courses › Finance & Money

CAGR & Doubling

level 45 course

Average growth that compounds, the rule of 72, and logarithms for 'how long?'

Pen and paper is fine · no calculator needed why?

Opens at level 45.

Builds on: Compound Growth (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

The compound annual growth rate (CAGR) is the one steady yearly rate that links a start and an end. Revenue that goes from $100,000 to $121,000 in two years grew 10% a year, because 1.1 × 1.1 = 1.21. It hides the path in between, and it is not a forecast.

Averaging yearly changes misleads: up 100% then down 50% ends where it started, a CAGR of 0%, not 25%.

For how long, count doublings, estimate with the rule of 72, or get the exact time from logarithms, which find the power that gives a number.

Techniques

Find the steady factor

You know a start and an end, or two yearly changes.

Find the growth multiple: end ÷ start, or the yearly factors multiplied.
Find the factor that gives it when used once per year: 1.1 × 1.1 × 1.1 = 1.331.
Subtract 1 and write it as a percent.

worked example

Example: A food truck's revenue goes from $50,000 to $72,000 over 2 years. What is the compound annual growth rate, as a percent?

$72,000 ÷ $50,000 = 1.44.
1.2 × 1.2 = 1.44, so the CAGR is 20% a year.

Answer: 20%

Rule of 72 and doublings

How long money takes to double, or to grow 8 times.

Years to double ≈ 72 ÷ the rate (8 for 8%). Backward, the rate ≈ 72 ÷ the years.
Count the doublings in the multiple: 8 is 2 × 2 × 2, three doublings.
Multiply the doublings by the doubling time.

worked example

Example: Assume an investment doubles every 7 years. How many years does it take to grow from $3,000 to $24,000?

$24,000 ÷ $3,000 = 8.
8 = 2 × 2 × 2: three doublings.
3 × 7 = 21 years.

Answer: 21

Exact doubling time with logs

You are given ln 2 and ln of the growth factor.

Doubling means the growth factor, used t times, reaches 2.
Logs turn a power into a multiple, so t = ln 2 ÷ ln of the factor.

worked example

Example: Assume money grows 10% a year. Using ln 2 ≈ 0.6931 and ln 1.1 ≈ 0.09531, how many years does it take to double, allowing fractional years? Round to one decimal place.

t = ln 2 ÷ ln 1.1
0.6931 ÷ 0.09531 ≈ 7.3 years.

Answer: 7.3

Tips by skill

TipCompound annual growth rate: Divide the end by the start, then find the yearly factor that gives that multiple.
TipAverage rate vs. CAGR: Multiply the two yearly factors first. The CAGR's factor, times itself, must equal that product.
TipRule of 72: Divide 72 by the rate (8 for 8%) to get the years, or 72 by the years to get the rate.
TipCount the doublings: Count the doublings in the multiple (4 times is two, 8 times is three), then multiply by the doubling time.
TipExact doubling time with logs: Divide ln 2 by the ln value you are given, then round to one decimal place.

Watch out for

Dividing total growth by the years: 21% over two years is 10% a year, not 10.5%.
Using 100 instead of 72: at 8%, money doubles in about 9 years, not 12.5.
Multiplying the multiple by the doubling time. Growing 8 times is three doublings.
Treating ln 1.05 as 0.05, or using the rule of 72 for an exact answer.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Compound annual growth rate par 20 s not tried yet
worked example
Revenue goes from $300,000 to $432,000 over 2 years. What is the compound annual growth rate, as a percent? (Use a minus sign for a decline.)

Answer: 20%
1. Growth multiple = $432,000 ÷ $300,000 = 1.44.
2. Which yearly factor, used 2 times, gives 1.44? 1.2, since 1.2 × 1.2 = 1.44.
3. So the CAGR is 20% a year.
Average rate vs. CAGR par 22 s not tried yet
worked example
Revenue rises 125% in year 1 and stays flat in year 2. What is the compound annual growth rate over the two years, as a percent? (Use a minus sign for a decline.)

Answer: 50%
1. Multiply the yearly factors: 2.25 × 1 = 2.25.
2. The yearly factor that gives 2.25 over two years is 1.5 (1.5 × 1.5 = 2.25), so the CAGR is 50%.
Rule of 72 par 11 s not tried yet
worked example
Assume money grows 8% a year. Using the rule of 72, about how many years does it take to double?

Answer: 9
1. Years to double ≈ 72 ÷ rate = 72 ÷ 8 = 9.
2. It's an estimate: closest for rates near 8%, and further off at very low or very high rates.
Count the doublings par 13 s not tried yet
worked example
Assume an investment doubles every 5 years. How many years does it take to grow from $9,000 to $72,000?

Answer: 15
1. $72,000 ÷ $9,000 = 8 = 2 × 2 × 2: 3 doublings.
2. 3 doublings × 5 years = 15 years.
Exact doubling time with logs par 20 s not tried yet
worked example
Assume money grows 15% a year. Using ln 2 ≈ 0.6931 and ln 1.15 ≈ 0.1398, how many years does it take to double, allowing fractional years? Round to one decimal place.

Answer: 5
1. Solve 1.15ᵗ = 2 with logs: t = ln 2 ÷ ln 1.15 = 0.6931 ÷ 0.1398 ≈ 5 years.
2. If growth is only credited once a year, the first year-end at or above double is year 5.

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