Compound annual growth rate (CAGR) — the yearly rate from a start value and an end value

On the left are the start value, the end value and the number of years. The right-hand column shows, from the top, Total growth, Growth ÷ years and the Annual growth rate. Total growth is how much the value grew from start to end, as a percentage; Growth ÷ years divides that by the number of years. The Annual growth rate at the bottom is the rate this page is about. Changing a number on the left updates the values on all three cards.

Sales, users, population, output — whatever the number counts, the calculation is the same as long as you have two values at two points in time. This example uses 100 → 150.

50% ÷ 5 = 10% adds the same amount every year

100 became 150. It grew by 50, which is 50% of the original 100: the Total growth card on Board 1 shows 50%. Divided by 5 years, that gives the 10% on the Growth ÷ years card.

But this 10% means adding 10% of the original 100 every year — 10 a year. 10 × 5 = 50, which lands exactly on 150. What was added in earlier years never enters the next year's calculation.

Growing 10% a year gives 161.051 after five years

"It grew 10% a year" normally means each year's value is multiplied by 1.1. Board 2 does that one year at a time.

The left card, At 10%, starts from 100 and applies 10% as many times as the Times field says. The right card does the same with a rate that comes up later. With Times on the left card set from 1 to 5, the values are:

• Year 1 — 110
• Year 2 — 121
• Year 3 — 133.1
• Year 4 — 146.41
• Year 5 — 161.051

At 10% a year, five years gives 161.051, not 150. What was added also gets the next year's 10%. So the rate that lands exactly on 150 must be lower than 10%.

Working the rate back from 150

Suppose the value grew at the same rate r every year and reached 150 in five years:

100 × (1 + r)⁵ = 150

Multiplying by (1 + r) five times has to make 1.5. Solving for r:

r = (150 ÷ 100)^1/5 − 1

The ^1/5 means the fifth root: the number that, multiplied by itself five times, gives 1.5. That is about 1.0845, so r is about 0.0845, or about 8.45%.

In terms of the start, the end and the years:

CAGR (%) = ((end ÷ start)^{1 ÷ years} − 1) × 100

The Annual growth rate card on Board 1 uses this formula. Here it shows 8.447177%.

Growing 8.45% a year for five years gets back to 150

One way to check the rate is to grow the start value at that rate and see where it ends up. Board 3 works out the rate and then applies it for the given number of years.

Top right is the annual rate, from the same formula as on Board 1. Bottom right, Start grown at that rate, applies it to the starting 100 five times. It comes to 150, the end value.

The right card on Board 2, At 8.447177%, shows the same growth year by year, with Times set from 1 to 5:

• Year 1 — 108.447177
• Year 2 — 117.607902
• Year 3 — 127.54245
• Year 4 — 138.316186
• Year 5 — 149.999999

It ends at 149.999999 because the card's rate is typed to six decimal places; the difference from 150 is 0.000001. The 10% card on the left ends at 161.051, so the difference between the two rates grows to about 11 over five years.

An "average" rate is not the average of each year

The compound annual growth rate is the one steady rate that, applied every year, would give the same end value. It does not mean the value actually grew 8.45% each year.

The path could go up and down — 100 → 130 → 110 → 140 → 120 → 150, say. As long as it starts at 100, ends at 150 and takes five years, the rate is the same 8.45% or so. What happened in between does not show up in this number.

"Compound" in the name says that each year's growth also earns the next year's rate, as with compound interest.

Other numbers

The table changes the start, end and years on Board 1. The last column is Start grown at that rate on Board 3 with the same numbers.

100 → 150 over 5 years and 100 → 200 over 10 years both give 10% for growth ÷ years. But their annual growth rates are about 8.45% and about 7.18%. The same 10% for growth ÷ years does not mean the same year-on-year rate; of these two, the longer 10-year case sits further below 10%.

The same 100 → 150 over 10 years gives 4.137974%, a little less than half of the 8.447177% over 5 years.

200 → 300 gives 8.447177%, the same as 100 → 150, because both grow 1.5 times. With the same multiple and the same number of years, the rate is the same whether you start at 100 or 200.

When the value falls, as in 150 → 100, the rate is negative: −7.789209%, a bigger drop than the −6.666667% from growth ÷ years. The rate applies to a value that keeps getting smaller, so it takes a larger percentage each year to lose the same total. In every row, Board 3 gets back to the end value.

What this calculation covers, and what it does not

The compound annual growth rate uses only the first and the last value. A year with a big fall in between does not show up in it; the path shows up in the percentage change from year to year.

With a start value of 0, the formula would divide by 0, so there is no rate. With Start set to 0 on Board 1, the cards show that they cannot calculate. The formula also does not work as it is for numbers that can go below zero, such as profit.

A growth rate from the past sums up how a value grew so far. It does not say the value will keep growing at that rate.

Read next — Compound interest — the future value from an annual rate and a number of years It calculates the future value from a principal, an annual rate and a number of years, and how long the amount takes to double.

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