Some
Landmarks
•
In
Actuarial
Science
Catalogue of an exhibition at Staple Inn Hall
November 1985
Introduction
Up to 1700
People will not look forward to posterity, who never look backward to
their ancestors.
Valuation techniques were known in ancient Ro'man times, but came
increasingly to the fore as world trade developed after 1500 AD.
Commercial needs gave rise to transactions involving compound interest,
and life annuities came into existence. During the 16th Century, some of
the Continental writers on arithmetic, such as Jan Trenchant, devoted a
little space to elementary problems in compound interest. There were
even the first glimmerings of life assurance. There were no actuaries in
those days: the nearest equivalent one can find is the mathematical
practitioner who would tackle all kinds of arithmetical problems on
request, from commercial matters to navigation. Richard Witt (1568-1623)
was one such man; he practised in London and was the author of the first
comprehenSive book in English on compound interest.
Witt's book did not venture into life contingencies and we have no
evidence that he considered such problems. It was not until later in the
17th Century that the necessary tools became available. One of these tools
was the developing science of probability, on which Christian Huygens,
the Dutch mathematician, published an important paper in 1657.
Another tool was the concept of the life table based on mortality
investigations, the first example of which was published by John Graunt
ef London in 1662.
Thus, by 1670 the three main foundations of actuarial science were
firmly in place: compound interest, probability and the life table. These
tools were employed almost immediately by the Dutch prime minister,
Jan de Wit, to investigate the value of government life annuities.
However, his treatise remained unpublished for many years and so did
not influence the development of actuarial science. It was not until 1693
that Edmund Halley published his paper to the Royal Society which set
out the method for valuing life annuities which is essentially that used
today.
(Edmund Burke, Reflections on the Revolution in France, 1790)
As part of the celebration of the 75th anniversary of the Institute of
Actuaries' Students' Society, a small exhibition of books and papers
forming landmarks in actuarial science is being displayed at Staple Inn in
1985.
Selection of landmarks must be subjective, and consequently several
members of the Historic Texts Sub-Committee of the Library Committee
were severally given a free hand to choose and describe items from one
past period. My own task was to reduce their descriptions to fit into this
small booklet, printed for the Students' SOCiety by Canada Life as their
contribution to the celebrations. The original full texts are available, and
could form the starting point for further research.
For a profession which examines the experience of the past in order to
project the future, it is surprising how little interest has previously been
taken in the history of its scientific achievement. The Students' SOCiety
and Institute are now pursuing a deliberate policy of improving the
Library's holdings of historic material: for example, the second item in
this catalogue is a unique copy of a sixteenth century book on arithmetic,
purchased under this policy.
Some of the authors whose works are displayed did not originate the
ideas they describe: their achievement was to refine and demonstrate the
landmarks in print to their contemporaries and successors. I invite you to
take a backward look at these landmarks since they can help to guide the
profession forward.
Derek Renn
2
C G Lewin
3
Ulpian's Table
About 211 A.D.
Arithmeticall Questions
First edition, 1613
Domitiu5 Ulpianus
Richard Witt
The table was intended for the valuation of annuities, to meet the legal
requirement that a testator had to leave at least one-quarter of his
property to his rightful heirs. The figures probably relate to expectations
of life with no allowance for interest. A companson wIth the Stockholm
mortality table, which was based on actual experience in the 18th century,
suggests that Ulpian may perhaps have denved hIS fIgures 10 some way
from actual experience rather than guesswork alone.
It is a curious fact that the Tuscan Government authorised the use of
Ulpian's Table of the valuation of life annuities as recently as 1814!
The book delves deeply into compound interest in a very practical way. It
is evident from the clarity of expression and the care which has been taken
that the author thought in much the same way as modern actuaries. Many
of the tables are based on 10%, which was then the legal maximum rate
of interest.
Witt understood his subject thoroughly. For example, he dealt not only
with annual payments but also with payments at half-yearly and
quarterly intervals. Witt was well known as a mathematical practitioner,
and he then lived in the parish of St. Mary Woolchurch near the site of the
present Mansion House.
L' Arithmetique
1558
De Ratiociniis in Aleae Ludo
1657
lan Trenchant
Christianus Huygens
The book is a general treatise on arithmetic, with a co~mercial fl~vour.
There is a chapter on simple and compound interest whIch deals wIth the
subject in a somewhat mathematical way. Only a few tab~e~l were ~iven:
these were (1 + i)" and s"at 4%, and a table of (1 + 1) at 10 Yo for
periods of less than a year (in complete months). As an example. he
dicusses whether it is better to receIve 4% per quarter Interest on a loan
or an annuity of 5% per quarter for 41 quarters. The latter is shown to be
marginally worse, because (5 - 4) s"l at 4% is slightly less than 100.
Trenchant points out that to receive 4% per quarter is better than 16% per
annum.
Exhibited is the only known copy of the first edition.
This work is not specifically actuarial but its methods, systematically
stated here for the first time, constitute one of the foundation stones of
actuarial science. Huygens puts forward 14 propositions. For example, he
asserts that if a player has p chances of gaining a and q chances of gaining
b, his expectation is (pa + qb)/(p + q). He then proceeds to some
questions relating to dice, e.g. in how many throws a player may
undertake to throw a six with a single die.
The author was a Dutch mathematician and astronomer. In 1663 he was
elected a Fellow of the Royal Society.
5
Natural and Political Observations
upon the Bills of Mortality
First edition, 1662
An estimate of the degrees of the mortality
First edition, 1693
of mankind
John Graunt
Edmund Halley
The author examines the London Bills of Mortality for many years and
derives numerous interesting conclusions from the statistics thus
obtained, including an estimate of the city's population. In particular he
produces the first ever life table arguably based on actual experience.
He then applies this table, in a manner foreshadowing the method of
stationary populations, to estimate the numbers of men now alive at
different ages.
The fourth edition appeared in 1665, when the Great Plague was at its
height. It included a table showing the deaths week by week up to
September 26th, when mortality from the plague was just starting to
decrease.
