100% a year, compounded quarterly, how much do you get?

Question

You invest \1 at a 100\% annual rate, but the interest is credited four times a year, 25\% each quarter, compounding. Solve in your head. What is the balance after one year? Aim: under 30 seconds.

Accepted Answer

Each quarter multiplies by 1.25, four times: 1.25^{4} = (1.25^{2})^{2} = 1.5625^{2}. Square 1.5625: 1.5625^2 = 2.4414. So you end with about \2.44, noticeably more than the \2.00 that a single 100\% credit would give. Takeaway. More frequent compounding of the same nominal rate always pays more: quarterly gives \2.44, monthly \2.61, and the limit of infinitely frequent compounding is e \approx \2.718. That limiting value is exactly where the constant e comes from.

100% a year, compounded quarterly, how much do you get?

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