Loan Terminology and Amortization

After 10 years on a $400,000 mortgage at 6%, the balance is over $335,000. Early payments are almost entirely interest.

HIGH-FREQUENCY: Amortization schedules and prospective/retrospective balance formulas are among the most heavily tested FM concepts.

The principal $L$ is the borrowed amount. A standard amortized loan requires level end-of-period payments $R$ such that:

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

The payment is:

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

Prospective: PV of all remaining payments:

$$B_t = R \cdot a_{\overline{n-t\vert i}}$$

Retrospective: accumulated loan minus accumulated payments:

$$B_t = L(1+i)^t - R \cdot s_{\overline{t\vert i}}$$

Both give the same answer for level payments. The retrospective method is essential when payments change (e.g., refinancing).

Each payment splits:

$$I_t = i \cdot B_{t-1}, \quad P_t = R - I_t$$

Closed form for level payments:

$$P_t = R \cdot v^{n-t+1}$$

Principal components form a geometric progression with ratio $(1+i)$. Each payment has $(1+i)$ times as much principal as the prior one.

Common mistakes

• Wrong balance method. Prospective uses remaining term $n - t$; retrospective uses elapsed $t$. Computing $R \cdot a_{\overline{t\vert}}$ instead of $R \cdot a_{\overline{n-t\vert}}$ gives wrong balance. On a 20-year loan, $B_8$ uses $a_{\overline{12\vert}}$, not $a_{\overline{8\vert}}$. Trap: $27,074 instead of $36,547.
• Using original principal for refinancing. The refinanced amount is $B_t$, not $L$. Using $50,000 gives $R_2 = 4817$. Trap: $4,817.
• Wrong exponent in $P_t$. Principal in payment $t$ is $R \cdot v^{n-t+1}$, not $R \cdot v^t$.

Bottom line

• Loan equation: $L = R \cdot a_{\overline{n\vert i}}$. Payment equals principal divided by the annuity factor.
• Outstanding balance: prospective $R \cdot a_{\overline{n-t\vert}}$ or retrospective $L(1+i)^t - R \cdot s_{\overline{t\vert}}$.
• Payment split: $I_t = i \cdot B_{t-1}$, $P_t = R - I_t = R \cdot v^{n-t+1}$. Total principal repaid $= L$; total interest $= nR - L$.
• Principal grows at $(1+i)$ each period: early payments are mostly interest, later ones mostly principal.

Exam shortcut

For interest/principal in a specific payment, use $P_t = R \cdot v^{n-t+1}$ directly. Never build a full schedule. For refinancing, compute $B_t$ first, it becomes the new principal. "PIB": Payment = Interest + Principal. Interest first ($i \cdot B_{t-1}$). Balance drops by principal. Principal grows like compound interest: $P_t = P_1(1+i)^{t-1}$.

The full lesson (about 3,738 words, 25 min read) adds 6 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 3a

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