 |
|
23 |
2-464716 |
2.752166 |
3.071524 |
3.819750 |
4.740530 |
5.871464 |
|
24 |
2.563304 |
2.876014 |
3.225100 |
4.048935 |
5.072367 |
6.341181 |
|
25 |
2.665836 |
3.005434 |
3.386355 |
4.291871 |
5-427433 |
6.848475 |
|
2G |
2.772470 |
3.140679 |
3.555673 |
4-549383 |
5-807353 |
7.396353 |
|
27 |
2.883369 |
3.282010 |
3-733456 |
4.822346 |
6.213868 |
7.988061 |
|
28 |
2.998703 |
3.429700 |
3.920129 |
5.111687 |
6.648838 |
8.627106 |
|
29 |
3.118651 |
3.584036 |
4.116136 |
5.418388 |
7.114257 |
9.317275 |
|
30 |
3.243398 |
3.745318 |
4-321942 |
5.743491 |
7.612255 |
10.062657 |
|
31 |
3-373133 |
3.913857 |
4.538039 |
6.088101 |
8.145113 |
10.867669 |
|
32 |
3.508059 |
4.089981 |
4.764941 |
6.453387 |
8.715271 |
11.737083 |
|
33 |
3.648381 |
4.274030 |
5.003189 |
6.840590 |
9.325340 |
12.676050 |
|
34 |
3-794316 |
4.466362 |
5.253348 |
7.251025 |
9.978114 |
13.6901.34 |
|
35 |
3-946089 |
4.667348 |
5.516015 |
7.686087 |
10.676581 |
14.78.5344 |
|
36 |
4.103933 |
4.877378 |
5.791816 |
8.147252 |
11.423942 |
15.968172 |
|
37 |
4.268090 |
5.096860 |
6.081407 |
8.636087 |
12.22.3618 |
17.245626 |
|
38 |
4.438813 |
5.326219 |
6.385477 |
9.154252 |
13.079271 |
18.625276 |
|
39 |
4-616366 |
5.565899 |
6.704751 |
9.703507 |
13.994820 |
20.115298 |
|
40 |
4-801021 |
5.816365 |
7.039989 |
10.285718 |
14.974458 |
21.724522 |
|
41 |
4.993061 |
6.078101 |
7.391988 |
10.902861 |
16.022670 |
23--162483 |
|
42 |
5.192784 |
6.351615 |
7.761588 |
11.557033 |
17.144257 |
25-339482 |
|
43 |
5-400495 |
6.637438 |
8.149667 |
12.250455 |
18.344355 |
27-366640 |
|
44 |
5.616515 |
6.936123 |
8.557150 |
12.985482 |
19.628460 |
29-555972 |
|
45 |
5.841176 |
7.248248 |
8.985008 |
13.764611 |
21.002452 |
31.920449 |
|
46 |
6.074823 |
7.574420 |
9.434258 |
14.590487 |
22.472623 |
34.474085 |
|
47 |
6.317816 |
7.915268 |
9.905971 |
15.465917! |
24.045707 |
37.232012 |
|
48 |
6.570528 |
8.271456 |
10.401270 |
16.393872 |
25.728907 |
40-210573 |
|
49 |
6.833349 |
8.643671 |
10.921333 |
17.377504 ' |
27.529930 |
43.427419 |
|
50 |
7.106683 |
9.032636 |
11.407400 |
18.420154 |
29.457025 46.9016131 |
TABXiZ: X.
COMPOUND INTEREST,
Showing the Amount of £1 improved at Compound Interest, for anj number of years not exceeding 100.
|
Years. |
1 #• Cent. |
li f Cent. |
2 #• Cent. 21 #" Cent. |
3 W Cent. 3i f Cent. 1 |
|||
|
51 |
1.661078 |
2.136818 |
2.745420 |
3.523036 |
4.515423 |
5.780399 |
|
|
52 |
1.677689 |
2.168870 |
2.800328 |
3.611112 |
4.650886 |
5.982713 |
|
|
63 |
1.694466 |
2.201404 |
2.856335 |
3.701390 |
4.790412 |
6.192108 |
|
|
54 |
1.711411 |
2.234425 |
2.913461 |
3.793925 |
4.934125 |
6.408832 |
|
|
55 |
1.728525 |
2.267946 |
2.971731 |
3.888773 |
5.082149 |
6.633141 |
|
|
56 |
1.745810 |
2.301964 |
3.031165 |
3.985992 |
5.234613 |
6.8^5301 |
|
|
57 |
1.763268 |
2.336494 |
3.091789 |
4.085642 |
5.391651 |
7.105587 |
|
|
58 |
1.780901 |
2.371541 |
3.153624 |
4.187783 |
5.553401 |
7.354282 |
|
|
59 |
1.798710 |
2.407114 |
3.216697 |
4.292478 |
5.720003 |
7.611682 |
|
|
60 |
1.816697 |
2.443220 |
3.281031 |
4.399790 |
5.891603 |
7.878091 |
|
|
61 |
1.834864 |
2.479868 |
3.346651 |
4.509784 |
6.068351 |
8.153824 |
|
|
62 |
1.853213 |
2.517067 |
3.413584 |
4.622529 |
6.250402 |
8.439208 |
|
|
63 |
1.871745 |
2.554823 |
3.481856 |
4.738092 |
6.437914 |
8.734580 |
|
|
64 |
1.890462 |
2.593145 |
3.551493 |
4.856545 |
6.631051 |
9.040291 |
|
|
65 |
1.909367 |
2.632042 |
3.622523 |
4.977958 |
6.829983 |
9.356701 |
|
|
66 |
1.928461 |
2.671522 |
3.694974 |
5.102407 |
7.034882 |
9.684185 |
|
|
67 |
1.947746 |
2.711594 |
3.768873 |
5.229967 |
7.245929 |
10.023132 |
|
|
68 |
1.967223 |
2.752267 |
3.844251 |
5.360717 |
7.463307 |
10.373941 |
|
|
69 |
1.986895 |
2.793550 |
3.921136 |
5.494734 |
7.687206 |
10.737029 |
|
|
70 |
2.006764 |
2.835454 |
3.999558 |
5.632103 |
7.917822 |
11.112825 |
|
|
71 |
2.026832 |
2.877986 |
4.079549 |
5.772905 |
8.155357 |
11.501774 |
|
|
72 |
2.047100 |
2.921156 |
4.161140 |
5.917228 |
8.400017 |
11.904336 |
|
|
73 |
2.067571 |
2.964974 |
4.244363 |
6.065159 |
8.652018 |
12.320988 |
|
|
74 |
2.088247 |
3.009449 |
4.329250 |
6.216788 |
8.911578 |
12.752223 |
|
|
75 |
2.109129 |
3.054590 |
4.415835 |
6.372207 |
9.178926 |
13.198550 |
|
|
76 |
2.130220 |
3.100409 |
4.504152 |
6.531513 |
9.454293 |
13.660500 |
|
|
77 |
2.151522 |
3.146913 |
4.594235 |
6.694800 |
9.737922 |
14.138617 |
|
|
78 |
2.173037 |
3.194117 |
4.686120 |
6.862170 |
10.030060 |
14.633469 |
|
|
79 |
2.194767 |
3.242029 |
4.779842 |
7.033725 |
10.330962 |
15.145640 |
|
|
80 |
2.216715 |
3.290659 |
4.875439 |
7.209568 |
10.640891 |
15.675738 |
|
|
81 |
2.238882 |
3.340020 |
4.972948 |
7.389807 |
10.960117 |
16.224388 |
|
|
82 |
2.261271 |
3.390120 |
5.072407 |
7.574552 |
11.288921 |
16.792242 |
|
|
83 |
2.283884 |
3.440971 |
5.173855 |
7.763916 |
11.627588 |
17.379970 |
|
|
84 |
2.306723 |
3.492586 |
5.277332 |
7.958014 |
11.976416 |
17.988269 |
|
|
85 |
2.329790 |
3.544975 |
5.382879 |
8.156964 |
12.335709 |
18.617859 |
|
|
86 |
^ 2.353088 |
3.598150 |
5.490536 |
8.360888 |
12.705780 |
19.269484 |
|
|
1 87 |
2.376619 |
3.652123 |
5.600347 |
8.569911 |
13.086953 |
19.943916 |
|
|
i 88 |
2.400385 |
3.706905 |
5.712354 |
8.784158 |
13.479562 |
20.641953 |
|
|
89 |
2.424389 |
3.762509 |
5.826601 |
9.003762 |
13.883949 |
21.364421 |
|
|
90 |
2.448633 |
3.818947 |
5.943133 |
9.228856 |
14.300467 |
22.112176 |
|
|
91 |
2.473119 |
3.876231 |
6.061996 |
9.459578 |
14.729481 |
22.886102 |
|
|
92 |
2.497850 |
3.934374 |
6.183236 |
9.696067 |
15.171366 |
23.687116 |
|
|
93 |
2.522828 |
3.993390 |
6.306900 |
9.938469 |
15.626507 |
24.516165 |
|
|
94 |
2.548056 |
4.053291 |
6.433038 |
10.186931 |
16.095302 |
25.374230 |
|
|
95 |
2.573537 |
4.114090 |
6.561699 |
10.441604 |
16.578161 |
26.262329 |
|
|
96 |
2.599272 |
4.175800 |
6.692933 |
10.702644 |
17.075506 |
27.181510 |
|
|
97 |
2.625265 |
4.238437 |
6.826792 |
10.970210 |
17.587771 |
28.132863 |
|
|
98 |
2.651518 |
4.302013 |
6.963328 |
11.244465 |
18.115404 |
29.117513 |
|
|
99 |
2.678033 |
4.366543 |
7.102594 |
11.525577 |
18.658866 |
30.136626 |
|
|
100 |
2.704813 |
4.432041 |
7.244646 |
11.813716 1 19.218632 1 31.191408 |
TABZ«E X.
