How much do you need to invest semiannually into an ordinary annuity earning an annual interest rate of 6.12% compounded semiannually so that you will have $5,116.19 after 9 years?
Answer (2)
You need to invest approximately $217.33 semiannually into an ordinary annuity to have $5,116.19 after 9 years with an annual interest rate of 6.12% compounded semiannually. This is calculated by using the future value of an annuity formula and substituting the relevant values. The detailed calculations show how the semiannual payment was derived step-by-step.
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Define the variables: future value (FV), semiannual interest rate (r), number of semiannual periods (n), and semiannual payment (PMT).
State the future value of an ordinary annuity formula and rearrange it to solve for PMT: PMT = ( 1 + r ) n − 1 F V × r .
Substitute the given values into the formula: PMT = ( 1 + 0.0306 ) 18 − 1 5116.19 × 0.0306 .
Calculate the semiannual payment amount: $217.33 .
Explanation
Understanding the Problem We are asked to find the semiannual investment amount needed to reach a future value of $5,116.19 after 9 years, with an annual interest rate of 6.12% compounded semiannually. This is a future value of an ordinary annuity problem.
Defining the Variables Let's define the variables:
FV = Future Value = $5,116.19
r = Semiannual interest rate = 6.12% / 2 = 0.0306
n = Number of semiannual periods = 9 years * 2 = 18
PMT = Semiannual payment amount (what we want to find)
Stating the Formula The formula for the future value of an ordinary annuity is: F V = PMT × r ( 1 + r ) n − 1 We need to rearrange this formula to solve for PMT: PMT = r ( 1 + r ) n − 1 F V PMT = ( 1 + r ) n − 1 F V × r
Calculating the Semiannual Payment Now, we substitute the values into the formula: PMT = ( 1 + 0.0306 ) 18 − 1 5116.19 × 0.0306 PMT = ( 1.0306 ) 18 − 1 5116.19 × 0.0306 PMT = ( 1.0306 ) 18 − 1 156.5555 Calculating ( 1.0306 ) 18 :
( 1.0306 ) 18 ≈ 1.7196 PMT = 1.7196 − 1 156.5555 PMT = 0.7196 156.5555 PMT ≈ 217.33
Final Answer Therefore, you need to invest approximately $217.33 semiannually to have $5,116.19 after 9 years.
Examples
Understanding annuities can be incredibly useful in planning for your future. For instance, if you aim to save a specific amount for a down payment on a house in 5 years, you can calculate how much you need to deposit each month into an annuity account. By knowing the interest rate and the time frame, the annuity formula helps you determine the necessary regular contributions to reach your goal. This approach provides a structured savings plan, ensuring you meet your financial targets effectively.
