# Loans and amortization > How a payment splits into interest and principal, months to repay, a flat fee against interest on the balance, and the real annual cost of a loan as its internal rate. - Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md) - Module: Money over time - Lesson 34/36, ~27 min - Source: https://productovi.com/courses/math-basics/loans-and-amortisation ## One step of a loan **Alex Tutor:** Klasna is buying laptops for its support team on an equipment loan: 60,000 UAH at 2% a month, repaid in six equal monthly payments of 10,711.55 UAH. **Sofia:** How does the lender get 10,711.55 out of 60,000 and 2%? **Alex Tutor:** From the annuity formula in module 6. But a loan is easier to understand one month at a time. Each month the balance grows by the interest and shrinks by the payment. $$ B_{t+1} = B_t \times (1 + r) - \text{payment} $$ *B_t: the balance after month t; r: the monthly rate.* **Alex Tutor:** Month 1: interest is 60,000 × 0.02 = 1,200. The payment covers that first, and the rest, 10,711.55 − 1,200 = 9,511.55, repays the loan. The new balance is 50,488.45. > **Exercise:** A loan has a balance of 50,000 UAH at 2% a month, and this month’s payment is 8,000. What is the balance after the payment? ## Interest shrinks, principal grows **Alex Tutor:** Repeat that six times. The payment stays the same, but the interest is always 2% of a smaller balance: 1,200, then 1,010, 816, 618, 416, 210. So each month more of the same payment repays the loan itself. ![Each 10,711.55 UAH payment first covers 2% interest on the balance, from 1,200 down to 210, and the rest repays principal until the balance reaches 0.](https://productovi.com/courses/math-basics/diagrams/m9-l2-amortization.png) *An amortization schedule: same payment, a changing split.* > **Exercise:** In the laptop loan, how much of the first 10,711.55 UAH payment repays principal? **Alex Tutor:** Your turn on month 2 of the laptop loan. > **Exercise:** Klasna’s laptop loan is 60,000 UAH at 2% a month, repaid in payments of 10,711.55 UAH. After month 1 the balance is 50,488.45 UAH. How much of the second payment is interest, in UAH, to the kopiyka? **Sofia:** And what does the loan cost in total? **Alex Tutor:** Everything paid minus what was borrowed: 6 × 10,711.55 − 60,000 = 4,269.30 UAH. That’s the "price" of the loan, and it’s the sum of the six interest parts. > **Exercise:** A 20,000 UAH course loan is repaid in 12 monthly payments of 1,950 UAH. What is the total price of the loan, in UAH? ## The minimum-payment trap **Alex Tutor:** Now the same rule on a credit card. Sam, a support coordinator at Klasna, owes 20,000 UAH at 3% a month. Interest is 600 a month. What happens if Sam pays exactly 600? **Sofia:** 20,000 × 1.03 − 600 = 20,000. The balance never moves. **Alex Tutor:** Forever. Paying 1,000 a month clears it in about 31 months, and the months come from the ln trick in module 6: n = −ln(1 − r × B ÷ payment) ÷ ln(1 + r). Over 31 months Sam pays about 31,000 for 20,000 of debt. > **Exercise:** A card balance of 15,000 UAH at 3% a month with a 400 UAH monthly payment will slowly go down. **Alex Tutor:** And how long would it take at 1,500 a month? > **Exercise:** Sam owes 20,000 UAH on a card at 3% a month and decides to pay 1,500 a month. Using n = −ln(1 − r × B ÷ payment) ÷ ln(1 + r), how many months does it take to clear the debt? Give one decimal. **Alex Tutor:** With two debts and some spare money, put the extra on the highest rate first, the **avalanche**: it saves the most interest. Paying the smallest balance first, the **snowball**, costs a bit more but gives quick wins that keep people going. > **Exercise:** Sam owes 20,000 on a card at 3% a month and 6,000 on an installment plan at 1.5% a month, and has 2,000 extra each month. Which choice saves the most interest? ## A flat fee and the real rate **Alex Tutor:** Back to last lesson’s "pay in parts": 12,000 in four parts with 1.5% a month of the original amount, 180 every month. Interest on what is left would be 180, 135, 90 and 45: 450 instead of 720. ![A flat 1.5% of the original 12,000 is 180 every month, 720 in total; 1.5% of what is left is 180, 135, 90, 45, only 450.](https://productovi.com/courses/math-basics/diagrams/m9-l2-flat-vs-declining.png) *The fee doesn’t shrink when the debt does.* **Sofia:** So what is the real monthly rate of that offer? **Alex Tutor:** The rate r at which four payments of 3,180, discounted, add back up to the 12,000 you received. That’s the loan’s **internal rate**, its IRR from module 6. A spreadsheet finds it: RATE(4, −3180, 12000) ≈ 2.37% a month, about 28% a year. **Sofia:** Could I check that rate without a spreadsheet? **Alex Tutor:** Roughly. Try 2% and 3% a month, discount the four payments at each, and see on which side of 12,000 you land; then narrow it down. It’s the same trial and error as IRR in module 6. The spreadsheet just does it faster. **Alex Tutor:** That is what a real annual cost of credit measures: every fee counted, against the money you actually got. Upfront fees push it up further, because you receive less but repay the same. **Alex Tutor:** Before your turn, put a number on that offer. > **Exercise:** A shop sells 12,000 UAH of goods in four monthly parts of 3,180 UAH: 3,000 of principal plus a flat 1.5% of the original 12,000 (180) each month. What is the real monthly rate of this offer? ## Your turn > **Exercise:** A shop advertises: "Only 1.5% a month, that’s 18% a year!" for paying 12,000 UAH in four monthly parts, the fee taken on the original amount. In two or three sentences, explain why 18% is wrong and how to find the true rate. ## Where you will meet this > **In short** > > - Each month: **B_{t+1} = B_t × (1 + r) − payment**. Interest comes first; the rest repays principal. > - Interest shrinks with the balance, so later payments repay more of the loan. > - **Price of a loan** = n × payment − principal. A payment at or below the interest never clears the debt. > - **Avalanche** (highest rate first) saves the most; **snowball** (smallest first) buys motivation. > - A **flat fee** on the original amount costs far more than it sounds. The real rate is the loan’s IRR: RATE in a spreadsheet. > **These lessons use it** > > - [The real annual cost of a loan](/courses/personal-finance/real-cost-of-a-loan): every fee counted, as a yearly rate. > - [Credit cards: grace period and the minimum-payment trap](/courses/personal-finance/credit-cards): a payment that only covers interest. > - ["Pay in parts" and microloans](/courses/personal-finance/installments-and-microloans): a flat fee on the original amount. > - [Case: Sam’s way out of debt](/courses/personal-finance/way-out-of-debt-case): avalanche against snowball. **Alex Tutor:** Next time: inflation, and why a 14% deposit doesn’t make you 14% richer. - Previous lesson: [Simple interest, days and annualizing](https://productovi.com/courses/math-basics/simple-interest-and-day-counts.md) - Next lesson: [Real and nominal rates](https://productovi.com/courses/math-basics/real-and-nominal-rates.md) --- Productovi Learning. Free, no sign-up needed to read.