How each system amortizes
Every loan installment has two parts: amortization, which reduces the outstanding balance, and interest, charged on the balance still owed. Each month's interest is always that month's balance times the rate. This is identical in both systems, what separates them is which of the two parts stays constant.
SAC, constant amortization
- You pay down the same slice of the balance every month (A = PV/n).
- Because the balance falls at a fixed pace, interest falls with it and the installment decreases.
- Higher first installment; less total interest; equity in the property grows faster.
PRICE, constant installment
- The installment is computed to be equal from the first month to the last.
- Early on almost all of it is interest and amortization is tiny; this reverses at the end.
- Predictable, lower installment at the start; more total interest; the balance falls slowly.
- Outstanding balance
- What is still owed of the principal. The month's interest is charged on it, and only on it.
- Amortization
- The slice of the installment that reduces the balance. It is the piece that behaves differently in SAC and PRICE.
- Installment
- Amortization + the month's interest. In SAC it falls; in PRICE it is fixed.
Hold on to the core idea before any arithmetic: the rate is the same and applies to the same balance in both systems. SAC is not "cheaper" because it has a lower rate, it does not. It costs less interest because it returns the principal sooner, so there is less balance accruing interest in the months that follow. Everything else in this guide is the proof of that single sentence.
The two formulas, derived from scratch
Start with PRICE, because the formula looks like it fell from the sky and it did not. The system's rule is: find a fixed installment PMT that, paid for n months, exactly clears a debt PV at a rate i per month. Each installment paid in month k is worth, in present value, PMT ÷ (1+i)ᵏ. Summing the present value of the n installments and setting it equal to the amount borrowed, PV = PMT × [(1+i)⁻¹ + (1+i)⁻² + … + (1+i)⁻ⁿ]. The bracket is a geometric series; its closed sum is (1 − (1+i)⁻ⁿ) ÷ i. Isolating PMT gives the annuity formula.
PMT = PV × i ÷ ( 1 − (1 + i)^(−n) )- PMT
- the fixed installment amount
- PV
- present value: the financed balance (price − down payment)
- i
- interest rate per period, in decimal (0.8%/mo = 0.008)
- n
- number of installments (term in months)
SAC is more direct and needs no present value. The rule is: amortize the same amount every month. If you clear PV in n installments, the constant amortization is simply A = PV ÷ n. Each month's installment is that amortization plus the interest on the remaining balance. In month k, (k−1) installments have already been amortized, so the balance is PV − (k−1)·A and the installment is A + [PV − (k−1)·A]·i.
A = PV ÷ n • P(k) = A + [ PV − (k − 1) × A ] × i- A
- constant amortization = PV divided by n
- P(k)
- installment value in month k
- k
- month number, from 1 to n
- A × i
- the progression step: how much the installment falls each month
Notice what the SAC formula reveals: since A and i are fixed, the installment always falls by the same amount, A·i. It is a perfect arithmetic progression. That makes the total interest trivial to sum. The sum of the balances over the n months is an arithmetic progression, and multiplying it by i yields a surprisingly clean closed formula for SAC's total interest. For PRICE, total interest is whatever remains after subtracting the principal from everything paid: n·PMT − PV.
J(SAC) = i × PV × (n + 1) ÷ 2 • J(PRICE) = n × PMT − PV- J(SAC)
- total SAC interest over the n months
- J(PRICE)
- total PRICE interest = total paid minus the principal
These two expressions settle the "which is cheaper" debate. The (n+1)/2 term in SAC is the average number of months the principal stays owed; in PRICE, the principal stays owed longer because amortization starts tiny. Same rate, same principal, different exposure times. In the next sections, we put numbers into these formulas.
Worked example: R$ 100k over 120 months
Finance PV = R$ 100,000.00 at i = 1% per month (0.01) for n = 120 months. In SAC, amortization is A = 100,000 ÷ 120 = R$ 833.33. First-month interest is 1% of 100,000 = R$ 1,000.00, so the first installment is 833.33 + 1,000.00 = R$ 1,833.33. Each month the installment falls by the step A·i = 833.33 × 0.01 = R$ 8.33, down to the last, about R$ 841.67. By the closed formula, SAC total interest is 0.01 × 100,000 × 121 ÷ 2 = R$ 60,500.00.
In PRICE, the installment comes from the annuity: (1+0.01)¹²⁰ = 3.300387, so (1.01)⁻¹²⁰ = 0.302995 and the denominator 1 − 0.302995 = 0.697005. The installment is PMT = (100,000 × 0.01) ÷ 0.697005 = 1,000 ÷ 0.697005 = R$ 1,434.71, fixed. PRICE total interest is 120 × 1,434.71 − 100,000 = 172,165.20 − 100,000 = R$ 72,165.20. Same 1%/mo rate, yet PRICE charges about R$ 11,665 more in interest, the price of amortizing slowly.