Graunt was born in London and became a member of the Drapers'
Company.
Halley, one of Britain's greatest ever astronomers and mathematicians,
took the Bills of Mortality for Breslau and derived a mortality table. He
then used this to calculate a table of a, at 6%, using the method which, in
principle, is the same as that used today. He drew the conclusion that
British Government life annuities, which were being sold on the basis of
7 years' purchase, were very cheap, as the true value of the annuity for a
young life was over 13 years' purchase. He went on to give the method for
the valuation of annuities on more than one life, with geometrical
diagrams by way of explanation, and emphasised the benefit of using
logarithms to reduce the volume of calculation.
The Amicable Society Prospectus
Report to the States of Holland
1671
Jan de Wit
This report was submitted to the States of Holland by its long serving
prime minister, as the basis for the terms on which Government life
annuities should be granted. These annuities were being sold on the basis
of 14 years' purchase, and de Wit calculated that at least 16 years'
purchase should be paid, based on interest at 4% p.a.
The report took account of the fundamental proposition pointed out by
Huygens a few years earlier, that the probability of an event happening
could be expressed by the ratio of the number of ways it could happen or
not happen. The mortality basis underlying the calculations was founded
on an arithmetical assumption, taking account of the fact that lives would
be carefully selected by purchasers of annuities.
De Wit had also examined the actual mortality experienced by several
thousand Government annuitants and had confirmed that on average
they had received annuities having an initial present value of more than
16 years' purchase.
The report was not published and it was lostto view for many years but
it was eventually unearthed in the State archives of Holland by Frederick
Hendriks, Actuary to the Globe Insurance Company, and published for
the first time (in JrA) in 1852/53.
6
Early life assurance ventures had been based on the idea of charging
premiums to the members as and when claims arose, i.e. no reserves were
set aside for the future. The Amicable, however, specifically aimed to build
up reserves, as this prospectus makes clear. The premiums collected each
year, after deducting expenses and prescribed amounts set aside towards
the building up of reserves, were added to the interest earned in the year
and the total sum was divided out among the claimants. The SOCiety had a
long and honourable history; it was eventually taken over by the Norwich
Union in 1866.
The prospectus states that the office of the Society was 'kept at Mr
Hartley's, a bookseller over against SI. Dunstan's Church in Fleet Street'. It
seems that Mr Hartley was both the originator and first Registrar: long-term
life assurance thus started in a book shop!
7
Annuities on Lives
1700-1850
1725
Abraham De Moivre
At the start of the eighteenth century Halley's paper on the mortality
experience of Breslau and the value of annuities had been published
barely seven years and was little heeded. Life annuities of various types
were bought and sold freely, although in the main their values were not
properly calculated. Life assurance consisted mainly of short term risk
only contracts by single premium. There were no life assurance
companies, all risks being accepted by individual underwriters, except for
one ill-fated widows' annuity scheme run by the Mercers. Probability
theory was at a rudimentary stage of development. Gaming and
wagering were common. Compound Interest was understood, and well
used in land transactions and the mathematics were partially developed.
Some of the pioneers were Fellows of the Royal SOCiety and eminent in
other scientific fields. Many papers on actuarial matters were read before
the Royal Society and published in the Philosophical Transactions, there
being no other formal forum for discussion.
By 1850 life assurance companies were plentiful and were established
on a scientific basis. A wide range of level annual premium contracts was
available to the public. Annuities were calculated by proper methods and
the dangers of inappropriate mortality tables known. Life contingencies,
compound interest and probability theory were at an advanced state of
development, although there were still areas where more work was
necessary. Mortality tables had developed from crude tabulations of
deaths to form the I " column to properly calculated and considered
works. The first table of life assurance mortality from pooled data and
English Life Table No.1 were published coinCidentally in the same year
(1843). The collection and analysis of sickness statistics were under way,
but the level of attainment was still fairly crude. Finally, the Institute of
Actuaries was formed to bring together all the threads constituting the
actuarial profession.
The texts displayed are a personal choice, like other sections of the
exhibition. Corbaux's work on mortality is usually overlooked, possibly
because of its rarity (not more than 250 copies were published) and
because in parts it is quirky. Nevertheless some of the comments in it
seem to be similar to remarks attributed to William Farr some years later.
De Moivre knew how to calculate annuity values for single or for several
lives exactly both in possession and in reversion. His hypothesis that 'the
probabilities of life decrease in arithmetic progreSSion' was 'taken to make
the calculation of lives easy'. The hypothesis was intended to be a
satisfactory approximation to the mortality table published by Halley in
1693 and served its purpose reasonably well. His fictitious life hypothesis
for two or more lives was rather inaccurate.
a, = vp, (1
+ a,+I) is given for the first time.
Expectation of life is defmed correctly: many later writers, up to well
into the last Century defined expectation of life as that period of years for
which the probability of survival is 50%.
A new method for valuing of annuities
upon lives
1727
Richard Hayes
Hayes published his first book of tables, 'upon suppositions of the
various degrees of probability, which lives of different ages have to
continue in being'. He did not give details of his methods of computation.
His tables of annuities upon lives at 5% interest give results very close to
those given by De Moivre in 1725. Some of the other tables may have been
calculated by theoretically unsound approximate methods, but the tables
are a major advance over the rough and ready compound interest
methods then commonly used.
Trevor A Sibbett
8
9
. The Gentleman's Steward and Tenants
of Manors Instructed
1730
John Richards of Exon
The Doctrine of Annuities and
Reversions, Deduced from General
and Evident Principles
1742
Thomas Simpson
This work is written for persons dealing with income arising from the
possession of land. The calculation using the 'v -;.. a' method of an annual
payment to produce a lump sum at the end of 7 years appears to be the
first calculation of this kind - and Richards surmises that it may be a little
too difficult for some people.
The author calculates annuity values for one, two and three lives on the
foundation laid by Halley and the calculation performed by the method
set forth by de Moivre.
