Suppose you want to earn an effective rate of 14.73% and you are looking at an account that compounds on a monthly basis. What is the corresponding APR (in %) to quote? Note: Express the % answer in the form of ab.cd. Keep two (2) numbers after the decimal point. For example, if your answer is 0.3245 or 32.45%, write 32.45 as your answer. Do NOT write 0.3245.
The Correct Answer and Explanation is:
To find the corresponding Annual Percentage Rate (APR) from the effective annual rate (EAR), we use the relationship between the EAR and the APR for monthly compounding. The formula to calculate the effective annual rate based on the APR is:EAR=(1+APRn)n−1EAR = \left(1 + \frac{APR}{n}\right)^n – 1EAR=(1+nAPR)n−1
Where:
- EAR is the effective annual rate.
- APR is the annual percentage rate.
- n is the number of compounding periods per year (since compounding is monthly, n=12n = 12n=12).
In this case, you’re given an EAR of 14.73%, or 0.1473 in decimal form, and we need to solve for the APR.
Step-by-Step Calculation:
- Start with the equation:
0.1473=(1+APR12)12−10.1473 = \left(1 + \frac{APR}{12}\right)^{12} – 10.1473=(1+12APR)12−1
- Rearrange to isolate the term involving APR:
1+APR12=(1+0.1473)1121 + \frac{APR}{12} = \left(1 + 0.1473\right)^{\frac{1}{12}}1+12APR=(1+0.1473)121
- Simplify:
1+APR12=1.14731121 + \frac{APR}{12} = 1.1473^{\frac{1}{12}}1+12APR=1.1473121
- Take the 12th root of 1.1473:
1+APR12=1.0116161 + \frac{APR}{12} = 1.0116161+12APR=1.011616
- Subtract 1 from both sides:
APR12=0.011616\frac{APR}{12} = 0.01161612APR=0.011616
- Multiply both sides by 12 to find APR:
APR=0.011616×12=0.1394APR = 0.011616 \times 12 = 0.1394APR=0.011616×12=0.1394
- Convert to percentage:
APR=13.94%APR = 13.94\%APR=13.94%
Final Answer:
The corresponding APR is 13.94%.
