Advertised interest rates are typically nominal rates that are the annual interest rates disregarding any add-on fees and compounding. Effective interest rates, on the other hand, are what you actually pay. To calculate annual effective interest rates, consider the nominal or stated interest rate and how the lender calculates interest -- the effect of compounding. The effective interest rate can be used to figure actual interest paid on a personal loan or mortgage by using a simple formula.
Using a Formula
To understand how to calculate effective annual interest rates, it can help to practice the formula on small numbers. Suppose you borrow $1,000 to be paid back at 5 percent interest over a year in which the interest will be compounded monthly. Use the formula: r = (1 + i/n)^n - 1 and solve for "r"; in this formula, "i" represents the nominal rate and "n" stands for the number of compounding periods in one year. In this example, the effective interest rate is approximately 5.11 percent. This simplistic illustrates the power of compounding, as the effective interest rate will be more than the stated rate.
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Auto Loan Example
To figure the effective rate on an auto loan, plug the information into the formula. Suppose you purchase a new car with the final invoice totaling $22,339. You pay a down payment of $2,339 along with the tax and license and finance the remaining $20,000 at 6 percent interest compounded monthly for 48 months. Using the nominal 6 percent fixed rate, solving for r = (1 + 0.06/12) ^12 - 1, gets r = 1.0616778 - 1, or 0.061678; when changed to a percentage, it equals 6.1678 percent. This effective rate would mean that the monthly car payment would be $469.70 to be paid off in 48 installments. You would pay back a total of $22,545.60.
Why It's Important
Knowing the effective rate of interest is important because it shows the borrower exactly what he will pay just as it shows the investor actual earnings. In loans, to reduce the power of compounding, shorter terms are required. For example, working from the auto loan example of $20,000 at 6 percent nominal interest, if the term is reduced to 36 months, the monthly payments increase but the total to be paid back decreases to $21,888. By contrast, if the term is extended to 60 months, the monthly installment payment decreases to $387. However, although the nominal interest rate remains the same, it takes longer for the principal to decrease, thereby increasing overall interest paid on the loan. Over a 60-month period, the total to be paid back would be $23,220.
In real-world situations, such as evaluating the life of a mortgage contract, finding the effective interest rate requires knowing the principal amount, or the amount to be financed; the nominal interest rate; any additional loan fees or charges; the number of times each year the loan is compounded; and the number of payments to be made each year. For example, suppose a homeowner borrows $100,000 at 4 percent interest, compounded monthly, for a 15-year term and no fees were added on. The effective annual interest rate becomes 4.0742 percent. The homeowner would pay back a total of $133,144 in 180 monthly installments of $740 each.
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- The Mortgage Professor: Tutorial on Annual Percentage Rate (APR)
- University of Nebraska -- Lincoln Extension: Figuring the True Cost (APR) of Credit
- Washington State University: Installment Buying