Question

-/0.5 points HarMathAp12 6.4.043. My Notes Ask Your Teacher 11 A young couple wants to have a college fund that will pay $20,000 at the end of each half-year for 8 years (a) If they can invest at 9%, compounded semiannually, how much do they need to invest at the end of each 6-month period for the next 18 years to begin making their college withdrawals 6 months after their last investment? (Round your answer to the nearest cent.) $ (b) Suppose 8 years after beginning the annuity payments, they receive an inheritance of $36,000 that they contribute to the account, and they continue to make their regular payments as found in part (a). How many college withdrawals will they be able to make before the account balance is $07 (Round your answer to the nearest whole number) withdrawals Noed Help? ed Talk to a Tuter

Answer 1

(a) Required Semi-Annual Withdrawals = $ 20000 and Withdrawal Tenure = 8 years or (8 x 2) = 16 half-years

Deposit Rates = 9%, Compounding Frequency: Semi-Annual, Deposit Tenure = 18 years or (18 x 2) = 36 half-years

Applicable Half-Yearly Rate = 9/2 = 4.5 %

Let the required semi-annual deposits be $ k

Now, as per the laws of the time value of money, Total Future Value of Semi-Annual Deposits at the end of Year 18 = Total Present Value of College Withdrawals at the end of Year 18

PV of Withdrawals = 20000 x (1/0.045) x [1-{1/(1.045)^(16)}] = $ 224680.301

Future Value of Deposits = k x (1.045)^(35) + k x (1.045)^(34) +.............+ k = 224680.301

k x 86.163966 = 224680.301

k = $ 2607.59

(b) The depositors receive an inheritance worth $ 36000 at the end of Year 8 which implies that this inheritance along with the withdrawals are compounded for another 10 years or 20 half-years.

Total Future Value of Deposits = $ 224680.301 and Future Value of Inheritance = 36000 x (1.045)^(20) = $ 86821.705

Total Future Value at the end of Year 18 = 86821.705 + 224680.301 = $ 311502.0059

Let the possible number of withdrawals be N

Therefore, 311502.0059 = 20000 x (1/0.045) x [1-{1/(1.045)^(N)}]

0.700879 = 1-{1/(1.045)^(N)}

0.29912 = 1/(1.045)^(N)

(1.045)^(N) = 3.343134

N = = 27.419 or 27.42 ~ 27 half-yearly withdrawals before the account reaches $ 0.

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