Loan or Investment Formulas

Compound interest may work for you or against you. Whether you are taking a personal loan or making an investment, in any case it is the same set of formulas. This page gives you the formulas, shows where they come from and works through many examples. Excel books are also provided.

All formulas apply when payments are made at the end of each period, and please understand that the results are approximate. The names of the variables should be quite simple, but I explain them below.

The Formulas

Balance of loans after n payments have been derived
B (n) = A (1 + i) ^ n – (P / i) [(1 + i) ^ n – 1]
(For a savings account or other investment, simply change the first minus sign to a plus sign).

 Amount of the payment in a loan derivation
P = iA / [1 – (1 + i) ^ – N]

Number of payments in a loan derivation
N = -log (1-iA / P) / log (1 + i)

Derivation of the original loan amount
A = (P / i) [1 – (1 + i) ^ – N]
(The original loan amount is also called the present value of an annuity or present value of a payment stream.)

Amount of payment to reach a derivation of investment goal
P = iF / [(1 + i) ^ N – 1]

Number of payments to reach an investment goal derivation
N = log (1 + iA / P) / log (1 + i)

Interest rate deduction
i = 2u + 2v ^ 2 (N-1) [- 1/3 + (2N + 1) u / 9 – (2N + 1) (11N + 7) u ^ 2/135 + (2N + 1) ^ 2 (13N + 11) u ^ 3/405 – …]
where u = [(PN / A) – 1] / (N + 1)

But this problem is usually solved by iterative methods.

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