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 $228.91 (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 $400/year, and

a 50% probability that business will be bad and the project will earn $0/year.

(Note that the expected annual cash flows will be: [0.5 * $400] + [0.5 * $0] = $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 $400/year for 10 years is $1457.83.

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 $0 per year is - $1000.

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 $500 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 $500 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 is easy: the gain is $500. Your competitor is offering $500, and you would earn $0 by staying open otherwise.

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)?