Question

Austin needs to purchase a new heating/cooling system for his home. He is thinking about having a geothermal system installed, but he wants to know how long it will take to recoup the additional cost of the system. The geothermal system will cost $20,500. A conventional system will cost $6,000. Austin is eligible for a 30% tax credit to be applied immediately to the purchase only if he chooses the geothermal system. He estimates that he will save $1,500 per year in utility bills with the geothermal system. These cash outflows can be assumed to occur at the end of the year. The cost of capital (or interest rate) for Austin is 6%. How long will Austin have to use the system to justify the additional expense over the conventional model? In other words, what is the DISCOUNTED payback period in years? Discount future cash flows before calculating payback rounded UP to a whole year. (Enter just the number without comma and round off decimals.)

Answer 1

Total cost of conventional system = $6,000

Cost of capital (discount rate) = 6%

Total cost of geothermal system = $20,500

Tax credit availed by Austin = $6,150

Net cost incurred for acquiring geothermal system= $20,500 - $6,150

= $14,350

Additional expense (excess cash outflow) incurred by Austin over the conventional model = $14,350 - $6,000

= $8,350

Amount saved by Austin on yearly utility bills (cash inflows), if the geothermal system is installed = $1,500

Discounted Cash Flow formula = FV (Cash inflow/outflow) / (1+r)^t

FV = Future Value

r = Discount rate

t = Time period

Discount Rate	6%
	Today
Period (in years)	0	1	2	3	4	5	6	7	8	9
Cash Flows	$8,350	$1,500	$1,500	$1,500	$1,500	$1,500	$1,500	$1,500	$1,500	$1,500
Discounted Cash flows	$8,350	$1,415	$1,335	$1,259	$1,188	$1,121	$1,057	$998	$941	$888
NPV	$8,350	$6,935	$5,600	$4,340	$3,152	$2,031	$974	$24	$965	$1,853

NOTE:

Discounted payback period can be calculated when Net Present Value of discounted cash flows is equal to zero. We know that discounted payback period will be between 6 years and 7 years.

Discounted Cashflow in year 7 = $998

Discounted Cashflow required for NPV to be equal to zero = $974

Discounted Payback Period = 6 + ($974/$998) years ------ Linear Equation

= 6 + 0.97 years

= 6.97 years = 7 years (approx.)

Hence, Austin will have to use the geothermal system for 7 years to justify the additional expense over the conventional model.