Smart's mortality table of 1738 was adjusted at ages under age 25 to allow
for population movements used to calculate annuity tables for 1, 2 and 3
lives on a first and last death basis at 3%,4% and 5% per annum. There
are also empirical rules which allow annuity values to be calculated
approximately at other rates of interest with a reasonable degree of
A letter to George Heathcote, Esq
inclosing tables extracted from ye Bills of
Mortality ... in order to estimate annuities etc.
Simpson also examined (and this is original thought) the adjustments to
be made to life annuity vlaues if payments are to be made every half year
or quarterly and came up with the correct practical additions.
This work is also a text book on the mathematics of annuities and
reversions. De Moivre's hypothesis of equal decrements and fictitious
lives are examined and the inaccuracies revealed are considered. De
Moivre protested that Simpson had plagiarised and mutilated De
Moivre's propositions. Simpson denied this promptly and claimed his
methods were different. Karl Pearson concluded that the mathematics in
Simpson contained nothing which could not be found in de Moivre. A
careful examination undertaken in 1985 of the text of Simpson's 1742
work with de Moivre's work of 1725 indicates that parts of Simpson's text
are drawn from de Moivre and are often re-written slightly. But for
practical purposes Simpson's work is more useful than de Moivre's.
John Smart
1738
The letter, gives a life table calculated from numbers of deaths
occurring in London in the years 1728-37. This is the table which Thomas
Simpson adjusted for his 1742 treatise on annuities.
In 1726, Smart on his Tables of Interest, Discount, Annuities, etc. did
not calculate the value of annuities on lives "as I could meet with very few
observations that could be depended upon". He made a plea for the
number dying at every age to be printed in the yearly bills of mortality.
About 18 months later, the parish clerks commenced inserting age at
death in the weekly bills. Smart's own copy of 1726 tables is in the
Institute library, with manuscript data on Bills of Mortality in his
handwriting.
10
accuracy_
11
Eerste (tweede, derde) verhandling tot een
proeve om te weeten de probable menigte des
volks in de provintie van Holland
en Westvrieslandt;
1742
(First, (second, third) treatise being an attempt to ascertain the probable
amount of population in the Province of Holland and Westfriesland.)
William Kersseboom
These three treatises contain an examination of the previously published
works on mortality and population estimates.
Kersseboom maintains that the population of a country can be obtained
by multiplying the number of births in a year by 35. He thus estimates the
population of the Province of Holland and Westfriesland as 980,000, and
similarly the populations of London in 1684 (496,700) and Paris c. 1670
(610,300).
Most of the work consists of observations based on a vast amount of
data collected by the author. Tables (probably obtained from data of tontines from 1640 onwards) give by age the complete mortality experience
of each of 11 groups of male and female lives together with the corresponding life expectations.
Essai sur Ies Probabilites de la duree
de La vie humaine
1746
M. Deparcieux
The author collected mortality data for various monastic orders mainly for
the period 1685-1745. He shows the raw data separately for each entry age
i.e. the ages at which the vow of profession was taken. Earlier writers had
noted the superior longevity of females, but before Deparcieux's work no
female life table had been produced.
Data of the French tontines of 1689 and 1695 were also collected and
used to produce a life table.
Deparcieux's technique was to form the L column of his tables solely
from observed deaths, derived from populations which were not
generally subject to loss of observations due to migration or withdrawal.
His data do not allow him to calculate the expectallon of hfe at bIrth and
many of his remarks are applicable to stationary populations.
Observations on the past growth and present
state of the City of London
1751
Corbyn Morris
Table II, 'An Account of the Christenings and Burials, and of the
respective Ages of the Persons buried, within the City and Suburbs of
London, from 1728 to 1750, both Years inclusive' was used by James
Dodson as a base from which to calculate the first table of level annual
premiums for whole life assurances.
Morris's main interest to the actuarial profession in future may well be
his Essay on the Science of Insurance, 1757. This uses the binomial
theorem to examine chances of loss in general insurance within a lnarine
insurance context.
12
13
Calculations deduced from first principles,
in the most familiar manner, by plain
arithmetic; etc
Observations on Reversionary Payments
Richard Price
1772
William Dale
From a table of mortality calculated from the London Bills of Mortality for
the years 1728-1771, Dale tabulates and then sums from age x = 93'12 to
age 50'12 values of Lv'·so at half-yearly intervals using an interest rate of
3'12% per annum. He then explains how his totals can be used to calculate
the value of an annuity of 1 every half-year at any age over 50 (194 = 0). the
same method is used for Halley'S, Simpson's and another mortality table
calculated from the London Bills of Mortality for the years 1759-68. The
commutation table method is not developed further.
An act for regulating insurances on lives,
and for prohibiting all such insurances, except
in cases where the persons insuring shall have
an interest in the life or death of
the persons insured
1774
In the eighteenth century, speculators were ready to sell and to buy
insurance on any life who was thought not to be in good health or who
was subject to other risks. Public reaction to this type of wagering affected
the development of legitimate life assurance.
The Life Assurance Act 1774 was a turning point. Although wagering
poliCies continued for many years, the successful rejection of claims on
the ground of lack of insurable interest slowly led to the extinction of the
abuses. The speculators were still able to satisfy their desire to wager by
means of (illegal) insurance of lottery tickets, until the Lottery Act 1802
reduced the period over which tickets were drawn from 42 to 8 days and
thus effectively put many out of business.
16
1812
The first edition of this work appeared in 1771, following Price's paper
Observations on the Expectations of Lives to the Royal Society in 1769. The
first chapter consisted of the solution of problems in the field of
Reversionary Annuities and Assurances. These solutions were then used
to analyse the unsound position of Societies selling reversionary
annuities and as a result some of these Societies closed.
Further matter was added to each new edition of the book and after
Price's death it was continued by his nephew William Morgan. The
edition on display contains the results of about 20 mortality investigations
including associated monetary functions in many cases, the most well
known original tables being in respect of Northampton and Sweden.
