For example, in the formula that shows the relationship between the nominal and effective interest rate shown below, 1 is first added to in/m and then 1 is subtracted from the result. What is the intuitive explanation for this?nominal and effective rate

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The formula is multiplicative because interests are compounded (re-invested).

Let's say you have 10% interest on a 1 USD deposit, compounded (calculated and paid) twice a year. That would be 5 cents of interest for the first semester. Then you have a deposit of 1.05 USD during the next semester. After one year, your interests are

1 * 0.05 + 1.05 * 0.05 = 0.05 + 0.0525 = 0.1025 = 10.25% per annum

Using the formula you provided

1.05 * 1.05 - 1 = (1 + 0.05) * (1 + 0.05) = 1 + 0.05 + 0.05 + 0.05*0.05 - 1
= principal + interest[principal, S1] + interest[principal, S2] + interest[interests of S1, S2]

Where interest[amount, period] denotes the interests on "amount" during the "period"

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