# Taxes and fees

> The after-tax rate r(1 − t), the tax-equivalent rate r ÷ (1 − t), a flat tax plus a fixed contribution and its effective rate, and how a yearly fee compounds against you.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Money over time
- Lesson 36/36, ~38 min
- Source: https://productovi.com/courses/math-basics/taxes-and-fees

## Where 10.78% came from

**Alex Tutor:** Still at Klasna with Dana, the English tutor: last lesson her 14% deposit grew at 10.78%. Here’s why. As of September 2026, Ukraine taxes deposit interest at 18% income tax plus a 5% military levy, 23% in total, and only the interest, not the deposit itself.

**Sofia:** So she keeps 77% of the interest: 14% × 0.77?

**Alex Tutor:** Exactly: 14% × (1 − 0.23) = 10.78%. On 150,000 that’s 21,000 of interest, 4,830 of tax and 16,170 kept, the same as 150,000 × 10.78%.

$$
r_{after} = r \times (1 - t)
$$

*r: the rate before tax; t: the tax rate on the interest.*

> **Exercise:** A deposit pays 12% a year, and interest is taxed at 23%. What is the after-tax rate, in percent, to two decimals?

**Sofia:** Does tax change which of two deposits is better?

**Alex Tutor:** Not the order: both lose the same 23%, so 14% still beats 12%. Tax matters when one option is taxed and the other isn’t, or when they’re taxed differently.

**Alex Tutor:** Check that with numbers.

> **Exercise:** Two deposits pay 14% and 12% a year, and both have interest taxed at 23%. On 100,000 UAH for a year, which leaves more after tax, and by how much?

## Comparing taxed and untaxed

**Alex Tutor:** Government bonds held by individuals were tax-free as of September 2026. So a 16% bond keeps all 16%. What deposit rate would match it? Turn the formula around: r = 16% ÷ (1 − 0.23) = 20.78%.

**Sofia:** That’s rearranging from module 2: divide by 0.77 instead of multiplying.

**Alex Tutor:** Right. It’s the **tax-equivalent rate**: a deposit would need about 20.8% before tax to leave what the bond leaves. Compare after tax, or compare tax-equivalent; never a taxed rate with an untaxed one.

> **Exercise:** A tax-free bond pays 15% a year. What pre-tax deposit rate leaves the same, with interest taxed at 23%? Percent, two decimals.

## A flat tax plus a fixed part

**Alex Tutor:** Dana is also a sole proprietor on the simplified system. As of September 2026, group 3 pays 5% of income plus a 1% military levy, and a fixed social contribution of 5,707.02 UAH a quarter whatever she earns.

**Alex Tutor:** That’s a line with an intercept, like in module 5: tax = 0.06 × income + 5,707.02. The **effective rate** is tax ÷ income. On 150,000 a quarter: 9,000 + 5,707.02 = 14,707.02, or 9.8%.

![With 6% of income plus a fixed 5,707.02 per quarter, the effective rate is 15.5% at 60,000, 9.8% at 150,000 and 7.9% at 300,000, never below 6%.](https://productovi.com/courses/math-basics/diagrams/m9-l4-effective-rate.png)

*The fixed part spreads thinner as income grows.*

**Sofia:** So in a slow summer quarter her rate jumps.

**Alex Tutor:** Yes: on 60,000 it’s 15.5%, and with no income she still owes 5,707.02. That’s why irregular earners set money aside in good quarters.

> **Exercise:** Dana earns 100,000 UAH in a quarter. With 6% of income plus 5,707.02 UAH fixed, what is her effective tax rate, in percent, to one decimal?

**Alex Tutor:** Now the other rate on the same line.

> **Exercise:** Dana pays 6% of income plus a fixed 5,707.02 UAH a quarter. Her income this quarter rises from 150,000 to 160,000 UAH. How much more tax does she pay, in UAH?

