Your R&D division has just synthesized a material that will superconduct electricity at room

temperature; you have given the go$-$ahead to try to produce this material commercially. It will

take five years to find out whether the material is commercially viable, and you estimate that

the probability of success is $25\%.$ Development will cost $$10$ million per year, paid at the

beginning of each year. If development is successful and you decide to produce the material,

the factory will be built immediately. It will cost $$1$ billion to put in place, and will generate

profits of $$100$ million at the end of every year in perpetuity. Assume that the current five$-$year

risk$-$free interest rate is $10\%$ per year, and the yield on a perpetual risk$-$free bond will be $12\%,$

$10\%,8\%,$ or $5\%$ in five years. Assume that the risk$-$neutral probability of each possible rate is

the same. What is the value today of this project?

$5.$ You own a wholesale plumbing supply store. The store currently generates revenues of $$1$

million per year. Next year, revenues will either decrease by $10\%$ or increase by $5\%,$ with equal

probability, and then stay at that level as long as you operate the store. You own the store

outright. Other costs run $$900,000$ per year. There are no costs for shutting down; in that case,

you can always sell the store for $$500,000.$ What is the business worth today if the cost of

capital is fixed at $10\%?$