# Simple interest, days and annualizing

> Interest on the original amount, counting days, the cost of paying late or skipping an early-payment discount, and turning a period rate into a yearly one.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Money over time
- Lesson 33/36, ~28 min
- Source: https://productovi.com/courses/math-basics/simple-interest-and-day-counts

## Money has a price per day

**Alex Tutor:** A new module: money over time. Interest, loans, inflation and taxes all come down to the formulas you already know, applied to dates. We start with a Klasna for Teams invoice: 576,000 UAH for a quarter of corporate English.

**Sofia:** The client has 60 days to pay. Why would the date matter?

**Alex Tutor:** Because while the money is with the client, Klasna borrows from its credit line at 24% a year. Every day of waiting has a price. **Simple interest** counts it on the original amount: principal × yearly rate × days ÷ 365.

$$
I = P \times r \times \frac{d}{365}
$$

*P: the principal; r: the yearly rate; d: the number of days.*

**Alex Tutor:** If the client pays 30 days late: 576,000 × 0.24 × 30 ÷ 365 = **11,362 UAH**. That’s almost 2% of the invoice gone, for one month of patience.

![A cat walks along a row of tiles with one identical blue coin on each, placing one more orange coin.](https://productovi.com/courses/math-basics/inline/m9-l1-i1.webp)

*Simple interest adds the same amount every day.*

> **Exercise:** A client pays a 240,000 UAH invoice 45 days late. Klasna’s money costs 20% a year. What does the delay cost, in UAH, to the nearest hryvnia?

**Alex Tutor:** Which year you divide by changes the bill too.

> **Exercise:** Klasna borrows 576,000 UAH for 30 days at 24% a year. One lender divides by a 360-day year, another by 365. How many hryvnias more does the 360-day lender charge, to the nearest hryvnia?

**Sofia:** Do I count months or days?

**Alex Tutor:** Days between the two dates. 1 March to 30 April is 60 days, not "two months". Contracts say which year to divide by: 365 is usual, some lenders use 360, which makes the same rate slightly dearer.

## From a day to a year

**Sofia:** And the other way: if I know a rate per day or per month, what’s the yearly rate?

**Alex Tutor:** The simple way multiplies by the number of periods. 1% a day is 365% a year; 3% a month is 36%. As of September 2026, Ukrainian law caps consumer loans at 1% a day, which is exactly that 365%.

> **Exercise:** A microloan charges 0.8% a day on the amount borrowed. What is its simple yearly rate?

**Alex Tutor:** But if unpaid interest joins the balance and earns interest too, a year grows faster. That gives the **effective annual rate**: (1 + period rate)^periods − 1. For 3% a month: 1.03¹² − 1 = 42.6%, not 36%.

![At 3% a month the simple yearly rate is 36%, while the effective rate with interest on interest is 1.03¹² − 1 = 42.6%.](https://productovi.com/courses/math-basics/diagrams/m9-l1-apr-vs-ear.png)

*Same monthly rate, two yearly numbers; ask which one a lender quotes.*

**Sofia:** So the nominal 36% is the friendlier-looking number.

**Alex Tutor:** Which is why the law asks lenders to show a real annual cost. Compounding was in module 6; here it just tells you which yearly number you’re looking at.

> **Exercise:** A card charges 2.5% a month on an unpaid balance, added to the balance each month. What is the effective annual rate, in percent, to one decimal?

> **Exercise:** A loan at 2% a month costs 24% a year, whichever way you count.

## The early-payment discount

**Alex Tutor:** Now flip the invoice. Klasna offers 2% off if the client pays on day 10 instead of day 60. Is that cheap money or expensive money?

**Alex Tutor:** Klasna receives 98 instead of 100, 50 days earlier. So it pays 2 to use 98 for 50 days: 2 ÷ 98 = 2.04% per 50 days. Annualized: × 365 ÷ 50 = **14.9%** a year, well below the 24% credit line.

$$
\text{annual cost} = \frac{d}{1 - d} \times \frac{365}{\text{days}_{net} - \text{days}_{discount}}
$$

*d: the discount as a share; the divisor 1 − d is what you actually receive.*

![2% off for paying on day 10 instead of day 60 costs 2/98 × 365/50 = 14.9% a year; 30 days late on 576,000 UAH at 24% costs 11,362 UAH.](https://productovi.com/courses/math-basics/diagrams/m9-l1-invoice-timeline.png)

*Two prices of time on one invoice.*

**Sofia:** Why 2 ÷ 98 and not 2 ÷ 100?

**Alex Tutor:** Because the money you get early is 98, not 100: the base is what you hold. It’s the same base rule from module 1.

> **Exercise:** A supplier offers Klasna 1% off for paying within 10 days instead of 30. What is the annual cost of not taking the discount, in percent, to one decimal?

**Alex Tutor:** Two slips hide in that formula: the base and the days.

> **Exercise:** Klasna offers a corporate client 3% off for paying on day 10 instead of day 40. What yearly rate does the client earn by taking the discount?

## A fee on the original amount

**Alex Tutor:** One more trap. A "pay in parts" offer: a 12,000 UAH course in four monthly parts of 3,000, plus 1.5% a month of the original 12,000, so 180 a month and 720 in total. It sounds like 18% a year.

**Sofia:** But the debt shrinks every month, and the fee doesn’t.

**Alex Tutor:** Exactly. The debt goes 12,000, 9,000, 6,000, 3,000: about 7,500 on average. 720 on 7,500 for four months is 9.6%, about 29% a year, not 18%. Next lesson does this properly.

**Alex Tutor:** Put a number on that "pay in parts" offer.

> **Exercise:** A 12,000 UAH course is paid in four monthly parts of 3,000, plus a fee of 1.5% a month on the original 12,000 (180 a month, 720 in total). The debt during the four months is 12,000, 9,000, 6,000 and 3,000, 7,500 on average. What simple yearly rate does the 720 come to on the average debt, in percent, to one decimal?

## Your turn

**Alex Tutor:** Hanna isn’t sure about offering clients the 2% discount for paying on day 10 instead of day 60: "2% of 576,000 is 11,520 hryvnias we’d just give away."

> **Exercise:** Hanna, Klasna's head of finance, isn’t sure about offering a Klasna for Teams client 2% off a 576,000 UAH invoice for paying on day 10 instead of day 60: "2% of 576,000 is 11,520 hryvnias we’d just give away." In two or three sentences, tell Hanna whether the discount is worth offering while Klasna borrows at 24% a year, with the numbers.

## Where you will meet this

> **In short**
>
> - **Simple interest:** I = P × r × d ÷ 365, always on the original amount.
> - 30 days late at 24% a year ≈ 2% of the invoice. 1% a day ≈ 365% a year, simple.
> - **Effective annual rate:** (1 + period rate)^periods − 1. 3% a month: 36% simple, 42.6% effective.
> - **Early-payment discount:** d ÷ (1 − d) × 365 ÷ (days saved). 2% for 50 days ≈ 14.9% a year; compare it with what borrowing costs.
> - A **fee on the original amount** ignores that the debt shrinks: 1.5% a month on 12,000 repaid in four parts costs about 29% a year, not 18%.

> **These lessons use it**
>
> - [Receivables and payment terms](/courses/finance-for-product/receivables-and-terms): the cost of 60-day terms and the 2/98 × 365/50 discount.
> - [Case: can Klasna grow Teams without a loan?](/courses/finance-for-product/teams-growth-case): days of cash against the cost of a credit line.
> - [The real annual cost of a loan](/courses/personal-finance/real-cost-of-a-loan): a monthly rate turned into a yearly one.
> - ["Pay in parts" and microloans](/courses/personal-finance/installments-and-microloans): 1% a day and fees on the original amount.

**Alex Tutor:** Next time: loans that you repay month by month, and how each payment splits into interest and principal.

- Previous lesson: [Flows: customers and cash](https://productovi.com/courses/math-basics/flows-customers-and-cash.md)
- Next lesson: [Loans and amortization](https://productovi.com/courses/math-basics/loans-and-amortisation.md)

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