A company has licenses for a new cobalt mine. In year 1 a full exploration will be carried out at a cost of $5 million to determine the depth and concentration of the cobalt. This will enable a prediction of the rate at which cobalt can be extracted. This rate is uncertain but has expected value of 5000 tons per year. The profit per ton of cobalt after extraction costs is $1000(=0.001 \$million ) but the cost of developing the mine before extraction can start is estimated to be $20 million per year over a 3 year period (so $60 million in total). Development will start in year 2 if the cobalt resource is sufficiently rich to be profitable with actual production starting at the beginning of year 5. Assume a discount rate of 6% per year and that the mine will be in operation indefinitely. You can also assume that all money flows occur at the beginning of the year with the $5 million exploration cost paid at the beginning of year 1. (a) (5 marks) Write down an expression for the net present value at the beginning of year 1 for the profit, in terms of the cobalt extraction rate x tons per year if development goes ahead. (b) (4 marks) What is the net present value at the beginning of year 1 if it is known that the cobalt extraction rate is 5000 tons per year? Give your answer in millions of dollars and use 3 decimal places (equivalent to the nearest $1000.) (c) 5 marks) What is the lowest predicted rate of cobalt extraction that will make it worthwhile to go ahead with the development of the mine? (d) ( 8 marks) Now suppose that the rate at which cobalt can be extracted is uniformly distributed between 3000 and 7000 tons per year. Use real options to find the net present value of the mine at the beginning of year 1 given that there is an option not to go ahead with the development of the mine