Compounding is the calculation of interest on interest. A common example is the balance on a credit card. So long as there is an outstanding balance, the interest owed is calculated on the entire balance, including any part of it that was previously calculated as interest and added to the balance in earlier months.
We will be using two formulas from Chapter 21: the compound interest formula (page 875) and the savings formula (page 884), phrasing them for loans. Here is the compound interest formula, followed by an example.
Compound Interest Formula RULE
If a principal is loaned at interest rate per compounding period, then after compounding periods (with no repayment), the amount owed is
To make the connection to multiple compoundings per year, we give the formula in a slightly more elaborate and detailed version below.
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General Compound Interest Formula RULE
For a principal loaned
the amount owed after years (hence after compounding periods), with no payment meanwhile of interest or principal, is
EXAMPLE 2 What Happens If You Don’t Make the Payments on the Principal?
After you begin to repay your federal direct student loan, what happens? The loan is capitalized, meaning that any unpaid interest accrued while you were in school and during the grace period is added to the principal and becomes the new balance on which interest is to be paid. That balance will decline as you amortize (pay off) the loan. You will be responsible for fixed monthly payments. The interest is calculated monthly, and the proportion of your payment going to interest will decline as you pay off the loan.
What if you do not make the payments due for the repayment? Well, the interest accrues and is capitalized every quarter into the principal: The interest is added to the loan balance (the amount owed), and interest for the next payment is calculated on the new loan balance. In other words, the compounding period is one quarter.
Suppose that six months after you graduate, when the grace period expires and you have to begin making payments, you owe $10,000 at 4.29% interest per year. However, you haven’t been able to find a job, so you fail to make any payments for the next year. How much would you owe then?
The principal is $10,000. The quarterly interest rate is and there are compounding periods. The compound interest formula gives the new amount owed as .
(It would be very foolish to just ignore the payments due, since after 270 days of nonpayment, the loan would be in default and all kinds of bad things would happen! Instead, you should contact the loan server about possible reduction or postponement of payments. Also, by executive order of President Obama, after December 2015 student loan repayments will be capped at 10% of the borrower’s monthly income.)
Suppose that your federal student loans total $20,000 and you fail to make the first six monthly payments. How much does the loan grow to?
Terminology for Loan Rates
The interest on a loan depends on whether compounding is done and how the interest is calculated.
A nominal rate is any stated rate of interest for a specified length of time. For instance, a nominal rate could be a 1.5% monthly rate on a credit card balance. By itself, such a rate does not indicate or take into account whether or how often interest is compounded.
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The effective rate takes into account compounding. It is the rate of simple interest that would realize exactly as much interest over the same period of time.
As we saw, a student loan of $10,000 at a yearly interest rate of 4.29% (a nominal rate), calculated as 1.0725% per quarter compounded quarterly, yields $436.00 in interest owed at the end of the year, which is 4.36% of the original principal. Hence, the effective annual rate is 4.36%. In other words, a $10,000 loan at simple interest of 4.36% for one year would owe exactly the same interest.
When stated per year (“annualized”), the effective rate is called the effective annual rate (EAR). (In connection with savings, the effective annual rate is the annual percentage yield discussed in Section 21.3.) Thus, for the student loan example above, the EAR (rounded) is 4.36%.
To keep the rates straight, we use for a nominal rate for the specified compounding period—such as a day, a month, or a year—within which no compounding is done; this rate is the effective rate for that length of time. For a nominal rate compounded times per year, we have . For that $10,000 student loan at 4.29% compounded quarterly, we have and , so per quarter.
Just as the Truth in Savings Act (mentioned in Chapter 21, page 874) does for savings, the Truth in Lending Act establishes terminology and calculation methods for interest for loans. The Truth in Lending Act introduced the term annual percentage rate (APR).
Annual Percentage Rate (APR) DEFINITION
The annual percentage rate (APR) equals the number of compounding periods per year times the rate of interest per compounding period:
For the student loan in repayment, the interest is compounded quarterly, or times per year, and the interest rate for the compounding period is ; so the APR is . The APR is the rate that the Truth in Lending Act requires the lender to disclose to the borrower. The APR is usually smaller than the effective annual rate (EAR). In the case of the student loan of Example 2, the APR is 4.29%, while we calculated subsequently that the EAR is 4.36%. Spotlight 22.1 explains further.
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What’s the Real Rate? Spotlight 22.1
Financial experts agree that the real, “true” rate of interest for savings or loans is the effective annual rate.
The 1991 Federal Truth in Savings Act requires that savers be told the annual percentage yield (APY) (discussed on page 878 in Chapter 21), which is just the effective annual rate.
The 1968 Federal Truth in Lending Act, however, requires that borrowers be told the APR, which is not the same as the effective annual rate. The APR is the rate of interest per compounding period times the number of compounding periods per year. Thus, a credit card rate of 1.5% per month translates to an APR of 18%. The APR does not take into account compounding. Hence, it is not equivalent to—indeed, it understates—the true cost of borrowing; that is, the effective annual rate. For the credit card loan, with monthly compounding, the effective annual rate is
The APR also ignores costs that are sometimes involved in borrowing, such as a flat charge for making the loan in the first place (called a “loan-processing fee”), charges for late payments, and charges for failing to make a minimum payment.
In 2015–2016, Federal Direct Loans had a 1.073% origination fee, and one-half of the loan is disbursed at the start of each semester; both these factors raise the effective interest rate. For Federal Direct PLUS Loans to parents of students, the origination fee is 4%. The fees are intended to cover in part the cost of loans that default.
For home mortgage loans, however, the Truth in Lending Act requires that lenders include in the APR some of the upfront costs referred to as “closing costs”: any “loan origination” fee, “loan-processing” fee, and “points” (additional charges to get a reduced interest rate). The APR does not include title insurance, appraisal, credit-report fees, or transaction taxes.
Closing costs are paid at the closing of the sale, while interest is paid over the life of the loan. However, the APR treats the closing costs included in it as if they will be amortized over the term of the mortgage, despite the fact that they were paid beforehand. Here, too, the APR understates the true costs.
However, very few people hold a mortgage to its maturity. The median life of a 30-year mortgage is only about 5 years; that is, half of all mortgage holders pay off their mortgage before 5 years are up, usually because they sell their homes and move elsewhere. Thus, for almost all home loans, the APR also includes interest that will never be paid.
Also, we must take into account inflation. One advantage of buying a home with a fixed-rate mortgage is that your payment stays the same, but your earnings and the value of your home are likely to go up with inflation. You are thus paying back the loan with dollars of lesser value. For any loan in a time of inflation, Fisher’s effect comes into play: If your loan has an effective annual rate of 7% but inflation is running at 3.5% per year, the true cost to you of the loan is not exactly . Instead, for an effective annual rate of and an inflation rate of a, the cost of the loan at the beginning of the first year is indeed (3.5% in our example), but at the end of the first year, it is
For and , we get . The reason that this is less than the expected 3.5% is that at the end of the first year, you are paying back the loan with dollars that have been inflated for a year. As inflation mounts over the term of a mortgage, the cost goes down steadily each year. For example, at the end of 5 years of steady inflation at 3.5%, the total inflation has been , and we have .
A final—and major—consideration is that interest paid on your home mortgage is deductible from taxable income on federal, state, and some local income tax returns. Thus, your home ownership is subsidized by other taxpayers (just as you help subsidize other home buyers), and the cost to you of the loan is reduced further.