Calculating Amortized Loan Payments

Chapter Seven: Calculating Amortized Loan Payments

Learning Objectives

  • Solve for the interest and principal components of multiple months of amortized payments.
  • Master the calculations involved in amortization.

Introduction to Amortized Loans

  • Distinct from non-amortized loans where interest payments remain constant.
  • In amortized loans, the total payment stays constant, but the allocation between interest and principal varies.
  • Interest is calculated based on the current principal amount, leading to diminishing interest payments as principal decreases over time.

Definitions and Terms

  • Mortgage Payment Breakdown: Monthly payments can be divided into two distinct components:

    • Interest: The cost incurred by the borrower for borrowing money.
    • Principal: The portion of the payment that reduces the outstanding loan balance.

  • Abbreviations Used:

    • $P_1$: Portion of the first payment going towards principal.
    • $I_1$: Portion of the first payment going towards interest.
    • $P_2$: Portion of the second payment going towards principal.
    • $I_2$: Portion of the second payment going towards interest.

Understanding Diminishing Principal

  • Interest is calculated as:
    extInterest=extInterestRateimesextPrincipalext{Interest} = ext{Interest Rate} imes ext{Principal}
  • As the principal decreases, the interest payment also decreases.
  • Clients making larger monthly payments early can save on long-term interest costs.
  • Loan Buy Down: Paying additional upfront to lower monthly payments or interest rates.

Example: Jimmy's Loan

  • Loan Details:

    • Loan amount: $300,000
    • Interest rate: 4.25%
    • Monthly payment: $1,476
    • Loan term: 30 years (360 payments)

  • Total interest paid after 30 years:

    • Total payments: 1,476imes360=531,3601,476 imes 360 = 531,360
    • Total interest: 531,360−300,000=231,360531,360 - 300,000 = 231,360
Monthly Payment Breakdown
January Payment Breakdown
  • Payment Distribution:
    • Interest:
      I1=0.0425imes300,00012=1,062.50ext(roundedto1,063)I_1 = \frac{0.0425 imes 300,000}{12} = 1,062.50 ext{ (rounded to } 1,063)
    • Principal:
      P1=1,476−1,063=413P_1 = 1,476 - 1,063 = 413
  • Remaining principal after payment:
    • $300,000 - 413 = 299,587$
February Payment Breakdown
  • New Principal: $299,587
  • Payment Distribution:
    • Interest:
      I2=0.0425imes299,58712=1,061I_2 = \frac{0.0425 imes 299,587}{12} = 1,061
    • Principal:
      P2=1,476−1,061=415P_2 = 1,476 - 1,061 = 415
  • Remaining principal after payment:
    • $299,587 - 415 = 299,172$
March Payment Breakdown
  • New Principal: $299,172
  • Payment Distribution:
    • Interest:
      I3=0.0425imes299,17212=1,060I_3 = \frac{0.0425 imes 299,172}{12} = 1,060
    • Principal:
      P3=1,476−1,060=416P_3 = 1,476 - 1,060 = 416
  • Remaining principal after payment:
    • $299,172 - 416 = 298,756$

Future Payment Calculations

  • Repeat the steps above using the new principal balance each month:
    • New Balance for Month 3:
      299,587−415=299,172299,587 - 415 = 299,172
    • Continue calculating I_n and P_n using the same formulas as above.
  • Over time, the majority of payments will go towards principal as the interest portion declines.

Cumulative Interest Costs

Total Loan Cost Calculation
  1. Total Cost: Multiply monthly payment by total number of payments:
    extTotalLoanCost=extMonthlyPaymentimesextTotalPaymentsext{Total Loan Cost} = ext{Monthly Payment} imes ext{Total Payments}
  2. Total Interest Paid: Subtract original principal from total cost:
    extTotalInterest=extTotalLoanCost−extOriginalPrincipalext{Total Interest} = ext{Total Loan Cost} - ext{Original Principal}
Example: Savannah's Loan
  • Loan amount: $300,000, Monthly payment: $1,500, Loan term: 30 years
  • Total Payments:
    1,500imes360=540,0001,500 imes 360 = 540,000
  • Total Interest Paid:
    540,000−300,000=240,000540,000 - 300,000 = 240,000

Comparing Mortgages

Jacob's Loan Comparison
  • Two loan options:
    1. Fifteen-Year Loan: 5% interest, $791/month
    • Total Payments:
      791imes180=142,380791 imes 180 = 142,380
    • Total Interest:
      142,380−100,000=42,380142,380 - 100,000 = 42,380
    1. Thirty-Year Loan: 5% interest, $537/month
    • Total Payments:
      537imes360=193,320537 imes 360 = 193,320
    • Total Interest:
      193,320−100,000=93,320193,320 - 100,000 = 93,320
Cost Ratio Calculation

  • Cost Ratio:
    extCostRatio=extThirty−YearTotalextFifteen−YearTotalext{Cost Ratio} = \frac{ ext{Thirty-Year Total}}{ ext{Fifteen-Year Total}}

    =193,320142,380=1.36=\frac{193,320}{142,380} = 1.36

    • Converted to percentage: 1.36 imes 100 = 136 ext{%}

  • Summary: Jacob pays 136% of the fifteen-year loan cost, $50,940 more in interest.

Conclusion

  • The power of understanding amortized loans is crucial for both borrowers and financial advisors as it illustrates how payment structures can significantly affect long-term financial commitments and total costs.