CAGR & Doubling
Average growth that compounds, the rule of 72, and logarithms for 'how long?'
Pen and paper is fine · no calculator needed why?
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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
- 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
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
- 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
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
- 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
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
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.
practice
Compound annual growth rate
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%
- Growth multiple = $432,000 ÷ $300,000 = 1.44.
- Which yearly factor, used 2 times, gives 1.44? 1.2, since 1.2 × 1.2 = 1.44.
- So the CAGR is 20% a year.
Average rate vs. CAGR
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%
- Multiply the yearly factors: 2.25 × 1 = 2.25.
- 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
worked example
Assume money grows 8% a year. Using the rule of 72, about how many years does it take to double?
Answer: 9
- Years to double ≈ 72 ÷ rate = 72 ÷ 8 = 9.
- It's an estimate: closest for rates near 8%, and further off at very low or very high rates.
Count the doublings
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
- $72,000 ÷ $9,000 = 8 = 2 × 2 × 2: 3 doublings.
- 3 doublings × 5 years = 15 years.
Exact doubling time with logs
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
- Solve 1.15ᵗ = 2 with logs: t = ln 2 ÷ ln 1.15 = 0.6931 ÷ 0.1398 ≈ 5 years.
- If growth is only credited once a year, the first year-end at or above double is year 5.