Effective and Nominal Interest and Discount Rates (Finance)
Effective and Nominal Interest Rate
In practice, interest is paid more frequently than a year. However, interest rates are not quoted, for example, quarterly even if the interest is paid every three months. They are instead quoted per annum payable, for example, quarterly. This is the nominal interest rate. This is denoted by: \[i^{(p)}=pi_{[p]}\] where:
- $p =$the number of times you pay interest in a year
- $i_{[p]} =$ the interest rate per period
- $i^{(p)} =$ nominal rate per annum payable $p$ times a year
The relationship between the effective and nominal interest rate is: \[1+i=\left(1+\frac{i^{(p)}}{p}\right)^p\]
Worked Example 1
Worked Example - Finding The Nominal Interest Rate
What is the nominal rate payable monthly if the effective rate is $10\%$?
Solution
Re-arranging the formula to make $i^{(12)}$ the subject and substituting in the numbers: \begin{align} &1+0.1=\left(1+\frac{~i^{(12)}~}{12}\right)^{12}\\ &\Rightarrow i^{(12)}=0.09568968515\\ &\Rightarrow i^{(12)} \approx 9.57\% \end{align}
Worked Example 2
Worked Example - Finding The Effective Interest Rate
What is the effective rate if the nominal rate per annum payable semi-annually is $4.94\%$?
Solution
Re-arranging the formula to make $i$ the subject and substituting in the numbers: \begin{align} &1+i=\left(1+\frac{0.0494}{2}\right)^2\\ &\Rightarrow i\approx5.00\% \end{align}
Effective and Nominal Discount Rate
In a similar fashion, the nominal rate of discount is denoted by: \[d^{(p)}=pd_{[p]}\] where:
- $d_{[p]} =$ the discount rate per period
- $d^{(p)} =$ nominal rate of discount compounded $p$ times a year
The relationship between the effective and nominal discount rate is: \[1-d=\left(1-\frac{d^{(p)}}{p}\right)^p.\]
Worked Example 3
Worked Example = Finding The Nominal Discount Rate
What is the nominal rate payable quarterly if the effective rate is $10\%$?
Solution
Re-arranging the formula to make $d^{(4)}$ the subject and substituting in the numbers: \begin{align} &1-0.1=(1-\frac{~d^{(4)}~}{4})^4\\ &\Rightarrow d^{(4)}=0.1039850143\\ &\Rightarrow d^{(4)} \approx 10.40\% \end{align}
Worked Example 4
Worked Example - Finding The Effective Discount Rate
What is the effective rate if the nominal rate payable weekly is $22.27\%$?
Solution
Re-arranging the formula to make $d$ the subject and substituting in the numbers: \begin{align} &1-d=\left(1-\frac{0.2227}{52}\right)^{52}\\ &\Rightarrow d \approx 20.00\% \end{align}
