10.3 Compound Interest 609 Many banks compound interest daily. When computing the effective annual yield, they use 360 for the number of periods in a year. To determine the effective annual yield for any interest rate, calculate the amount using the compound interest formula where p is $1. Then subtract $1 from that amount. The difference, written as a percent, is the effective annual yield, as illustrated in Example 3. The sign in the margin shows interest rates and the corresponding annual percentage yields (APY) that were available for certificates of deposit (CDs) from Crestview Credit Union on August 21, 2023. Crestview Credit Union was compounding interest daily. To determine the effective annual yield (or APY) for the 48-month CD, calculate the amount of interest earned on $1 for 1 year compounded daily. A 1 1 0.0306 360 1.03107 (360 1) = ⎛ + ⎝⎜ ⎞ ⎠⎟ ≈ ⋅ From this result, subtract 1 to obtain 1.03107 1 0.03107, − = or 3.107%. Confirm that the other annual percentage yields shown on the sign are correct. When making financial decisions, investors usually want to choose the investment that provides the highest interest rate. However, it can sometimes be difficult to compare interest rates. For example, if one investment states a compound interest rate and another investment states a simple interest rate, it may be difficult to choose the better investment. To help make such comparisons, we can compute the effective annual yield. TECHNOLOGY TIP There’s a Compound Interest App for That! Many of the concepts that we discuss throughout this section and throughout this entire chapter can be further explored with the use of an app on your smartphone or tablet computer. Several of these apps allow you to calculate the accumulated amount in an account using the compound interest formula. Users enter the principal, annual interest rate, number of compounding periods, and the time in years, and the app will calculate the total amount accumulated. As always, consult with your instructor before using these apps when completing work related to your course. Suppose in Example 2, we had invested $1 at 4% compounded semiannually for 1 year. A 1 1 0.04 2 1.0404 (21) = ⎛ + ⎝ ⎞ ⎠ = ⋅ To get the interest earned for 1 year, subtract the initial investment of $1: i 1.0404 1 0.0404 = − = The amount of interest earned on $1 for 1 year, written as a percent, is 4.04%. For this investment we say that the effective annual yield is 4.04%. Most financial institutions refer to the effective annual yield as the annual percentage yield (APY) . An investment with an interest rate of 4% compounded semiannually has an APY of 4.04%. This means an investment with an interest rate of 4% compounded semiannually provides the same amount of interest as an investment with a simple interest rate of 4.04%. Crestview Credit Union CD Rates 12 mo 3.780% 3.852% 24 mo 3.450% 3.510% 36 mo 3.110% 3.159% 48 mo 3.060% 3.107% * Annual Percentage Yield Rate APY* Type Definition: Effective Annual Yield The effective annual yield or annual percentage yield (APY) is the simple interest rate that gives the same amount of interest as a compound rate over the same period of time.
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