Introductory Corporate Finance: Time Value of Money - Loan Amortization, Amortization Schedule Principal vs. Interest Components

What This Is

Loan amortization is the process of gradually paying off a loan by making regular payments that cover both the principal and interest components. This concept is crucial in corporate finance as it helps companies manage their debt obligations and understand the impact of interest rates on their cash flows. For instance, consider a company like Tesla, which has taken on significant debt to finance its expansion. Understanding loan amortization will help Tesla's management and investors assess the company's debt burden and make informed decisions about future borrowing.

Key Formulas & Models

• A = P × r × (1 + r)^n / ((1 + r)^n - 1) – loan amortization formula; calculates the periodic payment (A) required to pay off a loan with principal (P), interest rate (r), and number of periods (n).
  • A: periodic payment
  • P: principal amount
  • r: interest rate
  • n: number of periods
• PMT = P × r × (1 + r)^n / ((1 + r)^n - 1) – periodic payment formula; same as above, but rearranged to solve for PMT.
• IPMT = P × r × (1 + r)^n / ((1 + r)^n - 1) - P × (1 + r)^n / ((1 + r)^n - 1) – interest payment formula; calculates the interest paid in a given period.
• PPMT = PMT - IPMT – principal payment formula; calculates the principal paid in a given period.
• Total Interest Paid = IPMT × n – total interest paid formula; calculates the total interest paid over the life of the loan.
• Total Amount Paid = PMT × n – total amount paid formula; calculates the total amount paid over the life of the loan.
• Amortization Schedule = {PMT, IPMT, PPMT} for n periods – amortization schedule formula; generates a schedule of periodic payments, interest payments, and principal payments.

Step-by-Step Calculation

1. Determine the loan details: principal amount (P), interest rate (r), and number of periods (n).
2. Calculate the periodic payment (PMT) using the loan amortization formula.
3. Calculate the interest payment (IPMT) for each period using the interest payment formula.
4. Calculate the principal payment (PPMT) for each period by subtracting the interest payment from the periodic payment.
5. Generate an amortization schedule by listing the periodic payments, interest payments, and principal payments for each period.
6. Calculate the total interest paid and total amount paid over the life of the loan.

Common Mistakes

• Mistake: Assuming that the interest payment remains constant over the life of the loan.
• Correction: The interest payment decreases over time as the principal balance decreases.
• Counterexample: Consider a loan with a principal amount of $100,000, interest rate of 6%, and 5-year term. The interest payment in the first year is $6,000, but in the fifth year, it is only $4,500.
• Mistake: Failing to account for prepayment penalties or fees.
• Correction: These fees can significantly impact the total amount paid over the life of the loan.
• Counterexample: Consider a loan with a principal amount of $100,000, interest rate of 6%, and 5-year term, but with a prepayment penalty of 2% of the outstanding balance. If the borrower pays off the loan in 3 years, they will incur a penalty of $2,000.
• Mistake: Ignoring the impact of inflation on the loan's interest rate.
• Correction: Inflation can erode the purchasing power of the loan's interest payments, making it more difficult to pay off the loan.
• Counterexample: Consider a loan with a principal amount of $100,000, interest rate of 6%, and 5-year term, but with an inflation rate of 3%. The interest payment in the first year is $6,000, but in the fifth year, it is equivalent to only $4,500 in today's dollars.

Exam / CFA Tips

• Tip: Be careful when using the loan amortization formula to calculate the periodic payment. Make sure to use the correct formula and plug in the correct values.
• Tip: When generating an amortization schedule, make sure to list the periodic payments, interest payments, and principal payments for each period.
• Tip: Be prepared to answer questions about the impact of prepayment penalties or fees on the total amount paid over the life of the loan.
• Tip: Be careful when ignoring the impact of inflation on the loan's interest rate. Make sure to adjust the interest payments for inflation when calculating the total amount paid over the life of the loan.

Quick Practice Problem

A company has a loan with a principal amount of $500,000, interest rate of 8%, and 10-year term. What is the periodic payment (PMT) required to pay off the loan?

Answer: $63,919.41

Explanation: Using the loan amortization formula, we can calculate the periodic payment as follows:

PMT = $500,000 × 0.08 × (1 + 0.08)^10 / ((1 + 0.08)^10 - 1) = $63,919.41

Last-Minute Cram Sheet

• Loan amortization formula: A = P × r × (1 + r)^n / ((1 + r)^n - 1)
• Periodic payment formula: PMT = P × r × (1 + r)^n / ((1 + r)^n - 1)
• Interest payment formula: IPMT = P × r × (1 + r)^n / ((1 + r)^n - 1) - P × (1 + r)^n / ((1 + r)^n - 1)
• Principal payment formula: PPMT = PMT - IPMT
• Total interest paid formula: Total Interest Paid = IPMT × n
• Total amount paid formula: Total Amount Paid = PMT × n
• Amortization schedule formula: Amortization Schedule = {PMT, IPMT, PPMT} for n periods
• In M&M Proposition I (no taxes), firm value is independent of capital structure – but with taxes, value increases with debt due to the interest tax shield.
• The interest payment decreases over time as the principal balance decreases.
• Prepayment penalties or fees can significantly impact the total amount paid over the life of the loan.
• Inflation can erode the purchasing power of the loan's interest payments, making it more difficult to pay off the loan.