Chapters
- Simple interest
- Compound interest (compounding)
- Capitalisation: how often interest is added
- The rule of 72
- Saving every month: the power of time
- Inside a loan: the annuity and the APR
- The real interest rate: nominal minus inflation
- Today's euro and tomorrow's euro
- The mathematics of scams → The iron principle of the module
For the youngest · ages 6–7
🐣 The youngest: If you keep your coins for a long time, sometimes more are added to them. This adding is called interest. Then the new coins bring new ones too. Like a snowball that rolls and grows. The longer it rolls, the bigger it gets.
Play together (5 minutes): Put 2 buttons in a bowl. Every day add one new button for every two buttons in the bowl. Count together for 5 days: 2, 3, 4, 6, 9, 13 — the bowl grows faster and faster.
A question for the evening: "What would you wait a long time for, so that it becomes more?"
A game from the catalogue: LV-01 "The tree of patience" — https://pecuvo.eu/igri/katalog/#lv-01 (in Bulgarian)
Chapter 1 · Simple interest
Formula: interest = principal × interest rate × years.
👧 Child: you lend €100 and every year you get €5 on top — always €5. This is simple interest.
🧑 Teenager: €1,000 at 5 % a year for 3 years → interest €150 → total €1,150. The interest is always calculated only on the starting amount.
🧔 Adult: simple interest applies to some short-term loans and to bonds that pay out the interest every year (the coupon) instead of adding it to the principal. If the interest is not reinvested, it does not "give birth" to new interest.
Scene for 3D: a little pile of coins that grows by one identical cube every year.
Chapter 2 · Compound interest (compounding)
Formula: final amount = principal × (1 + interest rate)^years.
👧 Child: the interest is added to the money, and the next year it earns interest too. Like a snowball that rolls and keeps getting bigger.
🧑 Teenager: €1,000 at 5 % a year for 10 years → €1,628.89. With simple interest it would be €1,500. The difference of €128.89 is "interest on interest" — and it grows faster and faster over the years.
🧔 Adult: compounding is the basis of both saving and debt. The same formula that makes savings grow makes unpaid expensive debt double. Time is the strongest variable: not the percentage, but how long.
Myth → truth
- Myth: "The difference between 5 and 7 % is small."
- Truth: over 30 years €1,000 becomes €4,321.94 at 5 % and €7,612.26 at 7 % — almost double. Small percentages make a big difference over time.
Who profits? Those on the right side of the exponential: savers — when saving; lenders — when it is debt.
Scene for 3D: a snowball on a slope. The child chooses the gradient (the interest rate) and the length of the slope (years) and watches the ball become a house.
Your step in Pecuvo: in "Goals" — how much you will have at rate X for Y years, shown on a chart.
Chapter 3 · Capitalisation: how often interest is added
Formula: final amount = principal × (1 + annual rate / n)^(n × years), where n is how many times a year the interest is added.
👧 Child: if the interest is added every month and not once a year, money grows a little faster — because the interest starts "working" earlier.
🧑 Teenager: €1,000 at 12 % a year for 1 year: | How often interest is added | Final amount | Effective annual rate | |---|---|---| | once a year | €1,120.00 | 12.00 % | | quarterly | €1,125.51 | 12.55 % | | monthly | €1,126.83 | 12.68 % | | daily | €1,127.47 | 12.75 % | | continuously | €1,127.50 | 12.75 % |
🧔 Adult: that is why, when comparing, what counts is the effective annual rate, not the advertised (nominal) one. Two "12 %" offers can be different if one capitalises monthly and the other yearly. With a loan it is the same, but against you.
The second "capitalisation": on the stock market, "capitalisation" also means a company's market value — the price of one share times the number of shares. A different concept, the same word. In Pecuvo we make the context clear.
Myth → truth
- Myth: "12 % is 12 %."
- Truth: it depends on how often it is added. Compare the effective rates.
Who profits? Whoever knows the difference. Advertisements often show the nicer number.
Scene for 3D: a clock made of coins. The more often the clock "strikes" (capitalisation), the more small coins are added. The child compares clocks.
Chapter 4 · The rule of 72
Formula (approximate): years to double ≈ 72 ÷ the interest rate.
👧 Child: if you know the percentage, you can quickly say after how many years the money will double.
🧑 Teenager:
- at 6 % → 72 ÷ 6 = 12 years (exactly: 11.90);
- at 9 % → 8 years (exactly: 8.04).
It also works the other way round: with inflation at 3 %, prices double in about 24 years.
🧔 Adult: the rule is an approximation and is most accurate at rates of around 6–10 %. It is used as a quick check of offers: if someone promises "double in 2 years", that is 36 % a year. Ask where it comes from.
