Question

Five years ago, a company was considering the purchase of 62 new diesel trucks that were 15.23% more fuel-efficient than the ones the firm is now using. The company uses an average of 10 million gallons of diesel fuel per year at a price of $1.25 per gallon. If the company manages to save on fuel costs, it will save $1.875 million per year (1.5 million gallons at $1.25 per gallon). On this basis, fuel efficiency would save more money as the price of diesel fuel rises (at $1.35 per gallon, the firm would save $2.025 million in total if he buys the new trucks).

Consider two possible forecasts, each of which has an equal chance of being realized. Under assumption #1, diesel prices will stay relatively low; under assumption #2, diesel prices will rise considerably. The 62 new trucks will cost the firm $5 million. Depreciation will be 25.17% in year 1, 38.07% in year 2, and 36.4% in year 3. The firm is in a 40% income tax bracket and uses a 11% cost of capital for cash flow valuation purposes. Interest on debt is ignored. In addition, consider the following forecasts:

Forecast for assumption #1 (low fuel prices):

	Price of Diesel Fuel per Gallon
Prob. (same for each year)	Year 1	Year 2	Year 3
0.1	$0.8	$0.92	$1.01
0.2	$1	$1.12	$1.09
0.3	$1.13	$1.23	$1.31
0.2	$1.3	$1.47	$1.47
0.2	$1.4	$1.56	$1.62
Forecast for assumption #2 (high fuel prices):
	Price of Diesel Fuel per Gallon
Prob. (same for each year)	Year 1	Year 2	Year 3
0.1	$1.2	$1.49	$1.71
0.3	$1.31	$1.72	$2.01
0.4	$1.79	$2.31	$2.49
0.2	$2.2	$2.52	$2.82

Required: Calculate the percentage change on the basis that an increase would take place from the NPV under assumption #1 to the probability-weighted (expected) NPV.

Further Information (solution steps):

• Step (1): Calculate the annual expected price of diesel per gallon under each assumption, based on the probabilities outlined in the inputs section.

• Step (2): Using the annual expected fuel prices calculated in step (1), determine the increase in annual savings created by the proposed efficiency for each assumption.

• Step (3): Find the increased cash flow after taxes (CFAT) for both forecasts, based on the annual increase in fuel savings determined in step (2) as the increase in earnings before depreciation and taxes (EBDT), and the starting point from which profit is calculated for each assumption. As part of this step, you must establish annual depreciation (remember: depreciation is a noncash charge).

• Step (4): Considering the increased annual CFAT produced in step (3), calculate the NPV of the truck purchases for each assumption, based on the discount rate (cost of capital) indicated in the inputs section

• Step (5): In view of the outcomes produced in step (4), estimate the combined NPV weighed by the probability of each assumption.

• Step (6): Finally, calculate the percentage difference hypothesizing that an increase took place starting from the NPV for assumption #1 to the combined NPV worked out in step (5).

Answer 1

PV of cash flows = CFt/ (1+k)^t

where CFt is cash flow in year t, k = Cost of capital (WACC) & t is the year of Cash flow

NPV = Sum of PV of all future cash flows - Investment

Expected Price = Sum of Probability X Estimated Price for each year

For Year 1 = 0.1 x 0.80 + 0.2 X 1 + 0.3 X 1.13 + 0.2 X 1.30 + 0.2 X 1.40 = 1.16.

Similarly for other years

Assumption 1 - Working for NPV

	Year 0	Year 1	Year 2	Year 3
Expected Price (A)		$1.16	$1.29	$1.33
Fuel Used (B) in Gallons		10,000,000	10,000,000	10,000,000
Fuel cost(C = A x B)		11,590,000	12,910,000	13,300,000
Savings in Fuel cost (15.23%) (D = C x 15.23%)		1,765,157	1,966,193	2,025,590
Depreciation (E = H XL)		1,258,500	1,903,500	1,820,000
EBIT (F = D - E)		506,657	62,693	205,590
Taxes @40% (G = F x 40%)		202,663	25,077	82,236
Net Income (H = F - G)		303,994	37,616	123,354
Add : Depreciation (E )		1,258,500	1,903,500	1,820,000
CFAT (I = H+ E)		1,562,494	1,941,116	1,943,354
Investments	(5,000,000)
Discount Factor @11% J = 1/(1+11%)^t	1	0.9009	0.8116	0.7312
Discounted CF K = I x J	(5,000,000)	1,407,652	1,575,453	1,420,964
NPV	(595,931)
Depreciation Rate L		25.17%	38.07%	36.40%

NPV for Assumption 1 = - 595931

Assumption 2 - Working for NPV

	Year 0	Year 1	Year 2	Year 3
Expected Price (A)		$1.67	$2.09	$2.33
Fuel Used (B) in Gallons		10,000,000	10,000,000	10,000,000
Fuel cost(C = A x B)		16,690,000	20,930,000	23,340,000
Savings in Fuel cost (15.23%) (D = C x 15.23%)		2,541,887	3,187,639	3,554,682
Depreciation (E = H XL)		(1,258,500)	(1,903,500)	(1,820,000)
EBIT (F = D - E)		1,283,387	1,284,139	1,734,682
Taxes @40% (G = F x 40%)		(513,355)	(513,656)	(693,873)
Net Income (H = F - G)		770,032	770,483	1,040,809
Add : Depreciation (E )		1,258,500	1,903,500	1,820,000
CFAT (I = H+ E)		2,028,532	2,673,983	2,860,809
Investments	(5,000,000)
Discount Factor @11% J = 1/(1+11%)^t	1	0.9009	0.8116	0.7312
Discounted CF K = I x J	(5,000,000)	1,827,506	2,170,265	2,091,799
NPV	1,089,570
Depreciation Rate L		25.17%	38.07%	36.40%

NPV for Assumption 2 = 1,089,570

Weighted Average NPV = 50% X 1089570 + 50% x (-595931)

= 651,787.87

% Change from assumption 1 = (651787 - (-595931)) / 595931

= -203.33% (Negative would appear because base value being negative)