Loan Refinancing

Refinancing replaces a loan's remaining outstanding balance with a brand-new loan at a new rate. The old rate stops mattering the moment the balance is known; only the current balance carries forward into the new loan.

By the end you'll be able to compute a refinanced level payment from a mid-loan balance, and tell a balloon payment apart from a drop payment.

Predict: refinancing the remaining balance at a lower rate — does the new level payment go down? Drag the new rate and check.

A $10,000, 5-year loan at 8% is repaid by level payments of $2,504.56. Pick a year to refinance and a new rate below: the outstanding balance at that point becomes the principal of a fresh loan, re-amortized over whatever years remain. Watch the kink where the balance path bends — same balance, new slope.

Before: $2,504.56/yr at 8%. After refinancing $6,454.51 at 6%: $2,414.69/yr for the remaining 3 years.

Outstanding balance over time — old schedule, refinance point, new schedule
before refinance after refinance if never refinanced

Rounding the new payment to the nearest dollar leaves a small leftover once the last regular payment is made. Toggle how it's settled:

Rounded payments of $2,414/yr leave $2.21 outstanding after year 5. Balloon: the final payment becomes $2,414 + $2.21 = $2,416.21, paid on the same date as the last regular payment.

Refinancing has two moving pieces: find today's outstanding balance, then treat it as the principal of a fresh loan. Everything about the old loan — its original principal, its original rate — becomes irrelevant the instant the balance is known.

Applied

Picture a borrower two years into a 5-year loan who spots a lower rate elsewhere. The lender doesn't care what the original loan looked like — only what's still owed. Three steps: (1) compute the outstanding balance \(B_t\) at the refinance date, prospectively or retrospectively (see the amortization method); (2) that balance becomes the new loan's principal, \(L' = B_t\); (3) solve for the new level payment, \(R' = L'/a_{\overline{m}|i'}\), over whatever term \(m\) remains at the new rate \(i'\).

Worked example

A $10,000, 5-year loan at 8% has level payments of $2,504.56. After 2 payments the balance is $6,454.52. Refinancing the remaining 3 years at 6%: \(R' = 6454.52/a_{\overline{3}|6\%} = 6454.52/2.67301 = \) $2,414.68. The payment drops from $2,504.56 to $2,414.68 — the benefit of the lower rate.

Because payments are usually rounded to the cent (or dollar), a level schedule rarely clears a loan to exactly zero. Two devices settle the leftover so the loan closes exactly: a balloon payment enlarges the final scheduled payment to absorb the residual; a drop payment instead makes one smaller extra payment one period later, sized to \(B_{\text{last}} \cdot (1+i')\) — accumulated by one more period's interest because it's now overdue.

Your turn

A different loan: $8,000 over 4 years at 10% has level payments of $2,523.76. After 2 payments the balance (prospectively, \(B_2 = R \cdot a_{\overline{2}|10\%}\)) is $4,380.10. Refinance the remaining 2 years at 6%: \(a_{\overline{2}|6\%} = 1.83339\). Now finish it: \(R' = 4380.10 / 1.83339 = \) ____

Reveal the answer

\(R' = 4380.10/1.83339 = \) $2,389.07 — down from $2,523.76, again because the new rate (6%) is lower than the old one (10%). Try setting the demo's year slider to 2 and rate slider near 6% and compare the shape of the kink to this smaller loan's numbers.

More info — common refinancing pitfalls

Three mistakes show up often: refinancing off the original principal instead of the current outstanding balance (only the current balance is owed — the original amount is history); confusing a balloon (a larger final payment on the same date) with a drop payment (a smaller extra payment one period later); and forgetting to accumulate a leftover balance by \((1+i')\) when it's paid a period after the last regular payment — the leftover doesn't sit interest-free while it waits. See the Marcel Finan guide in Dive deeper for more worked balloon/drop problems.

Check your understanding

Question 1 of 4

You refinance a loan's remaining outstanding balance at a new rate \(i'\), keeping the same remaining term \(m\). If \(i'\) is lower than the old rate, what happens to the new level payment \(R'\)?

Question 2 of 4

A $1,000 loan at 10% annual is repaid by 2 level annual payments of $576.19. Using the retrospective method, what outstanding balance would you carry into a refinance after the first payment?

Question 3 of 4

After refinancing, rounded level payments leave a small positive balance once the last scheduled payment is made. Which pairing correctly distinguishes the two ways to clear it?

Question 4 of 4

A refinanced balance of $4,000 is amortized over the remaining 3 years at a new rate of 5% (\(a_{\overline{3}|}\) at 5% \(\approx 2.7232\)). What is the new level payment \(R'\)?

Recap

  • Refinancing swaps the remaining outstanding balance \(B_t\) — not the original principal — into a new loan at rate \(i'\): \(R' = B_t/a_{\overline{m}|i'}\).
  • The old rate stops mattering the moment the balance is known; only \(B_t\) carries forward.
  • Holding the remaining term fixed, a lower \(i'\) always produces a lower level payment, and a higher \(i'\) always produces a higher one.
  • A balloon payment enlarges the final scheduled payment to clear a rounding leftover on the same date; a drop payment clears it with a smaller extra payment one period later, accumulated by \((1+i')\).

Dive deeper

Sources

  • Refinancing, Balloon Payments, and Drop Payments