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Chapter 10: Problem 23

Find the APR of a bond that doubles its value in 12 years. Round your answer to the nearest hundredth of a percent.

Short Answer

Expert verified
The APR of the bond that doubles its value in 12 years is approximately 6.12%.

Step by step solution

01

Setting up the compound interest equation

The compound interest formula is \( A = P(1 + r/n) ^ {nt}\), where 'A' is the future value of the investment, 'P' is the principal amount, 'r' is the annual interest rate in decimal form, 'n' is the number of times that interest is compounded per year, and 't' is the time in years. The term APR is used in context of a year so 'n' would be 1 as the compounding here would be done annually. Since the bond doubles its value in 12 years, our 'A' would be 2P and 't' would be 12. So our equation becomes \(2P = P(1 + r) ^ {12}\)
02

Solving for 'r'

By dividing both sides of the equation by P, we can simplify it to \(2 = (1 + r) ^ {12}\). To solve for 'r', we need to take the 12th root of both sides. As such, the equation becomes \( (1 + r) = 2^{1/12}\) . Subtracting 1 from both sides, we get r to be \(2^{1/12} - 1\)
03

Getting the APR

After finding 'r' in the previous step, we now convert 'r' into a percentage terms to find the APR. Since 'r' is expressed in decimal form, we will multiply 'r' by 100% to convert it into a percentage. The final answer for APR then becomes \((2^{1/12} - 1) * 100%\)

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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 abbreviated as APR, represents the cost of borrowing or the profit made on investments. It is expressed as a percentage and includes both the interest rate and all the associated fees over a year. For students dealing with bonds, loans, or savings accounts, understanding APR is crucial.
  • APR allows you to compare different financial products by standardizing the cost of a loan or the yield of an investment annually.
  • For the bond in our example, the APR illustrates how much the investment will grow in one year.
  • In our exercise, converting the calculated annual interest rate into an APR provides a clear, comparable number.
We derived the APR from the decimal form of the interest rate by multiplying by 100, which makes it easier for comparison with other financial products.
Future Value
Future Value refers to the amount of money an investment will grow to after interest has been applied over a specific period. In our exercise, the future value was double the original investment, effectively showing the power of compound interest.
  • Future value is crucial in determining how much you can expect from your current investments.
  • For our bond, the future value being twice the principal indicates a 100% growth over 12 years.
  • This concept helps in planning financial goals, such as retirement savings.
Knowing the future value gives investors a clear view of potential earnings and aids in making informed financial decisions.
Interest Rate Calculation
Interest rate calculation is the process of determining the rate at which invested money will grow over time. This involves several steps and is foundational in finance for understanding investment growth.
  • Calculating the interest rate involves using the formula: \(2 = (1 + r)^{12}\), where the base of compound interest can significantly affect the growth of investments.
  • The exercise used the compound interest formula to solve for the rate \(r\), which revealed how effective the interest is over multiple periods.
  • To isolate \(r\), we took the 12th root and converted it into a percentage, showing the importance of both algebra and exponential functions in finance.
Mastering interest rate calculations empowers individuals to evaluate and optimize their investments effectively, ensuring sound financial planning.

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Most popular questions from this chapter

For an investment having an APY of \(6 \%\), estimate the number of years needed to double the principal.

Elizabeth went on a fabulous vacation in May and racked up a lot of charges on her credit card. When it came time to pay her June credit card bill, she left a balance of \(\$ 1200\). Elizabeth's credit card billing cycle runs from the nineteenth of each month to the eighteenth of the next month, and her interest rate is \(19.5 \% .\) She started the billing cycle June \(19-\) July 18 with a previous balance of \(\$ 1200 .\) In addition, she made three purchases, with the dates and amounts shown in Table 10-11. On July 15 she made an online payment of \(\$ 500.00\) that was credited to her balance the same day. (a) Find the average daily balance on the credit card account for the billing cycle June 19 -July 18 . (b) Compute the interest charged for the billing cycle June 19-July 18 . (c) Find the new balance on the account at the end of the June 19 -July 18 billing cycle. $$ \begin{array}{|c|c} \hline \text { Date } & \text { Amount of purchase/payment } \\ \hline 6 / 21 & \$ 179.58 \\ \hline 6 / 30 & \$ 40.00 \\ \hline 7 / 5 & \$ 98.35 \\ \hline 7 / 15 & \text { Payment } \$ 500.00 \end{array} $$

At the Happyville Mall, you buy a pair of earrings that are marked \$6.95. After sales tax, the bill was \(\$ 7.61\). What is the tax rate in Happyville (to the nearest tenth of a percent)?

Consider a CD paying a \(3 \%\) APR compounded continuously. Find the future value of the \(\mathrm{CD}\) if you invest \(\$ 1580\) for a term of three years.

Suppose you invest \(\$ P\) on a CD paying \(2.75 \%\) interest compounded continuously for a term of three years. At the end of the term you get \(\$ 868.80\) from the bank. Find the value of the original principal \(P\).
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