a. The balance in the retirement fund assuming all the payments were made on time is $1,76,791.86
We use the Future Value of an annuity formula to arrive at the answer.
Future Value of an Annuity = P * [ { (1+r)ⁿ -1 } / r]
Since interest is compounded semi-annually, an investment will earn interest twice a year at half the annual interest rate. Hence n = 60 (30×2) and r = 0.05 (10% ÷ 2). P is taken as $500.
b. The balance in the retirement fund after taking into account the $10,000 withdrawal is $1,50,258.88.
For this part of the question, we first calculate the balance at the end of 20 years. In the formula above, we use r = 0.05, n= 40 and P =50.
This gives us a balance of $60,399.89. After the withdrawal, the amount lying in the account is $50,399.89 (
$60,399.89 -$10,000).
This amount will continue to earn interest semiannually for the next 10 years. We will then calculate the Future Value of a lump sum ($50,399.89) using the formula :
Future Value = P × (1+r)ⁿ
By pugging in r = 0.05, n= 20 and P = $50,399.89, we arrive at the value of the lump sum as of today.
FV ( $50,399.89)₀.₀₅,₂₀ = $1,33,725.90
(A)
Finally we need to find the Future Value of all the deposits in the final 10 years by plugging in P = $500, r = 0.05 and n = 20 in the Future Value of an annuity formula.
Future Value of annuity = $16,532.98
(B)
Total Value in the retirement account = $ 1,33,725.90 + $ 16,532.98
= $ 1,50,258.88
