Loan Payment Calculations

The payment formula

For a fixed-rate, fully amortizing loan: $PMT = P \cdot \dfrac{r(1 + r)^n}{(1 + r)^n - 1}$

where

• $P$ = principal (amount borrowed),
• $r$ = periodic rate (e.g. monthly if payments are monthly),
• $n$ = total number of payments,
• $PMT$ = fixed payment per period.

The payment is exactly the amount that reduces the balance to $0 on the last payment, with interest accrued on the outstanding balance each period.

Step-by-step solution

Setup: $P = 25{,}000$, annual rate 5.5% ⟹ $r = 0.055/12 \approx 0.0045833$, 5 years ⟹ $n = 60$ payments.

Step 1 — Compute $(1 + r)^n$: $(1.0045833)^{60} \approx 1.31560$

Step 2 — Numerator of the formula: $r \cdot (1 + r)^n = 0.0045833 \cdot 1.31560 \approx 0.006030$

Step 3 — Denominator: $(1 + r)^n - 1 \approx 0.31560$

Step 4 — Divide: $\dfrac{0.006030}{0.31560} \approx 0.019105$

Step 5 — Multiply by principal: $PMT = 25{,}000 \cdot 0.019105 \approx \boxed{477.53}$

The monthly payment is about $477.53.

Total cost

• Total paid: $60 \cdot 477.53 \approx 28{,}651.80$
• Total interest: $28{,}651.80 - 25{,}000 = 3{,}651.80$

So you pay ~$3,652 in interest for the 5-year, 5.5% loan.

Rearrangements

You can solve for different unknowns:

• Affordability (how much can I borrow for a given monthly payment?): $P = PMT \cdot \dfrac{(1 + r)^n - 1}{r(1 + r)^n}$
• Term (how long to pay off at a given PMT?): $n = -\dfrac{\ln(1 - Pr/PMT)}{\ln(1 + r)}$

The rate $r$ generally requires numerical solution (no closed form).

Intuition: the payment factor

The multiplier $\dfrac{r(1+r)^n}{(1+r)^n - 1}$ is the loan-amortization factor. It is bigger than $r$ (you pay interest and return principal) but approaches $r$ as $n \to \infty$ (for very long loans, nearly all of each payment is interest).

How rate and term trade off

At $25K:

• 3 years @ 5.5%: $PMT \approx$754/mo — less interest, high payment.
• 5 years @ 5.5%: $PMT \approx$478/mo — middle ground.
• 7 years @ 5.5%: $PMT \approx$359/mo — low payment, ~$5.2K interest.

Longer terms reduce the monthly pinch but balloon total interest.

Common mistakes

• Treating $r$ as the annual rate when payments are monthly. Divide by 12 first.
• Using $n$ in years instead of periods.
• Forgetting fees. The formula gives the raw principal-and-interest payment; taxes, insurance, origination fees are on top.

Try it in the visualization

Adjust principal, rate, and term sliders. The monthly payment updates in real time, alongside a total-interest bar. Toggle between 3-, 5-, and 7-year views to compare.