COMPOUND INTEREST,
Showing the Amount of £1 improved at Compound Interest, for any number of years not exceeding 100.
Years
51 52 53 54 55
56
57 58 59 60
61
62 63
64 i
65 j
66 ! 67 68 69 70
71 72 73 74 75
76
77 78 79 80
81 82 83 84 85
86
87
4 4P' Cent. !4^ #* Cent,
89 90
91 92 93 94 95
9G 97 98 99 100
7.390951 7.686589 7.994052 8.313814 8.646367
8.992222
9.351910
9.725987
10.115026
10.519627
10.940413 11.378029 11.833150 12.306476 12.798735
13-310685 13.843112 14.396836 14.972710 15.571618
16.194483 16.842262 17.515953 18.216591 18.945255
19.703065 20.491187 21.310835 22.163268 23.049799
23.971791 24.930663 25.927889 26.965005 28.043605
29.165349 30.331963 31.545242 32.807051 34.119333
35.484107 .36.903471 38.379610 39.914794 41.511386
43.171841 I 44.898715 46.694664 48.562450 50.504948
9.439105
9.863865
10.307739
10.771587
11.256308
11.762842 12.292170 12.845318 13.423357 14.027408
5 ^ Cent. 6 ^ Cent
14.658641 15.318280 16.007603 16.727945 17.480702
18.267334 19.089364 19.948385 20.846063 21.784136
22.764422
23.788821 24.859318 25.977987 27.146996
28.368611 29.645199 30.979233 32.373298 33.830096
35.352451 36.943311 38.605760 40.343019 42.158455
44.055586 46.038087 48.109801 50.274742 52.537105
54.901275 57.371832 59.953565 62.651475 65.470792
68.416977 71.495741 74.713050 78.075137 81.588518
7 #' Cent. 8 ^ Cent
12.040770 12.642808 13.274949 13.938696 14.635631
15.367412 16.135783 16.942572 17.789701 18.679186
19.613145 20.593802 21.623493 22.704667 23.839901
25.031896 26.283490 27.597665 28.977548 30.426426
31.947747 33.545134 35.222391 36.983510 38.832686 j
40.774320 42.813036 44.953688 47.201372 49.561441
52.039513 54.641489 57.373563 60.242241 63.254353
66.417071 69.737925 73.224821 76.886062 80.730365
50 053742 54*706041 59'082524 63" 8091 26 68*91 3856
74.426965 80.381122 86.811612 93.756540 101.257064
31.5190171 33.725348 36.086122 38.612151 41.315001
44.207052 47.301545 50.612653 54.155539 57.946427
62.0026771109.357629 66.342864 118.106239 70.986865 127.554738 75.955945 137.759117 81.272861 148.779847
86.961962 160.682234
93.049299173.536813
99.562750 187.419758
106.532142 202.413339
113.989392 218.606406
121.968650 236.094918 130.506455 254.982512
19.525364] 20.6968851 21.9386981 23.255020 1 24.650322
26.129341 27.697101 29.358927 31.120463 32.987691
34.966952 37.064969 39.288868 41.646200 44.144972
46.793670 49.601290 52.577368 55.732010 59.075930
62.620486 66.377715 70.360378 74.582001 79.056921
83.800336 88.828356 94.158058 99.807541 105.795993
112.143753 118.872378 126.004721 133.565004 141.578904
150.073639 159.078057 168.622740 178.740105 189.46451l|441.102980ll018.91509
139.641907 149.416840 159.876019
171.067341 183.042054 195.854998 209.564848 224.234388
239.930795 256.725950 274.696767 293.925541
275.381113 297.411602 321.204530
346.900892 374.652964 404.625201 436.995217 471.954834
509.711221 550.488119 594.527168 642.089342
314.500328 693.456489
336.515351748.933008 360.071426 808.847649 385.276426;873.555461 412.2457761943.439897
84.7668831200.832382 471.980188 1100.42830
505.018802,1188.46256 540.370118 11283.53956 578.196026 1386.22273 618.669748 1497.12055
89.005227,212.882325 93.455489 225.655264 98.1282631239.194580 103.0346761253.546255
108.186410,268.759030 661.976630 1616.89019 1 13.595731 1284.884572 708.314994 1746.24141 119.275517 301.977646,757.897044 1885.94072 125.239293 320.096305 810.949837 2036.81.598 131.5012.58 3.39.302084'867. 716326 2199.76126
TikB£.£S IZ,
DEFERRED SUxMS CERTAIN,
Showing the Present Value of £1 to he received at the end of anY number of years not exceeding 100.
|
Years. 1 |
1 #" Cent. |
li<^Cent. |
2 ^ Cent. |
21 ^ Cent. |
3 f Cent.' |
3i #• Cent. |
|
.990099 |
.985222 |
.980392 |
.975610 |
.970874 |
.9661 8^ |
|
|
2 |
.980296 |
.970662 |
.9(51169 |
.951814 |
.942596 |
.933511 |
|
3 |
.970590 |
.956317 |
.942322 |
.928599 |
.915142 |
.901943 |
|
4 |
.960980 |
.942184 |
.923845 |
.90595] |
.888487 |
.871442 |
|
5 |
.951466 |
.928260 |
.905731 |
.883854 |
.862609 |
.841973 |
|
6 |
.942045 |
.914542 |
.887971 |
.862297 |
.837484 |
.813501 |
|
7 |
.932718 |
.901027 |
.870560 |
.841265 |
.813092 |
.785991 |
|
8 |
.923483 |
.887711 |
.853490 |
.820747 |
.789409 |
.759412 |
|
9 |
.914340 |
.874592 |
.836755 |
.800728 |
.766417 |
.733731 |
|
10 |
.905287 |
.861667 |
.820348 |
.781198 |
.744094 |
.708919 |
|
11 |
.896324 |
.848933 |
.804263 |
.762145 |
.722421 |
.684946 |
|
12 |
.887449 |
.836387 |
.788493 |
.743556 |
.701380 |
.661783 |
|
13 |
.878662 |
.824027 |
.773033 |
.725420 |
.680951 |
.639404 |
|
14 |
.869963 |
.811849 |
.757875 |
.707727 |
.061118 |
.617782 |
|
15 |
.861349 |
.799852 |
.743015 |
.690466 |
.641862 |
.596891 |
|
16 |
.852821 |
.788031 |
.728446 |
.673625 |
.623167 |
.576706 |
|
17 |
.844377 |
.776385 |
.714163 |
.657195 |
.605016 |
.557204 |
|
18 |
.836017 |
.764912 |
.700159 |
.641166 |
.587395 |
.538361 |
|
19 |
.827740 |
.753607 |
.686431 |
.625528 |
.570286 |
.520156 |
|
20 |
.819544 |
.742471 |
.672971 |
.610271 |
.553676 |
.502566 |
|
21 |
.811430 |
.731498 |
.659776 |
.595386 |
.537549 |
.485571 |
|
22 |
.803396 |
.720687 |
.646839 |
.580865 |
.521893 |
.469151 |
|
23 |
.795442 |
.710037 |
.634156 |
.566697 |
.506692 |
.453286 |
|
24 |
.787566 |
.699544 |
.621721 |
.552875 |
.491934 |
.437957 |
|
25 |
.779768 |
.689206 |
.609531 |
.539391 |
