Chapter 4: Mortgage Math: Payments, Amortization, DTI, LTV, APR, and the Calculations That Drive Every Decision

"A borrower will forgive you for a rate that moved. They will not forgive you for a payment that was wrong." — constructed; a working rule

Overview

This is the chapter every other chapter quotes.

None of the arithmetic here is difficult. There is one formula with an exponent in it and you will never derive it. Everything else is division. What makes this chapter matter is not the difficulty — it is that these seven or eight numbers are the entire vocabulary in which lending decisions get made, and a loan officer who is fluent in them can do something a competitor cannot: answer a borrower's real question, on the phone, correctly, in ninety seconds.

There is a second reason to take it seriously. Every one of these numbers has a failure mode — a specific way it misleads the person who trusts it. Debt-to-income is the most-quoted figure in origination and it routinely approves borrowers who cannot afford the house and declines borrowers who can. Loan-to-value looks like a simple division until the appraisal comes in low and you discover which of two numbers the denominator uses. The annual percentage rate was designed to let consumers compare loans and is arguably the most misunderstood disclosure in American consumer finance.

We will do all of it on the Linden Street file, and every figure will resolve. By the end of this chapter you will have derived, from scratch, every number in that file that the last three chapters have been quoting on credit.

In this chapter, you will learn to:

• Compute a monthly principal-and-interest payment and explain what each part of the formula does
• Build an amortization schedule by hand and explain the shape of it
• Assemble a complete PITI figure
• Compute LTV, CLTV, and HCLTV — and know which value the denominator uses
• Compute both qualifying ratios and state exactly what goes in each
• Explain at least four things DTI cannot measure
• Compute points, break-even, APR, and per-diem interest

Learning Paths

🎓 Exam — all of it, and §4.4, §4.5, and §4.8 in particular. The test asks you to compute. Know that LTV uses the lesser of price or appraised value, and know what APR includes. 🏠 New LO — §4.5, §4.6, §4.9, §4.10. §4.10 is the one that changes your phone calls. 🤝 Partner — §4.4, §4.5, §4.6. If you understand why a borrower "qualifies" for a payment they cannot live with, you will stop writing offers that die in underwriting. 📊 Operations — §4.3, §4.4, §4.5, §4.9. These are the figures you will be reconciling.

4.1 The payment formula and what each piece does

The monthly principal-and-interest payment on a fully amortizing fixed-rate loan is:

$$M = P \cdot \frac{i}{1-(1+i)^{-n}}$$

where

Two errors account for most wrong answers, and both are about units. $i$ is monthly, not annual. $n$ is months, not years. If your payment comes out absurd, check those two first.

What the pieces do

The numerator, $P \cdot i$, is one month's interest on the full balance. On this file:

$$\$365{,}750.00 \times 0.0055208333 = \$2{,}019.24$$

That is what the loan costs for one month at the beginning. If the borrower paid only that, the balance would never move — that is an interest-only payment.

The denominator, $1-(1+i)^{-n}$, is what converts an interest-only payment into an amortizing one. It is a discount factor: $(1+i)^{-n}$ is the present value of \$1 received 360 months from now, and subtracting it from 1 produces a number slightly less than 1. Dividing by a number less than 1 makes the payment larger than interest-only — and the extra is the principal.

$$(1.0055208333)^{-360} = 0.137789 \qquad 1 - 0.137789 = 0.862211$$

$$M = \frac{\$2{,}019.24}{0.862211} = \$2{,}341.94$$

🧮 Run the Numbers

The whole payment, in four lines.

Step	Arithmetic	Result
1. Monthly rate	0.06625 ÷ 12	0.0055208333
2. One month's interest	\$365,750.00 × 0.0055208333 | \$2,019.24
3. The factor	1 − (1.0055208333)−360	0.862211
4. Payment	\$2,019.24 ÷ 0.862211 | **\$2,341.94**

Do this once by hand. After that, use a calculator or a spreadsheet — but you should know what the machine is doing, because §4.10 is about sanity-checking its answer in front of a borrower.

The three levers

The formula has exactly three inputs, which means a payment can be changed in exactly three ways. Every conversation in Chapter 13 is about these:

That third row deserves a caution. Extending the term is the weakest of the three levers and the one borrowers ask for most.

🧮 Run the Numbers

Why "just go to a 40-year loan" is a worse idea than it sounds.

Same \$365,750 at 6.625%:

Term	Payment	Monthly saving vs. 30	Total P&I paid
15 years (180)	\$3,211.26 | — | \$578,026.80
30 years (360)	\$2,341.94** | — | **\$843,098.40
40 years (480)	\$2,173.96 | \$167.98	\$1,043,500.80

Ten additional years of obligation buys \$167.98 a month** and costs **\$200,402.40 in additional interest. The lever is real but it is nearly exhausted by year thirty — because by then the discount factor is already close to 1 and stretching it further does very little.

Look at the fifteen-year row while you are here. The payment is \$869.32 higher, and it saves \$265,071.60 in interest. That is the same lever pointed the other way, and it is the trade most borrowers have never had shown to them.

(40-year terms are not Qualified Mortgages; see Chapter 24. Shown here as arithmetic.)

4.2 Amortization: why the early years are almost all interest

An amortization schedule is the month-by-month record of how a payment splits. The rule generating it is one line:

Interest this month = current balance × monthly rate. Principal this month = payment − interest. New balance = old balance − principal.

That is the whole thing. Every amortization schedule ever printed is that rule, repeated.

THE FIRST SIX PAYMENTS — Linden Street                    [the Linden Street file]
$365,750.00 at 6.625%, 360 months, payment $2,341.94

  #    BALANCE       INTEREST    PRINCIPAL    NEW BALANCE
  ──────────────────────────────────────────────────────────
  1   365,750.00    2,019.24      322.70      365,427.30
  2   365,427.30    2,017.46      324.48      365,102.82
  3   365,102.82    2,015.67      326.27      364,776.55
  4   364,776.55    2,013.87      328.07      364,448.48
  5   364,448.48    2,012.06      329.88      364,118.60
  6   364,118.60    2,010.24      331.70      363,786.90
  ──────────────────────────────────────────────────────────
  Six payments = $14,051.64 paid.  Balance reduced by $1,963.10.
  86.0% of six months of payments was interest.

Interest is computed on the balance and rounded to the cent each month, which is what a servicer does and what you will get doing this by hand. Carry full precision instead and the balances drift by a few cents — see the callout at the end of this section.

The shape is the lesson. Interest falls by about \$1.80 a month at first; principal rises by the same \$1.80. It compounds — slowly at first, then not slowly.

WHERE THE PAYMENT GOES OVER TIME                          [the Linden Street file]
  each bar = one month's payment, split interest / principal

  month   1   ████████████████████████████████████░░░░░░  86.2% interest
  month  60   ███████████████████████████████░░░░░░░░░░░  80.9%
  month 120   ██████████████████████████░░░░░░░░░░░░░░░░  73.5%
  month 180   ███████████████████░░░░░░░░░░░░░░░░░░░░░░░  63.1%
  month 240   ████████████░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░  48.6%
  month 300   ██████░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░  28.5%
  month 360   ▏░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░   0.5%

  ██ interest    ░░ principal            (schematic, not to scale)

Why borrowers find this upsetting, and what to say

Told that their first payment of \$2,341.94 reduces the balance by \$322.70, borrowers commonly react as though something is wrong. Nothing is. Interest is rent on money, the balance is at its maximum on day one, so the rent is at its maximum on day one.

📞 On the Phone

Borrower: "Only three hundred dollars goes to the loan? That feels like a scam."

What works: "It's not, but I get why it looks that way. Interest is just rent on the money, and you're renting the most money you'll ever rent on the very first payment — three hundred sixty-five thousand dollars of it. As the balance drops, so does the rent, and every dollar the rent drops goes to principal instead. It compounds in your favor. Five years in you'll be putting almost four hundred and fifty a month toward the balance without paying a penny more."

Then hand them the useful number: "The part people actually care about — after five years you'll have paid the balance down about twenty-three thousand dollars, and that's before anything the house does in value."

The concrete figures, all verified against the schedule:

Total of 360 payments: \$843,098.40.** Total interest **\$477,348.40 — 1.31 times the amount borrowed. That figure is worth showing a borrower once, in the context of §4.7's break-even discussion, and never used to scare them.

⚠️ Where Deals Die

The 96-cent problem, and why you should know about it.

The note's payment is \$2,341.94 — rounded to the cent. The mathematically exact payment is \$2,341.9373. Multiply the rounded payment by 360 and you get **\$843,098.40; use the unrounded one and you get \$843,097.44. Ninety-six cents, across thirty years.**

Which is right? The rounded one, because it is what the borrower actually pays.

The same thing happens inside the schedule. Run all 360 rows the way the table above does — exact interest, rounded to the cent, principal is the remainder — and the balance does not land on zero. It lands \$3.07 past it. The borrower has slightly overpaid, because every month they paid a payment rounded up from \$2,341.9373.

In practice the servicer adjusts the final payment to clear the balance exactly. The last payment on this loan is about **\$2,338.87** rather than \$2,341.94, and no borrower in the history of mortgage lending has noticed.

Why it matters to you: dollars are canonical; percentages and totals are derived. If you back-compute a dollar figure from a rounded percentage, or a total from an unrounded payment, you will produce a number that disagrees with the document in front of the borrower.

The habit that prevents it: always derive from the figure that appears on the disclosure, and if two of your own numbers disagree by pennies, find out which one the borrower will see.

Negative amortization: the same rule, running backward

The one-line rule has a failure mode, and it is worth seeing once, because it explains an entire era of American lending and it is on the exam.

Principal this month = payment − interest. Nothing in that sentence guarantees the result is positive. If the payment is smaller than the month's interest, the principal figure comes out negative, and the shortfall is added to the balance rather than subtracted from it. The balance goes up. That is negative amortization, and the reason it alarms underwriters is that it feeds itself: a larger balance produces more interest next month, which produces a larger shortfall, which produces a larger balance again.

