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Chapter 2: Problem 38

Determine the interest rate needed to obtain an APY of (a) \(100 \%\) (b) \(10 \%\)

Short Answer

Expert verified
For an APY of \(100\%\), the interest rate (annual) required is \(0\) or \(0\%\). For an APY of \(10\%\), the interest rate required is \(0.10\) or \(10\% \).

Step by step solution

01

Rearrange the formula to solve for \(i\)

To find the interest rate, rearrange the formula for APY as follows: \( APY = (1 + \frac{i}{n})^{n} - 1 \) Rearranged, we get: \( \frac{i}{n} = (\sqrt[n]{APY + 1}) - 1 \) For simplicity, assume the interest is compounded annually, so \(n = 1\). This simplifies the equation to: \( i = (\sqrt[1]{APY + 1}) - 1 \)
02

Solve for \(i\) in case (a)

Plug the given APY value of \(100\%\) or 1 (since percentages must be written as decimals in calculations) into the rearranged equation: \( i = (\sqrt[1]{1 + 1}) - 1 = 1 - 1 = 0 \)
03

Solve for \(i\) in case (b)

Now plug the given APY value of \(10\%\) or 0.10 into the rearranged equation: \( i = (\sqrt[1]{0.10 + 1}) - 1 = 0.10 \)

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Key Concepts

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

Interest Rate Calculation
Calculating the interest rate is an essential skill in financial mathematics, as it helps in understanding how much your savings or loans will grow over time. To determine the interest rate needed for a desired Annual Percentage Yield (APY), we use a formula that relates the APY to the interest rate:
  • The formula for APY is: \( APY = \left(1 + \frac{i}{n}\right)^n - 1 \)
  • Where \(i\) is the interest rate per period, and \(n\) is the number of compounding periods per year.
To find the interest rate \(i\) when the interest is compounded once annually (i.e., \(n = 1\)), we simplify the equation to:\[ i = \left(1 + APY\right)^{1} - 1 \]This rearrangement makes it easy to plug in any APY value to find the necessary interest rate. Remember to convert APY percentages into decimals before using them in calculations.
Compounding Interest
Compounding interest is the concept of earning interest not only on your initial principal but also on the accumulated interest from previous periods. This means that over time, your interest earns interest, which can lead to exponential growth of your investment or savings.
  • When interest is compounded, it means the interest calculation involves multiple smaller periods within a year, leading to more frequent application of interest.
  • The core idea is that the more frequently interest is compounded, the more your investment will grow.
A common example is compounding annually, where interest is added once per year. However, interest can also be compounded semi-annually, quarterly, monthly, or even daily. The formula that reflects this growth is \( APY = \left(1 + \frac{i}{n}\right)^n - 1 \), which demonstrates how the number of compounding intervals \(n\) affects the overall yield.
Financial Mathematics
Financial mathematics involves using mathematical formulas and concepts to solve problems relating to the growth of money. It encompasses topics such as interest rates, compounding interest, annuities, loans, and investment appraisal.
  • Understanding these principles allows individuals to make more informed financial decisions, whether they are saving for the future, investing, or borrowing money.
  • Key to financial mathematics is understanding the time value of money, which suggests that a dollar today is worth more than a dollar in the future because of its potential earning capacity.
In practice, financial mathematics provides the tools to calculate future values, interest rates, and returns on investment. Calculating the interest rate for a desired APY, as in the given exercise, is just one of the many applications of financial mathematics. This discipline is crucial for both personal finance management and professional investment strategies.

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