NMLS Exam Math: Mortgage Formulas and Worked Examples

Math is the part of the SAFE MLO National Test with UST that many people dread, and it is also the part that rewards practice the fastest. The formulas are short, the numbers are usually round, and many wrong answers come from reading the stem too quickly rather than from hard arithmetic. The calculation topics sit mostly in Mortgage Loan Origination Activities, the largest domain at about 27% of the exam, and they show up again in General Mortgage Knowledge (about 20%) and in disclosure questions under Federal Mortgage-Related Laws (about 24%).

This guide walks through the formulas you should know cold, with a worked example for each: loan-to-value, monthly income, debt-to-income and maximum payment, discount points, interest and amortization, prepaid interest, escrow, ARM adjustments, and mortgage insurance cancellation.

Three habits before any math question

Find what is being asked. A stem can give you an annual figure and ask for a monthly one, or give you a rate and ask for a dollar amount. Underline the unit of the answer before you start.
Convert percentages to decimals. 6.5% is 0.065. 1.5 points is 0.015 of the loan amount.
Check the answer against common sense. A monthly interest payment on a $200,000 loan will not be $12,000. If your result looks wrong, it usually is.

Loan-to-value (LTV) and combined LTV

LTV compares the loan amount to the property value. On a purchase, the value is the lower of the sales price or the appraised value. That one rule is behind a large share of LTV traps.

LTV = loan amount / lower of price or appraised value

Example: the sales price is $300,000, the appraisal comes in at $290,000, and the loan amount is $261,000. Use $290,000. $261,000 / $290,000 = 90% LTV. If you divided by the price instead, you would get 87%, which is exactly the kind of distractor the test likes to offer.

The same rule works in reverse to find a loan amount or a down payment. A buyer is purchasing a home for $350,000 that appraises at $360,000, and the program allows 95% LTV. The maximum loan is 95% × $350,000 = $332,500, so the minimum down payment is $350,000 - $332,500 = $17,500. The higher appraisal does not let the borrower borrow more.

Combined LTV (CLTV) adds every lien on the property. A home is worth $400,000 with a $300,000 first mortgage and a $40,000 second. CLTV = ($300,000 + $40,000) / $400,000 = 85%. When the second lien is a HELOC, CLTV uses the amount drawn, while HCLTV uses the full credit line. With a $60,000 HELOC that has $20,000 drawn, CLTV is $320,000 / $400,000 = 80% and HCLTV is $360,000 / $400,000 = 90%.

Turning pay into gross monthly income

Qualifying ratios use gross monthly income, before taxes and deductions. The trap is the pay frequency.

Hourly: rate × hours per week × 52 / 12. $25 an hour at 40 hours = $25 × 40 × 52 = $52,000 a year, or $4,333.33 a month.
Weekly: pay × 52 / 12.
Biweekly (every two weeks): pay × 26 / 12. A $2,000 biweekly check = $52,000 a year, or $4,333.33 a month.
Semimonthly (twice a month): pay × 24 / 12. A $2,000 semimonthly check = $48,000 a year, or $4,000 a month.

Biweekly and semimonthly sound alike and produce different answers. Expect at least one answer choice built on mixing them up.

DTI and maximum housing payment

The housing (front-end) ratio is the total monthly housing payment divided by gross monthly income. The total (back-end) ratio adds recurring monthly debts to the housing payment. The ATR/QM guide covers how DTI fits the Ability-to-Repay rule. Here is the version of the question that asks you to work backward.

Example: a borrower earns $6,000 a month gross and pays $500 a month on a car loan and credit cards. The stem says the program allows a 28% housing ratio and a 36% total ratio. What is the maximum housing payment?

Housing limit: $6,000 × 0.28 = $1,680
Total limit: $6,000 × 0.36 = $2,160, minus $500 in debts = $1,660

The borrower must meet both limits, so the answer is the lower figure, $1,660. Picking $1,680 is the most common mistake on this question type. Use whatever ratios the stem gives you; do not assume a program limit that the question did not state.

Discount points and break-even

One discount point equals 1% of the loan amount, not the purchase price. Points are prepaid interest the borrower pays to buy a lower rate. Do not confuse a point with a basis point, which is one hundredth of a percentage point (0.01%).

Example: a borrower takes a $250,000 loan and pays 1.5 points. Cost = $250,000 × 0.015 = $3,750. If the lower rate saves $50 a month on the payment, the break-even is $3,750 / $50 = 75 months, or 6.25 years. A borrower who expects to sell or refinance before then would lose money buying the points.

Exam shortcut: points are a percent of the loan amount, LTV uses the lower of price or appraisal, and a maximum payment question takes the lower of the two ratio results.

Interest, payments and amortization

Simple annual interest is the balance times the rate. Divide by 12 for a month.

