To convert an effective annual interest rate of 12% per annum to a nominal rate per annum, compounded:
### (1) Quarterly:
1. Determine the effective annual rate: The given effective annual rate is 12%, or 0.12 in decimal form.
2. Identify the compounding frequency: Since we are dealing with quarterly compounding, there are 4 compounding periods in a year ([tex]\( n = 4 \)[/tex]).
3. Calculate the quarterly nominal rate placed within each period:
[tex]\[
\left(1 + \text{effective annual rate}\right)^{\frac{1}{n}} - 1
\][/tex]
Specifically:
[tex]\[
\left(1 + 0.12\right)^{\frac{1}{4}} - 1 = \left(1.12\right)^{0.25} - 1
\][/tex]
4. Calculate the annual nominal rate by multiplying the quarterly rate by the number of periods per year:
[tex]\[
\text{quarterly nominal rate} \times 4
\][/tex]
From these steps, the equivalent nominal annual interest rate compounded quarterly is approximately:
- 0.11494937888832091, or about 11.495% per annum.
### (2) Monthly:
1. Determine the effective annual rate: The given effective annual rate is 12%, or 0.12 in decimal form.
2. Identify the compounding frequency: Since we are dealing with monthly compounding, there are 12 compounding periods in a year ([tex]\( n = 12 \)[/tex]).
3. Calculate the monthly nominal rate placed within each period:
[tex]\[
\left(1 + \text{effective annual rate}\right)^{\frac{1}{n}} - 1
\][/tex]
Specifically:
[tex]\[
\left(1 + 0.12\right)^{\frac{1}{12}} - 1 = \left(1.12\right)^{\frac{1}{12}} - 1
\][/tex]
4. Calculate the annual nominal rate by multiplying the monthly rate by the number of periods per year:
[tex]\[
\text{monthly nominal rate} \times 12
\][/tex]
From these steps, the equivalent nominal annual interest rate compounded monthly is approximately:
- 0.11386551521499655, or about 11.387% per annum.
Therefore:
1. The nominal rate compounded quarterly is approximately 11.495%.
2. The nominal rate compounded monthly is approximately 11.387%.