.477606 |
.423147 |
|
26 |
.772048 |
.679020 |
.597579 |
.526235 |
.463695 |
.408838 |
|
27 |
.764404 |
.668986 |
.585862 |
.513400 |
.450189 |
.395012 |
|
1 *-28 |
.756836 |
.659099 |
.574375 |
.500878 |
.437077 |
.381654 |
|
1 29 |
.749342 |
.649359 |
.563112 |
.488661 |
.424346 |
.368748 |
|
1 ^^ |
.741923 |
.639762 |
.552071 |
.476743 |
.411987 |
.356278 |
|
1 31 |
.734577 |
.630308 |
.541246 |
.465115 |
.399987 |
.344230 |
|
1 32 |
.727304 |
.620994 |
.530633 |
.453771 |
.388337 |
.332590 |
|
1 33 |
.720103 |
.611816 |
.520229 |
.442703 |
.377026 |
.321343 |
|
34 |
.712973 |
.602774 |
.510028 |
.431905 |
.366045 |
.310476 |
|
35 |
.705914 |
.593866 |
.500028 |
.421371 |
.355383 |
.299977 |
|
36 |
.698925 |
.585090 |
.490223 |
.411094 |
.345032 |
.289833 |
|
37 |
.692005 |
.576443 |
.480611 |
.401067 |
.334983 |
.280032 |
|
38 |
.685153 |
.567924 |
.471187 |
.391285 |
.325226 |
.270562 |
|
39 |
.678370 |
.559531 |
.461948 |
.381741 |
.315754 |
.261413 |
|
40 |
.671653 |
.551262 |
.452890 |
.372431 |
.306557 |
.252572 |
|
41 |
.665003 |
.543116 |
.444010 |
.363347 |
.297628 |
'244031 |
|
42 |
.658419 |
.535089 |
.435304 |
.354485 |
.288959 |
•235779 |
|
43 |
.651900 |
.527182 |
.426769 |
.345839 |
.280543 |
•227806 |
|
44 |
.645445 |
.519391 |
.418401 |
.337404 |
.272372 |
•220102 |
|
45 |
.639055 |
.511715 |
.410197 |
.329174 |
.264439 |
•212659 |
|
46 |
.632728 |
.504153 |
•402154 |
•321146 |
.256737 |
.205468 |
|
47 |
.626463 |
.496702 |
.304268 |
•313-313 |
.249259 |
.198520 |
|
48 |
.620260 |
.489362 |
.386538 |
•305671 |
.241999 |
.191806 |
|
49 |
.614119 |
.482130 |
.378958 |
.298216 |
.234950 |
.185320 |
|
1 50 |
.608039 |
.475005 |
.371528 |
.290942 |
.228107 |
.179053 |
TABXiS XX.
DEFERllED SUMS CERTAIN,
Showing the Present Value of £1 to be received at the end of any
number of years not exceeding 100.
|
Years. 1 |
4 #■ Cent. A |
li #• Cent.i |
5 ^ Cent. |
6 W Cent. |
7 W Cent. |
8 W Cent. |
|
.961538 |
.956938 |
.952381 |
.943396 |
.934579 |
.92.5926 |
|
|
2 |
.924556 |
.91573;) |
.907029 |
.889996 |
.873439 |
.857339 |
|
3 |
.888996 |
.876297 |
.863838 |
.839619 |
.816298 |
.793832 |
|
4 |
.854804 |
.838561 |
.822702 |
.792094 |
.762895 |
.73.5030 |
|
5 |
.821927 |
.802451 |
.783526 |
.747258 |
.712986 |
.680583 |
|
G |
.790315 |
.767896 |
.746215 |
.704961 |
.666342 |
.630170 |
|
7 |
.759918 |
.734828 |
.710081 |
.665057 |
.622750 |
.583490 |
|
8 |
.730690 |
.703185 |
.676839 |
.627412 |
.582009 |
.540209 |
|
9 |
.702587 |
.672904 |
.644609 |
.591898 |
.543934 |
..500249 |
|
10 |
.675564 |
.643928 |
.613913 |
.558395 |
.508349 |
.463193 |
|
11 |
.649581 |
.616199 |
.584679 |
.526788 |
.475093 |
.428883 |
|
12 |
.624597 |
.589664 |
.556837 |
.496969 |
.444012 |
.397114 |
|
13 |
.600574 |
.564272 |
.530321 |
.468839 |
.414964 |
.367698 |
|
14 |
.577475 |
.539973 |
.505068 |
.442301 |
.387817 |
.340461 |
|
15 |
.555265 |
.516720 |
.481017 |
.417265 |
.362446 |
.315242 |
|
16 |
.533908 |
.494469 |
.458112 |
.393646 |
.338735 |
.291890 |
|
17 |
.513373 |
.473176 |
.436297 |
.371364 |
.316574 |
.270209 |
|
18 1 |
.493628 |
.452800 |
.415521 |
.350344 |
.295864 |
.250249 |
|
19 |
.474642 |
.433302 |
.395734 |
.330513 |
.276508 |
.231712 |
|
20 |
.456387 |
.414643 |
.376889 |
.311805 |
.258419 |
.214.548 |
|
21 |
.438834 |
.396787 |
.358942 |
.294155 |
.241513 |
.198656 |
|
22 |
.421955 |
.379701 |
.341850 |
.277505 |
.225713 |
.183941 |
|
23 |
.405726 |
363350 |
.325571 |
.261797 |
.210947 |
.170315 |
|
24 |
.390121 |
347703 |
.310068 |
.246979 |
.197147 |
.157699 |
|
25 |
.375117 |
.332731 |
.295303 |
.232999 |
.184249 |
.146018 |
|
26 |
.360689 |
•318402 |
.281241 |
.219810 |
.172195 |
.135202 |
|
27 |
.346817 |
.304691 |
.267848 |
.207368 |
.160930 |
.12.5187 |
|
28 |
.3.33477 |
.291571 |
.255094 |
.195630 |
.150402 |
.115914 |
|
29 |
.320651 |
.279015 |
.242946 |
.184557 |
.140563 |
.107328 |
|
30 |
.308319 |
.267000 |
.231377 |
.174110 |
.131367 |
.099377 |
|
31 |
.296460 |
.255502 |
.220359 |
.164255 |
.122773 |
.092016 |
|
32 |
.285058 |
.244500 |
.209866 |
.154957 |
.114741 |
.085200 |
|
33 |
.274094 |
.233971 |
.199873 |
.146186 |
.107235 |
.078889 |
|
tJKJ 34 |
.263552 |
,223896 |
.190355 |
.137912 |
.100219 |
.07.3045 |
|
35 |
.253415 |
.214254 |
.181290 |
.130105 |
.093663 |
.067635 |
|
36 |
.243669 |
.205028 |
.172657 |
.122741 |
.087535 |
.062625 |
|
37 |
.234297 |
.196199 |
.164436 |
,115793 |
.081809 |
.057986 |
|
tJ 9 38 |
.225285 |
.187750 |
.156605 |
.109239 |
.076457 |
.053690 |
|
39 |
.216621 |
.179665 |
.149148 |
.103056 |
.071455 |
.049713 |
|
40 |
.208289 |
.171929 |
.142046 |
.097222 |
.066780 |
.046031 1 |
|
41 |
.200278 |
.164525 |
.135282 |
.091719 |
.062412 |
.042621 |
|
42 |
.192.575 |
.157440 |
.128840 |
.086527 |
.058329 |
.039464 |
|
43 |
.185168 |
.150661 |
.122704 |
.081630 |
.054513 |
.036541 |
|
44 |
.178046 |
.144173 |
.116861 |
.077009 |
.050946 |
.033834 |
|
45 |
.171198 |
.137964 |
.111297 |
.072650 |
1 .047613 |
.031328 |
|
4fi |
.164614 |
.132023 |
.105997 |
.068538 |
.044499 |
i .029007 |
|
47 48 49 50 |
.158283 |
.126338 |
.100949 |
.064658 |
.041587 |
1 .026859 |
|
.1.52195 |
.120898 |
.096142 |
.060998 |
.038867 |
1 .024869 |
|
|
.146341 |
.115692 |
.091564 |
.057546 |
.036324 |
! .023027 |
|
|
.140713 |
.110710 |
.087204 |
.054288 |
' .033948 |
.021321 |
TABX.X: xz.
