To find the nominal interest rate given an effective rate of 29.9%, with the investment being compounded quarterly, follow these steps:
1. Identify the given values:
- Effective annual rate (EAR): 29.9% or in decimal form 0.299.
- The number of compounding periods per year (n): 4 (since it is compounded quarterly).
2. Recall the relationship between the effective rate and the nominal rate:
The effective interest rate can be related to the nominal interest rate using the following formula:
[tex]\[
(1 + \text{effective rate}) = \left(1 + \frac{\text{nominal rate}}{n}\right)^n
\][/tex]
3. Rearrange the formula to solve for the nominal rate:
We need to isolate the nominal rate in the equation. Here are the steps to isolate the nominal rate:
[tex]\[
(1 + \text{effective rate}) = \left(1 + \frac{\text{nominal rate}}{n}\right)^n
\][/tex]
[tex]\[
(1 + 0.299) = \left(1 + \frac{\text{nominal rate}}{4}\right)^4
\][/tex]
To solve for the nominal rate, take the 4th root of both sides:
[tex]\[
(1 + 0.299)^{1/4} = 1 + \frac{\text{nominal rate}}{4}
\][/tex]
[tex]\[
1.299^{\frac{1}{4}} = 1 + \frac{\text{nominal rate}}{4}
\][/tex]
Subtract 1:
[tex]\[
1.299^{\frac{1}{4}} - 1 = \frac{\text{nominal rate}}{4}
\][/tex]
Finally, multiply both sides by 4 to solve for the nominal rate:
[tex]\[
\text{nominal rate} = 4 \left(1.299^{\frac{1}{4}} - 1\right)
\][/tex]
4. Compute the nominal rate:
Substituting the effective rate and the number of compounding periods into the formula, we get:
[tex]\[
\text{nominal rate} = 4 \left((1 + 0.299)^{\frac{1}{4}} - 1\right)
\][/tex]
[tex]\[
\text{nominal rate} = 4 \left(1.299^{\frac{1}{4}} - 1\right)
\][/tex]
Which gives us:
[tex]\[
\text{nominal rate} \approx 0.27033827554592005
\][/tex]
5. Convert the nominal rate from decimal to a percentage:
[tex]\[
\text{nominal rate percentage} = 0.27033827554592005 \times 100 \approx 27.033827554592005 \%
\][/tex]
Therefore, the nominal interest rate, when compounded quarterly, is approximately 27.03%.
