As soon as she graduated from college, kay began planning for her retirement. her plans were to deposit $500 semiannually into an ira (a retirement fund) beginning six months after graduation and continuing until the day she retired, which she expected to be 30 years later. today is the day kay retires. she just made the last $500 deposit into her retirement fund, and now she wants to know how much she has accumulated for her retirement. the fund has earned 10 percent compounded semiannually since it was established. a. compute the balance of the retirement fund assuming all the payments were made on time. b. although kay was able to make all the $500 deposits she planned, 10 years ago she had to withdraw $10,000 from the fund to pay some medical bills incurred by her mother. compute the balance in the retirement fund based on this information.

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

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