Sure, most people know that interest is the cost of borrowing money or the income from saving money. And they know that they should earn a return on their investments. However, when put in a specific borrowing, saving, or investing situation, people aren’t sure how interest is calculated and exactly what different interest rates mean. The better you understand interest, the better economic citizen and intelligent investor you will be.
In this post, I explain how to calculate interest and how interest works. I think you’ll find more than a few surprises as you go through it. The inherent problem in this post is that, by their very nature, interest and return on investment require calculations. However, most people have their enthusiasm for number crunching under control (except for accountants and actuaries). I don’t throw a bunch of mathematical formulas at you in this post (that may turn you off). Instead, I use transparent examples that demonstrate how to calculate interest and how it works. In sports, becoming a better player takes practicing and scrimmaging. In accounting, the best means for improving your understanding is practicing and scrimmaging with realistic examples as I usually show you in most my posts. Enjoy!
Simple Interest Calculation Scenario
Any explanation of interest has to start with what’s called simple interest— although, it’s not as simple as the term implies. The idea behind simple interest is that a certain amount of interest is paid or earned on a certain amount of money for a certain period of time, say one year. Suppose you put $1,000 in a savings account at the start of the year. At the end of the year your savings account was credited (increased) $40 for the interest you earned. The simple interest rate you earned is calculated as follows:
$40 interest earned / $1,000 amount invested for one year = 4.0 percent simple interest rate
But, what if you earned $10 each quarter instead of $40 at the end of the year?
Things get more complicated, as I explain on the next scenario example. Follow on…
Suppose you’ve filled out all the forms and convinced a bank to loan your business $100,000. The bank has examined your three “Cs”: character, collateral, and cash flow. The terms of the loan are that the bank will put $100,000 in your checking account today; the maturity of the loan is one year later; and, the annual interest rate on the loan is 6 percent. (The loan is renewable, but that’s another topic).
What amount do you have to pay the bank one year later, on the maturity date, to pay off this loan?
You can probably solve this problem in your head without doing any calculations on a hand held business/finance calculator or using Excel. I modify this simple example later in the post in order to explore several other important features of interest, so I want to be very clear about how interest is calculated for this basic example.
In this example, interest is added to the amount borrowed to determine the maturity value of the loan, which is the amount payable to the bank on the maturity date.
The amount borrowed, which is called the “face value [or principal]” of the loan, is the basis for calculating interest. The amount of interest is calculated as follows:
$100,000 amount borrowed today × 6 percent annual interest rate = $6,000 interest on loan
$100,000 amount borrowed today + $6,000 interest on loan = $106,000 maturity value of loan one year later
How Do You Do Such Calculations?
You’re probably asking, “How do I actually do the calculations in this post?” Well, you have two basic choices for doing interest and return on investment calculations: You can use either a business/financial calculator or the Excel spreadsheet program from Microsoft. I use both of these tools, with the complexity of the calculation making the actual determination.
If you use a computer-based spreadsheet, Excel is basically your only option. Fortunately, Excel runs equally well on both Windows and Macintosh computer platforms.
With both a business/financial calculator and Excel, you have to spend a little time familiarizing yourself with the tool and how to use it — that’s the rub. As I proceed with the examples in this post — which get more and more complicated — I include useful hints about doing calculations either with a hand-held business/financial calculator or in Excel. Keep reading…
The legal instrument used for the contract between a borrower and a lender is fairly complicated and has many clauses and provisions. Generally, the borrower signs a promissory note to the lender. You need a lawyer to fully explain the terms, conditions, obligations, and rights of each party under the loan agreement.
Suppose the lender does not refer to an annual interest rate and does not refer to the amount you borrow. You sign a legal instrument (probably a promissory note) that calls for the payment of $106,000 at the maturity date of the loan (one year later). The bank puts $100,000 in your checking account today.
Is the annual interest rate on this loan still 6 percent?
Yes, the annual interest rate is still 6 percent. Why? Well, the interest rate is calculated on the basis of the amount received at the start of the loan. So, the interest rate is calculated as follows:
$106,000 maturity value of loan one year later – $100,000 received today when signing the note payable = $6,000 interest for one year
$6,000 interest for one year / $100,000 received at start of loan = 6 percent annual interest rate
One year later you pay back to the bank 6 percent more than the bank loaned to you today, which means that the annual interest rate is 6 percent. Whether this interest rate is too high or not is up to you. You may want to shop around at different banks and other lenders to see if you can get a better interest rate.
