SKETCH OF DE MOIVRE.--HIS DOCTRINE OF CHANCES.--KERSSEBOOM.--DE
PARCIEUX.--HODGSON.--DODSON.--FIRST FRAUD IN LIFE ASSURANCE--ITS
ROMANTIC CHARACTER.--THOMAS SIMPSON.--CALCULATIONS OF DE BUFFON.
To the same year which witnessed the proposition for the new companies
we are indebted for the work entitled the “Doctrine of Chances,”
written by Abraham de Moivre, who, owing to the revocation of the Edict
of Nantes, was compelled to seek shelter in England, where he perfected
the studies he had commenced in his own country. In his boyhood he
had neglected classics for mathematics, to the great surprise of
his master, who often asked “what the little rogue meant to do with
those ciphers.” In 1718, he published the first edition of the above
book; and a few extracts from this, which led him afterwards to his
hypothetical application of those chances to the survivorship of life,
may not be unacceptable; as, though the author deemed it wise to
apologise in his dedication for publishing a work which “many people
in the world might think had a tendency to promote play,” yet his
volume will prove the best apology. The book is very entertaining in
its character, and is an evidence of an inquiring and mathematical mind
employing itself upon trifling questions rather than remain idle. Thus,
Case 1. is “To find the probability of throwing an ace in two throws of
one die.” And this kind of problem he varied to almost every possible
form. There is “the probability of throwing an ace in three throws,” of
“throwing an ace in four throws,” of “two aces in two throws,” of “two
aces in three throws,” worked out in a most exact and elaborate manner.
From dice he proceeded to lotteries, and showed how many tickets
ought to be taken to secure the probability of a prize. The volume, a
considerable quarto, was nothing more than an amusing book on gambling
and its various chances. But it produced a better effect. A few years
later, he published something more worthy of him, in his “Doctrine of
Chances, applied to the Valuation of Annuities on Lives,” in which he
says, with some appearance of surprise, “Two or three years after the
publication of the first edition of my ‘Doctrine of Chances,’ I took
the subject into consideration; and consulting Dr. Halley’s tables of
observation, I found that the decrements of life, for considerable
intervals of time, were in arithmetical progression; for instance, out
of 646 persons of 12 years of age, there remain 640 after 1 year; 634
after 2 years; 628, 622, 616, 610, 604, 598, 592, and 586, after 3,
4, 5, 6, 7, 8, 9, and 10 years respectively, the common difference of
those numbers being 6. Examining afterwards other cases, I found that
the decrements of life for several years were still in arithmetical
progression, which may be observed from the age of 54 to the age of 71,
where the difference for 17 years is constantly 10.”
The greatest difficulty which occurred to him was to invent practical
rules that might readily be applied to the valuation of several lives,
“which was, however, happily overcome, the rules being so easy that, by
the help of them, more can be performed in a quarter of an hour, than
by any method before extant in a quarter of a year.”
It was first published in 1725; and finding thus from Halley that, for
several years together the decrement of life was uniform, it being only
in youth and old age any considerable deviation took place, he founded
an hypothesis that it was uniform from birth to extreme old age; in
other words, that out of a given number of persons living at any age,
“an equal number die every year until they are all extinct.” On this
he gave a general theorem, by which the values of annuities on single
lives might be easily determined. This was of great use at the time, no
table of the real value of annuities having then been published, except
a very contracted one founded on Halley’s paper; and if subsequent
investigations proved that De Moivre was utterly wrong, his conclusions
formed the basis of many a future calculation.
Although the ability of De Moivre was recognised by the Royal Society
when it appointed him arbitrator in the contest betwixt Newton and
Leibnitz, and although Newton, when applied to for an explanation of
his own works, would often say “Go to De Moivre, he knows better than I
do,” yet it is to be feared that golden opinions were won by him more
freely than guineas.
It is sufficiently known that the coffee-houses of the eighteenth
century were the resort of all who sought intelligence or loved the
company of the wits and fine men about town. To one of these, in St.
