Question $6[5$ marks $-$free credit$$ given$]$

Suppose that there is a cash$-$generating machine that produces a cash flow every year that grows at a rate of $3\%$ each year. You are considering purchasing this machine today. The machine has just generated a cash flow of $$10$ yesterday, which you just missed, and will generate another one in one year from now. The discount rate applicable to this machine is $13\%$ per year. If you wanted to buy this machine now, how much would you be willing to pay for? $($Lecture notes pp$\mathrm{.}22-23)$

Answer $($show the steps$/$calculation toward your results$)$:

Draw the timeline:

t $=0123...$ t $=$ N $...$

$-10*(1\mathrm{.}03)=10\mathrm{.}310\mathrm{.}3(1\mathrm{.}03)10\mathrm{.}3(1\mathrm{.}03)210\mathrm{.}3(1\mathrm{.}03)$N$-1...$

This is a growing perpetuity with D$\_1=$$$10\mathrm{.}3.$ Note that CF at t $=0$ is $$10,$ and this will have grown to $$10\mathrm{.}3$ by t$=1,$ which is the initial CF for the perpetual stream. $$10\mathrm{.}3$ will then grow further at a rate of $3\%$ each year indefinitely.

Want to find the pv of a growing perpetuity with the following inputs: D$\_1=10\mathrm{.}3$;r$=13\%$;g$=3\%$

PV$=$C$/($r$-$g$)=10\mathrm{.}3/(\mathrm{.}13-\mathrm{.}03)=$$$103$

Question $7[5$ marks$]$

Suppose that the cash$-$generating machine described in Question $6$ has stopped for improvement. It will resume generating cash flows in two years from now, with an initial cash flow of $$10.$ Cash flows will then grow at a rate of $6\%($improved$!)$ each year thereafter. The discount rate is $13\%$ per year. You are still interested in purchasing this machine, and the current owner of the machine asks you to pay today$$i$.$e$.,$ two years before it resumes its production$$if you want to buy it$.$ At what price should the machine be bought? $($Lecture notes pp$\mathrm{.}22-23$ and pp$\mathrm{.}19)$

Answer $($show the steps$/$calculation toward your results$)$: