$400 \times \frac{\left[\left(1+\frac{0.065 \%}{12}\right)^{(12 \cdot 18)}-1\right]}{\frac{0.065}{12}}
Answer (2)
Define the variables: periodic payment, annual interest rate, number of compounding periods per year, and number of years.
Apply the future value of an ordinary annuity formula.
Calculate the future value using the given values.
The value of the expression is 86905.05 .
Explanation
Understanding the Expression We are asked to evaluate the expression: 400 × 12 0.065% [ ( 1 + 12 0.065% ) ( 12 ⋅ 18 ) − 1 ] This expression represents the future value of an ordinary annuity, where:
400 is the periodic payment.
0.065% is the annual interest rate.
12 is the number of compounding periods per year.
18 is the number of years.
Defining the Variables Let's define the variables:
Periodic Payment (PMT) = $400
Annual Interest Rate (r) = 0.065% = 0.00065
Number of Compounding Periods per Year (n) = 12
Number of Years (t) = 18
Applying the Future Value Formula The future value (FV) of an ordinary annuity formula is given by: F V = PMT × n r [ ( 1 + n r ) ( n ⋅ t ) − 1 ] Plugging in the values, we get: F V = 400 × 12 0.00065 [ ( 1 + 12 0.00065 ) ( 12 ⋅ 18 ) − 1 ]
Calculating the Future Value Now, we calculate the future value using the given formula. The result of the calculation is: F V = 86905.04954243804
Final Answer Therefore, the value of the expression is approximately $86905.05.
Examples
Annuities are commonly used in retirement planning. Suppose you deposit $400 each month into an account that earns 0.065% annual interest, compounded monthly. After 18 years, the annuity formula helps you calculate the total amount you'll have saved. This calculation is crucial for estimating your retirement income and making informed financial decisions. Understanding annuities assists in planning for long-term financial goals.
The expression represents the future value of an ordinary annuity, where $400 is paid monthly at an interest rate of 0.065% over 18 years. After performing the calculations, the future value is approximately $89,040. Understanding these values is essential in financial planning for investments and savings.
;
