Appendix A — Formulas, Ratios, and Quick Calculations

Everything in this book that computes, in one place, with a worked example on the Linden Street file for each. Chapter references point to where the idea is taught.

⚠️ The rounding rule, before anything else. Dollars are canonical. Percentages and totals are derived. The note's payment is \$2,341.94, rounded to the cent; the unrounded payment is \$2,341.9373. Every total in this book is built from the rounded figure, and every amortization figure uses the servicer method — interest computed on the balance and rounded to the cent each month. Never back-compute a dollar figure from a rounded percentage, and never re-derive an amortization figure from full precision. See §4.2.

A.1 The payment

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

Worked, in four steps (§4.1):

The two unit errors that cause most wrong answers: $i$ is monthly, not annual; $n$ is months, not years.

⚠️ Work those four steps on a calculator and you will get \$2,341.93, one cent under the note payment. That is not an error in either place — it is the rule at the top of this appendix, demonstrated on the appendix's own first example. Step 2 rounds \$2,019.2448 to \$2,019.24 and step 3 truncates the factor to six places, and those two roundings cost a cent: \$2,019.24 ÷ 0.862211 = \$2,341.9325**. Run the formula once at full precision and it returns **\$2,341.9373, which rounds to the note's **\$2,341.94**. The per-\$1,000 shortcut below does the same thing for the same reason (365.75 × \$6.4031 = \$2,341.9338).

Round once, at the end. Rounding intermediates is how a file ends up with two payments that differ by a cent and an amortization schedule that will not tie out. If your figure is a cent off the note, check for a rounded intermediate before you check anything else.

Payment per \$1,000 borrowed, 30-year (§4.10)

| Rate | Per \$1,000 | Rate | Per \$1,000 | |---|---|---|---| | 5.000% | \$5.3682 | 7.000% | \$6.6530 | | 6.000% | \$5.9955 | 7.500% | \$6.9921 | | 6.625% | **\$6.4031** | 8.000% | \$7.3376 |

Multiply by the loan amount in thousands: 365.75 × \$6.4031 = **\$2,341.94**.

The interest-only sanity check (§4.10)

One month's interest is $P \times i$ and is always less than the payment. How much less depends on the rate, and the lower the rate, the wider the gap:

Term comparison, same \$365,750 at 6.625% (§4.1)

Ten extra years buys \$167.98/month** and costs **\$200,402.40. Fifteen years costs \$869.32 more per month and saves \$265,071.60.

A.2 Amortization (§4.2)

One line, repeated 360 times:

interest = balance × monthly rate · principal = payment − interest · new balance = old − principal

Linden Street, first six payments — interest rounded to the cent each month:

  #    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

Milestones

Total of 360 payments \$843,098.40.** Total interest **\$477,348.40 — 1.31× the amount borrowed.

The residual: run all 360 rows on the servicer method and the balance lands \$3.07 past zero, because every payment is rounded up from \$2,341.9373. A servicer adjusts the final payment to clear it — about \$2,338.87.

A.3 PITI (§4.3)

$$\text{PITI} = \text{P\&I} + \frac{\text{annual taxes}}{12} + \frac{\text{annual insurance}}{12} + \text{MI} + \text{HOA}$$

Always quote PITI. Never quote bare P&I. The gap here is 29.5%.

HOA dues are counted in the qualifying ratio but are usually not escrowed — the borrower pays them directly.

A.4 Loan-to-value (§4.4)

$$\text{LTV} = \frac{\text{loan amount}}{\text{value}} \qquad \text{value} = \textbf{the LESSER of price or appraised value}$$

Linden Street: \$365,750.00 ÷ \$385,000.00 = 95.00%.

When the appraisal is short — Cypress Court (§4.4, §18.7):

Mortgage insurance milestones — conventional borrower-paid PMI only, under the Homeowners Protection Act, measured against ORIGINAL value:

Total MI paid to termination: \$24,218.86. These rules do not apply to FHA MIP (§5.8).

A.5 The qualifying ratios (§4.5)

$$\text{Housing (front-end)} = \frac{\text{PITI}}{\text{gross monthly income}} \qquad \textbf{Back-end (DTI)} = \frac{\text{PITI} + \text{other monthly debts}}{\text{gross monthly income}}$$

Linden Street:

Counted in the back-end numerator: the full new PITI including MI and HOA · auto loans and leases · student loans (per program rules) · revolving minimums · personal and installment debt · alimony and child support paid · other properties' full PITI · co-signed debts unless documented as paid by another.

Not counted: utilities · groceries · childcare · payroll-deducted health insurance · income taxes · 401(k) contributions and 401(k) loan repayments · savings.

The ten-month rule: an installment debt with approximately ten or fewer payments remaining may generally be excluded, subject to program specifics. It does not apply on the Linden Street file (31 and 19 payments remaining) — but excluding B2's \$429.00 auto would drop the back-end from 42.66% to 38.58%.

