Lesson 8 of 25 · 6 min

The blended effective rate, computed after the fact

The rate that reproduces the month is none of the rates that applied during the month.

Once the dollars are on the page you can ask a different question. What single rate, applied to the opening balance for all 31 days on an Act/360 basis, would have produced $11,343.47? That number is the effective rate for the period. It is computed backwards from the charge. It is not an input to the charge.

effective rate = charge × 360 ÷ (balance × days)

charge: The printed interest total. $11,343.47 on this March.
balance: The constant balance the charge sat on. $1,847,220, because there was no draw or repayment.
days: The length of the whole period, 31, not the length of either leg.
360: The same divisor the products used, so the inversion is exact.

$11,343.47 × 360 ÷ ($1,847,220 × 31) = $4,083,649.20 ÷ $57,263,820 = 0.07131290…, which is 7.1313 percent to four decimal places. Feed that rate forward: $1,847,220 × 0.07131290… × 31 ÷ 360 returns $11,343.47 on the nose. The rate that reproduces the month is 7.1313 percent. The rates that applied were 6.97 and 7.22.

On a flat balance the effective rate and the day-weighted average of the two legs coincide, which is why 7.1313 percent appeared in both lessons. The coincidence ends when the balance moves. A draw on 19 March weights the 7.22 percent stretch more heavily in the dollars than in the day count, and the effective rate then exceeds the day-weighted average. The formula above still works, provided the balance you put in is the one that was actually constant — which, after a draw, it was not. Lesson 11 is the honest split.

Key terms

Effective rate for the period: The single Act/360 rate that would have produced the printed charge on the balance and days you put into the inversion.
Inversion: Rearranging the four-factor product to solve for the rate. Charge × 360 ÷ (balance × days).

Worked example

March, inverted

Printed total $11,343.47 on a flat $1,847,220 for 31 days, Act/360.

Charge × 360: $4,083,649.20
Balance × days: $57,263,820
Quotient: 0.07131290…
Forward check: $11,343.47
Rates that actually applied: 6.97% and 7.22%

7.1313 percent is a property of the dollars after the month has closed. It is not a rate Halden & Wren charged on any morning.

Check yourself

Using $11,343.47, $1,847,220 and 31 days on an Act/360 basis, what is the effective rate for March, and which printed rate does it match?

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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.