Question

Five years ago, a company was considering the purchase of 72 new diesel trucks that were 14.56% 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 72 new trucks will cost the firm $5 million. Depreciation will be 24.84% in year 1, 38.39% in year 2, and 36.46% in year 3. The firm is in a 40% income tax bracket and uses a 10% 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 $0.81 $0.89 $1.01 0.2 $1.02 $1.11 $1.11 0.3 $1.11 $1.23 $1.32 0.2 $1.3 $1.48 $1.46 0.2 $1.4 $1.58 $1.61 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 $1.22 $1.52 $1.69 0.3 $1.3 $1.7 $2.01 0.4 $1.81 $2.32 $2.52 0.2 $2.21 $2.53 $2.83 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. Answer % Do not round intermediate calculations. Input your answer as a percent rounded to 2 decimal places (for example: 28.31%). Note: The educational purpose of this problem targets the students’ ability to read + follow instructions. 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

Part (1) Calculations for Assumption #1(Low fuel price)

Step (1)
Calculation of annual expected price of diesel per gallon under assumption #1 (low fuel prices)
Probability (1)	Year 1 (2)	weighted price (3) = (1 x 2)	Year 2 (4)	weighted price (5) = (1 x 4)	Year 3 (6)	weighted price (7) = (1 x 6)
0.1	0.81	$0.081	$0.89	$0.09	$1.01	$0.10
0.2	1.02	$0.204	$1.11	$0.22	$1.11	$0.22
0.3	1.11	$0.333	$1.23	$0.37	$1.32	$0.40
0.2	1.33	$0.266	$1.48	$0.30	$1.46	$0.29
0.2	1.4	$0.280	$1.58	$0.32	$1.61	$0.32
Weighted price	$1.164		$1.292		$1.333

Step (2)
Calculation of increase in annual savings
Year	Expected price of diesel	Annual savings(1.5 million)
1	$1.16	$1.75
2	$1.29	$1.94
3	$1.33	$2.00
Note:Annual savings 1.5 million gallons,it is given in the question itself

Step (3)
Calculation of increase in cashflows
Year	1	2	3
Annual savings	1.75	1.94	2
Depreciation %	24.84%	38.39%	36.46%
Less: deprecaition on (cost $5 million)	1.242	1.9195	1.823
Saving after depreciation	0.508	0.0205	0.177
Less:Tax @ 40%	0.2032	0.0082	0.0708
Savings after tax	0.3048	0.0123	0.1062
Add:depreciation           (non cash expense)	1.242	1.9195	1.823
Cash flow after taxes (CAFT)	1.5468	1.9318	1.9292

Step (4)
Calculation of NPV
Year	1	2	3
CAFT	1.5468	1.9318	1.9292
Discount rate @ 10%	0.909	0.826	0.751
Discounted cashflow	1.406	1.597	1.449
Total of discounted cashflow	$4.45
Cost of investment	$ 5
Hence NPV is ($4.45-$5)	-$0.55

Part (2) Calculations for Assumption #2(High fuel price)

Step (1)
Calculation of annual expected price of diesel per gallon under assumption #2 (high fuel prices)
Probability (1)	Year 1 (2)	Weighted price (3) = (1 x 2)	Year 2 (4)	Weighted price (5) = (1 x 4)	Year 3 (6)	Weighted price (7) = (1 x 6)
0.1	1.22	$0.122	$1.52	$0.15	$1.69	$0.17
0.3	1.3	$0.390	$1.70	$0.51	$2.01	$0.60
0.4	1.81	$0.724	$2.32	$0.93	$2.52	$1.01
0.2	2.21	$0.442	$2.53	$0.51	$2.83	$0.57
Weighted price		$1.68		$2.10		$2.35

Step (2)
Calculation of increase in annual savings
Year	Expected price of diesel	Annual savings(1.5 million)
1	$1.68	$2.52
2	$2.10	$3.14
3	$2.35	$3.52
Note:Annual savings 1.5 million gallons,it is given in the question itself

Step (3)
Calculation of increase in cashflows
Year	1	2	3
Annual savings	$2.52	$3.14	$3.52
Depreciation %	24.84%	38.39%	36.46%
Less: deprecaition on (cost $5 million)	1.242	1.9195	1.823
Saving after depreciation	$1.28	$1.22	$1.70
Less:Tax @ 40%	$0.51	$0.49	$0.68
Savings after tax	$0.77	$0.73	$1.02
Add:depreciation           (non cash expense)	1.242	1.9195	1.823
Cash flow after taxes (CAFT)	$2.01	$2.65	$2.84

Step (4)
Calculation of NPV
Year	1	2	3
CAFT	$2.01	$2.65	$2.84
Discount rate @ 10%	0.909	0.826	0.751
Discounted cashflow	1.825	2.194	2.134
Total of discounted cashflow	$6.15
Cost of investment	$ 5
Hence NPV is ($6.15-$5)	$1.15

Part(3) calculation of combined NPV

Calculation of combined NPV weighed by the probablity of each assumption
	Assumption #1	Assumption #2
NPV	-$0.55	$1.15
Probabity	0.5	0.5
Combined NPV	-$0.27	$0.58
Total of combined NPV	$0.30
Note:Both assumption have equal chance of being realized,hence probablity is 50% each assumption

Calculation of percentage of difference between combined NPV and NPV of Assumption #1
NPV of assumption #1	-0.55
Combined NPV	0.30
Increase	0.85
% of increase (0.85/.55)*100	-155.16
Or it is (ignore minus symbol)	155%