Monthly Loan Payment Calculator
Solve the classic amortising-loan formula M = P · r(1+r)ⁿ / ((1+r)ⁿ − 1) for any of its four unknowns — monthly payment, principal, monthly rate, or term — with step-by-step math, a live amortisation schematic, and companion metrics including total interest, total cost, and the amortisation split.
Each payment covers interest plus principal
Every monthly payment has two components: the interest owed for that month (computed on the remaining balance) and the principal repayment (whatever is left over). Because interest is charged on the outstanding balance, and that balance shrinks over time, the interest portion of each payment declines while the principal portion grows. Early in a 30-year mortgage, almost the entire payment is interest; late in the loan, almost all of it is principal. This is called amortisation, and the formula is designed so that after exactly n payments the balance is precisely zero.
The annuity factor concentrates the cost
The term (1+r)ⁿ / ((1+r)ⁿ − 1) is the "annuity factor" — it converts a lump-sum principal into an equivalent stream of equal payments. As n grows large, (1+r)ⁿ dominates and the factor approaches r; so the payment approaches P·r, the interest-only payment. This is why long-term loans have much lower monthly payments but never pay off the principal: a hypothetical infinitely-long mortgage would require only P·r per month and would never amortise. Any finite term adds a small "principal kicker" on top of the interest-only payment.
Small rate changes have big effects
Because r appears both linearly and inside the (1+r)ⁿ factor, small rate changes produce large payment changes over long horizons. On a $200,000 30-year mortgage, going from 6 % to 7 % raises the monthly payment from $1,199 to $1,331 — an 11 % increase, or about $158/month. Over 30 years that is $56,880 of extra interest. This is why borrowers refinance when rates fall by even 0.5 %. The sensitivity is strongest for long loans; on a 5-year auto loan, a 1 % rate change moves the payment by only about 3 %.
Term length and total cost trade off
Longer terms lower the monthly payment but raise the total interest dramatically. On a $200,000 loan at 6 %, going from 15 to 30 years drops the monthly payment from $1,687 to $1,199 (−29 %) but raises the total interest from $103,788 to $231,676 (+123 %). The ratio isn't linear: 30-year interest is about 2.2× the 15-year interest on the same principal. Borrowers must weigh the cash-flow relief of the longer term against the higher lifetime cost — usually by comparing the "effective" interest rate or the breakeven point at which the extra interest paid exceeds the value of the extra cash flow held.
How to Use This Calculator
The loan payment formula links four quantities: M, P, r, and n. Pick which one you want the tool to solve, then fill in the other three. Rate is in percent per year, term in years, money in dollars.
1. Choose what to solve for
Pick M (monthly payment), P (principal), r (annual rate), or n (term in years) from the toggle. The calculator rearranges the formula for whichever quantity you select and greys out that field.
2. Enter the loan amount
Use the slider, the number input, or the presets ($10k to $1M). For a mortgage, this is the price minus the down payment. For an auto loan, it is the negotiated price minus trade-in.
3. Enter the annual rate
Enter the APR as a percentage per year. Presets range from 2 % (subsidised auto) to 15 % (credit-card-level). The calculator converts to a monthly rate (r/12) internally when applying the formula.
4. Enter the term
Term is in years. Common values: 3–6 yr for an auto loan, 10 yr for a personal loan, 15 or 30 yr for a mortgage, up to 40 yr for some international mortgages.
5. Read the result and verdict
The result card shows the solved value plus derived metrics — the total interest, total cost, effective APR, and a colour-coded verdict classifying the rate level from low to very high.
6. Check the schematic and chart
The schematic shows the P + I stacked bar and the declining balance curve over the term. The result-card chart plots remaining balance and cumulative interest paid over time.
7. Switch to Advanced for depth
Advanced mode adds a per-year principal vs interest split chart, a sensitivity panel showing how M changes with P, r, and n, and companion metrics including the interest-to-principal ratio and the interest saved by a shorter term.
What This Tool Offers
Built around the classic amortising-loan formula — the standard way to compute a fixed monthly payment.
Four-way solving
Solve for monthly payment, principal, rate, or term. Solving for rate uses numerical search; the other three are closed-form.
Live amortisation schematic
A stacked P + I cost bar and a declining balance curve over the loan term, with the current payoff point marked.
Rate-class verdict
Each result is classified by the effective annual rate: low (under 4 %), moderate (4–8 %), high (8–15 %), very high (over 15 %), with a colour-coded badge.
Dual comparison charts
Balance and cumulative interest over time, plus an advanced per-year principal-vs-interest split bar chart.
Companion loan metrics
Total interest, total cost, interest-to-principal ratio, effective annual rate, interest per dollar borrowed, and monthly interest on first payment.
