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Containing Observations on the Expectations of Lives; the Increase of Mankind; the Number of Inhabitants in LONDON; and the Influence of great Towns, on Health and Population.
IN A LETTER TO BENJAMIN FRANKLIN, ESQ. LL.D. AND F. R.S.
I Beg leave to submit to your perusal the following observations. If you think them of any importance, I shall be obliged to you for communicating them to the Royal Society. You will find, that the chief subject of them is the present state of the city of London, with respect to healthfulness and number of inhabitants, so far as it can be collected from the bills of mortality. This is a subject which has been considered by others; but the but the proper method of calculating
This Essay was read to the ROYAL SOCIETY, April 27th, 1769, and has been published in the Philosophical Transactions, Vol. 59. It is here republished with corrections; and with several additions, particularly the Postcript.
from the bills has not, I think, been sufficiently explained.
No competent judgment can be formed of the following observations, without a clear notion of what the writers on Life-Annuities and Reversions have called the Expectation of Life. Perhaps this is not properly understood; and Mr. De Moivre's manner of expressing himself about it is very liable to be mistaken.
The most obvious sense of the expectation of a given life is, "That particular number "of years which a life of a given age has "an equal chance of enjoying." This is the time that a person may reasonably expect to live; for the chances against his living longer are greater than those for it; and, therefore, he cannot entertain an expectation of living longer, consistently with probability. This period does not coincide with what the writers on Annuities call the expectation of life, except on the supposition of an uniform decrease in the probabilities of life, as Mr. Simpson has observed in his Select Exercises, p. 273. It is necessary to add, that, even on this supposition, it does not coincide with what is called the expectation of life, in any case of joint lives. Thus, two lives of 40 have an even chance, according to Mr. De Moivre's hypothesis, of continuing together only 13 years. But the expectation of
See the Notes in page 2 and 39, Vol, I,
two equal joint lives, being (according to the same hypothesis) always a third of the common complement; it is, in this case, 15years. It is necessary, therefore, to observe, that there is another sense of this phrase, which ought to be carefully distinguished from that now mentioned. It may signify, The mean continuance of any given single, "joint, or surviving lives, according to any given Table of Observations:" that is, the number of years which, taking them one with another, they actually enjoy, and may be considered as sure of enjoying; those who live or survive beyond that period, enjoying as much more time in proportion to their number, as those who fall short of it enjoy less. Thus; supposing 46 persons alive, all 40 years of age; and that, according to Mr. De Moivre's hypothesis, one will die every year till they are all dead in 46 years; half 46, or 23, will be their expectation of life: That is, The number of years enjoyed by them all, will be just the same as if every one of them had lived 23 years, and then died; so that, supposing no interest of money, there would be no difference in value between annuities payable for hfe to every single person in such a set, and equal annuities payable to another equal set of persons of the same common age, supposed to be all sure of living just 23 years and no more.
In like manner; the third of 46 years, or 15 years and 4 months, is the expectation of two joint lives both 40; and this is also the expectation of the survivor. That is; supposing a set of marriages between persons all 40, they will, one with another, last just this time; and the survivors will last the same time. And annuities payable during the continuance of such marriages would, supposing no interest of money, be of exactly the same value with annuities to begin at the extinction of such marriages, and to be paid, during life, to the survivors. In adding together the years which any great number of such marriages and their survivorships have lasted, the sums 'would be found to be equal.
One is naturally led to understand the expectation of life in the first of the senses now explained, when, by Mr. Simpson and Mr. De Moure, it is called, the number of years which, upon an equality of chance, a person may expect to enjoy; or, the time which a person of a given age may justly expect to continue in being; and, in the last sense, when it is called, the share of life due to a person. But, as in reality it is always used in the last of these senses, the former language should not be applied to it: And it is in this last sense that it coincides with the sums of the present probabilities, that any given single or joint lives shall attain to the end of the
* See Note (K) at the end of Volume I.
ist, 2d, 3d, &c. moments, from this time to the end of their possible existence; or (in the case of survivorships) with the sum of the probabilities, that there shall be a survivor at the end of the 1st, 2d, 3d, &c. moments, from the present time to the end of the possible existence of survivorship. This coincidence every one conversant in these subjects must see, upon reflecting, that both these senses give the true present value of a lifeannuity, secured by land, without interest of money d
This period in joint lives, I have observed, is never the same with the period which they have an equal chance of enjoying; and in single lives, I have observed, they are the same only on the supposition of an uniform decrease of the probabilities of life. If this decrease, instead of being always uniform, is accelerated in the last stages of life; the former period, in single lives, will be less than the latter; if retarded, it will be greater.
It is necessary to add, that the number expressing the former period, multiplied by the number of single or joint lives whose expectation it is, added annually to a society or town, gives the whole number living together, to which such an annual addition would in time grow. Thus; since 19, or the third of 57, is the expectation of two
See Note (K) at the end of Volume I.