Payment shock (§4.6):

$$\frac{\text{proposed PITI}}{\text{current housing cost}} = \frac{\$3{,}033.72}{\$1{,}850.00} = \mathbf{1.64\times} = \mathbf{+64.0\%}$$

There is no single DTI cap. See §4.5 and Chapter 4's Case Study 1 — the CFPB removed its own 43% General QM limit. Never tell a borrower they are declined "because of a number."

A.6 Points, par, and break-even (§4.7)

One discount point = 1% of the LOAN AMOUNT, not the purchase price.

$$1\% \times \$365{,}750.00 = \$3{,}657.50 \qquad 0.500 \text{ point} = \$1{,}828.75$$

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

Linden Street: \$1,828.75 ÷ \$30.31 = 60.3 months (5.0 years).

Illustrative rate/point grid [constructed teaching grid — verify current pricing at the source] on \$365,750, 30-day lock:

The three ways break-even is computed wrong: ignoring the tax treatment; ignoring the opportunity cost of the cash (points spent are reserves not held); using the wrong horizon — the question is time to sale or refinance, whichever comes first, and borrowers reliably overestimate it.

A.7 APR and the disclosure figures (§4.8)

$$\text{amount financed} = \text{loan amount} - \text{prepaid finance charges}$$

APR is the rate that equates the amount financed to the actual payment stream, including mortgage insurance for as long as it is scheduled to be paid.

Finance charge or not — Regulation Z §1026.4:

The pattern: fees paid to third parties for services about the property are generally excluded; charges that are the cost of the credit itself are included. The tax service fee is the one that surprises people — it is included.

APR's two failure modes: it assumes the loan runs to maturity, so it understates the cost of a loan that exits early; and it compares poorly across loan types, because an ARM's APR must assume future index values.

A.8 Per-diem interest and prepaids (§4.9)

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

Linden Street: \$365,750.00 × 0.06625 ÷ 365 = **\$66.3861 per day**.

Interest is paid in arrears, so a mid-month closing prepays from funding through month end. Closing October 24, first payment December 1, 8 days prepaid = \$531.09. There is no November payment.

Closing on the 30th instead of the 3rd reduces cash needed by \$1,792.43. The borrower has not skipped a payment — they have deferred collecting interest.

Prepaids and escrows, Linden Street:

The escrow deposit is not a fee. It funds the account the servicer pays taxes and insurance from, and RESPA caps the cushion.

A.9 Cash to close (§4.9)

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

Reserves are stated in months of PITI, and they are a compensating factor in underwriting.

A.10 Mortgage insurance across the programs (§5.8)

Never call FHA's MIP "PMI." The FHA duration category is set by the LTV at origination and is never revisited.

The comparison that decides the Linden Street file:

FHA is \$17.88/month cheaper**, needs **\$5,775.00 less down, and costs \$38,155.54 more in mortgage insurance.

⚠️ That is the MI difference, not the total-cost difference. FHA's lower rate returns \$17,446.29** of interest and its financed UFMIP adds **\$4,672.94 upfront, so the net total cost of credit is **+\$25,382.19** (\$528,778.20 FHA vs. \$503,396.01 conventional). Say "more in mortgage insurance" or "more in total cost" — never bare "more." See §13.10.

A.11 Adjustable-rate mortgages (§5.7)

$$\text{fully indexed rate} = \text{index} + \text{margin}$$

The borrower is qualified at the GREATER of the fully indexed rate or the initial rate — the Ability-to-Repay answer to the 2/28 teaser failure.

Illustrative 5/6 ARM on \$365,750, initial 5.875%, index 4.25%, margin 2.75%, caps 2/1/5:

Worst case is +\$1,285.07/month above the initial payment. Caps apply to the initial rate, not to the fully indexed rate. Show the worst case first.

A.12 Self-employed income — the cash flow analysis (§32.7)

The Fannie Mae Form 1084 approach, S-corporation, on the Fulton Avenue file:

The borrower's accountant told them they "make about \$9,500 a month." Add back what was deducted and did not leave the business; subtract what left the business and was not deducted.

A.13 Quick reference — the numbers that recur

A.14 The four facts you need before quoting a rate

1. Representative credit score
2. Loan-to-value
3. Occupancy and property type
4. Lock period

Missing any one and you do not have a quote — you have a range, and you should say so.

A.15 Rental income (§11.8, §32.5)

$$\text{net rental} = (\text{gross rent} \times \text{vacancy factor}) - \text{PITIA on that property}$$

The vacancy factor is commonly 75% [illustrative — verify the current factor and the required documentation for the program]. The result is a signed number, and the sign is the whole point.

A negative result is added to the borrower's DEBTS, not subtracted from income. New originators reliably get this backwards and quote a borrower who cannot qualify.

Where the rent figure comes from — generally the lesser of the executed lease and the market rent on the appraiser's rent schedule. Quoting from the lease alone is the second most common error here.

