You are planning to save for retirement over the next 25 years. To do this, you will invest $820 per month in a stock account and $420 per month in a bond account. The return of the stock account is expected to be 10.2 percent, and the bond account will pay 6.2 percent. When you retire, you will combine your money into an account with a return of 7.2 percent. How much can you withdraw each month from your account assuming a 20-year withdrawal period

You will contribute 25 x 12 = 300 monthly payments to your savings accounts. In order to determine their future value, we must first determine the effective interest rates:

stock account = 1.102 = (1 + r)¹²

¹²√1.102 = ¹²√(1 + r)¹²
1.008127 = 1 + r
r = 0.008127 = 0.81% monthly rate

bond account = 1.102 = (1 + r)¹²

¹²√1.062 = ¹²√(1 + r)¹²
1.0050 = 1 + r
r = 0.005 = 0.5% monthly rate

In 25 years, you will have:

stock account = $820 x 1,265.21433 (PV annuity factor, 0.81%, 300 periods) = $1,037,475.75
bond account = $420 x 692.99396 (PV annuity factor, 0.5%, 300 periods) = $291,057.46
total = $1,328,533.21

using the payout annuity formula:

P₀ = [d (1 - (1 + r/x)⁻ⁿˣ)] / (r/x)

P₀ = $1,328,533.21
d = monthly withdrawal = ?
r = annual interest rate = 0.072
x = number of compounding periods = 12
n = number of years = 20

$1,328,533.21 = [d (1 - (1 + 0.072/12)⁻²⁴⁰)] / (0.072/12)

$7,971.20 = d (1 - 0.23795)

$7,971.20 = d (0.762)

d = $7,971.20 / 0.762 = $10,460