Discount points versus rate break-even

What buying the rate down actually purchases

Both instalments come from the same amortisation relation on an identical principal, term and payment frequency — only the rate differs between them. That symmetry is the point of the design: the gap between the two payments is exactly what the points purchased each month, with nothing else mixed in.

The break-even divides the cost of the points by that monthly gap. It is a payback period, counted in plain undiscounted months: every month the loan survives beyond it is gain, and every month short of it leaves part of the points unrecovered. A borrower who sells or refinances early has bought a discount they never finished collecting.

That makes the purchase a bet on the horizon. Points pay best on a loan that is held long and left alone — and the scenario that most tempts a refinance, falling rates, is precisely the one in which points bought earlier go to waste. The month count turns that bet into a threshold the household can hold its plans against.

The count also flatters the points slightly, because it treats money at closing and money saved years later as equal. Cash paid up front could have earned a return elsewhere, and nothing in this arithmetic charges for that. A break-even that only just fits inside the expected holding period is therefore thinner than it looks. A zero rate on either branch is handled as its own case, with the payment simply spreading the principal evenly.

When this calculation is used

• Deciding at the rate-lock whether a lender’s points quote is worth paying, given how long you expect to keep the loan.
• Comparing several point-and-rate combinations on the same loan by reducing each to its break-even month.
• Seeing what a fraction of a percentage point is actually worth per month on your own principal and term, rather than in a lender’s illustration.
• Fixing the longest break-even you would accept and using the declared reverse workflow to solve for the most the points should cost.

Worked example

A typical mortgage quote: one principal, one term, monthly payments, and a pair of rates — a baseline with no points, and a slightly lower rate offered for an upfront cost.

The engine reports both level payments and the break-even in months. The payment gap is what the points actually purchase each month; the break-even is how many of those months it takes to be repaid. Hold well past it and the points were cheap money; leave before it and part of the cost is simply gone.

The comparison also runs backwards: fix a break-even you could accept and the declared reverse workflow solves for the most the points should cost to stay inside it. Remember the count is undiscounted — the cash could have earned a return elsewhere, so treat a break-even that barely fits your horizon as a warning rather than a pass.

What each input represents

Assumptions and limits

• Both schedules share the principal, the term and the payment frequency; the rate is the only difference the comparison sees.
• Both rates are fixed for the whole term, with each periodic rate taken as the annual rate divided by the payments per year — a nominal convention.
• Break-even is counted in undiscounted months; the return the upfront cash could have earned elsewhere is not modelled.
• Tax treatment of points is not modelled, and neither are lender credits or the other closing costs of the loan.
• The loan is assumed to run at least to break-even; a sale or refinance before it leaves part of the cost unrecovered.

What the guards protect against

• A with-points rate that is not below the no-points rate is refused — points that do not lower the rate buy nothing, and no break-even exists.
• The principal and the term must be greater than zero, and the payments per year must be at least one, so that both schedules are real schedules.
• The points cost must be greater than zero — with nothing paid up front there is nothing to recover, and the question dissolves.
• Rates outside a realistic range are refused rather than answered, because the result would not describe any lending arrangement.

Provenance

Discount-points break-even relation over level-payment amortisation

Two level payments are computed from the same principal, term and payment frequency at the two rates; the upfront cost of the points divided by the difference between the payments gives the break-even in months.

Educational reference, not financial advice, and not a full cost-of-credit comparison. The signed pack carries its own citation; the page reports the verification state of the release it mounted rather than asserting one.