Put an \$1,800 payment on the Linden Street loan — nothing else changed, same rate, same balance — and watch it run.

NEGATIVE AMORTIZATION — Linden Street at an $1,800 payment    [constructed teaching example]
$365,750.00 at 6.625%. Month-1 interest is $2,019.24. The payment is $1,800.00.

  MONTH   INTEREST DUE    PAID     SHORTFALL   NEW BALANCE     vs. ORIGINAL
  ─────────────────────────────────────────────────────────────────────────
      1      2,019.24    1,800.00    219.24     365,969.24      +   219.24
      6      2,025.36    1,800.00    225.36     367,083.76      + 1,333.76
     12      2,032.93    1,800.00    232.93     368,462.31      + 2,712.31
     24      2,048.84    1,800.00    248.84     371,359.88      + 5,609.88
  ─────────────────────────────────────────────────────────────────────────
  Every payment made on time. Nothing missed. The balance rose $5,609.88.

Read the bottom line twice. This borrower did nothing wrong by any standard a servicer measures. They paid on the first of the month, twenty-four consecutive times, and they owe five thousand six hundred dollars more than they borrowed. On a 95% loan against a \$385,000 house, that one fact is close to enough to put them underwater without the market moving at all — and a borrower who is underwater cannot sell and cannot refinance, which are the only two exits an ordinary household has.

Where does a payment smaller than the interest come from? Not from a borrower deciding to send less; a servicer would reject a short payment. It comes from products that offer it as a choice. The payment-option adjustable-rate mortgage of the mid-2000s presented borrowers with a menu each month — a fully amortizing payment, an interest-only payment, and a "minimum" payment set below the accrued interest — and it was routinely marketed on the minimum. Graduated-payment structures do a gentler version of the same thing on purpose, starting below the amortizing payment and stepping up. The lesson the market took from the 2008 crisis is not that negative amortization is exotic. It is that a borrower can be qualified on a payment that was never going to retire the debt.

Three things look like negative amortization and are not. You will be asked about all three, by a borrower or by a test:

• An interest-only payment. Payment equals interest exactly, so the balance is flat, not rising. Flat is not falling, and the borrower has bought nothing but time — but the balance is not growing, and that distinction matters.
• An escrow shortage spread over twelve months. The borrower's monthly total rises to make up a tax or insurance underage. The mortgage balance does not move at all; a separate escrow ledger does. Chapter 23 works this.
• A deferral or partial claim after a forbearance, where missed payments are moved to a non-interest-bearing balance due at payoff. The amount owed rises, but it does not accrue interest, so it does not compound. That is the whole difference.

The diagnostic is one multiplication, and it takes four seconds. Any time somebody quotes you a payment on a balance, compute $P \times i$ first. On this file that is \$2,019.24. A quoted payment below \$2,019.24 is not a mortgage payment on this loan; it is a deferral with a marketing name, and your next question is what happens to the difference?

⚖️ Compliance Check

Negative amortization is not illegal, and it is close to it in the consumer market.

Under the Ability-to-Repay/Qualified Mortgage rule adopted by the Consumer Financial Protection Bureau under the Dodd-Frank Act, a loan with negative amortization cannot be a Qualified Mortgage. The same rule excludes interest-only features and terms longer than thirty years from QM status — which is why the forty-year row in §4.1 carries the note it does. QM status is not a licensing requirement; it is a lender's safe harbor from ability-to-repay liability, and the practical effect is that the mainstream secondary market will not buy these loans, so almost nobody offers them.

Regulation Z also imposes disclosure and, for high-cost mortgages, outright restrictions on features of this kind, and a number of states legislated against payment-option products directly after 2008. State law varies substantially here.

The rules in this area have been revised repeatedly since 2013 and will be revised again. Verify the current text of the ATR/QM rule and your state's provisions with your compliance department and your regulator before you describe any product to a borrower. Chapter 24 covers ATR/QM in full; nothing in this book is legal advice.

4.3 PITI and the things that are not P&I

The payment formula produces principal and interest. It is not the borrower's payment, and telling a borrower "your payment is \$2,341.94" is one of the more common ways to lose trust in month one.

PITI is principal, interest, taxes, and insurance — and in practice it is a container for everything collected monthly:

🧮 Run the Numbers

Building PITI from the source documents.

Line	Source	Annual	÷ 12
Property taxes	county assessor's record	\$4,620.00 | **\$385.00**
Homeowners insurance	the binder from the borrower's agent	\$1,560.00 | **\$130.00**
Mortgage insurance	0.58% annual factor × \$365,750 = \$2,121.35	\$2,121.35 | **\$176.78**

\$2,341.94 + \$385.00 + \$130.00 + \$176.78 = \$3,033.72

The tax figure is 1.20% of the purchase price, which is a useful sanity check but never a substitute for the actual assessment — rates vary enormously by jurisdiction and the assessed value is frequently not the purchase price.

The four traps in PITI

1. The tax figure is the future tax figure. In many jurisdictions a sale triggers a reassessment, so the seller's current tax bill can badly understate what the buyer will pay. A borrower qualified on the seller's taxes and closed at the new ones has a payment that is wrong from month one. Ask what the property will be assessed at after transfer, not what it is assessed at now.

2. Insurance is a quote until it is a binder. Estimated at application, real once the borrower has actually bought a policy. In markets with hardening insurance — coastal wind, wildfire, hail — the difference between the estimate and the binder can be hundreds of dollars a month, and it lands late. Chapter 21 covers this.

3. Mortgage insurance is not permanent, and borrowers should know when it ends. §4.4 and Chapter 5 cover the mechanism. On this file it terminates automatically at payment 137.

4. HOA dues are not collected in the escrow account in most cases — the borrower pays them directly — but they are counted in the qualifying ratio. A borrower comparing two houses where one has \$300 in monthly dues is comparing very different files, and the dues do not show up in the mortgage payment they will be quoted.

⚠️ Where Deals Die

"Your payment is \$2,341.94."

Say this to a borrower and one of two things happens. Either they budget on it and discover at closing that the real number is \$3,033.72 — 29.5% higher — or they hear it, mistrust it, and assume you are shading numbers to win their business.

The rule: quote PITI, always, and say the letters out loud. "Your payment, all in — principal, interest, taxes, insurance, and mortgage insurance — is about three thousand thirty-five a month." If you have to estimate taxes and insurance, say you are estimating and say what would change it.

4.4 Loan-to-value, CLTV, and HCLTV

Loan-to-value is the loan amount divided by the property's value:

$$\text{LTV} = \frac{\text{loan amount}}{\text{value}}$$

On Linden Street:

$$\frac{\$365{,}750.00}{\$385{,}000.00} = 0.9500 = \mathbf{95.00\%}$$

Simple — until you ask what "value" means.

The rule that decides files

LTV uses the lesser of the purchase price or the appraised value.

For a purchase. (On a refinance there is no purchase price, so the appraised value governs, subject to seasoning rules covered in Chapter 37.)

This single rule is the most consequential piece of arithmetic in this chapter, because it is what makes a low appraisal a crisis rather than an opinion. The lender will lend a percentage of the lower number. If the appraisal is below the contract price, the borrower makes up the entire difference in cash — the seller's price does not automatically fall, and the lender does not lend against a value that has not been supported.

🧮 Run the Numbers

The Cypress Court file — what a low appraisal actually costs.

A \$540,000 contract, conventional, 20% down (\$108,000), loan \$432,000. The appraisal returns at \$505,000** — \$35,000 low, 6.48%** under contract — eleven days before closing.

	On the contract price	On the appraised value
Value used	\$540,000 | **\$505,000** (the lesser)
Maximum loan at 80% LTV	\$432,000 | **\$404,000**
Purchase price (unchanged)	\$540,000 | \$540,000
Required down payment	\$108,000 | **\$136,000**
The gap	—	\$28,000

The buyers must produce \$28,000 they did not plan for, or renegotiate the price, or obtain a reconsideration of value, or terminate under the appraisal contingency. Nothing about the house changed. Chapter 18 works all five options and their costs; Chapter 20 covers the contingency.

(Constructed teaching file.)

The Cypress Court arithmetic, all the way through

That callout gives you the answer. This subsection gives you the machinery, because the \$28,000 is not a fact about Cypress Court — it is an instance of an identity you can run on any file, in your head, while the appraiser's email is still open.

The identity. On a purchase where the appraisal comes in low, the borrower's required cash rises by the maximum LTV times the shortfall:

$$\text{cash gap} = \text{max LTV} \times (\text{contract price} - \text{appraised value})$$

Why: the price is unchanged, so the borrower still owes the seller \$540,000. The lender's maximum loan is a fixed percentage of the lesser number, which just fell. Every dollar the value fell cost eighty cents of borrowing capacity at 80% LTV, and required cash is price minus loan, so it rose by the same eighty cents.

$$0.80 \times \$35{,}000 = \mathbf{\$28{,}000}$$

Run the same identity on Linden Street and it reproduces a figure this chapter's Loan File checkpoint states independently: at 95% LTV, an appraisal \$13,000 light would cost $0.95 \times \$13{,}000 = \$12{,}350$. The higher the LTV, the more a short appraisal hurts — which is the opposite of most people's intuition, because a low-down-payment borrower feels like they have less exposure to the appraisal, not more.

WHAT A SHORT APPRAISAL COSTS — the cash gap, by LTV       [constructed teaching example]
  cash gap = max LTV x (contract price - appraised value)

  SHORTFALL      at 80% LTV    at 90% LTV    at 95% LTV    at 96.5% LTV
  ──────────────────────────────────────────────────────────────────────
   $ 5,000         $ 4,000       $ 4,500       $ 4,750        $ 4,825
   $10,000         $ 8,000       $ 9,000       $ 9,500        $ 9,650
   $20,000         $16,000       $18,000       $19,000        $19,300
   $35,000         $28,000  <--  $31,500       $33,250        $33,775
   $50,000         $40,000       $45,000       $47,500        $48,250
  ──────────────────────────────────────────────────────────────────────
  Cypress Court is the marked cell: $540,000 contract, $505,000 appraisal,
  80% conventional. The gap is not the shortfall. It is a fraction of it --
  and the fraction is the LTV.

