Calculate Monthly Mortgage Payment

Finance & Interest 11th-12th Grade
PROBLEM
Craig decides to purchase a property that's been valued at $475,000. He has $80,000 available as a deposit, and will require a mortgage for the remaining amount. The bank offers him a 25-year mortgage at 2% interest. Calculate his monthly payment. Round your answer to the nearest cent. Do NOT round until you have calculated the final answer.

What This Problem Teaches

  • Mortgage amortization — understanding how fixed monthly payments cover both interest and principal reduction over time
  • Present value of annuities — recognizing that a mortgage payment stream has a present value equal to the loan amount
  • Compound interest in reverse — applying compound interest formulas to find payments rather than future values
  • Financial precision — maintaining accuracy through multi-step calculations before final rounding
  • Loan structure fundamentals — distinguishing between purchase price, down payment, and financed amount

Solution: The Amortization Formula

This mortgage problem requires the standard amortization formula, which calculates the fixed monthly payment needed to fully pay off a loan over a specified term.

Step 1 — Calculate the loan amount

The mortgage covers the purchase price minus the down payment:

Loan Amount = Purchase Price - Down PaymentLoan Amount = $475,000 - $80,000 = $395,000

Step 2 — Convert the annual interest rate to monthly

Mortgages compound monthly, so we need the monthly interest rate:

Monthly Rate = Annual Rate ÷ 12r = 0.02 ÷ 12 = 0.001666666...

Keep the full precision for now — we'll round only the final answer.

Step 3 — Calculate the total number of payments

A 25-year mortgage has:

Total Payments = Years × 12n = 25 × 12 = 300 payments

Step 4 — Apply the amortization formula

The monthly payment formula is:

M = P × [r(1 + r)ⁿ] ÷ [(1 + r)ⁿ - 1]

Where P = $395,000, r = 0.02/12, and n = 300.

Let's calculate the components step by step:

(1 + r)ⁿ = (1 + 0.02/12)³⁰⁰(1 + 0.001666666...)³⁰⁰ = 1.64361946...
Numerator = r(1 + r)ⁿ= (0.02/12) × 1.64361946...= 0.002739365...
Denominator = (1 + r)ⁿ - 1= 1.64361946... - 1= 0.64361946...

Step 5 — Complete the calculation

M = $395,000 × (0.002739365... ÷ 0.64361946...)M = $395,000 × 0.004256725...M = $1,681.406...

Rounding to the nearest cent:

Monthly Payment = $1,681.41

Solution: Method 2 — Present Value Factor Approach

Instead of computing the exponential directly, we can use the Present Value Interest Factor for an Annuity (PVIFA), which represents how much $1 per period is worth today.

Step 1 — Calculate PVIFA

The PVIFA formula is the denominator of our payment calculation:

PVIFA = [(1 + r)ⁿ - 1] ÷ [r(1 + r)ⁿ]PVIFA = [1.64361946... - 1] ÷ [0.002739365...]PVIFA = 0.64361946... ÷ 0.002739365...PVIFA = 235.0327...

Step 2 — Find the payment

The monthly payment is simply the loan amount divided by PVIFA:

Monthly Payment = Loan Amount ÷ PVIFAM = $395,000 ÷ 235.0327...M = $1,681.41

This method shows the economic meaning clearly: we're finding what payment, when discounted back 300 months at 2%/12 per month, equals exactly $395,000 today.

Verification

Let's verify our answer makes sense by checking the first month's payment breakdown:

Month 1 Interest and Principal

Interest = Outstanding Balance × Monthly RateInterest = $395,000 × (0.02/12) = $658.33
Principal Payment = Total Payment - InterestPrincipal Payment = $1,681.41 - $658.33 = $1,023.08

Verification check: The payment covers the month's interest ($658.33) plus reduces the principal by $1,023.08. The new balance would be $393,976.92. ✓

Reasonableness Check

For a $395,000 loan at 2% over 25 years:

  • Interest-only payment would be: $395,000 × (0.02/12) = $658.33
  • Our payment of $1,681.41 is reasonably higher, allowing principal paydown
  • Total payments: $1,681.41 × 300 = $504,423, which includes $109,423 in interest over 25 years — reasonable for this rate and term

Common Pitfalls

✗ Using the annual rate instead of monthly rate

If you mistakenly use r = 0.02 instead of r = 0.02/12, you'd get a payment around $17,500 per month — clearly wrong. Always convert annual rates to the payment frequency.

✗ Calculating payments on the full purchase price

Some students use $475,000 instead of $395,000 as the loan amount. Remember: the bank only lends the amount after your down payment. The mortgage doesn't cover money you're paying upfront.

✗ Rounding intermediate calculations

If you round the monthly rate to 0.0017 or the exponential to 1.64, your final answer will be off by several dollars. Financial calculations require precision until the final step.

✗ Confusing loan years with total payments

Using n = 25 instead of n = 300 in the formula. The exponent must match the payment frequency — monthly payments means monthly periods.

The Math Beneath the Surface

The amortization formula comes from the present value of an annuity. Each monthly payment M is worth less today due to interest, and we need the sum of all discounted payments to equal the loan amount:

P = M/(1+r) + M/(1+r)² + M/(1+r)³ + ... + M/(1+r)ⁿ

This geometric series simplifies to:

P = M × [(1+r)ⁿ - 1] / [r(1+r)ⁿ]

Solving for M gives us the payment formula. The beauty is that this works for any loan: auto loans, personal loans, or mortgages — just change the rate and term.

