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Many other fuppofitions might be made with regard to the improvement of money by compound intereft. The interest might be fupposed to be joined to the capital, and along with it to bear intereft at the end of every month, at the end of every day, or even at the end of every inftant, and fuitable calculations might be formed; but thefe fuppofitions, being feldom ufed in practice, are omitted.
II. Of Annuities.
An annuity is a payment made annually for a certain term of years, and the chief problem, with regard to it is, to determine
its prefent worth.' The fuppofition on which the folution proceeds is, that the money received by the feller, being improved by him in a certain manner during the continuance of the annuity, amounts to the fame fum as the feveral payments received by the purchafer, improved in the fame The fuppofitions with regard to
the improvement may be various. be various. What is called the method of fimple intereft, in which fimple intereft only is reckoned upon the purchase-money, and fimple intereft on each annuity from the time of payment, is fo manifeftly unequitable, as to be univerfally rejected; and the fuppofition which is now generally admitted in practice is the highest improvement poffible on both fides, viz. by compound intereft. As the taking compound intereft is prohibited by law, the realizing of this fuppofed improvement requires punctual payment of intereft, and `therefore the intereft in fuch calculations is ufually made low. Even with this advantage, it can hardly be rendered effectual in its full extent; it is however univerfally acquiefced
acquiefced in, as the moft proper foundation of general rules; and when peculiar circumstances require any different hypothefis, a suitable calculation may be made.
Let then the annuity be called a, and let p be the prefent worth of it or purchasemoney, the time of its continuance, and let the other letters denote as formerly.
The feller, by improving the price received p, at compound intereft, at the time the annuity ceases, has pR'.
The purchaser is fuppofed to receive the first annuity a at the end of the first year, which is improved by him for t―1 years; it becomes therefore (Th. 2.) aR11.
He receives the 2d annuity at the end of the 2d year, and when improved 7-2, it becomes a R-2.
The 3d annuity becomes aR-3, &c.
The last annuity is fimply a, therefore the whole amount of the improved annuities is the geometrical feries a+aR+aR2, &c. aR'. The fum of the series by
The fame conclusion results from calculating the present worth of the feveral annuities, confidered as fums payable in reverfion.
Cor. I. Of these four p, a, R, t, any three being given, the fourth may be found, by the folution of equations; t is found eafily by logarithms, R or r can be found only by refolving an adfected equation of the t order.
Cor. 2. If an annuity has been unpaid for the term t, the arrear, reckoning com
pound intereft, will be ax
Cor. 3. The prefent worth of an annuity in reverfion, that is to commence after a certain time (n), and then to continue t years,
years, is found by fubtracting the present worth for n years from the present worth for nt years, and then,
Also of R, t, n, a, p, any four being gi
ven, the fifth may be found.
Cor. 4. If the annuity is to continue for
ever, then R-1 and R
as the fame; and p=ax:
may be confidered
Cor. 5. A perpetuity in reversion (by
Cor. 3.) fince R'—1=R', is p=
Prob. When 12 years of a leafe of 21 were expired, a renewal for the fame term. was granted for 1000 l.; 8 years are now expired, and for what fum muft a corresponding renewal be made, reckoning 5 per cent. compound interest?
From the first tranfaction the yearly profit rent must be deduced; and from this the proper fine in the fecond may be computed.