Luis has $100,000 in his retirement account at his present company. Because he is assuming a position with another company, Luis is planning to "roll over" his assets to a new account. Luis also plans to put $2000/quarter into the new account until his retirement 25 years from now. If the new account earns interest at the rate of 5.5%/year compounded quarterly, how much will Luis have in his account at the time of his retirement? Hint: Use the compound interest formula and the annuity formula. (Round your answer to the nearest cent.)
To find out how much Luis will have in his account at the time of his retirement, we can break it down into two parts: the amount he already has ($100,000) and the amount he will contribute over the next 25 years.
1. Calculating the future value of the initial amount:
Let's use the compound interest formula:
FV = P(1 + r/n)^(nt)
Where:
FV = future value
P = principal amount (initial amount Luis has)
r = annual interest rate (5.5%)
n = number of times the interest is compounded per year (quarterly compounded means n = 4)
t = number of years (25)
Using the formula:
FV = $100,000(1 + 0.055/4)^(4*25)
Calculating this value, we get:
FV = $100,000(1 + 0.01375)^(100)
FV = $100,000(1.01375)^(100)
FV ≈ $100,000(2.864732654)
FV ≈ $286,473.27
So, the initial amount Luis has will grow to approximately $286,473.27 after 25 years.
2. Calculating the future value of the regular contributions:
By using the annuity formula:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
Where:
FV = future value (amount Luis will contribute)
P = periodic payment (amount Luis contributes per quarter, $2,000)
r = annual interest rate (5.5%)
n = number of times the interest is compounded per year (quarterly compounded means n = 4)
t = number of years (25)
Using the formula:
FV = $2,000 * [(1 + 0.055/4)^(4*25) - 1] / (0.055/4)
Calculating this value, we get:
FV = $2,000 * [(1 + 0.01375)^(100) - 1] / (0.01375)
FV = $2,000 * [(1.01375)^(100) - 1] / (0.01375)
FV ≈ $2,000 * (2.864732654 - 1) / 0.01375
FV ≈ $2,000 * 1.864732654 / 0.01375
FV ≈ $286,472.13
So, the regular contributions Luis makes over 25 years will grow to approximately $286,472.13.
3. Calculating the total future value:
To find the total future value, we need to add the future values of the initial amount and the regular contributions:
Total FV = Initial FV + Regular Contributions FV
Total FV = $286,473.27 + $286,472.13
Total FV ≈ $572,945.40
Therefore, Luis will have approximately $572,945.40 in his retirement account at the time of his retirement.
To find the amount Luis will have in his retirement account at the time of his retirement, we can break down the problem into two parts: the initial amount already in the account and the quarterly contributions he plans to make.
1. Initial Amount:
Luis currently has $100,000 in his retirement account. We don't need to calculate anything for this part since it's already given.
2. Quarterly Contributions:
Luis plans to contribute $2,000 every quarter for the next 25 years. To calculate the future value of this annuity, we can use the annuity formula:
Future Value = Payment × [(1 + r)^n - 1] / r
Where:
Payment = $2,000 (quarterly contribution)
r = interest rate per quarter = 5.5% / 4 (since it's compounded quarterly)
n = total number of quarters = 25 years × 4 quarters/year = 100 quarters
Now, let's calculate the future value of the annuity:
Future Value = $2,000 × [(1 + 0.055 / 4)^100 - 1] / (0.055 / 4)
Calculating this expression will give us the future value of the annuity contributions.
Finally, to find the total amount in the account at the time of Luis's retirement, we need to add the initial amount and the future value of the annuity contributions.
Total Amount = Initial Amount + Future Value of Annuity Contributions
Calculating this expression will give us the final result, which is the amount Luis will have in his account at the time of his retirement.
The question said:
"Hint: Use the compound interest formula and the annuity formula. "
and .... ? , did you do that?
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