The Rule of 72 And How It Helps
Posted on December 23, 2019
by Don Shaughnessy
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People don’t understand compound growth at an intuitive level so they need a tool to help them. The rule of 72 is such a tool.
Counting the doubles
You get a sense of exponential growth, compounding, if you understand the idea of doubles. How long until you have twice as much money.
It will help if you learn what the various powers of 2 are.
2^2 = 4, 2^3 = 8, 2^4 =16, 2^5 = 32 and so on. 2^10 = 1024, 2^20 = 1,048,576
Suppose you are watching the Antiques Road Show and someone brings in a treasure their great-grandparents bought in 1919 for $12. The price sticker is still on it. Maybe a painting by an unfamiliar artist. The appraiser is quite excited and after their normal education talk assesses its value at between $90,000 and $100,000. What is the annual rate of return on the $12 invesstment?
It must be pretty high $12 becomes $100,000 must be pretty unusual. But a 100 years is a long time too. Put a guess in your mind. High – maybe 25% or lower maybe 12%, you decide.
Counting the doubles leads to the solution. Suppose we use $96,000 as the value now. So the painting is worth 8,000 times cost.
We know 8 is three doubles and 1,000 is about 10 doubles, so 13 doubles altogether make 8,000. 2^13 is 8192. dividing 13 into 100 we know each double took 7.7 years.
The rule of 72
The rule says the rate of return is approximately 72 divided by the number of years it takes to double. So 72/7.7 = 9.35%. How close was your original guess? For many, not close at all.
The other use of the rule is to find how long it takes to double. it works the same way. 72 divided by yield = time to double.
If someone says 8% you know it will take 9 years to double your money.
All very nice, but the rule of 72 can help you understand the investment world a little.
Everything is about meaning
It doesn’t realy matter what you know until you know what it means. There is a risk in knowing facts that don’t connect to you life.
The meaning of the rule helps you understand things.
- The most important thing. The last double is worth as much as all of the doubles that came before. With our painting, each double took 100/13 = 7.7 years. So if jump back to the middle of 2011, and assume growth is constant, it would have been worth only $46,000. The key point. Your retirement plan will work the same way. Starting a little early will have a huge impact on where you end up. Or how much you must save.
- Income taxes matter – a lot. Taxes compound too, just like interest. If you pay tax at a marginal rate of 40%, 8% yield turns into 4.8%. At 8% you will double your money in 9 years. At 4.8% it takes about 15 years. So if you have 45 years to invest, you can have five doubles or three. If losing one double costs half the money then losing two costs three quarters of the money. Big loss. If you are missing 3/4 of the money and there is only one other player in the game, the government, they must have it. Managing your taxes, even a little, will provide a meaningful win for you.
- You can work out how long to halve something too. If it takes 8 years to double, it takes 8 years to halve the value too. Even low rates of deterioration turn out significant if there is a long time. Suppose your health begins to deteriorate at age 20 at 2% a year. You probably wouldn’t notice that each year, but at 70, you are at 36.4 % of where you were at 20. Maybe that’s about normal. But what if you worked at being healthy. No smoking, decent diet, a bit of exercise, moderate drinking, good driving, and all those things we pay no attention to day-to-day. Now the deterioration is 1.5% instead of 2%. Again you won’t notice it and you won’t notice the difference between 2% and 1.5%. But at 70, you’ll be at 47% of what you were at 20. About a third better.
- Time matters. The outcome is a function of three variables. Time, yield, and capital. The double your saving to get twice as much is easy to understand, but the other two don’t work like that. You will need to think about all three together. The results are often surprising. Take very long times and prepare to be amazed. In 2010, the estimate for the number of Christians in the world was 2.2 billion. In the year 33, we know there were at least 12 and likely more. Must have been that many for sure. What is the growth rate over the 1,977 years to 2010? 183,333,333 times as many. We know 2^20 is about 1.05 million and we know the other 174 times is between 2^7 = 128 and 2^8 = 256. So 27.5 doubles. Each one must have taken 71.9 years so the average growth rate of Christianity is about 1% annually. You get very large number if you have any growth at all for a very long time. The acutal rate of growth based on the outcome and start above is 1.009%. You could test yourself and assume there were 70 Christians in the beginning. The growth rate then is barely more than half of 1%. You could work it out.
It will work better for you if you understand rate of growth and time as a single idea. How many doubles. Work at a bit and pay no attention to graphs that are linear instead of logarithmic. Both these graphs describe 8% growth for 200 years.
While both graphs display the same information,The logarithmic graph more precisely tells the story of what happens at 8% over long periods.
We are not intuitive about exponential growth. Once you know that you can use tools like the log graphs and the rule of 72 to help you come to well reasoned positions.
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