Interest: meaning, types and calculation – Financial Management (Senior 4 Entrepreneurship) | Ganzaa

Financial Management

Lessons

Senior 4 · Entrepreneurship

Financial Management

5 lessons

Lesson 4 of 5

Interest: meaning, types and calculation

25 min

Case: two offers at "18%"

Uwimana is a tailor in Muhanga who sews dresses on a treadle machine. To take school-uniform orders she wants three electric machines costing Frw 3,000,000, repaid over twelve months. A lending company offers the money "at 18% a year, flat". Her SACCO offers it "at 18% a year on the reducing balance". The two rates look identical. (Uwimana and both offers are illustrative.)

Think about it: Will Uwimana pay the same interest on both loans? What does "flat" mean, and why might a lender prefer to quote it?

What interest is

Interest is the price of using someone else's money for a period. A borrower pays it; a saver or lender earns it. Lenders ask for interest for four reasons: they give up using the money now (time preference), they might not be repaid (risk), prices may rise meanwhile (inflation), and they could have used the money elsewhere (opportunity cost).

In Rwanda the starting point for interest rates is the Central Bank Rate set by the National Bank of Rwanda: 6.75% a year from August 2025, kept there in November 2025. Banks lend above it and pay savers less than they lend at. In 2025 the average lending rate was 15.84% and the average deposit rate 9.20% (BNR, March 2026).

Definition

Principal $(P)$ : the amount borrowed or invested. Rate of interest $(r)$ : the interest for one period as a fraction or percentage of the principal, usually quoted per year (per annum, p.a.). Term $(t \text{ or } n)$ : how long the money is borrowed or invested.

Simple interest: interest calculated on the original principal only. Compound interest: interest calculated on the principal plus all the interest already added, so that interest earns interest.

Definition

Flat rate: interest charged on the whole original loan for the whole term, even though the borrower repays part of it each month. Reducing-balance rate: interest charged each period only on the amount still owed.

Fixed rate: stays the same for the whole loan. Variable (floating) rate: changes with market rates. Real interest rate: the nominal rate adjusted for inflation, showing the gain in buying power.

The formulas

Type	Formula	Symbols
Simple interest	$I = P r t$ , $A = P (1 + r t)$	$A$ = amount at the end; $t$ in years
Compound, once a year	$A = P (1 + r)^{n}$	$n$ = number of years
Compound, $m$ times a year	$A = P\left(1 + \frac{r}{m}\right)^{mn}$	$m = 12$ for monthly, 4 for quarterly
Effective annual rate	$\text{EAR} = \left(1 + \frac{r}{m}\right)^{m} - 1$	the true yearly rate when interest is added $m$ times a year
Flat-rate loan	$I = P r n$ , instalment $= \frac{P + I}{N}$	$N$ = number of instalments
Reducing-balance loan	$M = P \times \frac{i}{1 - (1 + i)^{-N}}$	$M$ = equal instalment; $i$ = rate per instalment period
Real interest rate	$1 + r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \text{inflation}}$	approximately $r_{\text{nominal}} - \text{inflation}$

The graph shows why compounding matters over long periods. In the first year the two lines are equal; after ten years compound interest has added Frw 2.11 million against simple interest's Frw 1.20 million, because each year's interest is itself earning 12%.

Tip

The rule of 72. Money compounding at $r %$ a year doubles in roughly $\frac{72}{r}$ years. At 12% that is about 6 years; the exact answer is $\frac{\ln 2}{\ln 1.12} \approx 6.1$ years. Use it to check that a compound-interest answer is reasonable.

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Worked example 1: simple, compound and effective rates

A cooperative saves Frw 2,000,000 for 3 years at 9% a year. Find the amount at the end if interest is (a) simple; (b) compounded yearly; (c) compounded monthly. (d) Find the effective annual rate in (c).

(a) $I = 2\,000\,000 \times 0.09 \times 3 = 540\,000$ , so $A = 2\,540\,000$ Frw.

(b) $A = 2\,000\,000 \times 1.09^{3} = 2\,000\,000 \times 1.295029 = 2\,590\,058$ Frw.

(c) $A = 2\,000\,000 \left(1 + \frac{0.09}{12}\right)^{12 \times 3} = 2\,000\,000 \times 1.0075^{36} \approx 2\,617\,291$ Frw.

(d) $\text{EAR} = 1.0075^{12} - 1 \approx 0.0938 = 9.38\%$ .

Interpretation: the more often interest is added, the more the saver earns: 9% compounded monthly is really 9.38% a year. A borrower should compare loans by their effective rate, not the headline rate.

Worked example 2: Uwimana's two offers

Loan: Frw 3,000,000 for 12 months, repaid in 12 equal monthly instalments, at 18% a year.

Flat rate: interest on the full sum for the full year.

I = 3\,000\,000 \times 0.18 \times 1 = 540\,000\ \text{Frw}, \qquad \text{instalment} = \frac{3\,000\,000 + 540\,000}{12} = 295\,000\ \text{Frw}

Reducing balance: $i = \frac{0.18}{12} = 0.015$ a month, $N = 12$ .