Martin’s Lane, De Moivre went, where it was customary to apply to him
for the solution of many questions connected with annuities, and for
answers to queries concerning games of hazard, which were propounded to
him by those who hoped to turn the chance of loss into a certainty of
gain. The payment of these questions was his chief mode of subsistence;
and there is something unpleasant in the memory of this man, compelled,
in his old age, to be at the bidding of gamesters, and to consort with
men who lived on the town by their wits.
The opinion of posterity is divided upon his merits. “By the most
simple and elegant formulæ,” says Francis Baily, “he pointed out the
method of solving all the most common questions relative to the value
of annuities on single and joint lives, reversions, and survivorships.”
The subsequent editions of his works prove that he was aware of his
errors of detail, by correcting them. He enlarged the boundaries of the
science which he loved, and encouraged others to follow in the same
path. Although his hypothesis may not be applicable to all occasions
and circumstances, and though later discoveries proved that it could
not be always safely adopted, “nevertheless it is still of great use
in the investigation of many cases connected with this subject, and
will ever remain a proof of his superior genius and ability.” Such is
the opinion of Baily on the merits of De Moivre; but it has been added
by Morgan, that “on the whole the hypothesis of De Moivre has probably
done more harm than good, by turning the attention of mathematicians
from investigating the true laws of mortality.”
In 1737, an attempt was made to calculate the number of the people,
which was estimated at 6,000,000, an amount probably not very far from
the mark; as in 1688 the population was reckoned at a little over
5,000,000. Some important assistance was rendered in 1738, by the
publication of Kersseboom’s tables, taken from the records of life
annuities in Holland[10]; and as the ages of the annuitants had been
there recorded for 125 years, they proved a considerable aid to those
interested. So small was the progress made in England by 1746, that
Dr. Halley’s Breslau Tables and those of M. Kersseboom were the only
ones which gave anything like a representation of the true laws of
mortality. In this year, however, the “Essai sur les Probabilités de
la Durée de la Vie Humaine” of M. De Parcieux, with several valuable
tables deduced from the mortuary registers of religious houses
in France, and from the nominees in the French tontines, were an
additional contribution to our information.
The first effort to show the value of annuities on lives from the
London Bills of Mortality is attributable to James Hodgson. Nor was
this endeavour uncalled for or unnecessary. Many assurance offices
had arisen, undertaking to grant these annuities; and the tables
principally in use were founded on the decrease of life at Breslau. But
by the Breslau Tables, half the people lived till they were about 41
years of age, while in London half did not reach the age of 10. This
was a vast difference in the estimate of mortality, and affected the
price of annuities in a proportionate degree. But if the Breslau Tables
calculated life at too high a rate, it was equally evident that the
London Tables made them too low; it is obvious, therefore, that the
value of a life annuity founded on any confined observations would be
unsuitable to the general annuitant; and it is evident that a scale of
prices should have been based on a more enlarged foundation.
The work of Mr. Hodgson deserves very great attention, and the notice
of the reader is called to its investigation, as the conclusions were
arrived at after great labour, and are a specimen of the time and
trouble bestowed on the subject. “The easy way of raising money for
public uses,” says Mr. Hodgson, “by granting annuities upon lives, has
met with so great encouragement that there is no room to doubt it will
be carried down to future times.” The following statements of this
gentleman will be read with surprise by those who are acquainted with
the chances of life as calculated at the present day. He estimated that
“1000_l._ would purchase an annuity of 70_l._ per annum for a life of
29 years 10 months, when money is valued at 3 per cent. per annum; that
the same sum will purchase the same annuity for a life of 23 years,
when money is valued at 4 per cent. per annum; and that the same sum
will purchase the same annuity for a life of 23 years, when money is
valued at 5 per cent. per annum; and that it will purchase the same
annuity for a life of 16 years 2 months, when money is valued at 6 per
cent.
“It appears that the highest value of a life is when the person is
about 6 years of age, and that from the birth to that time the value of
lives decrease, as they do from that time to the utmost extremity of
old age; that a life of 1 year old is nearly equal in value to a life
of 7 years old; that a life of 3 years old is nearly equal in value to
a life of 12 years old; that a life of 4 years old is nearly equal in
value to a life between 9 and 10 years; and that a life of 5 years is
nearly equal in value to a life of 7 years of age. And hence arose the
custom of putting the value of the lives of minors upon the same value
with those of a middling age, which at the best is but a bold guess,
and made use of for no better reason, than that they knew of no better
way to find the true value.”
