Ear formula

Definition

The ear formula, or Effective Annual Rate formula, is used to calculate the true interest rate on an investment or loan over a year, taking compounding into account. It helps in understanding how much interest is earned or paid when compounding occurs more frequently than annually. This formula is crucial for comparing different financial products with varying compounding periods and understanding the actual cost or return of an investment.

5 Must Know Facts For Your Next Test

1. The ear formula is expressed as $$EAR = (1 + \frac{r}{n})^{nt} - 1$$ where 'r' is the nominal interest rate, 'n' is the number of compounding periods per year, and 't' is the number of years.
2. Unlike the APR, which does not consider compounding, the ear provides a more accurate reflection of what you earn or pay on loans and investments.
3. The effective annual rate increases as the frequency of compounding increases, making it essential to compare rates across different financial products.
4. In practical terms, when comparing loans or investments, always look for the ear to make informed financial decisions.
5. The ear can also be calculated using the formula $$EAR = e^{r} - 1$$ when considering continuous compounding.

Review Questions

• How does the ear formula improve your understanding of financial products compared to just using APR?
  • The ear formula provides a clearer picture of the true cost or return of financial products by accounting for the effects of compounding. While APR gives a simple annual rate, it doesn't reflect how often interest is applied. The ear allows you to see how much you actually earn or pay over time, especially when comparing products with different compounding frequencies.
• If an investment offers a nominal interest rate of 6% compounded quarterly, what would be its effective annual rate according to the ear formula?
  • Using the ear formula, you would plug in 'r' as 0.06 and 'n' as 4 (for quarterly compounding). The calculation would be: $$EAR = (1 + \frac{0.06}{4})^{4*1} - 1$$ which results in an effective annual rate of approximately 6.14%. This shows that due to quarterly compounding, you end up earning slightly more than the stated nominal rate.
• Evaluate how changing the frequency of compounding from monthly to daily impacts the ear for a given nominal interest rate.
  • When you change the frequency of compounding from monthly to daily for a fixed nominal interest rate, the ear will increase. This is because daily compounding leads to interest being calculated and added to the principal more frequently, allowing interest to accrue on previously earned interest more often. As a result, you'll see a higher effective annual rate compared to monthly compounding, emphasizing the importance of understanding compounding frequency in financial decisions.

Related terms

Compounding: The process of earning interest on both the original principal and the accumulated interest from previous periods.

Annual Percentage Rate (APR): The annual rate charged for borrowing or earned through an investment, not accounting for compounding.

Discount Rate: The interest rate used to determine the present value of future cash flows.