A few of the notable items are: strong support for a scheme of old age
pensions, a crude attempt at contributions for sickness insurance and a
note on how to construct a mortality table properly instead of summing
observed deaths to form a table of I,.
The Doctrine of Life Annuities and Assurances
Analytically Investigated and Explained 1813
Francis Baily
A = 1-dii is obtained in a problem concerning three jOint lives. This result
is then generalised for the first time in respect of 1 or more lives. The
method and explanation of surrender values for whole life assurances is
given. Expenses are ignored.
Baily championed George Barrett who submitted an original paper on
commutation tables to the Royal SOCiety. The paper was not published by
the Society and its rejection was attributed to William Morgan, Actuary to
the Equitable. Barrett's paper is explained in an appendix to the second
volume.
17
A Treatise on the Valuation of Annuities
and Assurances on lives and survivorships
1815
On the Nature of the Function Expressive
of the Law of Human Mortality
1825
Joshua Milne
Benjamin Gompertz
The Carlisle mortality table (published here) was constructed from sparse
data -e.g. 406 observed deaths between ages 20 and 60. The table was little
heeded for some years after its publication but was later adopted
enthusiastically by actuaries and became a standard table.
The work contains an extensive survey of previous investigations into
mortality and various examples of life contingency problems.
Gompertz noticed that the I , function over large portions of different
mortality tables decreased approximately in geometrical progression,
especially for small intervals of time. This led him to his law of mortality,
which in modern notation can be expressed as IL x= Be x or Ix = kg eX.
Gompertz was by no means the first author to propose a law of
mortality, but this paper was well received because he demonstrated how
his formulae could be applied with a good degree of accuracy to the
Carlisle, Northampton and other mortality tables.
Changes in the shape of mortality tables led to Makeham modifying
Gompertz's work in 1889 and to Perks generalising Makeham's formula
in 1931.
Report on Friendly or Benefit Societies
Exhibiting the Law of Sickness as deduced from
returns by Friendly Societies in different
parts of Scotland
1824
Highland Society of Scotland
A Comparative View of the Various
Institutions for the Assurance of Lives
1826
Charles Babbage
This detailed report is the first mathematical investigation into sickness
rates. At the time Friendly Societies were in difficulties as a result of
miscalculation rather than mismanagement. In some societies the capital
funds increased annually and progressively for a long period of years;
benefit payments might then be increased without it being realised that
the original benefit levels were too high.
The sickness rates published were the average number of weeks sick in
a year, calculated in 10 year age bands and interpolated to give rates for
each year of age. (An average of the mortality rates published elsewhere
was assumed.) Sickness by occupational group was also analysed.
There are four main tables, showing the benefits secured for ages 21-45
at entry for 10 contribution levels, ceasing at age 70.
18
Babbage's work was written for the benefit of the public in general to
expose the disgraceful practices which prevailed in some companies. The
paying of commission to agents, which was then very controversial, is
covered in forthright language and unsatisfactory areas in the matter of
division of profits are also discussed.
The premium rates charged by companies individually, the profit
margins, mortality tables used in calculating premium rates, various
policy conditions, capital and directors are some of the other items
covered. Babbage also published a table of mortality he deduced from the
experience of the Equitable SOCiety.
In 1827 this book was translated into German, the German edition
being dedicated to E W Arnoldi. Arnoldi took Babbage's Equitable
experience and modified it to calculate the premiums for the 'Gotha'. The
'Gotha' was anxious to extend strict justice to its policyholders: it
distributed all its surplus instead of only a proportion as was common in
England.
19
An investigation of the bases for calculation of
life contingencies, of the profits on
Life Assurances and of an equitable method of
apportioning those profits by way of
bonuses among the assurers
1821
Griffith Davies
The novel proposition is made that the cash value of reversionary
bonuses should be in proportion to the accumulated amount of the
premiums paid less the policy values.
Surrender values then offered were of such trifling amounts that Davies
feels some legislative enactment on the subject might be called for. He
also refers to the 'reasonable expectations' of policyholders. The
valuation of policies and division of profits between the Company and
Shareholders is covered in some detail, and a diagram shows how policy
reserves change during a policy year and increase as each premium is
paid.
The rates of mortality amongst civilians and the military in India and
the terms on which assurances and annuities can be granted to lives
resident in that country are also investigated.
Natural and Political Laws concerning
population, vitality and mortality
1833
Francis Corbaux
Corbaux argued that population mortality tables are constructed from an
aggregate of different mortality rates for lives of several separate classes.
He identified a number of factors including occupation which lead to class
selection. He recommended graduation of third or fourth differences of
log q, in order 'that a rectification of any irregularities, incidentto the data
supplied by experience, may thus be arrived at.'
His discussion covered increasing, stationary and decreasing
populations, complete expectations of life, mortality rates, initial and
class selection of assured lives, births per marriage for the general
population as a whole and also according to age of the female at marriage
etc.
He published 10 mortality tables (five for each sex) in respect of Perfect
Lives, Life-Annuitants, Assurable Lives, Indiscriminate Population and
Inferior Lives. Twenty-five mortality tables published previously ate also
discussed.
A series of tables of annuities and assurances
calculated from a new rate of mortality amongst
assured lives.
1843
Jenkin Jones
20
The 17 Offices' Table was the first mortality experience to be derived from
the combined data from a number of insurance companies. The original
Report and Tables of the 17 Offices' Experience contained no monetary
functions. Before the Committee had corne to any conclusion over the
calculation and publication of monetary tables, Jenkin Jones seized the
initiative and published this work.
A number of sub-divisions of data were investigated, but the final table
was a combination of experience of life offices and included both male and
female lives - the latter on this occasion showing generally a heavier
mortality experience than the assured males. The Committee commented
that the average duration of policies in the investigation was 51/2 years
and that the tables represented a lower rate of mortality than could be
expected to prevail in the long run. The final table, calculated from the
data of 62,537 policies, showed life expectations similar to but slightly less
than those of Milne's Carlisle table of 1815.
