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£249.1445——£101.3401, which are the respective présent values of the 1st, 2nd, 3rd, and 4th presentations; and the same part that the second term is of the first, the third term will be of the second, and so on; now the sum of all the terms of such a decreasing series as this, ad infinitum, is as easily found by having the two first terms, as if there were 2000 terms; for the rule is, to divide the square of the first term by the difference between the first and second terms; and the square of 1491.12 divided by 81-6127 gives 25221. for the value of the perpetual ad
And the sum of the present values of annuities of 801. during the lives of the present and each incumbent in succession, ad infinitum, will be equal to the worth of a perpetual income of 801. per annum, or to 20001, itself, for,
The present value (during 1st life aged
To al present value of 4 successive lives, £1898 1598
It should be observed here, that the regular decreasing series, does not begin until the successive lives are of equal ages; as in the above, the first term of the decreasing series is the value of the first life of 30 years of age, and the sum of the terms of this series, found as before, and which is £1491.12 being added to the value of the annuity during the life of the incumbent aged 70, (forming no part of the series) or 508.88 amount together to the 20001. itself, the worth of the perpetual income of 801. per Annum, (computing at the same rate of interest).
All the foregoing calculations suppose that the respective sums are due, immediately on the extinction of Life. Dr.
Price and Mr. Morgan have considered that a sum to be paid on death, will not be due until a year after such life drops; and therefore they make a distinction between the present value of a reversionary sum, and a reversionary estate; the one beginning to produce an income immediately on death, but the other not until one year after death. If such were the general practice, in paying reversionary sums, certainly a distinction ought to be made, by reducing each of the present values now found in proportion to the discount for 1 year, or, ( at 4 per cent. ) As 104 is to 100, so will the present value found, be to the true value: but I think the more general practice is to pay the reversionary sum immediately on death, and in the cases of copyhold tenure, and presentations to a living, such is the practice, therefore no difference whatever ought to be made; for a sum paid now, as 1001., must be of equal value to its corresponding income of 41. per annum, to commence now, each producing 41. at the end of the year.
I have thus been led on, my dear sir, from one value to another, until I have increased a few lines to many sheets; and I fear that the total sum of my values, ad infinitum, will not be an equivalent to the value of your time, in perusing what I have written, but your approbation will afford sufficient pleasure to the mind of
Dear Sir, Your's truly,
To James Pulham, Esq.
P. S. The tables showing the progression of mortality (and on which the valuation of lives must be grounded ), greatly vary in different places; the continual discoveries. medical philosophy to lessen the mortality of diseases, seem to have rendered the tables formed 50 or 100 years ago,
too incorrect for the calculations of the present time; indeed, a French author has computed that the mean duration of human life is increased at least three years, by the vaccine inoculation alone. But an increased expectation of life, if applied to the calculations in this letter, would have the effect of lessening the values of either an enfranchisement of copyhold land, or of a perpetual advowson, because the fine, &c. would not be so often received.
It appears from the tables which have been formed to show the relative mortality of the sexes, that females, in general, live longer than males; and nature seems wisely to have provided for such a difference of mortality (whether caused by a greater exposure to casualties, or from any other circumstance), as it appears (according to the best writers on the subject) that the proportion of births are about 18 males to 17 females: Providence thus acting has preserved an equality of sexual life in existence, for nearly 6000 years.
From a table formed at Sweden, it appears, that out of 10,000 males and 10,000 females, who were there born, the following numbers were alive of each, at the respective ages against the numbers.
Females living out of 10,000
The following extract, from a table given in Dr. Hutton's Mathematical Dictionary, shows the number of persons living at the respective ages in the column (without distinction of sex, out of 1000, who had been born at the several places mentioned on the top of the respective.