Scene for 3D: two sand towers. The child chooses a percentage and watches when the tower becomes twice as tall.
Chapter 5 · Saving every month: the power of time
👧 Child: if you put in a little every week, what you collect grows not only from your coins but also from the interest on them.
🧑 Teenager: €100 a month at 5 % a year (interest added every month): | How many years | Paid in | At the end | |---|---|---| | 10 years | €12,000 | €15,528.23 | | 20 years | €24,000 | €41,103.37 |
Doubling the years more than doubles the final amount.
🧔 Adult: that is why "pay yourself first" (Module 8, principle 1) works best when it starts early. Ten years earlier means not "a bit more" but often much more. This is mathematics, not product advice. The return depends on where the money is and with what risk (Module 3).
Your step in Pecuvo: a "savings" goal with a monthly contribution — Pecuvo shows the curve.
Chapter 6 · Inside a loan: the annuity and the APR
Formula for the equal monthly instalment (annuity): instalment = amount × i ÷ (1 − (1 + i)^−n), where i is the monthly interest rate and n is the number of months.
👧 Child: when you borrow and pay back in parts, each part is a little money for the interest and a little for the loan itself.
🧑 Teenager: €10,000 over 36 months: | Annual interest rate | Monthly instalment | Total repaid | Of which interest | |---|---|---|---| | 6 % | €304.22 | €10,951.90 | €951.90 | | 20 % | €371.64 | €13,378.89 | €3,378.89 |
At the beginning most of the instalment is interest, at the end — principal.
🧔 Adult: the interest rate is not the whole price. The APR — the annual percentage rate of charge — includes the interest and the fees and commissions. Under EU law it is compulsory in advertising and in offers of consumer credit ✔. The new Consumer Credit Directive (EU) 2023/2225 requires it in advertising (Article 8(3)), sets how it is calculated (Article 30) and obliges Member States to have measures against excessive costs, for example caps on the interest rate, the APR or the total cost of credit (Article 31); it applies from 20.11.2026 ✔. Compare the APR and the total amount to be repaid, not the monthly instalment. Extending the term makes the instalment smaller but the total price bigger.
Example: Bulgaria — ✔ Under the Consumer Credit Act in force, the APR may not be higher than five times the statutory interest on late payments (Article 19(4)). Caution: on 23.09.2026 the parliament adopted at first reading a new law (transposing Directive 2023/2225). It keeps the cap, but not for loans of up to three minimum monthly wages ⚠ — the final text is to be checked after the second reading.
Check in your country — Does your country cap the APR, the interest rate or the total cost of credit, and how is the cap calculated? Look at the consumer protection authority or the financial supervisor. Ask the lender for the standard European form (SECCI): it shows the APR and the total amount payable in the same way in every EU country.
Myth → truth
- Myth: "The instalment is low, so the loan is cheap."
- Truth: a low instalment often means a long term and a lot of interest. Look at the total repaid.
Who profits? The lender — from the term and the fees. The advert shows the instalment, not the total.
Scene for 3D: a puzzle made of monthly instalments. Each piece is split into red (interest) and blue (principal). At the start there is more red.
Your step in Pecuvo: every loan is a "debt" account. A "repayment" goal shows how much is left and for how many months.
Sources
- ✔ Directive (EU) 2023/2225 — the APR in advertising (Article 8(3)), its calculation (Article 30), measures against excessive costs, e.g. caps (Article 31); applies from 20.11.2026: https://eur-lex.europa.eu/eli/dir/2023/2225/oj
- ✔ Example Bulgaria: Consumer Credit Act, Article 19(4) — "The annual percentage rate of charge may not be higher than five times the statutory interest on late payments in euro and in foreign currency…" (consolidated text with amendments up to State Gazette No. 87 of 24.09.2026, opened 21.10.2026) · ⚠ new bill, first reading 23.09.2026 — exception for loans of up to 3 minimum wages: only from a media report (https://taralej.bg/politika/parlamentat-ogranichi-gpr-po-kreditite-do-pet-pati-razmera-na-zakonnata-lihva, in Bulgarian); the bill's page on parliament.bg did not open — to be checked after the second reading.
Chapter 7 · The real interest rate: nominal minus inflation
Formula: real rate = (1 + nominal) ÷ (1 + inflation) − 1 (approximately: nominal − inflation).
👧 Child: if the bank gives you €2 extra, but ice cream goes up by €3, you can actually buy less.
🧑 Teenager: a deposit at 2 % with inflation at 3 % → real interest rate −0.97 %. The amount grows on paper but buys less.
