Nine traps in
mortgage arithmetic

Floor four

The nine, worked

Every figure below was produced by the same functions the calculators run, at a placeholder interest rate you can change. Nothing here is a rate, a quote or an average.

1. 01What does FHA insurance cost over the loan?

   Illustration 01 of 9

   The 3.5%-down purchase where the insurance never stops

   Applying the conventional cancellation rule to an FHA loan hides $15,036.

   A buyer has $11,000 saved and is looking at a $310,000 house. An FHA loan at 3.5% down needs $10,850, which fits, and the payment works on paper. The mortgage insurance line looks like every other mortgage insurance line: a number that will go away when enough of the loan is paid off. On this loan it will not.

   The working

   1. The down payment is $10,850, so the base loan is $299,150 and the loan-to-value at origination is 96.5%. Up-front MIP is 1.75% of the base loan — $5,235.13 — financed into the mortgage, bringing the amount amortised to $304,385.13. At a placeholder 6.5% over 30 years, principal and interest are $1,923.92 a month.
   2. The annual MIP rate for this loan size, term and LTV is 0.55% under HUD Mortgagee Letter 2023-05. Charged on the average balance for each year, that is $138.80 in month one and falls slowly as the balance does.
   3. Here is the part that decides the cost. HUD Mortgagee Letter 2013-04 fixes the duration on the LTV at origination: above 90%, the premium runs for the full mortgage term. This loan started at 96.5%, so it is charged in all 360 months. Total annual MIP: $32,850.
   4. Now apply the rule people expect instead. The scheduled balance reaches 80% of the original $310,000 in month 139 — eleven years and seven months in. A calculator that cancelled there, as the conventional Homeowners Protection Act rule would, stops counting at $17,814.
   5. The borrower goes on to pay another $15,036 after that point. That is the size of the error, and it runs in the direction that makes the loan look cheaper than it is.

   What it means

   FHA annual MIP has no balance test. Its duration is set on closing day by the loan-to-value on closing day: the full term above 90%, eleven years at or below. The 80% threshold belongs to conventional PMI and to nothing else. If a comparison shows FHA cheaper than conventional over the life of the loan, check which duration rule it used before you believe it.

   • FHA
   • Mortgage insurance
   • First purchase

   Run the FHA MIP calculator
2. 02When does a refinance pay back?

   Illustration 02 of 9

   The refinance that lowers the payment and raises the cost

   A $296 monthly saving that adds $34,229 to the loan, because the clock reset.

   Six years into a 30-year loan at 6.75%, $268,000 is outstanding and 288 payments — twenty-four years — are left to run. A new 30-year loan is available at 5.875% with $5,400 of closing costs paid at closing. The break-even looks excellent, so the conversation stops there. It should not.

   The working

   1. The payment on the existing balance over its remaining 288 months is $1,881.55 in principal and interest. The new 30-year loan pays $1,585.32.
   2. That is a saving of $296.23 a month. Divide the $5,400 of costs by it and the break-even is 19 months. On the cash question this refinance is unambiguously good.
   3. But the old loan had 288 payments left and the new one has 360. The refinance adds 72 payments — six years — to the end of the mortgage.
   4. Interest still to pay on the existing loan over its own remaining life: $273,887. Interest on the new loan over its 30 years: $302,716. Add the $5,400 of costs and the refinance costs $34,229 more in total.
   5. Now run the same rate into a 24-year term, so the loan ends when the old one would have. The payment is $1,737.82 — still $143.73 a month less than before. The break-even stretches to 38 months. Interest over the new term is $232,493, and net of costs the refinance saves $35,993.

   What it means

   Same rate, same lender, same paperwork; a seventy-thousand-dollar swing decided by the term box. A longer break-even can be the better loan. If the only number anyone says out loud is the monthly saving, the term has been reset and nobody has priced it.

   • Refinance
   • Break-even
   • Term

   Run the break-even both ways
3. 03How much house is defensible?