Nominal and Effective Interest Rates
Assume a business borrows $100,000 for one year at an annual interest rate of 6 percent. One alternative is that the bank charges 6 percent simple interest that is payable at the end of the year. In this case the business pays the bank $106,000 (which includes $6,000 interest) at the maturity date one year later.
Now here’s a twist that happens all the time: Instead of 6 percent simple interest that is figured once a year, assume that the bank quotes a 6 percent annual interest rate that is compounded quarterly, which means it wants to be paid interest every three months.
Does this make a difference?
It sure does, and it makes the calculation of interest more complicated. Here how it works. Follow on…
When used in reference to an annual rate of interest, compounding refers to the frequency of charging (or earning) interest during the year. The annual rate has to be converted into the interest rate per period. Assuming the lender charges interest quarterly, the 6 percent annual rate is divided by four to get the 1.5 percent interest rate per quarter. In the example just introduced, the business could pay $1,500 at the end of each quarter ($100,000 × 1.5 percent interest rate per quarter = $1,500 interest per quarter). This way, the bank collects interest income earlier and can put the money to work sooner. On the other hand, the main reason for quarterly compounding may be to raise the effective annual interest rate.
Assuming quarterly compounding of the quoted 6 percent annual interest rate, determine the amount the business has to pay the bank at the maturity date of the note one year from now.
The 6 percent annual interest rate quoted by the bank is often referred to as the “nominal rate“. Nominal means in name only. In the example, the lender insists on quarterly compounding, which means that it charges a 1.5 percent rate each quarter. If the business doesn’t pay interest at the end of each quarter, the unpaid interest is added into the loan balance for the next quarter.
Accordingly, the amount owed to the bank one year later is calculated as follows:
The business owes the bank $106,136.36 at the maturity date of the note. Therefore, interest on the note is $6,136.36, which is $136.36 higher than the $6,000.00 simple interest the bank would have charged if the entire year had been treated as just one interest period.
So, there are two interest rates at work in this loan: the nominal rate (6 percent per year in this example) and the effective annual interest rate that gives effect to the compounding of the nominal rate:
. The effective annual interest rate on the loan in the example is 6.13636 percent, and is calculated as follows:
$6,136.36 interest for one year / $100,000.00 borrowed at the start of year = 6.13636 percent effective annual interest rate on loan.
. The true interest rate is the effective interest rate because this rate determines the actual payment of interest. The nominal interest rate is simply the point of departure for calculating the effective interest rate.
If the business in this example had agreed to a 6.13636 percent annual interest rate on the loan in the first place, the bank would have been willing to compound annually, which means once a year. At this higher rate, the bank would have ended up with the same amount of money. Essentially, the bank should be indifferent about whether it charges a 6.13636 percent annual interest rate that is compounded annually, or a nominal 6 percent annual rate that is compounded quarterly. It’s six of one, or a half dozen of the other.
Is it misleading to quote a nominal annual interest rate of 6 percent that in fact is compounded quarterly?
The “effective” or “real” annual interest rate isn’t 6 percent, but 6.13636 percent. You could argue that compounding is a sleight of hand trick for jacking up the true annual interest rate.
Financial institutions have to be careful to abide with federal and state laws for truth in lending in this regard. But these laws still leave lenders a fair amount of wiggle room in advertising interest rates. All I can do is to caution you that featuring nominal annual interest rates is common practice. If you’re loan-shopping, be sure to find out whether the nominal annual interest rate is compounded more frequently than once a year.
Banks (and other lenders) make loans to businesses on the discount basis. The business signs a note to the lender that calls for a certain amount, say $100,000, to be paid at the maturity date of the loan. But the note contains no mention of an interest rate and there is no mention of how much money the business receives. To determine the amount loaned to the business, the lender discounts the maturity value of the loan by deducting a certain amount from the maturity value. The difference between the maturity value and the amount loaned to the business is the interest.
A business signs a note to its bank that calls for $100,000.00 to be paid on the maturity date one year later. The bank deposits $93,295.85 in the company’s checking account on the day the note is signed. The note doesn’t refer to an interest rate.
Determine the effective annual interest rate on this loan and the nominal annual interest rate assuming that interest is compounded quarterly.
A problem like this isn’t that technical. Nevertheless, the calculations required to solve the problem are demanding. I recommend using a hand-held business/financial calculator to get an accurate answer. Here are step-by-step instructions for using a hand-held calculator to answer this problem:
Step-1. Select the TVM (time value of money) function.
Step-2. To find the effective annual interest rate on the loan enter 1 for N, which is the number of periods (one year).