DEFERRED SUMS CERTAIN,
Showing the Present Value of £1 to be received at the end of any number of years not exceeding 100.
|
Years. |
1 #* Cent. |
l^c^Cent. |
2#^Cent. 1 |
2^ ^ Cent. |
1 3 W Cent. |
3i W Cent |
|
51 |
,602019 |
.467985 |
.364243 |
.283846 |
.221463 |
.172998 |
|
52 |
.596058 |
.461069 |
.357101 |
.276923 |
.215013 |
.167148 |
|
53 |
.590156 |
.454255 |
.350099 |
.270169 |
.208750 |
.16149C |
|
54 |
.584313 |
.447542 |
.343234 |
.263579 |
.202670 |
.156035 |
|
55 |
.578528 |
.440928 |
.336504 |
.257151 |
.196767 |
.150758 |
|
56 |
.572800 |
.434412 |
.329906 |
.250879 |
.191036 |
.145660 |
|
57 |
.567129 |
.427992 |
.323437 |
.244760 |
.185472 |
.140734 |
|
58 |
.561514 |
.421661 |
.317095 |
.238790 |
.180070 |
.135975 |
|
59 |
.555954 |
.415435 |
.310878 |
.232966 |
.174825 |
.131377 |
|
60 |
.550450 |
.409296 |
.304782 |
.227284 |
.169733 |
.126934 |
|
61 |
.545000 |
.403247 |
.298806 |
.221740 |
.164789 |
.122642 |
|
62 |
.539604 |
.397288 |
.292947 |
.216332 |
.159990 |
.118495 |
|
63 |
.534261 |
.391417 |
.287203 |
.211055 |
.155330 |
.114487 |
|
64 |
.528971 |
.385632 |
.281572 |
.205908 |
.150806 |
.110616 |
|
65 |
.523734 |
.379933 |
.276051 |
.200886 |
.146413 |
.106875 |
|
66 |
.518548 |
.374318 |
.270638 |
.195986 |
.142149 |
.103261 |
|
67 |
.513414 |
.368787 |
.265331 |
.191206 |
.138009 |
.099769 |
|
68 |
.508331 |
.363337 |
.260129 |
.186542 |
.133989 |
.096395 |
|
69 |
.503298 |
.357967 |
.255028 |
.181992 |
.130086 |
.093136 |
|
70 |
.498315 |
.352677 |
.250028 |
.177554 |
.126297 |
.089986 |
|
71 |
.493381 |
.347465 |
.245125 |
.173223 |
.122619 |
.086943 |
|
72 |
.488496 |
.342330 |
.240319 |
.168998 |
.119047 |
.084003 |
|
73 |
.483659 |
.337271 |
.235607 |
.164876 |
.115580 |
.081162 |
|
74 |
.478871 |
.332287 |
.230987 |
.160855 |
.112214 |
.078418 |
|
75 |
.474130 |
.327376 |
.226458 |
.156931 |
.108945 |
.075766 |
|
76 |
.469435 |
.322538 |
.222017 |
.153104 |
.105772 |
.073204 |
|
77 |
.464787 |
.317771 |
.217664 |
.149370 |
.102691 |
.070728 |
|
78 |
.460185 |
.313075 |
.213396 |
.145726 |
.099700 |
.068337 |
|
79 |
.455629 |
.308449 |
.209212 |
.142172 |
.096796 |
.066026 |
|
80 |
.451118 |
.303890 |
.205110 |
.138705 |
.093977 |
.063793 |
|
81 |
.446651 |
.299399 |
.201088 |
.135322 |
.091240 |
.061636 |
|
82 |
.442229 |
.294975 |
.197145 |
.132021 |
.088582 |
.059551 |
|
83 |
.437851 |
.290616 |
.193279 |
.128801 |
.086002 |
.057538 |
|
84 |
.433516 |
.286321 |
.189490 |
.125659 |
.083497 |
.055592 |
|
85 |
.429223 |
.282089 |
.185774 |
.122595 |
.081065 |
.053712 |
|
86 |
.424973 |
.277920 |
.182132 |
.119605 |
.078704 |
.051896 |
|
87 |
.420766 |
.273813 |
.178560 |
.116687 |
.076412 |
.050141 |
|
88 |
.416600 |
.269767 |
.175059 |
.113841 |
.074186 |
.048445 |
|
89 |
.412475 |
.265780 |
.171627 |
.111065 |
.072026 |
.046807 |
|
90 |
.408391 |
.261852 |
.168261 |
.108356 |
.069928 |
.045224 |
|
91 |
.404348 |
.257983 |
.164962 |
.105713 |
.067891 |
.043695 |
|
92 |
.400344 |
.254170 |
.161728 |
.103135 |
.065914 |
.042217 |
|
93 |
.396380 |
.250414 |
.158556 |
.100619 |
.063994 |
.040789 |
|
94 |
.392456 |
.246713 |
.155448 |
.098165 |
.062130 |
.039410 |
|
95 |
.388570 |
.243067 |
.152400 |
.095771 |
.060320 |
.038077 |
|
96 |
.384723 |
.239475 |
.149411 |
.093435 |
.058563 |
.036790 |
|
97 |
.380914 |
.235936 |
.146482 |
.091156 |
.056858 |
.035546 |
|
98 |
.377142 |
.232449 |
.143610 |
.088933 |
.055202 |
.034344 |
|
99 |
.373408 |
.229014 |
.140794 |
.086764 |
.053594 |
.033182 |
|
100 |
.369711 |
.225629 |
.138033 |
.084647 |
.052033 |
.032060 |
TABXaS ZZ.
DEFERRED SUMS CERTAIN,
Showing the Present Value of £1 to be received at the end of any number of years not exceeding 100.
|
Years. |
4^ Cent. |
4i ^ Ceut. |
5 #* Cent. |
6 ^ Cent. |
7 ^ Cent. |
8 ^ Cent. |
|
51 |
.135301 |
.105942 |
.083051 |
.051215 |
.031727 |
.019742 |
|
52 |
.130097 |
.101380 |
.079096 |
.048316 |
.029651 |
.018280 |
|
53 |
.125093 |
.097014 |
.075330 |
.045582 |
.027711 |
.016925 |
|
54 |
.120282 |
.092837 |
.071743 |
.043001 |
.025899 |
.015672 |
|
55 |
.115656 |
.088839 |
.068326 |
.040567 |
.024204 |
.014511 |
|
56 |
.111207 |
.085013 |
.065073 |
.038271 |
.022621 |
.013436 |
|
57 |
.106930 |
.081353 |
.061974 |
.036105 |
.021141 |
.012441 |
|
58 |
.102817 |
.077849 |
.059023 |
.034061 |
.019758 |
.011519 |
|
59 |
.098863 |
.074497 |
.056212 |
.032133 |
.018465 |
.010666 |
|
60 |
.095060 |
.071289 |
.053536 |
.030314 |
.017257 |
.009876 |
|
61 |
.091404 |
.068219 |
.050986 |
.028598 |
.016128 |
.009144 |
|
62 |
.087889 |
.065281 |
.048558 |
.026980 |
.015073 |
.008467 |
|
63 |
.084508 |
.062470 |
.046246 |
.025453 |
.014087 |
.007840 |
|
64 |
.081258 |
.059780 |
.044044 |
.024012 |
.013166 |
.007259 |
|
65 |
.078133 |
.057206 |
.041946 |
.022653 |
.012304 |
.006721 |
|
66 |
.075128 |
.054743 |
.039949 |
.021370 |
.011499 |
.006223 |
|
67 |
.072238 |
.052385 |
.038047 |
.020161 |
.010747 |
.005762 |
|
68 |
.069460 |
.050129 |
.036235 |
.019020 |
.010044 |
.005336 |
|
69 |
.066788 |
.047971 |
.034509 |
.017943 |
.009387 |
.004940 |
|
70 |
.064219 |
.045905 |
.032866 |
.016927 |
.008773 |
.004574 |
|
71 |
.061749 |
.043928 |
.031301 |
.015969 |
.008199 |
.004236 |
|
72 |
.059374 |
.042037 |
.029811 |
.015065 |
.007662 |
.003922 |
|
73 |
.057091 |
.040226 |
.028391 |
.014213 |
.007161 |
.003631 |
|
74 |
.054895 |
.038494 |
.027039 |
.013408 |
.006693 |
.003362 |
|
75 |
.052784 |
.036836 |
.025752 |
.012649 |
.006255 |
.003113 |
|
76 |
.050754 |
.035250 |
.024525 |
.011933 |
.005846 |
.002883 |
|
77 |
.048801 |
.033732 |
.023357 |
.011258 |
.005463 |
.002669 |
|
78 |
.046924 |
.032280 |
.022245 |
.010620 |
.005106 |
.002471 |
|
79 |
.045120 |
.030890 |
.021186 |
.010019 |
.004772 |
.002288 |
|
80 |
.043384 |
.029559 |
.020177 |
.009452 |
.004460 |
.002119 |
|
81 |
.041716 |
.028287 |
.019216 |
.008917 |
.004168 |
.001962 |
|
82 |
.040111 |
.027069 |
.018301 |
.008412 |
.003895 |
.001817 |
|
83 |
.038569 |
.025903 |
.017430 |
.007936 |
.003640 |
.001682 |
|
84 |
.037085 |
.024787 |
.016600 |
.007487 |
.003402 |
.001557 |
|
85 |
.035659 |
.023720 |
.015809 |
.007063 |
.003180 |
.001442 |
|
86 |
.034287 |
.022699 |
.015056 |
.006663 |
.002972 |
.001335 |
|
87 |
.032969 |
.021721 |
.014339 |
.006286 |
.002777 |
.001236 |
|
88 |
.031701 |
.020786 |
.013657 |
.005930 |
.002596 |
.00114.5 |
|
89 |
.030481 |
.019891 |
.013006 |
.005595 |
.002426 |
.001060 |
|
90 |
.029309 |
.019034 |
.012387 |
.005278 |
.002267 |
.000981 |
|
91 |
.028182 |
.018215 |
.011797 |
.004979 |
.002119 |
.000909 |
|
92 |
.027098 |
.017430 |
.011235 |
.004697 |
.001980 |
.000841 |
|
93 |
.026056 |
.016680 |
.010700 |
.004432 |
.001851 |
.000779 |
|
94 |
.025053 |
.015961 |
.010191 |
.004181 |
.001730 |
.000721 |
|
95 |
.024090 |
.015274 |
.009705 |
.003944 |
.001616 |
.000668 |
|
96 |
.023163 |
.014616 |
.009243 |
.003721 |
.001511 |
.000618 |
|
97 |
.022272 |
.013987 |
.008803 |
.003510 |
.001412 |
.000573 |
|
98 |
.021416 |
.013385 |
.008384 |
.003312 |
.001319 |
.000530 |
|
99 |
.020592 |
.012808 |
.007985 |
.003124 |
.001233 |
.000491 |
|
100 |
.019800 |
.012257 |
.007604 |
.002947 |
.001152 |
.000455 |
TABX.E XZZ.
