4

Given annual interest rate r_ann and n compounds per year, why do we use the formula

(1+r_ann/n)^tn

rather than (1+r)^tn given r = (1+r_ann)^(1/n) - 1.

8

Your rate r_ann in (1+r_ann/n)^tn is a nominal rate compounded over a specific period, which means for example, a monthly rate has been simply multiplied by 12 to produce a "nominal annual rate compounded monthly", in order to make the calculation of the monthly rate easier, (just divide by 12).

The alternative is to use an effective annual rate, which indeed is the legal requirement in the EU when quoting an annual percentage rate (APR).

Converting between Nominal and Effective APR is described here:-

Wikipedia: Effective interest rate - Calculation

For example, 10% Nominal APR compounded monthly is 10.4713% Effective APR

(1 + 0.10/12)^12 - 1 = 0.104713

and 10% Effective APR is 9.56897% Nominal APR compounded monthly

((0.10 + 1)^(1/12) - 1)*12 = 0.0956897

Finally, reference 3 on the Wikipedia APR page is quite illuminating:-

The "Truth in Lending Act" passed in 1968 did not incorporate the mathematically-true annual percentage rate, because the true calculation used compounding (sometime fraction compounding), which was not readily available. The result on expression of the APR on credit cards uses a Nominal (simple interest) method ... which can far from the truth. The Truth in Lending Act should be changed to the mathematically-true (EFFECTIVE) APR from the untrue (NOMINAL) APR, merely by changing the word in act from "multiplied by" to "compounded for".

"the true calculation used compounding ... which was not readily available." - i.e. difficult to calculate, especially in 1968.

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