Such was a portion of Mr. Hodgson’s contribution in 1747 to vital
statistics. This work was followed in 1751 by the “Observations on the
past Growth and present State of the City of London” of Corbyn Morris,
containing tables of burials and christenings from 1601 to 1750. The
tables were important in themselves, and the book is noticeable as
containing a proposal to remodel the Bills of Mortality.
The topic was particularly interesting to mathematical men. In 1753,
Mr. James Dodson pursued the subject, and solved in his “Repository”
an immense variety of questions. Hitherto a table deduced by Simpson
from the London Bills of Mortality, was the only one taken from real
observation. But it need not be said that London was a very limited
theatre on which to found the payment of premiums. The number of
persons who died there in a given time, doubled that of other and more
healthy cities. It was impossible to separate the casual visitors from
the natives, in the record of deaths. It was equally difficult to
divide those who had been born there, from those who were naturalised
by virtue of a long and continued residence. The city, which has ever
been the land of promise to the country, brought adventurers from the
rural districts in a continued stream. The difficulties which prevented
correct information from spreading may be judged by the statement that,
from 1759 to 1768 a third more deaths than births were registered,
the average annual burials being 22,956 to 15,910 of births. In the
previous 10 years, the excess had been 10,500, or near half the
burials. The baptismal registries were also very deficient in that
large class denominated sectarians; Jews, Quakers, Roman Catholics,
and all who refused to recognise the rites of the English Church being
excluded. It required, therefore, care and calculation of no ordinary
character to make any approximation to the truth; and Mr. Dodson
believed he would be nearer it, by adopting the opinions of De Moivre
as the ground work of his tables, rather than by entering on a sea of
uncertain and hypothetical calculations.
In 1754, a further “valuation of annuities on lives,” deduced also from
the London Bills of Mortality was published. By this it appeared that
the work of Mr. Hodgson had not produced much effect in sending the
Breslau Tables out of general use; for, says the author, “I think it
very unreasonable that a poor citizen of London should be made to pay
for an annuity according to the probability of the duration of life at
Breslau, where, as appears from the bills of mortality, one-half of the
people that are born live till they are about 41 years of age, whereas
at London one-half die before they arrive at the age of 13.”
The first known fraud in assurance is one of the most singular in
its annals. The reader must judge for himself of the circumstances
attending it; but there is no doubt that others far more fearful in
their results have since been practised.
About 1730, two persons resided in the then obscure suburbs of St.
Giles’s, one of whom was a woman of about twenty, the other a man
whose age would have allowed him to be the woman’s father, and who was
generally understood to bear that relation. Their position hovered on
the debatable ground between poverty and competence, or might even be
characterised by the modern term of shabby genteel. They interfered
with no one, and they encouraged no one to interfere with them. No
specific personal description is recorded of them, beyond the fact that
the man was tall and middle aged, bearing a semi-military aspect, and
that the woman, though young and attractive in person, was apparently
haughty and frigid in her manner. On a sudden, at night time, the
latter was taken very ill. The man sought the wife of his nearest
neighbour for assistance, informing her that his daughter had been
seized with sudden and great pain at the heart. They returned together,
and found her in the utmost apparent agony, shrinking from the approach
of all, and dreading the slightest touch. The leech was sent for; but
before he could arrive she seemed insensible, and he only entered the
room in time to see her die. The father appeared in great distress,
the doctor felt her pulse, placed his hand on her heart, shook his
head as he intimated all was over, and went his way. The searchers
came, for those birds of ill-omen were then the ordinary haunters of
the death-bed, and the coffin with its contents was committed to the
ground. Almost immediately after this the bereaved father claimed from
the underwriters some money which was insured on his daughter’s life,
left the locality, and the story was forgotten.