ANNUITIES CERTAIN-AMOUNTS,
Showing the Amount of £1 per Annum forborn and improved for any number of years not exceeding 100.
Years.
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
36 37 38 39 40
41 42 43 44
45
46 47 48 49 50
1 W Cent.
Cent,
2 #" Cent, 121 ^ Cent. 3 <f Cent. 3^ f Cent
1.000000 2.010000 3.030100 4.060401 5.101005
6.152015 7.213535 8.285670 9.368526 10.462211
11.566833 12.682501 13.809326 14.947419 16.096893
17.257862 18.430441 19.614746 20.810894 22.019003
23.239193 24.471585 25.716301
26.973464 28.243199
29.525631
30.820887 32.129096 33.450387 34.784891
36.132740 37.494067 38.869007 40.257696 41.660272
43.076874 44.507642 45.952718 47.412245
48.886307
50.375230 51.878982 53.397772 54.931750 5G.481068
58.045879 59.626338 61.222602 62.834829 64.463178
1 .000000 2.015000 3.045225 4.090903 5.152266
6.229550 7.322994 8.432839 9.559331 10.702720
11.863260 13.041208 14.236824 15.450374 16.682128
17.932359 19.201343 20.489362 21.796701 23.123640
24.470500 25.837555 27.225117 28.633493 30.062995
i 31.513940 32.986649 34.481449 35.998671 37.538651
39.101731
40.688258 42.298582 43.933061 45.592058
47.275940 48.985081 50.719858 52.480657 54.267868
56.081887 57.923116 59.791963 61.688842 63.614175
65.568387 67.551912 69.565189 71.608666 73.682795
1.000000 2.020000 3.060400 4.121608 5.204040
6.308121 7.434283 8.582969 9.754628 10.949721
12.168715 13.412090 14.680332 15.973938 17.293417
18.639285 20.012071 21.412312 22.840559 24.297370
25.783317 27.298984 28.844963 30.421862 32.030300
33.670906 35.344324 37.051210 38.792235 40.568079
42.379441 44.227030 46.111570 48.033802 49.994478
51.994367 54.034255 56.114940
58.237238 60.401983
62.610023 64.862223 67.159468 69.502657 71.892710
74.330564 76.817176 79.353519 81.940590 84.579401
1.000000 2.025000 3.075625 4.152516 5.256329
6.387737 7.547430 8.736116 9.954519 11.203382
12.483466 13.795553 15.140442 16.518953 17.931927
19.380225 20.864730 22.386349 23.946007 25.544658
27.183274 28.862856 30.584427 32.349038 34.157764
36.011708 37.912001 39.859801 41.856296 43.902703
46.000271 48.150278 50.354034 52.612885 54.928207
57.301413 59.733948 62.227297 64.782979 67.402554
70.087617 72.839808 75.660803 78.552323 81.516131
84.554034 87.667885 90.859582 94.131072 97.484349
1.000000 2.030000 3.090900 4.183627 5.309136
6.468410
7.662462
8.892336
10.159106
11.463879
12.807796 14.192030 15.617790 17.086324 18.598914
20.156881 21.761588 23.414435 25.116868 26.870374
28.676486 30.536780 32.452884 34.426470 36.459264
38.553042 [40.709634 142.930923 4.5.218850 47.575416
50.002678 52.502759 55.077841 57.730177 60.462082
63.275944 66.174223 69.159449 72.234233 75.401260
78.663298 82.023196 85.483892 89.048409 92.719861
96.501457 100.39650 104.40840 108.54065 112.79687
1.000000 2.035000 3.106225 4.214943 5.362466
6.550152
7.779408
9.051687
10.368496
11.731393
13.141992 14.601962 16.1130.30 17.676986 19.295681
20.971030 22.705016 24.499691 26.357181
28.279682
30.269471 32.328902 34.460414 36.666528
38.949857
41.313102 43.759060 46.290627 48.910799 51.622677
54.429471 57.334.502 60.341210 63.453152 66.674013
70.007603 73.457869 77.028895 80.724906 84.550278
88.509537 92.607371 96.848629 101.23833 105.78167
110.48403 115.35097 120.38826 125.60185 130.99791
TABXiS ZIZ.
ANNUITIES CERTAIN— AMOUNTS,
Showing the Amount of i'l per Annum forborn and improved for any number of years not exceeding 100.
