Question - Calculating Future Value and Annual Percentage Yield of Savings Account
mathfuture value calculationcompound interest formulasavings account interestannual percentage yield
Solution:
The image contains a question which says:"A bank features a savings account that has an annual percentage rate of 3.8% with interest compounded monthly. Jace deposits $2,000 into the account.a) How much money will Jace have in the account in 1 year?Round answer to the nearest penny.b) What is the annual percentage yield (APY) for the savings account?Round to the nearest hundredth of a percent."Let's solve part a) first.We are given that the annual interest rate is 3.8% and it is compounded monthly. To find the total amount in the account after one year, we will use the formula for compound interest:A = P(1 + r/n)^(nt)Where:A = the future value of the investment/loan, including interestP = the principal investment amount ($2,000 in this case)r = the annual interest rate (decimal) (3.8% or 0.038)n = the number of times that interest is compounded per year (12 for monthly)t = the time the money is invested or borrowed for, in years (1 year in this case)Plugging in the values:A = 2000(1 + 0.038/12)^(12*1)A = 2000(1 + 0.0031667)^(12)A = 2000(1.0031667)^(12)Now use a calculator to evaluate the expression:A ≈ 2000(1.0031667)^12A ≈ 2000 * 1.038726A ≈ 2077.45So, after one year, Jace will have approximately $2077.45 in the account, rounding to the nearest penny.For part b), the annual percentage yield (APY) takes into account the effect of compound interest over the year. It is calculated by the formula:APY = (1 + r/n)^(n) - 1Again, substitute the values:APY = (1 + 0.038/12)^(12) - 1APY = (1 + 0.0031667)^(12) - 1Now evaluate the expression using the calculator:APY ≈ (1.0031667)^12 - 1APY ≈ 1.038726 - 1APY ≈ 0.038726To express this as a percentage, multiply by 100:APY ≈ 0.038726 * 100APY ≈ 3.8726%Rounding to the nearest hundredth of a percent, we get an APY of approximately 3.87%.
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