# Quoted vs Periodic Interest Rate

Quoted interest rate (also called nominal interest rate or annual percentage rate) is the non-compounded interest rate for a period of one year. It can be converted to periodic interest rate by dividing it with the number of compounding periods per year.

Let us say you obtain a $100,000 bank loan at 8% annual interest rate compounded annually. It means that the bank will charge you interest of$8,000 (=$100,000 × 8%) per annum. If you repay your loan after a six-month period, the interest rate applicable to you would be 4% (=8% divided by 2). The 8% interest rate quoted by the bank for the annual period is called the annual percentage rate (APR) or nominal interest rate or quoted interest rate and the interest rate of 4% applicable to a fraction of a year is called the periodic interest rate. ## Formula The following equation shows the relationship between quoted and periodic interest rates: $$\text{r}\ =\ \frac{\text{APR}}{\text{m}}$$ Where r is the periodic interest rate, APR is the annual percentage rate (quoted rate) and m is the number of periods per year. Please note that the periodic interest rate can be obtained only when we start with the annual percentage rate. If we have to work with the effective interest rate, we would first need to convert it to (nominal) annual percentage rate and the use the expression above to obtain the periodic interest rate. ## Example You have received proposals from two banks for a loan of$20 million. Bank A quotes an annual percentage rate of 6% and Bank B quotes an effective interest rate of 6.5%. If both banks require semi-annual compounding, work out the interest expense for the first six-monthly period.

### Solution

The interest expense in case of Bank A is straight-forward. We need to find periodic interest rate and apply it to the principal amount to find interest expense.

$$\text{r} _ \text{A}\ =\ \frac{\text{6%}}{\text{2}}=\text{3%}$$

$$\text{I} _ \text{B}\ =\ \text{P}\times \text{r}=\text{\20M}\times\text{3%}\ =\ \text{\0.6M}$$

In case of Bank B, we have the effective annual rate (EAR) instead of the (nominal) annual percentage rate. We need to first convert EAR to APR and then work out the periodic interest rate.

The relationship between EAR and APR is given by the following expression

$$\text{EAR}=\left(\text{1}+\frac{\text{APR}}{\text{m}}\right)^\text{m}-\text{1}$$

Which can be rearranged as follows:

$$\text{APR}=\text{m}\times\left(\left(\text{1}+\text{EAR}\right)^\frac{\text{1}}{\text{m}}-\text{1}\right)$$

In case of Bank B, APR is 6.11%.

$$\text{APR}=\text{2}\times\left(\left(\text{1}+\text{6.20%}\right)^\frac{\text{1}}{\text{2}}-\text{1}\right)=\text{6.13%}$$

This translates to periodic (nominal) interest rate of 3.053% and six-monthly interest expense of \$0.61 million.