Interest rates and compounding are crucial concepts in finance. They determine how quickly money grows over time, whether you're saving or borrowing. Understanding the difference between stated and effective rates helps you make smarter financial decisions.

Compounding frequency plays a big role in the actual interest earned or paid. More frequent compounding leads to higher effective rates, impacting both investors and borrowers. Knowing how to calculate effective annual rates helps you compare different financial products accurately.

Interest Rates and Compounding

Stated vs effective interest rates

• Stated interest rate (nominal rate) represents the annual interest rate quoted by lenders or paid by borrowers but does not account for the effect of compounding
• Effective interest rate (annual percentage yield) calculates the true annual rate of interest earned or paid after considering compounding and accounts for the frequency of compounding
• Compounding adds earned interest to the principal balance, increasing the principal balance for the next compounding period, and more frequent compounding results in a higher effective interest rate (daily vs monthly)
• Higher effective interest rates lead to higher total interest paid over the life of a loan, so borrowers should consider the effective rate when comparing loan options (mortgage, car loan)
• The periodic rate is the interest rate applied to each compounding period, derived by dividing the stated annual rate by the number of compounding periods per year

Calculation of effective annual rates

• Compound interest formula: $A = P(1 + r/n)^{nt}$ where $A$ is final amount, $P$ is initial principal balance, $r$ is annual stated interest rate (as a decimal), $n$ is number of compounding periods per year, and $t$ is number of years
• Effective annual rate (EAR) formula: $EAR = (1 + r/n)^n - 1$
• Steps to calculate EAR:
  1. Identify the stated annual interest rate and compounding frequency (5% compounded monthly)
  2. Divide the stated annual rate by the number of compounding periods per year (5% / 12 = 0.4167%)
  3. Add 1 to the result (1 + 0.004167 = 1.004167)
  4. Raise the result to the power of the number of compounding periods per year (1.004167^12 = 1.0511)
  5. Subtract 1 from the result to obtain the EAR as a decimal (1.0511 - 1 = 0.0511)
  6. Multiply by 100 to express the EAR as a percentage (0.0511 100 = 5.11%)

Impact of compounding frequency

• Compounding frequency refers to the number of times interest is calculated and added to the principal balance per year, with common frequencies being annually, semi-annually, quarterly, monthly, daily, and continuously
• More frequent compounding results in higher effective interest rates, causing borrowers to pay more in total interest over the life of the loan (credit card debt compounded daily vs monthly)
• More frequent compounding leads to higher effective returns for investors, allowing them to earn more in total interest over the investment period (savings account compounded daily vs quarterly)
• Continuous compounding involves interest being compounded infinitely often, leading to the highest effective rate, with the effective annual rate calculated as $EAR = e^r - 1$, where $e \approx 2.71828$ (continuously compounded 5% rate = 5.13% EAR)

Time Value of Money Concepts

• Future value represents the amount an investment will grow to over time, considering compound interest
• Present value is the current worth of a future sum of money, given a specified rate of return
• Simple interest is calculated only on the principal amount, without compounding, and is less common in practice than compound interest