AI Math
āđ€āļ›āļīāļ”āđāļ­āļ›

Mortgage Rate Calculator with Points

Price a rate change, discount points and DSCR limits with AI-powered step-by-step solutions
Payment on $300,000 at 6% over 30 years, and at 6.5%
Break-even on 2 points costing $6,000 to cut 6.75% to 6.25%
Maximum loan at a 1.25 DSCR on $186,000 of NOI at 6.25% over 30 years
Total interest difference between 6% and 7% on $300,000 over 30 years

What a Rate Does to a Payment

The payment on a fixed-rate mortgage is

M=L⋅r(1+r)n(1+r)n−1,r=annual rate12M = L\cdot\frac{r(1+r)^n}{(1+r)^n - 1}, \qquad r = \frac{\text{annual rate}}{12}

and the total interest over the full term is nM−LnM - L. The payment is not linear in the rate: because (1+r)n(1+r)^n appears on both sides of the fraction, each additional percentage point costs a little more than the last, and the effect is magnified by a long term.

On \300{,}000over30years,ahalf−pointstepfromover 30 years, a half-point step from6%toto6.5%addsaboutadds about$97.55amonth—arounda month — around$35{,}119$ across the full term. That sensitivity is what makes points worth computing rather than guessing.

Discount points

A point costs 1%1\% of the loan and buys a lower rate. The comparison is a simple ratio:

break-even months=cost of pointsMhigh−Mlow\text{break-even months} = \frac{\text{cost of points}}{M_{\text{high}} - M_{\text{low}}}

Below the break-even the points lose; beyond it they win. How much rate a point buys is set by the lender and moves with the market, so it is an input here, never a constant.

Quoted rates change daily and depend on term, credit and programme. This page prices whatever rate you enter.

Income Property: Sizing a Loan from DSCR

Multifamily and other income-property loans are usually sized by debt service coverage rather than by borrower income:

DSCR=NOIAnnual debt serviceDSCR = \frac{NOI}{\text{Annual debt service}}

Fix a minimum coverage and the arithmetic inverts:

Max annual debt service=NOIDSCR,Lmax⁡=ADS12⋅1−(1+r)−nr\text{Max annual debt service} = \frac{NOI}{DSCR}, \qquad L_{\max} = \frac{\text{ADS}}{12}\cdot\frac{1 - (1+r)^{-n}}{r}

Two properties of this deserve attention. First, the amortisation period drives the answer as much as the rate: the same debt service over 25 years instead of 30 supports a materially smaller loan. Second, many such loans amortise over a long schedule but mature much earlier, leaving a balloon — the payment formula does not show that.

Net operating income is itself defined by the underwriting: gross rent less vacancy and operating expenses, before debt service and before depreciation. Related:

Cap rate=NOIValue\text{Cap rate} = \frac{NOI}{\text{Value}}

Coverage minimums, expense assumptions and reserve requirements are lender-specific and change. Use the figures in your own term sheet.

Common Mistakes to Avoid

  • Comparing points on payment alone: the saving only matters relative to how long the loan is actually held. Divide the cost by the monthly saving.
  • Using the annual rate as rr: the formula wants the monthly rate. Everything downstream is wrong otherwise.
  • Assuming the rate change scales linearly: going from 6%6\% to 7%7\% costs more than twice what 6%6\% to 6.5%6.5\% costs on the same loan.
  • Sizing an income-property loan on payment rather than coverage: with DSCR, the debt service is the constraint and the loan is the output.
  • Ignoring the amortisation-versus-term distinction: a 30-year amortisation with a 10-year maturity produces the payments of the former and a balloon at the latter.
  • Forgetting what sits outside MM: taxes, insurance and escrow are real cash but never appear in the principal-and-interest formula.

Examples

Step 1: At 6%6\%: r=0.005r = 0.005, (1.005)360≈6.0225752(1.005)^{360} \approx 6.0225752, M = 300{,}000\dfrac{0.005 \times 6.0225752}{5.0225752} \approx \1{,}798.65$
Step 2: Total interest: 360 \times 1{,}798.65 - 300{,}000 \approx \347{,}514.57$
Step 3: At 6.5%6.5\%: r≈0.0054167r \approx 0.0054167, (1+r)360≈6.9917980(1+r)^{360} \approx 6.9917980, M \approx \1{,}896.20$
Step 4: Total interest: 360 \times 1{,}896.20 - 300{,}000 \approx \382{,}633.47$
Step 5: Monthly difference: 1{,}896.20 - 1{,}798.65 \approx \97.55$
Step 6: Lifetime difference: 382{,}633.47 - 347{,}514.57 \approx \35{,}118.90$
Answer: About \97.55amonthanda month and$35{,}119$ over the full term

Step 1: At 6.75%6.75\%: r=0.005625r = 0.005625, (1+r)360≈7.5332455(1+r)^{360} \approx 7.5332455, M \approx \1{,}945.79$
Step 2: At 6.25%6.25\%: r≈0.0052083r \approx 0.0052083, (1+r)360≈6.4891664(1+r)^{360} \approx 6.4891664, M \approx \1{,}847.15$
Step 3: Monthly saving: 1{,}945.79 - 1{,}847.15 \approx \98.64$
Step 4: Break-even: 6,000/98.64≈60.86{,}000 / 98.64 \approx 60.8 months
Step 5: Held the full 30 years, the saving is 360 \times 98.64 - 6{,}000 \approx \29{,}510$
Answer: Break-even at about 61 months — the points pay off only if the loan is kept past roughly five years

Step 1: Maximum annual debt service: 186{,}000/1.25 = \148{,}800$
Step 2: Monthly: 148{,}800/12 = \12{,}400$
Step 3: r≈0.0052083r \approx 0.0052083, (1+r)−360≈0.1541030(1+r)^{-360} \approx 0.1541030, factor =1−0.15410300.0052083≈162.4122= \dfrac{1 - 0.1541030}{0.0052083} \approx 162.4122
Step 4: L_{\max} = 12{,}400 \times 162.4122 \approx \2{,}013{,}911.58$
Step 5: On a 25-year amortisation the factor is 151.5911151.5911, giving \1{,}879{,}729.51—about— about$134{,}182$ less
Step 6: At a \2{,}400{,}000valuationthecaprateisvaluation the cap rate is186{,}000/2{,}400{,}000 = 7.75%$
Answer: About \2{,}013{,}912overa30−yearamortisation,orover a 30-year amortisation, or$1{,}879{,}730$ over 25 years

Frequently Asked Questions

On $300,000 over 30 years, moving from 6% to 6.5% raises the payment from about $1,798.65 to $1,896.20 — $97.55 a month, or roughly $35,119 in extra interest over the full term. The effect scales with the loan size and the term.

That depends entirely on how long the loan is held. Divide the cost of the points by the monthly payment saving to get the break-even month; $6,000 buying a $98.64 saving breaks even at about 61 months. Whether you will still hold the loan then is the real question.

By debt service coverage: maximum annual debt service = NOI ÷ the required DSCR. Convert that to a monthly payment and invert the mortgage formula, L = M(1 − (1+r)^−n)/r, using the amortisation period rather than the maturity.

The annuity factor (1 − (1+r)^−n)/r grows with n, so the same debt service supports a bigger loan over a longer schedule. At 6.25%, 30 years gives a factor of 162.41 against 151.59 for 25 years — about 7% more loan for the same payment.

Try AI-Math for Free

Get step-by-step solutions to any math problem. Upload a photo or type your question.

Start Solving