# Real and nominal rates

> Inflation and purchasing power, the real rate as a ratio of growth factors, why n − i is only a shortcut, deflating a future sum, and exchange spreads.

- Course: [Math for Statistics and Unit Economics](https://productovi.com/courses/math-basics.md)
- Module: Money over time
- Lesson 35/36, ~26 min
- Source: https://productovi.com/courses/math-basics/real-and-nominal-rates

## Richer by how much?

**Alex Tutor:** Dana, an English tutor at Klasna, is saving 150,000 UAH for a new kitchen in two years. A deposit pays 14% a year; after tax, which is next lesson, that’s 10.78%. In two years: 150,000 × 1.1078² = 184,083 UAH.

**Sofia:** So 34,083 more. Is that 34,083 more kitchen?

**Alex Tutor:** No, because the kitchen gets dearer too. As of August 2026 Ukrainian inflation was about 8% a year. A price grows like a deposit: in n years it’s p × (1 + i)ⁿ. The kitchen will cost 150,000 × 1.08² = 174,960.

**Alex Tutor:** First, a quick check on what "richer" means.

> **Exercise:** Dana’s 150,000 UAH grows to 184,083 UAH in two years, while the kitchen she is saving for goes from 150,000 to 174,960 UAH. So in two years she can buy 34,083 UAH more kitchen than she could today.

> **Exercise:** A 32-lesson package costs 20,000 UAH today. If prices rise 8% a year, what will it cost in three years, to the nearest hryvnia?

## The real rate

**Alex Tutor:** So compare growth with growth. Each year money grows ×1.1078 and prices ×1.08. What you can buy grows by their ratio: 1.1078 ÷ 1.08 = 1.0257. That’s a **real rate** of 2.57% a year.

$$
r_{real} = \frac{1 + n}{1 + i} - 1
$$

*n: the nominal rate you are paid; i: inflation over the same period.*

![The deposit grows from 150,000 to 184,083 while the kitchen price grows to 174,960; in today’s money the savings are 157,821, a real gain of 5.2% over two years.](https://productovi.com/courses/math-basics/diagrams/m9-l3-money-vs-prices.png)

*Real growth is the gap between the two ladders.*

**Sofia:** Why not just 10.78 − 8 = 2.78%?

**Alex Tutor:** That’s the shortcut n − i, and here it’s off by only 0.2 pp. But it always overshoots, and the gap grows with inflation. At 30% nominal and 25% inflation the shortcut says 5%; the truth is 1.30 ÷ 1.25 − 1 = 4%.

![The shortcut n − i always sits above the exact real rate: 2.78% vs 2.57%, 8% vs 7.41%, 5% vs 4.00%, 10% vs 7.69%; the gap grows with inflation.](https://productovi.com/courses/math-basics/diagrams/m9-l3-shortcut-error.png)

*Fine for a quick look at low inflation; divide when it matters.*

> **Exercise:** A tax-free government bond pays 16% a year, and inflation is 8%. What is the exact real rate, in percent, to one decimal?

> **Exercise:** With a 40% nominal rate and 30% inflation, the real rate is 10%.

**Sofia:** Can a real rate be negative?

**Alex Tutor:** Easily. A 6% deposit with 8% inflation gives 1.06 ÷ 1.08 − 1 = −1.85%. The balance grows every month, and what it buys shrinks. That’s the usual fate of money on a card or under a mattress.

**Alex Tutor:** Put a number on that card-or-mattress fate.

> **Exercise:** A deposit pays 6% a year and inflation is 8%. What is the exact real rate, in percent, to two decimals? Use a minus sign if it is negative.

## Back to today’s money

**Alex Tutor:** Another way to say the same thing: turn the future sum into today’s money by dividing out inflation. That’s **deflating**: 184,083 ÷ 1.08² = 157,821. In today’s prices, Dana ends up with 157,821, not 184,083.

**Sofia:** That’s discounting from module 6, with inflation as the rate.

**Alex Tutor:** Exactly the same move. And 157,821 ÷ 150,000 = 1.052: a 5.2% real gain in two years, which is 1.0257². Both routes agree, as they should.

