Simple and compound interest · RiskAtoms Academy

Know these first

L1.3.01 The time value of money

Math Atoms

∑ Exponents

By the end you will be able to

1Calculate future values under different compounding frequencies.
2Tell simple interest from compound interest, and work out both.
3Turn a quoted (nominal) annual rate into an effective annual rate, to compare offers fairly.
4Estimate how long money takes to double with the rule of 72.

Why it matters

Two savers each put £1,000 away at 5% a year for 30 years. One ends with £2,500, the other with £4,322. Same money, same rate, same time. The £1,822 difference is interest earned on interest.

Core explanation

Interest on interest

In L1.3.01 we moved money forward by one year: multiply by $1 + r$ . Over several years there are two ways interest can work.

With simple interest, you earn interest on your original amount only. £1,000 at 5% earns £50 every year, and after 30 years you have £1,000 + 30 × £50 = £2,500. Some short-term loans and many money market deals work like this.

With compound interest, each year's interest is added to the balance, and next year you earn interest on the larger balance. Year 1 earns £50. Year 2 earns 5% of £1,050, which is £52.50. The extra £2.50 is interest on interest. It looks tiny, but it grows every year:

Year	Simple: interest	Simple: balance	Compound: interest	Compound: balance
1	50.00	1,050.00	50.00	1,050.00
2	50.00	1,100.00	52.50	1,102.50
3	50.00	1,150.00	55.13	1,157.63
4	50.00	1,200.00	57.88	1,215.51
5	50.00	1,250.00	60.78	1,276.28

How often is interest added? Savings accounts, loans and bonds add interest at different compounding frequencies. A rate of 6% “compounded monthly” means 0.5% is added each month. Because each month's interest starts earning straight away, 6% paid monthly grows money a little faster than 6% paid once a year.

That makes quoted rates tricky to compare. The quoted figure, 6% a year, is the nominal rate. What you actually earn over a year, once compounding is included, is the effective annual rate (EAR). UK banks show it on savings as the AER (annual equivalent rate); US banks show it as the APY (annual percentage yield). Compare offers on the effective rate, not the headline.

Which account pays more?

Decide, then see why

Account A pays 6.00% a year, added once a year. Account B pays 5.90% a year, added monthly. Which grows your money faster?
Now add account C: 5.95% a year, added quarterly. Which of A, B and C is best?

The formal version

We start from the one-year move, $\mathrm{FV} = \mathrm{PV}(1+r)$ , and repeat it. Exponents (powers) are the new tool here: $(1+r)^3$ means $(1+r)\times(1+r)\times(1+r)$ .

Next, let interest be added $m$ times a year. The quoted rate $r$ is split into $m$ equal pieces, and there are $m$ pieces every year.

The effective annual rate follows in two lines. Grow 1 for one year with $m$ periods, then take away the 1 you started with:

1 + \mathrm{EAR} = \Big(1 + \frac{r}{m}\Big)^{m}

\mathrm{EAR} = \Big(1 + \frac{r}{m}\Big)^{m} - 1

The formulas, for reference

\text{Simple: } \mathrm{FV}_T = \mathrm{PV}(1 + rT)

\text{Compound: } \mathrm{FV}_T = \mathrm{PV}\Big(1 + \frac{r}{m}\Big)^{mT}

$r$ is the nominal annual rate as a decimal, $m$ the number of compounding periods a year and $T$ the number of years. With $m = 1$ the compound formula becomes $\mathrm{PV}(1+r)^T$ .

The rule of 72. To estimate how many years money takes to double, divide 72 by the rate in percent. At 6%: 72 ÷ 6 = 12 years. The exact answer, from $(1.06)^T = 2$ , is 11.90 years. The rule works well for rates between about 2% and 15%.

Try it

Start at 6% for 20 years with interest added annually. Then switch to monthly and daily: the effective rate creeps up, but by less each time. Now stretch the years to 40 and watch the shaded gap, interest on interest, take over.

