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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

### 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).