|
Years. |
4 ^ Cent. |
4^ f Cent. |
S^-Cent. |
6^ Cent. |
7 #" Cent. |
8 ^ Cent. |
|
1 |
1.000000 |
1.000000 |
1.000000 |
1.000000 |
1.000000 |
1 .000000 |
|
2 |
2.040000 |
2.045000 |
2.050000 |
2.060000 |
2.070000 |
2.080000 |
|
3 |
3.121000 |
3.137025 |
3.152500 |
3.183600 |
3.214900 |
3.246400 |
|
4 |
4.246464 |
4.278191 |
4.310125 |
4.374016 |
4.439943 |
4.506112 |
|
5 |
5.416323 |
5.470710 |
5.525631 |
5.G37093 |
5.750739 |
5.866601 |
|
G |
6.632975 |
6.716892 |
6.801913 |
C.975319 |
7.153291 |
7.335929 |
|
7 |
7.898294 |
8.019152 |
8.142008 |
8.393838 |
8.654021 |
8.922803 |
|
8 |
9.214226 |
9.380014 |
9.549109 |
9.897468 |
10.259803 |
10.636628 |
|
9 |
10.582795 |
10.802114 |
11.0265f)4 |
11.491316 |
11.977989 |
12.487558 |
|
10 |
12.006107 |
12.288209 |
12.577893 |
13.180795 |
13.816448 |
14.486562 |
|
11 |
13.486351 |
13.841179 |
14.206787 |
14.971643 |
15.783599 |
16.645487 |
|
12 |
15.025805 |
15.464032 |
15.917127 |
16.869941 |
17.888451 |
18.977126 |
|
13 |
16.626838 |
17.159913 |
17.712983 |
18.882138 |
20.140643 |
21.495297 |
|
14 |
18.291911 |
18.932109 |
19.598632 |
21.015066 |
22.550488 |
24.214920 |
|
15 |
20.023588 |
20.784054 |
21.578564 |
23.275970 |
25.129022 |
27.152114 |
|
16 |
21.824531 |
22.719337 |
23.657492 |
25.672528 |
27.888054 |
30.324283 |
|
17 |
23.697512 |
24.741707 |
25.840366 |
28.212880 |
30.840217 |
33.750226 |
|
18 |
25.645413 |
26.855084 |
28.132385 |
30.905653 |
33.999033 |
37.450244 |
|
19 |
27.671229 |
29.063562 |
30.539004 |
33.759992 |
37.378965 |
41.446263 |
|
20 |
29.778079 |
31.371423 |
33.065954 |
36.785591 |
40.995492 |
45.761964 |
|
21 |
31.969202 |
33.783137 |
35.719252 |
39.992727 |
44.865177 |
50.422921 |
|
22 |
34.247970 |
36.303378 |
38.505214 |
43.392290 |
49.005739 |
55.456755 |
|
23 |
36.617889 |
38.937030 41.430475 |
46.995828 |
53.436141 |
60.893296 |
|
|
24 |
39.082604 |
41.689196144.501999 |
50.815577 |
58.176671 |
66.764759 |
|
|
25 |
41.645908 |
44.565210 |
47.727099 |
54.864512 |
63.249038 |
73.105940 |
|
26 |
44.311745 |
47.570645 |
51.113454 |
59.156383 |
68.676470 |
79.954415 |
|
27 |
47.084214 |
50.711324 |
54.669126 |
63.705766 |
74.483823 |
87.350768 |
|
28 |
49.967583 ' |
53.993333 |
58.402583 |
68.528112 |
80.697691 |
95.338830 |
|
29 |
52.966286 |
57.423033 |
62.322712 |
73.639798 |
87.346529 |
103.96594 |
|
30 |
56.084938 |
61.007070 |
66.438848 |
79.058186 |
94.460786 |
113.28321 |
|
31 |
59.328335 |
64.752388 |
70.760790 |
84.801677 |
102.07304 |
123.34587 |
|
32 |
62.701469 |
68.666245 |
75.298829 |
90.889778 |
110.21815 |
134.21354 |
|
33 |
66.209527 |
72.756226 80.063771 |
97.343165 |
118.93343 |
145.95062 |
|
|
34 |
69.857909 |
77.030256 : 85.066959 |
104.18376 |
128.25877 |
158.62667 |
|
|
35 |
73.652225 |
81.496618! 90.320307 |
111.43478 |
138.23688 |
172.31680 |
|
|
36 |
77.598314 |
86.163966 |
95.836323 |
119.12087 |
148.91346 |
187.10215 |
|
37 |
81.702246 |
91.041344 |
101.62814 |
127.26812 |
160.33740 |
203.07032 |
|
38 |
85.970336 |
96.138205 |
107.70955 |
135.90421 |
172.56102 |
220.31595 |
|
39 |
90.409150 |
101.46442 |
114.09502 |
145.05846 |
185.64029 |
2.38.94122 |
|
40 |
95.025516 |
107.03032 |
120.79977 |
154.76197 |
199.63511 |
259.05652 |
|
41 |
99.826536 |
112.84669 |
127.83976 |
165.04768 |
214.60957 |
280.78104 |
|
42 |
104.81960 |
118.92479 |
135.23175 |
175.95055 |
230.63224 |
304.24352 |
|
43 |
110.01238 |
125.27640 |
142.99334 |
187.50758 |
247.77650 |
329.58301 |
|
44 |
115.41288 |
131.91384 |
151.14301 |
199.75803 |
266.12085 |
356.9496.5 |
|
45 |
121.02939 |
138.84997 |
159.70016 |
212.74351 |
285.74931 |
386.50562 |
|
46 |
126.87057 |
146.09821 |
168.68516 |
226.50812 |
306.75176 |
418.42607 |
|
47 |
132.94539 |
153.07263 |
178.11942 |
241.09861 |
329.22439 |
452.90015 |
|
48 |
139.26321 |
161.58790 |
188.02539 |
256.56453 |
353.27009 |
490.13216 |
|
49 |
145.83373 |
169.85936 |
198.42066 |
272.95840 |
378.99900 |
530.34274 |
|
50 |
152.66708 |
178.50303 '209.34800 |
290.33590 |
406.52893 |
573.77016 |
c z
TABZiS III.
, ANNUITIES CERTAIN— AMOUNTS,
Showing the Amount of £1 per Annum forborn and improved for any number of years not exceeding 100.
|
Years. 51 |
1 ^ Cent. |
lic^Cent. |
2 ^ Cent. |
2i ^ Cent. |
3 W Cent. |
3i ^ Cent. |
|
66.107810 |
75.788035 |
87.270989 |
100.92146 |
117.18077 |
136.58284 |
|
|
52 |
67.768888 |
77.924853 |
90.016409 |
104.44449 |
121.69620 |
142.36324 |
|
53 |
69.446577 |
80.093723 |
92.816737 |
108.05561 |
126.34708 |
148.34595 |
|
54 |
71.141043 |
82.295127 |
95.673072 |
111.75700 |
131.13749 |
154.53806 |
|
55 |
72.852454 |
84.529552 |
98.586534 |
115.55092 |
136.07162 |
160.94689 |
|
56 |
74.580979 |
86.797498 |
101.55826 |
119.43969 |
141.15377 |
167.58003 |
|
57 |
76.326789 |
89.099462 |
104.58943 |
123.42569 |
146.38838 |
174.44533 |
|
58 |
78.090057 |
91.435956 |
107.68122 |
127.51133 |
151.78003 |
181.55092 |
|
69 |
79.870958 |
93.807497 |
110.83484 |
131.69911 |
157.33343 |
188.90520 |
|
60 |
81.669668 |
96.214611 |
114.05154 |
135.99159 |
163.05344 |
196.51688 |
|
61 |
83.486365 |
98.657831 |
117.33257 |
140.39138 |
168.94504 |
204.39497 |
|
62 |
85.321229 |
101.13770 |
120.67922 |
144.90116 |
175.01339 |
212.54880 |
|
63 |
87.174442 |
103.65477 |
124.09281 |
149.52369 |
181.26379 |
220.98801 |
|
64 |
89.046187 |
106.20959 |
127.57466 |
154.26179 |
187.70171 |
229.72259 |
|
65 |
90.936649 |
108.80273 |
131.12616 |
159.11833 |
194.33276 |
238.76288 |
|
66 |
92.846016 |
111.43478 |
134.74868 |
164.09629 |
201.16274 |
248.11958 |
|
67 |
94.774477 |
114.10630 |
138.44365 |
169.19870 |
208.19762 |
257.80376 |
|
68 |
96.722223 |
116.81789 |
142.21253 |
174.42866 |
215.44355 |
267.82689 |
|
69 |
98.689446 |
119.57016 |
146.05678 |
179.78938 |
222.90686 |
278.20084 |
|
70 |
100.67634 |
122.36371 |
149.97791 |
185.28411 |
230.59406 |
288.93786 |
|
71 |
102.68311 |
125.19916 |
153.97747 |
190.91622 |
238.51189 |
300.05069 |
|
72 |
104.70994 |
128.07715 |
158.05702 |
196.68912 |
246.66724 |
311.55246 |
|
73 |
106.75704 |
130.99831 |
162.21816 |
202.60635 |
255.06726 |
323.45680 |
|
74 |
108.82461 |
133.96328 |
166.46252 |
208.67151 |
263.71928 |
335.77779 |
|
75 |
110.91286 |
136.97273 |
170.79177 |
214.88830 |
272.63086 |
348.53001 |
|
76 |
113.02198 |
140.02732 |
175.20761 |
221.26050 |
281.80978 |
361.72856 |
|
77 |
115.15220 |
143.12773 |
179.71176 |
227.79202 |
291.26407 |
375.38906 |
|
78 |
117.30373 |
146.27464 |
184.30600 |
234.48682 |
301.00200 |
389.52768 |
|
79 |
119.47676 |
149.46876 |
188.99212 |
241.34899 |
311.03206 |
404.16115 |
|
80 |
121.67153 |
152.71079 |
193.77196 |
248.38271 |
321.36302 |
419.30679 |
|
81 |
123.88825 |
156.00145 |
198.64740 |
255.59228 |
332.00391 |
434.98252 |
|
82 |
126.12713 |
159.34147 |
203.62034 |
262.98209 |
342.96403 |
451.20691 |
|
83 |
128.38840 |
162.73159 |
208.69275 |
270.55664 |
354.25295 |
467.99915 |
|
84 |
130.67228 |
166.17256 |
213.86661 |
278.32056 |
365.88054 |
485.37913 |
|
85 |
132.97901 |
169.66514 |
219.14394 |
286.27857 |
377.85695 |
503.36739 |
|
86 |
135.30880 |
173.21012 |
224.52682 |
294.43553 |
390.19266 |
521.98525 |
|
87 |
137.66188 |
176.80827 |
230.01735 |
302.79642 |
402.89844 |
541.25474 |
|
88 |
140.03850 |
180.46039 |
235.61770 |
311.36633 |
415.98539 |
561.19865 |
|
89 |
142.43889 |
184.16730 |
241.33006 |
320.15049 |
429.46496 |
581.84061 |
|
90 |
144.86328 |
187.92980 |
247.15666 |
329.15425 |
443.34890 |
603.20503 |
|
91 |
147.31191 |
191.74875 |
253.09979 |
338.38311 |
457.64937 |
625.31720 |
|
92 |
149.78503 |
195.62498 |
259.16179 |
347.84269 |
472.37885 |
648.20331 |
|
93 |
152.28288 |
199.55936 |
265.34502 |
357.53875 |
487.55022 |
671.89042 |
|
94 |
154.80571 |
203.55275 |
271.65192 |
367.47722 |
503.17672 |
696.40659 |
|
95 |
157.35376 |
207.60604 |
278.08496 |
377.66415 |
519.27203 |
721.78082 |
|
96 |
159.92730 |
211.72013 |
284.64666 |
388.10576 |
535.85019 |
748.04314 |
|
97 |
162.52657 |
215.89593 |
291.33959 |
398.80840 |
552.92569 |
775.22465 |
|
98 |
165.15184 |
220.13436 |
298.16638 |
409.77861 |
570.51346 |
803.35752 |
|
99 |
167.80335 |
224.43638 |
305.12971 |
421.02308 |
588.62887 |
832.47503 |
|
100 |
170.48139 |
228.80292 |
312.23231 |
432.54865 |
607.28773 |
862.61166 |
TABXiB ZZZ.
