courses › Finance & Money

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

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Builds on: Compound Growth (not open yet)

skills · practice stats

From rounds of this course only: box, review and trial 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.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

1. 1 day
2. 3 days
3. 7 days
4. 14 days
5. 30 days
6. 60 days
7. mastered · every 90 days

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