Four ways the arithmetic can move, and what each one is worth

Chapter 18 works the options as a decision — who to call, what a reconsideration of value actually requires, how to talk to a listing agent. What belongs here is the price of each one in dollars, because you will be asked for those numbers on the phone before anybody decides anything.

1. Bring the cash. \$28,000. The simplest and usually the least available. Note what it does beyond the closing table: \$28,000 that was going to be reserves is now equity, and a borrower who had six months of reserves may have two. Chapter 12 puts a price on that.

2. Renegotiate the price to the appraised value. If the seller agrees to \$505,000, the arithmetic is not "the gap closes." It is better than that:

The buyer needs seven thousand dollars less than they budgeted before the appraisal ever came in. That is worth saying out loud, because a borrower in the middle of a short-appraisal crisis hears "renegotiate" as a long shot and does not realize it is the only outcome on the list that leaves them better off than they started.

3. Split the difference. Say the parties meet at \$522,500 — exactly halfway. Most people assume this halves the gap, to \$14,000. It does not:

$$\$522{,}500 - \$404{,}000 = \$118{,}500 \qquad \$118{,}500 - \$108{,}000 = \mathbf{\$10{,}500}$$

The gap falls from \$28,000 to **\$10,500, better than half. Here is why, and it is the single most useful thing in this subsection: once the value is the binding constraint, the maximum loan stops moving.** It is frozen at 80% of \$505,000 no matter what the price does. So every dollar taken off the price is a dollar taken off the required cash — not eighty cents. A \$17,500 price reduction removes \$17,500 of cash need, and \$28,000 − \$17,500 = \$10,500.

Contrast that with the other direction. A reconsideration of value that raises the appraisal moves the loan cap, so it returns only eighty cents on the dollar:

A dollar of price concession is worth more than a dollar of appraised value. That is not intuitive, it is arithmetically certain, and it should shape which phone call you make first.

4. Keep the loan and accept mortgage insurance. The borrower wanted \$432,000 and can still have \$432,000 — just not at 80% loan-to-value:

$$\frac{\$432{,}000}{\$505{,}000} = 85.54\%$$

Above 80%, so mortgage insurance appears. At an illustrative annual factor of 0.40% [constructed teaching example — MI factors are set by the insurer and change; verify current factors at the source], that is $\$432{,}000 \times 0.0040 \div 12 = \$144.00$ a month.

Now the part that makes this option legible to a borrower. To cancel that mortgage insurance under the Homeowners Protection Act schedule set out later in this section, the balance has to reach 80% of the original value — 80% of \$505,000 is \$404,000 — which means paying the balance down by \$432,000 − \$404,000 = \$28,000.

The same \$28,000. The borrower does not escape it; they finance it, at \$144 a month plus interest, until the schedule retires it. That is a complete and honest answer to "can't we just do MI instead?", and it takes about fifteen seconds to deliver.

📄 Read the File

text FIGURE 4.1 — "The number that moved" [the Cypress Court file] THE DOCUMENT Uniform Residential Appraisal Report (Form 1004), completed by a state- licensed appraiser through an appraisal management company, delivered eleven days before the scheduled closing. THE CONTEXT A $540,000 contract on a four-bedroom in an appreciating neighborhood. Conventional, 20% down ($108,000), loan $432,000, 80% LTV. Everything else in the file is clean. WHAT IT SHOWS Opinion of market value: $505,000. Sales comparison approach, three closed comparables with adjustments for gross living area, garage, and date of sale. $35,000 below contract -- 6.48% under. That single figure resets the LTV denominator, because LTV uses the LESSER of price or appraised value. Maximum 80% loan falls from $432,000 to $404,000. Required down payment rises from $108,000 to $136,000. Gap: $28,000. WHAT IT DOESN'T It does not show that the appraiser was wrong, and it does not show that the appraiser was right. It does not show whether a closed sale on the subject's street was available and unused, whether an adjustment was mis-signed, or whether the contract itself was above market -- which is the possibility the buyers will least want to hear and the one that is true most often. An appraisal is one licensed opinion, not a fact. THE DECISION Today: read the grid before you call anybody. Confirm the comparables are the ones you would have chosen and check whether a better sale exists. Then call the borrower with the $28,000 already computed and all four arithmetic paths priced -- cash, price, split, or MI -- because the call where you deliver a problem without options is the call that loses the file. THE LESSON The appraisal did not change the house and it did not change the price. It changed the denominator. Everything downstream is division.

Constructed teaching file. Form 1004 is a real, current agency form; the values here are illustrative.

📞 On the Phone

The call, eleven days out. You make it, immediately, and you make it with numbers already in hand. What you must not do is forward the appraisal with "let me know your thoughts."

You: "The appraisal came in this morning at five-oh-five. Contract's five-forty, so it's thirty-five thousand light. Here's what that does and here's what we can do about it — I have four options and I've run the numbers on all of them."

Borrower: "Wait — so the house isn't worth what we're paying?"

What works: "It's worth what somebody will pay for it. What the appraiser is saying is that the three closest recent sales support five-oh-five, and the lender will only lend against the lower of the two numbers. That's a rule, not a judgment about your house. Practically, your loan caps at four-oh-four instead of four-thirty-two, and that's twenty-eight thousand more cash — if nothing else changes. Four things could change."

Then walk the four, in this order: the seller comes down (best outcome, and at five-oh-five you actually need seven thousand less than you planned); you split it (at five-twenty-two-five the gap is ten-five, not fourteen); we ask for a reconsideration of value with better comparables if better comparables exist; or you keep the loan at four-thirty-two, add about a hundred forty-four a month in mortgage insurance, and pay it off over time. "And there's a fifth, which is that your contract probably has an appraisal contingency and you can walk. I'm not recommending it. I'm telling you it's there, because you should know what your options are before your agent calls."

The failure mode: waiting a day to "get more information" before calling. There is no more information coming, the closing date does not move on its own, and every hour you sit on it is an hour the borrower and the agents do not have to negotiate.

CLTV and HCLTV

When there is more than one lien, two more ratios appear:

The HCLTV distinction catches people. A home equity line with a \$100,000 limit and a \$5,000 balance counts as \$100,000 in the HCLTV calculation, because the borrower can draw the rest tomorrow.

🧮 Run the Numbers

The Harlow Street file — layering a down-payment assistance second.

A \$215,000 purchase. FHA financing with a \$10,000 forgivable county DPA second lien.

	Amount	Ratio
Base FHA loan (96.5%)	\$207,475.00	LTV = 207,475 ÷ 215,000 = 96.50%
DPA second lien	\$10,000.00
Combined	\$217,475.00	CLTV = 217,475 ÷ 215,000 = 101.15%

A CLTV above 100% looks alarming and is entirely ordinary in assisted lending — it is the point of the assistance. What matters is whether the program permits it, which is a Chapter 33 question.

Note also that the FHA base loan LTV of 96.50% is computed before the financed upfront mortgage insurance premium is added; the total loan of \$211,105.81 is not the LTV numerator. Chapter 16 explains why.

(Constructed teaching file. Illustrative FHA factors — verify current figures with HUD.)

What LTV drives

And on the way back down, for a borrower already in a loan:

Note original value, not current value. A borrower whose house appreciated may be able to cancel earlier based on a new appraisal, but that is a separate request under the servicer's rules, not the automatic HPA schedule. Chapter 23 covers the mechanics.

🎓 NMLS Exam Watch

Three testable items here, and the exam mixes them deliberately:

• LTV uses the lesser of price or appraised value on a purchase.
• 80% = request; 78% = automatic, both measured against the original value, both under the Homeowners Protection Act, and both applying to borrower-paid private mortgage insurance on conventional loans. FHA's MIP follows entirely different rules (Chapter 16) — a favorite trap.
• HCLTV uses the full HELOC line, not the drawn balance.

4.5 The two qualifying ratios

Two ratios, same denominator.

$$\text{Housing (front-end) ratio} = \frac{\text{PITI}}{\text{gross monthly income}}$$

$$\text{Total debt (back-end) ratio} = \frac{\text{PITI} + \text{all other monthly debts}}{\text{gross monthly income}}$$

On the Linden Street file:

$$\text{Housing} = \frac{\$3{,}033.72}{\$10{,}500.00} = 0.288926 = \mathbf{28.89\%}$$

$$\text{Back-end} = \frac{\$3{,}033.72 + \$1{,}446.00}{\$10{,}500.00} = \frac{\$4{,}479.72}{\$10{,}500.00} = 0.426640 = \mathbf{42.66\%}$$

When this book says "DTI" without qualification, it means the back-end ratio. That is the industry convention and it is what declines files.

The denominator: what "gross monthly income" actually contains

Both ratios divide by the same figure, which has a useful consequence and a dangerous one. The useful consequence is that anything which moves the denominator moves both ratios at once, in the same proportion — so a borrower who picks up a documentable second job improves the housing ratio and the back-end ratio simultaneously. The dangerous one is that the denominator is the number most often wrong, and when it is wrong, every ratio in the file is wrong with it.

Three things about it are worth fixing in your head now, because Chapters 11 and 14 build an entire apparatus on top of them.

It is gross, not net. Before taxes, before withholding, before health premiums, before retirement contributions. This is a deliberate choice by the agencies — gross pay is documentable from a paystub and a W-2, while net pay depends on elections the borrower can change next week — and it is also the source of the ratio's first blind spot, which §4.6 takes apart.

It is qualifying income, not earnings. The number in the denominator is what an underwriter can document, average, and reasonably expect to continue, which is frequently less than what the household actually receives. Overtime that has run for four months does not count. A bonus paid once does not count. Income from a job the borrower started three weeks ago may not count. The borrower experiences this as the lender refusing to acknowledge money that is visibly landing in their account, and it is worth saying plainly at application rather than at underwriting.

Variable income is averaged, and the averaging costs money when income is rising. This is the one that surprises loan officers, so work it on the file in front of you.