   Illustration 03 of 9

   The ratio that was never the binding one

   Paying off the car raised the ceiling by nothing — until the debts crossed $787 a month.

   A household earns $118,000, has $40,000 for a down payment, and is told to pay down debt before applying. Whether that advice is worth anything depends entirely on which of the two debt-to-income tests is actually limiting them, and nobody has told them which one it is.

   The working

   1. Gross income is $9,833.33 a month. At the classic 28% front-end guideline, the housing budget is $2,753.33. At the 36% back-end guideline, the budget is $3,540.00 less whatever other debt they carry.
   2. With $640 a month of other debt, the back-end budget is $2,900.00 and the front-end budget is $2,753.33. The front-end test is tighter, so the housing budget is $2,753.33 and the maximum price is $366,506 — a $326,506 loan at 89.1% LTV, with $2,063.74 of principal and interest and $157.81 of PMI.
   3. At that debt level the back-end test has $146.67 a month of slack. Clearing the car would raise the ceiling by exactly nothing.
   4. Now raise the other debt to $940 a month, changing nothing else. The front-end budget is still $2,753.33. The back-end budget falls to $2,600.00 and takes over. The ceiling drops to $346,962.
   5. From there, clearing $500 a month of that debt restores the ceiling to $366,506 — $19,544 more house. Leaving the debt and finding a rate half a point cheaper instead lifts the ceiling only to $360,235, a gain of $13,273.

   What it means

   The crossover on these inputs is $786.67 a month of other debt. Below it the front-end ratio binds and debt is irrelevant to the ceiling; above it the back-end ratio binds and retiring one payment can beat half a point of interest rate. The advice to pay down debts before applying is right about half the time and a waste of effort the rest, and which half you are in takes thirty seconds to establish.

   • Affordability
   • Debt ratios
   • First purchase

   See which constraint binds you
4. 04What does paying extra buy me?

   Illustration 04 of 9

   The same $20,000, worth four times as much in year one

   An overpayment does not save interest. It cancels every future interest charge on that money.

   A $300,000 loan at a placeholder 6.5% over 30 years, and a $20,000 windfall. The question is whether to put it against the mortgage now or later, and the usual reasoning — the money is the same either way — is the wrong model of what an overpayment does.

   The working

   1. The baseline: $1,896.20 a month in principal and interest, 360 payments, $382,633 of total interest.
   2. Apply the $20,000 in month 12. The loan retires in 302 months instead of 360 — 58 months early — and total interest falls by $91,623.
   3. Apply the same $20,000 in month 204, seventeen years in. The loan retires in 337 months, 23 months early, and total interest falls by $23,737.
   4. Same amount, same loan, same rate. The early payment is worth 3.9 times the late one.
   5. For comparison, a standing $200 a month from the first payment removes 83 months and $103,449 of interest — more than the $20,000 lump sum, because it starts sooner and keeps going.

   What it means

   A dollar of principal cancels every interest charge that dollar would have attracted for the rest of the term. In year one that is 29 years of charges; in year seventeen it is 13. This is also the argument against holding cash to make one large payment later: the loan compounds against you while you save.

   • Overpayment
   • Interest
   • Amortisation

   Move the month and watch it collapse
5. 05Are discount points worth it?

   Illustration 05 of 9

   The discount point that breaks even two months after you move

   A buy-down is a bet on your own holding period, and the odds are close to even.

   A $300,000 loan over 30 years is offered at a placeholder 6.5%, or at 6.25% for one discount point. A point is 1% of the loan — $3,000 in cash at closing — and the pitch is that it pays for itself. It does, eventually. Whether eventually arrives before you move is the entire question.

   The working

   1. At 6.5% the payment is $1,896.20. At 6.25% it is $1,847.15. The saving is $49.05 a month.
   2. The point costs $3,000. Divide: the break-even is month 62 — five years and two months.
   3. Hold the loan seven years and you are $1,120 ahead. Hold it five years and you are $57 behind. Two months of holding period separates a modest win from a modest loss.
   4. Two points behave identically, because the cost and the saving both double: $6,000 buys 6.0% and saves $97.55 a month, breaking even at the same month 62. At seven years that is $2,194; at five years it is $147 in the red.
   5. Note that a refinance ends the bet early and does not refund the point. Rates falling — the thing that would make you refinance — is the scenario in which buying points loses.

