Loan Payoff Calculator

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Reviewed by: Dr. Emily White, Ph.D. in Economics
Dr. White is an expert in financial mathematics and compound interest calculations, providing assurance on the complex amortization logic used in this module.

Use the authoritative **Loan Payoff Calculator** to find any missing loan variable: Principal Amount, Annual Interest Rate, Loan Term (Years), or Monthly Payment. Simply enter any three values to instantly solve for the fourth and generate a full amortization analysis.

P: Principal Loan Amount ($)

R: Annual Interest Rate (%)

N: Loan Term (Years)

M: Monthly Payment ($)

Loan Payment Formula

The standard amortization formula for Monthly Payment (M) is:

$$ M = P \frac{i(1 + i)^n}{(1 + i)^n – 1} $$

Note: Calculations for P, R, or N require algebraic manipulation or iterative solving.

Formula Source: The Balance

Formula Variables

P (Principal): The initial amount of money borrowed.
R (Annual Rate): The nominal annual interest rate (e.g., 6.5%).
N (Term in Years): The length of time over which the loan must be repaid.
M (Monthly Payment): The fixed amount paid each month towards principal and interest.
i (Monthly Rate): R / 1200
n (Total Payments): N $\times$ 12

Related Calculators

What is a Loan Payoff Calculation?

A loan payoff calculation determines the fixed, periodic payment required to fully pay off a debt over a specified term, accounting for the interest rate. This calculation is a fundamental component of financial analysis and applies to mortgages, car loans, personal loans, and more. It uses the compound interest amortization formula, which ensures that early payments are weighted heavily toward interest, while later payments pay down more principal.

The ability of this calculator to perform **inverse calculations** (solving for Principal, Rate, or Term) is incredibly valuable. For instance, if you know your budget allows for a certain Monthly Payment (M), you can solve for the maximum Principal (P) you can afford. Similarly, if you want to pay off a loan faster, you can solve for the new Term (N) by inputting a higher Monthly Payment (M).

How to Calculate Loan Term (Example)

Let’s find the Loan Term (N) given $P = \$10,000$, $R = 5\%$, and $M = \$300$.

Step 1: Calculate Monthly Rate ($\mathbf{i}$)
Monthly Rate ($i$) = 5.0% / 12 / 100 = 0.0041667.
Step 2: Calculate Total Payments ($\mathbf{n}$) using Logarithms
The formula for $n$ is: $n = \frac{\ln(1 – \frac{P \times i}{M})}{\ln(1 + i)}$
Step 3: Substitute and Solve for $\mathbf{n}$
Substituting the values yields $n \approx 35.79$ months.
Step 4: Convert Total Payments ($\mathbf{n}$) to Years ($\mathbf{N}$)
Term (N) = $35.79 / 12 \approx 2.98$ years. The loan will be fully paid off in 36 months.

Frequently Asked Questions (FAQ)

What is Amortization?

Amortization is the process of paying off debt over time in regular installments. Each payment includes both principal and interest, but the ratio shifts over time: initially, more of the payment goes toward interest; later, more goes toward principal.

Why does the interest rate calculation take longer?

Unlike solving for Payment or Principal, the Interest Rate (R) is embedded in the amortization formula in a non-linear way, making it impossible to isolate algebraically. Therefore, the calculator must use an iterative approximation method (like the Newton-Raphson method) to find a highly accurate rate.

How does compounding frequency affect the payment?

Most consumer loans and mortgages compound monthly, which is the standard assumption here. If a loan compounded daily or annually, the Monthly Rate ($i$) component of the formula would need to be adjusted accordingly.

What happens if my Monthly Payment (M) is too low?

If $M$ is less than the monthly interest accrued ($P \times i$), the payment will not even cover the interest, and the loan balance will never decrease, leading to an infinite term. The calculator will flag this as an error.

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