$1$ Net Present Value

The goal of this exercise is for you to remember the concept of net present value, which is a key tool to understand dynamic models like the ones we will see in class.

A perpetuity. Suppose there is an asset that pays a flow of $$x$ every period $($including the period when you purchased it$.)$ Suppose that the interest rate at which you can borrow and lend money is $r.$

$($a$)$ What is the NPV of the asset's flows? Call it $S_0$

$($b$)$ How would your answer change if the asset only started paying the next period? Call it $S_1$

$($c$)$ Consider the asset in $($b$).$ What is the NPV of the asset if the flow is $x=75$ and the interest rate is $1\%?$ How does your answer change when the flow increases? And when the interest rate increases?

$($d$)$ Now consider again the asset in question $($b$).$ How much would you be willing to pay for it today? Why?

$($e$)$ Consider again the asset in $($b$).$ How much would you be willing to pay for it today, if the sequence of interest rates is $r_0,r_1,r_2,$dots $?$

Now consider two projects, A and B$,$ described by the following table.

$\backslash$table$[[$time$,$A$,$B$],[0,-10,-20],[1,0,10],[2,0,10],[3,30,10]]$

Table $1$: Projects' Payoff Profiles

$1$

For example, option A means you invest $10$ dollars today and receive nothing for $2$ years and $30$ dollars at the end of the third year. Suppose that the interest rate is constant at $5\%.$

$($a$)$ Assuming you can invest equal amounts in either project, which project would you choose and why? $[$Hint you should consider rates of ret urn properly discounted.$]$

$($b$)$ At what interest rate would you be indifferent between the two? Just set up the equation. Do not solve it$.$