You are planning to save for retirement over the next 25 years. To do this, you will invest $880 per month in a stock account and $480 per month in a bond account. The return of the stock account is expected to be an APR of 10.8 percent, and the bond account will earn an APR of 6.8 percent. When you retire, you will combine your money into an account with an APR of 7.8 percent. All interest rates are compounded monthly. How much can you withdraw each month from your account assuming a withdrawal period of 20 years

$14,143.86 can be withdrawn each month from the account for 20 years.

Explanation:

To determine this, the first step is to use the formula for calculating the future value (FV) of ordinary annuity to calculate the FV of both stock and bond as follows:

Calculation of Future Value of Stock

FVs = M × {[(1 + r)^n - 1] ÷ r} (1)

Where,

FVs = Future value of the amount invested in stock after 25 years =?

M = Monthly investment = $880

r = Monthly interest rate = 10.8% ÷ 12 = 0.9%, or 0.009

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

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

FVs = $880 × {[(1 + 0.009)^360 - 1] ÷ 0.009}

FVs = $880 × 1,522.3445923122

FVs = $1,339,663.24

Calculation of Future Value of Bond

FVd = M × {[(1 + r)^n - 1] ÷ r} (1)

Where,

FVd = Future value of the amount invested in bond after 25 years =?

M = Monthly investment = $480

r = Monthly interest rate = 6.8% ÷ 12 = 0.566666666666667%, or 0.00566666666666667

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

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

FVd = $480 × {[(1 + 0.00566666666666667)^300 - 1] ÷ 0.00566666666666667}

FVd = $480 × 784.895879465925

FVd = $376,750.02

Calculation of the amount that can be withdrawn monthly for 20 years

To calculate this, the formula for calculating the present value of an ordinary annuity is used as follows:

PV = P × [{1 - [1 ÷ (1+r)]^n} ÷ r] …………………………………. (3)

Where;

PV = Combined present values of stock and bond investments after retirement = FVs + FVb = $1,339,663.24 + $376,750.02 = $1,716,413.26

P = Monthly withdrawal = ?

r = Monthly interest rate = 7.8% ÷ 12 = 0.65%, or 0.0065

n = number of months = 20 years * 12 months = 240

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

PV = P × [{1 - [1 ÷ (1+r)]^n} ÷ r]

$1,716,413.26 = P × [{1 - [1 ÷ (1 + 0.0065)]^240} ÷ 0.0065]

$1,716,413.26 = P × 121.353915567094

P = $1,716,413.26 / 121.353915567094

P = $14,143.86

Therefore, $14,143.86 can be withdrawn each month from the account for 20 years.