How to Deal With Compound Interest
Why Does It Matter to Me?
You have a $1,000 sitting in your non-interest bearing checking account.
You know that your money will lose buying power if you continue to leave that $1,000 in there. The year is looming to an end, and you haven’t made any gains to beat inflation.
You want to transfer that money into a savings account that will generate interest, but how much exactly will you gain from doing this?
Cue compound interest.
Last week I mentioned that interest is usually compounded rather than computed simply whether you are investing money or borrowing a loan. Meaning, compound interest is both your best friend and also your enemy. So what is compound interest?
To better understand what compound interest means, first break this word pair into two separate sets of definitions. Compound means something that is made up of two or more elements. Interest means an amount of money paid in regular intervals for the use of someone else’s money at a certain rate. A rate is a fixed or variable price paid for borrowing that money. When you borrow a loan (and that could be credit cards, car loans, mortgages, etc.), you are charged with either simple or compound interest. Lenders are required by law to disclose whether interest will be accrued simply or it will be compounded. The same concept goes for investing money into accounts, and the amount of interest you receive in return will depend greatly on whether the interest is sustained simply or is compounded. In the cases of compound interest, you will either pay more or earn more with each transaction.
Compound interest means the going interest rate is calculated on the original principal and on the accrued interest from earlier periods of time when the money was borrowed or deposited. Let’s say you invested that $1,000 sitting in your checking account into a savings account with interest at 1.20%. Now that your worries of losing buying power has gone, you wanted to know your annual rate of return within 2 years. You do some calculations, and you come up with $24.14 as your gain over the course of two years, and your new balance is $1,024.14. But how do you find out the answer?
How to Calculate Compound Interest
There are plenty of available online calculators that can do the work for you, but it doesn’t hurt to know the algorithm behind those formulas. You can use both online and manual calculations to know for sure that the numbers are correct.
The formula for interest compounded annually can also be referred to as the formula for finding the future value of a single amount (a sum). The equation is expressed as follows:
So let’s calculate how you came up with $24.14 as your gain from moving your $1,000 into a savings account. Our principal would be $1,000 at a rate of 1.20% [.0120] with interest compounded once a year over the course of 2 years. Our equation would be: 1000(1 + .0120)²
First add the numbers within the parentheses to get 1.012. Then calculate 1.012² to get 1.024144. Finally, multiply the interest accumulated [1.024144] with your present amount [1000], and you should get $1,024.14 as your answer. Subtract your original amount from the new amount, and that is how you find your rate of return [$1,024.14 — $1,000.00 = $24.14].
Now what if you had an interest-bearing savings account that compounded interest more than once a year? The formula for this is expressed as:
After you put in your $1000 in this other savings account, you want to know how much interest you will gain over the course of two years when it’s compounded. For simplicity, the interest rate stays the same (.0120), and the interest is compounded 12 times a year (monthly).
Your equation should look like: 1000(1 + .0120/12)¹²*²
First compute the numbers within the parentheses. Divide the rate with the number of times interest is compounded, and you should get .001 as your answer. Then add .001 with 1, and you should get 1.001. Before you calculate 1.001 with the exponent, multiply the number of times interest is compounded with the amount of years, and you should get 24. Then calculate 1.001²⁴, and you should get 1.0242780347. In some calculators, you may have to close the parentheses when you are calculating exponents and any other mathematical input that requires parentheses. Once you get the accumulated interest [1.0242780347], you can finally multiply that with the principal invested [$1000] to get a total of $1,024.28. Subtract the original amount from the new one, and you can see that your return on investment is $24.28. That doesn’t seem too much of an increase from interest compounded annually, but the longer that money gets to sit in the account, the more significant the returns will be when you invest into a savings account that compounds interest more than once a year. So if your money is compounded monthly, you earn more money each time (assuming you don’t withdraw). You will have a little more money than you did before, and when it compounds again, your new amount will be calculated with the interest — not your original principal. In other words, you will accumulate wealth more quickly. The beauty of this is that the longer you have to wait, the bigger your rates of return will grow.
