Question
Find the effective rate of interest per annum equivalent to the nominal interest rate of 9.8% per annum compounded quarterly.
Answers
GPT-4o mini
To find the effective interest rate equivalent to a nominal interest rate compounded quarterly, we can use the formula for the effective interest rate (EIR):
\[
EIR = \left(1 + \frac{r}{n}\right)^n - 1
\]
where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year.
In this case:
- The nominal interest rate \( r = 9.8\% = 0.098 \),
- The number of compounding periods \( n = 4 \) (since it is compounded quarterly).
Now, we can plug in the values:
\[
EIR = \left(1 + \frac{0.098}{4}\right)^4 - 1
\]
First, we calculate \( \frac{0.098}{4} \):
\[
\frac{0.098}{4} = 0.0245
\]
Now, substitute back into the formula:
\[
EIR = \left(1 + 0.0245\right)^4 - 1
\]
Next, calculate \( 1 + 0.0245 \):
\[
1 + 0.0245 = 1.0245
\]
Now raise \( 1.0245 \) to the power of 4:
\[
1.0245^4 \approx 1.100601
\]
Finally, subtract 1 to find the effective interest rate:
\[
EIR = 1.100601 - 1 \approx 0.100601
\]
Convert this back to a percentage:
\[
EIR \approx 0.100601 \times 100 \approx 10.0601\%
\]
Thus, the effective rate of interest per annum equivalent to a nominal interest rate of 9.8% per annum compounded quarterly is approximately **10.06%**.
\[
EIR = \left(1 + \frac{r}{n}\right)^n - 1
\]
where:
- \( r \) is the nominal interest rate (as a decimal),
- \( n \) is the number of compounding periods per year.
In this case:
- The nominal interest rate \( r = 9.8\% = 0.098 \),
- The number of compounding periods \( n = 4 \) (since it is compounded quarterly).
Now, we can plug in the values:
\[
EIR = \left(1 + \frac{0.098}{4}\right)^4 - 1
\]
First, we calculate \( \frac{0.098}{4} \):
\[
\frac{0.098}{4} = 0.0245
\]
Now, substitute back into the formula:
\[
EIR = \left(1 + 0.0245\right)^4 - 1
\]
Next, calculate \( 1 + 0.0245 \):
\[
1 + 0.0245 = 1.0245
\]
Now raise \( 1.0245 \) to the power of 4:
\[
1.0245^4 \approx 1.100601
\]
Finally, subtract 1 to find the effective interest rate:
\[
EIR = 1.100601 - 1 \approx 0.100601
\]
Convert this back to a percentage:
\[
EIR \approx 0.100601 \times 100 \approx 10.0601\%
\]
Thus, the effective rate of interest per annum equivalent to a nominal interest rate of 9.8% per annum compounded quarterly is approximately **10.06%**.