# Methodology | The Math Says Canonical: https://themathsays.com/methodology/ > The exact formulas, assumptions, and verdict thresholds behind every calculator on The Math Says. All of it documented, nothing hidden. ## Core formulas All calculators on this site use the same shared math library (`assets/calc.v2.js`). Single source of truth for these standard finance formulas: ### Monthly payment for an amortizing loan (PMT) For principal *P*, monthly rate *r*, and total payments *n*: PMT = P × (r × (1 + r)^n) / ((1 + r)^n − 1) Used for: every mortgage, refi, auto loan, home equity loan calculation. This is the standard amortizing-loan formula. ### Remaining loan balance after k payments Balance(k) = P × (1 + r)^k − PMT × ((1 + r)^k − 1) / r Used for: figuring out what you'd still owe at any given month, needed for hold-period comparisons (ARM vs fixed) and total-cost analyses. ### Future value of a series of payments (annuity) FV = PMT × ((1 + r)^n − 1) / r Used for: the pay-off-vs-invest comparison. Projecting what regular investments would grow to over a horizon. ### Months to payoff if you increase the payment months = ln(PMT / (PMT − P × r)) / ln(1 + r) Used for: bi-weekly mortgage and extra-principal-payment calculations. Requires the payment to be high enough to cover monthly interest, otherwise loan never pays off. ### Break-even point Break-even months = upfront cost / monthly savings Used for: every refinance scenario, deductible-optimizer, and similar "is this worth the upfront cost" comparison. ## Per-calculator assumptions ### 01. Insurance deductible optimizer - Annual premium savings = (old premium − new premium). User-supplied. - Expected-value calculation: annual savings minus probability-of-claim × extra deductible cost. - Claim-frequency options: 0%/yr (never), 10%/yr (once in 10), 20%/yr (twice in 10), 33%/yr (3+ in 10). Historical pattern is the best predictor. - **Typical premium-savings ranges** (referenced on the calc page): $500→$1,000 deductible = 5-10% premium drop. $1,000→$2,500 = 8-15%. $1,000→$5,000 = 20-30%. Source: NAIC rate-factor industry publications. - **State carveouts:** CA, HI, MD, MA, MI restrict how credit-based insurance scores may be used in pricing. The calculator does not model these state-specific rules. - **Verdict thresholds:** ✅ if break-even < 4 yrs AND claim freq ≤ 10% AND can cover from savings. ⚠️ if 4-8 yrs OR moderate claim history. ❌ if > 8 yrs OR can't cover. ### 02. ARM vs fixed mortgage - ARM "worst case" = lifetime cap rate applied after the fixed period ends, re-amortized over remaining months. - Total cost over hold = (ARM monthly × months at start rate) + (worst-case monthly × months after adjustment) for stays past the fixed period. - Net wealth = total paid + remaining balance at exit. - **"Average homeowner stays ~13 years"** anchor (used in the hold-period default + 12-month-buffer rationale): National Association of Realtors Profile of Home Buyers and Sellers, multi-year average. - **Verdict thresholds:** ✅ if hold ≤ fixed period − 12 months buffer. ⚠️ if hold right at fixed period. ❌ if hold > fixed period AND worst-case payment unaffordable. ### 03. Pay off mortgage early vs invest - Effective mortgage rate: nominal × (1 − marginal tax rate) if itemizing, else nominal rate. - After-tax investment return: full return for tax-advantaged (Roth) accounts. Taxable: return × 0.85 (assumes 15% LTCG, the most common bracket for filers with taxable income up to ~$500k MFJ). Traditional: return × (1 − marginal × 0.85), the drag at withdrawal. - Wealth at horizon: pay-down path = FV of freed-up cashflow after mortgage paid off + interest saved. Invest path = FV of equivalent monthly contribution. - **Investment-return defaults** (used in the calc's hint text + verdict logic): S&P 500 long-term ~10% nominal / ~7% real (inflation-adjusted). To compare against a nominal mortgage rate on equal terms, we recommend 7-8%. Source: Robert Shiller S&P 500 historical data, widely-cited. - **Verdict thresholds:** Tie if within 5%. ✅ Invest if > 15% advantage. ⚠️ Invest if 5-15%. ✅ Pay down if > 10% advantage to paydown. ⚠️ Pay down if 5-10%. ### 04. Bi-weekly mortgage - Bi-weekly equivalent monthly payment = monthly × 13/12 (26 half-payments per year = 13 full payments). - Total fees over loan life = setup fee + (per-payment fee × biweekly payment count). - **Verdict thresholds:** ✅ if fees = $0. ❌ if fees ≥ interest saved. ❌ if fees > 20% of interest saved. ⚠️ if fees modest but DIY captures same. ### 05. 