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An amortizing loan keeps the monthly payment constant, but the split between interest and principal shifts every month. The fixed-payment formula is M = P × r × (1+r)^n / ((1+r)^n − 1), where P is the principal, r is the periodic rate (annual rate / 12 for monthly payments), and n is the total number of payments. This formula derives from setting the present value of an annuity equal to the borrowed amount; it is not arbitrary and not negotiable. Consider a 30-year, $300,000 mortgage at 6.5% APR. Monthly payment is $1,896.20. In month 1, interest accrues on the full $300,000 at 6.5%/12 = 0.5417% per month, so the interest portion is $1,625.00 — meaning only $271.20 of the first payment reduces principal. By month 360 (final payment), the remaining principal is so small that the interest portion is just $10.22 and the principal portion is $1,885.98. The same payment dollar buys 7× more principal reduction at the end as at the start. Aggregate the first 5 years vs the last 5 years and the asymmetry becomes visceral. Months 1–60 of that loan: total interest paid = $94,738.60, total principal paid = $19,033.40 — interest is 4.98× principal. Months 301–360: total interest paid = $4,025.30, total principal paid = $109,746.70 — principal is 27.3× interest. Over the full 30 years you pay $382,633 in interest on a $300,000 loan, or 127.5% of principal. This is not a flaw or hidden fee — it is the mathematical consequence of borrowing for a long time at a meaningful rate. The intuition that helps: interest is always charged on the outstanding balance. Early in the loan, the balance is huge, so interest dominates. As principal works down, the same rate generates less interest, so more of each payment kills principal. The acceleration is exponential, not linear. This is why a single extra payment in year 2 saves dramatically more total interest than the same payment in year 25 — the early prepayment removes principal that would otherwise generate interest for decades. Quick sanity check: a 30-year loan's "break-even" point (where principal portion finally exceeds interest portion in a single payment) at 6.5% APR is around month 224, or year 18.7. At 4.0% it shifts earlier to month 154 (year 12.8). Lower rates compress the asymmetry; higher rates exaggerate it. The Consumer Financial Protection Bureau (consumerfinance.gov) provides standardized amortization disclosures lenders must give borrowers under TILA/RESPA — read them. Educational only — not financial advice. Consult a licensed lender or housing counselor for your specific loan.

// Standard amortization
function monthlyPayment(P, annualRate, years) {
  const r = annualRate / 12, n = years * 12;
  return P * r * Math.pow(1+r, n) / (Math.pow(1+r, n) - 1);
}

// Build amortization schedule
function schedule(P, annualRate, years) {
  const M = monthlyPayment(P, annualRate, years);
  const r = annualRate / 12;
  let balance = P;
  const rows = [];
  for (let m = 1; m <= years*12; m++) {
    const interest = balance * r;
    const principal = M - interest;
    balance -= principal;
    rows.push({ m, interest, principal, balance });
  }
  return rows;
}

const s = schedule(300000, 0.065, 30);
// s[0]:   interest = 1625.00, principal = 271.20
// s[359]: interest =   10.22, principal = 1885.98

