Loan Amortization: Fixed vs. Declining
Compare traditional fixed installments side-by-side with decreasing installments (constant principal) to see exactly how much you can save on total interest depending on the system
Loan Details
Understanding the Amortization Systems
When you take out a loan, the way you pay back the principal affects your total interest. The Fixed-Payment system keeps your monthly payment exactly the same throughout the entire loan. The Constant Principal system (like SAC in Brazil or Tilgungsdarlehen in Germany) starts with higher payments that decrease over time.
Why compare them? While fixed systems offer predictable payments, constant principal pays down the debt faster. Because interest is always calculated on the remaining balance, the declining method almost always results in a lower total cost for the borrower.
Fixed-Payment Amortization
Constant Principal (SAC)
Payment schedule, month by month
| Month | Payment | Interest | Principal | Balance left |
|---|
How to use this tool
- Choose the country or currency preset, then enter the loan amount, the annual interest rate and the term in months.
- Read the fixed-installment result (Price/annuity table) and the constant-principal result (SAC) side by side.
- Compare total interest and total paid for both systems; the first-installment gap shows how much cash flow you need early on.
- Check the “saves you” line to see how much the declining system saves, and copy the summary if you need it.
Worked example
Take a $10,000 loan at 12% a year over 36 months. With fixed installments you pay $332.14 every month, for $1,957.15 of interest and $11,957.15 in total. With constant principal the first installment is higher ($377.78) but falls to $280.56 by the last month, so total interest is $1,850.00 and the total paid is $11,850.00 – a saving of $107.15. The trade-off is a heavier start: you pay about $45 more in month one to save roughly $107 overall.
Why this tool exists and where its numbers come from
Borrowers choosing between the Price system (equal installments) and SAC (equal principal, falling installments) need to see both side by side. The formulas are the standard ones taught in financial mathematics; fees, insurance and taxes of a real loan are not included. The fixed-payment formula and the columns of an amortization schedule follow the open textbook OpenStax Contemporary Mathematics (section 6.8); the equal-principal system (SAC) follows the two Brazilian references.
Sources
- OpenStax Contemporary Mathematics, 6.8: The Basics of Loans
- Sistemas de amortização Price e SAC (artigo, IFRS)
- Marinha do Brasil (CCCPM): Sistemas de amortização e saldo devedor
For the full method, see the Methodology section just below.
Methodology
This tool relies on the standard closed-form formulas for compound-interest amortization, with a monthly rate equal to the annual rate divided by 12 (a nominal rate, not an effective annual rate). It does not account for bank administrative fees, mandatory insurance, taxes or index-linked adjustments often attached to real-world loans, and it is a comparison aid, not a loan offer.
Fixed vs. declining installments: what changes with the term
Same loan, 200,000 at 8% a year (nominal, rate / 12), only the term changes. Amounts are in the currency you select, shown as the tool rounds them:
| Term (months) | Fixed installment | Declining: first / last | Interest saved by declining |
|---|---|---|---|
| 5 years (60) | 4,055.28 | 4,666.67 / 3,355.56 | 2,650.06 |
| 10 years (120) | 2,426.55 | 3,000.00 / 1,677.78 | 10,519.56 |
| 20 years (240) | 1,672.88 | 2,166.67 / 838.89 | 40,824.57 |
| 30 years (360) | 1,467.53 | 1,888.89 / 559.26 | 87,643.83 |
How to read it: the saving grows much faster than the term. Five years saves about 2,650; twenty years saves about 40,825. The first declining installment is 15% to 30% above the fixed one, so the cash-flow test matters as much as the interest gap.
A case: Daniel is offered 200,000 over 20 years. In month one the declining system asks for 2,166.67 against 1,672.88, which is 493.79 more. It only drops below the fixed installment from payment 90 on, and by then he has paid 22,191.45 more in cash. Over the full term he would pay 40,824.57 less interest. Because his budget is tight in the early years, Daniel picks the fixed installment and accepts the higher total cost.
For comparison only; fees, insurance and index-linked adjustments are not included, and this is not a loan offer.
Tools for Smart Financial Planning
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Frequently asked questions
Why does the declining system charge less total interest? ▼
Because in the declining (constant principal) system, you amortize the debt faster during the first few months. Since interest is calculated strictly on the remaining debt balance, a lower balance means less interest accumulates. With the same rate and term, the constant principal total is always lower than the fixed-payment total.
Which system is better for me? ▼
If you need predictability and lower initial payments to fit your budget, the fixed system is easier to manage. If you can afford higher initial payments and want to save money on the total cost of the loan, the declining system is mathematically cheaper. In the example above, the first SAC installment is about 5% higher than the fixed payment, so check that your budget survives month one.
Is the rate I type nominal or effective? ▼
This tool divides the annual rate you enter by 12. If your offer quotes an effective annual rate (common for Brazilian credit, where the monthly rate is (1 + annual)^(1/12) − 1) or an APR that includes fees, the monthly rate that results will differ, so convert it first or the installment will be slightly off.
Why don't my bank's actual installments match these results? ▼
This tool applies only the loan amount, the annual rate divided by 12, and the term in months. Bank fees, mandatory insurance, taxes and any index-linked adjustments are not included, so a real contract can show higher installments. Use the figures to compare the two systems, not as a binding quote.