THE DENOMINATOR, BUILT FROM THE DOCUMENTS                     [the Linden Street file]

  BORROWER 1 -- registered nurse, W-2, 3 years, same employer
    base            $33.00/hr x 2,080 hrs / 12                      $5,720.00   stable
    differential
      + overtime    24-month average                                $  580.00   VARIABLE
                                                                    ─────────
                                                                    $6,300.00

  BORROWER 2 -- outside sales, W-2 base + commission, 4 years
    base            $28,800/yr / 12                                 $2,400.00   stable
    commission      ($19,800 + $23,400) / 24 months                 $1,800.00   VARIABLE
                                                                    ─────────
                                                                    $4,200.00
  ────────────────────────────────────────────────────────────────────────────
  TOTAL QUALIFYING INCOME                                          $10,500.00
  of which VARIABLE                     $2,380.00  =  22.67% of the denominator
  ────────────────────────────────────────────────────────────────────────────

Look at Borrower 2's commission. Year one was \$19,800; year two was \$23,400. That is a rise of 18.18%, and the borrower knows it — they will tell you their commission is "about two thousand a month now," and they are right. The 24-month average does not care:

$$\frac{\$19{,}800 + \$23{,}400}{24} = \$1{,}800.00 \qquad \text{versus} \qquad \frac{\$23{,}400}{12} = \$1{,}950.00$$

The convention costs this borrower \$150.00 a month of qualifying income. Follow it through to the ratio, because \$150 sounds small and the effect is smaller still:

Sixty basis points. That is the whole difference, and it is why arguing with an underwriter about a rising commission is almost never worth the relationship — but knowing the number means you can tell the borrower exactly what the conservatism cost them instead of shrugging.

The convention runs the other way too, and that direction is not optional. When variable income is declining rather than rising, the underwriter does not average — the lower, most-recent figure governs, and the burden shifts to explaining why the decline will not continue. Chapters 11 and 14 work that case in full, on a file built for it.

One last point about that 22.67%. Nearly a quarter of this household's qualifying income is variable, which is ordinary and approvable and also a fact worth carrying forward: every dollar of it depends on a shift differential and a commission plan that neither employer is contractually obliged to maintain. The ratio does not know that. You do.

🎓 NMLS Exam Watch

The ratio questions on the SAFE test are arithmetic wrapped in a distractor, and the distractor is almost always in the numerator or the denominator rather than in the division. Four traps, in rough order of how often they appear:

• Gross, never net. A stem that hands you take-home pay is testing whether you noticed. If the question gives you both, use gross.
• The front-end numerator is the entire housing payment — principal, interest, taxes, insurance, mortgage insurance, and HOA dues — not principal and interest. A stem that lists \$2,341.94 alongside taxes, insurance, and MI and asks for the housing ratio is checking whether you added them.
• The back-end numerator includes the housing payment. Candidates routinely divide only the other debts by income and produce a number far too low. Back-end = front-end + everything else.
• Only counted obligations count. Utilities, groceries, phone, childcare, and payroll-deducted health premiums are not debts. A stem that lists a \$600 daycare bill is testing that, and the answer is that it changes nothing.

And know both names for each ratio. The exam uses housing ratio, front-end, and top ratio interchangeably, and total debt ratio, back-end, bottom ratio, and DTI for the other.

What is actually in the numerator

The back-end numerator is monthly obligations that appear on the credit report or are otherwise required to be counted — not everything the household spends.

Two of those deserve emphasis because they surprise people. A 401(k) loan is generally not counted as a debt, on the reasoning that the borrower is repaying themselves and the loan is secured by their own balance. And childcare is not counted, which is a real limitation we return to in §4.6.

The ten-month rule

An installment debt with a small number of payments remaining may generally be excluded from the ratio — commonly ten or fewer payments, subject to program specifics and to the debt not being large enough to affect the borrower's ability to pay in the near term.

On the Linden Street file, neither auto loan qualifies: B1's has 31 payments remaining and B2's has 19. But it is worth seeing what it would be worth:

🧮 Run the Numbers

What the ten-month rule would be worth on this file.

If B2's auto loan (\$429.00/month) had nine payments left instead of nineteen:

	Obligations	Back-end
As the file stands	\$4,479.72	42.66%
Excluding the auto	\$4,050.72	38.58%

Four full points of DTI, for free, on a debt that will be gone in nine months either way.

This is why you read the credit report for remaining terms, not just for payments — and why, when a borrower is over a limit, "which of these debts is nearly finished?" is one of the first questions to ask. Chapter 10 covers reading a tradeline; Chapter 14 covers the program rules.

(Illustrative. Verify current guideline treatment; the ten-month convention and its conditions vary by agency and program.)

What the limits are

The honest answer is that there is no single number, and any book that gives you one is teaching you something that will cost you a file.

Two rules follow, and they are the practical content of this section:

Never tell a borrower they are declined because of a DTI number. The file is evaluated as a whole. A 47% back-end ratio with twelve months of reserves, a 780 score, and a documented twenty-year employment history is not the same file as a 47% ratio with no reserves and a 640 score, and the automated underwriting system does not treat them the same.

Never tell a borrower they are approved because of a DTI number, either. Which is §4.6.

4.6 What DTI cannot tell you

This is the most important section in the chapter, and it is not on the exam.

Debt-to-income is a good ratio. It is cheap to compute, uses data that can be verified, is comparable across files, and correlates with default well enough to be worth using. It is also structurally blind to most of what determines whether a household can afford a house, and a loan officer who does not know exactly where the blindness lies will do real harm while following every rule correctly.

1. It uses gross income, so it ignores taxes. Two borrowers with identical \$10,500 gross incomes can have materially different take-home pay — different filing statuses, different state income tax, different pre-tax deductions, different numbers of dependents. DTI treats them as identical. In a high-tax state the same 42.66% back-end ratio can consume a much larger share of what actually arrives in the account.

2. It ignores household size and the costs that come with it. Childcare is frequently the second or third largest line in a young family's budget and is not counted. A household with two children in full-time care and a household with none, at identical incomes and identical debts, have identical debt-to-income ratios and very different lives.

3. It ignores expenses that are not debts. Health insurance, medical care for a chronic condition, tuition, supporting a parent, a long commute, utilities in a large old house. None of these appear. All of them are as real as a car payment and several are less avoidable.

4. It treats all debts as equivalent. A \$500 student loan payment and a \$500 boat payment are the same number to the ratio. One of them can be sold on Saturday.

5. It is a snapshot of a household that is about to change. The ratio is computed on the day of the file. It says nothing about the raise, the layoff, the pregnancy, the parent moving in, or the fact that the borrower's overtime is entirely dependent on one contract at their employer.

6. And running the other way — it declines people who can obviously afford it. The most frustrating version: a borrower who has paid \$2,800 in rent, on time, for six years, applying for a loan with a \$2,400 PITI, and failing the ratio because of student loan debt. They have demonstrated the exact behavior the ratio is proxying for, and the proxy declines them.

🧮 Run the Numbers

Payment shock — the number DTI does not compute.

The Linden Street borrowers currently pay \$1,850.00 a month in rent.

$$\frac{\$3{,}033.72}{\$1{,}850.00} = 1.64$$

Their housing cost is increasing 64.0%. Their back-end ratio of 42.66% is comfortable. Both statements are true, and the second one is the one that will be tested every month for thirty years.

Payment shock is not a guideline limit on a conventional file. It is a compensating factor in manual underwriting and it appears in some program overlays — but mostly it is a conversation. Chapter 8 is built around this number.

The honest framework

The ratio answers the underwriter's question: will this file perform, on the evidence available?

It does not answer the borrower's question: can we live in this house and still have a life?

You are the only person in the transaction positioned to ask the second one. That is not sentiment — it is the practical reason the book's first theme is you don't sell rates, you solve problems. A borrower who is talked out of the top of their approval and into a payment they can carry is a borrower who refinances with you in four years and refers their sister.

📞 On the Phone

Borrower: "So we're approved up to about \$460,000? Let's look at houses in that range."

The wrong answer: "Yes." True, and it makes the next twelve months harder for them.

What works: "You qualify there — that's real, and if you find the right house at that number I'll close it. But let me show you the other number. At \$460,000 your all-in payment is about thirty-six hundred a month, against the eighteen-fifty you pay now. That's roughly double. At \$385,000 it's about three thousand. Both get approved. Only one of them leaves you able to replace the furnace in year two. Where do you want to be?"

Notice the structure: do not withhold the qualification, and do not make the decision. Give them both numbers and ask. Borrowers almost always choose well when they can see the comparison, and they remember who showed it to them.

4.7 Points, price, and the rate/cost trade-off

A discount point is 1% of the loan amount, paid at closing, to obtain a lower interest rate. One point on the Linden Street loan is:

$$\$365{,}750.00 \times 0.01 = \$3{,}657.50$$

Note the base: the loan amount, not the purchase price. This is a common error and it is always in the borrower's disfavor to get it wrong.

Distinguish two charges that both look like a percentage of the loan:

A borrower is entitled to know which is which, and the Loan Estimate separates them (Chapter 22).

Reading the trade-off

Here is the pricing the Linden Street borrowers were shown. This is a constructed teaching grid — modeled on the structure of a real rate sheet, not on current pricing. Verify actual pricing at the source.

Three things to read off it.

The par rate is the reference point. Par is the rate at which no points are paid and no credit is given. Above par the borrower buys the rate down; below par the lender pays the borrower a lender credit in exchange for a higher rate. Chapter 29 explains where these numbers come from.

The trade is roughly linear over a small range and stops being so. Each 0.125% of rate costs roughly 0.4 to 0.6 of a point here and saves roughly \$30 a month. Push far enough and the cost per increment rises.

A negative number is a real option. Taking 6.875% generates a \$1,371.56 lender credit that can be applied to closing costs. For a borrower short on cash, that is sometimes the correct structure even though the rate is higher, and Chapter 13 covers when.

Break-even

$$\text{break-even (months)} = \frac{\text{cost of the points}}{\text{monthly payment saved}}$$

🧮 Run the Numbers

Should the Linden Street borrowers buy the half point?

