A project that costs $$1000$ today is expected to have cash flows of $$200/$year for $10$ years.

Under these projections, this is a positive NPV project: if your cost of capital is $10\%/$year$,$ investing $$1000$ today and earning $$200$ per year for $10$ years yields an NPV of $$228\mathrm{.}91($you can verify this yourself, or take my word for it$).$

However, those annual cash flows of $$200$ are based on the following expectations:

A $50\%$ probability that business will be good and the project will earn $$300/$year$,$ and

a $50\%$ probability that business will be bad and the project will earn $$100/$year$.$

$($Note that the expected annual cash flows will be: $[0\mathrm{.}5*$ $$300]+[0\mathrm{.}5*$ $$100]=$ $$200/$year$,$ as stated above.$)$

Unfortunately, you will not know which state of the world you will be in until you spend the $$1000$ and start the business; in year $1,$ after the business opens, you will find out.

Obviously, if you open for business and business is good, you will want to stay open for the entire $10$ years; if your cost of capital is $10\%/$year$,$ the NPV of investing $$1000$ today and earning $$300/$year for $10$ years is $$843\mathrm{.}37.$

And just as obviously, if you open for business and business is bad, you will want to shut down at the end of year $1$ instead of staying open until year $10$; your NPV of investing $$1000$ and earning $$100$ per year for $10$ years is $-$$$385\mathrm{.}54.$

While this is indeed a positive NPV project overall, based on the above information, the $50\%$ probability that you will find out that business is bad is making you nervous to take this project.

As you are mulling this over, a competitor, who knows you are considering this investment, tells you that they will buy you out for $$750$ at the end of year $1$ if you decide not to continue in the business. $($They may think they can run the business better than you; they may have a different projection of what the cash flows will be$,$ or have other reasons for making this offer.$)$

This gives you what is called an "abandonment option": you will stay open if you find out that business is good, and you will sell to your competitor for $$750$ at the end of year $1$ if business is bad.

Calculate the value of this abandonment option $($at t$=0)$ by following this $3-$step process:

$1.$ Calculate the gain from abandonment. In this case, this will be the difference between the $$750$ offer and the present value of earning $$100/$year for the remaining $9$ years of the project. This difference must be a positive number, otherwise you will reject the offer.

$2.$ Multiply this gain by the probability of you being the situation of needing to abandon the project.

$3.$ Since you make this decision at t$=1,$ discount this value back to t$=0$ using your cost of capital.

What is the value of the abandonment option today $($at t$=0)?$