# Compound Interest

An interest rate is the price of credit. Interest can be considered from two standpoints. It is both the compensation an investor receives for lending his money; and the price the borrower must pay for credit.

The amount that is invested today is referred to as the *principal*, or *present value*, while the amount that the investor will receive in the future is called the *future value*. The investment may be structured to pay either simple interest or compound interest. Simple interest means that interest will be paid on the principal only. If you deposit $1,000 into an account that pays 10% simple interest, you will earn $1,000 x 0.10 = $100 in interest the first year and $1,000 x 0.10 = $100 in interest the second year and every year after that.

Most accounts pay compound interest, which means you also earn interest on the interest you have accumulated. Therefore, if you deposit $1,000 into an account that pays 10% compound interest, you will earn $1,000 x 0.10 = $100 the first year, just as you would if the account paid simple interest. However, in the second year, you will earn 10% on the $1,100that is now in your account: interest earned = $1,100 x 0.10 = $110. Thus, your account balance at the end of the second year will total $1,000 + $100 + $110 = $1,210. You can imagine how cumbersome it would be to determine how much you would have in the account at the end of 5, 10, or 20 years if you had to calculate the interest earned on the new principal balance each year. Fortunately, there is a formula that calculates it easily:

FV = PV x (1 + i)^{n}, where

FV = future value;

PV = present value, or principal;

i = interest rate per period; and

n = number of periods

To illustrate, if you deposit $1,000 in an account that pays 5% annual interest and leave it in for 20 years, you will have $2,653.30 in the account at the end of the 20 years.

FV = $1,000 x (1.05)^{20} = $1,000 x 2.6533 = $2,653.30

You will have earned a total of $1,653.30 in interest over the 20-year period. Had the account paid simple interest, you would have earned only $1,000 (= $50 annual interest x 20 years).

Note that in the formula provided, i is defined as the interest rate *per period*, and n is the number of periods. Some accounts pay interest more than once a year. They may pay interest daily, monthly, quarterly, semiannually, or even continuously. All else equal, the more frequently interest is compounded, the greater the amount of interest earned. For example, assume that instead of paying 5% compounded annually, the account pays 5% compounded quarterly. In this case, the 5% is referred to as the stated annual rate. It doesn’t mean you will earn 5% each quarter. Instead, you will earn 5% ÷ 4 = 1.25% each quarter. Nevertheless, you will accumulate more in that account over time since you will be earning interest on your interest each quarter. If you deposit $1,000 into the account and leave it in for 20 years, you will earn 1.25% each quarter for 80 quarters. Applying the formula, we use i = 1.25% and n = 80:

FV = $1,000 x (1.0125)^{80 }= $1,000 x 2.7015 = $2,701.50

You will have earned $1,701.50 in interest over the 20-year period instead of $1.653.30. Obviously, the higher the interest rate earned, the greater the difference in the dollar amount of interest earned will be. Of course, if you are the borrower, the more frequently interest is compounded, the more interest you will be paying on the loan.

When interest is compounded continuously, it is compounded at the smallest possible time intervals, and we must use the mathematical constant e to calculate the future value, where e = 2.71828. The formula becomes:

FV = PV x e^{(i x n)}, where i is the stated annual interest rate and n is the number of years.

If we deposit $1,000 into an account that pays 5%, compounded continuously, and leave it in for 20 years, we will have $2,718.28 in our account at the end of 20 years:

FV = $1,000 x 2.71828^{(0.05 x 20) }= $1,000 x 2.71828^{1.0 }= $1,000 x 2.71828 = $2,718.28

Of this amount, $1,718.28 is interest earned.

**Compound annual growth rate**

The compound interest formula is also often used to calculate compound annual growth rates (CAGR). We simply solve the equation for i, which also happens to be the compound annual growth rate:

FV = PV x (1 + i)^{n}, so

(1 + i)^{n}= FV/PV, and

1 + i = (FV/PV)^{1/n}, thus

i = (FV/PV)^{1/n} – 1

For example, suppose a start-up company had $500,000 in sales in its first year of operation. Ten years later, sales had grown to $1,500,000.

CAGR = i = ($1,500,000/$500,000)^{1/10} – 1 = 11.6%

Thus, the sales of the company grew at an average annual rate of 11.6% over the 10-year period. This can be useful information, but bear in mind that this calculation ignores all the sales in between the first and tenth year of operation. It’s possible that the firm even experienced a decline in sales in a couple of years, but had significant sales growth in others.

All of the above calculations are very easy to do using a financial calculator or a spreadsheet, such as Excel. However, should you have neither on hand, you can always resort back to the original formula, in which case, even the calculator built into your smartphone will be able to produce the correct answer.

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