📄 LOAN REPAYMENT — QUICK REFERENCE CARD

Print this. Scan 10 min before exam. Solve 3 traps immediately.

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FORMULA BANK (Memorize These Cold)

Level Payment & OB

$R = \frac{L}{a_{\overline{n}|i}} = \frac{L \cdot i}{1 - v^n}$ $\text{OB}_t^{\text{pro}} = R \cdot a_{\overline{n-t}|i} = R \cdot \frac{1 - v^{n-t}}{i}$ $\text{OB}_t^{\text{retro}} = L(1+i)^t - R \cdot s_{\overline{t}|i}$

Interest & Principal Components

$I_t = i \cdot \text{OB}_{t-1} = R(1 - v^{n-t+1})$ $PR_t = R - I_t = R \cdot v^{n-t+1}$ $PR_{t+s} = PR_t \cdot (1+i)^s \quad \text{(geometric growth)}$

Aggregate

$\sum_{t=1}^{n} I_t = nR - L$ $\sum_{t=1}^{n} PR_t = L$

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🚨 TOP 3 JEBAKAN (Do These First When Panicked)

TRAP #1: Exponent $n - t + 1$ vs $n - t$

Wrong: $PR_t = R \cdot v^{n-t}$ ❌
Right: $PR_t = R \cdot v^{n-t+1}$ ✓

When: “Pokok pada pembayaran ke-$t$” always uses $n - t + 1$.

Example: $PR_3$ di 10-year loan = $R \cdot v^{10-3+1} = R \cdot v^8$, bukan $v^7$.

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TRAP #2: Geometric Relationship

Formula: $PR_{t+s} = PR_t \cdot (1+i)^s$

Equivalently: $\frac{PR_b}{PR_a} = (1+i)^{b-a}$

When: Soal memberi 2 principal values di periode berbeda, minta rate atau principal di period ketiga.

Fast Check:

• $PR_{20} = 3 \times PR_5$ dan jarak 15 periode? → $(1+i)^{15} = 3$ → $i = 3^{1/15} - 1$ ✓

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TRAP #3: Rate Conversion (Annual ↔ Monthly ↔ Quarterly)

Nominal $r$, converted $m$ times per year: $i_{\text{per period}} = \frac{r}{m}$

Example: 6% nominal converted quarterly → $i_q = 6\% / 4 = 1.5\%$ per quarter.

Annual Effective from Monthly: $(1 + i_{\text{monthly}})^{12} - 1 = i_{\text{annual}}$

When: Problem gives “7.5% per year dikonversi bulanan” → always use $7.5\%/12$ per bulan.

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🔴 RED FLAGS — STOP & REREAD

Flag	What to Check
”Pada pembayaran ke-$t$”	Is it asking for $I_t$, $PR_t$, or $\text{OB}_t$? Each is different.
”Saldo setelah pembayaran ke-$t$”	This is $\text{OB}_t$ (AFTER payment), not $\text{OB}_{t-1}$ (before).
”Nominal X% dikonversi Y”	Divide by Y. Period rate = X% / Y.
”Pokok yang dibayarkan adalah…”	This is $PR_t$, triggers geometric formula.
”Principal = Interest”	Sets up equation $PR_t = I_t$; solve for unknown.
”Refinance / Rate change”	Multi-step: OB at refi point, new loan with new rate.

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DECISION TREE (30 sec to pick right formula)

Soal asks for...?
│
├─ "Cicilan tetap" (level payment)?
│  └─ L = R · a_n  →  R = L / a_n  ✓
│
├─ "Saldo pinjaman" / OB?
│  ├─ Prospective: OB_t = R · a_{n-t}  (faster, default)
│  └─ Retrospective: OB_t = L(1+i)^t - R·s_t  (backup check)
│
├─ "Bunga / Pokok pada pembayaran ke-t"?
│  ├─ I_t = R(1 - v^{n-t+1})  ← bunga
│  └─ PR_t = R · v^{n-t+1}  ← pokok
│
├─ "Perbandingan PR di dua periode"?
│  └─ PR_b / PR_a = (1+i)^{b-a}  → solve for i or PR
│
├─ "Saldo setelah 5 tahun, terus pembayaran baru"?
│  └─ Multi-step: (1) hitung OB_5; (2) treat sebagai L_new; (3) hitung R_new
│
└─ "Rate dalam per-tahun, pembayaran per-bulan"?
   └─ Convert: i_monthly = i_annual / 12

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FORMULAE CHEAT SHEET (Copy to Exam Paper First 30 sec)

$$a_n = \frac{1-v^n}{i}, \quad s_n = \frac{(1+i)^n - 1}{i}, \quad v = \frac{1}{1+i}$$ $$v^{n-t+1}, \quad 1 - v^{n-t+1}$$ $$\text{Total Interest} = nR - L$$

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SAMPLE PANIC SOLVE (1 Min)

Problem: “Pinjaman 100jt, 10 pembayaran tahunan, 6% per tahun. Pokok pada pembayaran ke-3?”

Solve:

1. $$v = 1/1.06 = 0.943396$$
2. $$a_{10} = (1 - v^{10})/0.06 = 7.3601$$
3. $$R = 100 / 7.3601 = 13.587$$
4. $$PR_3 = 13.587 \times v^{10-3+1} = 13.587 \times v^8$$
5. $$v^8 = 0.6302$$
6. $PR_3 = 13.587 \times 0.6302 = 8.56$ ✓

Total time: < 2 min.

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PRE-EXAM CHECKLIST (Do night before)

• Memorize $a_n = \frac{1-v^n}{i}$ formula
• Know $PR_t = R \cdot v^{n-t+1}$, not $v^{n-t}$
• Know $\text{OB}_t = R \cdot a_{n-t}$ (prospective, faster)
• Practice 1 “rate from PR” problem (use ratio method)
• Practice 1 “refinancing” problem (OB midpoint + new rate)
• Do 1 conversion problem (annual→monthly or vice versa)

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Good luck. You got this. 💪