_CONCERNING THE BENEFITS OF INSTITUTIONS WHICH DEPEND UPON THE
PROBABILITY OF EVENTS._
Let us recall here what has been said in speaking of hope. It has been
seen that in order to obtain the advantage which results from several
simple events, of which the ones produce a benefit and the others a
loss, it is necessary to add the products of the probability of each
favorable event by the benefit which it procures, and subtract from
their sum that of the products of the probability of each unfavorable
event by the loss which is attached to it. But whatever may be the
advantage expressed by the difference of these sums, a single event
composed of these simple events does not guarantee against the fear of
experiencing a loss. One imagines that this fear ought to decrease when
one multiplies the compound event. The analysis of probabilities leads
to this general theorem.
By the repetition of an advantageous event, simple or compound, the real
benefit becomes more and more probable and increases without ceasing; it
becomes certain in the hypothesis of an infinite number of repetitions;
and dividing it by this number the quotient or the mean benefit of each
event is the mathematical hope itself or the advantage relative to the
event. It is the same with a loss which becomes certain in the long run,
however small the disadvantage of the event may be.
This theorem upon benefits and losses is analogous to those which we
have already given upon the ratios which are indicated by the indefinite
repetition of events simple or compound; and, like them, it proves that
regularity ends by establishing itself even in the things which are most
subordinated to that which we name _hazard_.
When the events are in great number, analysis gives another very simple
expression of the probability that the benefit will be comprised within
determined limits. This is the expression which enters again into the
general law of probability given above in speaking of the probabilities
which result from the indefinite multiplication of events.
The stability of institutions which are based upon probabilities depends
upon the truth of the preceding theorem. But in order that it may be
applied to them it is necessary that those institutions should multiply
these advantageous events for the sake of numerous things.
There have been based upon the probabilities of human life divers
institutions, such as life annuities and tontines. The most general and
the most simple method of calculating the benefits and the expenses of
these institutions consists in reducing these to actual amounts. The
annual interest of unity is that which is called _the rate of interest_.
At the end of each year an amount acquires for a factor unity plus the
rate of interest; it increases then according to a geometrical
progression of which this factor is the ratio. Thus in the course of
time it becomes immense. If, for example, the rate of interest is 1/20
or five per cent, the capital doubles very nearly in fourteen years,
quadruples in twenty-nine years, and in less than three centuries it
becomes two million times larger.
An increase so prodigious has given birth to the idea of making use of
it in order to pay off the public debt. One forms for this purpose a
sinking fund to which is devoted an annual fund employed for the
redemption of public bills and without ceasing increased by the interest
of the bills redeemed. It is clear that in the long run this fund will
absorb a great part of the national debt. If, when the needs of the
State make a loan necessary, a part of this loan is devoted to the
increasing of the annual sinking fund, the variation of public bills
will be less; the confidence of the lenders and the probability of
retiring without loss of capital loaned when one desires will be
augmented and will render the conditions of the loan less onerous.
Favorable experiences have fully confirmed these advantages. But the
fidelity in engagements and the stability, so necessary to the success
of such institutions, can be guaranteed only by a government in which
the legislative power is divided among several independent powers. The
confidence which the necessary coöperation of these powers inspires,
doubles the strength of the State, and the sovereign himself gains then
in legal power more than he loses in arbitrary power.
It results from that which precedes that the actual capital equivalent
to a sum which is to be paid only after a certain number of years is
equal to this sum multiplied by the probability that it will be paid at
that time and divided by unity augmented by the rate of interest and
raised to a power expressed by the number of these years.
It is easy to apply this principle to life annuities upon one or several
persons, and to savings banks, and to assurance societies of any nature.
Suppose that one proposes to form a table of life annuities according to
a given table of mortality. A life annuity payable at the end of five
years, for example, and reduced to an actual amount is, by this
principle, equal to the product of the two following quantities, namely,
the annuity divided by the fifth power of unity augmented by the rate of
interest and the probability of paying it. This probability is the
inverse ratio of the number of individuals inscribed in the table
opposite to the age of that one who settles the annuity to the number
inscribed opposite to this age augmented by five years. Forming, then, a
series of fractions whose denominators are the products of the number of
persons indicated in the table of mortality as living at the age of that
one who settles the annuity, by the successive powers of unity augmented
by the rate of interest, and whose numerators are the products of the
annuity by the number of persons living at the same age augmented
successively by one year, by two years, etc., the sum of these fractions
will be the amount required for the life annuity at that age.