ANNUITIES CERTAIN— AMOUNTS,
Showing the Amount of £1 per Annum forborn and improved for any number of years not exceeding 100.
|
Years. 51 |
4 f Cent. |
4^ #• Cent. |
5 ^ Cent. |
6 ^ Cent. |
: 7 <^ Cent. |
8 #- Cent. |
|
159.77377 |
187.53566 |
220.81540 |
308.75606 |
435.98595 |
620.67177 |
|
|
52 |
167.16472 |
196.97477 |
232.85617 |
328.28142 |
467.50497 |
671.32551 |
|
53 |
174.85131 |
206.83863 |
245.49897 |
348.97831 |
501.23032 |
726.03155 |
|
54 |
182.84536 |
217.14637 |
258.77392 |
370.91701 |
537.31644 |
785.11408 |
|
55 |
191.15917 |
227.91796 |
272.71262 |
394.17203 |
575.92859 |
848.92320 |
|
56 |
199.80554 |
239.17427 |
287.34825 |
418.82235 |
617.24359 |
917.83706 |
|
57 |
208.79776 |
250.93711 |
302.71566 |
444.95169 |
661.45065 |
992.26402 |
|
58 |
218.14967 |
263.22928 |
318.85144 |
472.64879 |
708.75219 |
1072.6451 |
|
59 |
227.87566 |
276.07460 |
335.79402 |
502.00772 |
759.36484 |
1159.4568 |
|
60 |
237.99069 |
289.49795 |
353.58372 |
533.12818 |
813.52038 |
1253.2133 |
|
61 |
248.51031 |
303.52536 |
372.26290 |
566.11587 |
871.46681 |
1354.4704 |
|
62 |
259.45073 |
318.18400 |
391.87605 |
601.08282 |
933.46949 |
1463.8280 |
|
63 |
270.82875 |
333.50228 |
412.46985 |
638.14779 |
999.81235 |
1581.9342 |
|
64 |
282.66190 |
349.50989 |
434.09334 |
677.43666 |
1070.7992 |
1709.4890 |
|
65 |
294.96838 |
366.23783 |
456.79801 |
719.08286 |
1146.7552 |
1847.2481 |
|
66 |
307.76712 |
383.71853 |
480.63791 |
763.22783 |
1228.0280 |
1996.0279 |
|
67 |
321.07780 |
401.98587 |
505.66981 |
810.02150 |
1314.9900 |
2156.7102 |
|
68 |
334.92091 |
421.07523 |
531.95330 |
859.62279 |
1408.0393 |
2330.2470 |
|
69 |
349.31775 |
441.02362 |
559.55096 |
912.20016 |
1507.6020 |
2517.6667 |
|
70 |
364.29046 |
461.86968 |
588.52851 |
967.93217 |
1614.1342 |
2720.0801 |
|
71 |
379.86208 |
483.65382 |
618.95494 |
1027.0081 |
1728.1236 |
2938.6865 |
|
72 |
396.05656 |
506.41824 |
650.90268 |
1089.6286 |
1850.0922 |
3174.7814 |
|
73 |
412.89882 |
530.20706 |
684.44782 |
1156.0063 |
1980.5987 |
3429.7639 |
|
74 |
430.41478 |
555.06638 |
719.67021 |
1226.3667 |
2120.2406 |
3705.1450 |
|
75 |
448.63137 |
581.04436 |
756.65372 |
1300.9487 |
2269.6574 |
4002.5566 |
|
76 |
467.57662 |
608.19136 |
795.48640 |
1380.0056 |
2429.5334 |
4323.7612 |
|
77 |
487.27969 |
636.55997 |
836.26072 |
1463.8059 |
2600.6008 |
4670.6620 |
|
78 |
507.77087 |
666.20517 |
879.07376 |
1552.6343 |
2783.6428 |
5045.3150 |
|
79 |
529.08171 |
697.18440 |
924.02745 |
1646.7924 |
2979.4978 |
5449.9402 |
|
80 |
551.24498 |
729.55770 |
971.22882 |
1746.5999 |
3189.0627 |
5886.9354 |
|
81 |
574.29478 |
763.38779 |
1020.7903 |
1852.3959 |
3413.2971 |
6358.8903 |
|
82 |
598.26657 |
798.74025 |
1072.8298 |
1964.5396 |
3653.2279 |
6868.6015 |
|
83 |
623.19723 |
835.68356 |
1127.4713 |
2083.4120 |
3909.9538 |
7419.0896 |
|
84 |
649.12512 |
874.28932 |
1184.8448 |
2209.4167 |
4184.6506 |
8013.6168 |
|
85 |
676.09012 |
914.63234 |
1245.0871 |
2342.9817 |
4478.5761 |
8655.7061 |
|
86 |
704.13373 |
956.79079 |
1308.3414 |
2484.5606 |
4793.0764 |
9349.1626 |
|
87 |
733.29908 |
1000.8464 |
1374.7585 |
2634.6343 |
5129.5918 |
10098.096 |
|
88 |
763.63104 |
1046.8845 |
1444.4964 |
2793.7123 |
5489.6632 |
10906.943 |
|
89 |
795.17628 |
1094.9943 |
1517.7212 |
2962.3351 |
5874.9397 |
11780.499 |
|
90 |
827.98333 |
1145.2690 |
1594.6073 |
3141.0752 |
6287.1854 |
12723.939 |
|
91 |
862.10267 |
1197.8061 |
1675.3377 |
3330.5397 |
6728.2884 |
13742.854 |
|
92 |
897.58677 |
1252.7074 |
1760.1045 |
3531.3721 |
7200.2686 |
14843.282 |
|
93 |
934.49024 |
1310.0792 |
1849.1098 |
3744.2544 |
7705.2874 |
16031.745 |
|
94 |
972.86985 |
1370.0328 |
1942.5653 |
3969.9097 |
8245.6575 |
17315.284 |
|
95 |
1012.7846 |
1432.6843 |
2040.6935 |
4209.1042 |
8823.8535 |
18701.507 |
|
96 |
1054.2960 |
1498.1551 |
2143.7282 |
4462.6505 |
9442.5233 |
20198.627 |
|
97 |
1097.4679 |
1566.5720 |
2251.9146 |
4731.4095 |
10104.500 |
21815.518 |
|
98 |
1142.3666 |
1638.0678 |
2365.5103 |
5016.2941 |
10812.815 |
23501.759 |
|
99 |
1189.0613 ! 1712.7808 |
2484.7859 |
5318.2718 |
11570.712 |
25447.700 |
|
|
100 |
1237.6237 1790.8560 ' 2610.0252 ! 5638.3681 |
12381.662 |
27484.516 |
TABZiB IV.
