You are considering investing in a scholarship fund for your daughter. The fund operates in the following way:

Starting in one month's time you would make regular monthly deposits of $50 for a period of eight years into a scholarship fund. Immediately after

your final deposit the fund would pay $10,000 to your daughter to help her with expenses when she commences university study.

a) Illustrate this scenario as a fully labelled time line diagram.

[2 marks]

To be able to compare this fund with other possible investments, you need to know the effective interest rate, j12, that this fund returns.

b) Derive and present the equation of equivalence that is appropriate for this scenario. Do NOT seek to simplify the expression, but rather leave $10,000

as the value of the left hand side (LHS). [Hint: This is a "Price is Right" problem but with a FV.]

[4 marks]

c) Using an estimate of i=1%, determine the value of the right hand side (RHS) of the equation of equivalence.

[2 marks]

d) Present you next three estimates of i and the corresponding value of the right hand side of the equation of equivalence. Ensure that for each of these

three steps you explain why you increased or decreased the value of i.

[6 marks]

e) Determine a value of i, for which the LHS and RHS of the equation of equivalence vary by no more than $1.00. NB You do not need to show any

working for this question part, but merely present your answer, and the values of the RHS and LHS.

[2 marks]

f) Thus determine the effective interest rate, j12, that this scholarship returns.

[2 marks]

Question 2 [18 marks]

A retiree deposits $500,000 in a bank account which pay interest j12 = 5.4% p.a.

Starting in a month's time, they intend to withdraw $4000 to cover their living expenses until run out of money. [NB It is likely that their final withdrawal will

be a partial withdrawal].

a) Illustrate this scenario as a fully labelled time line diagram.

[2 marks]

b) Determine how many full sized withdrawals they can mark.

[2 marks]

c) Determine the size of the final partial withdrawal made one month after the last full sized withdrawal.

[4 marks]

d) Construct an amortization schedule showing the last three withdrawals (two full and one partial withdrawals. Ensure that you show how you found

the outstanding principal. Ensure that you have shown example calculations for one line of the table. {NB this scenario is essentially the same as a fully

amortized loan repayment, except the person and the institution have swapped roles.)

[4 marks]

e) After exactly one year, they had to have hip replacement surgery, and had to withdraw an extra $15000 from the account to cover the cost. That is,

their 12th withdrawal was $19000 rather than the usual $4000. Determine how many more full sized withdrawals they can make after this.

[6 marks]

Question 3: [18 marks]

You wish to save $2000 to go on a holiday when you finish university in 2.5 year's time. by making regular deposits of the same size into a

savings account that pays interest j52 = 4.68% p.a., starting in one week's time.

a) Illustrate this scenario as a fully labelled time line diagram.

[2 marks]

b) Determine the deposit size required.

[2 marks]

c) Explain and perform a sanity check on your answer to (b)

[1+1=2 marks]

d) Immediately after your 50th deposit, the interest rate increases to j52=4.78 % p.a. Draw an appropriate time line of the scenario and then determine

what deposit size will now be needed for deposits 51 to 130 in order to still meet your original target of $2000.

[2+6=8 marks]

e) Determine how much will be accumulated at the end of the savings scheme if you continue with the old weekly deposit size after the interest rate

changes.

[4 marks]