12.2. The XYZ firm can invest in a new DRAM chip factory for $425 million. The factory, which must be invested in today, has cash flows two years from now that depend on the state of the economy.

The cash flows when the factory is running at full capacity are described by the following tree diagram:

In year 1, the firm has the option of running the plant at less than full capacity. In this case, workers are laid off, production of memory chips is scaled down, and the subsequent cash flows are half of what they would be when the plant is running at full capacity.
An alternative use for the firm’s funds is investment in the market portfolio. In the states that correspond to the branches of the tree above, $1 invested in the market portfolio grows as follows:

Assume that the risk-free rate is 5 percent per year, compounded annually. Compute the project’s present value

(a) with the option to scale down and

(b) without the option to scale down. Compute the difference between these two values, which is the value of the option.

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Year 0 Year 1 Year 2 Very good $1 billion Good Bad Medium $200 million Very bad -$500 million