Intro to Finance: Time Value of Money - Loan Amortization, Principal vs. Interest Amortization Schedule

What This Is

Loan amortization is the process of gradually paying off a loan by making regular payments that cover both the interest and principal amounts. This concept is crucial in finance as it helps investors and borrowers understand the repayment schedule and the total cost of borrowing. For example, consider a $100,000 mortgage with a 5% annual interest rate and a 20-year repayment period. The monthly payment would be approximately $625, with the first few payments covering mostly interest and the later payments covering more principal.

Key Formulas & Symbols

• A = P × r × (1 + r)^n / ((1 + r)^n - 1) where A = monthly payment, P = principal amount, r = monthly interest rate, n = number of payments.
• PMT = P × r × (1 + r)^n / ((1 + r)^n - 1) where PMT = monthly payment, P = principal amount, r = monthly interest rate, n = number of payments.
• PV = PMT × (((1 + r)^n - 1) / r) where PV = present value, PMT = monthly payment, r = monthly interest rate, n = number of payments.
• FV = PV × (1 + r)^n where FV = future value, PV = present value, r = periodic interest rate, n = number of periods.
• I = P × r × n where I = total interest paid, P = principal amount, r = annual interest rate, n = number of years.
• S = P + I where S = total amount paid, P = principal amount, I = total interest paid.
• Amortization Schedule = | Period | Payment | Interest | Principal | Balance | where Period = payment period, Payment = monthly payment, Interest = interest paid, Principal = principal paid, Balance = remaining balance.

Step-by-Step Calculation

1. Determine the monthly interest rate by dividing the annual interest rate by 12.
2. Calculate the number of payments by multiplying the number of years by 12.
3. Use the formula A = P × r × (1 + r)^n / ((1 + r)^n - 1) to calculate the monthly payment.
4. Create an amortization schedule by calculating the interest and principal paid for each payment period.
5. Use the formula PV = PMT × (((1 + r)^n - 1) / r) to calculate the present value of the loan.
6. Use the formula FV = PV × (1 + r)^n to calculate the future value of the loan.

Common Mistakes

• Mistake: Forgetting to calculate the interest paid for each payment period.
• Correction: Create an amortization schedule to track the interest and principal paid for each period.
• Mistake: Using the wrong formula to calculate the monthly payment.
• Correction: Use the formula A = P × r × (1 + r)^n / ((1 + r)^n - 1) to calculate the monthly payment.
• Mistake: Not considering the compounding frequency when calculating the interest paid.
• Correction: Use the correct compounding frequency when calculating the interest paid.

Exam / CFA Tips

• Tip: Be careful when using the formula A = P × r × (1 + r)^n / ((1 + r)^n - 1), as it can be easily misapplied.
• Tip: Make sure to create an amortization schedule to track the interest and principal paid for each period.
• Tip: Be aware of the compounding frequency when calculating the interest paid.

Quick Practice Problem

A company issues a 5-year bond with a $1,000 face value and a 5% annual coupon rate. The market interest rate is 8%. What is the bond's yield to maturity?

Answer: 8.03% Explanation: Use the formula YTM = (C + (FV - PV) / n) / PV, where C = coupon payment, FV = face value, PV = present value, n = number of periods.

Last-Minute Cram Sheet

• The monthly payment formula is A = P × r × (1 + r)^n / ((1 + r)^n - 1).
• The present value formula is PV = PMT × (((1 + r)^n - 1) / r).
• The future value formula is FV = PV × (1 + r)^n.
• The total interest paid formula is I = P × r × n.
• The total amount paid formula is S = P + I.
• The amortization schedule must be created to track the interest and principal paid for each period.
• The compounding frequency must be considered when calculating the interest paid.
• The yield to maturity formula is YTM = (C + (FV - PV) / n) / PV.