How much does 7% for 30 years multiply your money?

Question

A portfolio compounds at 7\% a year for 30 years. Solve in your head, no calculator. Roughly how many times larger is the money at the end? Aim: under 30 seconds.

Accepted Answer

Find the doubling time, then see how many doublings fit. 	ext{doubling} \approx \frac{72}{7} \approx 10.3 	ext{ years} \;\Rightarrow\; \frac{30}{10.3} \approx 2.9 	ext{ doublings.} Now 2^{2.9} = 2^{2} 	imes 2^{0.9} \approx 4 	imes 1.87 \approx 7.5. So the money grows about 7.5	imes (the exact 1.07^{30} = 7.6). Takeaway. Convert years into a count of doublings by dividing by 72/	ext{rate}, then evaluate 2^{n}. For the fractional part, remember 2^{0.5} \approx 1.41 and 2^{0.9} \approx 1.87, a couple of anchors let you finish any compounding-factor estimate in your head.

How much does 7% for 30 years multiply your money?

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