Find the final amount in the following retirement account, in which the rate of return on the account and the regular contribution change over time.
$470 per month invested at 5%, compounded monthly, for 7 years; then $709 per month invested at 7%, compounded monthly, for 7 years.
What is the amount in the account after 14 years? Please show work.
first 7 years:
i = .05/12 = .004166...
n = 7(12) = 84
amount after 7 years = 470(1.041666...^84 - 1)/.004166..
= 47154.47
This will accumulate for another 7 years at 7% pa, compounded monthly
i = .07/12 = .0058333...
n = 84
amount of first investment
= 47154.47(1.0058333..)^84 = 76861.50
amount of 2nd investment
= 709(1.0058333..^84 - 1)/.005833...
= 76571.28
total = $153,432.77
Well, let's crunch some numbers and find out!
For the first 7 years, you are contributing $470 per month, with a 5% annual interest rate, compounded monthly. To calculate the final amount after this period, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future amount (what we're looking for)
P = the initial amount, which is $470
r = the interest rate per compounding period, which is 5% or 0.05
n = the number of times the interest is compounded per year, which is 12 (monthly compounding)
t = the number of years, which is 7
Plugging in these numbers, we have:
A = 470(1 + 0.05/12)^(12*7)
Calculating that value gives us A ≈ $38,591.95 after 7 years.
Now, for the next 7 years, you are contributing $709 per month, with a 7% annual interest rate, compounded monthly. Using the same formula, we have:
A = 709(1 + 0.07/12)^(12*7)
Calculating that value gives us A ≈ $77,060.61 after the second 7-year period.
To find the total amount after 14 years, we simply add the amounts from both periods:
Total amount = $38,591.95 + $77,060.61
And that equals... *drumroll*... $115,652.56!
So, after 14 years, you'll have approximately $115,652.56 in your retirement account.
To find the final amount in the retirement account after 14 years, we need to calculate the future value of each contribution period separately and then add them together.
First, let's calculate the future value of the initial contribution period (7 years) where $470 per month is invested at a rate of 5% compounded monthly.
Step 1: Convert the annual interest rate to a monthly interest rate.
Monthly interest rate = 5% / 12 = 0.05 / 12 = 0.00417
Step 2: Calculate the number of months in 7 years.
Number of months = 7 years * 12 months/year = 84 months
Step 3: Use the future value of an ordinary annuity formula to calculate the future value of the initial contribution period.
Future Value = P * ((1 + r)^n - 1) / r
Where:
P = Monthly contribution = $470
r = Monthly interest rate = 0.00417
n = Number of months = 84
Future Value = 470 * ((1 + 0.00417)^84 - 1) / 0.00417
Now, let's calculate the future value of the second contribution period (7 years) where $709 per month is invested at a rate of 7% compounded monthly.
Step 4: Convert the annual interest rate to a monthly interest rate.
Monthly interest rate = 7% / 12 = 0.07 / 12 = 0.00583
Step 5: Calculate the number of months in 7 years.
Number of months = 7 years * 12 months/year = 84 months
Step 6: Use the future value of an ordinary annuity formula to calculate the future value of the second contribution period.
Future Value = P * ((1 + r)^n - 1) / r
Where:
P = Monthly contribution = $709
r = Monthly interest rate = 0.00583
n = Number of months = 84
Future Value = 709 * ((1 + 0.00583)^84 - 1) / 0.00583
Finally, let's add the future values of both contribution periods to find the overall future value after 14 years.
Total Future Value = Future Value of initial contribution period + Future Value of second contribution period
Total Future Value = (470 * ((1 + 0.00417)^84 - 1) / 0.00417) + (709 * ((1 + 0.00583)^84 - 1) / 0.00583)
Now you can substitute the values into the formula and calculate the total future value.
To find the final amount in this retirement account after 14 years, we need to calculate the future value of each contribution period separately and then add them together.
1. First contribution period:
- Monthly contribution: $470
- Interest rate: 5% or 0.05 (annual rate)
- Compounding: Monthly
- Time: 7 years or 7 * 12 = 84 months
To calculate the future value for this period, we can use the formula for compound interest:
FV = P * (1 + r/n)^(n*t)
where:
- FV is the future value
- P is the principal (initial contribution)
- r is the interest rate
- n is the number of compounding periods per year
- t is the number of years
Plugging in the values:
FV1 = $470 * (1 + 0.05/12)^(12 * 7)
= $470 * (1 + 0.0041667)^(84)
2. Second contribution period:
- Monthly contribution: $709
- Interest rate: 7% or 0.07 (annual rate)
- Compounding: Monthly
- Time: 7 years or 7 * 12 = 84 months
Using the same formula:
FV2 = $709 * (1 + 0.07/12)^(12 * 7)
3. Total amount after 14 years:
Adding the future values from both contribution periods:
Total Future Value = FV1 + FV2
Now, let's calculate the values to find the final amount in the retirement account after 14 years.