Financial insight: Early payments are mostly interest because the outstanding balance is large. As you pay down principal, more of each payment goes toward reducing the debt. This is why mortgage amortization schedules show the interest portion decreasing over time.

Real-World Context

This exact calculation determines affordability for millions of homebuyers. Lenders use it to:

  • Qualification ratios — Your total monthly debt payments (including this mortgage) typically shouldn't exceed 36-43% of gross income
  • Amortization schedules — Banks generate payment calendars showing exactly how each month's payment splits between interest and principal
  • Rate shopping — A 0.25% rate difference on a $400,000 mortgage changes payments by about $60/month, or $18,000 over 25 years

The same mathematical structure applies to business equipment financing, where companies need predictable monthly payments for cash flow planning.

What If?

1
Higher Interest Rate
If the bank instead offers a 3.5% annual interest rate on the same 25-year mortgage with the same $80,000 deposit, what is Craig's new monthly payment? Round to nearest cent.
Step 1 — Same loan amount

Loan amount remains $475,000 - $80,000 = $395,000

Step 2 — New monthly rate

r = 0.035/12 = 0.002916667

Step 3 — Apply formula

(1 + 0.002916667)^300 = 2.40794
M = $395,000 × [0.002916667 × 2.40794] / [2.40794 - 1]
M = $395,000 × 0.005020 = $1,982.85

Verification

First month interest: $395,000 × 0.002916667 = $1,152.08
Principal payment: $1,982.85 - $1,152.08 = $830.77 ✓

Answer: $1,982.85

2
Reverse the Unknown
Craig can only afford a monthly payment of $1,500 on a 25-year mortgage at 2% interest. He still has $80,000 as a deposit. What is the maximum property value he can afford? Round to nearest dollar.
Step 1 — Calculate affordable loan amount

Use the payment formula in reverse: P = M × PVIFA
With r = 0.02/12 and n = 300:
PVIFA = [(1.02/12)^300 - 1] / [0.02/12 × (1.02/12)^300] = 235.0327

Step 2 — Find loan amount

P = $1,500 × 235.0327 = $352,549

Step 3 — Add down payment

Maximum Property Value = Loan + Down Payment
= $352,549 + $80,000 = $432,549

Verification

Check: Monthly payment on $352,549 loan:
M = $352,549 ÷ 235.0327 = $1,500 ✓

Answer: $432,549

3
Shorter Loan Term
If Craig chooses a 15-year mortgage (instead of 25 years) at 2% interest with the same $80,000 deposit, how much more does he pay per month? Round to nearest cent.
Step 1 — New parameters

Loan amount: $395,000 (same)
Monthly rate: 0.02/12 (same)
Payments: n = 15 × 12 = 180

Step 2 — Calculate new PVIFA

(1 + 0.02/12)^180 = 1.34489
PVIFA = (1.34489 - 1) / [0.02/12 × 1.34489] = 154.2915

Step 3 — Find new payment

M = $395,000 ÷ 154.2915 = $2,560.89

Step 4 — Calculate difference

Difference = $2,560.89 - $1,681.41 = $879.48

Verification

Total 15-year payments: $2,560.89 × 180 = $460,960
Total 25-year payments: $1,681.41 × 300 = $504,423
Shorter term saves $43,463 in total interest ✓

Answer: $879.48 more per month

4
Biweekly Payments
Instead of monthly payments, Craig makes biweekly payments (26 payments per year) on the 25-year, 2% mortgage. Each biweekly payment is half of the monthly payment. How much total interest does he save over the full term compared to monthly payments?
Step 1 — Biweekly payment amount

Biweekly payment = $1,681.41 ÷ 2 = $840.71
Annual payments = $840.71 × 26 = $21,858.46

Step 2 — Find payoff time

With 26 payments/year, the biweekly rate = 0.02 ÷ 26 = 0.000769
Using loan balance formula: n = ln(1 + P×r/M) ÷ ln(1+r)
n = ln(1 + 395000×0.000769/840.71) ÷ ln(1.000769) = 570 payments

Step 3 — Total biweekly payments

Total paid = $840.71 × 570 = $479,205
Interest paid = $479,205 - $395,000 = $84,205

Step 4 — Compare to monthly

Monthly total interest = ($1,681.41 × 300) - $395,000 = $109,423
Interest savings = $109,423 - $84,205 = $25,218

Verification

Biweekly pays off in 570 ÷ 26 = 21.9 years vs 25 years
Extra principal each year = $21,858 - ($1,681×12) = $1,686 ✓

Answer: $25,218 interest savings

Frequently Asked Questions

How do you calculate monthly mortgage payments? +
Use the amortization formula: M = P[r(1+r)^n]/[(1+r)^n - 1], where P is the loan amount, r is the monthly interest rate (annual rate ÷ 12), and n is total payments (years × 12). In this problem, P = $395,000, r = 0.02/12 = 0.001667, and n = 300 payments, giving M = $1,681.41.
What's the difference between purchase price and loan amount in a mortgage? +
The loan amount equals the purchase price minus the down payment. Lenders finance only the remaining balance after your deposit. Here, the $475,000 property with an $80,000 down payment requires a mortgage of $395,000.
Why convert annual interest rate to monthly for mortgage calculations? +
Mortgages compound monthly, not annually. The monthly rate is the annual rate divided by 12. A 2% annual rate becomes 0.02/12 = 0.001667 per month. This monthly rate is what actually applies to your outstanding balance each month.
DJ

Dr. Neven Jurkovic

Mathematics Professor & Financial Modeling Expert

NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-11-05

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