## Fees compound too

**Alex Tutor:** Last: fees. A fund that charges 2% of your money every year takes a share of the whole balance, not of the gain. If it earns 5%, you keep 1.05 × 0.98 = 1.029: 2.9% a year.

$$
r_{net} = (1 + r)(1 - f) - 1
$$

*r: the return before fees; f: the yearly fee as a share of the balance.*

**Sofia:** Why multiply, not just 5 − 2 = 3%?

**Alex Tutor:** Because the fee is taken from the grown balance: 100 becomes 105, then 2% of 105 goes, leaving 102.90. For one year 3% is close; over many years only the product compounds correctly.

**Alex Tutor:** Over twenty years, 100,000 at 5% becomes 1.05²⁰ × 100,000 = 265,330. At 2.9% it becomes (1.05 × 0.98)²⁰ × 100,000 = 177,136. The fee took 88,194, more than half of the growth.

![100,000 at 5% a year becomes 265,330 in 20 years; with a 2% yearly fee it becomes 177,136, so fees take 88,194, about 53% of the growth.](https://productovi.com/courses/math-basics/diagrams/m9-l4-fee-drag.png)

*The shaded gap is what the fee costs.*

**Sofia:** And a one-off entry fee instead?

**Alex Tutor:** A 3% entry fee costs 3% once: you keep 0.97. A 1% yearly fee keeps 0.99ⁿ: after 3 years they’re even, after 20 years you keep only 0.818. For long horizons the yearly fee is the one to watch.

> **Exercise:** A fund earns 7% a year before a 1.5% yearly fee on the balance. What is the net yearly return, in percent, to one decimal?

**Alex Tutor:** Now the twenty-year picture.

> **Exercise:** Over 20 years, 100,000 UAH at 5% a year grows to 265,330 UAH. With a 2% yearly fee on the balance it grows to only 177,136 UAH. What share of the growth without fees does the fee take, in percent, to one decimal?

**Alex Tutor:** And the last question: a one-off fee or a yearly one?

> **Exercise:** A 1% yearly fee on the balance keeps 0.99ⁿ of the money after n years. What share does it keep after 20 years, in percent, to one decimal? Compare it with a one-off 3% entry fee, which keeps 97%.

> **Exercise:** A friend says a 2% yearly fee "is nothing, the fund earns 5% anyway." In two or three sentences, explain with numbers what 2% a year does to 100,000 UAH over twenty years.

## Case: three places for 100,000

**Alex Tutor:** To close the module, put taxes, fees and inflation together. Same money, three options, three years. Compare them in today’s money.

> **Exercise:** Write a short comparison of the three options for Dana.

## Where you will meet this

> **In short**
>
> - **After tax:** r × (1 − t). 14% with 23% tax → 10.78%. Tax applies to the interest, not the deposit.
> - **Tax-equivalent:** r ÷ (1 − t). A 16% tax-free bond equals a 20.78% taxed deposit.
> - **Flat tax + fixed part:** tax = rate × income + fixed; effective rate = tax ÷ income, highest when income is low.
> - **Fee drag:** (1 + r)(1 − f) − 1 per year. 5% with a 2% fee → 2.9%; over 20 years the fee takes about half the growth.
> - A one-off entry fee hurts less than a yearly fee over long horizons.

> **These lessons use it**
>
> - [Taxes first: the FOP set-aside](/courses/personal-finance/taxes-first): a flat tax plus a fixed contribution per quarter.
> - [How big a cushion: months of essentials](/courses/personal-finance/how-big-a-cushion): what a cushion earns after tax.
> - [Deposits and government bonds](/courses/personal-finance/deposits-and-bonds): r(1 − t) and the tax-equivalent rate.
> - [Fees and time: what 2% a year costs](/courses/personal-finance/fees-and-time): (1 + r)(1 − f) over many years.
> - [Case: "just buy crypto" vs a regular plan](/courses/personal-finance/hot-tip-case): returns with a range, after fees.

**Alex Tutor:** That closes the course’s math toolkit: from percent and rounding to probability, curves and money over time.

- Previous 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.