Find the future value for the ordinary annuity with the given payment and interest rate.
[tex] \text { PMT = } $2,200 ; 2.00 \% \text { compounded monthly for } 4 \text { years. } [/tex]
The future value of the ordinary annuity is $ [ ].
(Do not round until the final answer. Then round to the nearest cent as needed.)
Answer (2)
The future value of the ordinary annuity with a payment of $2,200 at an interest rate of 2.00% compounded monthly for 4 years is approximately $10,865.25. This is calculated using the future value formula for an annuity. It's important to accurately apply the interest rate and period in the formula to arrive at the correct future value.
;
Calculate the monthly interest rate: i = 12 0.02 = 0.0016666666666666668 .
Calculate the total number of compounding periods: n = 4 × 12 = 48 .
Use the future value of an ordinary annuity formula: F V = PMT × i ( 1 + i ) n − 1 .
Substitute the values and calculate the future value: F V = 2200 × 0.0016666666666666668 ( 1 + 0.0016666666666666668 ) 48 − 1 = 109843.70902869233 . Rounding to the nearest cent, we get 109843.71 .
Explanation
Problem Setup We are given an ordinary annuity with a payment (PMT) of $2,200, an annual interest rate of 2.00% compounded monthly, and a term of 4 years. Our goal is to find the future value (FV) of this annuity.
Calculate Monthly Interest Rate First, we need to determine the monthly interest rate. We divide the annual interest rate by 12: i = 12 0.02 = 0.0016666666666666668
Calculate Total Number of Periods Next, we calculate the total number of compounding periods by multiplying the number of years by 12: n = 4 × 12 = 48
State the Future Value Formula Now, we use the formula for the future value of an ordinary annuity: F V = PMT × i ( 1 + i ) n − 1
Substitute Values into Formula Substitute the given values into the formula: F V = 2200 × 0.0016666666666666668 ( 1 + 0.0016666666666666668 ) 48 − 1
Calculate Future Value Calculating the future value, we get: F V = 2200 × 0.0016666666666666668 ( 1.0016666666666668 ) 48 − 1 = 2200 × 0.0016666666666666668 1.10477276 − 1 = 2200 × 0.0016666666666666668 0.10477276 = 2200 × 62.86365591722139 = 109899.99999999999
Final Answer Rounding to the nearest cent, the future value of the ordinary annuity is $109,843.71.
Examples
Annuities are commonly used in retirement planning. For example, an individual might make regular payments into an annuity account over their working life. Upon retirement, the accumulated future value can then be used to provide a steady stream of income. Understanding how to calculate the future value of an annuity helps individuals plan and save effectively for their financial goals, ensuring they have sufficient funds available when they need them.
