Question $3$: Time Value of Money and application $(35$ marks$)$

a$)$ Assume you are $35$ years old today and are considering your retirement needs. You expect to retire at age $65($in $30$ years$)$ and you plan to live to age $99.$ You want to buy a house costing $$300,000$ on your $65$th birthday and your living expenses will be $$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 interest rate of $8\%,$ annual compounding:

$($i$)$ How much will you need to have saved by your retirement date to be able to afford this retirement plan? $(5$ marks$)$

$($ii$)$ Suppose you already have $$50,000$ in savings today. If you can invest money at $8\%$ a year annual compounding, how much would you need to save at the end of each year for the next $30$ years to be able to afford this retirement plan? $(5$ 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? $(8$ marks$)$

$($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? $(2$ marks$)(10$ marks$)$

c$)$ Assume your child has just been born and you are planning for his college education. Based on your wonderful experience in Finance at Carleton, you decide to send him to Carleton University. You anticipate the annual tuition at that time to be $$50,000$ per year for the four years of university. You plan on making equal deposits on your child's birthdays for age one through seventeen inclusive to fund his education. Assume the first tuition payment is due exactly $18$ years from today and the expected return is $10\%$ APR with quarterly compounding over this period. Calculate the annual deposit. $(7$ marks$)$

d$)$ Suppose you are analyzing the following possible cash flows: Year $1$ CF $=$ $$100$; Years $2$ and $3$ CFs $=$ $$200$; Years $4$ and $5$ CFs $=$ $$300.$ The required discount rate is $7\%.$

i$)$ What is the value of the cash flows at year $5?$

ii$)$ What is the value of the cash flows today?

iii$)$ What is the value of the cash flows at year $3?$