   What it means

   The break-even on a typical buy-down lands somewhere in the fifth or sixth year, which is uncomfortably close to how long people actually keep a mortgage. Points are a good deal for a borrower who is certain they will stay and will not refinance, and a coin toss for everyone else. Buying more points does not change the break-even month; it only raises the stake.

   • Points
   • Break-even
   • Closing costs

   Set your own holding period
6. 06Where does each payment go?

   Illustration 06 of 9

   A 15-year loan you can stop paying in a bad month

   The 30-year loan plus the difference retires in exactly 180 months, for exactly the same interest.

   A borrower can afford the 15-year payment on a $300,000 loan but is nervous about committing to it. The received answer is that the 15-year loan is cheaper and the 30-year loan is safer, and that you have to pick one. On the arithmetic, you do not.

   The working

   1. At a placeholder 6.5%, a 30-year loan on $300,000 costs $1,896.20 a month and $382,633 in interest. A 15-year loan costs $2,613.32 a month and $170,398 in interest.
   2. So the payment is 37.8% higher — $717.12 more — and the interest is less than half. That is the trade as normally stated.
   3. Now take the 30-year loan and pay exactly $717.12 of extra principal every month from the first payment. It retires in 180 months, having paid $170,398 in interest.
   4. Not approximately. Identically. The two schedules are the same schedule, because a level payment against a balance does not care which box on the note it came out of.

   What it means

   The 15-year loan's advantage over a self-imposed schedule is the discipline, not the arithmetic — and its disadvantage is that the higher payment is contractual. The 30-year loan overpaid to a 15-year schedule gives you the same outcome with an option to fall back to $1,896.20 in a month with a boiler in it. What it costs is a slightly higher rate, since 15-year money usually prices below 30-year money; that spread is the real price of the option and it is worth pricing rather than assuming.

   • Term
   • Overpayment
   • Amortisation

   Compare the two schedules
7. 07What is the VA fee on this loan?

   Illustration 07 of 9

   $20,000 down that removes $7,500 of VA funding fee

   On a second use of the VA benefit, 5% down cuts the fee rate from 3.30% to 1.50%.

   A veteran using the VA benefit for the second time is buying a $400,000 house. The programme allows nothing down, so nothing down is what gets modelled. The funding fee is quoted as a percentage, which makes it easy to read as a fixed cost of using the benefit. It is not — it is a step function with a large step in it.

   The working

   1. The fee schedule is in 38 U.S.C. 3729, at the rates in VA Circular 26-23-06. On a subsequent use with less than 5% down, the rate is 3.30%. On $400,000 that is $13,200.
   2. Put 5% down — $20,000 — and two things happen. The base loan falls to $380,000, and the rate falls to 1.50%. The fee is $5,700.
   3. That is $7,500 of fee removed by $20,000 of cash, before counting the interest on the smaller loan or on the $7,500 that would otherwise have been financed for thirty years.
   4. The first-use tiers are gentler: 2.15% below 5% down ($8,600), 1.50% at 5% ($5,700), 1.25% at 10% or more ($4,500). The subsequent-use penalty applies only below 5% down; at 5% and above, first and subsequent uses pay the same rate.
   5. At a placeholder 6.5% over 30 years with the fee financed, the zero-down subsequent-use loan amortises $413,200 at $2,611.71 a month; the 5%-down version amortises $385,700 at $2,437.89. Neither carries any monthly mortgage insurance, which is the benefit doing its work.

   What it means

   The 5%-down threshold is the single largest step in the VA fee schedule and it exists only on a subsequent use. If you are using the benefit again and have the cash, price both. And if you receive VA compensation for a service-connected disability, are a Purple Heart recipient on active duty, or are an eligible surviving spouse, the fee is zero under 38 U.S.C. 3729(c) and none of this applies to you.