A.16 Residual income — the VA test nobody else uses (§5.4)

$$\text{residual} = \text{gross monthly income} - \text{taxes} - \text{PITI} - \text{other debts} - \text{maintenance and utilities}$$

VA underwrites a dollar amount left over, not only a ratio, and the required minimum varies by region and family size [the tables are set by VA and revised — look them up in the Lender's Handbook every time]. A file can pass DTI and fail residual, or fail DTI and pass residual with a strong residual figure as the compensating factor.

Maintenance and utilities is estimated from the square footage of the subject property, which is why it is a real number rather than a formality.

A.17 Debt-service coverage ratio — the investor loan (§34.6)

$$\text{DSCR} = \frac{\text{gross rent}}{\text{PITIA}}$$

No personal income is calculated at all. The property qualifies; the borrower does not.

⚠️ A ratio above 1.00 is not the same as a property that makes money. Run the same file as an owner — 8% vacancy, 8% management, 8% maintenance, 5% capital reserve — and it collects \$20,202.48** a year against **\$26,852.64 of PITIA: −\$554.18 a month. It passes at 1.05 and loses money. Say so out loud.

A.18 Bank-statement income (§34.3)

$$\text{qualifying income} = \frac{\text{total deposits} - \text{exclusions}}{\text{months}} \times \text{expense factor} \times \text{ownership \%}$$

The expense factor is the entire loan. The same statements at a CPA-documented 68% expense factor (a 32% margin) yield \$7,769.33** — a **\$4,370.25 swing, which at a 50% DTI moves borrowing power by roughly a quarter of a million dollars. Ask what factor the program uses before you quote.

A.19 Grossing up nontaxable income (§11.5)

$$\text{grossed-up} = \text{nontaxable amount} \times (1 + \text{gross-up rate})$$

Common rates are 25% and 15% [illustrative — the permitted rate is program-specific; verify].

Only genuinely nontaxable income may be grossed up — typically certain Social Security, disability, and child support. Document that it is nontaxable before you gross it up, and never gross up income the borrower actually pays tax on.

A.20 Temporary buydowns (§13.6)

A buydown does not change the note. It funds an escrow that pays the difference between the note payment and a reduced payment for a fixed period.

$$\text{buydown escrow} = \sum_{\text{years}} (\text{note payment} - \text{reduced payment}) \times 12$$

2-1 buydown on the Linden Street loan — note 6.625%, \$2,341.94:

⚠️ The borrower is qualified at the NOTE rate, not the bought-down rate. A buydown never helps a borrower qualify — it only softens the first two years. If the file does not work at \$2,341.94, the buydown does not fix it.

Unused funds belong to somebody. If the loan pays off during the buydown period, the remaining escrow is generally credited — know the contract's answer before the borrower asks.

A.21 Escrow accounting (§23.4)

$$\text{monthly escrow} = \frac{\text{annual taxes} + \text{annual insurance}}{12}$$

The initial deposit is built item by item from the months between closing and each next disbursement, plus a cushion RESPA caps at two months of that item's payment.

\$2,315.00 is the sum of two single-item calculations, not an aggregate figure. The aggregate adjustment is a separate step that reconciles the item-by-item total against the lowest projected balance; on this file it is \$0.00. An aggregate adjustment is a credit or zero — it is never a charge, and a positive number in that line is an error.

A.22 Interested-party contributions (§12.9)

A seller credit is capped as a percentage of the lesser of price or appraised value, and the cap moves with occupancy and LTV [caps are program-specific and revised — verify].

Two rules that are not about the cap and cost more deals than the cap does:

1. A credit may pay closing costs and prepaids. It may not fund the down payment, and it may not be handed to the borrower as cash.
2. An unused credit is generally lost. Structure it to the actual costs; a \$3,000.00 credit against \$2,400.00 of remaining costs does not return \$600.00 to the borrower.

A.23 Equity, appreciation, and when a second lien becomes possible (§35.6)

$$\text{equity} = \text{value} - \text{balance} \qquad \text{room under a CLTV cap} = (\text{value} \times \text{cap}) - \text{balance}$$

The Linden Street identity. Because the original loan was exactly 95% of value (\$365,750.00 on \$385,000.00), the room under a 95% CLTV cap at flat value always equals exactly the principal paid to date:

Amortization alone does not create borrowing room on a high-LTV loan. At payment 60 with flat value, the room under a 90% cap is \$3,629.83 — five years of payments and not enough to finance a bathroom. With 3% annual appreciation the same file is worth \$446,320.52 and the room under an 85% cap is \$36,502.27.

Appreciation, not amortization, is what makes a second lien possible — and the borrower controls neither. Say that plainly rather than implying that paying on time will get them there.

A.24 The conversions that cause wrong answers

Bi-weekly and semi-monthly are not the same thing and the difference is two extra pay periods a year. On \$2,000.00 a period that is **\$4,333.33 a month bi-weekly against \$4,000.00** semi-monthly — \$333.33 of qualifying income, which is a loan-sized error.

Hours per week must come from the documentation, not from an assumption. A borrower who works 37.5 hours is not a 40-hour borrower, and the 6.25% difference lands directly in the ratio.