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I have seen several conflicting sources on the internet about APR. Some say the effective rate is (1+APR/n)^n -1 while others say this formula is the APR since APR represents the rate over all compounding periods within a year. What is the textbook definition of APR typically used in contracts and other formal documents?

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If the compounding periods are shorter than a year, Annual Percentage Rate (APR) is calculated by ignoring the compounding. Annual Percentage Yield (APY) includes the effect of compounding.

In the original post's math:

  • Assume there are n compounding periods per year. For example, 12 equal-length months.
  • Assume the annual percentage rate is APR. For example, 6 percent per year.
  • The periodic rate is APR / n. For example, (6%/year) / (12 months/year) = 0.5%/month.
  • The annual percentage yield is APY = (1 + APR/n)^n - 1. For example, (1.005)^12 - 1 ~ 1.0617 - 1 ~ 6.17 percent per year.

Some (very one-sided) contracts calculate the daily interest by dividing a (notional) APR by 360 days per year -- and then compound that daily rate every day of the 365 or 366 day long year. The only place I have seen such a contract was for a "loan" by the new stockholders (in a leveraged buyout) to a holding company they set up. The purpose of the loan was to let the holding company pay de facto dividends to the (mostly tax-exempt) shareholders, but let the company take a tax deduction for the interest. If the holding company could not afford to pay the interest in cash, it could "pay in kind", by increasing the amount it owed. The holding company then invested the "loan" as equity in a company that had actual revenues, expenses, and profits. In this situation the "lenders" and "borrowers" were basically the same people, and the only restraint on the interest rate was "what will the I.R.S. let us get away with?"

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