Cost of 0.500 point	\$1,828.75
Payment at par (6.750%)	\$2,372.25
Payment at 6.625%	\$2,341.94
Monthly saving	\$30.31
Break-even	\$1,828.75 ÷ \$30.31 = 60.3 months (5.0 years)

The answer depends entirely on a question about the borrower, not about the loan: how long will they keep this mortgage? Not how long will they own the house — how long until they sell or refinance, whichever comes first.

Beyond five years the half point pays. Under five years it does not. And rates falling by a point in year three would make it a bad decision retroactively, because the refinance ends the saving while the \$1,828.75 stays spent.

⚠️ Where Deals Die

Three ways break-even is computed wrong, all of which make points look better than they are:

1. Ignoring the tax treatment. Points on a purchase may be deductible in the year paid; interest saved is also deductible. Both sides move, and for a borrower taking the standard deduction neither may matter at all. Do not give tax advice — say the effect exists and send them to their preparer.
2. Ignoring the opportunity cost of the cash. \$1,828.75 spent on points is \$1,828.75 not in reserves. On a file with 4.16 months of reserves, that is not a rounding error.
3. Using the wrong horizon. The relevant question is time to sale or refinance. Borrowers reliably overestimate this. The honest framing is "if rates drop a point, you refinance and the points are gone" — which is also a fair description of what actually happens.

4.8 APR: what it includes, what it does not, and why it confuses everyone

The annual percentage rate expresses the cost of credit as a yearly rate, including certain costs beyond the interest itself. It was designed to let consumers compare offers. It does that imperfectly, and it generates more confused phone calls than any other disclosure.

How it is actually computed

APR is the rate that makes the amount financed equal the present value of the payment stream.

Step 1 — identify the prepaid finance charges. Not all closing costs are finance charges. Regulation Z excludes several real-estate-related fees when they are bona fide and reasonable.

On the Linden Street file:

Step 2 — compute the amount financed.

$$\$365{,}750.00 - \$6{,}095.34 = \mathbf{\$359{,}654.66}$$

Step 3 — solve for the rate that equates \$359,654.66 to the actual payment stream: \$2,518.72 (P&I plus MI) for months 1 through 137, then \$2,341.94 for months 138 through 360.

🧮 Run the Numbers

The Linden Street APR.

Note rate	6.625%
Amount financed	\$359,654.66
APR	7.253%
Spread over the note rate	0.628 percentage points

Where the spread comes from: roughly a quarter of it is the \$6,095.34 in prepaid finance charges, and the rest — most of it — is the mortgage insurance. Excluding MI would produce an APR of 6.789%, which is wrong and is not what appears on the disclosure.

This is why an FHA loan's APR often looks dramatically worse than its note rate: the upfront and annual mortgage insurance premiums are both finance charges. It is also why comparing a conventional loan's APR to an FHA loan's APR is genuinely informative in a way that comparing note rates is not.

🎓 NMLS Exam Watch

Finance charges are tested more heavily than the APR arithmetic itself, because the test can ask about them without making you solve for a rate. The framing that survives the exam and the desk:

A finance charge is a cost imposed on the consumer as a condition of getting the credit, and would not be imposed in a comparable cash transaction. Work that sentence against any fee on a settlement statement and you will get most of them right.

• Would exist in a cash sale → not a finance charge. A buyer paying cash still wants a survey, a pest inspection, an owner's title policy, and a recording of the deed. Those are excluded.
• Exists only because there is a loan → finance charge. Origination, discount points, prepaid interest, a tax service fee, mortgage insurance premiums, and a loan-related assumption fee.
• The appraisal is the classic trap. It is a real-estate-related fee that Regulation Z excludes when it is bona fide and reasonable in amount, even though the lender required it. Candidates reason "the lender ordered it, so it must be a finance charge" and get it wrong. The same exclusion covers the credit report, flood determination, title insurance, and the settlement fee.
• Know the four TILA figures and what each one is, because a stem will offer you the wrong one: the amount financed (loan amount less prepaid finance charges), the finance charge (the total dollar cost of credit), the total of payments, and the APR. On this file: \$359,654.66, \$507,662.60, \$867,317.26, and 7.253%.
• The APR is never lower than the note rate on a loan with costs, and it is never the payment rate. A question asking which rate the payment is computed from is asking for the note rate.

One caution the exam will not give you: which fees are finance charges has been amended more than once, and lenders apply the "bona fide and reasonable" test through their own compliance policy. Learn the principle for the test; verify the current treatment of any specific fee with Regulation Z, its commentary, and your compliance department before you tell a borrower anything. Chapter 24 is where this material is built out.

What APR does badly

It assumes the loan runs to maturity. The APR on a thirty-year loan amortizes the upfront costs over 360 months. A borrower who sells in six years paid all of those costs and got one-fifth of the term, so their actual effective rate was much higher than the disclosed APR. APR systematically understates the cost of a loan that is not held to term, which is most loans.

It cannot compare loans of different types well. A 30-year fixed against a 5/1 ARM: the ARM's APR must make assumptions about future index values, and those assumptions are not what will happen.

Borrowers think it is the rate. It is not, and this is the single most common APR conversation:

📞 On the Phone

Borrower: "Wait — the paperwork says 7.253 percent. You told me 6.625."

What works: "Both are on there and both are correct — they measure different things. Your interest rate is 6.625 percent; that's what your payment is calculated from, and it's what you'll pay for thirty years. The APR is a government-required disclosure that folds the closing costs and the mortgage insurance back into a single rate so you can compare us against another lender on more than just the headline number. It's higher because your loan has mortgage insurance and you're paying about six thousand in costs that count.

Here's the useful part: if another lender quotes you 6.5 percent, ask them for their APR. If theirs is 7.6, their costs are higher than mine and their better rate isn't better."

Never say "don't worry about the APR." It is the one comparison tool the borrower has been given, and teaching them to use it against you is how you win the ones who shop.

4.9 Per-diem interest, prepaids, and the closing-date question

Mortgage interest is paid in arrears — the December 1 payment covers November. A loan that funds mid-month therefore has a stub period at the start, and the borrower pays that interest at closing.

$$\text{per-diem interest} = \frac{\text{loan amount} \times \text{annual rate}}{365}$$

$$\frac{\$365{,}750.00 \times 0.06625}{365} = \mathbf{\$66.3861 \text{ per day}}$$

The Linden Street file closes October 24 and the first payment is due December 1. The borrower prepays interest from closing through the end of October — 8 days:

$$\$66.3861 \times 8 = \mathbf{\$531.09}$$

November's interest is covered by the December 1 payment, in arrears. There is no payment in November.

🧮 Run the Numbers

Why closing at the end of the month saves cash — and why it is not free money.

Same loan, three closing dates, first payment December 1 in every case:

Close	Days of prepaid interest	Prepaid interest due
October 3	29	\$1,925.20
October 24	8	\$531.09
October 30	2	\$132.77

Closing on the 30th instead of the 3rd reduces cash needed at closing by \$1,792.43.

What the borrower has not saved is a payment. They have deferred it. The loan accrues interest from the day it funds either way; closing late simply means fewer days of it are collected up front. A borrower who believes they "skipped a month" has misunderstood, and it is worth thirty seconds to correct.

What they have actually bought is timing — and for a borrower who is tight on cash to close, moving the closing four days later is one of the cheapest levers available.

The rest of the prepaids

The escrow deposit is not a fee. It funds the account from which the servicer will pay the borrower's taxes and insurance, and RESPA limits how large a cushion may be collected. Chapter 23 covers escrow accounts and the aggregate adjustment; the number of months collected depends on when the next tax bill is due, which is why it varies by closing date and jurisdiction.

⚠️ Where Deals Die

The escrow deposit is the most-misunderstood line on a Closing Disclosure, and the confusion is predictable: the borrower reads \$2,315.00 and hears "another fee."

The correction takes one sentence: "That's your money — it's going into the account the servicer pays your taxes and insurance out of, and it comes back to you if you ever refinance or sell."

Say it at application, not at closing. A cost explained on day 5 is information; the same cost explained on day 49 is a surprise, and surprises at closing are how a borrower decides you were not straight with them.

4.10 Doing this on a calculator, in your head, and in front of a borrower

Three modes, and you need all three.

On the machine

A financial calculator or a spreadsheet payment function takes rate-per-period, number of periods, and present value. Use them. Nobody computes exponents by hand in front of a customer.

What matters is that you sanity-check the output, because the most common error is not arithmetic — it is an input. Rate entered annually instead of monthly. Term entered in years instead of months. Price entered instead of loan amount.

In your head

Two approximations that are good enough to catch an error on a call.

The interest-only floor. One month's interest is $P \times i$. It is always less than the payment and, on a thirty-year loan at ordinary rates, it is most of it.

$$\$365{,}750 \times 0.0055 \approx \$2{,}010$$

Actual P&I is \$2,341.94 — about 16% above interest-only.

The size of that gap depends on the rate, and it is worth knowing the shape rather than a single number, because the relationship is not intuitive: the lower the rate, the wider the gap.

At high rates almost the entire payment is interest, so amortizing adds little. At low rates the interest is small and the principal component is proportionally much larger. Either way, if somebody quotes you \$2,050 on this loan, you know instantly it is wrong.

**The per-\$1,000 factor.** Memorize a handful. At 6.625%, thirty years, the payment per \$1,000 borrowed is:

$$\frac{\$2{,}341.94}{365.75} = \$6.4031 \text{ per \$1,000}$$

At 6% it is almost exactly \$6.00 per thousand, which is a useful anchor. Multiply by the loan amount in thousands: 365.75 × \$6.40 ≈ **\$2,341**. That is a five-second mental estimate accurate to within a dollar.

🧮 Run the Numbers

A ninety-second phone estimate, start to finish.

A caller says: \$420,000 purchase, 10% down, taxes about 1.2%, decent credit.