21
_
1850-1900
-----------------------.
History of the Formation of the Institute
of Actuaries
May 1848 to July 1851
Much of the fundamental work had been done before the latter half of the
nineteenth century. What then followed was a very rapid advance in the
refinement and practical application of actuarial theory. By then the life
assura~ce business had expanded enormously, and employed many
actuanes full time. Many of the advances in actuarial science may well
have occurred to different people at about the same time.
These years saw the development of the Institute and Faculty of
Actuanes through whose Journal and Transactions respectively a stream
of p~pers on actuarial topiCS was published, as well as the publication of
the flrst text books to aSSIst students studying for the examinations. The
LIfe Assurance Policies Act 1870, was the first statutory framework
regulating the lIfe assurance business, Parliament having chosen the
'freedom with responsibility' principle; this led to the arguments (which
have not yet been settled) on the relative merits of the bonus reserve and
the net premium valuation methods, and also to the invention of the
contribution method of the distribution of surplus. Technical advances
included
(a) the appearance of Makeham's well known formula for mortality,
whIch was found able to express much published mortality data for
the next 100 years or so;
(b) the publication of premium conversion tables, which eased the
burden. of calculating life assurance premiums in the days before
calculatmg machines became generally available;
(c) the invention of summation graduation formulae;
(d) the development of analysis of surplus, whereby the causes of the
surplus or defiCIency revealed in a valuation can be analysed;
(e) the ratification by the Second International Actuarial Congress of the
famIlIar InternatIOnal Actuarial Notation.
Alex McKinnell
22
. . . _-
These reports, minutes and press cuttings collected by Peter Hardy
record in detail the discussions (and some of the dissensions) which led
to the formation of the Institute of Actuaries.
The objects of the Institute included 'protection generally to members
of the profession and the public. The examinations for a certificate of
competency were to include 'Mathematical Theory - Vital Statistics Computation and Construction of Tables - and Book-keeping and Office
Routine'.
The formation of the Institute was a landmark not only for the
introduction of a formal examination system, with its Fellowship status as
a mark of competence, but also because the regular meetings of members
and the foundation of the Journal provided means for the exchange of
knowledge.
Single and Annual Assurance Premiums
1850
WOrchard
'Orchard's Tables' were the first published set of tables making use of the
relationship
1
P = a-d ,and A = 1- dli
to give a simple method of calculating single or annual premiums for
whole life or endowment assurances making use only of the appropriate
annuity value.
In the days before calculating machines were available the saving in
labour occasioned by premium conversion tables was enormous - only
one life contingency function (Ii) was needed to calculate a premium, as
the other functions required could be looked up in a premium conversion
table.
23
Examination Papers - Monday 10th June 1850
and Tuesday 11th June 1850
1850
On the Law of Mortality and the Construction
1860
of Annuity Tables
Solutions of the questions proposed to candidates
WMMakeham
These are the first papers ever set in actuarial science. One may note that
the examinations lasted two days. If two three-hour papers can be sat per
day, the 1985 examinations may be said to last 8 ' /2 days, although the 1850
candidates had to suffer a viva voce examination as well as the written one.
It would appear that 6 candidates presented themselves in 1850 and all
passed. All the successful candidates got was a 'Certificate of
Competency' - it was necessary to obtain an appointment as Actuary to
Makeham proposed generalising Gompertz's law of mortality
the Government or Actuary to a Life Assurance, Annuity or Reversionary
Interest Society in order to attain F.I.A.
fLx= Be
x
into the form
A + Be
He demonstrated at the end how convenient the assumption that
mortality follows his proposed law as in the calculation of joint life
annuity values.
At the time it was proposed Makeham's law appeared to fit existing
data well, and numerous mortality tables have been graduated using
Makeham's law: the most recent major table was probably the CSO 1941
table, which follows Makeham's law from age 15 to age 95.
J.L
x=
X
Contributions to the history of insurance and
of the theory of life contingencies
1854
F Hendriks
This contains a detailed history of insurance from the earliest times and is
interesting in containing the comparison of a marine insurance contract
quoted in Demosthenes and current around 350 BC with a marine
insurance contract current in 1850.
Also included is an investigation into the origins of the Commutation
Column method of calculating life assurance or annuity functions in life
contingencies. Hendriks advances the claim of Tetens as having first
published the method in Hanover in 1785, thus rivalling the claims of
Barrett to have invented the method.
Beitriige zur theorie der priimiem-reserve
bei lebens-versicherungs-anstalten
(Contribution to the theory of premium
reserves for life assurance companies)
1863
A Zillmer
This work shows how initial expenses and commission can be accurately
allowed for in the calculation of life assurance policy reserves. All the
calculations are based on the 17 Offices Table of 1843 with interest at 3' /2%
p.a.
The net premium for a whole life assurance with premiums limited to
age 90 for a life aged 40 is shown, together with the additions which have
to be made in order to allow for acquisition costs of 1 % and 1'14% ofthe
sum assured. Another table gives (by age at entry) the maximum initial
expenses as a percentage of sum assured (and as a percentage of the first
year's premium in the footnote) if the policy reserve at the end of the first
year is not to be negative.
A discussion of the problems of negative reserves follows and the
consequences of a policy going off the books while the reserve is negative
are mentioned.
24
25
On the equitable distribution of surplus
1863
Shephard Homans
On interpolation, summation, and the
adjustment of numerical tables
1865
W S B Woolhouse
This paper contains the first published description of the' contribution
method' for distributing surplus which the author and JParks Fackler had
evolved.
Mortality experience at the Mutual Life Insurance Company of New
York (of which Homans was actuary) had been much lower than had been
assumed in the premium basis, and interest earnings were at a high rate
(6 1/2% net). Consequently the surplus was large, and the author records
his view that any of the usual methods current at the time would have led
to an inequitable distribution, which would apparently have been in
violation of the Company's Charter.
