You are planning to save for retirement over the next 30 years. To do this, you will invest $780 per month in a stock account and $380 per month in a bond account. The return of the stock account is expected to be 9.8 percent, and the bond account will earn 5.8 percent. When you retire, you will combine your money into an account with an annual return of 6.8 percent. Assume the returns are expressed as APRs. How much can you withdraw each month from your account assuming a 25-year withdrawal period

The total amount saved for 30 years after retirement can be estimated by employing the formula for calculating the future value (FV) of ordinary annuity for both stock and bond as follows:

Future Value of Stock

FVs = M * (((1 + r)^n - 1) / r) ................................. (1)

Where,

FVs = Future value of the amount invested in stock after 30 years =?

M = Monthly investment = $780

r = Monthly return rate = 9.8% / 12 = 0.098 / 12 = 0.00816666666666667

n = number of months = 30 years * 12 months = 360

Substituting the values into equation (1), we have:

FVs = $780 * (((1 + 0.00816666666666667)^360 - 1) / 0.00816666666666667)

FVs = $780 * 2,166.28572458476

FVs = $1,689,702.87

Future Value of Bond

FVb = M * (((1 + r)^n - 1) / r) ................................. (2)

Where,

FVb = Future value of the amount invested in bond after 30 years =?

M = Monthly investment = $380

r = Monthly interest rate = 5.8% / 12 = 0.058 / 12 = 0.00483333333333333

n = number of months = 30 years * 12 months = 360

Substituting the values into equation (2), we have:

FVb = $380 * (((1 + 0.00483333333333333)^360 - 1) / 0.00483333333333333)

FVb = $380 * 966.933721691683

FVb = $367,434.81

Calculation of the amount that can be withdrawn monthly for 25-year withdrawal period

This can be calculated by employing the formula for calculating the present value of an ordinary annuity as follows:

PV = P * ((1 - (1 / (1 + r))^n) / r) …………………………………. (3)

Where;

PV = Sum of present values of stock and bond investments after retirement = FVs + FVb = $1,689,702.87 + $367,434.81 = $2,057,137.68

P = Monthly withdrawal = ?

r = Monthly interest rate = APR / 12 = 6.8% ÷ 12 = 0.068 / 12 = 0.00566666666666667

n = number of months = 25 years * 12 months = 300

Substitute the values into equation (3) and solve for P, we have:

$2,057,137.68 = P * ((1 - (1 / (1 + 0.00566666666666667))^300) / 0.00566666666666667)

$2,057,137.68 = P * 144.077250670093

P = $2,057,137.68 / 144.077250670093

P = $14,278.02

Therefore, the amount that can be withdrawn each month from your account assuming a 25-year withdrawal period is $14,278.02.