Question

John is going to establish a University Fund for his daughter Mary, who has just been born. He plans to make the first deposit of $20,000 today and then annual deposits of $5,000 will be made until Mary’s 17th birthday. Given the long term nature of the investment, John anticipates a 5% pa return. The money is transferred to an account for Mary on her 17th birthday and she will then withdraw the money in equal annual amounts for 5 years starting on her 18th birthday. Mary will only be able to earn 3% pa on her money. (i) How much money will be available on Mary’s 17th birthday? (ii) How much will Mary be able to spend each year? (Your answers should be accurate to the nearest dollar)

Answer 1

The fund available on Mary's 17th birthday is calculated using compound interest for the initial deposit and the future value of an annuity for subsequent deposits, both at a 5% annual return. The amount Mary can spend each year for five years is calculated using the present value of an annuity formula, considering a 3% annual return.

Step-by-step explanation:

To calculate the university fund available on Mary's 17th birthday, we will use the formula for compound interest, considering John's initial deposit and his annual deposits of $5,000 at a 5% per annum return rate. The initial $20,000 will compound for 17 years, and each subsequent $5,000 deposit will compound starting from the year it was deposited until the 17th year. The compound interest formula is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.

To find the total available funds:

1. First deposit: A = $20,000(1 + 0.05)^17 = $20,000(1.05)^17
2. Subsequent deposits (17 in total), we'll use the future value of an ordinary annuity formula: FV = PMT[((1 + r)^t - 1) / r], where PMT is the annual deposit.

The total will be the sum of the compounded initial deposit and the future value of the annuity.

To find the amount Mary can spend each year, we will assume the full amount will be withdrawn in equal parts over 5 years at a 3% return. We'll use the formula for the present value of an annuity: PV = PMT[1 - (1 + r)^-t] / r to calculate what amount that can be withdrawn annually.