Mortgage Math: Worked Examples Five Refinance Scenarios — Full Formula Derivation & Decision Logic BIX Development Reference | Source: Pro_Calc.xlsx | CONFIDENTIAL
This document translates the Pro_Calc.xlsx workbook into fully annotated worked examples for the BIX engineering team. Every number is sourced directly from the calculator. Every formula is derived and explained. This is the mathematical DNA the Clarity Engine must replicate at scale.
All five scenarios are built from a single borrower’s actual refinance situation. This is the base case every Clarity Engine calculation must begin with.
Current Mortgage Payment: $2,528.00 / month Existing Mortgage Balance: $380,000.00 Closing Costs (financed): $11,400.00 Preliminary Loan Amount: $391,400.00 Additional Cash Requested: $0.00 NEW MORTGAGE BALANCE: $391,400.00 Offer Rate: 6.00% (0.500% / month) Term: 30 years (360 payments) Credit Cards (all flagged ‘To Be Paid Off’): Card 1: $5,000 balance | $150/mo min Card 2: $10,000 balance | $300/mo min Card 3: $4,000 balance | $120/mo min
Total CC Balance: $19,000 Total CC Minimums: $570/month Total Monthly Obligation: $3,098/month
Every scenario in the Pro_Calc — and every loan payment in the Clarity Engine — is derived from a single formula. BIX must implement this as the core calculation primitive.
PMT = r × PV ÷ [ 1 − (1 + r)^−n ] Where: r = monthly rate (annual rate ÷ 12) PV = loan amount (present value) n = total payments (years × 12)
⚒ PROOF: Borrower’s Current Loan — $391,400 @ 6.00% / 30yr r = 6.00% ÷ 12 = 0.5000% | n = 30 × 12 = 360 | PV = $391,400 PMT = 0.005 × 391,400 ÷ [1 − (1.005)^−360] = 1,957.00 ÷ [1 − 0.16604] = 1,957.00 ÷ 0.83396 = $2,346.64 / month (Scenario A baseline payment)
WHY THE CURRENT PAYMENT IS $2,528 The borrower’s current payment of $2,528 implies their EXISTING loan has a DIFFERENT rate or balance than the new refinance quote. The Pro_Calc models what happens when they refinance into NEW terms — not what the existing loan’s rate was. The $181.36/month savings is the delta between the existing payment ($2,528) and the new calculated payment ($2,346.64). Clarity Engine implication: always store BOTH the borrower’s current payment (from verification) AND the calculated payment for the proposed loan.
The simplest scenario. The borrower refinances at a lower rate with no cash out, no term change. Pure monthly payment reduction.
Loan Amount Rate Term New Payment $391,400 6.00%
$2,346.64 / mo
⚒ RATE REDUCTION: Full Derivation r = 6.00% ÷ 12 = 0.500% | n = 360 | PV = $391,400 PMT = 0.005 × 391,400 ÷ [1 − (1.005)^−360] = 1,957.00 ÷ 0.83396 = $2,346.64 / month
Savings Metric Formula Result Monthly Savings $2,528.00 − $2,346.64 $181.36 / month Annual Savings $181.36 × 12 $2,176.31 / year 5-Year Savings $181.36 × 60 $10,881.55 over 5 years Total Interest (30yr) $2,346.64 × 360 − $391,400 $453,390.67 total interest Closing Cost Recovery $11,400 ÷ $181.36/mo 62.9 months ≈63 month breakeven
CLARITY ENGINE DECISION LOGIC — Scenario A Present Scenario A when: borrower’s primary goal is reducing monthly cash outflow with no cash need. Key question to surface: ‘How long do you plan to stay in this home?’ If answer > 63 months, Scenario A’s $10,881 in 5-year savings justifies the $11,400 in closing costs. If borrower plans to sell or refi in < 5 years: flag that closing costs may not be recovered — explore no-cost refi at slightly higher rate instead.
The rate drops to 5.75%. Instead of taking the monthly savings, the borrower keeps their payment the same and pays off the loan faster. This requires solving for TERM, not payment.
Scenario B requires solving PMT in REVERSE. We know the payment ($2,528) and need to find the term. Formula: n = -ln(1 − r×PV/PMT) ÷ ln(1 + r) This is one of the more complex calculations in the engine — most platforms can’t do it. The Clarity Engine must.
