Loan Calculations and Refinancing

A company refinances a $200,000 loan after 6 years, the new payment, total interest, and principal in the 30th payment all flow from the same amortization engine.

$$L = R \cdot a_{\overline{n\vert i}}$$

For $R$: $R = L/a_{\overline{n\vert}}$.

For $L$: $L = R \cdot a_{\overline{n\vert}}$.

$$n = \frac{-\ln(1 - iL/R)}{\ln(1+i)}$$

For $i$: no closed form. Iterate or use answer choices.

HIGH-FREQUENCY: Problems giving four parameters and asking for the fifth are bread-and-butter FM.

$$P_t = R \cdot v^{n-t+1}, \quad I_t = R(1 - v^{n-t+1})$$

Total interest over payments $j+1$ through $k$:

$$\sum_{t=j+1}^k I_t = (k-j)R - (B_j - B_k)$$

HIGH-FREQUENCY: Refinancing problems combine balance computation with a new loan setup.

• Compute $B_t$, this becomes the new principal.
• Apply new rate $i'$ and term $n'$: $R' = B_t / a_{\overline{n'\vert i'}}$.

Common mistakes

• Using annual rate instead of periodic. 6% convertible monthly means $j = 0.005$, not 0.06. Using 0.06 in the annuity formula gives monthly payment of $6,044. Trap: $6,044.
• Wrong number of periods. 30-year monthly = 360, not 30. Using 30 gives $R = 3598$. Trap: $3,598.
• Balance at wrong time. After 12 years of monthly payments = 144 payments, not 12. Remaining term = 216 months, not 348.

Bottom line

• Loan equation: $L = R \cdot a_{\overline{n\vert i}}$. Given any three of $L, R, i, n$, solve for the fourth.
• Solving for $n$ uses logarithms; a non-integer $n$ means the final payment is a drop or balloon.
• Prospective method gives the PV of remaining payments; retrospective accumulates the payment history.
• Refinancing: compute $B_t$ first (the new principal), then $R' = B_t / a_{\overline{n'\vert i'}}$.

Exam shortcut

On refinancing, draw a timeline: original start, refinancing date, new maturity. Compute $B_t$ immediately, everything depends on it. "BRP": Balance at refinancing, Rate (new), Payment = $B_t / a_{\overline{n'\vert i'}}$. Period conversion: "rate / m, years x m." Total interest: "payments out minus money in."

The full lesson (about 3,800 words, 25 min read) adds 10 worked examples, all 8 common mistakes, a self-check, free in the app.

Learning objectives

• 3b

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