162 lines
5.6 KiB
Markdown
162 lines
5.6 KiB
Markdown
---
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created_at: '2016-06-23T06:33:08.000Z'
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title: The Rule of 72 (2007)
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url: http://betterexplained.com/articles/the-rule-of-72/
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author: shubhamjain
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points: 323
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story_text:
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comment_text:
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num_comments: 92
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story_id:
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story_title:
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story_url:
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parent_id:
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created_at_i: 1466663588
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_tags:
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- story
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- author_shubhamjain
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- story_11959230
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objectID: '11959230'
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year: 2007
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---
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The Rule of 72 is a great [mental math
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shortcut](https://betterexplained.com/articles/mental-math-shortcuts/)
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to estimate the effect of any growth rate, from quick financial
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calculations to population estimates. Here’s the formula:
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Years to double = 72 / Interest Rate
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This formula is useful for **financial estimates** and understanding the
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nature of compound interest. Examples:
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- At 6% interest, your money takes 72/6 or 12 years to double.
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- To double your money in 10 years, get an interest rate of 72/10 or
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7.2%.
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- If your country’s
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GDP
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grows at 3% a year, the economy doubles in 72/3 or 24 years.
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- If your growth slips to 2%, it will double in 36 years. If growth
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increases to 4%, the economy doubles in 18 years. Given the speed at
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which technology develops, shaving years off your growth time could
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be very important.
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You can also use the rule of 72 for **expenses like inflation or
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interest**:
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- If inflation rates go from 2% to 3%, your money will lose half its
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value in 24 years instead of 36.
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- If college tuition increases at 5% per year (which is faster than
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inflation), tuition costs will double in 72/5 or about 14.4 years.
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If you pay 15% interest on your credit cards, the amount you owe
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will **double** in only 72/15 or 4.8 years\!
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The rule of 72 shows why a “small” 1% difference in inflation or GDP
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expansion has a huge effect in forecasting models.
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By the way, the Rule of 72 applies to anything that grows, including
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population. Can you see why a population growth rate of 3% vs 2% could
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be a huge problem for planning? Instead of needing to double your
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capacity in 36 years, you only have 24. Twelve years were shaved off
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your schedule with one percentage point.
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## Deriving the Formula
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Half the fun in using this magic formula is seeing how it’s made. Our
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goal is to figure out how long it takes for some money (or something
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else) to double at a certain interest rate.
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Let’s start with $1 since it’s easy to work with (the exact value
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doesn’t matter). So, suppose we have $1 and a yearly interest rate R.
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After one year we have:
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`1 * (1+R)`
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For example, at 10% interest, we’d have $1 \* (1 + 0.1) = $1.10 at the
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end of the year. After 2 years, we’d have
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`1 * (1+R) * (1+R) = 1 * (1+R)^2`
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And at 10% interest, we have $1 \* (1.1)2 = $1.21 at the end of year 2.
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Notice how the dime we earned the first year starts earning money on its
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own (a penny). Next year we create another dime that starts making
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pennies for us, along with the small amount the first penny contributes.
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As Ben Franklin said: “The money that money earns, earns money”, or “The
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dime the dollar earned, earns a penny.” Cool, huh?
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This deceptively small, cumulative growth makes compound interest
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extremely powerful – Einstein called it one of the most powerful forces
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in the universe.
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Extending this year after year, after N years we have
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`1 * (1+R)^N`
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Now, we need to find how long it takes to double — that is, get to 2
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dollars. The equation becomes:
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`1 * (1+R)^N = 2`
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Basically: How many years at R% interest does it take to get to 2? Not
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too hard, right? Let’s get to work on this sucka and find N:
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1: 1 * (1+R)^N = 2
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2: (1+R)^N = 2
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3: ln( (1+R)^N ) = ln(2) [natural log of both sides]
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4: N * ln(1+R) = .693
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5: N * R = .693 [For small R, ln(1+R) ~ R]
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6: N = .693 / R
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There’s a little trickery on line 5. We use an approximation to say that
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ln(1+R) = R. It’s pretty close – even at R = .25 the approximation is
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10% accurate ([check accuracy here](http://instacalc.com/22281)). As you
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use bigger rates, the accuracy will get worse.
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Now let’s clean up the formula a bit. We want to use R as an integer (3)
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rather than a decimal (.03), so we multiply the right hand side by 100:
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`N = 69.3 / R`
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There’s one last step: 69.3 is nice and all, but not easily divisible.
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72 is closeby, and has many more factors (2, 3, 4, 6, 12…). So the rule
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of 72 it is. Sorry 69.3, we hardly knew ye. (We could use 70, but again,
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72 is nearby and even more divisible; for a mental shortcut, go with the
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number easiest to divide.)
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## Extra Credit
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Derive a similar rule for tripling your money – just start with
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`1 * (1+R)^N = 3`
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Give it a go – if you get stuck, [see the rule of 72 for any
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factor](http://instacalc.com/22282).
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Happy math.
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## A Note On Accuracy
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From Colin’s [comment](https://news.ycombinator.com/item?id=11959464) on
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Hacker News, the Rule of 72 works because it’s on the “right side” of
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`100*ln(2)`.
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`100*ln(2)` is ~69.3, and 72 rounds up to the bigger side. This is a
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great choice because the [series
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expansion](https://www.wolframalpha.com/input/?i=taylor+series+ln\(2\)+%2F+ln\(1+%2B+x\))
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of `r * ln(2) / ln(1 + r/100)`
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is:
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![taylor\_series\_r\_\_\_ln\_2\_\_\_\_ln\_1\_\_\_r\_100\_\_-\_Wolfram\_Alpha](https://betterexplained.com/wp-content/uploads/2007/01/taylor_series_r___ln_2____ln_1___r_100__-_Wolfram_Alpha.png)
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This series expansion is the Calculus Way of showing how far the initial
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estimate strays from the actual result. The first correction term
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frac(1)(2) r log(2) is small but grows with r. 72 is on the “right side”
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because it helps us stay in the accurate zone for longer. Neat insight\!
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## Other Posts In This Series
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