The Pre-Approval Math: Millage, PMI, and an Appraisal That Comes Up Short
The Whitakers earn $6,800 gross per month and carry a $315 car payment and a $100 student loan payment. Their lender qualifies conventional loans at 28% front-end and 36% back-end ratios. They have 10% down; the quoted rate is 6.5% on a 30-year fixed, with principal and interest of $6.32 per $1,000 borrowed. Private mortgage insurance runs 0.5% of the original loan amount per year. Property taxes in the county are 12.5 mills on the county's assessed value of $184,000 (not the purchase price). Hazard insurance is $1,150 per year, the HOA is $45 per month, and the agreement of sale calls for the parties to split the 2% Pennsylvania realty transfer tax equally. Estimated buyer closing costs are 2.5% of price, and the loan will fund on the 19th of a 30-day month.
1. What is the maximum purchase price the Whitakers qualify for under the stated ratios? Show the monthly breakdown.
Read the worked answer
Step 1 - Ratio ceilings.
- Front-end: 28% × $6,800 = $1,904
- Back-end: 36% × $6,800 = $2,448; less existing debt of $415 = $2,033
- The binding limit is the lower figure: $1,904 per month for housing.
Step 2 - Fixed housing costs.
- Property tax: 12.5 mills on $184,000 = 0.0125 × $184,000 = $2,300/yr ÷ 12 = $191.67
- Hazard insurance: $1,150 ÷ 12 = $95.83
- HOA: $45.00
- Subtotal: $332.50
Note the tax was computed on the county assessment, not the sales price. In a county with an old base-year assessment the tax bill may bear little relation to the purchase price, and a client who assumes "taxes will be about 1% of price" will be surprised either way.
Step 3 - What's left for principal, interest, and PMI. $1,904 − $332.50 = $1,571.50
Step 4 - Solve for the loan. Cost per dollar of loan:
- P&I: $6.32 ÷ $1,000 = 0.00632
- PMI: 0.005 ÷ 12 = 0.0004167
- Total: 0.0067367 per dollar
Loan = $1,571.50 ÷ 0.0067367 ≈ $233,100
Step 5 - Price. Loan ÷ 0.90 = $233,100 ÷ 0.90 ≈ $259,000
Check: P&I $1,473 + PMI $97 + taxes $192 + insurance $96 + HOA $45 = $1,903, which is 28.0% front-end. Total debt $1,903 + $415 = $2,318 = 34.1% back-end - under 36%, so the front-end ratio controls.
The working answer for the clients: a price near $259,000, not a penny of it justified by "the payment felt okay."
2. Assuming a $259,000 price, estimate the cash the Whitakers need at closing, including the transfer tax and prepaid items.
Read the worked answer
Line by line:
| Item | Calculation | Amount |
|---|---|---|
| Down payment | 10% × $259,000 | $25,900 |
| Loan amount | 90% × $259,000 | $233,100 |
| Buyer closing costs | 2.5% × $259,000 | $6,475 |
| Realty transfer tax (buyer's half) | 1% × $259,000 | $2,590 |
| Prepaid interest, 12 days | $233,100 × 0.065 ÷ 365 = $41.51/day × 12 | $498 |
| Tax escrow | 3 × $191.67 | $575 |
| Insurance escrow | 3 × $95.83 | $288 |
| Total | ≈ $36,326 |
Judgment points to explain to the client:
- Why 12 days of interest. Mortgage interest is paid in arrears; the first payment covers interest from the funding date to the end of that month. Closing on the 19th means roughly 12 days are collected at closing. Closing late in the month reduces this line and increases the first payment's interest component - a real reason to compare closing dates.
- The transfer tax split is a contract term, not a law. Pennsylvania imposes 1% state plus 1% local realty transfer tax (2% in most counties); local custom on who pays varies by county and the agreement of sale governs. Here the contract says 50/50, so the buyer's half is 1% of price. If the county or municipality has a different rate or an additional local levy, the number changes - confirm the actual rate for the property's location before quoting it.
- Escrows are the buyer's money, held for the buyer. They are not fees.
- Appraisal and inspection fees are typically paid outside closing and should be added to the cash plan.
3. The appraisal comes back at $248,000. The seller will not reduce the price. What are the Whitakers' options and what does the shortfall actually cost them?
Read the worked answer
Why the appraisal matters: lenders base the loan on the lesser of the contract price or the appraised value. At a $248,000 appraised value and 90% LTV:
- Maximum loan = 90% × $248,000 = $223,200
- Loan needed at the $259,000 price = $233,100
- Gap = $9,900 the buyer must supply in cash, above the original plan
Options, with trade-offs:
- Cover the gap. New loan $223,200 (down payment becomes $35,800 against the $259,000 price). PMI is now 0.5% of $223,200 = $1,116/yr → $93/mo; P&I = $223,200 × 0.00632 = $1,411. Total housing ≈ $1,838 - still inside the 28% ceiling, so the ratios are not the obstacle; liquidity is. Cash to close rises by roughly $9,900.
- Renegotiate. Ask the seller to meet the appraisal at $248,000–$250,000, with the appraisal report attached. Sellers resist this emotionally but respond to arithmetic - the next buyer's appraiser will see the same data.
- Challenge the appraisal through the lender's reconsideration process with better comps, if the licensee genuinely believes the appraiser missed a superior sale. Weak or self-serving comps damage credibility.
- Change the loan structure (higher LTV) - which raises PMI and the payment, and may not fix an LTV-driven shortfall. Model it before recommending it.
- Walk, if the contract has an appraisal contingency or a financing contingency that is still open. Check the deadlines and don't let one lapse by inaction.
Advisor framing: the appraisal is not a verdict on the house's worth - it is a lender's risk number. The clients need to know the gap in dollars, not adjectives, and they should hear it while the contingencies are still alive.