Let us suppose that a person wishes by means of a life annuity to assure
to his heirs an amount payable at the end of the year of his death. In
order to determine the value of this annuity, one may imagine that the
person borrows in life at a bank this capital and that he places it at
perpetual interest in the same bank. It is clear that this same capital
will be due by the bank to his heirs at the end of the year of his
death; but he will have paid each year only the excess of the life
interest over the perpetual interest. The table of life annuities will
then show that which the person ought to pay annually to the bank in
order to assure this capital after his death.
Maritime assurance, that against fire and storms, and generally all the
institutions of this kind, are computed on the same principles. A
merchant having vessels at sea wishes to assure their value and that of
their cargoes against the dangers that they may run; in order to do
this, he gives a sum to a company which becomes responsible to him for
the estimated value of his cargoes and his vessels. The ratio of this
value to the sum which ought to be given for the price of the assurance
depends upon the dangers to which the vessels are exposed and can be
appreciated only by numerous observations upon the fate of vessels which
have sailed from port for the same destination.
If the persons assured should give to the assurance company only the sum
indicated by the calculus of probabilities, this company would not be
able to provide for the expenses of its institution; it is necessary
then that they should pay a sum much greater than the cost of such
insurance. What then is their advantage? It is here that the
consideration of the moral disadvantage attached to an uncertainty
becomes necessary. One conceives that the fairest game becomes, as has
already been seen, disadvantageous, because the player exchanges a
certain stake for an uncertain benefit; assurance by which one exchanges
the uncertain for the certain ought to be advantageous. It is indeed
this which results from the rule which we have given above for
determining moral hope and by which one sees moreover how far the
sacrifice may extend which ought to be made to the assurance company by
reserving always a moral advantage. This company can then in procuring
this advantage itself make a great benefit, if the number of the assured
persons is very large, a condition necessary to its continued existence.
Then its benefits become certain and the mathematical and moral hopes
coincide; for analysis leads to this general theorem, namely, that if
the expectations are very numerous the two hopes approach each other
without ceasing and end by coinciding in the case of an infinite number.
We have said in speaking of mathematical and moral hopes that there is a
moral advantage in distributing the risks of a benefit which one expects
over several of its parts. Thus in order to send a sum of money to a
distant part it is much better to send it on several vessels than to
expose it on one. This one does by means of mutual assurances. If two
persons, each having the same sum upon two different vessels which have
sailed from the same port to the same destination, agree to divide
equally all the money which may arrive, it is clear that by this
agreement each of them divides equally between the two vessels the sum
which he expects. Indeed this kind of assurance always leaves
uncertainty as to the loss which one may fear. But this uncertainty
diminishes in proportion as the number of policy-holders increases; the
moral advantage increases more and more and ends by coinciding with the
mathematical advantage, its natural limit. This renders the association
of mutual assurances when it is very numerous more advantageous to the
assured ones than the companies of assurance which, in proportion to the
benefit that they give, give a moral advantage always inferior to the
mathematical advantage. But the surveillance of their administration can
balance the advantage of the mutual assurances. All these results are,
as has already been seen, independent of the law which expresses the
moral advantage.
One may look upon a free people as a great association whose members
secure mutually their properties by supporting proportionally the
charges of this guaranty. The confederation of several peoples would
give to them advantages analogous to those which each individual enjoys
in the society. A congress of their representatives would discuss
objects of a utility common to all and without doubt the system of
weights, measures, and moneys proposed by the French scientists would be
adopted in this congress as one of the things most useful to commercial
relations.
Among the institutions founded upon the probabilities of human life the
better ones are those in which, by means of a light sacrifice of his
revenue, one assures his existence and that of his family for a time
when one ought to fear to be unable to satisfy their needs. As far as
games are immoral, so far these institutions are advantageous to customs
by favoring the strongest bents of our nature. The government ought then
to encourage them and respect them in the vicissitudes of public
fortune; since the hopes which they present look toward a distant
future, they are able to prosper only when sheltered from all inquietude
during their existence. It is an advantage that the institution of a
representative government assures them.
