Amortization Key Concepts • Loan Principal: The original amount borrowed. • Interest Rate: The cost of borrowing money, expressed as a percentage. • Payment Amount: The fixed amount paid periodically to the lender. • Amortization Schedule: A table showing the breakdown of each payment into principal and interest over the life of the loan. — Formula for Loan Repayments The formula to calculate the fixed payment amount for a fully amortized loan is: P MT = P · r(1 + r)n (1 + r)n − 1 where: • P M T : Payment amount per period. • P : Loan principal (amount borrowed). • r: Interest rate per period. • n: Total number of payments. — Amortization Schedule Each loan payment consists of two parts: 1 • Interest Payment: The portion of the payment that covers the interest on the remaining balance. • Principal Payment: The portion of the payment that reduces the loan balance. At any given payment t, the interest payment is calculated as: Interestt = Remaining Balancet−1 · r The principal payment is: Principalt = P M T − Interestt The remaining balance after the t-th payment is: Remaining Balancet = Remaining Balancet−1 − Principalt — Example: Calculating Loan Payments and Schedule You borrow $100,000 for 15 years at an annual interest rate of 6%, compounded monthly. Calculate: 1. The monthly payment. 2. The total interest paid over the life of the loan. 3. An amortization schedule for the first 3 months. — Solution: Step 1: Calculate Monthly Payment Convert the annual interest rate to a monthly rate: r= 6% = 0.005 12 Calculate the total number of payments: n = 15 · 12 = 180 2 Substitute into the payment formula: P M T = 100, 000 · 0.005(1 + 0.005)180 (1 + 0.005)180 − 1 Step-by-step calculations: 1. 1 + r = 1.005 2. (1.005)180 ≈ 3.058N umerator : 0.005 · 3.058 ≈ 0.01529 4. 3. Denominator: 3.058 − 1 = 2.058 5. Payment: P M T = 100, 000 · 0.01529 2.058 ≈ 741.57 The monthly payment is approximately $741.57. — Step 2: Calculate Total Interest Paid The total amount paid over the life of the loan is: Total Paid = P M T · n = 741.57 · 180 ≈ 133, 482.60 The total interest paid is: Total Interest = Total Paid − Loan Principal = 133, 482.60 − 100, 000 = 33, 482.60 — Step 3: Amortization Schedule (First 3 Months) Month Payment ($) 0 1 741.57 2 741.57 3 741.57 Interest ($) 500.00 498.79 497.58 Principal ($) 241.57 242.78 243.99 Remaining Balance ($) 100,000.00 99,758.43 99,515.65 99,271.66 — Applications of Amortization • Mortgages: Calculating monthly payments and total interest for home loans. 3 • Car Loans: Determining fixed payments over a specified term. • Student Loans: Managing repayment schedules and minimizing total interest paid. • Early Repayment Analysis: Exploring the impact of additional payments on reducing loan terms and interest. 4