Find the amount of the annuity if the deposit is $800 quarterly for 3 years at 3% compounded quarterly. Amount = $

Question

GinnyAnswer · Answer

Calculate the quarterly interest rate: i = 4 0.03 ​ = 0.0075 . 
 Calculate the total number of compounding periods: N = 4 × 3 = 12 . 
 Apply the future value of an ordinary annuity formula: F V = 800 × 0.0075 (( 1 + 0.0075 ) 12 − 1 ) ​ . 
 The amount of the annuity after 3 years is approximately $10006.07 ​ . 
 
 Explanation 
 
 Understanding the Problem We are asked to find the future value of an annuity where regular deposits are made. Let's break down the given information:

The deposit amount (PMT) is $800 per quarter. 
 The time period (t) is 3 years. 
 The annual interest rate (r) is 3%, compounded quarterly. 
 
 Our goal is to calculate the total amount accumulated after 3 years, considering the quarterly compounding. 
 
 Calculating Quarterly Interest Rate and Total Compounding Periods To find the future value, we'll use the future value of an ordinary annuity formula. First, we need to determine the quarterly interest rate and the total number of compounding periods. 
 
 Quarterly interest rate (i): The annual interest rate is 3%, so the quarterly interest rate is: i = n r ​ = 4 0.03 ​ = 0.0075 
 
 Total number of compounding periods (N): The annuity runs for 3 years, and interest is compounded quarterly, so the total number of compounding periods is: N = n × t = 4 × 3 = 12 
 
 Applying the Future Value Formula Now that we have the quarterly interest rate and the total number of compounding periods, we can use the future value of an ordinary annuity formula:

F V = PMT × i (( 1 + i ) N − 1 ) ​ 
 Where: 
 
 FV is the future value of the annuity 
 PMT is the periodic payment ($800) 
 i is the interest rate per period (0.0075) 
 N is the number of periods (12) 
 
 Substituting the values, we get: 
 F V = 800 × 0.0075 (( 1 + 0.0075 ) 12 − 1 ) ​ 
 F V = 800 × 0.0075 (( 1.0075 ) 12 − 1 ) ​ 
 F V = 800 × 0.0075 ( 1.093806898 − 1 ) ​ 
 F V = 800 × 0.0075 0.093806898 ​ 
 F V = 800 × 12.50758641 
 F V = 10006.06913 
 
 Final Answer Therefore, the amount of the annuity after 3 years is approximately $10,006.07. 
 
 Amount = $$10006.07 
 Examples 
 Annuities are commonly used in retirement planning. For example, an individual might make regular contributions to an annuity account, which grows over time due to compound interest. The future value of the annuity helps the individual estimate the total amount they will have saved by the time they retire. This calculation is essential for making informed decisions about retirement savings and financial planning, ensuring a comfortable and secure future.

Find the amount of the annuity if the deposit is $800 quarterly for 3 years at 3% compounded quarterly. Amount = $

Answer (1)

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