In the determination of the present worth of a sum of money at compound interest due a certain number of years hence, the following symbols will be used:
- A = the sum which placed at interest at the given rate will
- accumulate to the given amount in the given time.
- B = the given amount accumulated
- r = given rate of interest per period
- R = 1 + r
- n = the given number of periods
The present worth of B is manifestly A, the sum which, when placed at interest, amounts to B. At compound interest the amount to which A will accumulate in n periods is given by the formula:
- (1) B = A(1 + r)(1 + r) ... to n factors,
- which reduces to
- = ARⁿ,
- whence solving the equation for A
- B
- (2) A = ——, i.e., the present worth
- Rⁿ
- B
- = ——.
- Rⁿ
From this, the present worth of $1 is seen to be 1/Rⁿ .
Formulas for Annuities
An annuity is a given sum of money placed at interest usually at the end of each successive year or period, and allowed to accumulate at compound interest for a number of periods. For bond valuations it is necessary to know the amount of an annuity and the present worth of this amount. The following terminology will be used in the solution of the problem:
- A = the sum put at interest periodically
- B = the amount to which A accumulates
- r = the rate of interest per period
- R = 1 + r
- n = the number of periods
- P. W. = the present worth of the annuity
The first sum, A, placed at interest at the end of the first period will accumulate for n-1 periods and, according to formula (1) above, will amount to ARⁿ⁻¹. The second sum, A, placed at interest at the end of the second period will amount to ARⁿ⁻²; etc. The last sum, A, will not accumulate and so is worth just A.
The amount of the annuity is thus seen to be the sum of the amounts of the several periodic sums. Expressed as a formula: