Mortgage Payment Calculations

The setup

A mortgage is a long-term amortized loan secured by real estate. Each monthly payment covers interest on the current balance plus a portion of principal — the split shifts dramatically over time.

The formula (same as any amortizing loan)

$PMT = P \cdot \dfrac{r(1 + r)^n}{(1 + r)^n - 1}$

Step-by-step solution

Step 1 — Compute the loan principal.

Down payment: $20\%$ of $300{,}000 = 60{,}000$. Loan amount: $P = 300{,}000 - 60{,}000 = 240{,}000$

Step 2 — Monthly rate and term.

Annual 3.5% → monthly $r = 0.035/12 \approx 0.002917$. 30 years → $n = 360$.

Step 3 — Growth factor. $(1.002917)^{360} \approx 2.8536$

Step 4 — Payment. $PMT = 240{,}000 \cdot \dfrac{0.002917 \cdot 2.8536}{2.8536 - 1} = 240{,}000 \cdot \dfrac{0.008324}{1.8536} \approx 240{,}000 \cdot 0.004491$ $\boxed{PMT \approx 1{,}077.71/\text{month}}$

Step 5 — Lifetime total. $\text{Total paid} = 360 \cdot 1{,}077.71 \approx 387{,}977$ $\text{Total interest} = 387{,}977 - 240{,}000 \approx 147{,}977$

You pay about $148K in interest for the $240K loan — roughly 62% extra.

Amortization milestones

• Month 1: Interest $240{,}000 \cdot 0.002917 = 700.00$; Principal $1{,}077.71 - 700 = 377.71$.
• Month 180 (halfway in time): Outstanding balance \approx \153{,}500$. Interest$\approx 447.75$; Principal$\approx 629.96$. At this point you've still only paid off about 36% of the principal.
• Month 300: Outstanding balance \approx \73{,}500$. Interest$\approx 214$; Principal$\approx 864$.
• Month 360 (last payment): Interest $\approx 3.14$; Principal $\approx 1{,}074.57$.

Total housing cost (PITI)

Full monthly housing cost is usually called PITI:

• Principal & Interest → $1{,}077.71 (computed above)
• Taxes → property tax (varies, often 1–2% of home value annually)
• Insurance → homeowners insurance

For a $300K house with ~1.2$100/mo insurance: extra ~400/mo → true monthly cost closer to **\1,500**.

Prepayment impact

Adding just $200/month toward principal on this mortgage:

• Shortens the term by ~6.5 years.
• Saves about $40K in interest.

The early-payment interest savings are outsized because the extra principal compounds in reverse against a long future of interest.

Why rate matters enormously

Same $240K, 30-year loan:

• 3.0%: PMT $\approx 1{,}011.85; total interest$\approx 124{,}267$.
• 3.5%: PMT $\approx 1{,}077.71; total interest$\approx 147{,}977$ (our case).
• 5.0%: PMT $\approx 1{,}288.37; total interest$\approx 223{,}814$.
• 7.0%: PMT $\approx 1{,}596.73; total interest$\approx 334{,}822$.

A rate jump from 3.5% to 7% more than doubles the total interest paid.

Common mistakes

• Forgetting to subtract the down payment from the home price.
• Using the annual rate without dividing by 12.
• Comparing PMT alone when evaluating affordability — always include taxes and insurance.
• Assuming "30-year mortgage" means 30 years of interest. It means 30 years of payments that amortize down to zero.

Try it in the visualization

Stacked-area chart shows interest (warm) and principal (cool) portions of each monthly payment over the full 360-month timeline, with the outstanding-balance line curving down. Toggle a "with extra payment" overlay.