Differences between Effective Annual Interest Rate (EAR) and Nominal Interest Rate (ChatGPT++)

Differences between Effective Annual Interest Rate (EAR) and Nominal Interest Rate:

1. Definition:

   • Nominal Interest Rate: The interest rate stated by a financial institution or lender, typically not accounting for compounding within the year.
   • Effective Annual Interest Rate (EAR): The interest rate that reflects the true cost of an investment or loan after accounting for the effect of compounding during the year.

2. Compounding:

   • Nominal Interest Rate does not consider how often interest is compounded within a year.
   • Effective Annual Interest Rate (EAR) accounts for the effect of compounding. It reflects what you would earn or pay in interest if the nominal rate is compounded periodically (monthly, quarterly, etc.).

3. Usage:

   • Nominal Interest Rate is often used to describe loan or deposit rates when compounding frequency is not important.
   • EAR is a better representation of the actual return or cost because it factors in compounding, making it easier to compare different interest-bearing products with varying compounding periods.

Example:

Let’s say a loan offers a nominal interest rate of 12% per year, compounded monthly.

1. Nominal Interest Rate: 12% (This is the interest rate stated by the lender, with no regard for the compounding frequency.)

2. Effective Annual Interest Rate (EAR): To calculate the EAR, we apply the formula:

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

   Where:

   • $r = 0.12$ (nominal rate in decimal form),
   • $n = 12$ (compounded monthly, so 12 periods in a year).

   Substituting into the formula:

   $$\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = \left(1 + 0.01\right)^{12} - 1$$ $$\text{EAR} = (1.01)^{12} - 1 = 1.1268 - 1 = 0.1268 \quad \text{or} \quad 12.68\%$$

Thus, while the nominal interest rate is 12%, the effective annual interest rate is 12.68%. This shows that compounding monthly leads to a higher effective rate than the nominal rate, making the loan more expensive or the investment more profitable.

My own words of conclusion:
Differences between Effective Annual Interest Rate (EAR) and Nominal Interest Rate resulting higher rate result for EAR (12.68%>12%) and keep more higher as more intense frequency (i.e. more monthly) of same term (yearly)