Step	Mental arithmetic	Result
Loan amount	\$420,000 × 0.90 | \$378,000
P&I at ≈6.625%	378 × \$6.40 | ≈ **\$2,419**
Taxes	\$420,000 × 1.2% ÷ 12 | ≈ **\$420**
Insurance	estimate	≈ \$130
MI at 90% LTV	≈ 0.40% × \$378,000 ÷ 12 | ≈ **\$126**
PITI + MI		≈ \$3,095

Then say the sentence that makes it honest: "That's an estimate — call it thirty-one hundred a month, all in. It'll move once I have your actual credit score and a real tax figure and an insurance quote, and I'd expect it to move by fifty or a hundred dollars either way, not by five hundred."

A range with a stated basis and a stated uncertainty is a professional answer. A precise number you cannot support is not.

In front of a borrower

Three rules, and they are the practical content of this chapter.

1. Always quote PITI, and say the letters. §4.3.

2. Show the arithmetic. Not the formula — the arithmetic. "Your income is ten thousand five hundred. Your payment is three thousand thirty-four. Three thousand thirty-four divided by ten five is twenty-nine percent, and that's your housing ratio." Borrowers who can see the division trust the result. Borrowers handed a percentage do not.

3. Give the number and its uncertainty together. Every figure you quote before underwriting is an estimate. Say which ones are firm (the rate you have locked), which are close (taxes from an assessor's record), and which are genuinely soft (an insurance estimate before a binder). A borrower who has been told what can still move will forgive movement. One who was given eight decimal places on day one will not.

4.11 Working backward: from a payment to a purchase price

Every calculation so far has run in the same direction. Start with a loan amount, produce a payment, produce a ratio. That is the direction underwriting runs.

It is not the direction borrowers ask in. Nobody calls and says "I have a \$365,750 loan, what is my ratio?" They call and say "how much house can we buy?" — and answering that means running the entire chain in reverse, from a ratio limit back to a purchase price. This is the single most frequently performed calculation in origination and it is the one most often done badly, usually by guessing at a price and checking it, three or four times, while the borrower waits.

Run it forward once and backward once and you will never guess again.

THE CHAIN, IN BOTH DIRECTIONS

  FORWARD (how underwriting runs)          BACKWARD (how borrowers ask)
  ─────────────────────────────────        ─────────────────────────────────
  purchase price                           gross monthly income
    x (1 - down payment %)                   x the ratio limit you will use
  = loan amount                            = maximum total obligations
    x payment factor                         - documented monthly debts
  = P&I                                    = maximum PITI
    + taxes + insurance + MI + HOA           - taxes, insurance, MI, HOA
  = PITI                                   = maximum P&I
    + other monthly debts                    / payment factor
  = total obligations                      = maximum LOAN AMOUNT
    / gross monthly income                   / (1 - down payment %)
  = DTI                                    = maximum PURCHASE PRICE

The circularity, and the two ways around it

Read the backward column again and you will find a problem. To subtract taxes and mortgage insurance from the maximum PITI, you need to know the taxes and the mortgage insurance — and both of those depend on the price and the loan amount, which are what you are trying to solve for. The chain eats its own tail.

There are two honest ways out.

Iterate. Guess a price, compute the ratio, adjust, repeat. Two or three passes converge. This is what a loan officer with a calculator actually does and there is nothing wrong with it, but it is slow and it is hard to do while talking.

Collapse it into one number. Every line that scales with the price can be expressed as a fraction of the price, added together once, and used forever. On a 5%-down conventional file at 6.625% with taxes at 1.20% of price and mortgage insurance at a 0.58% annual factor:

Insurance is the one line that does not scale cleanly with price — it scales with replacement cost and with the carrier's view of the risk — so hold it as a flat dollar figure and subtract it separately. On this file that is \$130.00.

Now the backward chain is arithmetic you can do out loud:

$$\text{maximum price} = \frac{(\text{income} \times \text{ratio limit}) - \text{debts} - \text{insurance}}{0.007542}$$

\$7.54 of PITI per \$1,000 of price. Memorize that number for the structure you write most often and recompute it when rates move materially. It is the most useful single figure in this chapter.

Worked, at a 45% limit

🧮 Run the Numbers

How much house can the Linden Street borrowers buy?

Income \$10,500.00. Documented debts \$1,446.00. 5% down, conventional, 6.625%, taxes at 1.20%, insurance \$130.00, MI factor 0.58%.

Step	Arithmetic	Result
1. Maximum total obligations at 45%	\$10,500.00 × 0.45 | \$4,725.00
2. Less documented monthly debts	− \$1,446.00 | \$3,279.00 max PITI
3. Less homeowners insurance	− \$130.00 | \$3,149.00
4. Divide by PITI-per-dollar-of-price	÷ 0.007542	≈ \$417,500

Check it forward, which you must always do:

At a \$417,500 purchase price
Loan at 95%	\$396,625.00
P&I	\$2,539.63
Taxes (1.20% ÷ 12)	\$417.50
Insurance	\$130.00
MI (0.58% ÷ 12)	\$191.70
PITI	\$3,278.83
Plus debts	\$1,446.00
Back-end	\$4,724.83 ÷ \$10,500.00 = 45.00%

It closes. Now push the limit to 48% — which is above the conventional talking point and inside what an automated approval can return on a strong file — and the same arithmetic gives \$10,500.00 × 0.48 = \$5,040.00, less \$1,446.00 and \$130.00, divided by 0.007542 ≈ \$459,300. Round it the way you would say it on a phone: about \$460,000, which is the figure §4.6 quotes. At exactly \$460,000 the loan is \$437,000.00, P&I is \$2,798.16, taxes are \$460.00, MI is \$211.22, and PITI is **\$3,599.38 — a back-end of 48.05%**.

(Constructed. The ratio limits used here are illustrative talking points, not caps; §4.5 explains why there is no single number and Chapter 15 explains what actually decides.)

The second constraint: they also have to be able to close

The number you just computed is the income-constrained maximum. There is a second constraint, and on first-time-buyer files it binds more often than the first one does: the borrower has to produce the down payment and the closing costs and the prepaids, and still have something left.

Purchasing power is the lesser of the two answers, and only one of them is a ratio.

Estimate the cash side quickly by taking the closing costs and prepaids as a percentage of price. On the Linden Street file they total \$14,126.34 on a \$385,000 purchase — 3.67%. That fraction is stable enough to plan with on a conventional purchase, and it moves for reasons you can name: more discount points push it up, a lender credit pushes it down, and the escrow deposit swings by a month or two depending on when the next tax bill falls.

Against the file they actually wrote: \$25,376.34 to produce, **\$12,623.66 left, 4.16 months**.

That is the whole argument of §4.6, in figures. Both houses are approvable. One of them leaves this household with four months of cushion and the other leaves them with three weeks, and no ratio anywhere in the file reports that difference. Note also that the \$460,000 estimate assumes no seller credit — the \$3,000 on the actual contract was negotiated, not given — and that title premiums and the escrow deposit would both run higher on a larger purchase, so 0.87 months is generous rather than pessimistic.

Chapter 8 turns this comparison into a conversation and Chapter 12 puts a guideline value on reserves. Your job in this chapter is to be able to produce both columns in under two minutes, which you now can.

4.12 Paying it down early: extra principal, the thirteenth payment, and the recast

"What if we just pay a little extra every month?" is asked on roughly every third closing, and the honest answer is more interesting than borrowers expect, because the amortization rule from §4.2 does something disproportionate with unscheduled principal.

Here is the mechanism, and it is the whole section. A dollar of extra principal removes that dollar's interest for every remaining month of the term. Send \$100 in month one and you have not bought \$100 of anything — you have deleted \$100 from a balance that was going to be charged 0.55208333% a month for another 359 months. That is why the numbers look implausible.

Savings are measured against the base schedule built in §4.2 — interest computed on the balance and rounded to the cent each month, with the final payment adjusted to clear the balance. Measured on the note-payment convention instead, each figure is \$3.07 larger. §4.2 explains that gap and neither reading is wrong; what matters is that you say which one you used.

Two things in that table deserve to be said out loud to a borrower. One hundred dollars a month retires this loan forty-one months — nearly three and a half years — early. And one extra payment a year is worth nearly as much as two hundred dollars a month, because \$2,341.94 spread over twelve months is \$195.16.

What a biweekly plan actually is

A servicer or a third-party company offers to take half the payment every two weeks instead of the whole payment every month. It is presented as a clever restructuring of the interest and it is not.

$$\frac{\$2{,}341.94}{2} = \$1{,}170.97 \qquad \$1{,}170.97 \times 26 = \$30{,}445.22$$

$$\$2{,}341.94 \times 12 = \$28{,}103.28 \qquad \$30{,}445.22 - \$28{,}103.28 = \$2{,}341.94$$

There are twenty-six two-week periods in a year and only twenty-four half-months, so a biweekly plan collects exactly one extra monthly payment per year — the third row of the table. That is the entire effect. There is no additional interest magic, and there are two things to check before you endorse one:

• How the servicer applies the half payments. Many hold each half until the second arrives and then post one monthly payment. If they do, the borrower gets the thirteenth-payment benefit and nothing else — which is the whole benefit, so this is fine, but it is not what the marketing implied.
• Whether anyone is charging a setup or per-transfer fee. A third-party biweekly service that charges for a result the borrower can produce for free by sending \$195.16 extra each month is selling arithmetic that this chapter just gave away.

The recast — and the fork that catches people

Suppose a borrower comes into \$25,000 in year five and wants to put it against the mortgage. The balance after 60 payments is \$342,870.17**, so the new balance is **\$317,870.17. There are two completely different things that can happen next, and the borrower will not know there is a choice unless you tell them.

$25,000 AT MONTH 60 — THE FORK                              [the Linden Street file]
Balance after 60 payments $342,870.17.  Apply $25,000 -> $317,870.17.