Not very long after, the neighbourhood of Queen Square, then a
fashionable place, shook its head at the somewhat unequivocal
connection which existed between one of the inmates of a house in
that locality, and a lady who resided with him. The gentleman wore
moustaches, and though not young, affected what was then known as the
macaroni style. The lady accompanied him everywhere. The captain,
for such was the almost indefinite title he assumed, was a visitor
at Ranelagh, was an _habitué_ of the Coffee-houses, and being an
apparently wealthy person, riding good horses and keeping an attractive
mistress, he attained a certain position among the _mauvais sujets_ of
the day. Like many others at that period, he was, or seemed to be, a
dabbler in the funds, was frequently seen at Lloyd’s and in the Alley;
lounged occasionally at Garraway’s; but appeared more particularly to
affect the company of those who dealt in life assurances.
His house soon became a resort for the young and thoughtless, being
one of those pleasant places where the past and the future were alike
lost in the present; where cards were introduced with the wine, and
where, if the young bloods of the day lost their money, they were
repaid by a glance of more than ordinary warmth from the goddess of the
place; and to which, if they won, they returned with renewed zest. One
thing was noticed, they never won from the master of the house, and
there is no doubt, a large portion of the current expenses was met by
the money gambled away; but whether it were fairly or unfairly gained,
is scarcely a doubtful question.
A stop was soon put to these amusements. The place was too remote
from the former locality, the appearance of both characters was too
much changed to be identified, or in these two might have been traced
the strangers of that obscure suburb where as daughter, the woman was
supposed to die, and as father, the man had wept and raved over her
remains. And a similar scene was once more to be acted. The lady was
taken as suddenly ill as before; the same spasms at the heart seemed to
convulse her frame, and again the man hung over her in apparent agony.
Physicians were sent for in haste; one only arrived in time to see her
once more imitate the appearance of death, while the others, satisfied
that life had fled, took their fees, “shook solemnly their powdered
wigs,” and departed. This mystery, for it is evident there was some
collusion or conspiracy, is partially solved when it is said, that many
thousands were claimed and received by the gallant captain from various
underwriters, merchants, and companies with whom he had assured the
life of the lady.
But the hero of this tradition was a consummate actor; and though
his career is unknown for a long period after this, yet it is highly
probable that he carried out his nefarious projects in schemes which
are difficult to trace. There is little doubt, however, that the
_soi-disant_ captain of Queen Square was one and the same person who,
as a merchant, a few years later appeared daily on the commercial
walks of Liverpool; where, deep in the mysteries of corn and cotton,
a constant attender at church, a subscriber to local charities, and a
giver of good dinners, he soon became much respected by those who dealt
with him in business, or visited him in social life. The hospitalities
of his house were gracefully dispensed by a lady who passed as his
niece, and for a time nothing seemed to disturb the tenour of his
way. At length it became whispered in the world of commerce, that his
speculations were not so successful as usual; and a long series of
misfortunes, as asserted by him, gave a sanction to the whisper. It
soon became advisable for him to borrow money, and this he could only
do on the security of property belonging to his niece. To do so it
was necessary to insure their lives for about 2000_l._ This was easy
enough, as Liverpool, no less than London, was ready to assure anything
which promised profit, and as the affair was regular, no one hesitated.
A certain amount of secresy was requisite for the sake of his credit;
and availing himself of this, he assured on the life of the niece
2000_l._ with, at any rate, ten different merchants and underwriters in
London and elsewhere. The game was once more in his own hands, and the
same play was once more acted. The lady was taken ill, the doctor was
called in and found her suffering from convulsions. He administered a
specific and retired. In the night he was again hastily summoned, but
arrived too late. The patient was declared to be beyond his skill; and
the next morning it became known to all Liverpool that she had died
suddenly. A decorous grief was evinced by the chief mourner. There was
no haste made in forwarding the funeral; the lady lay almost in state,
so numerous were the friends who called to see the last of her they had
visited; the searchers did their hideous office gently, for they were,
probably, largely bribed; the physician certified she had died of a
complaint he could scarcely name, and the grave received the coffin.
