Question

Five years ago, a company was considering the purchase of 77 new diesel trucks that were 15.45% 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 77 new trucks will cost the firm $5 million. Depreciation will be 25.2% in year 1, 38.48% in year 2, and 36.34% 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

$0.79

$0.92

$1.01

0.2

$0.99

$1.13

$1.12

0.3

$1.12

$1.2

$1.3

0.2

$1.31

$1.44

$1.44

0.2

$1.4

$1.57

$1.6

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

$1.51

$1.7

0.3

$1.3

$1.71

$2.02

0.4

$1.82

$2.33

$2.49

0.2

$2.21

$2.5

$2.79

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

Assumption1

Computation of Yearly rate

Year	1	2	3
Probability	Rate	Probability weighted rate (Probability x rate)	Rate	Probability weighted rate (Probability x rate)	Rate	Probability weighted rate (Probability x rate)
0.1	0.79	0.079	0.92	0.092	1.01	0.101
0.2	0.99	0.198	1.13	0.226	1.12	0.224
0.3	1.12	0.336	1.2	0.36	1.3	0.39
0.2	1.31	0.262	1.44	0.288	1.44	0.288
0.2	1.4	0.28	1.57	0.314	1.6	0.32
Yearly rate (Aggregate of probability weighted rates)	1.155	1.280	1.323

Annual expected savings under assumption 1

Year	1	2	3
Annual savings (in Mn gallons)	1.5	1.5	1.5
Expected rate from above table	1.155	1.28	1.323
Annual savings (in Mn)	1.7325	1.9200	1.9845

Depreciation computation

Year	1	2	3
Purchase value of trucks (in Mn)	5
Depreciation rate	25.20%	38.48%	36.34%
Depreciation (Purchase Value x Depreciation rate)	1.26	1.924	1.817

Increased cashflow forecast for assumption 1

Year	0	1	2	3
Initial cashflows
Cost of new trucks	(5.0000)
Intermediate cashflows
Annual savings (from above table)		1.7325	1.9200	1.9845
Less : Depreciation		1.2600	1.9240	1.8170
Profit before taxes		0.4725	(0.0040)	0.1675
Less : Taxes @40%		0.1890	(0.0016)	0.0670
Profit after taxes		0.2835	(0.0024)	0.1005
Add : Depreciation		1.2600	1.9240	1.8170
Cashflow after taxes		1.5435	1.9216	1.9175
Net cashflows (Initial + Intermediate)	(5.0000)	1.5435	1.9216	1.9175
PV factor @ 10% --- > 1/ (1+10%)^year	1.0000	0.9091	0.8264	0.7513
Present value of cashflows	(5.0000)	1.4032	1.5881	1.4406
Net Present Value of cashflows under assumption 1	(0.5681)

PV factor computation --- > 1/ (1+discounting rate)^year --- Year 1 --> 1/(1+10%)^1; Year 2 ---> 1/ (1+10%)^2;

Year 3 ---> 1/(1+10%)^3

Depreciation is added back to profit to arrive at after tax cashflow as it is a non cash charge

Assumption 2

Computation of Yearly rate

Year	1	2	3
Probability	Rate	Probability weighted rate (Probability x rate)	Rate	Probability weighted rate (Probability x rate)	Rate	Probability weighted rate (Probability x rate)
0.1	1.220	0.122	1.510	0.151	1.700	0.170
0.3	1.300	0.390	1.710	0.513	2.020	0.606
0.4	1.820	0.728	2.330	0.932	2.490	0.996
0.2	2.210	0.442	2.500	0.500	2.790	0.558
Yearly rate (Aggregate of probability weighted rates)	1.682	2.096	2.330

Annual expected savings under assumption 2

Year	1	2	3
Annual savings (in Mn gallons)	1.5	1.5	1.5
Expected rate from above table	1.682	2.096	2.33
Annual savings (in Mn)	2.523	3.1440	3.495

Depreciation computation

Year	1	2	3
Purchase value of trucks (in Mn)	5
Depreciation rate	25.20%	38.48%	36.34%
Depreciation	1.26	1.924	1.817

Increased cashflow forecast under assumption 2

Year	0	1	2	3
Initial cashflows
Cost of new trucks	(5.0000)
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