Lesson 22 of 25 · 6 min

The effective annual rate versus the stated rate

The stated rate is a 360-day object; a 365-day year and daily compounding both make the money larger.

7.22 percent on this facility is an Act/360 rate. It is the number that goes into balance × rate × days ÷ 360. It is not the fraction of the balance that a 365-day year will take, and it is not the rate a daily-compounded year will produce. Those are different objects, and both are larger.

day-count annual = stated × 365 ÷ 360 daily-compounded = (1 + stated ÷ 360)^365 − 1

stated: The Act/360 rate as a decimal. 0.0722 for the sample's closing rate.
365 ÷ 360: The extra five days of charge at the 360-day daily addend.
(1 + stated ÷ 360)^365: A year of daily capitalisation at that addend. Not how this facility capitalises.

7.22 percent × 365 ÷ 360 = 7.320277… percent. On $1,847,220 that is $1,847,220 × 0.0722 × 365 ÷ 360 = $135,221.635…, which prints as $135,221.64. The stated 7.22 percent of $1,847,220 is $133,369.28, a 360-day year on a frozen base. The gap is $1,852.36, five daily addends of $370.470233….

Daily compounding at the same addend is (1 + 0.0722 ÷ 360)^365 − 1 = 0.07594080…, which is 7.5941 percent. On $1,847,220 that is $140,279.36. This facility capitalises monthly, not daily, so the live annual figure sits between $135,221.64 and $140,279.36 and is the chain from lesson 19 once the rate path is allowed to move. The stated 7.22 percent is smaller than all of them. The March effective 7.1313 percent is a 31-day object and is not an annual rate at all.

Key terms

Stated rate: The Act/360 annual percentage used in the four-factor product. 6.97 percent, then 7.22 percent, on the sample.
Day-count annual: Stated × 365 ÷ 360. 7.3203 percent when the stated rate is 7.22 percent. Five extra addends, no compounding.
Daily-compounded annual: (1 + stated ÷ 360)^365 − 1. 7.5941 percent at 7.22 percent stated. A different capitalisation schedule from this facility.

Worked example

Three annual figures from one stated 7.22 percent

Frozen $1,847,220, 7.22 percent stated, Act/360 daily addend. Illustrative year, not the lesson-19 path.

Stated, 360-day year: $133,369.28
Day-count annual, 7.3203%: $135,221.64
Daily-compounded, 7.5941%: $140,279.36
March effective rate: 7.1313%
Lesson-19 chain, moving rates: $131,610.03

7.22 percent is the input. 7.3203 percent, 7.5941 percent and $131,610.03 are three different years built from related but not identical operations.

Check yourself

A stated 6.40 percent, Act/360, on $1,512,600. What is the day-count annual rate, and what does a 365-day year charge without compounding?

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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.