🧔 Adult: this is the second of the "Big Three" questions (inflation). With moderate inflation of around 2 % (the ECB's target — Module 4), money that does not earn at least that much loses in real terms. It is not a reason to panic but a reason to know: every amount should be compared with inflation.
Scene for 3D: two trains on parallel tracks — "my money" and "prices". The child sees which one pulls ahead.
Chapter 8 · Today's euro and tomorrow's euro
Formula: present value = future amount ÷ (1 + r)^years.
👧 Child: €10 today is more than €10 you will get in a year — because today you can use it.
🧑 Teenager: €1,000 in 10 years, at 3 % a year, is worth about €744.09 today. That is why "I'll pay you later" has a price.
🧔 Adult: the time value of money is the basis of every valuation — of a loan, an investment, a project. It explains why a late payment on a project (Module 11) costs money and why "profit is not cash" (Module 8, principle 8).
Scene for 3D: a telescope pointed at the future. The further the child looks, the smaller the same coin appears.
Chapter 9 · The mathematics of scams
👧 Child: if someone promises you that your money will double every week, ask: "Where from?" If there is no answer — it is a trap.
🧑 Teenager: "Only 2 % a day" sounds small. Over one year: €1,000 × 1.02³⁶⁵ ≈ €1,377,408. No real business does that. "10 % a month" over one year is ×3.14. Compound interest shows that these promises cannot last. They are paid out only with the money of new investors — a Ponzi scheme (Module 17).
🧔 Adult: the rule of 72 and the compound interest formula are the fastest scam detector. A high promised return without a clear source, without risk and with "today only" pressure is the classic recipe. The motto of Module 17: "The book 'How to get rich quick' is sold together with the Criminal Code."
Scene for 3D: a balloon that the child inflates with "2 % a day". The balloon grows unrealistically fast and bursts. Inside — nothing.
→ Continue at the source
- Your country's financial supervisor usually publishes a list of firms that are not licensed and warnings about them — check there before you pay.
- Example: Bulgaria — Financial Supervision Commission (FSC), list of companies without a licence: https://www.fsc.bg/spisak-na-druzhestva-bez-litsenz/ ✔ · for crypto-assets: https://www.fsc.bg/za-potrebitelya/preduprezhdeniya-za-potrebitelite/ot-komisiata-za-finansov-nadzor/spisak-na-litsata-koyto-ne-sa-litsenzirani-i-nyamat-pravo-da-predostavyat-uslugi-s-kriptoaktivi/ (both in Bulgarian)
The iron principle of the module
Time multiplies everything — both savings and debts. Before you sign, work out the final amount, not just the percentage and the instalment.
- 👧 Child: "How much will I have at the end?"
- 🧑 Teenager: before a loan or "buy now, pay later" — how much do I pay back in total?
- 🧔 Adult: compare the effective annual rate and the APR. If a promised return doubles the money faster than the rule of 72 allows, with no clear source — run.
Your step in Pecuvo: in "Goals", look at the savings curve. In "Debt" — how much interest is left. Do the sum once, before you sign.
Check yourself (3 questions per level)
- 👧 1. What is the difference between simple and compound interest? 2. Why is it good to start saving early? 3. What will you ask if someone promises "double every week"?
- 🧑 1. How much does €1,000 become at 5 % for 10 years? 2. What does the rule of 72 say at 9 %? 3. Why can a loan with a lower instalment be more expensive?
- 🧔 1. Why do we compare the effective rate and not the nominal one? 2. What is the real interest rate at 2 % nominal and 3 % inflation? 3. Why is "2 % a day" mathematically impossible to sustain?
Sources
- ✔ Stango & Zinman, "Exponential Growth Bias and Household Finance", Journal of Finance 64 (2009), 2807–2849, DOI 10.1111/j.1540-6261.2009.01518.x
- ✔ GFLEC — the "Big Three": https://gflec.org/education/questions-that-indicate-financial-literacy/
- ✔ Directive (EU) 2023/2225 (the APR) — https://eur-lex.europa.eu/eli/dir/2023/2225/oj
- ✔ Example Bulgaria: Consumer Credit Act, Article 19(4) — APR cap of 5 × the statutory interest (the text of the law, opened 21.10.2026) · ⚠ the new law (first reading 23.09.2026, from a media report) — to be checked after adoption.
- The calculations — Pecuvo, 19.10.2026: 1,150; 1,628.89; 4,321.94 / 7,612.26; 1,120.00 / 1,125.51 / 1,126.83 / 1,127.47 / 1,127.50; 11.90 / 8.04 years; 15,528.23 / 41,103.37; 304.22 / 371.64 (951.90 / 3,378.89); −0.97 %; 744.09; 1,377,408; ×3.14.