Step-3. Enter $93,295.85 for the PV (present value).
Step-4. Enter $100,000.00 as a negative number for the FV (future value); this amount is negative because it has to be paid out to the bank at maturity.
Step-5. Press the INT (interest) key for the answer, which should be 7.19 percent (rounded). (On Hewlett-Packard calculators, this key is labeled I/YR, which stands for interest per year, but it really means interest per period.)
Step-6. After finding the effective annual interest rate, keep the same values in the registers for PV and FV, but enter 4 for N, the number of interest periods (the number of compounding periods). Then hit the INT key, and you get the quarterly interest rate, which should be 1.75 percent.
You multiply this percentage by 4 to get the nominal annual interest rate, which is 7 percent.
You can also use the RATE function in the financial set of functions in Excel to solve for the effective and nominal interest rates for this problem.
In the “nominal and effective interest rates” section earlier in this post, I explain how quarterly compounding increases the effective annual interest rate compared with the quoted nominal annual interest rate. In this and following sections, the effective interest rate is taken for granted. Instead of focusing on what happens within one year, the following discussion look at the effects of compounding over multiple years for investing and borrowing examples.
Quoted and Effective Interest Rates on Borrowing and Saving
When borrowing and saving, I would offer you the same advice concerning quoted interest rates and effective interest rates. For an example, assume a lender quotes you a 6 percent annual interest rate on your home mortgage loan. You make monthly payments, so the actual interest rate is 0.5 per- cent per month. This equals a 6.168 percent effective interest rate on your mortgage. In contrast, if you save money in your credit union and it pays 6 percent interest that is compounded monthly, you do not earn 6 percent, but rather 6.168 percent effective interest. If you had $10,000 in your account at the start of the year your balance is not $10,600.00 at the end of the year, but rather $10,616.80 — because of the monthly compounding effect.
In quoting mortgage interest rates, most lenders refer to the nominal annual rate (in this example, the 6 percent annual interest rate). They do not mention the effective annual interest rate, which takes into account the monthly compounding effect. The credit union in which you have a savings account probably advertises the 6.168 percent effective annual interest rate it pays on savings accounts. In general, lenders refer to their nominal annual interest rates, whereas institutions that want to attract your savings refer to their effective annual interest rates. But, you have to be careful out there and make sure you know which rate is being referred to.
Be careful: “compounding” in these longer-term contexts (which run 5,10, 20, or more years) takes on a different emphasis. Compounding in these long range settings refers to the exponential growth idea — that if something grows at a certain rate from year to year, over enough years its size will end up being two or more times larger than what you started with.
If you start with a population of, say, 10,000 persons in a town and its population grows 6 percent per year, its population will double to 20,000 in 12 years. You just invested your $10,000 year-end bonus in a 401(k) plan (a qualified tax-deferred retirement account). You don’t pay income tax on the $10,000 or on the earnings in your retirement account until you withdraw money from the account sometime in the future. You plan to retire in 20 years. Being conservative, you put the money in a savings account that pays 5 percent annual effective interest. (The interest rate could change in the future, but assume that the interest rate remains the same over all 20 years).
Assuming that your retirement account earns 5 percent annual interest for 20 years, what will be the balance in your account when you retire in 20 years?
In order to answer the question for the 20-year lifespan of the investment, you need to understand how year-to-year compounding works. Compounding means that you don’t withdraw your interest earnings each year. Instead, you reinvest the annual earnings. The result of compounding for, say, the first four years is shown as follows:
The total amount of your earnings over the first four years is $2,155.06. Someone may think that, at 5 percent, you earn $500.00 each year on your $10,000.00 investment, and over four years, you will have earned $2,000.00. But as you can see in the schedule, you earn $2,155.06. You reinvest your annual earnings, which means that, year-to-year, you have more money invested in your retirement account.
Over 20 years, your retirement account balance grows to $26,532.98. This amount assumes an annual 5 percent annual earnings rate and assumes that the financial institution you have your retirement investment account with doesn’t go belly up (The FDIC may insure your account, but that still doesn’t guarantee that you’ll get all your earnings).
Your total amount of earnings over 20 years is $16,532.98 (the future value less the $10,000.00 you started with). It may be useful to think of your total earnings as follows: At $500.00 per year interest, based on your initial investment, you earned $10,000.00 (=$500.00 per year × 20 years = $10,000.00). The other $6,532.98 of earnings over the 20 years comes from compounding (reinvesting) your earnings every year.
How did I get the answer?
