VII. Calculation of Annual Debt Payment, [ca. 15 November 1801]
VII. Calculation of Annual Debt Payment
[ca. 15 Nov. 1801]
A debt of 21,955,900. D bearing an interest of 1,310,401.50 is to be paid in 8. years, by eql. annl. paimts.
what is the annual paiment?
if the interest were uniform, it would be of 6. pr. cent wanting an insensible fraction.
| but | 6,481,700. | D. | bears an interest | of 8. | p. cent, | = | 518,536. |
| then | 15,474,200. | D. | must be at | 5 117⁄1000 | p. cent | = | 791,865.50 |
| 21,955,900. | 1,310,401.50 |
| in problems of this kind 4. things are material | the annuity | a. |
| rate of int. | r. | |
| time or no. of years. | t. | |
| amount to be paid | z. |
any three of these being given, the 4th. can be found. but1 from 2 only given, 2 cannt be found.
the present question divides itself into two. viz
what annual sum would pay 15,474,200. D @5 117⁄1000 p. cent so soon that
the same annual sum would pay 6,481,700. D @ 8 p. cent by the end of the 8th. year?
it is evident that here neither the time nor amount is fixed for either proposition.
in that form it is insoluble then. but we may solve it nearly enough for our purpose by assigning an uniform & equivalent interest, to wit, of 6. per cent to the whole, which gives us the time 8. years. and consequently we have then 3. things to wit the amount, rate, and time: required the 4th. which is the annual sum.
21,955,900. + 10,483,212. interest for 8. years at 6. p.c. makes the whole amt to be paid 32,439,112.
| then | z. | = | 32,439,112. |
| r | = | 1.06 | |
| t | = | 8. |
required a.
| the equation of the case is that a = | z × r–1 | or by Logarithms Log. a = Log. z + Log. r–1 – Log. rt–1 |
| rt–1 |
| Log. of 32,439,112 | is | 7.5110686 |
| Log. of r. or 1.06 | is | 0.0253059 |
| Log of r–1 or 0.06 | is | 8.7781513 |
Log of rt = Log. r × t = Log. r × 8 = 0.2024472 which is
| Log. of | 1.593848 | = | rt |
| –1. | |||
| .593848 | = | rt–1 |
Log. of rt–1 or .593848 = 9.7736753
the Logarithms being stated the operation of the theorem is
| Log. of | z. | 7.5110688 | ||
| + | Log. of | r–1 | 8.7781513 | |
| 16.2892201 | ||||
| – | Log. of | rt–1 | 9.7736753 | |
| gives Log. a. | 6.5155448 | which is the Log. of 3,277,516. Dol. the annual sum required. | ||
it is stated that 950,965. D. of the 21,955,900. have been paid out of the treasury, but without stopping the interest: but for how long the interest has been unstopped is not mentioned: consequently we cannot say how much of 950,965 D. will be absorbed by the interest of the remainder, which remainder alone with 8. years interest should have been deducted from 32,439,112. at the commencement of the operation. but ascertaining how much of the 950,965. is absorbed as interest for the residuum of that sum, the following operation of the rule of three will give the effect it should have on the annual sum 3,277,516. above stated. to wit
As 32,439,112 : 3,277,516 :: 950,965 – the portion of it absorbed by interest + 8. years int. on that2 : a 4th. number which will be the annual sum which the previous paiment of the 950,965 defalcates from the 3,277,516.
MS (NHi: Gallatin Papers); entirely in TJ’s hand; undated, but likely written after TJ received Document vi and before Gallatin wrote Document viii. PrC (DLC: TJ Papers, 118:20294).
TJ owned books that dealt with the problem of figuring an annuity. One of them, by his friend Richard Price, devoted a chapter to “Public Credit, and the National Debt” (Richard Price, Observations on Reversionary Payments, 2d ed. [London, 1772], 135–65; Francis Maseres, The Principles of the Doctrine of Life-Annuities [London, 1783]; , Nos. 3688, 3689).
1. TJ here canceled “in the present [case].”
2. TJ interlined “+ 8. years int. on that.”