The fully-indexed ARM rate, rounded as the note requires

What “fully indexed” actually names

Two ingredients, with completely different characters. The index is a published benchmark that moves without anybody’s permission and belongs to nobody in the transaction. The margin is a spread agreed once and then fixed for the life of the note — it does not adjust, it is not renegotiated at an adjustment, and it is the part of the rate the borrower actually shopped for. Only one of the two is ever news, which is why a rate that feels volatile is usually being driven by a benchmark and priced by a spread the borrower accepted years earlier.

The word “fully” is load-bearing. A fully-indexed rate is the rate the arrangement is entitled to charge on its own terms, with no discount applied and no cap yet consulted. It is emphatically not the introductory rate, and at a given adjustment it need not be the rate actually charged either, because the caps in the note can hold the charged rate below it. Read this page as the rate the note points at; what the note is permitted to reach at any one adjustment is a separate calculation.

Rounding is a term of the note, not a display choice. The increment defines a ladder of permissible rates, and the sum is moved to the nearest rung on it. The declared vectors show both outcomes: two of them supply sums that already sit exactly on a rung — one on an eighth-of-a-point ladder, one on the coarser quarter-point ladder — and the rounding leaves them untouched; the third supplies a sum that falls between rungs, and the rate steps to the neighbouring rung, a movement smaller than the spacing of the ladder itself. Nearest rung means exactly that: the step can go either way, and the declared vectors happen to illustrate the upward one. A sum landing exactly midway between rungs is an edge no declared vector exercises, and the note’s own rounding language is what settles it.

The two reported rates are meant to be read against each other. The unrounded sum is the arithmetic; the rounded rate is the arithmetic after the note’s ladder has had its say. When they differ, the difference is the rounding convention made visible — small enough to look like noise on the rate, large enough to move a payment on a large balance, and entirely legitimate because the borrower agreed to the ladder when they signed.

Nothing here is a forecast, and no benchmark value is asserted. The index figure is supplied by the reader, and a note fixes the date whose published value counts — a look-back rule this calculation has no calendar for. Feeding it a value is asking a conditional question: if the benchmark stood here on the day my note reads it, this is the rate the note would construct.

When this calculation is used

• Auditing an adjustment notice: does the index plus the margin, rounded on the note’s ladder, reproduce the rate the servicer says applies?
• Asking what the rate becomes if the benchmark stands at a level you want to test.
• Comparing two offers whose margins differ, at one common benchmark value, so the spread is the only thing being compared.
• Seeing what a coarser rounding ladder does to the same sum, when two notes round differently.
• Producing the fully-indexed figure that the payment-gap comparison takes as an input.

Worked example

The pack declares three benchmark-and-margin pairs read against two different ladders: two sums that already fall on a rung — one on the finer ladder, one on the coarser — and a third that falls between rungs on the finer ladder.

Compare the unrounded and rounded outputs on each run. On the first two the two figures agree exactly, which is what a sum that already sits on a rung looks like — rounding is present and simply has nothing to do. On the third they part company: the rounded rate sits on the next rung up, a distance smaller than the ladder’s own spacing. That third case is the whole reason this calculator exists as its own page rather than as a line inside another one.

Every figure is produced by the certified engine when the calculator loads; this page stores none. The pack also declares a refusal: an increment of nothing is declined rather than answered, because a ladder with no spacing has no rungs to round to.

What each input represents

Assumptions and limits

• The benchmark value entered is the one the note’s look-back rule would use; no date arithmetic happens here.
• The margin is a single figure fixed for the life of the note — step-margin and promotional-spread structures are not modelled.
• The sum is moved to the nearest multiple of the stated increment, in whichever direction is nearer.
• No caps are applied: this is the rate before any periodic or lifetime limit is consulted, and therefore not necessarily the rate charged at an adjustment.
• No rate floor is applied either. Notes commonly state a minimum — often the margin itself — and enforcing one is outside this calculation.
• Nothing here forecasts a benchmark. The output is conditional on the value supplied.

What the guards protect against

• The increment must be greater than nothing, and the pack declares a refusal vector on an increment of nought: dividing by no spacing names no ladder, so the request is declined rather than answered.
• The increment is bounded above as well, at a whole percentage point, so a ladder coarser than any note’s rounding rule is refused — the shape a decimal slip takes when a fraction is typed as a whole number.
• The benchmark is bounded to a band that extends a little below nought and well above any ordinary quoted rate: a genuinely negative published index is accepted rather than refused, and a value entered as a count of basis points lands far outside the band and is declined.
• One slip the bands cannot catch is worth naming: a rate typed as a decimal fraction rather than as a percentage sits comfortably inside every band here and will be answered. The refusals police impossible entries, not plausible ones in the wrong unit — the units on the fields are the only defence against that, and they are worth reading.
• The margin must not be negative and is bounded above: a spread is added to a benchmark, never subtracted from it, and a margin beyond the band would describe no note in circulation.

Provenance

Fully-indexed adjustable-rate construction with note rounding

The benchmark value and the note margin added, then divided by the stated rounding increment, rounded to the nearest whole multiple and multiplied back — the rounding convention ARM notes and plan riders state in words, performed as arithmetic.

Educational reference, not financial advice, and not a quotation of any benchmark: the index value, the margin and the rounding increment are all supplied by the reader, and the note’s own terms govern. The signed pack carries its own citation, which displays from the verified leaf once the calculator loads; the page reports the verification state of the release it mounted rather than asserting one.