# Compounding

> Growth factors, why −50% then +50% loses a quarter, (1 + r)ⁿ, simple vs compound interest, CAGR with a fractional power, and monthly ↔ yearly rates and churn.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Growth and time
- Lesson 21/36, ~27 min
- Source: https://productovi.com/courses/math-basics/compounding

## Changes multiply

**Alex Tutor:** A new module, same team: you're still on Klasna's product team. Leo is pleased: "Klasna Club lost 50% of its members over the summer, then grew 50% in September. So we're back where we were."

**Sofia:** Say there were 1,000. Half leave: 500. Then +50% of 500 is 250 more: 750. We're a quarter down.

**Alex Tutor:** Because each percent applies to a different base. Turn every change into a **growth factor**: +r becomes ×(1 + r), so −50% is ×0.5 and +50% is ×1.5. Then chain changes by multiplying: 0.5 × 1.5 = 0.75.

**Alex Tutor:** It works for big swings too. +300% and then −75% is ×4 × 0.25 = 1: right back to the start, although the two percentages "average" +112.5%.

> **Exercise:** A metric falls 20% one month and grows 20% the next. What is the total change, in percent? Use a minus sign for a decrease.

**Alex Tutor:** Now a big swing, the kind crypto charts show.

> **Exercise:** A metric that grows 300% and then falls 75% ends exactly where it started.

## Interest on interest

**Alex Tutor:** Repeat the same factor n times and you get a power. Money at rate r for n years grows to FV = PV × (1 + r)ⁿ. That's **compound** interest: each year earns on last year's interest too.

$$
FV = PV \cdot (1 + r)^n \qquad FV = PV \cdot (1 + r \cdot n)
$$

*Left: compound interest; right: simple interest. PV: the amount today; FV: the amount after n periods; r: the rate per period.*

**Sofia:** And **simple** interest only pays on the original amount. For 1,000,000 at 25% over two years: 1,500,000 simple, 1,000,000 × 1.25² = 1,562,500 compound.

**Alex Tutor:** The 62,500 gap is interest on the first year's interest. Over two years it's small; over twenty it dominates.

![At 25% a year, 1,000,000 grows to 1,500,000 with simple interest and 1,562,500 with compound interest after two years; after six years the gap is 2,500,000 vs 3,814,697.](https://productovi.com/courses/math-basics/diagrams/m6-l1-simple-vs-compound.png)

*Simple interest adds the same step each year; compound interest multiplies.*

**Alex Tutor:** Put the two formulas side by side on real money.

> **Exercise:** Klasna puts 1,000,000 UAH on deposit at 25% a year for two years, and each year's interest is added to the balance. What is the balance after two years?

> **Exercise:** A balance grows 2% every month. By how many percent does it grow over 12 months? Two decimals.

## The average yearly growth: CAGR

**Alex Tutor:** Now backwards. Klasna delivered 8,000 lessons in its first spring and 11,200 three springs later. What steady yearly growth gives that?

**Sofia:** Total growth is 11,200 ÷ 8,000 = 1.4, so +40%. Divided by 3, about 13.3% a year?

**Alex Tutor:** That ignores compounding. You need the factor x with x³ = 1.4, so x = 1.4^(1/3), the cube root. A power of 1/n is the n-th root. 1.4^(1/3) ≈ 1.1187: the **CAGR** is 11.87%.

$$
CAGR = \left(\frac{V_n}{V_0}\right)^{1/n} - 1
$$

*CAGR: compound annual growth rate; V₀: the start value; Vₙ: the value after n years. In a spreadsheet: =(end/start)^(1/n)-1.*

**Alex Tutor:** The personal-finance course has the same trap with returns: 20 years averaging 6% on paper turned 100,000 into 275,589, a CAGR of 5.2%. Losing years cost more than equal winning years add.

> **Exercise:** Klasna Club grew from 400 to 900 members in 4 years. What is the CAGR, in percent? Two decimals.

**Alex Tutor:** One more case where a plain average misleads.

> **Exercise:** A savings fund rose 50% in one year and fell 20% the next. What is its CAGR over the two years, in percent? Two decimals.