The contribution method was rapidly adopted almost universally in
North America. However this paper seems to have had almost no effect
on methods for distribution of surplus adopted in the United Kingdom.
Woolhouse's paper was the first detailed exposition of graduation of
mortality rates using summation formulae.
Woolhouse's graduation formula was used to graduate the H m
experience, the first set of mortality tables published by the Institute.
On an improved theory of annuities
and assurances
1869
W S B Woolhouse
Tables of lifetimes, annuities, and premiums
1864
with an introduction
William FaIT
English Life Table No.3 was derived from the 1841 and 1851 censuses and
deaths recorded during the years 1838 to 1854.
The interest of the extensive series of tables lies in the fact that the tables
were calculated and typeset by machines, of a type invented by George and
Edward Scheutz. The first Scheutz machine was exhibited and won a gold
medal at the Exhibition of 1855 in Paris. Its powers had been displayed in the
production of 35 pages of five-figure logarithms from 1,000 to 10,000. It is
interesting to compare the quality of typesetting achieved by the Scheutz,
machine with that produced by more recent electronic machines.
26
This was the first presentation of the theory of life contingenCies on a
continuous basis, i.e. assurances are assumed to be payable at the
moment of death, and annuities are assumed to be payable from day to
day.
In his paper Woolhouse derives
the value of a continous annuity a
the value of an assurance payable at the moment of decease Ii:
the premium conversion relationship Ii: = 1 - Sa
The author also goes into considerable details to derive the value of an
annuity payable m times a year, with a proportionate payment up to the
instant of decease.
27
On the practice of the eagle company with
regard to lives regarded as unsound
1900-1960
1874
G Humphreys
This paper is the first detailed published account of an investigation into
the experience of a group of lives accepted for life assurance at additional
rates because of medical impairments. It covers 3147 lives accepted
between 1808 and 1871.
Cases are classified by cause of additional premium as follows:Gout; Hernia; Affections of the Organs of Circulation; Affections of the
Organs of Respiration; Obesity; Intemperate Habits; Family History, or
General Want of Robustness; Miscellaneous Affections.
The author analyses the extent to which the true ages of the lives
accepted were rated up on account of their impediments, and constructs
a mortality table based on the lives of the experience, where the rate of
mortality is expressed as a function of the life's true age plus the number
of years he was rated up.
On the solution of problems connected with
loans payable by instalments
1874
WMMakeham
The author demonstrates his well known formula
A = K + g (C-Kl
1
for the valuation of redeemable securities.
Previous papers by P Gray and Makeham himself had dealt with the
subject of the valuation of redeemable securities, but the publication of
this paper placed the last brick in the edifice of classic compound interest
formulae on the subject.
28
The beginning of the present century was remarkable for the number of
mathematicians devoting themselves to actuarial research. G F Hardy
published his graduation of the British Offices' experience, W P Elderton
wrote a book applying the theory of frequency curves to mortality,
G J Lidstone refined his method for endowments, whilst H W Manly and
G King independently systematised the mathematics of pension fund
valuation.
Twenty five years later, life office investment carne under radical
srcutiny. Coutts argued that assets should always be selected for a
defined purpose, HE Raynes proposed that a portion of funds should be
invested in ordinary shares and C M Douglas' call for measurement of
performance of securities led to the foundation of the Actuaries' Index.
Perks' mathematical refinements in the thirties stood alongside
Phillips' work on binary calculation. Phillips not only designed a
mechanical device to calculate in this manner, but proposed an optical
variant, so going three parts of the way to inventing the electronic
computer.
Matching was foreshadowed by Coutts' paper of 1925, (and
immunisation even earlier by a cryptic letter in JIA in 1908 from a certain
Dr Moll ofthe Netherlands). In the past, the assets and the liabilities ofthe
business had been considered, in separate compartments. The main work
was to place a value on the liabilities - the assets would automatically be
taken at book value, or market value if lower. 1952 saw the publication of
papers by A T Haynes and R J Kirton on matching, and by F M Redington
on immunisation. These papers gave parallel and consistent treatment to
both assets and liabilities.
One more recent important development can already be distinguished:
the invention of profit-testing by J C H Anderson. The life assurance
industry is asked to show that it can offer a sound yield on capital
employed, as well as providing a safe home for policyholders' funds. The
bell of commercial enterprise is thus once again sounded loud and clear,
and provides a confident note on which to close.
Gary Chamberlin
29
On the valuation of staff pension funds
1901
HWManly
1906
W P Elderton
Manly was the first to publish a systematic treatment of staff pension
funds, (including the related topics of widows' and orphans' funds), with
ample formulae and tables to illustrate the principles he evolved.
Further remarks on the valuation of
endowment assurances in groups
Frequency curves and correlation
1903
G J Lidstone
The author derived a formula linked to the unexpired term of the policy
based upon Makeham's law: even when the mortality rates do not follow
the law, very accurate results are obtained from this Z method.
Elderton's first paper on 'Temporary Assurances' (1903) contained
extensive reference to curve-fitting, and the President of the time,
William Hughes, suggested he should expand the work on Pearsonian
frequency curves. Elderton published his book on the subject three years
later.
The work gives a detail€d account of frequency curves, plus instruction
in the method of moments and on special difficulties in the graduation of
actuarial functions. Such is its quality that it remained widely used in
universities for upwards of 60 years after publication, going eventually
into a fifth edition. In 1969, it was entirely revised by Professor N L
Johnson FIA, and re-published under the title 'Systems of Frequency
Curves'.
Bonus reserve valuations
1907
C R V Coutts
Graduation of the British Offices'
experience 1863-93
1903
GFHardy
The main idea underlying the graduation is Hardy'S own discovery in
1894 that Makeham's formula could be applied to select as well as to
aggregate tables.
In the practical working-out of the graduation the author evolved many
new methods. He adopted Pearson's basis of fitting by means of
moments, but brought in the great practical improvement of calculation
by successive summation. As a result, the O(m) select, O(nm) select,
O(am) select and Om, aggregate tables were all graduated on a strictly
Makeham basis, giving important advantages in the practical calculation
of jOint life annuities and other functions.