To prepare the four-year schedule, I grabbed my trusty HP calculator: I entered 20 for N, 5 for I/YR, negative 10,000 for PV, and then I punched the FV key to get the answer. (By the way, be sure that the PMT [payment] key has zero entered.)
Before computer spreadsheet programs and hand-held business/financial calculators came along, you had to use a table look-up method to solve problems like this one. Some of the biggest disadvantages of this method are that tables of future values and present values don’t cover every situation, they’re clumsy to use, and they require you to do pencil and paper calculations by hand. Surprisingly, many college accounting and finance textbooks still include these tables. For the life of me, I don’t know why — it’s like teaching the Morse code when everyone has a telephone.
Borrowing and Investing in Installments
Borrowing and investing are most commonly done in installments. With this method, payments are made regularly to pay off a loan or to build an investment. In this section, I stick with interest-based investments and examples of fixed income investments and loans.
Paying off a loan [With Case Example]
Your business borrows $100,000 from a bank. You and the bank negotiate an installment loan in which you will pay off the loan over four years. The effective annual interest rate is 6 percent. The bank wants your business to amortize one-fourth of the principal amount each year. Amortize means to pay down the principal value of the loan. At the end of the first year, for instance, your business has to pay $25,000 on the principal balance of the loan plus interest for that year, and so on for the following three years. You sign the note to the bank and receive $100,000, which is deposited in your business’s checking account.
How much is each annual payment to the bank?
Probably the best approach to answering this question is to prepare an Excel spreadsheet to do the year-by-year calculations (Of course you could do the calculations with pencil and paper by hand, but the Excel program is much faster and less prone to calculation mistakes). The loan payment schedule is as follows:
Next, let’s try with a different scenario. Follow on…
Using the basic premise of the preceding question, suppose the bank wants equal payments at the end of each year (In the preceding answer, the total payment varies year to year).
What is the annual payment on the loan under these terms?
A question like this shows the value of a hand-held business/financial calculator, which is designed for the express purpose of solving problems like this one. You enter 4 for N (the number of periods); 6 in INT (the I/YR key on HP calculators); 100,000 in PV (the present value of the loan, or the amount borrowed); and 0 in FV (future value). The reason for entering 0 in FV is that you want the loan completely paid off and reduced to a zero balance at the end of the fourth year. Press the PMT (payment) key, and the answer pops up on the screen — $28,859.15. Each payment to the bank should be $28,859.15. The following schedule shows the proof of this answer. Compared with the schedule in the answer to the preceding question note that the annual payments are equal in this schedule:
You can see that the principal balance reduces to zero at the end of the fourth quarter. In this schedule, as the amount of interest goes down each quarter, the amount of principal amortization goes up.
Investing in a Retirement Account
Many people invest in tax-deferred retirement accounts on the installment, or serial basis. They put some money in their retirement accounts at the end of each pay period, and their employers may or may not make matching payments. The federal income tax law encourages setting aside money from wages and other sources of steady income to build up a retirement fund, such as a 401(k), IRA, and many other plans.
Assume that your employer encourages employees to invest money from their monthly salaries in retirement accounts, and you’ve decided to do so. Each month, you put $250 into your retirement account, and your employer adds $150, so $400 is invested each month.
To be conservative, assume that your retirement account will earn 4.8 percent income per year, compounded monthly (because you make monthly contributions). Although you may increase your monthly contributions in the future if your salary increases, at the present time, you can’t forecast an increase. So you assume that $400 will be contributed into your retirement account at the end of each month.
What will the balance be in your retirement account at the end of 20 years?
One way you can do this computation is to go to one of many Web sites that have retirement calculators. You enter your monthly contribution, the assumed rate of income per period, the number of years, and presto — the answer comes up on the screen. You can also use a business/financial calculator or the Excel spreadsheet program. In Excel, the FV function asks you to enter the same variables as a Web site retirement calculator and a business/financial calculator. The balance in your retirement account after 20 years is $160,670.
How do you know this answer is correct? Does it pass the common sense test?
You invest $4,800 per year for 20 years, which is a total investment of $96,000. If the answer is correct, you will earn more than $64,000 income over the 20 years.
Does this amount seem reasonable?
Your intuition isn’t particularly helpful here. To be reasonably certain that $160,670 is correct, you could program an Excel spreadsheet to see how your retirement balance accumulates month by month for 20 years. Or you could do the calculation a second time, to see if you come up with the same answer. Frankly, there’s no easy way to prove the calculation is correct. I’m 99.9 percent sure that my answer here is correct, but if you come up with a different answer please feel free to spot it through the comment form below.
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