                          ┌──────────────────────────────┐
                          │  $25,000 to principal        │
                          └───────────────┬──────────────┘
              ┌───────────────────────────┴───────────────────────┐
              ▼                                                   ▼
  KEEP THE PAYMENT (curtailment)                    RECAST (re-amortize)
  payment stays      $2,341.94                      payment falls to $2,171.18
  term shortens      312 months (26.0 yr)           term unchanged  360 months
  48 payments never made                            monthly relief  $170.76
  interest, month 61 to payoff                      interest, month 61 to payoff
                    $270,697.43                                   $333,480.96
  ──────────────────────────────────────────────────────────────────────────────
  Same $25,000.  Difference in interest: $62,783.53.

Neither branch is the right answer. They answer different questions.

Curtailment — keep paying \$2,341.94 — retires the loan four years early and saves \$62,783.53 more in interest. It is the correct choice for a household whose problem is the total cost of the debt.

Recast — re-amortize the reduced balance over the remaining 300 months — cuts the payment by \$170.76 a month and changes nothing about the maturity date. It is the correct choice for a household whose problem is monthly cash flow: a job change, a new child, a spouse going back to school. The \$62,783.53 is the price of that relief, and a borrower is entitled to know the price before they choose.

Three practical notes. A recast is a servicer transaction, not a new loan: no application, no appraisal, no new closing costs, typically a modest processing fee — and it is not available on every product, so it must be confirmed rather than assumed. It requires a lump sum large enough to meet the servicer's minimum. And an extra principal payment does not reduce next month's payment on its own; without a recast, the payment is fixed by the note, and the borrower who assumes otherwise will be unpleasantly surprised in month sixty-one.

Finally, the question a borrower will eventually ask that this book will not answer: whether to prepay at all rather than do something else with the money. That is investment advice and you are not licensed to give it. What you can do is lay out the arithmetic — the guaranteed return on prepaying is the note rate, 6.625% here, on money that becomes illiquid the moment it goes into the house — and say plainly that reserves, higher-rate consumer debt, and an employer retirement match are the usual competing claims, and that their financial advisor and tax preparer own the decision.

One thing you should mention without being asked: whether the note carries a prepayment penalty. On a consumer purchase loan of this kind it almost certainly does not — Regulation Z permits a prepayment penalty on a covered transaction only where the loan is a fixed-rate qualified mortgage that is not higher-priced — but "almost certainly" is not the same as "I read the note." Chapter 34 covers where these do live.

4.13 Sensitivity: what a quarter point, a hundred dollars, and one new debt are worth

An experienced loan officer carries a small set of exchange rates in their head. They are not guidelines and they are not on the exam. They are the reason a good originator can answer a structuring question during the call instead of after it, and every one of them is a division you have already done in this chapter.

THE EXCHANGE RATES — Linden Street structure               [the Linden Street file]
$365,750 loan, 6.625%, 30-year fixed, $10,500 income, 5% down, 1.20% taxes, 0.58% MI

  ONE UNIT OF ...              IS WORTH ...
  ─────────────────────────────────────────────────────────────────────────
  $1 of monthly P&I            $156.1741 of loan amount        (call it $156)
  $100 of monthly P&I          $15,617.41 of loan amount
  $1 of monthly PITI           $132.5887 of purchase price     (call it $133)
  $100 of monthly PITI         $13,258.87 of purchase price
  $100 of monthly debt         $13,258.87 of purchase price -- given up
  1 point of back-end DTI      $105.00 a month (1% of $10,500)
  0.125% of rate               $30.31 a month of P&I
  0.250% of rate               $60.78 a month of P&I
  1.000% of rate               $246.82 a month of P&I
  ─────────────────────────────────────────────────────────────────────────
  The first four are just 1 / the factors built in 4.1 and 4.11: one dollar
  of P&I is 1 / 0.0064031 of loan, one dollar of PITI is 1 / 0.007542 of
  price. Carry four decimals if you are going to multiply by a big number;
  round to $156 and $133 for anything you say out loud.
  Recompute the rate rows when the market moves; the others hold.

The two most useful lines are the ones about debt.

One point of DTI is \$105 a month on this file — and that is not a metaphor, it is the answer to "how much debt can they add before this breaks?" A borrower at 42.66% with a program looking at 45% has 2.34 points of room, which is \$245.70 a month of new obligation, and you can say so.

One hundred dollars a month of debt costs \$13,258.87 of purchase price. This is the number to have ready when a borrower asks whether to pay off a card or keep the cash. The answer depends on which constraint is binding — cash to close or the ratio — and Chapter 12 works that trade properly. What matters here is that you can price it.

🧮 Run the Numbers

What the furniture actually cost.

On day 41 the Linden Street borrowers financed \$5,200.00 of furniture at **\$611.00 a month**, on a nine-month promotional plan, three days before their scheduled closing. Chapter 19 owns what happened next. Here is what it was worth in this chapter's terms.

In ratio: obligations go from \$4,479.72 to \$5,090.72.

$$\frac{\$5{,}090.72}{\$10{,}500.00} = \mathbf{48.48\%}$$

A jump of 5.82 points of back-end ratio, past the approved ratio the conditional approval was written against, on a file that was clear on everything else.

In purchase price: \$611.00 a month, at \$132.5887 of price per dollar of PITI, is

$$\$611.00 \times 132.5887 = \mathbf{\$81{,}011.70}$$

Eighty-one thousand dollars of purchasing power, for a nine-month furniture plan. Had they opened it before application rather than after approval, they would have been shopping in a completely different price band and would never have seen this house.

The sentence that prevents it, said at application and again at approval: "Between now and the day you get keys, do not open any new credit, do not finance anything, and do not let a store run your credit for a discount — not a couch, not a lawnmower, not a card at the register. If you think you need to, call me first and I will tell you in five minutes what it costs. It is never zero."

How much can the market move before this file breaks?

Here is the same tool pointed at the risk you actually carry between application and lock. Hold the loan amount, taxes, insurance, and mortgage insurance where they are, and move only the rate:

This file has roughly a full point of rate headroom before it reaches the 45% conventional talking point and about three points before it reaches the 50% ceiling an automated approval will not exceed. That is a comfortable file, and knowing it is comfortable is worth as much as knowing a tight file is tight — it tells you how hard to push on a lock, how much bad news the file can absorb, and whether an unlocked borrower shopping for another week is taking a real risk or an imaginary one.

Run the same table on a borrower who starts at 47% and the answer is different: a quarter point of market movement puts them out. That borrower does not get to shop. Chapter 20 is about the lock decision; this is the arithmetic underneath it.

4.14 Cash to close: the number the borrower actually has to produce

Everything in this chapter so far has been about the monthly payment. There is a second number, it is larger, it arrives sooner, and it kills more files than the payment does.

Cash to close is what the borrower must deliver at the closing table:

$$\text{cash to close} = \text{down payment} + \text{closing costs} + \text{prepaids} - \text{credits} - \text{earnest money already paid}$$

Four blocks and two subtractions. Build it in that order every time, on paper, and you will not lose a line.

🧮 Run the Numbers

The Linden Street cash to close, from the ground up.

Block 1 — the down payment. \$385,000.00 × 5% = **\$19,250.00**. This is not a cost; it is equity. It is the only line in the build that the borrower keeps.

Block 2 — closing costs.

Line	Amount	Paid to
Origination charge (1.000%)	\$3,657.50	lender
Discount points (0.500%)	\$1,828.75	lender
Appraisal	\$650.00	third party
Credit report	\$85.00	third party
Flood certification	\$14.00	third party
Tax service	\$78.00	third party
Lender's title policy	\$1,150.00	third party
Settlement fee	\$595.00	third party
Recording	\$212.00	government
Owner's title policy	\$875.00	third party
Survey	\$450.00	third party
Pest inspection	\$125.00	third party
Total closing costs	\$9,720.25

Block 3 — prepaids and the escrow deposit, built in §4.9: prepaid interest \$531.09 + twelve months of homeowners insurance \$1,560.00 + escrow deposit \$2,315.00 = \$4,406.09.

Blocks 2 and 3 together: \$9,720.25 + \$4,406.09 = \$14,126.34.

The subtractions.

Costs and prepaids	\$14,126.34
Plus down payment	+ \$19,250.00
Less earnest money already deposited	− \$5,000.00
Less seller credit toward closing costs	− \$3,000.00
CASH TO CLOSE	\$25,376.34

And the figure the borrower will ask about ten seconds later:

$$\$38{,}000.00 - \$25{,}376.34 = \$12{,}623.66 \qquad \frac{\$12{,}623.66}{\$3{,}033.72} = \textbf{4.16 months of reserves}$$

Three observations that are not obvious from the arithmetic.

The down payment is the biggest block and the least interesting one. At \$19,250.00 it is 76% of the cash to close, and it is the only part that does not disappear. When a borrower says closing costs are "outrageous," it is often worth separating the two: \$19,250 of what they are bringing is still theirs the moment they own the house.

A credit is not a discount. The \$3,000 seller credit and the \$5,000 earnest money both reduce what the borrower wires, and they do it for completely different reasons — one is somebody else's money, the other is the borrower's own money paid earlier. Chapter 22 shows how each appears on the disclosures, and they do not appear the same way.

The escrow deposit is in this number but is not a closing cost. It is in Block 3 because the borrower has to produce it, and it is not a cost because they still own it. §4.9 covers the conversation.

⚠️ Where Deals Die

The earnest money gets counted twice, and the error always runs in the same direction.

These borrowers had \$43,000.00. On day 4 they wrote a \$5,000.00 earnest money check, which cleared. The \$38,000.00 in verified assets is what is left — the earnest money is gone, and it is sitting in an escrow account with the title company.

So the \$5,000.00 appears in the cash-to-close build as a credit, reducing what they wire, precisely because they already paid it. Subtract it a second time from the \$38,000.00 — as though it were still in the savings account waiting to be spent — and you will report reserves \$5,000.00 too small, which is more than a month and a half of payments on a file where reserves are a compensating factor.

The error is easy to make because both treatments feel conservative. The test that catches it: ask what the bank statement says. The statement is dated after day 4, the check has cleared, and the balance already reflects it. If your asset figure and your credit figure both include the same \$5,000, you have spent it twice.

The same discipline applies to a gift. The \$10,000.00 gift is inside the \$38,000.00 because it was received and deposited. A gift that has been promised is not an asset and does not belong in any of these numbers until it is in an account and documented. Chapter 12 owns the sourcing rules.

