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# Compound interest

Compound interest is, in a nutshell, interest upon interest. That is, when an interest payment is added to the principal and then the whole thing (principal + interest) earns interest.

## Expanded Definition

Compound interest has been called "the eighth wonder of the world" and is the engine that drives supersized returns for investors who start young or invest for a very long time. If an investment is paying "simple" interest, then it is paying the same amount every time, like an annuity and the total value increases at a steady, plodding pace. If it is paying compound interest, on the other hand, then it is paying an increasingly larger amount each time, and the total value increases exponentially.

Two factors determine the power of compound interest:

• Time
• Compounding period

## Simplified explanation (by a 15 year old):

Double.

It’s a really simple word, and it might seem simple to comprehend as well. If you double one, it’s two, two doubled is four, four doubled is eight, and so on. But when you put it in a real life situation, it’s so much more complex and harder to fathom.

In elementary school, we were given a story about a girl who asked for one grain of rice, doubled every day for a month. And at the end of 30 days, she had over a billion grains of rice. Over a billion. Just by doubling a single grain of rice over the course of 30 days.

What’s even more unbelievable is that by folding a dollar bill in half just 50 times, it’d be the same distance as traveling from the earth to Mars – and back. If you folded it in half another time, it would be as long as a trip to Mars, back to earth, back to mars, and back to earth again. That’s two round trip tickets to Mars, the soon-to-be-inhabited planet.

Of course, it’s not possible to create a dollar bill large enough to be able to fold 50 times, not to mention the impossible task of folding something millions of miles thick all the way till it reaches another planet. However, numbers don’t lie and that’s how we know it’s possible.

It’s all really just math.. I mean, folding a dollar bill in half is really just doubling its thickness. Just think about that – something that’s not even 1/100 of an inch thick becomes 50 million miles after just 50 doubles. One more double would make it 100 million miles.

When the numbers are on a smaller scale like two, it doesn't seem like much. You just added two and it made four. Then you add four to four and it’s eight and that’s a double. When on a larger scale, however, the numbers seem more impossible. 500,000 is only halfway to a million, but that’s all you need. One more double and you’re a millionaire.

I’m pretty sure majority of people have a hard time comprehending the reality of it. I know I did. I guess that’s why people don’t really understand the stock market and how much money you can gain if you just wait for the right amount of doubles. Once you can wrap your head around that and partially understand it, time and patience (and a dash of luck) is all you really need to become a millionaire in the game of stocks. Well, those are the main stuff you’ll need.

Of course, there are numerous other factors to be taken into consideration such as how much money you put in and how often you put in. Consistently adding money to your stock will help aid it to become bigger and better. You should also find one person to follow. Someone with expertise in stock choosing and a reputable track record (like David Gardner, cough cough).

But really, I guess the ultimate message of this is that duplication is a very powerful thing, and in the stock market, it’s the reason why all long term investors are sitting on the edge of their seat – they’re just waiting for a another double. It’s just like folding a bill in half.

### Time

The longer the compounding has to work, the more impressive are the results. For instance, suppose you invest \$10,000 into a Certificate of Deposit paying 5%, compounded quarterly. By "compounded quarterly," I mean that the interest is added to the principal every 3 months and begins to earn interest upon itself.

The more time that passes, the greater the contribution is from the interest upon the interest. Here's a table that shows this. Pay attention to the rightmost column, where the amount contributed just from interest on interest is shown.

 Year Simple Compounding Additional that yearfrom compounding 0 \$10,000.00 \$10,000.00 \$0.00 1 \$10,500.00 \$10,509.45 \$9.45 2 \$11,000.00 \$11,044.86 \$35.41 3 \$11,500.00 \$11,607.55 \$62.68 4 \$12,000.00 \$12,198.90 \$91.35 5 \$12,500.00 \$12,820.37 \$121.48 7.5 \$13,750.00 \$14,516.13 \$203.68 10 \$15,000.00 \$16,436.19 \$296.76 20 \$20,000.00 \$27,014.85 \$805.56 30 \$25,000.00 \$44,402.13 \$1,652.43

### Compounding period

The more often interest is added to the principal, the greater the effect is from compounding. This makes sense because you are getting that "interest on interest" to work sooner.

Here's a table showing how compounding period affects the balance of an identical investment of \$10,000 earning 5% interest, but compounded annually, quarterly, monthly, or daily.

 Year Annually Quarterly Monthly Daily 0 \$10,000.00 \$10,000.00 \$10,000.00 \$10,000.00 1 \$10,500.00 \$10,509.45 \$10,511.62 \$10,512.68 2 \$11,025.00 \$11,044.86 \$11,049.41 \$11,051.63 3 \$11,576.25 \$11,607.55 \$11,614.72 \$11,618.22 4 \$12,155.06 \$12,198.90 \$12,208.95 \$12,213.86 5 \$12,762.82 \$12,820.37 \$12,833.59 \$12,840.03 10 \$16,288.95 \$16,436.19 \$16,470.10 \$16,486.65 20 \$26,532.98 \$27,014.85 \$27,126.40 \$27,180.96 30 \$43,219.42 \$44,402.13 \$44,677.44 \$44,812.29

### The math

OK, here is how it works. If you can do multiplication and division, you can figure out compounding interest, 'cause that's all it is. The only difficulty lies in figuring out how many times you do the multiplication.

First, you need to know what the annual percentage rate (APR) is. In the example above, this was 5%. Second, you need to know how often the compounding occurred every year. That is, the number of compounding periods. In the above table, this was once (annually), four times (quarterly), 12 times (monthly), or 365 times (daily). Now, to the math.

Step 1:
Take the APR and divide by the number of compounding periods. Let's work with quarterly, so that would be 5% / 4 = 1.25% = 0.0125. This is the periodic interest rate.

Step 2:
Now add the periodic interest rate (1.25%) to one. 1 + 0.0125 = 1.0125. By doing this, your answer will show what the new total value will be, not just how much interest is earned in a given period.

Step 3:
Then, multiply that number by itself a total number of times equal to the number of compounding periods. 1.0125 * 1.0125 * 1.0125 * 1.0125 = 1.0125^4 = 1.050945... To do this with a calculator (such as the one in Windows -- you need the scientific calculator found under "View"), type in 1.0125, hit the x^y key, type in 4, and hit enter.

Step 3a:
If you want to figure out for a longer time period, figure out how many total compounding periods there will be. For two years, quarterly, it's eight. For 10 years, it's 40, and so on. So 10 years at 5%, compounded quarterly, you'd have 1.0125^40 = 1.643619.

Step 4:
Now, multiply that by your original amount invested (the principal). For \$10,000, you should get \$10,509.45 after one year (four periods).