The prevailing system of life contingency calculation is one not of variable but of invariable quantities. At the very threshold the admission of two such important assumptions is asked for, as that the rate of mortality is always invariable at the same age, whether old or young, and that the rate of interest is equally invariable for all periods, whether long or short. Upon these admissions of invariability a system is formed for assessing the relative values of different cases, thereby necessarily in every instance indicating an invariable answer; and with such indications the system rests content. Whether such assessments, however logically fair in connection with agreed postulates of invariability, are themselves eventually justified by the same invariability of actual result as was à priori assumed, has not hitherto been commonly brought within the general scope of the actuary’s studies. Directly, however, he is called on to take upon himself the practical responsibility of upholding this theory of invariability, he is somewhat surprised to find that, good as the mere logic of his studies may have been, it is by no means an easy task to connect such logic with the nature of the events he may see passing around him.

1. The object of the following investigation is to show how the probabilities of an individual living any given number of years are to be deduced from any table of mortality. All writers (with the exception of Laplace) have considered the probability of an individual dying at any age to be the number of deaths at that age recorded in the table, divided by the sum of the deaths recorded at all ages. This would be the case if the observations on which the table is founded were infinite; but the supposition differs the more widely from the truth the less extended are the observations, and cannot, I think, be admitted where the recorded deaths do not altogether exceed a few thousand, as is the case in the tables used in England. The number of deaths on which the Northampton Tables are founded is 4,689 (Price, vol. i. p. 357). The tables of Halley are founded upon the deaths which took place at Breslau, in Silesia, during five years, and which amounted to 5,869.

It is not without some degree of pride, that we recall the fact that England has taken the lead in fostering and extending those social institutions which appear destined to carry out the beneficent design of procuring the greatest possible amount of happiness for the greatest possible number. She has not been afraid to encourage a spirit of self reliance in the mass of the people, and to allow her working classes to associate freely in the effort to equalize the uncertainties of life, so that those who may have a little better fortune than the average may assist those who have a little less. Now and then, indications may have been observed of fear amongst the ruling powers lest this free association should be used for political purposes, and doctrines dangerous to the Government or the good order of society be thereby enabled to circulate too easily amongst classes who may have real wrongs to redress or fancied rights to assert; but, in the long run, good sense has prevailed, and Government has fortunately perceived that Friendly Societies for mutual aid in sickness or want, and other associations for bettering the condition of the working classes, gave them a direct interest in the preservation of the public peace, and formed by their very principles the antidote to the fears they had excited.

When I began to direct my particular attention to the balance-sheets of the Hamburg Marine Insurance Societies, I hoped that a minute examination would lead to conclusions of general interest, and enable us to judge of the risks and state of navigation in general. I hoped that the collection of data and the comparison of the different balance-sheets would not offer great difficulties, and that my examinations might embrace a long period. My hopes have not been fulfilled. I perceived very soon that the work would be very troublesome, and that it would be difficult to form a complete collection of balance-sheets. There were none at the Public Commercial Library, and it was only by the aid of private friends that I at length succeeded in forming a complete collection of the balance-sheets since 1847. Thus I was obliged to limit my comparisons to the years 1847-53; and I think this is the principal reason why the law which governs the events cannot be deduced, and why no general conclusions can be drawn.

The subject proposed for discussion this evening is one of considerable importance to all who engage in the pursuits of an actuary. The old practice of valuing life contingencies by a pet table of mortality and at one fixed rate of interest has long since been found to work unprofitably; and parties seeking investments in securities of this description have been compelled, for their own safety and advantage, to adjust the valuation of the actuary before determining the price to be given for the particular property under consideration. If such a state of things is unsatisfactory to the public, it must be equally so to the actuary to find his calculations used only as a partial guide, instead of being, as in fact they ought to be, the index by which a proposed purchaser may know the rate of interest and the other advantages to be derived by obtaining the security in question at the price fixed by the actuary. In the ordinary transactions of a Life Assurance Office, where the fluctuation in the rate of mortality is protected by the admission of large numbers, it is necessary also to protect its pecuniary interests by limiting the amount to be assured on each life. Even with this precaution, it not unfrequently happens that particular years have proved less favourable than might have been expected—not so much from an increase in the number of deaths, as from the average amount of the sums insured by the policies so lapsed being greater than the average of the whole number. When, however, the average of a number of years is taken, and the amount insured upon any one risk is properly limited, these fluctuations become less apparent and generally disappear.