The following is Mr. SIMPSON's table of the probabilities of life.

By the help of this table we find what probability there is for a man of a certain age, 30 for example, fhall live 1 year; thus against 31 is 376 and against 30 is 385, the meaning of which is that out of 385 perfons living at 30, there is only 376 living at 31: Thus the probability of a perfon of 30 years of age fhall live one year is measured by this fraction 3, that is, he has the odds of 385 to 9, or nearly 43 to 1 that he does not die in one year.

376

So the odds that a man of 34 fhall live 7 years is found by fubtracting 284 in the table from 349; hence the odds is 284 to 65, or 4 to 1, that a man of 34 lives 7 years; and the like for any other number of years.

Hence it appears, that the value of infurance upon lives ought to be regulated, there being a great difference between infuring the life of a man of 20, and that of another of 50 years of age; fince 'tis 65 to 1 that the man of 20 dies not in an year, and but 24 to 1 for a man of 50 years of age.

The moft general method of valuing annuities on lives is from Mr. de Moivre; viz. Take the value of an annuity certain for fo many years as are denoted by the complement of life (which is what the given life wants of 86;) multiply this value by the rate of intereft, and divide the product by the complement of life; fubtract the quotient from 1, divide the remainder by the intereft of 1 pound, and this last quotient will exprefs the value of an annuity for any age given.

But I think we can get an eafier rule than this of Mr. de Moivre's from the following confideration. Suppose a perfon A was to receive

11. upon this condition, that another, B, 20 years of age fhell live 1 year; quere the value of A's expectation?

Against the ages of 20 and 21 are 462 and 455 now the prefent value of 11. due at the end of one year at 5 per cent. is .95238, which multiplied by this fraction 4 (that is, multiplying by the numerator and dividing by the denominator) gives .938 of a pound for the true value required.

In like manner the probability of a perfon of 20 years of age has of living 2 years is 41, and the prefent worth of 1 1. due at the end of 2 years at 5 per cent. is.907; therefore multiply .907 by 42, and it gives .8795 of a pound for the value of A's expectation to receive 11. at the end of the 22d year of B's life. And thus you may proceed for all the other years of his life to the extremity of age; and the fum of all thefe (from 20 and upwards) being found, and added together, will amount to 13 1. very nearly.

It is demonftrated by the writers on the value of lives, that the fum of the values of 11. for the fucceffive years of a perfon's life will be found by dividing I by the numbers in the table of the amount of 11. which numbers being added together will give the value of that life which is equal to the number of thefe numbers fo added, if there was no contingency in the cafe; but as we must allow for that, there arifes a long and tedious algebraic operation, which is too long to be inferted here; but the rule thence arifing is this;

To the value of the given life add one year's purchafe, difcount that fum, and multiply it by the probability of a life of one year younger than the given life, this laft product will be the value of an annuity upon this life.

For example, the life of 20 being 131. this increafed by one year's purchafe is 14, which difcounted at 5 per cent. is 13.3, this multiplied by the probability of a life of 19, gives 13.16 for the required value of a life of 19 years of age: And in this manner are the numbers in the following table computed, which fhews the value of an annuity on any fingle life from 1 to 80, at the rate of 3, 4, 5 and 6 per cent. compound interest.

N. B. The numbers in the following table, and alfo the anfwers to the questions on annuities on lives fhew the pounds that will purchafe one pound annuity on fuch lives; or they fhew the number of years purchafe that any life or lives (contained in the table) are worth; therefore, multiply the numbers in the table, or the anfwers to the queftions on annuities on lives by the annuity, and the product fhews the va lue of that annuity.

The following is a table fhewing the prefent value of an annuity of 11. (or, which is the fame, the number of years value which fuch annuity is worth) for a life of any age under 81, at 3, 4, 5, and 6 per cent. compound intereft.

Aget.