   • VA
   • Funding fee
   • Down payment

   Check your own tier
8. 08What would the payment actually be?

   Illustration 08 of 9

   The re-appraisal that did not cancel the PMI

   The Homeowners Protection Act measures against the original value. Appreciation does not count.

   A buyer puts 12% down on a $385,000 house. Two years later the market has moved and a neighbour's sale suggests the house is worth $430,000, which would put the loan comfortably under 80% of today's value. They call the servicer to cancel the PMI and are told no. Both parties are right, and the reason is in the statute.

   The working

   1. The loan is $338,800 at 88.0% of the purchase price. At a placeholder 6.5% over 30 years, principal and interest are $2,141.45, and PMI at 0.58% of the loan a year is $163.75 in month one.
   2. Under the Homeowners Protection Act of 1998 (12 U.S.C. 4901), the borrower may require cancellation when the scheduled balance reaches 80% of the ORIGINAL value, and the servicer must terminate automatically at 78%. Original value means the lesser of the purchase price and the appraised value at closing — $385,000 here, not what the house is worth now.
   3. 80% of $385,000 is $308,000. On the scheduled amortisation the balance reaches it in month 81 — six years and nine months in. Total PMI paid over that period: $13,264.
   4. The lever that does work is the balance, because the test is on the balance. Pay an extra $150 a month from the first payment and the balance reaches $308,000 in month 58 instead — four years and ten months — and total PMI falls to $9,498. A $150 overpayment bought $3,766 of insurance removal, on top of the interest it saved.

   What it means

   The statutory rights run against the original value and nothing else, so a rising market does not shorten them. Separately from the statute, loan investors such as Fannie Mae and Freddie Mac publish their own policies allowing cancellation based on a current appraisal after a seasoning period — that is a servicer and investor question, not a right, and it costs an appraisal to ask. If you want a date you control, overpay.

   • PMI
   • Conventional
   • Overpayment

   See when the PMI line stops
9. 09Which one leaves me better off?

   Illustration 09 of 9

   The rent-versus-buy answer that flips on one input

   Same house, same loan: buying wins at year six, or loses for nineteen years, depending on the rent.

   A household is choosing between a $385,000 house with 12% down and staying where they are. Both sides of this argument are normally made with slogans — rent is throwing money away; you cannot afford the maintenance. Run symmetrically, with both households spending the same budget every month and the cheaper one investing the difference, the answer turns out to depend almost entirely on two numbers nobody has checked.

   The working

   1. The purchase: $385,000 at a placeholder 6.5% over 30 years, 12% down, tax at 1.25%, insurance at $1,800, PMI at 0.58%, maintenance at 1% of value a year, $5,400 to buy and 6% to sell. Rent grows 3% a year and both households invest surplus cash at 5%.
   2. Against rent of $1,950 a month, buying loses for a long time. At seven years the renter is ahead by about $43,000; at ten years by about $40,000; at fifteen years still by about $26,000. Buying does not overtake renting until roughly month 234 — nineteen and a half years.
   3. Change one number. Against rent of $2,400 for the same house, buying overtakes renting at month 74 — a little over six years — and is ahead by about $40,000 at ten years.
   4. Now change a second. Hold that $2,400 rent and set home appreciation to zero instead of 3%. Buying never overtakes renting inside ten years, and at ten years the renter is ahead by about $65,000.
   5. At 2% appreciation the crossover is month 118; at 3% it is month 74; at 4% it is month 46. The whole answer sits on a growth rate nobody knows.

   What it means

   There is no general answer to rent versus buy, and anyone who gives you one has fixed the two inputs that decide it. What matters is the rent for the house you would actually otherwise occupy, and an appreciation assumption you should set to zero at least once to see how much of the case rests on it. The transaction costs are why short horizons favour renting: 6% to sell has to be earned back before anything else counts.

   • Rent vs buy
   • Appreciation
   • First purchase

   Set appreciation to zero and look again