Question

A project under evaluation has a first cost of $35,000. The firm’s MARR is 15%. Assume the probability distributions for annual benefit and life are unrelated and statistically independent. Calculate the expected value for the Present Worth.

Annual Benefit	Probability	Live	Probability
$17,000	0.2	5	0.27
19,000	0.3	5	0.27
22,000	0.5	5	0.27
$17,000	0.2	7	0.45
19,000	0.3	7	0.45
22,000	0.5	7	0.45
$17,000	0.2	9	0.28
19,000	0.3	9	0.28
22,000	0.5	9	0.28

Answer 1

Initial Investment = $35,000
MARR = 15%
NPV = PV of Cash Inflows – PV of Cash Outflows

Expected Annual Benefit = (P1 * Annual Benefit1) + (P2 * Annual Benefit2) + (P3 * Annual Benefit3)
where,
P1 = Probability of outcome 1
P2 = Probability of outcome 2
P3 = Probability of outcome 3

Annual Benefit1 = Annual Benefit given P1
Annual Benefit2 = Annual Benefit given P2
Annual Benefit3 = Annual Benefit given P3

Expected Annual Benefit = (0.2 * 17000) + (0.3 * 19000) + (0.5 * 22000)
Expected Annual Benefit = 3,400 + 5,700 + 11,000
Expected Annual Benefit = $20,100


Case 1 : Life is 5 Years
Calculation of NPV given Case 1 :

NPV = PV of Cash Inflows – PV of Cash Outflows
         = PV of Cash Inflows for Year 1 to 5 – PV of Cash Outflows
         = PVAF (15%,5) * 20,100] – 35,000
         = [3.3522 * 20,100] – 35,000
         = 67,378.32 – 35,000
= $32,378.32

PVAF (15%,5) is calculated by adding the PV Factor of 15% for 5 years

Where,
Present Value Factor have been calculated as = (1/1+r)n

Where
r= Cost of Capital (MARR)
n= No of Year

Case 2 : Life is 7 Years
Calculation of NPV given Case 2 :

NPV = PV of Cash Inflows – PV of Cash Outflows
         = PV of Cash Inflows for Year 1 to 7 – PV of Cash Outflows
         = PVAF (15%,7) * 20,100] – 35,000
         = [4.1604 * 20,100] – 35,000
= 83,624.44 – 35,000
= $48,624.44

PVAF (15%,7) is calculated by adding the PV Factor of 15% for 7 years

Case 3 : Life is 9 Years

Calculation of NPV given Case 3 :

NPV = PV of Cash Inflows – PV of Cash Outflows
         = PV of Cash Inflows for Year 1 to 9 – PV of Cash Outflows
         = PVAF (15%,9) * 20,100] – 35,000
         = [4.7716 * 20,100] – 35,000
         = 95,908.84 – 35,000
         = $60,908.84

PVAF (15%,9) is calculated by adding the PV Factor of 15% for 9 years.


Calculation of NPV of the project :

Probability of Case 1 (Life = 5 Years) = 0.27
Probability of Case 2 (Life = 7 Years) = 0.45
Probability of Case 3 (Life = 9 Years) = 0.28
NPV given Case 1 = $32,378.32
NPV given Case 2 = $48,624.44
NPV given Case 3 = $60908.84

NPV of the project = (Probability of Case 1 * NPV given Case 1) + (Probability of Case 2 * NPV given Case 2) + (Probability of Case 3 * NPV given Case 3)
                                      = (0.27 * $32,378.32) + (0.45 * $48,624.44) + (0.28 * $60908.84)
                                      = $8742.15 + 21881.00 + 17054.48
                                      = $47677.63