Reference tables
Typical rates by loan product, a payment lookup grid across rates and terms, and a full variable reference table.
Reference Tables
Typical rates by loan product, a monthly payment lookup grid, and the loan payment variable reference.
Typical Rates by Loan Product
Representative annual percentage rates (APR). Actual rates vary by credit score, collateral, term, and market conditions.
| Product | Typical APR | Typical term | Notes |
|---|---|---|---|
| 30-year fixed mortgage | 6–7.5 % | 30 years | Most common US mortgage |
| 15-year fixed mortgage | 5–6.5 % | 15 years | Lower rate, higher payment, less total interest |
| 5/1 ARM (initial rate) | 5–6.5 % | 30 years (adjusts after 5) | Rate risk after fixed period |
| Auto loan (new, 60 mo) | 4–8 % | 5 years | Best rates for excellent credit |
| Auto loan (used, 72 mo) | 6–12 % | 6 years | Higher rate on used cars |
| Personal loan (unsecured) | 8–20 % | 3–7 years | No collateral |
| Federal student loan (undergrad) | 5–7 % | 10–25 years | Subsidised during study |
| Private student loan | 6–14 % | 5–20 years | Credit-based pricing |
| Credit card (balance) | 18–30 % | Revolving | Not amortising — use for reference only |
| Home equity loan | 7–10 % | 5–30 years | Second mortgage |
| HELOC (initial draw) | 7–9 % | Revolving + 10–20 yr repayment | Variable rate |
| Business term loan (SBA) | 7–11 % | 5–25 years | Government-guaranteed |
Monthly Payment Lookup — $200,000 loan
Monthly payments for a $200,000 fixed-rate loan at different rates and terms. Values in dollars per month.
| Term \ Rate | 3 % | 4 % | 5 % | 6 % | 7 % | 8 % |
|---|---|---|---|---|---|---|
| 5 years | 3,593 | 3,683 | 3,774 | 3,867 | 3,961 | 4,056 |
| 10 years | 1,931 | 2,025 | 2,121 | 2,220 | 2,322 | 2,426 |
| 15 years | 1,381 | 1,479 | 1,581 | 1,687 | 1,798 | 1,912 |
| 20 years | 1,109 | 1,212 | 1,320 | 1,433 | 1,551 | 1,673 |
| 25 years | 948 | 1,056 | 1,169 | 1,289 | 1,414 | 1,543 |
| 30 years | 843 | 955 | 1,074 | 1,199 | 1,331 | 1,468 |
| 40 years | 716 | 836 | 965 | 1,101 | 1,245 | 1,395 |
Total Interest Cost — $200,000 loan
Total interest paid over the full term. Values in thousands of dollars.
| Term \ Rate | 3 % | 4 % | 5 % | 6 % | 7 % | 8 % |
|---|---|---|---|---|---|---|
| 10 years | 31.7k | 43.0k | 54.5k | 66.4k | 78.6k | 91.1k |
| 15 years | 48.6k | 66.3k | 84.6k | 103.8k | 123.6k | 144.1k |
| 20 years | 66.1k | 90.9k | 116.8k | 143.9k | 172.3k | 201.5k |
| 30 years | 103.6k | 143.7k | 186.5k | 231.7k | 279.0k | 328.6k |
| 40 years | 143.7k | 201.2k | 263.2k | 328.6k | 397.6k | 469.6k |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| M | Monthly payment | 100 – 20,000 | $ / month |
| P | Principal (amount borrowed) | 1,000 – 5,000,000 | $ |
| r | Annual interest rate (APR) | 2 – 25 | % per year |
| r/12 | Monthly interest rate | 0.0017 – 0.021 | decimal |
| n | Loan term in years | 1 – 40 | years |
| n·12 | Total number of monthly payments | 12 – 480 | count |
| A | Total amount paid = M × n × 12 | P + I | $ |
| I | Total interest = A − P | 0.05 – 3 × P | $ |
What Does M = P · r(1+r)ⁿ / ((1+r)ⁿ − 1) Measure?
The loan payment formula computes the fixed monthly payment required to fully amortise a loan over a specified number of months. It is the mathematical heart of every mortgage, auto loan, personal loan, and student loan in the world. Given three inputs — the amount borrowed, the interest rate, and the term — it produces the single monthly payment that will pay off the loan exactly at the end of the term, with the balance reaching zero after the final payment. The formula balances two opposing forces: the lender wants to be compensated for the time value of money, and the borrower wants predictable, level payments. The result is a stream of equal monthly payments whose internal split between interest and principal shifts gradually over the loan's life.
