Question

You are planning to save for retirement over the next 40 years. To do this, you plan to invest $5,000 a year in a stock account at the beginning of each year. In 10 years, you plan to diversify your portfolio by investing $3,000 a year in a bond account. The return of the stock account is expected to be 8 percent, and the bond account will pay 4 percent. When you retire, you will combine your funds into an account with a 7 percent return.

1. Estimate the value of your portfolio at the end of your saving period (in 40 years).  
2. How much can you withdraw each year from your account, assuming a 30-year withdrawal period?  

Answer 1

(a) Amount invested in stock account = P₁ = 5000

Number of Investment Periods = n₁ = 40

Return = r₁ = 8%

Value of stock account after 40 years = P₁(1+r₁)^n₁ +....+ P₁(1+r₁)² + P₁(1+r₁)
= P₁ [((1 + r₁)^n₁ - 1) / r₁])(1 + r₁)
= 5000 [((1 + 0.08)⁴⁰ - 1) / 0.08])(1 + 0.08)
= $1398905.20

Amount invested in Bond account = P₂ = 3000

Number of Investment Periods = n₂ = 30

Return = r₂ = 4%

Value of bond account after 30 years = P₂(1+r₂)^n₂ +....+ P₂(1+r₂)² + P₂(1+r₂)
= P₂ [((1 + r₂)^n₂ - 1) / r₂])(1 + r₂)
= 3000 [((1 + 0.04)³⁰ - 1) / 0.04])(1 + 0.04)
= $174985.00

Total Value in account after 40 years = 1398905.20 + 174985.00 = $1573890.2

(b) Let the amount withdrawn each year be X

Number of periods = t = 30

Interest Rate = i = 7%

Sum of Present Value of the future withdrawals = Value in account now

=> X/(1+i) + X/(1+i)² +....+ X/(1+i)^t = 1573890.2

=> X[1- (1+i)^-t]/i = 1573890.2

=> X[1- (1+0.07)^-30]/0.07 = 1573890.2

=> 12.409X = 1573890.2

=> X = $126834.57

Hence, $126834.57 can be withdrawn each year for next 30 years