A bank features a savings account that has an annual percentage rate of $5.9 \%$ with interest compounded semi-annually. Ricardo deposits $\$ 10,500$ into the account.
What is the annual percentage yield (APY) for the savings account?
$A P Y=$ $\square$ \%. Round to the nearest hundredth of a percent.
Answer (2)
Define the formula for annual percentage yield (APY): A P Y = ( 1 + n A PR ) n − 1 .
Substitute the given values: A PR = 0.059 and n = 2 .
Calculate the APY: A P Y = ( 1 + 2 0.059 ) 2 − 1 = 0.05987025 .
Convert to percentage and round to the nearest hundredth: A P Y = 5.99% .
Explanation
Understanding the Problem We are asked to find the annual percentage yield (APY) for a savings account with an annual percentage rate (APR) of 5.9% compounded semi-annually. This means the interest is calculated and added to the account twice a year. We need to use the formula for APY to find the effective annual rate, taking into account the compounding effect.
Stating the Formula The formula for APY when interest is compounded n times per year is: A P Y = ( 1 + n A PR ) n − 1 where A PR is the annual percentage rate (as a decimal) and n is the number of compounding periods per year.
Substituting the Values In this problem, the annual percentage rate (APR) is 5.9%, which as a decimal is 0.059. The interest is compounded semi-annually, which means n = 2 times per year. Substituting these values into the formula, we get: A P Y = ( 1 + 2 0.059 ) 2 − 1
Calculating APY Now, we calculate the APY: A P Y = ( 1 + 2 0.059 ) 2 − 1 = ( 1 + 0.0295 ) 2 − 1 = ( 1.0295 ) 2 − 1 = 1.05987025 − 1 = 0.05987025
Converting to Percentage and Rounding To express APY as a percentage, we multiply by 100: A P Y in percentage = 0.05987025 × 100 = 5.987025% We are asked to round to the nearest hundredth of a percent, so we round 5.987025 to 5.99.
Final Answer Therefore, the annual percentage yield (APY) for the savings account is 5.99%.
Examples
Understanding APY is crucial when comparing different savings accounts or investment options. For instance, if you're deciding between two accounts with slightly different APRs and compounding frequencies, calculating the APY will show you which account will actually yield a higher return over a year. This is particularly useful in long-term financial planning, such as saving for retirement or a down payment on a house, where even small differences in interest rates can significantly impact the final amount.
The annual percentage yield (APY) for the savings account with an APR of 5.9% compounded semi-annually is calculated to be 5.99%. This is determined using the formula for APY by plugging in the values and performing the calculations accordingly. The result is rounded to the nearest hundredth of a percent.
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