Suppose that the installed cost of a commercial solar hot water heating system is $$15,000$ and the annual fuel cost saving is $$1,000.$ Assume $($counter factually$)$ that fuel cost savings accrue as a single sum at the end of each year. The cost of the solar system is incurred at the end of year $0($or the very beginning of year$1).$ The system lasts for $25$ years, after which time, its value is zero.

$($a$)$ What is the net present value of the investment at $10\%?$ At $5\%?$

$($b$)$ As an alternative to Net Present Value, we can annualize the cost of the solar system and compare this figure with annual fuel cost savings. What is the annualized cost of the solar system at $10\%?$ At $5\%?$

$($c$)$ What is the net annualized value $($annualized benefits minus annualized costs$)$ of the solar system at $10?$ At $5\%?$ Note: If you now derive the present value of net annualized values, the answer should be the same as your answer in Part $($a$).$

$($d$)$ Go back to the fuel cost assumptions in part $($a$)$ and $($b$).$ We know that there are important social benefits to fuel oil displacement in the form of reduced CO$2,$

SOX, and NOX emissions. What level of annual air quality benefit $($measured in

$ terms, to be subtracted from fuel costs$)$ would be necessary to make the solar

system break even at $10\%?$ At $5\%?$

Hint: the answer can be found in Part $($b$).$

$($e$)$ Suppose we determine that the annual social cost of the fossil fuel the solar system displaces is $50\%$ of its market price. We place a $50\%$ tax on fossil fuel. Is the

solar system competitive $($now that we have internalized the social damage in the

fuel oil at $10\%?$ At $5\%.($Look at Part d$).$

NPV at $10\%\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$; NPV at $5\%\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Annualized cost at $10\%\_\_\_\_\_\_\_\_\_$; Annualized Costs at $5\%\_\_\_\_\_\_\_\_\_\_\_$

Net annualized value at $10\%\_\_\_\_\_\_\_\_\_$ Net annualized value at $5\%\_\_\_\_\_\_\_\_$

Check: PV of annualized value at $10\%\_\_\_\_\_\_\_.$ PV at $5\%\_\_\_\_\_\_\_\_\_\_\_\_\_$

Needed environmental damage savings for system to pay off at $10\%\_\_\_\_\_\_\_\_\_$

At $5\%\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

Is system now competitive at $10\%?\_\_\_\_\_\_\_\_\_\_\_\_$ At $5\%\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

SOULUTIONS MUST BE PRODUCED USING FORMULAS, OR GENERATED USING COMPUTED DISCOUNT FACTORS AS SHOWN IN THE CLASS VIDEOS FOR THE FINANCE CLASSES. YOU WILL GET NO CREDIT FOR SOLUTIONS PRODUCED USING EXCEL FORMULAS FOR NPV$,$ IRR, OR ANNUITY PRESENT VALUES.