> **Exercise:** In two years Dana’s deposit will hold 184,083 UAH. With 8% inflation each year, what is that in today’s money, to the nearest hryvnia?

![A cat holding a small purse sits between two baskets: the left one full of orange fish, the right one with fewer.](https://productovi.com/courses/math-basics/inline/m9-l3-i1.webp)

*The same purse buys less when prices rise.*

## Changing currency

**Alex Tutor:** Klasna’s EU families pay in euros, and video and AI costs are in dollars, so currency is a "price" too. An exchange point might buy dollars at 41.20 UAH and sell them at 41.80. The gap is the **spread**: 1 − buy ÷ sell.

> **Exercise:** An exchange point buys dollars at 41.20 UAH and sells them at 41.80. What share of the money is lost if you buy dollars and sell them back straight away, in percent, to two decimals?

**Sofia:** So the spread is a fee hidden in the rate.

**Alex Tutor:** Yes. Buying and selling back loses 1.44% even if the dollar doesn’t move at all. Any plan that switches currency starts 1.44% behind.

**Alex Tutor:** Saving in another currency is the same comparison as the real rate. A hryvnia deposit at 10.78% after tax beats a dollar deposit at 1.5% unless the dollar rises above 41.5 × 1.1078 ÷ 1.015 ≈ 45.29 within the year. That’s the **break-even rate**.

**Sofia:** But nobody knows the rate in a year.

**Alex Tutor:** Right, so the break-even rate doesn’t tell you what to do. It tells you what you’re betting on: keeping dollars bets that the hryvnia falls by more than 45.29 ÷ 41.5 − 1 ≈ 9.1% within the year, plus a little more to cover the spread.

**Alex Tutor:** One more number before your turn: the break-even rate.

> **Exercise:** A hryvnia deposit pays 10.78% a year after tax, a dollar deposit pays 1.5%, and a dollar costs 41.5 UAH today. Ignoring the spread, above what exchange rate in a year does the dollar deposit come out ahead? Give two decimals.

## Your turn

**Alex Tutor:** Dana says: "14% a year, so in two years I’ll be 28% richer." Help sort this out.

> **Exercise:** Dana says about her deposit: "14% a year, so in two years I’ll be 28% richer." In two or three sentences, correct Dana: what does the deposit really do to her purchasing power over two years? Use 10.78% after tax (what is left of 14% once 23% of the interest goes to tax) and 8% inflation.

## Where you will meet this

> **In short**
>
> - Prices grow like deposits: **p × (1 + i)ⁿ**.
> - **Real rate** = (1 + n) ÷ (1 + i) − 1. 10.78% and 8% give 2.57%. The shortcut n − i always overshoots, more so at high inflation.
> - **Deflating:** future sum ÷ (1 + i)ⁿ gives today’s money; 184,083 ÷ 1.08² = 157,821.
> - **Spread** = 1 − buy ÷ sell; a round trip at 41.20 / 41.80 loses 1.44%.
> - **Break-even exchange rate** = today’s rate × (1 + r_home) ÷ (1 + r_foreign).

> **These lessons use it**
>
> - [Inflation and the real rate](/courses/personal-finance/inflation-and-the-real-rate): (1 + n)/(1 + i) − 1 for a deposit.
> - [Hryvnia or dollars: the currency of a goal](/courses/personal-finance/currency-of-a-goal): exchange spreads and the break-even rate.
> - [Case: where to keep Dana’s savings for two years](/courses/personal-finance/two-year-savings-case): 184,083 against a kitchen priced at the date of payment.
> - [Goals with a price and a date](/courses/personal-finance/goals-with-a-price-and-a-date): a future price p(1 + i)ⁿ.
> - [A driver-based forecast](/courses/finance-for-product/driver-based-forecast): revenue and costs in different currencies.
> - [Case: plan vs actual](/courses/finance-for-product/plan-vs-actual-case): how much of a gap the exchange rate explains.

**Alex Tutor:** Next time: taxes and fees, where that 10.78% came from, and what a 2% yearly fee does over twenty years.

- Previous lesson: [Loans and amortization](https://productovi.com/courses/math-basics/loans-and-amortisation.md)
- Next lesson: [Taxes and fees](https://productovi.com/courses/math-basics/taxes-and-fees.md)

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