Where do the two installments cross? SAC starts at R$ 1,833.33 and falls R$ 8.33 per month; PRICE is R$ 1,434.71 fixed. Setting 1,833.33 − (k−1)·8.33 ≈ 1,434.71, the crossover lands around month 49: in installment 48 SAC still pays R$ 1,441.66 (above PRICE) and in installment 49 it pays R$ 1,433.33 (already below). From month 49 on, the SAC borrower pays less every month, and has already returned far more principal. Rebuild the full schedule in the loan simulator or compare it against a plain loan.
The installment crossover: R$ 300k over 240 months
Now a real-estate-sized case: PV = R$ 300,000.00, i = 0.8% per month (0.008), n = 240 months (20 years). In SAC, A = 300,000 ÷ 240 = R$ 1,250.00; the first installment is 1,250 + 0.008 × 300,000 = 1,250 + 2,400 = R$ 3,650.00 and falls by the step A·i = R$ 10.00 per month. In PRICE, (1.008)²⁴⁰ = 6.769050, (1.008)⁻²⁴⁰ = 0.147731, denominator 0.852269; PMT = 2,400 ÷ 0.852269 = R$ 2,816.01, fixed. The memory of the first installments lays the mechanics bare:
| Month | System | Installment | Interest | Amortization | Balance |
|---|---|---|---|---|---|
| 1 | SAC | R$ 3,650.00 | R$ 2,400.00 | R$ 1,250.00 | R$ 298,750.00 |
| 2 | SAC | R$ 3,640.00 | R$ 2,390.00 | R$ 1,250.00 | R$ 297,500.00 |
| 3 | SAC | R$ 3,630.00 | R$ 2,380.00 | R$ 1,250.00 | R$ 296,250.00 |
| 1 | PRICE | R$ 2,816.01 | R$ 2,400.00 | R$ 416.01 | R$ 299,583.99 |
| 2 | PRICE | R$ 2,816.01 | R$ 2,396.67 | R$ 419.34 | R$ 299,164.65 |
| 3 | PRICE | R$ 2,816.01 | R$ 2,393.32 | R$ 422.69 | R$ 298,741.96 |
View the data
| x | SAC | PRICE |
|---|---|---|
| 1 | 3,650 | 2,816.01 |
| 24 | 3,420 | 2,816.01 |
| 48 | 3,180 | 2,816.01 |
| 72 | 2,940 | 2,816.01 |
| 96 | 2,700 | 2,816.01 |
| 120 | 2,460 | 2,816.01 |
| 144 | 2,220 | 2,816.01 |
| 168 | 1,980 | 2,816.01 |
| 192 | 1,740 | 2,816.01 |
| 216 | 1,500 | 2,816.01 |
| 240 | 1,260 | 2,816.01 |
The chart shows the crossover that the average hides. The SAC installment starts R$ 833.99 above PRICE (3,650.00 versus 2,816.01), but shrinks R$ 10 per month. Setting 3,650 − (k−1)·10 = 2,816.01, they meet at k ≈ 84.4: in installment 84 SAC pays R$ 2,820 (still above) and in installment 85 it pays R$ 2,810 (already below). So from month 85 of the 240, the SAC installment is permanently smaller, and keeps falling to the last one, R$ 1,260.00.
The last statistic is the most underrated. At the contract's midpoint (month 120), the SAC borrower has already cut the balance to exactly R$ 150,000, half the principal, because amortization is linear. The PRICE borrower still owes R$ 216,708. That is R$ 66,708 more debt at the same point in time. Since your equity in the property is its value minus the outstanding balance, SAC makes you the owner of a bigger slice of the property far sooner, which matters if you plan to sell, refinance or pay it off partway. An honest note: simulators that round the installment cent by cent (ours included) leave a residue of a few reais at the end of PRICE; the exact interest comes from the formula n·PMT − PV.
Beyond the installment: CET, taxes and the mandatory insurance
So far we compared interest only. But the nominal rate, that advertised "0.8%/mo", is not what you actually pay. A mortgage carries taxes, fees and mandatory insurance that fatten every installment. The number that folds all of it into a single rate is the Total Effective Cost (CET). In Brazil, disclosing it before you sign is not a courtesy: it is a legal duty. The rule began with Resolution CMN No. 3,517/2007 and today runs under Resolution CMN No. 4,881/2020, which replaced it as of February 1, 2021.
- 1964Law 4,380: the SFH is born
Law 4,380/1964 creates the Housing Finance System (SFH), using savings (SBPE) and FGTS for mortgage credit.
- 1990Consumer Code, art. 52
The Consumer Defense Code guarantees early settlement with a proportional interest reduction (art. 52, § 2).
- 2007CET and the end of the payoff fee
Resolutions CMN 3,517 (mandatory CET disclosure) and 3,516 (bans early-settlement fees) take effect.
- 2020Resolution CMN 4,881
Consolidates the CET rule (in force since Feb 1, 2021), keeping the same components: interest, taxes, fees and insurance.
