- P = the principal sum to be redeemed at maturity
- n = the number of periods till redemption
- A = the amount paid into the sinking fund at the end of each period
- r = the rate per cent per period at which the moneys in the
- sinking fund are invested at compound interest
- R = 1 + r
Reference to [Chapter XV, page 271], gives the amount, A, of a sum of money put at compound interest at r% for a term of n years, as A(1 + r)ⁿ⁻¹. It is evident here that the sum, A, paid into the fund will accumulate for n-1 years and that each succeeding payment remains at interest one year less than the next preceding amount, the last amount earning no interest. Accordingly, the first A—we will denote it as A₁—will amount to A(1 + r)ⁿ⁻¹; A₂ will amount to A(1 + r)ⁿ⁻²; A₃ to A(1 + r)ⁿ⁻³; etc. The sum of all these amounts must be equal to P, the debt to be redeemed. Therefore, the equation may be formed:
A(1 + r)ⁿ⁻¹ + A(1 + r)ⁿ⁻² ... + A(1 + r) + A = P or
A(Rⁿ⁻¹ + Rⁿ⁻² ... + R + 1) = P, whence
| A |  | Rⁿ - 1 |  | = P and |
| R - 1 |
| A = | P(R - 1) | or | Pr | , |
| Rⁿ - 1 | Rⁿ - 1 |
which being interpreted means that the theoretical amount to be set aside at the end of each period can be found by multiplying the amount of the debt by the fraction
| r |
| Rⁿ - 1 |
If the annual payment is made into the fund at the beginning of each period instead of at the end, then A will be
| Pr |
| R(Rⁿ - 1) |