Answered

How could this function be rewritten to identify the monthly interest rate?

What is the approximate monthly interest rate?



Select the choices to correctly complete the table (if a value does not match, do not select it):



Options:

$A=1000(1+0.023)^{12 t}$

$A=1000\left(1.023^{\frac{1}{12}}\right)^{12 t}$

$A=1000\left(1.023^{12}\right)^{\frac{1}{12}}$

0.19\%

0.23\%

0.31\%

Verified Answer :

  • The original function \(A=1000(1+0.023)^{12t}\) is rewritten as \(A=1000(1.023^{\frac{1}{12}})^{12t}\) to highlight the monthly growth factor.
  • The monthly interest rate \(r_m\) is identified from the relationship \(1+r_m = 1.023^{\frac{1}{12}}\).
  • Calculating \(r_m = 1.023^{\frac{1}{12}} - 1\) gives approximately \(0.001897\), which corresponds to a percentage of \(\boxed{0.19\%}\).

Explanation

  1. Analyzing the Expression for Monthly Interest Rate

The given financial function is \(A=1000(1+0.023)^{12t}\), which represents the future value \(A\) of an investment of \(1000\) units, growing at a rate determined by the base \((1+0.023)\) over \(12t\) compounding periods. In compound interest, if there are \(12t\) periods, we can identify the growth factor per period as the base of the exponent. Here, the expression is already given as \(A=1000(1.023)^{12t}\). To rewrite this function to explicitly show the growth factor corresponding to a single monthly period, we can express the base \(1.023\) as \((1.023^{\frac{1}{12}})^{12}\). Substituting this into the original equation, we get \(A=1000(1.023^{\frac{1}{12}})^{12t}\). This form clearly highlights the monthly growth factor, \((1+r_m) = 1.023^{\frac{1}{12}}\), where \(r_m\) is the monthly interest rate.

  1. Calculating the Monthly Interest Rate

To find the monthly interest rate \(r_m\), we use the relation \(1+r_m = 1.023^{\frac{1}{12}}\). By solving for \(r_m\), we get \(r_m = 1.023^{\frac{1}{12}} - 1\). Calculating this value, we find \(r_m \approx 0.00189675\). Expressing this as a percentage, we multiply by 100: \(r_m \times 100 \approx 0.189675\%\). Rounding this to two decimal places, the approximate monthly interest rate is \(0.19\%\). Comparing this result with the provided options \(0.19\%\), \(0.23\%\), and \(0.31\%\), we confirm that \(0.19\%\) is the correct approximate monthly interest rate.

Examples

Understanding how to convert annual growth factors to monthly or other periodic rates is crucial in finance. For instance, if you are looking at a savings account offering a certain annual percentage yield, you might want to know how much your balance grows each month to plan your monthly budget or evaluate short-term savings goals. This mathematical process of base conversion allows you to compare different financial products on a consistent time basis, such as comparing a bank's monthly compound interest rate against a standard annual investment return.

Related Questions