calculate amortized loan

Chapter Seven: Calculating Amortized Loan Payments

Learning Objectives

• Understand how to solve for interest and principal in multiple months of amortized payments.
• Master amortization mathematics.

Concepts of Amortized Loans

• Definition: An amortized loan is structured so that payments remain constant, but the portions allocated to interest and principal change over time.
• Key Difference from Non-Amortized Loans:
  • Non-Amortized Loans: The interest paid remains constant over the life of the loan.
  • Amortized Loans: Payment amounts are stable, but the allocation between interest and principal changes as the principal balance declines.

Breakdown of Mortgage Payments

• Most borrowers make monthly payments on their mortgages.
• Each mortgage payment consists of:
  • Interest:
  • Definition: The fee the borrower pays to the lender for using borrowed funds.
  • Principal:
  • Definition: The amount of the monthly payment that goes towards reducing the borrower's remaining loan balance.

Notation for Payment Portions

• Abbreviations used for clarity in calculations:
  • P1: Portion of the first payment that goes towards principal.
  • I1: Portion of the first payment going towards interest.
  • P2: Portion of the second payment that goes towards principal.
  • I2: Portion of the second payment going towards interest.
  • P represents the portion towards principal, while I represents the portion towards interest.
  • The subscript numbers (1, 2) indicate the payment month.

Understanding Diminishing Principal

• Interest Calculation:
  • Formula: $ext{Interest} = ext{Interest Rate} imes ext{Principal}$
  • The interest amount depends on the remaining principal:
  • As the principal decreases, the interest portion of the payment diminishes.
  • Example Calculation: If the loan principal is high, the initial payments will be more heavily weighted towards interest.
• Strategic Payment Insight: Clients who can afford higher initial payments will reduce their principal faster, resulting in lower total interest paid over the loan's life.
• Buy Down Method: Paying upfront to lower the interest rate can significantly reduce total interest over the life of the loan.

Example Case: Jimmy's Loan

• Loan Details:
  • Loan Amount: $300,000
  • Interest Rate: 4.25%
  • Term: 30 years (360 payments)
  • Monthly Payment: $1,476
• Lifetime Payment Breakdown: Over 30 years, Jimmy will pay approximately $231,360 in interest, totaling $531,360, which includes the principal.

First Monthly Payment Breakdown

• Total Payment: $1,476
• Interest Portion (I1):
  • Calculation Steps:
  • Convert interest rate: 4.25% to decimal = 0.0425
  • Monthly interest: $\frac{0.0425 imes 300,000}{12} = \frac{12,750}{12} = 1,062.50$
  • Rounded Result: $1,063 to interest.
• Principal Portion (P1):
  • Calculation: $1,476 - 1,063 = 413$
  • Result: $413 goes towards principal.
• New Principal after Payment:
  • Previous Principal: $300,000
  • New Principal Calculation: $300,000 - 413 = 299,587$

Second Monthly Payment Breakdown

• New Principal for Month Two: $299,587
• Interest Portion (I2):
  • Calculation Steps:
  • Monthly interest: $\frac{0.0425 imes 299,587}{12} = \frac{12,732.45}{12} = 1,061$
• Principal Portion (P2):
  • Calculation: $1,476 - 1,061 = 415$
  • Result: $415 goes towards principal.

Third Monthly Payment Breakdown

• New Principal for Month Three: $299,172
• Interest Portion (I3):
  • Calculation Steps:
  • Monthly interest: rac{0.0425 imes 299,172}{12}
    ightarrow ext{Calculate}
• Principal Portion (P3):
  • Calculation:
  • $1,476 - I3 = P3$

Calculation of Total Interest Paid

Key Steps

• Total Loan Cost Formula:
  • Monthly payment times the number of payments:
  • $ext{Total Loan Cost} = ext{Monthly Payment} imes ext{Number of Payments}$
• Total Interest Calculation:
  • Total Loan Cost - Original Principal = Total Interest Paid

Example Case: Savannah's Loan

• Loan Amount: $300,000
• Monthly Payment: $1,500
• Total Payments Calculation:
  • $1,500 imes 360 = 540,000$
• Total Interest:
  • Total Cost - Principal:
  • 540,000 - 300,000 = 240,000 (Total Interest Paid)

Comparing Two Mortgages: Jacob's Situation

• Loan Options:
  • 5% Interest
  • Option 1 (15 years): Monthly Payment = $791
  • Option 2 (30 years): Monthly Payment = $537
• Total Loan Cost:
  • 15 Year Loan:
  • $791 imes 180 = 142,380$
  • 30 Year Loan:
  • $537 imes 360 = 193,320$
• Interest Paid Over Life of Loan:
  • 15 Year: Total Cost - Principal = 142,380 - 100,000 = 42,380
  • 30 Year: Total Cost - Principal = 193,320 - 100,000 = 93,320

Cost Ratio Calculation

• Cost Ratio:
  • Formula:
  • $ext{30-Year Cost} ext{ divided by 15-Year Cost}$
  • Calculation:
  • $\frac{193,320}{142,380} = 1.36$
• Percentage Representation:
  • 1.36 imes 100 = 136 ext{%}
• Conclusion:
  • The total payments of the thirty-year loan are 136% of the total payments of the fifteen-year loan.
  • Jacob would pay $50,940 less with the fifteen-year loan.

Conclusion

• Key Insight:
  • Understanding amortization and the relationship between principal repayment and interest over time helps borrowers make informed financial decisions and manage long-term costs effectively.

Recap of Concepts and Calculations

• Regularly reviewing the amortization process can elucidate how loan dynamics operate and empower borrowers with knowledge for budgeting and future commitments.