M = 3\,000\,000 \times \frac{0.015}{1 - 1.015^{-12}} \approx 275\,040\ \text{Frw}

\text{Total interest} = 12 \times 275\,040 - 3\,000\,000 \approx 300\,480\ \text{Frw}

Month	Interest (1.5% of balance)	Principal repaid	Balance owed
1	45,000	230,040	2,769,960
2	41,549	233,491	2,536,469
3	38,047	236,993	2,299,477
…	…	…	…
12	4,065	270,975	0

Conclusion: the flat loan costs Frw 540,000 in interest, the reducing-balance loan about Frw 300,480: nearly Frw 240,000 more for the same "18%". Under the flat rate Uwimana pays interest on money she has already repaid. Charged on the reducing balance, the flat loan's instalments of Frw 295,000 correspond to about 2.64% a month, roughly 31.7% a year.

Worked example 3: the real return on savings in 2025

In 2025 the average deposit rate was 9.20% and headline inflation averaged 7.0% (BNR, March 2026). Find the real interest rate.

1 + r_{\text{real}} = \frac{1.092}{1.070} \approx 1.0206, \qquad r_{\text{real}} \approx 2.06\%

The quick approximation $9.20\% - 7.0\% = 2.2\%$ is close. Interpretation: a saver's buying power grew by only about 2% a year. Money kept at home, earning nothing, lost about 7% of its buying power.

Warning

Common mistakes. Using $t$ in months in $I = P r t$ when $r$ is a yearly rate: 9 months is $t = 0.75$ . Writing $1.09 \times 3$ instead of $1.09^{3}$ for compound growth. Comparing a flat rate with a reducing-balance rate as if they were the same. Forgetting fees: a loan's true cost also includes application and insurance charges.

Khan Academy, "Introduction to interest" (about 10 minutes). Follow the simple and compound calculations on paper, then redo one with Uwimana's 18% to check you get the same pattern.

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Practice

Define principal, simple interest and compound interest, and give four reasons why lenders charge interest.
Find the simple interest on Frw 800,000 at 12% a year for 9 months.
Frw 1,500,000 is invested at 10% a year for 4 years. Find the amount if interest is compounded (a) yearly; (b) quarterly.
Find the effective annual rate of 12% a year compounded monthly.
A loan of Frw 1,200,000 is repaid in 12 monthly instalments at 20% a year. Find the monthly instalment and total interest (a) at a flat rate; (b) on the reducing balance.
A saver earns 8% a year when inflation is 7%. Find the real interest rate exactly and approximately. What happens to her buying power if inflation rises to 9%?
How many whole years does it take for money to at least double at 12% a year compounded yearly? Check with the rule of 72.
"A loan at 18% flat is as cheap as a loan at 18% on the reducing balance." Evaluate this statement.

Show answersHide answers

Principal: the amount borrowed or invested. Simple interest: on the principal only. Compound interest: on the principal plus interest already added. Reasons: time preference, risk of non-repayment, inflation, opportunity cost.
$I = 800\,000 \times 0.12 \times 0.75 = 72\,000$ Frw.
(a) $1\,500\,000 \times 1.1^{4} = 2\,196\,150$ Frw. (b) $1\,500\,000 \times 1.025^{16} \approx 2\,226\,758$ Frw.
$1.01^{12} - 1 \approx 0.1268 = 12.68\%$ .
(a) $I = 1\,200\,000 \times 0.20 = 240\,000$ ; instalment $= \frac{1\,440\,000}{12} = 120\,000$ Frw. (b) $i = \frac{0.20}{12}$ ; $M = 1\,200\,000 \times \frac{i}{1 - (1 + i)^{-12}} \approx 111\,161$ Frw; total interest $\approx 12 \times 111\,161 - 1\,200\,000 \approx 133\,937$ Frw.
Exact: $\frac{1.08}{1.07} - 1 \approx 0.93\%$ ; approximate: $8\% - 7\% = 1\%$ . At 9% inflation the real rate is negative (about $- 0.9 %$ ), so her savings lose buying power.
$1.12^{6} \approx 1.974$ (not yet double), $1.12^{7} \approx 2.211$ : 7 whole years. Rule of 72: $\frac{72}{12} = 6$ years, which agrees (exact 6.1 years).
False. A flat rate charges interest on the whole original loan for the whole term, although the balance falls each month; on a 12-month loan, 18% flat is roughly equal to 32% on the reducing balance. Compare loans by effective rate and total cost.

Quick check

Key points

Interest is the price of using money; it rewards waiting, risk, inflation and opportunity cost.
Simple: $I = P r t$ . Compound: $A = P (1 + r)^{n}$ , or $P\left(1 + \frac{r}{m}\right)^{mn}$ with $m$ compounding periods a year.
The effective annual rate $\left(1 + \frac{r}{m}\right)^m - 1$ shows the true yearly rate.
A flat rate costs much more than the same rate on the reducing balance.
Real rate: $\frac{1 + \text{nominal}}{1 + \text{inflation}} - 1$ ; in 2025 Rwandan savers earned about 2% in real terms.

Sources and credits

Image: African clothes for women, IGANZE, via Wikimedia Commons, CC BY-SA 4.0.
Video: Introduction to interest, Khan Academy, embedded from YouTube.
Data: Central Bank Rate, 2025 average lending and deposit rates and 2025 average inflation, from the Monetary Policy and Financial Stability Statement, March 2026, National Bank of Rwanda.

Written by Ganzaa: the Muhanga case, the formula table, the graph, the worked examples and the practice items. People, businesses and loan offers are illustrative (2026). This lesson uses CC BY-SA material and is shared under CC BY-SA 4.0.

1Case: two offers at "18%"

2What interest is

3The formulas

Case: two offers at "18%"

What interest is

The formulas