The "pay off the mortgage early" debate is a battle between behavioral finance and mathematical optimization. Mathematically, prepayment is a guaranteed risk-free return equal to your loan rate. If your mortgage is 4.0% fixed, every dollar of prepayment earns a guaranteed 4.0% pre-tax (less if mortgage interest is deductible). Compare that to expected returns elsewhere: long-run S&P 500 nominal return is roughly 10% with ~19% annual standard deviation; long-run high-grade corporate bonds yield 5–6% nominal. On expected-value terms, equities historically beat a 4% mortgage rate handily. The catch: that 10% expected return is not guaranteed and not stable. The S&P had decade-long stretches with near-zero returns (2000–2009 was −0.95% annualized including dividends, per Damodaran's NYU dataset). Sequence-of-returns risk matters: a bear market in years 1–3 of your "invest instead of prepay" plan can permanently destroy the math. Mortgage prepayment, by contrast, returns the rate with zero variance. Tax treatment changes the picture. In the US, the standard deduction since the Tax Cuts and Jobs Act of 2017 has risen high enough that fewer than 10% of households itemize, per IRS data — meaning the mortgage interest deduction does not apply to most borrowers. If you do itemize, the after-tax cost of a 6% mortgage at a 24% marginal bracket is 6% × (1 − 0.24) = 4.56%. Run your real comparison against this after-tax rate. Behavioral truth: most households do not actually invest the difference. Studies of 401(k) and brokerage flows show that consumers who choose lower mortgage payments overwhelmingly spend the freed cash, not invest it. If your alternative to prepayment is consumption, the mathematical argument collapses — prepayment becomes the strictly better choice. Prepayment is also psychologically valuable: a debt-free home reduces anxiety, increases willingness to take career risk, and provides a hard floor under retirement housing costs. These benefits do not show up in spreadsheets but are real. Practical decision framework: (1) Always max employer 401(k) match first — that is a 50–100% guaranteed return. (2) Pay off all debt above your mortgage rate (credit cards typically 18–24% APR, per Federal Reserve G.19 data). (3) Build 3–6 months emergency fund. (4) Then choose between additional retirement contributions and prepayment based on your discipline, marginal bracket, and risk tolerance. Either choice is defensible. Educational only. Consult a licensed financial planner or CPA for your specific tax and cashflow situation.

Lenders advertise three different rate concepts and consumers conflate them constantly. Understanding the gap protects you from overpaying. Note Rate (also called the contract rate or nominal rate) is the bare interest rate written on the loan agreement. A 30-year fixed mortgage with a 6.0% note rate will accrue interest at 0.5% per month on the outstanding balance. Note rate ignores fees, points, and the timing of cashflows. Annual Percentage Rate (APR) is mandated by the U.S. Truth in Lending Act (Regulation Z, 12 CFR Part 1026). APR rolls origination fees, discount points, mortgage insurance, and certain other charges into a single number expressing the true cost of borrowing as if it were just interest. The standard mortgage APR uses the actuarial method: solve for the rate that, applied to the actual amount you receive (note amount minus prepaid finance charges), equals the contractual payment stream. APR will always be higher than note rate when there are upfront costs. A 6.0% note rate with 1 discount point and $3,000 in fees on a $300,000 loan typically yields an APR around 6.15%–6.25%. The 25-basis-point gap is real money: $750/year on this loan. Annual Percentage Yield (APY) is mandated by the Truth in Savings Act (Regulation DD) for deposits. APY captures intra-year compounding: APY = (1 + r/n)^n − 1, where r is the nominal rate and n is the compounding frequency. A 5.00% nominal savings rate compounded daily yields APY ≈ 5.127%. For loans, the analogous concept is Effective Annual Rate (EAR), and lenders are not required to disclose it directly — but you can compute it from the APR and compounding frequency. Critical asymmetry: APR understates the true cost of high-frequency-compounded loans because APR is a simple-interest annualization. Credit cards illustrate this. A "23.99% APR" credit card that compounds daily has an EAR of (1 + 0.2399/365)^365 − 1 ≈ 27.10%. The 311 basis points of compounding effect are invisible in the APR disclosure. The Consumer Financial Protection Bureau has flagged this gap for years; consumer advocates argue EAR should be required disclosure on revolving credit. It is not, yet. When comparing loan offers, demand APR (not just note rate). When comparing deposit accounts, demand APY (not just nominal rate). When considering a credit card, compute the EAR yourself if balances will revolve. Federal regulations exist to standardize comparisons — use them. Educational only. The Federal Reserve, CFPB, and your state attorney general all publish guidance on rate disclosures.

// EAR from APR with daily compounding
function ear(apr, periodsPerYear) {
  return Math.pow(1 + apr/periodsPerYear, periodsPerYear) - 1;
}

ear(0.2399, 365);  // 0.27097 (credit card EAR ≈ 27.10%)
ear(0.06,   12);   // 0.06168 (mortgage 6% monthly → EAR 6.17%)

// APR from note rate + fees (simplified Newton-Raphson)
// Real APR calc requires solving for the rate that equates
// the discounted payment stream to (note - upfront fees).
// Lenders use TILA-mandated actuarial method — see CFPB Reg Z.