You are planning to save for retirement over the next 35 years. To do this, you will invest $770 a month in a stock account and $370 a month in a bond account. The return of the stock account is expected to be 9.7 percent, and the bond account will pay 5.7 percent. When you retire, you will combine your money into an account with a 6.7 percent return.How much can you withdraw each month from your account assuming a 30-year withdrawal period?
Assuming the rates are annual rates compounded monthly
amount of stock account = 770(1.0080833..^420 - 1)/.00808333.. = 2706320.70
amount from bonds = 370(1.00475^420 - 1)/.00475 = 492104.61
total available = 3198425.31
This will become the Present Value of the annuity
let the annuity payment be x
3198425.31 = x(1 - 1.00558333..^-360)/.00558333
x = 20638.73
check my arithmetic
To calculate how much you can withdraw each month from your retirement account, we need to follow these steps:
Step 1: Calculate the total amount accumulated in the stock account after 35 years.
The monthly investment in the stock account is $770, and the annual return is 9.7%. Since the investment is made monthly, we need to calculate the future value using monthly compounding.
n = number of years * 12 (since the investment is made monthly)
r = annual return / 12 (monthly return)
PMT = monthly investment in stock account = $770
Using the formula for the future value of an ordinary annuity:
FV_stock = PMT * ((1 + r)^n - 1) / r
n = 35 years * 12 months = 420 months
r = 9.7% / 12 = 0.00808333
FV_stock = $770 * ((1 + 0.00808333)^420 - 1) / 0.00808333
Step 2: Calculate the total amount accumulated in the bond account after 35 years.
The monthly investment in the bond account is $370, and the annual return is 5.7%. Using the same formula as before:
FV_bond = PMT * ((1 + r)^n - 1) / r
PMT = $370
n = 35 years * 12 months = 420 months
r = 5.7% / 12 = 0.00475
FV_bond = $370 * ((1 + 0.00475)^420 - 1) / 0.00475
Step 3: Combine the accumulated amounts in the stock and bond accounts.
Total accumulated amount = FV_stock + FV_bond
Step 4: Calculate the withdrawal amount each month for 30 years.
The withdrawal amount can be calculated using the formula for the present value of an ordinary annuity:
PMT = withdrawal amount each month
n = withdrawal period in years * 12 (since the withdrawal is made monthly)
r = annual return on the combined account / 12 (monthly return)
FV = total accumulated amount
PMT * ((1 - (1 + r)^-n) / r) = FV
PMT = FV * r / (1 - (1 + r)^-n)
n = 30 years * 12 months = 360 months
r = 6.7% / 12 = 0.00558333
PMT = (FV * r) / (1 - (1 + r)^-n)
Now, let's plug in the values and calculate the final result.
To calculate the amount that you can withdraw each month from your retirement account, we need to use the concept of present value and annuities.
Step 1: Calculate the future value of your contributions in the stock account over 35 years:
We can use the formula for the future value of an ordinary annuity to calculate the future value of your monthly contributions to the stock account:
FV_stock = P_stock * ((1 + r_stock)^n_stock - 1) / r_stock
Where:
P_stock = Monthly contribution to the stock account = $770
r_stock = Interest rate of the stock account = 9.7% = 0.097
n_stock = Number of contributions = 35 years * 12 months = 420
FV_stock = $770 * ((1 + 0.097)^420 - 1) / 0.097
Step 2: Calculate the future value of your contributions in the bond account over 35 years:
Similarly, we can calculate the future value of your monthly contributions to the bond account using the formula for the future value of an ordinary annuity:
FV_bond = P_bond * ((1 + r_bond)^n_bond - 1) / r_bond
Where:
P_bond = Monthly contribution to the bond account = $370
r_bond = Interest rate of the bond account = 5.7% = 0.057
n_bond = Number of contributions = 35 years * 12 months = 420
FV_bond = $370 * ((1 + 0.057)^420 - 1) / 0.057
Step 3: Calculate the total future value of your combined contributions:
To find the total future value of your combined contributions in both accounts, simply add the future values calculated in step 1 and step 2:
FV_total = FV_stock + FV_bond
Step 4: Calculate the monthly withdrawals from the combined account during the withdrawal period:
To determine the amount you can withdraw each month during the 30-year withdrawal period, we need to calculate the monthly withdrawals using the present value of an annuity formula:
Withdrawal = FV_total * r_withdrawal / ((1 + r_withdrawal)^n_withdrawal - 1)
Where:
r_withdrawal = Interest rate during the withdrawal period = 6.7% = 0.067
n_withdrawal = Number of withdrawals = 30 years * 12 months = 360
Withdrawal = FV_total * 0.067 / ((1 + 0.067)^360 - 1)
Now you can substitute the values into the formula and calculate the monthly withdrawal amount.