Refinancing and Loan Restructuring

A homeowner refinances a 6% mortgage at 5% but stretches the term three years longer. The lower rate looks like a win, but the total interest barely moves. Every restructure is a tradeoff between the payment and the lifetime cost.

Refinancing replaces the remaining balance of one loan with a new loan at a new rate and term. The balance is the only link between the two phases:

$$B_t=R\cdot a_{\overline{n-t|i}},R^{\prime }=\frac{B_t}{a_{\overline{n^{\prime }|i^{\prime }}}}$$

Stage 1: find the original payment and the balance at the refinance date. Stage 2: load that balance as the new principal, apply the new rate and term, and solve for the new payment.

HIGH-FREQUENCY: The refinanced principal is the balance $B_t$, never the original $L$. Using $L$ is the single most common refinancing error.

Interest is always total payments minus principal. Across two phases of $t$ then $n^{\prime }$ payments:

Common mistakes

• Refinancing on the original principal. The new loan is built on the balance $B_t$, not $L$. Using $200,000 instead of the $163,919 balance overstates the new payment badly. Trap: a payment far too high.
• Counting the wrong number of phase-1 payments. A refinance after 8 years on an annual loan is 8 payments; on a monthly loan it is 96. The remaining term is measured from there.
• Double-counting interest. Total interest is $tR+n^{\prime }{R}^{\prime }-L$, subtracting the principal once. Subtracting it from each phase double-counts.

Bottom line

• Refinance in two stages: compute the outstanding balance $B_t$, then treat it as a brand-new loan: $R^{\prime }=B_t/a_{\overline{n^{\prime }|i^{\prime }}}$.
• Total interest across a restructure = $tR+n^{\prime }{R}^{\prime }-L$, subtracting the original principal only once.
• Interest-only period: pay $iL$ each period (balance stays $L$), then amortize $L$ over the remaining term.
• Lower payment is not lower cost. A longer term cuts the payment but usually raises total interest.

Exam shortcut

Draw the timeline: original start, refinance date, new maturity. Compute the balance immediately; every later number depends on it. For a comparison, line up payment against total interest, they often point in opposite directions. "Balance, then build" for refinancing. "Lower payment, longer pain" for term extensions. "Interest-only is interest-more" over the life of the loan.

The full lesson (about 2,139 words, 14 min read) adds 3 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 3a
• 3b

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