Calculating Effective Annual Rate from a Periodic Invoice Credit Rate
Summary
The document clarifies how to annualize a rate when the quoted charge already applies to each compounding period. If invoice credit costs 5.26% per 20 days, that figure is the periodic rate, so it goes directly into the effective annual rate calculation; dividing it again by the number of periods would incorrectly treat it as a nominal annual rate.
The answer distinguishes a periodic rate from an annual percentage rate. A nominal annual rate is formed by multiplying the periodic rate by the number of periods per year, and it must be stated alongside its compounding frequency. The stated effective annual rate compounds the 20-day charge across a year, while the corresponding nominal annual rate is about 96%. The explanation depends on the invoice charge truly being quoted per 20-day period; the compounding convention and year length affect annualization.
Key ideas
- Use a quoted per-period rate directly in the effective annual rate formula.
- Divide a nominal annual rate by the number of compounding periods to find its periodic rate.
- A nominal annual rate requires a stated compounding frequency to be interpretable.
- The effective annual rate compounds the periodic charge over the periods in a year.
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Full text
# Effective Annual Rate (EAR) calculation from periodic rate of invoice credit
# Effective Annual Rate (EAR) calculation from periodic rate of invoice credit
I'm working on a finance class related problem, concerning the Effective Annual Rate (EAR) of an invoice credit rate.
The standard formula of the EAR is:
$ EAR = \big(1 + \frac{APR}{m}\big)^{m}-1$
where $APR$ is the Annual Percentage Rate and $ m $ is the number of compounding periods.
> The cost of delaying payment on the invoice is $ 0.0526\ (5.26\%) $ per 20 days.
I know that the answer is supposed to be:
> $ EAR = \big(1 + 0.0526\big)^{\frac{365}{20}}-1 = 1.5487 \approx 155\%$
As you can see, using the standard formula above, would yield another result:
$ EAR = \big(1 + \frac{0.0526}{365/20}\big)^{\frac{365}{20}}-1 = 0.0526 \approx 5.26\%$
My question: why is there no $ m $ denominator in the parenthesis of the suggested solution?
I'm new to this subject so I'm sure the reason for the divergence between the standard formula and the solution is blatantly obvious – probably having to do with the periodization of the EAR (20 days vs. annual rate)? However, I can't seem to grasp the concept properly.
## Answer by Chris Degnen (score 2, accepted)
https://quant.stackexchange.com/a/30985
Since "the cost of delaying payment on the invoice is 0.0526 (5.26%) 0.0526 (5.26%) per 20 days" there is no need to divide by anything to get the rate for the compounding period because it is given: 0.0526.
The literal APR (as an annual rate) would be a nominal rate of 95.995 % compounded every twenty days.
```
m = 365/20 = 18.25
periodic rate = 0.95995/m = 0.0526
```
Nominal annual rates are defined as the periodic rate * number of periods per year. It is necessary to specify the compounding period when describing a nominal annual rate. For examples see Wiki Calculation.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.