What is not in cash to close is worth naming too, because borrowers ask about all of it: the seller's own closing costs and payoffs, the real estate commissions, the first mortgage payment (December 1, more than a month after an October 24 closing), the monthly mortgage insurance, and any repairs negotiated after inspection unless they were rolled into the settlement statement. And on the other side, a figure that surprises borrowers on a refinance and never on a purchase: there is no down payment line at all. Chapter 37 covers what replaces it.

4.15 When your number and the system's number disagree

You will compute \$3,033.72 and the loan origination system will produce \$3,048.72, and one of you is wrong. This happens constantly, it happens to good loan officers, and the discipline for handling it is the last thing this chapter owes you.

Never send a number you cannot reconcile. Not to a borrower, not to an agent, not to a processor. A number you cannot explain is a number you will have to retract, and retracting a payment figure costs more trust than being twenty minutes slower.

There are only four places a payment discrepancy can come from. Work them in this order and you will find it in under two minutes every time.

1. A different input. Something in the six — loan amount, rate, term, taxes, insurance, MI factor — is not the same on both sides. This accounts for most of them. Two worked examples on the Linden Street structure:

Notice how each was found: subtract, then ask which single line could be exactly that size. A \$15.00 gap is not a rounding problem and it is not a rate problem; \$15.00 a month is \$180.00 a year, and the only annualized figures in the payment are taxes and insurance.

2. A different scope. You are comparing PITI to P&I, or PITI to PITI-plus-MI, or a payment with HOA dues to one without. This is the most embarrassing of the four because nothing is wrong — the two numbers are answers to different questions. It is also why §4.3 insists you say the letters out loud.

3. A different convention. Interest computed on a 360-day year rather than 365 will move a per-diem. A payment rounded rather than truncated will move a total, as §4.2 showed at length. An escrow deposit built on a different number of months will move cash to close by hundreds. None of these is an error; all of them require you to know which convention produced the figure in front of the borrower.

4. A different date. The rate is from Tuesday's sheet and the system repriced Wednesday morning. The tax figure is the current assessment and the system pulled the reassessed one. The lock expired. Date discrepancies are the most dangerous of the four because both numbers were correct when they were computed, and neither party is making a mistake.

🔍 Check Your Understanding

1. A borrower's Loan Estimate shows an estimated total monthly payment \$61 higher than the figure you quoted, on a \$365,750 loan at 6.625%. You confirm taxes, insurance, and MI all match. What is the most likely cause, and what is the one input you have not checked? (§4.13's exchange rates: \$60.78 is 0.250% of rate on this loan. The rate moved, or one of you is on a different rate sheet. Check the lock.)

2. You quote a payment of \$3,033.72 and the borrower's agent tells them their own lender quoted \$2,341.94 for the same house. Which of the four sources is this, and what do you say? (Scope, not price — the other quote is P&I only. Say so without implying bad faith: "That's the principal and interest only; mine includes taxes, insurance, and mortgage insurance. Ask them for their all-in number and we'll compare like for like.")

3. Your amortization schedule ends \$3.07 past zero. Is that an error? (No — §4.2. It is the accumulated effect of a payment rounded up to the cent, and the servicer resolves it by adjusting the final payment.)

4. A borrower adds a \$611 monthly obligation. Without recomputing anything, roughly how much purchase price did they give up on this structure? (About \$81,000 — \$611 × \$132.5887 of price per dollar of PITI, or \$611 × \$133 in your head.)

The habit underneath all four: name the source of every input, every time you record it. Taxes from the county assessor's record dated such-and-such. Insurance from the binder, not the quote. Rate from the sheet, locked at a specific time on a specific day. A file whose inputs are sourced reconciles in two minutes. A file whose inputs came from memory does not reconcile at all, and you will find that out in front of a borrower.

🗂️ The Loan File

Chapter 4 contribution: compute the file's core numbers from scratch.

Everything the last three chapters quoted, derived. Verify each one yourself.

The loan

The payment

The ratios

The disclosures

What this settles: every number in the file that is a consequence of the structure. If the loan amount, rate, term, taxes, insurance, and MI factor are what they are, all of the above follows necessarily.

What it does not settle: whether any of those six inputs is right. The rate is not locked until day 12. The MI factor depends on a credit score not yet pulled. The tax figure is the county's current assessment and may change on transfer. The insurance figure is an estimate. And the loan amount depends on a program decision not made until Chapter 13.

Open questions carried forward:

• Q1. Can they afford this, or only qualify for it? The ratios say qualify comfortably; the 64.0% payment shock says the second question is still open. (Chapter 8)
• Q2. Conventional or FHA? (Chapter 13)
• Q3. Will the appraisal support \$385,000? Note now what §4.4 established: if it comes in at \$372,000, the maximum 95% loan drops to \$353,400 and the borrowers need \$12,350 more in cash. (Chapter 18)

Your task. In the Appendix C workbook, build the first twelve rows of the amortization schedule by hand, using only the one-line rule in §4.2. Confirm that your month-12 balance is \$361,757.88. Then compute what the payment would be at 10% down instead of 5% — and note what happens to the MI line, which is the part most people forget.

Conclusion

One formula, and everything else is division.

The payment formula has three inputs, which means there are exactly three ways to change a payment: less principal, a lower rate, or a longer term. The third is the weakest and the one borrowers ask for. Amortization is one line repeated 360 times, and its shape — 86% interest in month one, 0.5% in month 360 — is the single fact borrowers find most surprising and most worth explaining well.

PITI is the number to quote, always. Loan-to-value uses the lesser of price or appraised value, which is what makes a low appraisal a cash problem rather than a debate. The two qualifying ratios share a denominator, and the back-end one — what this book calls DTI — is the one that declines files.

And DTI is blind to taxes, household size, childcare, non-debt expenses, the difference between a student loan and a boat, and the direction a household is heading. It answers the underwriter's question well and the borrower's question not at all. Payment shock — 64.0% on this file — is the number nobody computes and everybody lives with.

Points cost 1% of the loan amount and are worth buying only against an honest estimate of how long the loan will survive. APR folds the finance charges and the mortgage insurance back into a rate; it is 0.628 points above the note rate on this file, mostly because of MI, and it is the one comparison tool your borrower has been handed. Teach them to use it.

Next: Chapter 5 is the map of loan programs — conventional, FHA, VA, USDA, jumbo, portfolio, fixed versus adjustable — at the altitude you need for a first conversation. It is where Q2 finally gets its options, and where the mortgage insurance line you just computed gets its alternatives.

Key Terms

Principal — the amount borrowed, and thereafter the outstanding balance. (Ch.4)

Interest — the charge for the use of borrowed money, computed each month on the outstanding balance. (Ch.4)

Amortization — the repayment of principal over the life of a loan through scheduled payments, so the balance reaches zero at maturity. (Ch.4)

Amortization schedule — the month-by-month record of how each payment splits between interest and principal, and the resulting balance. (Ch.4)

PITI — principal, interest, taxes, and insurance; in practice, the full monthly housing payment including mortgage insurance and any HOA dues. (Ch.4)

Loan-to-value (LTV) — the loan amount divided by the value of the property, using the lesser of the purchase price or the appraised value on a purchase. (Ch.4)

Combined loan-to-value (CLTV) — all liens against the property, at their balances, divided by value. (Ch.4)

HCLTV — the combined ratio computed using a home equity line's full credit line rather than its drawn balance. (Ch.4)

Housing ratio (front-end) — PITI divided by gross monthly income. (Ch.4)

Back-end ratio / debt-to-income (DTI) — PITI plus all other counted monthly obligations, divided by gross monthly income. The book's default meaning of "DTI." (Ch.4)

Ten-month rule — the convention permitting exclusion of an installment debt with a small number of payments remaining, commonly ten or fewer, subject to program specifics. (Ch.4)

Payment shock — the increase from a borrower's current housing cost to the proposed one, expressed as a multiple or a percentage. (Ch.4)

Discount point — 1% of the loan amount, paid at closing to obtain a lower interest rate. (Ch.4)

Origination charge — the lender's compensation for making the loan, distinct from discount points. (Ch.4)

Par rate — the rate at which neither discount points are paid nor a lender credit is given. (Ch.4)

Lender credit — an amount the lender applies toward the borrower's costs in exchange for a rate above par. (Ch.4)

Break-even (points) — the cost of discount points divided by the monthly payment saved, expressed in months. (Ch.4)

Annual percentage rate (APR) — the cost of credit expressed as a yearly rate, computed by solving for the rate that equates the amount financed to the payment stream. (Ch.4)

Amount financed — the loan amount less prepaid finance charges. (Ch.4)

Finance charge — the total dollar cost of credit, including interest and those closing costs Regulation Z treats as finance charges. (Ch.4)

Prepaid finance charge — a finance charge paid at or before closing, deducted from the loan amount to produce the amount financed. (Ch.4)

Per-diem interest — one day's interest on the loan amount; loan amount × annual rate ÷ 365. (Ch.4)

Prepaid interest — interest collected at closing covering the period from funding through the end of that month, because mortgage interest is paid in arrears. (Ch.4)

Negative amortization — an increase in the balance occurring when a payment is less than the interest accrued. (Ch.4)

Spaced Review

1. (Ch. 1) The note names a payment of \$2,341.94 and the borrower will pay \$3,033.72. Using §1.2 and §4.3, explain why the note is not wrong.

2. (Ch. 3) A processor computes a payment for a borrower who asked. Using Chapter 3's functional definition, is that a licensing problem? Does your answer change if the processor merely reads back a figure the loan officer already disclosed?

3. A property under contract at \$400,000 appraises at \$381,000. The borrower is putting 10% down. Compute the new maximum loan, the new required down payment, and the gap.

4. State four things debt-to-income cannot measure. For each, describe a borrower it misjudges — two in each direction.

5. A borrower can buy 0.75 of a point for \$2,900 and save \$47 a month. Compute the break-even. Then state the one question you must ask before advising them.