what is the effective annual rate if a bank charges you 7.64ompounded quarterly? A)7.79% B)7.86% C)8.01% D)7.95% E)7.98%
The Correct Answer and Explanation is :
To find the effective annual rate (EAR) when interest is compounded quarterly, we use the formula:
[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n – 1
]
Where:
- ( r ) is the nominal interest rate (expressed as a decimal),
- ( n ) is the number of compounding periods per year.
Step-by-step Calculation:
Given:
- Nominal interest rate ( r = 7.64\% = 0.0764 ),
- Quarterly compounding means ( n = 4 ) periods per year.
Now, substitute these values into the formula:
[
\text{EAR} = \left(1 + \frac{0.0764}{4}\right)^4 – 1
]
First, calculate the value inside the parentheses:
[
\frac{0.0764}{4} = 0.0191
]
Now, add 1:
[
1 + 0.0191 = 1.0191
]
Next, raise this value to the power of 4:
[
(1.0191)^4 = 1.0779
]
Finally, subtract 1 to find the EAR:
[
1.0779 – 1 = 0.0779
]
Convert this back to a percentage:
[
0.0779 \times 100 = 7.79\%
]
Conclusion:
The effective annual rate is 7.79%, so the correct answer is A) 7.79%.
Explanation:
The effective annual rate (EAR) reflects the true rate of interest earned or paid over a year, accounting for compounding. Compounding more frequently than once per year increases the total interest accumulated. In this case, since the interest is compounded quarterly, the nominal rate of 7.64% is divided into four quarters, each earning 1.91%. After compounding quarterly for a full year, the overall rate becomes 7.79%. This is higher than the nominal rate due to the effect of compounding.