Resolution 3,517/2007, in art. 1 § 2, already defined what goes into the sum: "the CET must be computed considering the flows of disbursements and expected payments, including the interest rate to be agreed in the contract, taxes, fees, insurance and other expenses charged to the customer." Resolution 4,881/2020 kept that exact list. Meaning: two banks can advertise the same nominal rate yet have very different CETs, because insurance and fees vary. That is why the rate misleads and the CET is the number that matters.
| Component | What it is | In the CET? |
|---|---|---|
| Interest | The agreed rate on the outstanding balance. | Yes |
| Taxes (IOF) | IOF on credit, when it applies (housing is usually exempt). | Yes |
| Administration fee | Monthly charge for managing the contract. | Yes |
| MIP insurance | Death and Permanent Disability: clears the balance if the borrower dies or is disabled. | Yes |
| DFI insurance | Physical Damage to the Property: covers fire, collapse and the like. | Yes |
| Other expenses | Property appraisal, registration and third-party services charged to the customer. | Yes |
Both insurances are mandatory under sector rules: Susep confirms that "housing insurance is compulsory." MIP protects the borrower, if the holder dies or is disabled, the balance is cleared and the family keeps the property. DFI protects the collateral, the property itself, against fire, collapse and other physical damage. Both, plus the administration fee, are part of the CET under art. 1 § 2. Add it all up in the CET calculator to see the effective rate the nominal one hides; the same "real cost beyond the sticker" logic applies to a vehicle loan.
- Compare the annual Total Effective Cost (CET), not the nominal rate, it includes taxes, fees and the MIP and DFI insurance.
- Add up total interest and total cost paid at the end, not just the first installment.
- Value the down payment: the larger it is, the smaller the financed PV and all the interest.
- Check the term: stretching it eases the installment but raises the total per the interest formula.
- Confirm whether extra prepayment is allowed and how the interest discount is applied.
Extra prepayment and when each one makes sense
SAC tends to be worth more for those who can handle higher early installments and want to pay less interest overall, it is the predominant system in long-term housing contracts, and Caixa itself presents it as the standard for this credit. PRICE helps those who need a predictable, smaller early installment to fit the budget, accepting more interest. Since the interest effect over the term is large, it also helps to read the compound-interest guide and, if your contract is indexed, the monetary-restatement guide.
Extra prepayment: shorten the term or lower the installment?
An extra prepayment is a payment that hits the outstanding balance directly, outside the installment calendar. Since interest is charged on the balance, cutting it today wipes out all the future interest that slice would have generated, the proportional interest discount guaranteed by art. 52, § 2 of the Consumer Defense Code, and Resolution CMN 3,516/2007 also forbids the bank from charging a fee for it.
In practice you choose between two options. Shortening the term keeps the installment and brings the payoff forward: it saves the most interest, because it takes the balance out of circulation as early as possible. Lowering the installment keeps the term and eases the monthly amount: it saves less interest but gives immediate budget relief. Rule of thumb: if the goal is to pay less overall, shorten the term; if it is to breathe now, lower the installment.
Why is the CET higher than the nominal rate?
The nominal rate pays only for the money borrowed. The CET, by definition in Resolution CMN 4,881/2020, packs into the same annual percentage every cost in the flow: interest, taxes, the administration fee and the MIP and DFI insurance. Because those charges sit on top of interest, the CET almost always exceeds the advertised rate. Two contracts with the same nominal rate but different insurance and fees have different CETs, and it is by the CET, not the rate, that you should compare offers.
A caveat about the SFH: Resolution 3,516/2007 lists the Housing Finance System among operations with "directed resources or administered rates," which have their own rules for computing present value on early settlement. When in doubt, build both schedules in the loan simulator, add the charges in the CET calculator and decide by total cost, not by the most comfortable installment in month one.
Frequently asked questions
Does SAC or PRICE have the lower interest rate?
Why does SAC pay less total interest?
In which month does the SAC installment fall below PRICE?
What goes into the Total Effective Cost (CET)?
Extra prepayment: reduce the term or the installment?
Are MIP and DFI insurance mandatory?
SAC and PRICE charge the same rate on the same balance; the difference is only how amortization is spread over time. SAC amortizes at a fixed pace (A = PV/n), has a decreasing installment, pays less total interest and makes you the owner faster; PRICE fixes the installment (PMT = PV·i/(1−(1+i)⁻ⁿ)), starts lighter and charges more interest. The SAC installment starts larger and crosses below PRICE at a point you can compute. When deciding, compare the Total Effective Cost, interest, taxes, fees and the MIP and DFI insurance, not the nominal rate or the first installment.
Sources & references
- Resolution CMN No. 4,881/2020, calculation and disclosure of the CET (in force)
- Resolution CMN No. 3,517/2007, CET, art. 1 § 2 (components); revoked in 2021
- Resolution CMN No. 3,516/2007, bans early-settlement fee; present value
- Consumer Defense Code, Law 8,078/1990, art. 52, § 2 (proportional interest reduction)
- Law 4,380/1964, creates the Housing Finance System (SFH)
- Susep, Housing insurance (mandatory; MIP and DFI coverages)
- CAIXA, SAC or PRICE: differences between the amortization systems