Let us say a word about loans. It is clear that in order to borrow
perpetually it is necessary to pay each year the product of the capital
by the rate of interest. But one may wish to discharge this principal in
equal payments made during a definite number of years, payments which
are called _annuities_ and whose value is obtained in this manner. Each
annuity in order to be reduced at the actual moment ought to be divided
by a power of unity augmented by the rate of interest equal to the
number of years after which this annuity ought to be paid. Forming then
a geometric progression whose first term is the annuity divided by unity
augmented by the rate of interest, and whose last term is this annuity
divided by the same quantity raised to a power equal to the number of
years during which the payment should have been made, the sum of this
progression will be equivalent to the capital borrowed, which will
determine the value of the annuity. A sinking fund is at bottom only a
means of converting into annuities a perpetual rent with the sole
difference that in the case of a loan by annuities the interest is
supposed constant, while the interest of funds acquired by the sinking
fund is variable. If it were the same in both cases, the annuity
corresponding to the funds acquired would be formed by these funds and
from this annuity the State contributes annually to the sinking fund.
If one wishes to make a life loan it will be observed that the tables of
life annuities give the capital required to constitute a life annuity at
any age, a simple proportion will give the rent which one ought to pay
to the individual from whom the capital is borrowed. From these
principles all the possible kinds of loans may be calculated.
The principles which we have just expounded concerning the benefits and
the losses of institutions may serve to determine the mean result of any
number of observations already made, when one wishes to regard the
deviations of the results corresponding to divers observations. Let us
designate by _x_ the correction of the least result and by _x_ augmented
successively by _q_, _q´_, _q´´_, etc., the corrections of the following
results. Let us name _e_, _e´_, _e´´_, etc., the errors of the
observations whose law of probability we will suppose known. Each
observation being a function of the result, it is easy to see that by
supposing the correction _x_ of this result to be very small, the error
_e_ of the first observation will be equal to the product of _x_ by a
determined coefficient. Likewise the error _e´_ of the second
observation will be the product of the sum _q_ plus _x_, by a determined
coefficient, and so on. The probability of the error _e_ being given by
a known function, it will be expressed by the same function of the first
of the preceding products. The probability of _e´_ will be expressed by
the same function of the second of these products, and so on of the
others. The probability of the simultaneous existence of the errors _e_,
_e´_, _e´´_, etc., will be then proportional to the product of these
divers functions, a product which will be a function of _x_. This being
granted, if one conceives a curve whose abscissa is _x_, and whose
corresponding ordinate is this product, this curve will represent the
probability of the divers values of _x_, whose limits will be determined
by the limits of the errors _e_, _e´_, _e´´_, etc. Now let us designate
by _X_ the abscissa which it is necessary to choose; _X_ diminished by
_x_ will be the error which would be committed if the abscissa _x_ were
the true correction. This error, multiplied by the probability of _x_ or
by the corresponding ordinate of the curve, will be the product of the
loss by its probability, regarding, as one should, this error as a loss
attached to the choice _X_. Multiplying this product by the differential
of _x_ the integral taken from the first extremity of the curve to _X_
will be the disadvantage of _X_ resulting from the values of _x_
inferior to _X_. For the values of _x_ superior to _X_, _x_ less _X_
would be the error of _X_ if _x_ were the true correction; the integral
of the product of _x_ by the corresponding ordinate of the curve and by
the differential of _x_ will be then the disadvantage of _X_ resulting
from the values _x_ superior to _x_, this integral being taken from _x_
equal to _X_ up to the last extremity of the curve. Adding this
disadvantage to the preceding one, the sum will be the disadvantage
attached to the choice of _X_. This choice ought to be determined by the
condition that this disadvantage be a _minimum_; and a very simple
calculation shows that for this, _X_ ought to be the abscissa whose
ordinate divides the curve into two equal parts, so that it is thus
probable that the true value of _x_ falls on neither the one side nor
the other of _X_.
Celebrated geometricians have chosen for _X_ the most probable value of
_x_ and consequently that which corresponds to the largest ordinate of
the curve; but the preceding value appears to me evidently that which
the theory of probability indicates.