Monday, September 8, 2014

The Magic of Compounding

While Albert Einstein may never have said that compound interest is "the most powerful source in the universe", the exponential growth implied by the magic of compounding can lead to spectacular outcomes.

Example 1: As as example, consider the following: You start with a penny on day one. On day two, you double it. So you have $0.02. On day 3, you double it again ($0.04), and so on.

Without actually carrying out the math, can you guess how much money you will end up with at the end of a 30 day month?

The answer of course is 2^29 pennies, which is over 5 million dollars!

Example 2: The idea is also enshrined in legend. Consider, for example, the story of the chess-board and grains of rice. Essentially, a king was asked to set one grain of rice in the first square, two in the next, and to keep on doubling until all the 64 squares on the chessboard were used up.

A quick calculation shows that the total number of grains would be \[2^0 + 2^1 + ... + 2^{63} = 2^{64} - 1.\] Assuming each grain weights 25 mg, this corresponds to more than 450 billion tonnes of rice, which is about 1000 times larger than the annual global production.

Example 3: What makes Warren Buffett fabulously wealthy? If you start with an amount \(P\) and grow it at an annual growth rate of \(i\) for \(n\) years, you end up with,\[A = P (1 + i)^n.\] Two ways to get compounding to work its magic is to have large growth rates and/or long incubation times. In Buffett's case, he has managed both; he's compounded money at more than \(i = 0.20\), for a long time, \(n=60\) years. With this $100 becomes, \[A = $100  (1+0.2)^{60} = $5,634,514.\]
Example 4: This is in someways my favorite example, because it doesn't deal with material things. It is an insight that comes from this essay that I wrote about a while ago. I love the following quote:
What Bode was saying was this: "Knowledge and productivity are like compound interest.'' Given two people of approximately the same ability and one person who works ten percent more than the other, the latter will more than twice outproduce the former. The more you know, the more you learn; the more you learn, the more you can do; the more you can do, the more the opportunity - it is very much like compound interest. I don't want to give you a rate, but it is a very high rate. Given two people with exactly the same ability, the one person who manages day in and day out to get in one more hour of thinking will be tremendously more productive over a lifetime.

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