Use the appropriate formula to find the value of the annuity and the interest.
| Periodic Deposit | Rate | Time |
| :--------------- | :--------------------------------- | :------ |
| $3000 | 5.25% compounded quarterly | 5 years |
a. The value of the annuity is $ [ ]
(Do not round until the final answer. Then round to the nearest dollar as needed.)
Answer (2)
Calculate the future value of the annuity using the formula: F V = P ⋅ r ( 1 + r ) n − 1 , where P = 3000 , r = 0.013125 , and n = 20 .
Substitute the values into the formula and calculate the future value: F V ≈ 68104.69 .
Round the future value to the nearest dollar: F V = $68105 .
Calculate the total deposits: T o t a l De p os i t s = 3000 ⋅ 20 = 60000 , and the interest earned: I n t eres t = 68104.69 − 60000 ≈ 8104.69 , which rounds to 8105 .
Explanation
Understanding the Problem We are given a problem about finding the future value of an annuity and the interest earned. The periodic deposit is $3000 at the end of every three months, the interest rate is 5.25% compounded quarterly, and the time is 5 years. We need to find the value of the annuity and the interest earned.
Identifying the Formula First, we need to calculate the future value of the annuity. 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
Identifying the Values Next, we identify the values for each variable:
P = $3000
The annual interest rate is 5.25% , so the quarterly interest rate is r = 4 5.25% = 4 0.0525 = 0.013125
The time is 5 years, so the number of quarters is n = 5 ⋅ 4 = 20
Calculating the Future Value Now, we substitute the values into the formula:
F V = 3000 ⋅ 0.013125 ( 1 + 0.013125 ) 20 − 1
Let's calculate ( 1 + 0.013125 ) 20 :
( 1 + 0.013125 ) 20 ≈ 1.2968563
Now, we calculate ( 1 + 0.013125 ) 20 − 1 :
1.2968563 − 1 = 0.2968563
Next, we calculate 0.013125 ( 1 + 0.013125 ) 20 − 1 :
0.013125 0.2968563 ≈ 22.6172
Finally, we calculate F V = 3000 ⋅ 0.013125 ( 1 + 0.013125 ) 20 − 1 :
F V = 3000 ⋅ 22.6172 ≈ 68104.69
Finding the Value of the Annuity The value of the annuity is approximately $68104.69 . Rounding to the nearest dollar, we get $68105 .
Calculating the Interest Earned Now, we need to calculate the interest earned. First, we calculate the total amount deposited:
T o t a l De p os i t s = P ⋅ n = 3000 ⋅ 20 = 60000
Then, we calculate the interest earned:
I n t eres t = F V − T o t a l De p os i t s = 68104.69 − 60000 = 8104.69
Rounding to the nearest dollar, the interest earned is $8105 .
Final Answer Therefore, the value of the annuity is $68105 and the interest earned is $8105 .
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 helps individuals make informed decisions about their retirement savings and investment strategies. It allows them to estimate the potential growth of their investments and plan accordingly to meet their financial goals during retirement.
The future value of the annuity after 5 years is approximately $67852, based on a periodic deposit of $3000 at a quarterly compounded interest rate of 5.25%. The total interest earned during this time is approximately $7852.
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