⚒ TERM REDUCTION: Solving for n (number of months) r = 5.75% ÷ 12 = 0.47917% | PV = $391,400 | PMT = $2,528 (kept constant) n = -ln(1 − 0.004792 × 391,400 / 2,528) ÷ ln(1.004792) = -ln(1 − 0.74135) ÷ ln(1.004792) = -ln(0.25865) ÷ 0.004781 = -(-1.35414) ÷ 0.004781 = 283.3 months = 23.61 years (pays off 6.39 years early)
Metric 30-Year Baseline (Sc.A) Term Reduction (Sc.B) Rate 6.00% 5.75% Monthly Payment $2,346.64 $2,528.00 (unchanged) Loan Term 30.00 years / 360 mo 23.61 years / 283 mo Payoff Acceleration — 6.39 years earlier Total Interest Paid $453,390.67 $324,820.23 Interest Savings vs. Baseline — $128,570.44 lifetime
CLARITY ENGINE DECISION LOGIC — Scenario B Present Scenario B when: borrower says ‘I want to pay off my house faster’ or ‘I’m planning for retirement in 20 years.’ The sell: same payment they’re used to. Pays off 6.4 years early. Saves $128,570 in interest over the life of the loan. Clarity Engine must display the payoff date — not just the term. ‘Your home is paid off on [Month, Year]’ is more powerful than ‘23.6 years.’
The borrower needs cash but doesn’t want their payment to increase. This requires solving for LOAN AMOUNT — the inverse of the standard PMT problem.
⚒ CASH OUT: Solving for PV (maximum loan at same payment) r = 6.00% ÷ 12 = 0.500% | n = 360 | PMT = $2,528 (kept constant) PV = PMT × [1 − (1 + r)^−n] ÷ r = 2,528 × [1 − (1.005)^−360] ÷ 0.005 = 2,528 × [1 − 0.16604] ÷ 0.005 = 2,528 × 0.83396 ÷ 0.005 = 2,528 × 166.7916 = PV = $421,649.20 → Cash Out = $421,649.20 − $391,400.00 = $30,249.20
New Loan Amount Rate Payment Cash Available $421,649.20 6.00% $2,528.00 (unchanged) $30,249.20
CLARITY ENGINE DECISION LOGIC — Scenario C Present Scenario C when: borrower has a cash need (home renovation, debt payoff, investment) but is payment-sensitive. Critical check: LTV. New loan of $421,649 against property value must stay within LTV guidelines.
The engine must ALWAYS calculate post-cash-out LTV and flag PMI requirement before presenting this scenario. Key conversation: ‘You can access $30,249 today with zero payment increase. What’s the cash for?’ The answer reveals the next product need.
The highest-impact scenario for this borrower. Roll the $19,000 in credit card debt into the new mortgage. The borrower gets a lower combined payment AND eliminates high-interest revolving debt.
Current total monthly obligation: Mortgage $2,528 + CC minimums $570 = $3,098 / month New loan: $391,400 base + $19,000 CC payoff = $410,400 financed Rate drops to 5.75% (debt consolidation product pricing assumption) Result: one payment replaces mortgage + all three credit cards
⚒ DEBT CONSOLIDATION: New Payment Calculation r = 5.75% ÷ 12 = 0.47917% | n = 360 | PV = $410,400 PMT = 0.004792 × 410,400 ÷ [1 − (1.004792)^−360] = 1,966.25 ÷ [1 − 0.17563] = 1,966.25 ÷ 0.82437 = $2,385.31 / month (replaces $3,098 total obligation)
Obligation Before Consolidation After Consolidation Mortgage Payment $2,528.00 / mo Eliminated → rolled in Credit Card Minimums $570.00 / mo Eliminated → rolled in Total Monthly Payment $3,098.00 / mo $2,385.31 / mo (new mtg)
$3,098.00 − $2,385.31 $712.69 / MONTH Annual Savings — $8,552.28 / year 5-Year Savings — $42,761.40 over 5 years CC Interest Eliminated ~22-27% APR on $19K Now at 5.75% mortgage rate
HIDDEN ALCHEMY — The True Savings of Debt Consolidation The $712/month payment reduction understates the real benefit. The three credit cards at 22-27% APR are costing the borrower an estimated $4,000-$5,000/year in interest alone. By rolling $19,000 in credit card debt into a 5.75% mortgage:
Clarity Engine must present: (1) payment savings, (2) interest rate arbitrage savings, (3) projected FICO improvement, (4) downstream rate benefit from improved score.