Example: a $200,000 balance at 6% costs $200,000 × 0.06 = $12,000 a year in interest, or $1,000 a month. On an interest-only loan, $1,000 is the monthly payment and the balance does not go down.

On a fully amortizing loan, each payment covers that month's interest first and the rest reduces principal. The fully amortizing principal and interest payment on that same $200,000 loan at 6% for 30 years is $1,199.10. In month one, $1,000 goes to interest and $199.10 goes to principal, leaving a balance of $199,800.90. Next month's interest is a little lower because the balance is lower, so a little more goes to principal. That is why early payments are mostly interest.

When a stem gives you the payment, split it the same way: interest is the balance times the rate divided by 12, and the rest is principal. Also know that a shorter term means a higher payment and much less total interest over the life of the loan.

Prepaid (per diem) interest at closing

Mortgage interest is paid in arrears. The payment due December 1 pays the interest for November. So at closing, the borrower prepays interest from the closing date through the end of that month, and the first regular payment is due on the first day of the month after next. This is one of the prepaid items on the Loan Estimate and Closing Disclosure, which the TRID guide covers.

Daily interest = loan amount × rate / 365 (some stems tell you to use 360; follow the stem).

Example: a $300,000 loan at 6.5% closes on October 20. Annual interest is $19,500, so daily interest is $19,500 / 365 = about $53.42. Counting October 20 through October 31 gives 12 days, so prepaid interest is about $641. The first payment is due December 1.

Count the closing day itself unless the stem says otherwise. The number of days is where this item is usually won or lost.

Escrow, PITI and the escrow cushion

PITI is principal, interest, taxes, and insurance. Taxes and insurance are usually given as annual figures, so divide by 12.

Example: P&I is $1,199.10, annual property taxes are $3,600, and annual homeowners insurance is $1,200. Monthly escrow = ($3,600 + $1,200) / 12 = $400. PITI = $1,199.10 + $400 = $1,599.10. Add monthly mortgage insurance or HOA dues if the stem includes them in the housing payment.

Under RESPA's Regulation X, the servicer may hold a cushion in the escrow account of no more than one-sixth of the estimated total annual escrow disbursements, which works out to two months of escrow payments. With $4,800 a year in taxes and insurance, the maximum cushion is $4,800 / 6 = $800.

ARM adjustments and caps

An adjustable rate resets to the fully indexed rate, which is the index plus the margin, limited by the caps. The margin stays the same for the life of the loan; the index moves.

Caps are often written as three numbers, such as 2/2/5: the first is the most the rate can change at the first adjustment, the second is the most it can change at each later adjustment, and the third is the most it can ever rise above the start rate.

Example: a loan starts at 5.00% with 2/2/5 caps and a 2.75% margin. At the first adjustment the index is 4.50%, so the fully indexed rate is 7.25%. The first-adjustment cap limits the increase to 2 points, so the new rate is 7.00%, not 7.25%. Over the life of the loan the rate can never go above 5.00% + 5 = 10.00%.

Mortgage insurance cancellation

Under the Homeowners Protection Act, borrower-paid private mortgage insurance on a conventional loan follows two LTV triggers, both measured against the original value of the property (the lower of the sales price or appraised value at origination):

80%: the borrower may request cancellation once the balance is scheduled to reach, or actually reaches, 80% of original value, if the borrower has a good payment history and meets the other conditions.
78%: PMI terminates automatically when the balance is scheduled to reach 78% of original value under the original amortization schedule, as long as the borrower is current.

Example: a home bought for $250,000 that appraised at $255,000 has an original value of $250,000. The borrower can request cancellation at a $200,000 balance (80%), and PMI ends automatically when the scheduled balance hits $195,000 (78%). A rise in today's market value does not change these numbers.

How the exam baits math questions

Did you use the lower of the sales price or the appraised value for LTV?
Is the income annual, monthly, biweekly, or semimonthly? Convert before you divide.
Did you use gross income rather than take-home pay?
For a maximum payment, did you take the lower of the housing and total ratio results?
Are points figured on the loan amount rather than the price?
For prepaid interest, how many days, and does the stem say 365 or 360?
For an ARM, is the new rate limited by the first, later, or lifetime cap?
For PMI, is the percentage measured against original value rather than current value?

The national test has 120 items (115 scored) in a 190-minute window inside a 225-minute appointment, and you need 75% to pass. A one-time $99 unlock of MLO Pass gives you about 1,154 questions across all five domains: Mortgage Loan Origination Activities 27%, Federal Mortgage-Related Laws 24%, General Mortgage Knowledge 20%, Ethics 18%, and Uniform State Content 11%. Pay once and keep it. There is no subscription.

Practice this on MLO Pass

Work LTV, income, DTI, points, prepaid interest, and ARM cap problems until each one takes you under a minute, then mix them into timed sets the way the content outline mixes topics. Sign up to pass your test

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