Compound interest — the future value from an annual rate and a number of years

On the left are the starting amount, the annual rate and the years. The right-hand column shows, from the top, Simple growth, the difference and Compound growth. Simple growth adds 5% of the original 100 every year; compound growth applies 5% to whatever the balance is at that point. The difference in the middle is compound growth minus simple growth. Changing a number on the left updates all three cards on the right.

Adding the same amount each year gives 150

Adding 5% a year for 10 years gives 5% × 10 = 50%, which takes 100 to 150.

That adds 5% of the original 100 every year — 5 a year, every year. Working out each year's increase from the original amount only is called simple interest. Board 1 labels it Simple growth to keep the comparison easy to read, and in this example it comes to 150.

With compound growth, each year starts from the last year's result

With compound growth, the 5% for the next year applies to the result of the year before.

• Year 1 — 100 × 1.05 = 105
• Year 2 — 105 × 1.05 = 110.25
• Year 3 — 110.25 × 1.05 = 115.7625

1.05 is "the original 100% plus 5%" written as a decimal. The amount is multiplied by 1.05 once a year, as many times as there are years:

Future value = principal × (1 + annual rate ÷ 100)^years

Here that is 100 × 1.05¹⁰, which after 10 years is 162.889463, about 162.89. The Compound growth card on Board 1 uses this formula. The rate is 5% every year, but the amount it applies to gets bigger every year.

Board 2 does the same calculation one year at a time.

The left card, Year by year, starts from 100 and applies the 5% rate as many times as the Times field says. With Times set to 1, 2 and 3, the card shows 105, 110.25 and 115.7625; with 10 it shows 162.889463, the same as Compound growth on Board 1. The line in the card traces the amount year by year. The card on the right comes up in a later section.

The gap comes from growth on what has already grown

In this example simple growth gives 150 and compound growth about 162.89, a difference of about 12.89.

With simple growth, the yearly increase stays at 5 for ever. With compound growth, the 5 added in year 1 also earns 5% in year 2. With the years on Board 1 set to 2, simple growth is 110 and compound growth is 110.25, a difference of 0.25 — which is 5% of the 5 added in year 1.

What has been added earns the next 5%, and what that adds earns the 5% after it. Year after year this stacks up, and after 10 years the gap is about 12.89.

At 5% a year, the amount doubles in about 14.21 years

The right-hand card on Board 2, Doubling time, works out how many periods it takes to reach a given multiple when the amount grows by the same percentage each period. Here one period is one year: at 5%, the formula gives a doubling time of about 14.21 years.

The left card adds interest only once a year. On it, 14 times gives 197.99316 and 15 times gives 207.892818, so the balance first reaches or exceeds 200 after the 15th credit. With simple growth it reaches 200 in year 20, so compound growth gets there five years sooner.

The card also shows the rule of 72: 72 divided by the rate gives a rough number of years to double. At 5% that is 72 ÷ 5 = 14.4 years, close to the exact 14.21.

Other numbers

The first table keeps the starting amount on Board 1 at 100 and the rate at 5%, and changes the years.

After one year both are 105. The longer it runs, the faster the gap grows: about 12.89 after 10 years, about 65.33 after 20 and about 182.19 after 30. Doubling or tripling the years makes the gap about 5 or about 14 times as large, because growth on growth keeps repeating for every extra year.

The next table puts the years on Board 1 back to 10 and changes the rate. The two columns on the right come from the card on the right of Board 2, with its rate set to the same annual rate.

Doubling the rate from 5% to 10% takes the 10-year gap from about 12.89 to about 59.37, more than four times as much: a higher rate also means more growth on what has already grown. From 3% to 10%, the rule of 72 is within a year of the exact figure; at 1% it says 72 years against 69.66, a little over two years out.

With the rate back at 5% and a starting amount of 1,000,000, compound growth comes to 1,628,894.626777, exactly 10,000 times the result for 100. Whatever currency or scale the 100 stands for, the proportions stay the same.

With ± each time on the left card of Board 2 set to 10, 1 time gives 115: 100 grown by 5% is 105, and then 10 is added. 10 times gives 288.668388. In that case the total paid in is 200: the first 100 plus 10 a year for 10 years.

What this calculation covers, and what it does not

The boards assume the same annual rate every year, with each year's growth added once a year.

In practice the rate can change along the way, and how often interest is added during the year also changes the result a little. The boards show the amount before any tax or fees.

Read next — Compound annual growth rate (CAGR) — the yearly rate from a start value and an end value It calculates the steady yearly rate from a start value, an end value and the number of years, and checks that the rate leads back to the end value.

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