30-to-15 refinance - New monthly = PMT(balance + costs, new rate, 180 months) if costs rolled in. - Front-end ratio = new monthly ÷ gross monthly income. Industry standard: lender max 28%. - Back-end DTI = (new monthly + other debt) ÷ gross monthly income. Industry standard: lender max 36-43%. - **Verdict thresholds:** ❌ if front-end > 32% (affordability risk). ❌ if back-end > 43%. ⚠️ if rate spread < 0.25% (refi math weak). ✅ if front-end ≤ 25% AND interest saved > $30k. ### 06. PMI removal refinance - Current monthly outlay = P&I + PMI. New monthly = P&I only (assumes 20% equity to avoid new PMI). - **PMI typical range** (referenced on the calc page): $50-$400/month for the average homeowner. Source: industry-published rate factors. Varies by LTV, credit score, and loan size. - Break-even = closing costs ÷ (current outlay − new P&I). - Lifetime interest delta includes 5 years of PMI you would have otherwise paid (conservative estimate of years before automatic termination under HPA 1998). - **Verdict thresholds:** ❌ if savings ≤ 0. ⚠️ if conventional and rate alone doesn't save (try free cancellation first). ✅ if break-even < 24 months and rate also drops. ✅ if break-even < 36 months. ⚠️ 36-60 months. ❌ > 60 months. ### 07. HELOC vs cash-out refi - HELOC total cost = (HELOC rate × amount × years) + setup fees, amortized. - HELOC stress test = rate + 1.5% (typical rate-rise scenario over a multi-year payback). - Cash-out refi cost includes the hidden cost of re-rating the existing first mortgage at today's rate. - **Verdict thresholds:** ✅ HELOC if current rate < cash-out rate − 1% AND payback ≤ 7 years. ✅ Cash-out if current rate > cash-out rate + 0.5% AND payback ≥ 10 years. ⚠️ if margin < 10%. ### 08. Home equity loan vs cash-out - HEL total cost = (HEL rate × amount × years amortized) + closing costs. - Cash-out refi cost includes total interest on new full loan + closing costs − interest you'd have paid on existing first mortgage anyway. - **Verdict thresholds:** ✅ HEL if current rate < refi rate − 1%. ✅ Cash-out if current rate > refi rate + 0.5%. ⚠️ Close call if within 10%. ### 09. FHA to conventional - Equity check: LTV = balance ÷ home value. If > 80%, new conventional loan would still have PMI (defeats purpose). - Current outlay = P&I + FHA MIP. New outlay = conventional P&I (assumes ≤ 80% LTV). - **FHA annual MIP default** (used in the calc's hint text + verdict logic): 0.55% of loan balance for most modern FHA loans. Source: HUD Mortgagee Letter 2023-05 (Feb 2023) which reduced annual MIP by 30bp. - Conventional PMI assumption if > 80% LTV: 0.5% annually for sensitivity. - Lifetime MIP eliminated = current MIP × 12 × years remaining on original loan. - **Verdict thresholds:** ❌ if LTV > 80%. ❌ if savings ≤ 0. ✅ if break-even < 24 months. ✅ if < 36 months. ⚠️ if 36-60 months. ### 10. Lease vs buy - Buy total = loan total + maintenance after warranty ($1,500/yr) − vehicle value at horizon. - Lease total = (downcost + monthly × term + disposition fee + excess mileage cost) × number of cycles in horizon. - Excess mileage charge: $0.20/mile (industry typical). - Disposition fee: $400/cycle (industry typical). - **Verdict thresholds:** ⚠️ if horizon < 5 years (comparison unfair). ✅ Buy if margin > 25% over horizon. ✅ Buy if any positive margin and horizon ≥ 5 years. ### 11. Auto loan refi - LTV = balance ÷ vehicle value. Underwater flag if > 100%. - Total savings = (current total payments) − (new total payments + fees). - Term-extension trap: flagged if new term > current term by > 12 months AND total savings negative. - **FICO rate-shopping window:** 14 days. Multiple auto-loan rate inquiries within this window count as a single hard inquiry for FICO scoring. Shop aggressively within the window, not across months. - **Vehicle-age cutoff:** cars older than 10 years are often hard to refinance regardless of credit. Lender approval typically requires ≤ 10 years, ≤ 100k-125k miles. Source: industry-published lender guidelines (LendingTree, RateGenius, credit unions). - **Verdict thresholds:** ❌ if underwater. ❌ if term extended AND total cost higher. ✅ if rate drop ≥ 2% AND term not extended AND savings > $1,000. ### 12. Mortgage refi break-even (general) - Cash costs = closing costs if paid out of pocket, $0 if rolled in. - If lender-paid: assumes rate rises 0.25% to compensate. - Break-even = cash costs ÷ monthly savings (P&I delta). - Net effect over stay = monthly savings × stay months − cash costs. - **Verdict thresholds:** ❌ if savings ≤ 0. ✅ strong if < 12 months break-even, no term reset. ✅ if 12-36 months. ⚠️ if 36-60 months. ❌ if stay < break-even. ## What the calculators don't model - **Tax-loss harvesting** in the invest-vs-paydown comparison. Real after-tax returns on taxable accounts are often a bit better than the flat 0.85 multiplier we use. - **State income tax.** Federal marginal only. State tax adds another 0-13% drag depending on where you live. - **AMT, NIIT, phase-outs.** Edge cases for high earners. - **Refinancing costs paid via points** (specifically). The "no-cost" toggle approximates this but doesn't model points precisely. - **Insurance state-specific rules.** CA, HI, MD, MA, MI restrict credit-based pricing. The deductible optimizer doesn't model these. - **Lender-specific quirks.** Some lenders cap the rate buydown, won't refi certain LTV bands, or require seasoning. We assume vanilla terms. ## Updating cadence Calculators are updated: - **When the rate environment shifts** enough to move the assumptions (30-year fixed rates changing meaningfully). - **When tax law changes** (TCJA sunsets, bracket updates, MIP rule changes). - **Immediately** if a calculation error is identified. Last methodology review: 2026-05-12. If you spot something wrong, file a correction at the contact address on the [about page](https://themathsays.com/about/).