Lesson 14 of 25 · 6 min

When the printed figure does not match: rounding

The unrounded product is $3,934.065483, and the printed line is that figure taken to the nearest cent.

Period 1 is not $3,934.07 until someone rounds it. The four-factor product is $1,847,220 × 0.0697 × 11 ÷ 360 = $3,934.0654833…. The third decimal is 5 with a 4 after it, so the figure sits above the half-cent and prints as $3,934.07. The statement and the reconstruction meet at the rounded cent, not at the unrounded quotient.

Period 2 is $7,409.404666…, which sits below the half-cent and prints as $7,409.40. Add the two unrounded products and you have $11,343.47015…. Add the two printed legs and you have $11,343.47. This month they agree. They do not have to. Rounding each leg, then adding, is a different operation from adding the unrounded legs and rounding once.

A one-cent gap that appears only after you have the four inputs right is usually this. Take a 29-day stretch at 6.97 percent on $1,847,220: the single product is $10,371.627183…, which prints as $10,371.63. Split it into a 28-day product of $10,013.984866… ($10,013.98) plus a one-day product of $357.642316… ($357.64) and the rounded parts add to $10,371.62. The same stretch, two presentations, one cent apart.

A gap of a cent or two, after the legs otherwise reconcile, is the rounding of the products. A gap of $141.11 is the naive closing-rate product from lesson 6. A gap of about 1.39 percent is the 365-divisor from lesson 3. Size and direction identify the cause. Rounding is the smallest of those signatures.

Key terms

Unrounded product: The exact quotient of the four-factor formula. Period 1 is $3,934.0654833… before anyone takes it to the cent.
Round then add: Each leg taken to the cent, then summed. The usual statement presentation.
Add then round: Unrounded legs summed, then a single rounding. Can differ from round-then-add by a cent.

Worked example

The 5 that becomes a 7

Period 1 of the sample, written with more decimals than the statement keeps.

Balance × rate: $128,751.234
Times eleven: $1,416,263.574
Divide by 360: $3,934.0654833…
To the nearest cent: $3,934.07
29-day product, one rounding: $10,371.63
28-day plus 1-day, each rounded: $10,371.62

The printed $3,934.07 is $3,934.0654833… taken to the cent. A one-cent disagreement after that is a rounding convention, not a fourth-input error.

Check yourself

The four-factor product for period 1 is $3,934.0654833…. Why does the statement print $3,934.07 rather than $3,934.06, and when does a one-cent gap still appear?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.