This Loan Amortization Calculator is based on the standard monthly compounding annuity formula used by banks and financial institutions worldwide. The calculations account for the time value of money to ensure accurate payment and balance projections.
Welcome to the **Loan Amortization Calculator**. This sophisticated tool allows you to solve for any missing variable in a fixed-rate loan: the Principal (P), the Monthly Payment (A), the Annual Rate (r), or the Time in Years (t). By providing any three of these four inputs, you can gain deep insights into your loan structure, future commitments, and total cost of borrowing.
Loan Amortization Calculator
Loan Amortization Formula
The core formula for Monthly Payment (A) is:
A = P × [ i(1 + i)n / ((1 + i)n – 1) ]
Where i is the monthly interest rate (r/12) and n is the total number of periods (t × 12).
1. Solve for Monthly Payment (A):
A = P × [ i(1 + i)n / ((1 + i)n – 1) ]
2. Solve for Principal (P):
P = A × [ ((1 + i)n – 1) / (i(1 + i)n) ]
3. Solve for Loan Term in Years (t):
t = ln( A / (A – P × i) ) / (12 × ln(1 + i))
(where i = r/12)
4. Solve for Annual Rate (r):
r = Iterative Numerical Approximation (e.g., Newton’s Method)
Formula Source: Investopedia – Amortization
Variables Explained
- P – Principal Loan Amount: The total amount of money borrowed.
- A – Monthly Payment: The fixed amount paid each month towards principal and interest.
- r – Annual Interest Rate: The annual interest rate (expressed as a decimal, e.g., 0.05).
- t – Loan Term in Years: The total length of the loan in years.
- i – Periodic Interest Rate: The monthly interest rate (r / 12).
- n – Number of Periods: The total number of payments (t × 12).
Related Calculators
What is Loan Amortization?
Loan amortization is the process of paying off debt over time in fixed installments. Each scheduled payment consists of two components: a portion that covers the accrued interest and a portion that reduces the outstanding principal balance. In the initial years of the loan, the interest portion is high, but as the principal decreases, a larger share of each payment goes toward the principal.
Amortization schedules show exactly how much of each payment goes to interest and how much goes to principal, allowing borrowers to see their debt reduce over the life of the loan. This systematic reduction ensures the loan is fully paid off by the end of the specified term, provided all payments are made on time.
The fixed monthly payment (A) is calculated based on the principal, the interest rate, and the loan term, ensuring a level payment stream that is predictable for the borrower.
How to Calculate Monthly Payment (Example)
Let’s calculate the Monthly Payment (A) for a $100,000 loan, 10-year term (t=10), at a 6% annual rate (r=0.06):
- Determine Periods (n) and Monthly Rate (i): $n = 10 \times 12 = 120$ periods. $i = 0.06 / 12 = 0.005$.
- Calculate the Compounding Factor: $(1 + i)^n = (1.005)^{120} \approx 1.8193967$.
- Calculate the Payment Factor: $\frac{i(1 + i)^n}{(1 + i)^n – 1} = \frac{0.005 \times 1.8193967}{1.8193967 – 1} \approx \frac{0.00909698}{0.8193967} \approx 0.011102$.
- Determine the Monthly Payment (A): $A = P \times \text{Factor} = \$100,000 \times 0.011102 \approx \$1,110.20$.
Frequently Asked Questions (FAQ)
The Principal (P) is the original amount of money you borrowed. The Monthly Payment (A) is the fixed sum you pay every month, which includes both interest and a principal reduction component.
What is the amortization period?The amortization period (or term, t) is the total length of time, usually expressed in years, over which the loan is scheduled to be fully repaid.
How does making extra payments affect amortization?Making extra payments directly reduces the principal balance. Since the interest is calculated on the remaining principal, this reduces the total interest paid and shortens the overall loan term.
Why is solving for the Interest Rate (r) difficult?The interest rate is embedded within exponential terms of the amortization formula, making it impossible to isolate algebraically. It requires iterative numerical methods (like bisection or Newton-Raphson) to approximate the solution.