The merchant retained his position in Liverpool, and bore himself with
a decent dignity; made no immediate application for the money, scarcely
even alluding to the assurances which were due, and when they were
named, exhibited an appearance of almost apathetic indifference. He
had, however, selected his victims with skill. They were safe men, and
from them he duly received the money which was assured on the life of
the niece.
From this period he seemed to decline in health, expressed a loathing
for the place where he had once been so happy; change of air was
prescribed, and he left the men whom he had deceived, chuckling at the
success of his infamous scheme.
It need not be repeated, that the poverty-stricken gentleman of the
suburbs, the gambling captain of Queen Square, and the merchant of
Liverpool, were identical. That so successful a series of frauds was
practised appears wonderful at the present day; but that the woman
either possessed that power of simulating death, of which we read
occasional cases in the remarkable records of various times, or that
the physicians were deceived or bribed, is certain. There is no other
way of accounting for the success of a scheme which dipped so largely
into the pockets of the underwriters.
The next movement in the scientific annals of life assurance was made
by Thomas Simpson, a natural and self-taught mathematician, whose
life prior to throwing himself on the world of London for support had
been somewhat of a vagrant one. He had cast rustic nativities, told
fortunes, advanced courtships, and occasionally varied his vagabondism
by undertaking to raise the devil, an attempt in which he was so
successful, that he sent his pupil mad, and was obliged himself to
leave the village. In 1740, he produced a volume “On the Nature and
the Laws of Chance;” in 1742, this was followed by his “Doctrine of
Annuities and Reversions,” deduced from general and evident principles,
with tables showing the value of joint and single lives. In 1752, he
made an additional contribution to the statistics of annuities, as
he published in his “Select Exercises” a supplement, wherein he gave
new tables of the values of annuities on two joint lives, and on the
survivor of two lives, more copious than hitherto. He first attempted
to compute the value of joint lives; but as these were still taken from
the London Bills of Mortality, they were by no means fit for general
acceptance. He treated his subject, however, more broadly and clearly
than it had been previously treated, giving some of the best tables
of the values of life annuities, which were published for many years.
Though the manner in which they might be computed had been shown by
Dr. Halley, it is to the self-taught Simpson we are indebted for their
practical application.
In 1760, M. Buffon published a further contribution to the statistics
of assurance, in a table of the probabilities of life, estimated
from the mortality bills of three parishes in Paris, and two country
parishes in its neighbourhood.
The following are some of his calculations:--“By this table,” says the
author, “we may bet 1 to 1 that a new-born infant will live 8 years;
that a child of one year old will live 33 years more, that a child of
full two years old will live 33 years and 5 months more, that a man of
thirty will live 28 years more; that a man of forty will live 22 years
longer, and so through the other ages.”
Buffon adds, “The age at which the longest life is to be expected is 7,
because we may lay an equal wager, or 1 to 1, that a child of that age
will live 42 years and 3 months longer. That at the age of twelve or
thirteen, we have lived a fourth part of our life, because we cannot
reasonably expect to live 38 or 39 years longer; that in like manner
at the age of 28 or 29, we have lived one-half of our life, because we
have but 28 years more to live; and lastly, that before fifty we have
lived three-fourths of our life, because we can hope but for 16 or 17
years more.”
Some profound moral reflections followed these estimates; and as a
critic of the day “thought all serious remarks out of place in an
arithmetical calculation, and that M. Buffon had better reserve them
for his book on beasts,” the reader will not be troubled with their
repetition. He will not, however, be displeased to read the remarks on
this table, by one of the annotators of the day.
“For insuring for 1 year the life of a child of three years old we
ought to pay 10 per cent., for as it has by M. Buffon’s table an
equal chance of living 40 years, it is 40 to 1 that it does not die
in a year. In the same manner we ought to pay but 3 per cent. for
insuring for 1 year the life of a lad of nineteen or twenty; but 4
per cent. for insuring for 1 year the life of a man of thirty-five;
and 5 per cent. per annum for insuring for 1 year the life of a man
of forty-three; after which the insurances ought to rise above 5 per
cent. in proportion to the advance of a person’s age above forty-three.
So that a man of seventy-seven ought to pay 25 per cent., and a man of
eighty-five 33-1/2 per cent. for insuring his life for 1 year.”