Intro to Finance: Time Value of Money - Effective Annual, Rate EAR vs. Annual Percentage Rate APR

What This Is

The Effective Annual Rate (EAR) and Annual Percentage Rate (APR) are two related but distinct concepts in finance. The EAR represents the true annual rate of return on an investment, taking into account compounding, while the APR is the nominal interest rate charged on a loan or investment. Understanding the difference between EAR and APR is crucial for making informed investment decisions and calculating interest rates accurately.

For example, consider a savings account with a 5% annual interest rate compounded monthly. If you deposit $1,000, how much will you have after one year? The EAR will be higher than the APR, reflecting the compounding effect.

Key Formulas & Symbols

• EAR = (1 + r/n)^(n) - 1 where EAR = effective annual rate, r = nominal interest rate, n = number of compounding periods per year.
• APR = r where APR = annual percentage rate, r = nominal interest rate.
• r = nominal interest rate (e.g., 5% per annum).
• n = number of compounding periods per year (e.g., 12 for monthly compounding).
• FV = PV × (1 + r/n)^(n) where FV = future value, PV = present value, r = nominal interest rate, n = number of compounding periods per year.
• PV = FV / (1 + r/n)^(n) where PV = present value, FV = future value, r = nominal interest rate, n = number of compounding periods per year.
• C = P × r where C = interest payment, P = principal amount, r = nominal interest rate.
• A = P × (1 + r/n)^(n) where A = future amount, P = principal amount, r = nominal interest rate, n = number of compounding periods per year.

Step-by-Step Calculation

1. Determine the nominal interest rate (r) and the number of compounding periods per year (n).
2. Calculate the EAR using the formula EAR = (1 + r/n)^(n) - 1.
3. Calculate the APR, which is equal to the nominal interest rate (r).
4. Use the EAR to calculate the future value (FV) using the formula FV = PV × (1 + EAR).
5. Use the APR to calculate the interest payment (C) using the formula C = P × r.
6. Use the APR to calculate the future amount (A) using the formula A = P × (1 + r/n)^(n).

Common Mistakes

• Mistake: Confusing EAR and APR.
• Correction: Remember that EAR takes into account compounding, while APR is the nominal interest rate.
• Mistake: Not considering the number of compounding periods per year.
• Correction: Make sure to use the correct value for n in the EAR formula.
• Mistake: Using the APR to calculate the future value or interest payment.
• Correction: Use the EAR to calculate the future value or interest payment.

Exam / CFA Tips

• Tip: Be careful with question wording – some questions may ask for the APR, while others may ask for the EAR.
• Tip: Make sure to consider compounding when calculating interest rates.
• Tip: Use the correct formula for EAR, which takes into account the number of compounding periods per year.

Quick Practice Problem

A savings account has a 5% annual interest rate compounded monthly. If you deposit $1,000, how much will you have after one year?

Answer: $1,051.17 (using the EAR formula). Explanation: The EAR is 5.12%, which is higher than the APR due to compounding.

Last-Minute Cram Sheet

• The EAR formula is EAR = (1 + r/n)^(n) - 1, not EAR = r.
• The APR is equal to the nominal interest rate (r).
• Use the EAR to calculate the future value or interest payment.
• Don't confuse EAR and APR – they have different meanings.
• The number of compounding periods per year (n) affects the EAR.
• Use the correct formula for FV: FV = PV × (1 + EAR).
• Don't use the APR to calculate the future value or interest payment.