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Effective percentage rate for various compounding periods: We have seen that in spite of its name, the annual percentage rate (APR) does not generally tell directly how much interest accrues on a loan in a year. That value, known as the effective annual rate, \({ }^{17}\) or EAR, depends on how often the interest is compounded. Consider a loan with an annual percentage rate of \(12 \%\). The following table gives the EAR, \(E=E(n)\), if interest is compounded \(n\) times each year. For example, there are 8760 hours in a year, so that column corresponds to compounding each hour. $$ \begin{array}{|c|l|} \hline n=\text { Compounding periods } & E=\text { EAR } \\ \hline 1 & 12 \% \\ \hline 2 & 12.36 \% \\ \hline 12 & 12.683 \% \\ \hline 365 & 12.747 \% \\ \hline 8760 & 12.750 \% \\ \hline 525,600 & 12.750 \% \\ \hline \end{array} $$ a. State in everyday language the type of compounding that each row represents. b. Explain in practical terms what \(E(12)\) means and give its value. c. Use the table to calculate the interest accrued in 1 year on an \(\$ 8000\) loan if the APR is \(12 \%\) and interest is compounded daily. d. Estimate the EAR if compounding is done continuously - that is, if interest is added at each moment in time. Explain your reasoning.

Step by step solution

01

Understanding Part (a)

For part (a), we need to identify the compounding frequency based on the values in the column for compounding periods \(n\).- **\(n = 1\)**: Compounded annually- **\(n = 2\)**: Compounded semi-annually- **\(n = 12\)**: Compounded monthly- **\(n = 365\)**: Compounded daily- **\(n = 8760\)**: Compounded hourly- **\(n = 525,600\)**: Compounded every minute

02

Explaining Part (b)

In part (b), \(E(12)\) refers to the effective annual rate when interest is compounded monthly. This means the interest is calculated and added to the principal 12 times in a year. The table gives \(E(12) = 12.683\%\).

03

Calculation for Part (c)

In part (c), we calculate the interest accrued when interest is compounded daily using \(E = 12.747\%\). Given the principal is \$8000, the interest, \(I\), for one year is calculated using the formula:\[I = P \times \left(\frac{E}{100}\right) = 8000 \times \left(\frac{12.747}{100}\right) = 8000 \times 0.12747 = 1019.76\]

04

Estimating Part (d)

The effective annual rate for continuous compounding can be estimated using the limit of the compounding formula as \(n\) approaches infinity. The formula is \(E = e^r - 1\), where \(r\) is the annual interest rate (as a decimal). Given \(r = 0.12\), we approximate\[E \approx e^{0.12} - 1 \approx 1.1275 - 1 = 0.1275\,\]which translates to an EAR of approximately \(12.75\%\).

05

Conclusion

We've analyzed the table to understand the compounding types at various frequencies, detailed what \(E(12)\) represents, calculated interest accrued with daily compounding, and estimated continuous compounding.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Annual Percentage Rate (APR)

The Annual Percentage Rate, commonly known as the APR, is a financial term that describes the yearly interest rate charged on a loan or earned through an investment without accounting for the effects of compounding. Imagine you take out a loan with a stated APR of 12%. At first glance, you might think this means you would pay exactly 12% in interest at the end of one year. However, APR doesn't paint the whole picture when it comes to how much you actually pay or earn.

APR is calculated by multiplying the periodic interest rate by the number of periods in a year. It gives you a sense of how much interest you might end up paying or earning annually, but it does not include compounding. Therefore, it's essential to understand that while APR gives you a starting point, you need to know how often the interest is compounded to understand the true cost or earning potential fully. This is where the Effective Annual Rate (EAR) comes into play to give more context.

Compound Interest

Compound interest occurs when the interest you earn upon or pay adds back to the principal, causing you to earn or pay interest on interest in the subsequent periods. This is a cycle wherein each period new interest is calculated based on the total amount (principal + accumulated interest) rather than just the original principal amount.

Compound interest can be calculated using the formula:\[A = P imes igg(1 + \frac{r}{n}\bigg)^{nt}\]Where:

• A is the amount of money accumulated after n periodic compounding over t time,
• P is the principal amount,
• r is the annual interest rate,
• n is the number of times interest is compounded per year,
• t is the time in years.

Compound interest benefits savers because it results in more profit over time compared to simple interest, which does not reinvest earned interest.

Continuous Compounding

Continuous compounding is a concept where the frequency of compounding reaches an extreme. Instead of being compounded annually, monthly, daily, or even hourly, it's compounded at every possible moment. Hence, theoretically, it's as if the compounding activities happen all the time.

While it may sound theoretical, continuous compounding is a useful concept in fields like calculus and finance. The formula used to calculate the amount with continuous compounding is:\[A = P imes e^{rt}\]Where:

• A is the future value,
• P is the principal amount,
• e is the base of the natural logarithm (approximately 2.71828),
• r is the annual interest rate (decimal),
• t is the time in years.

Using continuous compounding provides a higher value at the end than any other compounding frequency for the same interest rate, demonstrating the power of compounded growth.

Interest Accrued

Interest accrued is the total amount of interest earned or payable over a certain period. It is important to differentiate between simple interest and compound interest when calculating accruals, as compound interest involves adding interest to the principal, which then earns interest in the subsequent periods.

To calculate interest accrued using Effective Annual Rate (EAR) for compound interest, use:\[I = P imes \frac{E}{100}\]Where:

• I is the interest accrued,
• P is the principal,
• E is the effective annual rate (as percentage),

Understanding the concept of interest accrued is essential for making informed financial decisions, whether it's taking a loan or investing money. It tells you the actual interest cost or earnings after considering the compounding effect, allowing for better financial planning.