Try itCompounding explorer

Open the compound interest calculator →

Deposit today£10,000

Nominal annual rate6.0%

Years20

Interest added

Compound, after 20 years

£32,071

Simple interest£22,000

Interest on interest£10,071

Effective annual rate6.000%

Balance over time compound simple interest on interest

Same rate, every frequency

Annual	£32,071
Semi-annual	£32,620
Quarterly	£32,907
Monthly	£33,102
Daily	£33,198
Simple	£22,000

Worked example

One rate, three frequencies

A saver in Germany deposits €10,000 for 3 years at a nominal rate of 4.8% a year. The bank offers three versions: interest added annually, quarterly or monthly. Try each step before revealing it.

€10,000 at 4.8% for 3 years

Step 1 of 6

1
Annual compounding
$10{,}000 \times 1.048^{3} = 11{,}510.23$

Regulatory lens

Because the same headline rate can mean different things, consumer rules require an effective rate. In the US, the Truth in Savings Act and its Regulation DD require banks to quote deposit rates as an annual percentage yield (APY), which includes compounding. UK savings are advertised with an AER. For consumer loans, the EU Consumer Credit Directive requires an annual percentage rate of charge (APRC), an effective rate that also includes most fees.

Looking ahead: inside a bank, every product's rate must be put on one basis before its interest rate risk can be measured (IRRBB, Level 4).

Sparks

SparkEinstein probably never said it

You will often read that Albert Einstein called compound interest “the eighth wonder of the world” or “the most powerful force in the universe”. There is no reliable record of him saying either. The maths is impressive enough without the celebrity.

SparkThe rule of 72 is over 500 years old

The rule appears in Luca Pacioli's Summa de arithmetica, printed in Venice in 1494, where he gives it without proof as a quick way to find doubling time. Pacioli is better known for setting out double-entry bookkeeping in the same book.

Key takeaways

Simple interest is paid on the original amount only: FV = PV(1 + rT).
Compound interest adds each period's interest to the balance, so interest earns interest: FV = PV(1 + r)^T.
With m compounding periods a year, use rate r/m and mT periods.
Compare offers on the effective annual rate, (1 + r/m)^m − 1, not the nominal headline rate.
The rule of 72: money doubles in roughly 72 ÷ (rate in %) years.

Reaction

Check what stuck

Question 1 · objective 1

You invest $5,000 at 6% a year, compounded annually, for 4 years. What is it worth at the end, in $?

Question 2 · objective 3

A card charges 12% a year, compounded monthly. What is the effective annual rate, in %, to two decimals?

Question 3 · objective 2

What is the difference between simple and compound interest?

ACompound interest uses a higher rateBCompound interest also earns interest on interest already addedCSimple interest is only paid at the endDThere is none over one year or more

Question 4 · objective 2

£2,000 earns simple interest of 3.5% a year for 4 years. What is the balance at the end, in £?

Question 5 · objective 4

Using the rule of 72, roughly how many years does money take to double at 8% a year?

years

Glossary

Simple interest: Interest paid only on the original amount, never on interest already earned.
Compound interest: Interest paid on the original amount and on all interest added so far.
Compounding frequency (m): How many times a year interest is added to the balance: once, twice, four times, monthly or daily.
Nominal rate: The quoted annual rate before allowing for compounding within the year, for example 6% a year paid monthly.
Effective annual rate (EAR): The rate that, added once a year, gives the same growth as the quoted rate with its compounding. Called AER in the UK and APY in the US.
Growth factor: What 1 grows into over a period: (1 + r)^T with annual compounding.
Rule of 72: A shortcut: doubling time in years is about 72 divided by the rate in percent.

Deep Dives & Origins

Deep Dives

Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill), chapter 2. Beginner.
John C. Hull, Options, Futures, and Other Derivatives (Pearson), chapter 4, “Interest rates”, on measuring rates and compounding frequency. Intermediate.

Origins

Hull, Options, Futures, and Other Derivatives (Pearson).
Brealey, Myers and Allen, Principles of Corporate Finance (McGraw Hill).
US Consumer Financial Protection Bureau, Regulation DD (Truth in Savings), 12 CFR Part 1030.
Directive 2008/48/EC on credit agreements for consumers (EU Consumer Credit Directive).
Luca Pacioli, Summa de arithmetica, geometria, proportioni et proportionalita (Venice, 1494).

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The time value of money

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Present value and discounting