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) for an effective annual rate of 14.73% compounded monthly, we can use the formula that relates the APR to the effective rate.
The formula for the effective rate reffr_{\text{eff}}reff is: reff=(1+rAPRn)n−1r_{\text{eff}} = \left(1 + \frac{r_{\text{APR}}}{n}\right)^n – 1reff=(1+nrAPR)n−1
Where:
- reffr_{\text{eff}}reff is the effective annual rate (14.73% in this case, or 0.1473),
- rAPRr_{\text{APR}}rAPR is the annual nominal rate (the APR we are looking for),
- nnn is the number of compounding periods per year (12 for monthly compounding).
Step-by-step solution:
- Rewrite the formula to solve for APR:
1+rAPR12=(1+reff)1121 + \frac{r_{\text{APR}}}{12} = (1 + r_{\text{eff}})^{\frac{1}{12}}1+12rAPR=(1+reff)121 rAPR12=(1+reff)112−1\frac{r_{\text{APR}}}{12} = \left(1 + r_{\text{eff}}\right)^{\frac{1}{12}} – 112rAPR=(1+reff)121−1 rAPR=12((1+reff)112−1)r_{\text{APR}} = 12 \left( \left(1 + r_{\text{eff}}\right)^{\frac{1}{12}} – 1 \right)rAPR=12((1+reff)121−1)
- Substitute the values:
reff=0.1473r_{\text{eff}} = 0.1473reff=0.1473 rAPR=12((1+0.1473)112−1)r_{\text{APR}} = 12 \left( \left(1 + 0.1473\right)^{\frac{1}{12}} – 1 \right)rAPR=12((1+0.1473)121−1) rAPR=12((1.1473)112−1)r_{\text{APR}} = 12 \left( (1.1473)^{\frac{1}{12}} – 1 \right)rAPR=12((1.1473)121−1)
- Calculate:
(1.1473)112≈1.011585(1.1473)^{\frac{1}{12}} \approx 1.011585(1.1473)121≈1.011585 rAPR=12(1.011585−1)r_{\text{APR}} = 12 \left( 1.011585 – 1 \right)rAPR=12(1.011585−1) rAPR=12×0.011585≈0.13902r_{\text{APR}} = 12 \times 0.011585 \approx 0.13902rAPR=12×0.011585≈0.13902
- Convert to percentage:
rAPR≈13.90%r_{\text{APR}} \approx 13.90\%rAPR≈13.90%
Thus, the corresponding APR to quote is 13.90%.
Explanation:
The APR represents the nominal annual interest rate without considering compounding within the year. To find the APR from an effective rate, we reverse the calculation by accounting for monthly compounding. By doing this, we ensure that the rate compounded monthly will result in the given effective annual rate of 14.73%.