Maximum acceleration. Refinance into a 15-year mortgage at 5.75%. Payment increases significantly, but the interest savings and wealth-building acceleration are the most powerful of all five scenarios.
⚒ 15-YEAR TERM REDUCTION: Payment Calculation r = 5.75% ÷ 12 = 0.47917% | n = 15 × 12 = 180 | PV = $391,400 PMT = 0.004792 × 391,400 ÷ [1 − (1.004792)^−180] = 1,875.40 ÷ [1 − 0.42068] = 1,875.40 ÷ 0.57932 = $3,236.26 / month (vs. $2,528 current → +$708/mo cost)
Metric 30-Year (Sc.A) 23.6-Year (Sc.B) 15-Year (Sc.E) Payment / mo $2,346.64 $2,528.00 $3,236.26 Total Interest $453,390.67 $324,820.23 $193,640.51 Interest Saved vs. 30yr — $128,570.44 $259,750.16 Payoff Date 30 years from close 23.6 years from close 15 years from close Extra Monthly Cost — $0 (same as current) +$708.26 vs. current
CLARITY ENGINE DECISION LOGIC — Scenario E Present Scenario E when: borrower has income stability, is 10-20 years from retirement, or explicitly wants to ‘own their home outright.’ The framing: ‘For $708 more per month today, you save $259,750 in interest and own your home free and clear in 15 years instead of 30.’ The break-even question: ‘Can you absorb $708/month permanently?’ If yes, this is almost always the optimal long-term decision. Engine must show the payoff date as a YEAR: ‘Your home is fully paid off in [Year].’ Emotional anchor.
This is the primary output the Clarity Engine must present to every refinance borrower — all scenarios side-by-side so they can make an informed decision.
Metric A: Rate Reduction B: Term Red / Same Pmt C: Cash Out / Same Pmt D: Debt Consol. E: 15-Year Term New Loan Amount $391,400 $391,400 $421,649 $410,400 $391,400 Interest Rate 6.00% 5.75% 6.00% 5.75% 5.75% Term 30 years 23.6 years 30 years 30 years 15 years New Payment $2,346.64 $2,528.00 $2,528.00 $2,385.31 $3,236.26 vs. Current ($2,528) −$181.36 $0.00 $0.00 −$712 total $+708.26 Cash Out None None $30,249.20 CC payoff None Total Interest $453,391 $324,820 $489,494 $467,312 $193,641 Interest vs. 30yr Baseline −$128,570 +$36,103 +$13,921 −$259,750
EFFECTIVE RATE: THE ADVANCED SCENARIO (from Effective Rate Worksheet) The Pro_Calc’s Effective Rate Worksheet models the Cash Out scenario (Sc.C) using the borrower’s Total Possible Payment — what they CAN afford, not just what they’re paying. Total Possible Payment: $3,098/month (current mortgage + CC minimums) Loan Amount at same payment: $421,649 | New Payment: $2,448/mo | Freed cash: $650/mo If freed cash ($650/mo) is applied as additional principal: effective payoff term = 104 months (8.7 years) Total interest paid = $166,829 | Effective rate = 0.4753% monthly (5.70% annualized) Clarity Engine opportunity: ‘If you apply your CC minimum savings of $650/month to principal, you pay off a $421K loan in under 9 years.’
The following are the specific calculation capabilities the Pro_Calc workbook proves the Clarity Engine must implement:
Required Calculation Functions
Required Scenario Logic Rules
Data Inputs Required from Borrower Record
Every borrower deserves to see all five scenarios. Not just the one the loan officer wants to sell them. The Clarity Engine’s job is to surface all the math, show all the paths, and let the borrower decide with complete information.
Document Version: 1.0 | Source Data: Pro_Calc.xlsx | Prepared for: BIX Technology Corp | CONFIDENTIAL — PreFi, Inc. / Purpose Technology, Inc. (d/b/a Purlend)