Follow the loan balance
A loan can be repaid with a balloon payment, interest-only payments followed by principal, equal principal reductions, or equal total payments. Compare them at one rate and date.
Period 0: +$8,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
| Year | Opening balance ($) | Interest ($) | Principal ($) | Closing balance ($) |
|---|---|---|---|---|
| 1 | 8,000.00 | 800.00 | 1,723.77 | 6,276.23 |
| 2 | 6,276.23 | 627.62 | 1,896.14 | 4,380.09 |
| 3 | 4,380.09 | 438.01 | 2,085.76 | 2,294.33 |
| 4 | 2,294.33 | 229.43 | 2,294.33 | 0.00 |
Total interest = NA − P = $2,095.07. A smaller payment with a longer loan can mean more total interest.
Choose the borrower viewpoint: loan receipt positive at time 0, repayments negative.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
What the model means
An equal-payment loan has payment A=P(A/P,i,N). Each payment first covers interest i times the opening balance; the remainder reduces principal. Under equal principal reductions, principal payment is P/N and the total payment decreases with time.
Build the calculation
For interest-only repayment, payments are Pi until the final period, when principal P is also repaid. A balloon accumulates to P(1+i)ᴺ. These plans can be equivalent at i even though their undiscounted payment totals differ. Fees and a different rate would change equivalence.
Set up. Calculate. Explain.
Rebuild the starting scenario
Loan principal: 8000 $; Annual effective rate: 10 % per year; Number of years: 4 years. Calculate the level annual loan payment.
Setup hint
Choose the borrower viewpoint: loan receipt positive at time 0, repayments negative.
Open the visual solution
- Choose the borrower viewpoint: loan receipt positive at time 0, repayments negative.
- Use one effective rate per payment period and payments at period end.
- Each payment = interest on opening balance + principal reduction; the last balance should be zero.
Period 0: +$8,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
| Year | Opening balance ($) | Interest ($) | Principal ($) | Closing balance ($) |
|---|---|---|---|---|
| 1 | 8,000.00 | 800.00 | 1,723.77 | 6,276.23 |
| 2 | 6,276.23 | 627.62 | 1,896.14 | 4,380.09 |
| 3 | 4,380.09 | 438.01 | 2,085.76 | 2,294.33 |
| 4 | 2,294.33 | 229.43 | 2,294.33 | 0.00 |
Total interest = NA − P = $2,095.07. A smaller payment with a longer loan can mean more total interest.
Choose the borrower viewpoint: loan receipt positive at time 0, repayments negative.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
Transfer the idea to a new scenario
Borrow $18,000 at 6% effective annual interest for six end-of-year equal payments. Find the annual payment.
Setup hint
Choose the borrower viewpoint: loan receipt positive at time 0, repayments negative.
Open the visual solution
- Choose the borrower viewpoint: loan receipt positive at time 0, repayments negative.
- Use one effective rate per payment period and payments at period end.
- Each payment = interest on opening balance + principal reduction; the last balance should be zero.
Period 0: +$18,000.00. This is the time-zero cash flow. Periods use the payment interval stated in the model.
| Year | Opening balance ($) | Interest ($) | Principal ($) | Closing balance ($) |
|---|---|---|---|---|
| 1 | 18,000.00 | 1,080.00 | 2,580.53 | 15,419.47 |
| 2 | 15,419.47 | 925.17 | 2,735.36 | 12,684.11 |
| 3 | 12,684.11 | 761.05 | 2,899.48 | 9,784.63 |
| 4 | 9,784.63 | 587.08 | 3,073.45 | 6,711.18 |
| 5 | 6,711.18 | 402.67 | 3,257.86 | 3,453.33 |
| 6 | 3,453.33 | 207.20 | 3,453.33 | 0.00 |
Total interest = NA − P = $3,963.16. A smaller payment with a longer loan can mean more total interest.
Choose the borrower viewpoint: loan receipt positive at time 0, repayments negative.
Cash-flow dates and signs are part of the answer. Model values retain precision; displays round money to cents. Illustrative assumptions exclude taxes and inflation unless stated.
Checks allow 0.1% numerical rounding (minimum $0.05 or 0.001 in the stated nonmonetary unit). Interpolated factors and rates are approximations; keep table precision consistent.
In a level-payment amortizing loan, which statement is true?
Can you defend your setup?
Can you place each amount on the timeline, match the interest period, calculate the equivalent quantity, and state the decision or limitation?
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