$(60$ points$)$ Consider a project of manufacturing and selling high$-$end

smartwatches to wealthy customers. The length of the project is $10$ years. The

industry demand for high$-$end smartwatches next year will be $1$ million units, and

every year it will either increase by $3\%($with $50\%$ probability$)$ or decrease by $3\%$

$($with $50\%$ probability$).$ Your market share next year will be $5\%,$ and every year it will

either increase by $1\%-$point $($with $50\%$ probability$)$ or decrease by $1\%-$point $($with

$50\%$ probability$),$ but your maximum capacity of annual production is $100,000$

watches per year. $($I$.$e$.,$ if your market share would grow to a level which would

require producing more than $100,000$ watches per year, it will be capped at the

level corresponding to $100,000$ watches per year. Then, next year, it will again

either increase by $1\%-$point or decrease by $1\%-$point, with the same capacity

constraint being in effect.$)$ The initial cost of the project is $$10$ million, which is

used to purchase the equipment necessary to manufacture smartwatches. This

initial cost is amortized over a $5-$year period using the straight$-$line method. Every

year, the annual inflation will be either $1\%($with $50\%$ probability$)$ or $2\%($with $50\%$

probability$).$ In the first year, fixed costs will be $$100,000,$ and every year they will

increase by the inflation. In the first year, variable costs will be $$200$ per unit, and

every year they will increase by the inflation. If your market share drops to $0\%,$ you

abandon the project and sell your equipment for its book value or $20\%$ of its initial

price $($whichever is higher$).$ At the end of the ${10}^{th}$ year, you sell the equipment for

$20\%$ of its initial price $($if you have not sold it earlier already$).$ The price at which you

can sell the watches depends on the industry$-$wide demand in that year $($a higher

demand leads to a higher price$).$ Namely, the price per unit is equal to

$\frac{\text{i}\text{n}\text{d}\text{u}\text{s}\text{t}\text{r}\text{y}\text{d}\text{e}\text{m}\text{a}\text{n}\text{d}}{4000}+x,$ where $x$ is a random number taking a value in the interval

$-50,50$ and it follows a uniform distribution $($i$.$e$.,$ it takes all value within the given

interval with equal likelihood$).$ In any case, the unit price never drops below the

actual variable costs per unit. $($I$.$e$.,$ if the unit price would be lower than the variable

costs per unit based on the above formula, then the unit price will be equal to the

variable costs per unit.$)$ The corporate tax rate is $25\%.$ Assume that every payment

and cash flow happens at the end of the year.

a $(42$ points$)$ Assume that the project is financed by equity only. Simulate a

random outcome where you calculate the total cash flow of the project for

each year $($from the $1^{st}$ to the ${10}^{th}$ year$).$ Then, simulate altogether $30,000$

such random outcomes. Then, calculate the expected cash flow for each

year. Using these expected cash flows, calculate the internal rate of return

$($IRR$)$ of the project.

b $(18$ points$)$ Assume that the IRR calculated in the previous question is the

unlevered firm's expected return on the project. I.e$.,$ the project value

$($without considering the cost of the initial investment$)$ for an unlevered

company is exactly equal to $$10$ million. Now, assume that you can partially

or fully finance the project by debt. Namely, you can take out a loan for a

$10-$year period at an interest rate of $10\%.$ The principal must be paid back

in one lump sum at the end of the ${10}^{th}$ year. The flotation fee of the loan is

equal to $5\%$ of the total loan amount, and it must be paid up front. $($E$.$g$.,$ if

you borrow $$100,000,$ then the $$100,0000\mathrm{.}05=$$$5,000$ flotation fee is

paid immediately to the bank, and you receive only $$95,000.$ But you still

owe the bank $$100,000,$ you need to pay $$100,0000\mathrm{.}1=$$$10,000$ interest

every year, and you need to repay $$100,000$ at the end of the ${10}^{th}$ year.$)$ The

flotation fee is amortized over