30
The real origins of the Bonus Reserve method go back before Coutts'
paper. It was foreshadowed in a paper by Sprague in 1857 alA 7,61), and
it was actually in use in certain offices for their official valuations before
Coutts' paper was published.
In 1901 Manly expressed the idea (in a letter to the Insurance Record) that,
in the valuation balance sheet, the present value of the bonus at the rate
to be declared should be treated as a liability for the future existence of the
policies. But it was left to Coutts to draw out the full conclusions.
3]
The fundamental principles of pension
funds
1908
1912
JJM'Lauchlan
W P Elderton and R C Fippard
M'Lauchlan was seeking to explain the characteristics of a pension fund
in simple terms. In particular, he wished to show that the large reserves
brought out by actuarial methods were really necessary.
His projections are straightforward and deal with 2 cases: i) a Fund of
1000 members entering at age 20, with pension and other benefits
balanced against contributions of 5% of salaries, but with no new
entrants, and ii) a similar Fund in which new entrants enter at the rate of
1000 with each succeeding year. M'Lauchlan found that the latter fund
would take 80 years to reach full maturity, by which point it would
amount to slightly less than 3 times the total annual salaries.
The paper shows how mortality rates can be found from Life Office data
by methods as used for rates from Censuses and Death Records data in
the general population. It also shows how the census method can be
modified to give a continuous mortality investigation.
Elderton and Pippard also published a book on the subject in 1914,
which later ran into 5 editions, entitled 'The Construction of Mortality &
Sickness Tables'.
A plan was agreed by the Institute and Faculty in 1914, but its
implementation was delayed by the War. In 1924, the CM.I. finally came
into being.
Report of the actuaries in relation to the
scheme of insurance against sickness,
disablement, etc, embodied in the
National Insurance Bill, 1911
Notes on Life Assurance Investment
Policy
1925
C R V Coutts
1911
G F Hardy and F B Wyatt
When the National Insurance Bill of 1911 was drawn up, advice was
required on finanCing and costs, and for this the Government naturally
approached the Institute of Actuaries. Hardy, together with Wyatt,
another former President, made a thorough study of the population base,
and the relevant rates of mortality, marriage, issue, sickness and
unemployment, etc. They then made financial projections of
contributions and benefits for the period 1912-28. These showed that the
required State subSidy for the Scheme was likely to rise from £1.9m in
1912-13 to £5.Sm in 1927-28.
32
Notes on the construction of
mortality tables
This paper clearly expresses the principle of parallel treatment for the
assets and liabilities. Coutts does not use the term 'matching' , but his
primary and universal canon is that no investment should be made
without due regard to the purpose to be achieved. Examining the nature
of life assurance business, he concludes that interest income is a more
vital factor than the capital values of assets. From this, it follows that a
period of rising yields and falling capital values is in fact advantageous to
the business, and that security is best served by a policy of investing long
rather than short. Further, on the valuation side, Coutts argues that the
assets, along with the liabilities, should be valued on an interest basis, ie.
on the income which they produce.
The paper is particularly notable for its readable style and the clarity of
its logic.
33
The place of ordinary stocks and shares in the
investment of life assurance funds
1927
On some experiments in the graduation of
mortality statistics
HERaynes
WPerks
This paper gave reasons for placing a part of the funds of a Life Office into
equity shares. Thus, such holdings would counterbalance a fall in the
value of money, and it could be shown historically that many advantages
were to be gained.
In applying the Gompertz and Makeham formulae, Perks noticed
discrepancies in the shape of the mortality curves at the older ages. This
led him to a significant extension of the formulae, producing a very
elegant and adaptable series of curves nowadays known as the Logistic
family - but still sometimes referred to in actuarial circles as 'Perks'
curves'.
Later experiments by Beard in fitting the curves to CMI data have
shown their capability for representing assured lives' mortality. As a
footnote, Perks was also able to show how his curves were linked with the
Pearsonian curves, through a simple mathematical transformation
(x~c" ).
The statistical groundwork of
investment policy
1929
1931
CMDouglas
Douglas considered the correct extenr of life office investment in equities.
He studied the relative variations in equity and debenture prices, and in
business activity over the period 1924-28. For this purpose, index
numbers were reqUired.
From his work, Douglas concluded that share prices were indeed
related to the level of business activity. In terms of the trade cycle, equity
prices were directly related to it and debentures inversely, although with
a different tempo and interval. Thus, if one could judge the future trend
of business activity, a sound investment policy would be pursued.
Douglas' work led directly to the establishment of the Actuaries'
Investment Index in the following year.
34
Binary Calculation
1936
E W Phillips
Phillips was a gifted man with a highly original cast of thought, who
practised as a barrister as well as an actuary. But his most astonishing
contribution is this paper which is now recognized as the earliest paperafter Babbage - in the chronology of computer development.
Phillips' purpose was to find a means of relieving the arithmetical
labour of mortality graduations and life office valuations. To this end, he
advocated the use of octonary arithmetic, and designed an elegant
machine for performing binary multiplication. Its principles are described
in the paper, and the machine itself has been put on display in the Science
Museum. But Phillips further envisaged an electrical development,
which would operate using light beams focusing on to selenium cells. It
would be capable, he thought, of performing up to 40,000 multiplications
an hour. Once in operation, it would enable the arithmetic of a valuation
to be accomplished in a morning (on several bases), and the results
presented in the afternoon.
35
The rate of interest in the valuation of a
pension fund
The Actuarial Principles of Investment
1947
1948
J B H Pegler
C E Puckridge
The usual (unspoken) assumption had previously been that assets would
be taken at the lower of book or market value. For liabilities, the rate of
interest would be determined by the relative importance of the yield on
the eXIstmg fund and the probable yield on future investments.
Puckridge, however, advocated an approach under which assets and
liabilities would be valued at the same long-term rate - ie: the rate of
Interest which it is anticipated can be earned on new money over a long
period into the future.
