🗓️Periodic Compounding & APR

In Simple & Compound Interest the interest settles once a year. The real world settles more often: some accounts monthly, some daily. Change the tempo and the same money at the same headline rate ends the year at different amounts.

The same 10%, three endings

Park 1000 for one year at a nominal annual rate of 10%, changing only the tempo:

• Yearly: one settlement, $1000 \times 1.10 = 1100.00$;
• Monthly: rate $\frac{0.10}{12}$ per period, compounded 12 times, $1000 \times (1 + \frac{0.10}{12})^{12} \approx 1104.71$;
• Daily: rate $\frac{0.10}{365}$ per period, compounded 365 times, $1000 \times (1 + \frac{0.10}{365})^{365} \approx 1105.16$.

The rule in one line: each period earns the annual rate ÷ the number of periods, and the interest compounds that many times. The sooner interest joins the principal, the sooner the snowball starts rolling.

Principal10000Annual rate 5%5Years10

Simple

15,000

Compound

16,289

Compound earns extra: +1,289

In the chart, simple interest is a straight line while compound interest bends ever upward. Read each bend as one settlement: the more frequent the settlements, the denser the bends, the earlier the curve climbs.

Nominal rate vs effective rate

The 10% on the sign is the nominal annual rate: it says the year's interest is priced at 10%, but not how often it settles. The share you actually grow in a year is the effective annual rate: about 10.47% with monthly compounding, 10.52% with daily. When comparing two offers, compare effective rates, not signs; the tool for comparing growth is in Percent Change.

Saving and borrowing are not the same game

Check yourself

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Quick quiz

↺ Try again

1. 1. 1000 at a nominal 10%, compounded yearly — the balance after one year?

   11001104.711200

2. 2. Same nominal rate, monthly vs yearly compounding — which ends the year ahead?

   MonthlyYearlyIdentical

3. 3. Switch daily compounding to hourly — does the gain grow noticeably?

   It doublesBarely movesIt shrinks