Monthly Loan Payment Formula
The defining relationship is:
where r is the MONTHLY interest rate (annual rate ÷ 12 ÷ 100) and n is the TOTAL number of monthly payments (years × 12). Rearranged to solve for the other three unknowns:
n = ln( M / (M − P·r) ) / ln( 1 + r )
r = numerical solution (no closed form)
The total amount paid and total interest follow directly:
Total interest I = A − P = M × n × 12 − P
The amortisation schedule — the split between interest and principal in each payment — is computed recursively. Let Bk be the balance after k payments. Then:
Interestₖ = Bk−1 × r
Principalₖ = M − Interestₖ
Bₖ = Bk−1 − Principalₖ
Loan Payment Variables
| Symbol | Meaning | Notes |
|---|---|---|
| M | Monthly payment | The constant amount paid every month |
| P | Principal | The amount borrowed at origination |
| r | Monthly interest rate | Annual rate ÷ 12 ÷ 100; expressed as a decimal in the formula |
| n | Number of monthly payments | Years × 12; the total number of payments over the loan life |
| A | Total amount paid | A = M × n; the sum of all payments |
| I | Total interest | I = A − P; the total cost of borrowing |
| Bₖ | Balance after k payments | Shrinks monotonically to zero at k = n |
How Each Variable Affects the Monthly Payment
The monthly payment is linear in P — doubling the loan amount doubles the payment. But it is highly nonlinear in r and n. Here are the three derivatives evaluated at typical values (P = $200,000, r = 0.5 % per month, n = 360):
- ∂M/∂P ≈ $0.006 — each extra dollar borrowed adds about 0.6 cents to the monthly payment.
- ∂M/∂r ≈ $48,000 — each 1 % (0.01) of monthly rate adds about $480 per month. Equivalently, each 0.5 % of annual rate adds about $200/month.
- ∂M/∂n ≈ −$2.50 — each extra month reduces the payment by about $2.50, but adds $1,199 of extra interest.
In practice the rate sensitivity dominates for long-term loans. On a 30-year $200k mortgage, a 0.5 % annual rate change moves the monthly payment by about $65 — roughly 5 % — which is why refinancing decisions hinge on rate movements of that magnitude. On short-term loans (auto, personal), rate sensitivity is much smaller in absolute terms but can still matter for the total cost.
Loan Payment Calculation Examples
r = 0.06 / 12 = 0.005; n = 30 × 12 = 360
M = 200,000 × 0.005 × (1.005)³⁶⁰ / ((1.005)³⁶⁰ − 1) = $1,199.10
n = 180; M = 200,000 × 0.005 × (1.005)¹⁸⁰ / ((1.005)¹⁸⁰ − 1) = $1,687.71
r = 0.05/12 = 0.0041667; n = 60; M = 30,000 × 0.0041667 × 1.0041667⁶⁰ / (1.0041667⁶⁰ − 1) = $566.14
P = 1,500 × ((1.005)³⁶⁰ − 1) / (0.005 × (1.005)³⁶⁰) = $250,187
n = ln(1,500 / (1,500 − 200,000 × 0.005)) / ln(1.005) ≈ 220 months ≈ 18.3 years
A = 1,199.10 × 360 = $431,676; I = A − P = $231,676
Frequently Asked Questions
Because interest is calculated on the outstanding balance, and at the start of the loan the balance is at its largest. In month 1 of a $200,000 mortgage at 6 %, the interest owed is 200,000 × 0.005 = $1,000. Since the payment is $1,199, only $199 goes to principal. By month 180, the balance has fallen to about $160,000, so the interest is $800 and principal is $399. By month 359, the balance is tiny and nearly the entire payment is principal. The crossover point — where principal exceeds interest — occurs around month 230 of a 30-year loan, about 64 % of the way through. This "back-loaded" structure is why paying extra early in the loan has an outsized effect on total interest: every extra dollar paid early reduces the balance on which future interest accrues.
It depends strongly on the loan size and term. On a $200,000 30-year loan, a 1 % rate increase from 6 % to 7 % raises the monthly payment from $1,199 to $1,331 — a $132/month increase, or 11 %. Over 30 years, the total interest rises from $231,676 to $279,017 — an extra $47,341. On a $30,000 5-year auto loan, the same 1 % increase raises the payment from $566 to $580 — a $14/month increase, or 2.5 % — and the total interest rises from $3,968 to $4,820 — an extra $852. The lesson: rate sensitivity scales with the loan term. Short-term loans are relatively insensitive; long-term loans are highly sensitive and worth refinancing at even small rate reductions.
