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Chapter 5: Problem 73

For what kind of compound interest investments is the effective rate greater than the nominal rate? When is it smaller? Explain your answer.

Short Answer

Expert verified
The effective annual rate (EAR) is greater than the nominal annual rate (NAR) for investments with multiple compounding periods per year (n > 1), as compounding more frequently generates more interest. The EAR will be equal to the NAR only when there is one compounding period per year (n = 1) or annual compounding. The EAR will never be lower than the NAR for investments involving compound interest.

Step by step solution

01

Understanding the terms: Effective Annual Rate (EAR) and Nominal Annual Rate (NAR)

The effective annual rate (EAR) is the actual annual rate an investment yields after accounting for compounding. The nominal annual rate (NAR) is the stated annual interest rate without considering compounding within the year. To compare these rates, let's understand their formulas: EAR = \((1+\frac{NAR}{n})^{n}\) - 1 Where: EAR = Effective Annual Rate NAR = Nominal Annual Rate n = Number of compounding periods per year
02

Case 1: EAR is greater than NAR

The EAR will be greater than the NAR when there are multiple compounding periods per year (n > 1). In this case, the interest is compounded more frequently (i.e., quarterly, monthly, or daily), and that leads to a higher overall effective interest rate. Example: Let's consider a 10% NAR with semi-annual compounding (n = 2): EAR = \((1+\frac{0.1}{2})^{2}\) - 1 EAR = (1.05)^{2} - 1 EAR 鈮� 0.1025 or 10.25% In this case, the effective annual rate of 10.25% is greater than the nominal annual rate of 10%.
03

Case 2: EAR is equal to NAR

The EAR and NAR will be the same when there is only one compounding period per year (n = 1). In this case, the interest compounds annually, and the effective annual rate equals the stated nominal annual rate. Example: Let's consider a 10% NAR with annual compounding (n = 1): EAR = \((1+\frac{0.1}{1})^{1}\) - 1 EAR = (1.1)^{1} - 1 EAR = 0.1 or 10% Here, the effective annual rate of 10% is equal to the nominal annual rate of 10%.
04

Case 3: EAR is lower than NAR

The EAR will never be lower than the NAR for investments with compound interest. As the compounding frequency increases (greater n), the EAR will only be greater than or equal to the NAR. This is because compounding helps to generate more interest, which results in a higher effective annual rate. In conclusion, the effective annual rate (EAR) is greater than the nominal annual rate (NAR) for investments with multiple compounding periods per year. The EAR will be equal to the NAR only when there is one compounding period per year (annual compounding). The EAR will never be lower than the NAR for investments with compound interest.

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Most popular questions from this chapter

Are based on the following chart, which shows monthly figures for Apple Inc. stock in \(2008 .^{14}\) Marked are the following points on the chart: $$\begin{array}{|c|c|c|c|c|c|} \hline \text { Jan. 2008 } & \text { Feb. 2008 } & \text { Mar. 2008 } & \text { Apr. 2008 } & \text { May 2008 } & \text { June 2008 } \\ \hline 180.05 & 125.48 & 122.25 & 153.08 & 183.45 & 185.64 \\ \hline \text { July 2008 } & \text { Aug. 2008 } & \text { Sep. 2008 } & \text { Oct. 2008 } & \text { Nov. 2008 } & \text { Dec. 2008 } \\ \hline 172.58 & 169.55 & 160.18 & 96.80 & 98.24 & 94.00 \\ \hline \end{array}$$ Suppose you bought Apple stock in April. If you later sold at one of the marked dates on the chart, which of those dates would have given you the largest annual loss (assuming annual compounding), and what would that loss have been?

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