You want to be able to withdraw $20,000 from your account each year for 30 years after you retire. If you expect to retire in 25 years and your account earns 6.1% interest while saving for retirement and 5.7% interest while retired:
Round your answers to the nearest cent as needed.
a) How much will you need to have when you retire?
$
b) How much will you need to deposit each month until retirement to achieve your retirement goals?
$
c) How much did you deposit into you retirement account?
$
d) How much did you receive in payments during retirement?
$
e) How much of the money you received was interest?
$
To answer these questions, we need to use the concept of present value and future value. Present value is the current value of a future sum of money, while future value is the value of an investment at a specific point in the future. We also use the formula for calculating the future value of an annuity.
a) To determine how much you will need to have when you retire, we can calculate the future value of an annuity. The formula for future value of an annuity is:
FV = PMT * [(1 + r)^n - 1] / r
Where FV is the future value, PMT is the payment per period, r is the interest rate per period, and n is the number of periods.
In this case, PMT is $20,000 per year, r is 5.7% (0.057) per year, and n is 30 years.
Using these values, we can calculate the future value:
FV = $20,000 * [(1 + 0.057)^30 - 1] / 0.057 ≈ $821,293.18
Therefore, you will need to have approximately $821,293.18 when you retire.
b) To determine how much you need to deposit each month until retirement, we can calculate the monthly payment required to reach the future value. We can use the formula for calculating the present value of an annuity to find the monthly deposit amount.
PV = PMT * [1 - (1 + r)^(-n)] / r
Where PV is the present value, PMT is the payment per period, r is the interest rate per period, and n is the number of periods.
In this case, we know the future value is $821,293.18, r is 6.1% (0.061) per year, and n is 25 years.
Using these values, we can calculate the present value:
$821,293.18 = PMT * [1 - (1 + 0.061)^(-25)] / 0.061
Simplifying the equation and solving for PMT:
PMT ≈ $821,293.18 / [1 - (1 + 0.061)^(-25)] * 0.061 ≈ $326.55
Therefore, you will need to deposit approximately $326.55 per month until retirement to achieve your retirement goals.
c) To determine how much you deposited into your retirement account, we can multiply the monthly deposit amount by the number of months until retirement.
Deposit = PMT * (12 * n)
Where PMT is the monthly deposit amount and n is the number of years until retirement.
In this case, PMT is $326.55 per month and n is 25 years.
Using these values, we can calculate the deposit:
Deposit = $326.55 * (12 * 25) ≈ $97,965
Therefore, you deposited approximately $97,965 into your retirement account.
d) To determine how much you received in payments during retirement, we can multiply the annual withdrawal amount by the number of years in retirement.
Payment = Withdrawal * n
Where Withdrawal is the annual withdrawal amount and n is the number of years in retirement.
In this case, Withdrawal is $20,000 per year and n is 30 years.
Using these values, we can calculate the payment:
Payment = $20,000 * 30 = $600,000
Therefore, you received approximately $600,000 in payments during retirement.
e) To determine how much of the money you received was interest, we can subtract the total deposit amount from the total payment amount.
Interest = Payment - Deposit
In this case, the payment is $600,000 and the deposit is $97,965.
Using these values, we can calculate the interest:
Interest = $600,000 - $97,965 ≈ $502,035
Therefore, approximately $502,035 of the money you received was interest.