## Monthly and yearly

**Alex Tutor:** The fractional power also converts rates. 25% a year is not 25 ÷ 12 per month: it's 1.25^(1/12) − 1 ≈ 1.877% a month, because twelve monthly steps must multiply to 1.25.

> **Exercise:** A deposit pays 20% a year with monthly compounding of the same total. What is the equivalent monthly rate?

**Sofia:** And churn? If 5% of Club members leave each month, that's 60% a year?

**Alex Tutor:** No: each month 5% leave from those still there. 0.95¹² ≈ 0.5404 stay, so yearly churn is 1 − 0.5404 ≈ 46%, not 60%.

![Of 100 members, with 5% monthly churn 54.04 remain after 12 months, so yearly churn is about 46%, while the straight-line guess 12 × 5% would leave only 40.](https://productovi.com/courses/math-basics/diagrams/m6-l1-churn-staircase.png)

*Each month loses 5% of a smaller group.*

![A cat rides a big snowball at the bottom of a hill; along the track the same snowball grows from tiny to huge.](https://productovi.com/courses/math-basics/inline/m6-l1-i1.webp)

*Compound growth: each turn adds to what is already there.*

> **Exercise:** Monthly churn is 3%. What share of members leave within a year, in percent? Two decimals.

**Sofia:** So any time a rate is quoted, I should ask: per month or per year, and compounded or not?

**Alex Tutor:** Exactly. A loan at "3% a month" sounds small, yet 1.03¹² ≈ 1.426: about 42.6% a year. Lenders know which version sounds better.

**Sofia:** And going down from yearly to monthly, I take the 12th root instead of dividing by 12.

**Alex Tutor:** Yes. In a spreadsheet: =(1+R)^(1/12)-1. Dividing by 12 slightly overstates the monthly rate.

## Your turn: the Club's year

**Alex Tutor:** Leo is planning next year's Club: "Monthly churn is 5%, so we lose 60% a year. Of 1,000 members, 400 stay. We need 600 new ones just to stand still."

**Sofia:** The 400 is wrong, and so is the 600. Though new members also churn during the year…

**Alex Tutor:** Keep it to his main claim: how many of today's 1,000 stay, and why his shortcut is wrong.

> **Exercise:** Reply to Leo, Klasna's CEO, in 3–4 sentences: with 5% monthly churn, how many of today's 1,000 Club members stay after a year, and why is "12 × 5% = 60%" wrong?

## Where you will meet this

> **In short**
>
> - A change of r is a **growth factor** of (1 + r); chained changes multiply. −50% then +50% is ×0.75; +300% then −75% is ×1.
> - **Compound** growth: FV = PV × (1 + r)ⁿ. **Simple** interest: PV × (1 + r·n).
> - **CAGR** = (end ÷ start)^(1/n) − 1; a power of 1/n is the n-th root. It is lower than the simple average of yearly returns when returns swing.
> - Monthly ↔ yearly: (1 + R)^(1/12) − 1 and (1 + r)¹² − 1. Yearly churn = 1 − (1 − c)¹², not 12 × c.

> **These lessons use it**
>
> - [Why a hryvnia next year is worth less](/courses/finance-for-product/time-value-of-money): 1,000,000 × 1.25² = 1,562,500.
> - [Discounted LTV and CAC payback in cash](/courses/finance-for-product/ltv-and-cac-payback-in-cash): a yearly rate turned into a monthly one.
> - [The real annual cost of a loan](/courses/personal-finance/real-cost-of-a-loan): R = (1 + r)¹² − 1.
> - [Risk and return: no "high and safe"](/courses/personal-finance/risk-and-return): 6% on average, 5.2% compounded.
> - [Case: "just buy crypto" vs a regular plan](/courses/personal-finance/hot-tip-case): +300% then −75%.
> - [Goals with a price and a date](/courses/personal-finance/goals-with-a-price-and-a-date): a price in n years, p × (1 + i)ⁿ.
> - [MRR and churn](/courses/bootstrapped-product/mrr-and-churn): monthly churn and what it does over a year.

**Alex Tutor:** Next time we run compounding in reverse: what money in the future is worth today.

- Previous lesson: [Splitting a total](https://productovi.com/courses/math-basics/splitting-a-total.md)
- Next lesson: [Discounting](https://productovi.com/courses/math-basics/discounting.md)

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