a$)$ Your $35-$year old uncle is considering his retirement needs. He expects to retire at age $65($in $30$

years$)$ and plans to live to age $99.$ He wants to buy a house costing $$300,000$ on his $65$th birthday

and his living expenses will be $$30,000$ a year after that $($starting at the end of year $65$ and

continuing through the end of year $99,$ i$.$e$.$ for $35$ years$).$ Assume an annual interest rate of $8\%,$

annual compounding:

i$)$ How much will he need to have saved by his retirement date to be able to afford

this plan?

ii$)$ Suppose he already has $$50,000$ in savings today. If he can invest money at $8\%$ a

year, how much would he need to save at the end of each year for the next $30$

years to be able to afford this retirement plan?

$(5+10=15$ marks$)$

b$)$ You have been hired to run a pension fund for Mackay Inc, a small manufacturing firm. The

firm currently has $$5$ million in the fund and expects to have cash inflows $($receipts$)$ of $$2$

million a year for the first $5$ years followed by cash outflows $($payments$)$ of $$3$ million a year

for the next $5$ years. Assume that interest rates are at $8\%.$

$($i$)$ How much money will be left in the fund at the end of the tenth year?

$($ii$)$ If you were required to pay a perpetuity after the tenth year $($starting in year $11$ and

going through infinity$)$ out of the balance left in the pension fund, how much could you

afford to pay every year?