The interest rate is the nominal rate charged on the loan balance. The APR (Annual Percentage Rate) is a broader measure that includes the interest rate PLUS certain lender fees — origination fees, points, mortgage insurance premiums, and some closing costs — expressed as an equivalent annual rate. By law in the US, lenders must disclose the APR under the Truth in Lending Act, so borrowers can compare loans apples-to-apples. If two loans have the same interest rate but different fees, the loan with higher fees will have a higher APR, correctly indicating that it costs more in total. For simple shopping, the APR is the more reliable number to compare. Note that this calculator uses the "interest rate" (not the APR) in the payment formula, because the payment itself depends only on the interest rate — the fees are a separate cost paid upfront.
Neither is universally better — it depends on your cash flow and investing alternatives. The 15-year has a lower interest rate (typically 0.5–1 % lower), a much higher monthly payment (about 40 % more), and far less total interest (about 45 % less). The 30-year has a lower monthly payment, freeing cash for investment, emergencies, or accelerated payments if you choose. The math: on a $200,000 loan at 6 %, the 30-year costs $231,676 in interest; the 15-year at 5.5 % costs $98,914 — a saving of $132,762 over the loan's life, in exchange for a monthly payment $488 higher. If you invest the $488/month difference at a 7 % return over 15 years, you would accumulate about $155,000 — more than the interest saved, after accounting for taxes on the investment. So the decision depends on your investing discipline, tax situation, and risk tolerance. Many borrowers choose the 30-year for flexibility and then make extra principal payments to accelerate payoff when cash allows.
Extra principal payments have a leveraged effect because they reduce the balance on which all future interest accrues. On a $200,000 30-year mortgage at 6 %, an extra $100/month from the start shortens the loan to about 25.5 years and saves roughly $54,000 in interest. An extra $500/month shortens it to about 18 years and saves roughly $110,000. The earlier the extra payments come, the bigger the impact: $100/month in year 1 saves far more than $100/month starting in year 20, because the early dollars reduce the interest base for many more months. To model extra payments precisely, you would need to modify the amortisation schedule and re-solve for the payoff date. This calculator does not include extra payments, but the amortisation logic is straightforward to extend.
The formula computes principal + interest only — the "P&I" portion of a mortgage payment. Real loans typically include several other costs: (1) Property taxes — usually escrowed monthly at 1–3 % of the property value per year. (2) Homeowner's insurance — about 0.3–1 % of property value per year. (3) PMI (Private Mortgage Insurance) — required if the down payment is under 20 %, typically 0.3–1.5 % of the loan per year. (4) HOA fees — for condos and planned communities. (5) Closing costs — 2–5 % of the loan amount, paid upfront at origination. (6) Prepaid interest and escrow funding — due at closing. Together, these can add 30–50 % to the monthly cost compared to P&I alone. When shopping for a mortgage, always ask for the "full PITI" (principal, interest, taxes, insurance) number, and compare that to your monthly budget — not just the P&I portion.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| M | Monthly payment | currency |
| P | Loan principal | currency |
| r | Monthly interest rate (annual rate/12) | |
| n | Total number of monthly payments |
What it means
This formula calculates the fixed monthly payment for a fully amortizing loan (e.g., mortgage, car loan) where the interest rate is constant and the loan is paid off over a fixed term. The payment is derived from the present value of an annuity. It ensures that each payment covers interest and a portion of principal, resulting in a zero balance after n payments. This calculation is essential for borrowers to understand affordability and for lenders to structure loans. It is also used in reverse to determine the maximum loan amount for a given payment. Understanding this formula is fundamental for personal financial planning and for real estate investment analysis.
Worked example
Loan Monthly Payment – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| P | 300000 |
| Annual Rate | 6.5% |
| n (months) | 360 |
| Parameter | Value |
|---|---|
| P | 25000 |
| Annual Rate | 7% |
| n | 60 |
Common mistakes
- Monthly payment M: The fixed payment amount for each period.
- Principal P: The loan amount.
- Monthly interest rate r: Annual rate divided by 12 – in decimal form.
- Number of payments n: Total number of monthly payments (loan term in years × 12).
- Sign: Payments are typically outflows – but the formula gives the positive payment amount.
- Ensure r and n are consistent (both monthly).
Applications
The loan/mortgage monthly payment formula, M = P·[r(1+r)^n]/[(1+r)^n − 1], determines the fixed periodic payment required to fully amortise a loan over a given term. This is the basis for most home mortgages, auto loans, and student loans. Banks, brokers, and consumers use it to understand affordability, to compare loan offers, and to plan budgets. By adjusting the interest rate, term, or principal, borrowers can see the effect on monthly payments. This formula also helps in refinancing decisions and in evaluating the total cost of borrowing. Understanding it is essential for personal financial management and for professionals in banking, real estate, and financial planning.
- Mortgage payment calculation for home buyers
- Auto loan and personal loan budgeting
- Loan comparison and refinancing analysis
- Amortisation schedule preparation
- Financial literacy and consumer protection education
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