ANNUITIES CERTAIN— PRESENT VALUES,
Showing the Present Value of £1 per Annum for any number of years not exceeding 100.
|
Years. |
l^Ceut. |
1 i #> Cent. |
2 ^ Cent. |
2i #■ Cent. |
3 #* Cent. |
3i#' Cent. |
|
1 |
.990099 |
.985222 |
.980392 |
.975610 |
.970874 |
.966184 |
|
2 |
1.970395 |
1.955884 |
1.941561 |
1.927424 |
1.913470 |
1.899694 |
|
3 |
2.940985 |
2.912201 |
2.883883 |
2.856024 |
2.828611 |
2.801637 |
|
4 |
3.901965 |
3.854385 |
3.807729 |
3.761974 |
3.717098 |
3.673079 |
|
5 |
4.853431 |
4.782645 |
4.713460 |
4.645828 |
4,579707 |
4.515052 |
|
6 |
5.795476 |
5.697187 |
5.601431 |
5.508125 |
5.417191 |
5.328553 |
|
7 |
6.728194 |
6.598214 |
6.471991 |
6.349391 |
6.230283 |
6.114544 |
|
8 |
7.651677 |
7.485925 |
7.325481 |
7.170137 |
7.019692 |
6.873956 |
|
9 |
8.566017 |
8.360517 |
8.162237 |
7.970866 |
7.786109 |
7.607687 |
|
10 |
9.471304 |
9.222184 |
8.982585 |
8.752064 |
8.530203 |
8.316605 |
|
11 |
10.367628 |
10.071117 |
9.786848 |
9.514209 |
9.252624 |
9.001551 |
|
12 |
11.255077 |
10.907504 |
10.575341 |
10.257765 |
9.954004 |
9.663334 |
|
13 |
12.133739 |
11.731531 |
11.348374 |
10.983185 |
10.634955 |
10.302738 |
|
14 |
13.003702 |
12.543380 |
12.106249 |
11.690912 |
11.296073 |
10.920520 |
|
15 |
13.865051 |
13.343232 |
12.849264 |
12.381378 |
11.937935 |
11.517411 |
|
16 |
14.717872 |
14.131263 |
13.577709 |
13.055003 |
12.561102 |
12.094117 |
|
17 |
15.562249 |
14.907648 |
14.291872 |
13.712198 |
13.166118 |
12.651321 |
|
18 |
16.398266 |
15.672560 |
14.992031 |
14.353364 |
13.753513 |
13.189682 |
|
19 |
17.226006 |
16.426167 |
15.678462 |
14.978891 14.323799 |
13.709837 |
|
|
20 |
18.045550 |
17.168638 |
16.351433 |
15.589162 |
14.877475 |
14.212403 |
|
21 |
18.856980 |
17.900136 |
17.011209 |
16.184549 |
15.415024 |
14.697974 |
|
22 |
19.660376 |
18.620823 |
17.658048 |
16.765413 |
15.936917 |
15.167125 |
|
23 |
20.455818 |
19.330860 |
18.292204 |
17.332110 |
16.443608 |
15.620410 |
|
24 |
21.243384 |
20.030404 |
18.913926 |
17.884986 |
16.935542 |
16.058368 |
|
25 |
22.023152 |
20.719610 |
19.523456 |
18.424376 |
17.413148 |
16.481515 |
|
26 |
22.795200 |
21.398630 |
20.121036 |
18.950611 |
17.876842 |
16.890352 |
|
27 |
23.559604 |
22.067616 |
20.706898 |
19.464011 |
18.327031 |
17.285365 |
|
28 |
24.316440 |
22.726715 |
21.281272 |
19.964889 |
18.764108 |
17.667019 |
|
29 |
25.065782 |
23.376074 |
21.844385 |
20.453550 |
19.188455 |
18.035767 |
|
30 |
25.807705 |
24.015836 |
22.396456 |
20.930293 19.600441 |
18.392045 |
|
|
31 |
26.542282 |
24.646144 |
22.937702 |
21.395407 |
20.000428 |
18.736276 |
|
32 |
27.269586 |
25.267138 |
23.468335 |
21.849178 |
20.388766 |
19.068865 |
|
33 |
27.989689 |
25.878954 |
23.988564 |
22.291881 |
20.765792 |
19.390208 |
|
34 |
28.702662 |
26.481728 |
24.498592 |
22.723786 |
21.131837 |
19.700684 |
|
35 |
29.408576 |
27.075594 |
24.998619 |
23.145157 |
21.487220 |
20.000661 |
|
36 |
30.107501 |
27.660684 |
25.488842 |
23.556251 |
21.832252 |
20.290494 |
|
37 |
30.799506 |
28.237127 |
25.969453 |
23.957318 22.167235 |
20.570525 |
|
|
38 |
31.484659 |
28.805051 |
26.440641 |
24.348603 22.492462 |
20.841087 |
|
|
39 |
32.163029 |
29.364582 |
26.902589 |
24.730344 22.808215 |
21.102500 |
|
|
40 |
32.834682 |
29.915844 |
27.355479 |
25.102775 23.114772 |
21.355072 |
|
|
41 |
33.499685 |
30.458960 |
27.799489 |
25.466122 23.412400 |
21.599104 |
|
|
42 |
34.158104 |
30.994049 |
28.234794 |
25.820607 23.701359 |
21.834883 |
|
|
43 |
34.810004 |
31.521231 |
28.661562 |
26.166446 23.981902 |
22.062689 |
|
|
44 |
35.455449 |
32.040622 |
29.079963 |
26.503849 24.254274 |
22.282791 |
|
|
45 |
36.094504 |
32.552337 |
29.490160 |
26.833024 24.518713 |
22.495460 |
|
|
46 |
36.727232 |
33.056490 |
29.892314 |
27.154170 24.775449 |
22.700918 |
|
|
47 |
37.353695 |
33.553192 |
30.286582 |
27.467483 25.024708 |
22.899438 |
|
|
48 |
37.973955 |
34.042554 |
30.673120 |
27.773154 25.266707 23.091244 1 |
||
|
49 |
38.588074 |
34.524684 |
31.052078 |
28.071369 25.501657 |
23.276564 |
|
|
50 |
39.196113 34.999689 |
31.423606 |
28.362312 25.729704 23.455618 j |
ANNUITIES CERTAIN— I'RESENT VALUES,
Showing the Present Valve of £1 jjcr Annum for any number of years not exceeding 100.
|
Years. |
4 f Cent. |
4i#'Cent. |
5 #* Cent. |
6<a^Cent. |
7 f Cent. |
8 #• Cent. |
|
1 |
.961538 |
.956938 |
.952381 |
.943396 |
.934579 |
.925926 |
|
2 |
1 .8800i)5 |
1.872668 |
1.859410 |
1.833393 |
1.808018 |
1.783205 |
|
3 |
2.775091 |
2.748964 |
2.723248 |
2.673012 |
2.624316 |
2.577097 |
|
4 |
3.629895 |
3.587526 |
3.545951 |
3.4G5106 |
3.387211 |
3.312127 |
|
5 |
4.451822 |
4.389977 |
4.329477 |
4.212364 |
4.100197 |
3.992710 |
|
6 |
5.242137 |
5.157872 |
5.075692 |
4.917324 |
4.766540 |
4.622880 |
|
7 |
6.002055 |
5.892701 |
5.786373 |
5.582381 |
5.389289 |
5.206370 |
|
8 |
6.732745 |
6.595886 |
6.463213 |
6.209794 |
5.971299 |
5.746639 |
|
9 |
7.435332 |
7.268790 |
7.107822 |
6.801692 |
6.515232 |
6.246888 |
|
10 |
8.110896 |
7.912718 |
7.721735 |
7.360087 |
7.023582 |
0.710081 |
|
11 |
8.760477 |
8.528917 |
8.306414 |
7.886875 |
7.498674 |
7.138964 |
|
12 |
9.385074 |
9.118581 |
8.863252 |
8.383844 |
7.942086 |
7.530078 |
|
13 |
9.985648 |
9.682852 |
9.393573 |
8.852683 |
8.357051 |
7.903770 |
|
14 |
10.563123 |
10.222825 |
9.898641 |
9.294984 |
8.745408 |
8.244237 |
|
15 |
11.118387 |
10.739546 |
10.379658 |
9.712249 |
9.107914 |
8.559479 |
|
16 |
11.652296 |
11.234015 |
10.837770 |
10.105895 |
9.446649 |
8.851369 |
|
17 |
12.165669 |
11.707191 |
11.274066 |
10.477260 |
9.763223 |
9.121638 |
|
18 |
12.659297 |
12.159992 |
11.689587 |
10.827603 |
10.059087 |
9.371887 |
|
19 |
13.133939 |
12.593294 |
12.085321 |
11.158116 |
10.335595 |
9.603599 |
|
20 |
13.590326 |
13.007936 |
12.462210 |
11.469921 |
10.594014 |
9.818147 |
|
21 |
14.029160 |
13.404724 |
12.821153 |
11.764077 |
10.835527 |