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Front-end vs back-end DTI

The back-end debt-to-income test caps every required monthly debt. The front-end test caps housing alone. On $9,000 a month, 43 percent leaves $2,720.00 for principal and interest and lends $430,333.43. A 28 percent housing test leaves $2,020.00 and lends $319,585.86. The lower loan governs.

 Back-end DTIFront-end DTI
What is cappedTotal monthly debt, including other loans.Housing only: principal, interest, tax and insurance.
On \$9,000 a month43 percent is $3,870.00 of total debt. After $650 of other debts and $500 of tax and insurance, $2,720.00 is left for principal and interest.28 percent is $2,520.00 of housing. After $500 of tax and insurance, $2,020.00 is left for principal and interest.
Loan at 6.5 percent over 30 years$430,333.43, and with $60,000 down a price of $490,333.43.$319,585.86, and with $60,000 down a price of $379,585.86.
Which one governsBoth have to pass. Here this test would allow more.The lower loan governs. Here this test is the binding one.
What the rate doesHold the $2,720.00 budget and raise the rate to 7.5 percent: the loan falls to $389,007.95.The front-end budget is $2,020.00 either way. The rate still turns that budget into a smaller loan.
What it is notA household budget. It uses gross pay and ignores tax, childcare and a saving rate.The amount a household should borrow. It is a ceiling.
On this page

Two ceilings, one applicant

On $9,000 of gross monthly pay, the 43 percent back-end test allows $3,870.00 of total debt. Subtract $650 of other debts and $500 of tax and insurance and $2,720.00 remains for principal and interest, which is a $430,333.43 loan at 6.5 percent over 30 years.

The 28 percent front-end test allows $2,520.00 of housing. Other debts play no part. After $500 of tax and insurance, $2,020.00 remains, which is a $319,585.86 loan. Both tests have to pass, so $319,585.86 governs.

How mortgage affordability works is the chain, with the calculator under the answer. Principal and interest are what the loan formula can support at the rate and term you type.

The rate moves the ceiling with no change to the applicant

Hold the back-end budget at $2,720.00 and raise the rate from 6.5 percent to 7.5 percent. The loan falls to $389,007.95. Income, debts and the $60,000 deposit did not move. The factor that turns a payment into a loan did.

The mortgage affordability calculator is that factor. How mortgages work is the loan once a size exists. This is educational material, not financial advice.

Worked examples

The ceiling on \$9,000 a month

Gross pay is $9,000 a month. Other debts take $650: a car loan, student loans and card minimums. Property tax and insurance on the home you want run $500 a month. Your lender works to a 43 percent back-end limit, the loan would be 30 years at 6.5 percent, and you have $60,000 for the deposit. What is the most it would lend?

  1. Apply the limit to gross pay: 0.43×9000=0.43 \times 9000 = $3,870.00 of total monthly debt allowed.
  2. Subtract the debts you already carry: $3,870.00 minus $650 leaves $3,220.00 for housing.
  3. Subtract tax and insurance, which sit inside the housing payment: $3,220.00 minus $500 leaves $2,720.00 for principal and interest.
  4. Turn that payment into a loan. The period rate is i=0.065/12i = 0.065/12, a decimal that does not terminate, and n=360n = 360. Keep ii unrounded: 1−(1+i)−360i≈158.21082\frac{1 - (1 + i)^{-360}}{i} \approx 158.21082. Round ii to 0.00541667 before raising it to the power and the factor arrives at 158.21075 instead, which is how two calculators come to disagree over the same loan.
  5. Multiply the budget by the factor: 2720×158.210822720 \times 158.21082, which is $430,333.43.
  6. Add the deposit to read it as a price: $430,333.43 plus $60,000.

The rule stops at a loan of $430,333.43, which with $60,000 down is a price of $490,333.43. Every figure in that chain is a maximum: $3,870.00 is the most debt the rule allows, $3,220.00 the most housing, and $2,720.00 the most principal and interest.

The front-end test on the same income

The same applicant is measured by the housing test instead. A 28 percent front-end rule caps the housing payment on its own, so the other debts play no part. Same $9,000 of gross pay, same $500 of tax and insurance, same 30 years at 6.5 percent, same $60,000 deposit. Where does this test stop?

  1. The front-end test looks at housing alone, so nothing is set aside for other debts: 0.28×9000=0.28 \times 9000 = $2,520.00.
  2. There is nothing else to take off at this stage, so the whole $2,520.00 is the housing budget.
  3. Tax and insurance still come out of it: $2,520.00 minus $500 leaves $2,020.00 for principal and interest.
  4. Apply the same factor as before: 2020×158.210822020 \times 158.21082, which is $319,585.86.
  5. Add the deposit: $319,585.86 plus $60,000 gives $379,585.86.

The housing test allows a loan of $319,585.86 and a price of $379,585.86, against $430,333.43 from the back-end test in the first example. Both tests have to pass, so the lower one governs and this applicant is capped by housing, not by total debt. Compare the two on the same footing to see where it goes: the back-end test left $3,220.00 for housing, the front-end test allows only $2,520.00, and that gap in the monthly budget is the whole distance between $430,333.43 and $319,585.86 of borrowing power.

The same applicant after a one percentage point rate rise

Nothing about the applicant changes. Gross pay is still $9,000, other debts still $650, tax and insurance still $500, the limit still 43 percent, the term still 30 years and the deposit still $60,000. The mortgage rate goes from 6.5 percent to 7.5 percent. What happens to the ceiling?

  1. The budget is untouched, because it comes from income and debts rather than from the rate: $3,870.00 of total debt, $3,220.00 for housing, $2,720.00 for principal and interest.
  2. Only the factor changes. At 7.5 percent the period rate is exactly i=0.075/12=0.00625i = 0.075/12 = 0.00625, so nothing is lost to rounding this time, and 1−(1.00625)−3600.00625≈143.01763\frac{1 - (1.00625)^{-360}}{0.00625} \approx 143.01763.
  3. Multiply the same budget by the smaller factor: 2720×143.017632720 \times 143.01763, which is $389,007.95.
  4. Add the deposit: $389,007.95 plus $60,000 gives $449,007.95.

The same $2,720.00 a month now borrows $389,007.95 rather than $430,333.43, and the price ceiling falls from $490,333.43 to $449,007.95. One percentage point on the rate took about 9.6 percent off what the rule would lend and about 8.4 percent off the price, the smaller share because the deposit does not move, with no change to income, to debts or to savings. It is the single fastest way for a buyer to lose ground while doing nothing wrong.

Common questions

Why does the front-end test ignore other debts?

Because it is a housing cap, not a total-debt cap. On $9,000 a month it allows $2,520.00 of housing. The back-end test is the one that subtracts the $650 of other debts. Both have to pass.

Is the higher loan the one I can take?

No. Both tests have to pass, so the lower loan governs. On this sheet that is $319,585.86 from the front-end test, not $430,333.43 from the back-end test.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.