The textbook by John Hull (9/e) describes "quarterly compounding" on page 80:
If the interest rate is measured with annual compounding, the bank’s statement that the interest rate is 10% means that $100 grows to
$100 x 1.1 = $110
at the end of 1 year...When the interest rate is measured with quarterly compounding, the bank’s statement [that the interest rate is 10%] means that 2.5% is earned every 3 months, with the interest being reinvested. The $100 then grows to
$100 x (1.025)^4 = $110.38
at the end of 1 year
Then in example 4.2 on p. 82, he gives the following calculation which seems at odds with the definition of quarterly compounding (it doesn't reinvest the interest):
Suppose that a lender quotes the interest rate on loans as 8% per annum with continuous compounding, and that interest is actually paid quarterly. ..[T]he equivalent rate with quarterly compounding is
4 x (e^(0.08/4) - 1) = 0.0808
or 8.08% per annum. This means that on a $1,000 loan, interest payments of $20.20 would be required each quarter.
Given the earlier definition of quarterly compounding, at a rate of 8.08% with quarterly compounding I would expect my balance at the end of one year to be
$1000 x (1.0202)^4 = $1083.28
However, my balance is actually $1000 + $20.20*4 = $1080.80. Then, using the earlier definition of quarterly compounding, my actual quarterly compounded rate is
$1000 x (1 + R/4)^4 = $1080.80 => R = 7.85%
which is not 8.08% as described.
Is this example indeed inconsistent with his definition, or am I missing something?