Let us consider again an annual percentage rate of \(5 \%\). If we invest \(\$ 1000\), with no compounding, wealth after one year is

Now, what if interest is earned semiannually? The typical convention is that if the annual rate \(\mathrm{APR}_{2}\) is compounded semiannually, it means that a rate  applies to each semester. Hence,

While the quoted rate \(\mathrm{APR}_{2}\) is \(5 \%\), the equivalent effective annual rate \(\mathrm{EAR}_{2}\), with semiannual compounding, is a bit larger and can be found as follows:

By a similar token, with quarterly compounding we find

which corresponds to an effective annual rate

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W = 1000 x 1.05 = $1050.