If you want to be paid from an account with a guaranteed rate of 2.458% compounded annually, how much should be in the account if you want to receive annual payments of $7,000.00 over the 12 year period?
Answer (2)
Define the variables: annual payment $PMT = 7 , 000.00 , in t eres t r a t e r = 0.02458 , an d n u mb ero f ye a rs n = 12$.
Apply the present value of an annuity formula: P V = PMT × r 1 − ( 1 + r ) − n .
Substitute the values into the formula: P V = 7000 × 0.02458 1 − ( 1 + 0.02458 ) − 12 .
Calculate the present value: P V ≈ 65986.9 . Therefore, the initial amount that should be in the account is 65986.9 .
Explanation
Understanding the Problem We are asked to find the initial amount (present value) needed in an account to receive annual payments of $7,000.00 over a 12-year period, given an annual interest rate of 2.458% compounded annually. This is a present value of an annuity problem.
Defining Variables Let's define the variables:
P V = Present value (initial amount)
PMT = Annual payment = $7,000.00
r = Annual interest rate = 2.458% = 0.02458
n = Number of years = 12
Applying the Formula The formula for the present value of an annuity is:
P V = PMT × r 1 − ( 1 + r ) − n
Now, we substitute the given values into the formula:
P V = 7000 × 0.02458 1 − ( 1 + 0.02458 ) − 12
Calculating the Present Value Calculating the present value:
P V = 7000 × 0.02458 1 − ( 1.02458 ) − 12
P V = 7000 × 0.02458 1 − 0.7683
P V = 7000 × 0.02458 0.2317
P V = 7000 × 9.4267
P V = 65986.9
Final Answer After performing the calculation, the present value (initial amount) that should be in the account is approximately $65,986.90.
Examples
Imagine you want to set up a college fund for your child. You want to be able to withdraw $7,000 each year for 12 years to help with tuition and expenses. If the savings account guarantees a 2.458% annual interest rate, this calculation helps you determine how much money you need to deposit initially to meet your goal. Understanding present value calculations is crucial for financial planning, retirement savings, and investment decisions, ensuring you have enough funds to cover future expenses.
To receive annual payments of $7,000 for 12 years at an interest rate of 2.458% compounded annually, you need to deposit approximately $65,986.90 in the account. This is calculated using the present value of an annuity formula. It is important to understand this calculation for effective financial planning.
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