Rex v Seal
1947
H L Seal was brought to a mock trial by the Students' SOCiety, on the
charge of mIsusmg funds entrusted to him as Editor of JSS. He was
accused of publishing 'a highly abstruse mathematical and statistical
periodical ... of no interest or value to members of the Society'. The trial
was held, before 'Justice' E W Phillips, and apparently resulted in a
convIchon although Seal continued to publish advanced statistical papers
inJSS
The importance of the trial is that it highlights the change in attitudes
towards statistical training.
It was not until the 1951 examinations that an adequate treatment of
stahshcs entered the syllabus. Many of the papers published by Seal then
became required reading, or were reflected in the material of the new
official textbook.
36
The investment principles enunciated by Bailey in 1862 were effectively
modernized by Pegler nearly a century later. His principles were
designed to:
i) maximise the overall yield in the long term
ii) spread the investment risk
iii) vary the portfolio for expected future trends
iv) assist socially and economically desirable ends
Whereas Bailey had emphasised the absolute security of the capital,
with Pegler it is the overall return from both capital and interest that
becomes paramount.
The Financial Structure of a Life Office
1952
A T Haynes and R J Kirton
The authors explain that the essence of matching is to maintain the
interest yield for the right future term, and the capital security at the right
future date. Present capital security is of far lesser importance, and safety
lies in holding medium or long-dated assets, according to the nature of
the liabilities. Judgement may be exercised to go short or long from the
matched position, but the departure must be covered by sufficient free
reserves to meet the losses which may thereby arise. Other corollaries
relate to options and to balance sheet presentation. Finally, it is pointed
out that the real danger for an increasing fund is that of a fall in interest
rates.
37
Review of the principles of Life Office
valuations
1952
F M Redington
Gross premium calculations and profit
r:zeasurement for non participating
Insurance
1959
J C H Anderson
~he
first ]Jar: of. this paper contained a startling new theory
( unmumsallon ) whIch gave full mathematical substance to Haynes and
Kirton's ideas on matching.
The question Redington asked was whether a fund could be made
secure against a change in the rate of interest? His theory showed the
answer to be 'yes', provided that the centre of gravity with respect to term
of the assets was equal to that of the liabilities, and that the spread of the
assets was wIder than that of the liabilities.
The growth of pension rights and their impact
on the National Economy
1954
The paper's analysis is from the point of view of the shareholder, or
financial entrepreneur. Previously, actuarial methods had tended to
obscure the idea that life assurance could be assessed, like all other
commercial contracts, to show the return given on the capital invested.
The method is to make financial projections for specimen policies on
the company's books. Assumptions are made for mortality, interest,
taxation and expense, and lapses and surrenders are brought in explicitly.
Finally (and this is the crux of Anderson's innovation), the statutory
valuation basis is also incorporated. Thus the returns assessed on a policy
allow for the necessary increases in valuation reserves, and genuinely
represent the distributable surplus, or profit, arising. Finally, the profits
are discounted back using a risk rate of return, and compared against a
suitable capital criterion.
F W Bacon, M D W Elphinstone & B Benjamin
The Report's main purpose is to make actuarial estimates for pension
outgo over the 30 year period from 1951. It shows a probable doubling
from 4% of national product to 8%, with possible ultimate growth to 14%
prognosllcated. It goes on to give a full economic view of the place and
mfluence of pensions in the life of the country. There is the principle, for
example, that whatever scheme of financing is used, the needs of the
elderly can in fact only be supplied out of current production. Further key
Ideas are: 1) that there 15 no necessary connection between pension
savings and the growth of investment in new capital resources, and ii)
that the principle of funding, so apt for occupational schemes, is
inappropriate when applied to the State Scheme itself.
38
39
Modern Landmarks?
You may be wondering why there are no exhibits post 1960. We are living
in exciting times; surely there must have been some modern landmarks?
True - but the committee organising the exhibition felt that we are so close
to modern events that it would not be possible to achieve an objective
selection. Had we nevertheless decided to proceed, further difficulties
would have presented themselves.
The appearance of profit testing using computer techniques must be a
modern landmark, but how should we have dated and illustrated it? It
cannot be attributed to one man and a paper, more likely a span of years
and a series of papers. (However, Anderson lTIUst warrant a mention and
one of his papers is included as the final item in this catalogue.) Our
Simplest solution would have been to illustrate the technique by
exhibiting a copy of the relevant actuarial text-book: we are still, however,
awaiting its appearance. Similarly in the case of other important new
ideas, there is as yet no obvious definitive work illustrating the event.
Another modern landmark must surely be the introduction of unitlinked assurance. How could its introduction have been identified by us?
Did it start with the first such plan marketed by a company in the U.K.
(assuming we are confining ourselves to the U.K.)? Most historical
introductions to unit-linked assurance identify the first unit-linked
contract offered by a life office in Britain as a deferred annuity for the selfemployed, approved under s22 of the Finance Act 1956 (now s226 of the
Income and Corporation Taxes Act 1970), which was introduced early in
1957 linked to Investment Trust Units. In the autumn of the same year
another company issued a unit-linked life contract; an endowment linked
to a Unit Trust. Yet, other contracts with their benefits linked to the value
of an investment predated both these plans by at least 40 years.
(Correspondence regarding these early policies is contained in the issues
of Post Magazine for 24th November and 1st December 19l7). Few people
took note of unit-linked assurance until specialist companies devoted to
selling such contracts were established. Should the establishment of the
first of these specialist companies have been used as a landmark rather
than the dates of any of these somewhat earlier policy launches? Difficult
decisions which the committee decided should not be made.
Many other events considered candidates for depicting as modern
landmarks can also be predated by earlier ideas and/or events. This is not
meant in any way to detract from the merits of the person who finally
finds a practical use for a theoretical idea.
Study the past, if you would divine the future.
(Confucius Analects, 551-479B. C.)
Geraldine Kaye
40