What if you wanted to know how much money you need to deposit now so that you will receive a certain amount in the future? The formula for that is referred to as the present value of a sum:
Let’s assume you want to earn $100 based on a 1.20% interest rate 3 years from now. Your formula should look like this: 100/(1 + .0120)³
First add the numbers within the parentheses to get 1.012. Cube it to get 1.036433728 [1.012³ = 1.036433728]. Divide the future amount desired with the accumulated interest to get the total of $96.48 [100/1.036433728 = 96.4847025897]. You would have to invest $96.48 to get $100 from a savings account that bears 1.20% interest for three years.
Now you know the formulas for calculating interest on a sum, how do you calculate interest for a series of payments? You make a regular fixed income, and you wanted to invest a portion of that. What if you wanted to know how much interest you will get from regular additions to your savings? The formula for the future value of a series of equal payments (an annuity) is shown as follows:
In order to perform this formula correctly, we must assume that:
- Each deposit is the same amount.
- The interest rate remains the same.
- The deposits are made at the end of each time period.
Let’s say you want to deposit $1000 at the end of each year for the next 3 years in a savings account earning 1.20% interest. Your formula should look like: 1000[(1 + .0120)³ — 1/.0120]
First add the numbers within the parentheses, which should be 1.012. Calculate 1.012³ to get 1.036433728. Subtract that with 1 to get 0.036433728, and then divide that by the interest rate (.0120) to get 3.036144. Finally, multiply that with the annuity amount to get $3,036.14 as your total over the course of three years [1000 * 3.036144 = 3036.14]. Each year you contributed $1000 to your savings, and you accumulated $36.14 in interest at a rate of 1.20%.
There is, of course, a formula for the present value of an annuity in case you wanted to know how to calculate the amount you need to save to receive a certain amount in the future. The equation is expressed as:
The rules stated for the future value of an annuity also applies to this formula. If you wanted to know how much money you will be left with after withdrawing money from your savings, this is the formula to use. Let’s say you withdraw $50 at the end of each year over the course of 3 years with a rate at 1.20%. Your formula should look like: 50[1–1/(1 + .0120)³/.0120
First add the numbers in the parentheses, and you should get, of course, 1.012. Calculate 1.012³ to get 1.036433728. Divide 1 with 1.036433728 to get 0.9648470259, then use that to subtract from 1 to get 0.0351529741. Divide that with the interest rate to get 2.9294145083, and then multiply that to get $146.47 as the total amount leftover.
What About Interest Compounded Continuously?
There is one last formula I wanted to go over, and that formula is used to calculate interest compounded continuously. Interest of this kind is compounded faster than every second, and causes your wealth to grow exponentially. This is how the rich quickly become even more rich, by using continuously compounded interest to their advantage.
This calculation involves the use of natural logarithms and exponential functions, and is expressed as follows:
The number “e” is used as a limit as “n” takes on larger and larger values, approaching infinity. “e” is the limiting value for when the amounts of the number of times interest is compounded gets too big to calculate. If you were to invest your $1000 into an account that compounds interest continuously at a rate of 1.20% over 3 years, you should get $1,036.66. Continuously compounded interest is almost, if not completely, impossible to calculate in your head, so go ahead and use a calculator. Usually, you put in the principal amount first (1000) times “e,” followed by an exponent of (.0120 * 3). Thus, you will earn $36.66 over three years with interest compounded continuously. And if you let that money sit there for a long period of time, you should be able to see your investment grow exponentially.
Continuously compounded interest is used quite frequently because it measures easily over multiple time periods and it is consistent with time.
Applaud this to your network so they can become financially literate too!
Check out my other articles:
Never Let Yourself Work Forever
Should You Jump On the Bitcoin Bandwagon?
