APR is the annual stated or proportional rate. Banks use it to quote deposits and loans in most countries (Portugal is named TANB; in Germany, it is Fester Sollzinssatz p. a.). Always assume proportional if the exercise does not mention whether it is an effective or proportional rate.
Why is it used? Convenience. Consider you are a bank and have two options to offer a 1,000 loan with a maturity of 1-year and monthly frequency for compounding and payments to a client.
Option A: We offer you an interest rate of 12%
Option B: We offer you an effective rate of 12.68%
Most clients understand with option A that they will pay 120 euros a year (10 each month). However, if we use option B, they have no idea what they will pay each month; moreover, they might think they will pay 12.68 a year (which is not true; both options refer to the same loan).
For all calculations aimed at discounting, compounding, etc., we need the effective rate. To transform proportional to effective there is a simple rule:
If we consider the frequency of compounding, proportional and effective are equal
This makes sense since the difference between the two is precisely that effective considers the compounding within the period, and proportional does not. Therefore, with monthly compounding, monthly proportional and monthly effective are the same. With annual compounding APR and EAR are the same. However, with monthly compounding APR is lower than EAR.
Different frequencies
When we work with the perpetuity and annuity formulas, we need to realize that the periods mean two things. For instance, in the case of annuities:
\(\dfrac{C}{r} \left(1-\left(\dfrac{1}{1+r}\right)^n\right)\)
n means: the number of periods that you receive (or pay) the payment but also the distance to the last payment (and assumes distance between payments is 1). This means:
- n needs to be an integer (5/6 payments make no sense).
- r needs to be in the frequency of payments.
How to go from APR to annuity formula
I consider we compound m times a year, and we receive n payments a year
- We transform the annual proportional rate to the frequency of compounding: \(r_m=\dfrac{APR}{m}\)
- Now we can call \(r_m\) effective rate because at that frequency, both rates are the same
- Finally, we transform \(r_m\) to the payment frequency: \((1+r) = (1+r_m)^{\frac{m}{n}}\)
- We can now use r in the annuity and perpetuity formula. For the annuity formula, the exponent will be \(n\times T\), where T is the number of years.