Question Video: Solving Word Problem Involving Percentages and Compound Interests Mathematics

A savings account offers an annual interest rate of 6.03%, compounded quarter-yearly (once every 3 months). Write the explicit formula for the return 𝑅 after 𝑛 years on a deposit of 𝑅₀ dollars. What annual percentage rate (compounded once a year) would give the same yield? Give your answer to 4 decimal places.

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Video Transcript

A savings account offers an annual interest rate of 6.03 percent, compounded quarter-yearly, which means once every three months. Write the explicit formula for the return 𝑅 after 𝑛 years on a deposit of 𝑅 sub zero dollars. What annual percentage rate compounded once a year would give the same yield? Give your answer to four decimal places.

In order to answer this question, we need to use the general formula for a value of an investment compounded 𝑛 times per year. This states that 𝑉 is equal to 𝑃 multiplied by one plus 𝑟 over 100 divided by 𝑛 all raised to the power of 𝑦 multiplied by 𝑛. 𝑃 is the principal value or initial investment. 𝑟 is the interest rate as a percentage. 𝑛 is the number of periods per year. And finally, 𝑦 is the number of years. In this question, we are told that the return is 𝑅. The initial deposit or principal value is 𝑅 sub zero. The interest rate is 6.03 percent. And our value of 𝑛 is four as the interest is compounded quarterly.

We are told that the number of years is 𝑛. Therefore, the exponent is 𝑛 multiplied by four or four 𝑛. 𝑅 is equal to 𝑅 sub zero multiplied by one plus 6.03 divided by 100 divided by four all raised to the power of four 𝑛. We can simplify the numerator inside the parentheses by dividing 6.03 by 100. This gives us 0.0603. We could calculate the multiplier by dividing this value by four and then adding it to one. However, in this question, we will leave the formula in the form 𝑅 is equal to 𝑅 sub zero multiplied by one plus 0.0603 over four all raised to the power of four 𝑛.

In the second part of our question, we want to find the annual percentage rate that would give the same yield. This means that our value of 𝑛, the number of years, must be equal to one. This means that our multiplier is equal to one plus 0.0603 divided by four all raised to the power of four multiplied by one. The value inside the parentheses simplifies to 1.015075. We need to raise this to the power of four. This is equal to 1.06167728, and so on.

To calculate the annual percentage rate, we need to multiply the decimal part of this answer by 100. To four decimal places, this is equal to 6.1677 percent. An annual percentage rate equal to 6.1677 percent would give the same yield.

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