a. Use the appropriate formula to find the value of the annuity.
b. Find the interest.
| Periodic Deposit | Rate | Time |
|---|---|---|
|$1000 at the end of every three months | $6.25 \%$ compounded quarterly | 6 years |
(i) Click the icon to view some finance formulas.
a. The value of the annuity is $29,289.
$\square$
(Do not round until the final answer. Then round to the nearest dollar as needed.)
Answer (1)
Calculate the future value of the annuity using the formula: F V = P ⋅ r ( 1 + r ) n − 1 , where P = 1000 , r = 4 0.0625 = 0.015625 , and n = 6 ⋅ 4 = 24 .
Substitute the values into the formula: F V = 1000 ⋅ 0.015625 ( 1 + 0.015625 ) 24 − 1 ≈ 27488 .
Calculate the total deposits: T o t a l = 1000 ⋅ 24 = 24000 .
Calculate the interest earned: I n t eres t = 27488 − 24000 = 3488 . The value of the annuity is 27488 and the interest earned is 3488 .
Explanation
Problem Overview We are given the periodic deposit, the interest rate, and the time period for an annuity. Our goal is to find the value of the annuity and the interest earned.
Formula Introduction The formula for the future value of an ordinary annuity is: F V = P ⋅ r ( 1 + r ) n − 1 where:
F V is the future value of the annuity
P is the periodic deposit
r is the interest rate per period
n is the number of periods
Given Values We are given:
Periodic Deposit, P = $1000
Annual interest rate = 6.25% = 0.0625
Time = 6 years
Compounded quarterly, so the number of compounding periods per year is 4.
Calculating r and n First, we calculate the interest rate per period: r = 4 0.0625 = 0.015625 Next, we calculate the number of periods: n = 6 ⋅ 4 = 24
Calculating Future Value Now, we substitute the values of P , r , and n into the future value formula: F V = 1000 ⋅ 0.015625 ( 1 + 0.015625 ) 24 − 1 F V = 1000 ⋅ 0.015625 ( 1.015625 ) 24 − 1 F V ≈ 1000 ⋅ 0.015625 1.4295 − 1 F V ≈ 1000 ⋅ 0.015625 0.4295 F V ≈ 1000 ⋅ 27.488 F V ≈ 27488 So, the value of the annuity is approximately $27 , 488 .
Calculating Total Deposits Next, we calculate the total amount of deposits: Total Deposits = Periodic Deposit * Number of periods Total Deposits = $1000 ∗ 24 = $24000
Calculating Interest Earned Now, we calculate the interest earned: Interest = Future Value - Total Deposits Interest = $27488 − $24000 = $3488
Final Answer a. The value of the annuity is $27 , 488 .
b. The interest earned is $3 , 488 .
Examples
Annuities are commonly used in retirement planning. For example, an individual might make regular deposits into an annuity account over their working life. Upon retirement, the accumulated value of the annuity can then be used to provide a steady stream of income. Understanding how to calculate the future value of an annuity and the interest earned is crucial for making informed decisions about retirement savings and investment strategies. This knowledge helps individuals plan effectively for their financial future and ensure a comfortable retirement. Annuities can also be used for other long-term savings goals, such as funding a child's education or purchasing a home.
