Different compounding periods, are used for different types of investments. In order to properly compare investments or loans with different compounding periods, we need to put them on a common basis. In order to do this, you need to understand the difference between the nominal interest rate $($INom$)$ and the effective annual rate $($EAR$).$ The $$ interest rate is quoted by borrowers and lenders, and it is also called the annual percentage rate $($APR$).$ If the compounding periods for different securities is the same, then you $$ use the APR for comparison. If the securities have different compounding periods, then the $$ must be used for comparison.

Here, $M$ is the number of compounding periods per year and $\frac{I_{\text{N}\text{O}\text{M}}}{M}$ is equal to the periodic rate $(I_{\text{P}\text{E}\text{R}}).$ If a loan or investment uses interest rate is also its effective annual rate. However, if compounding occurs more than once a year, EAR is $$ INOM.

Quantitative Problem: Bank $1$ lends funds at a nominal rate of $10\%$ with payments to be made semiannually. Bank $2$ requires payments to be made quarterly. If Bank $2$ would like to charge the same effective annual rate as Bank $1,$ what nominal interest rate will they charge their customers? Do not round intermediate calculations. Round your answer to three decimal places.

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