An engineer is considering the size of a reservoir for flood control in the nort
ID: 3123299 • Letter: A
Question
An engineer is considering the size of a reservoir for flood control in the northern part of Germany. The size of the reservoir is closely related to the annual rainfall. In addition, there will be reparation costs from damages when the amount of rainfall exceeds the design capacity. If such damage occurs, it is estimated that the annual reparation cost will be 2.5% of the capital investment. The probabilities of specific amounts of rainfall per year and the estimated investment costs of the reservoir are given below. Using an interest rate of 10% per year and a study period of 11 years, determine the appropriate investment based on AW.
What is the AW (Annual Worth) for annual rainfall 110?
Annual Rainfall, M3 Probability of Greater Rainfall Estimated Capital Investment 110 0.6 2,500,000 120 0.2 2,550,000 130 0.1 2,600,000 140 0.055 2,650,000 150 0.045 2,700,000Explanation / Answer
Annual Rainfall
Probability of higher rainfall
Estimated investment
110
0.6
2500000
120
0.2
2550000
130
0.1
2600000
140
0.055
2650000
150
0.045
2700000
Firstly, we need to find out expected investment value using the probabilities.
Expected investment is 0.6*2500000+0.2*2550000+0.1*2600000+0.055*2650000+0.045*2700000
= 2537250
Now, whenever there is an exceeding rainfall, we incur an expense of 2.5% of investment value.
I.e. 2.5% of 2537250 = 63431.25
This expense happens annually over a period of 11 years. We have to find the present value of each of these future expenses. Rate of interest = 10%
For example, PV of 63431.25 from one year = 63431.25/(1+0.1)^1 = 57664.77
Similarly, from 2 years = 63431.25/(1+0.1)^2 = 52422.52
We calculate PV for all 11 years and add up the amount to find out the PV of the future cash out flows.
Year
Expense
PV of expense
1
63431.25
57664.77
2
63431.25
52422.52
3
63431.25
47656.83
4
63431.25
43324.39
5
63431.25
39385.81
6
63431.25
35805.28
7
63431.25
32550.26
8
63431.25
29591.14
9
63431.25
26901.04
10
63431.25
24455.49
11
63431.25
22232.26
Total expense
411989.79
So, the total investment that can be put up on this project = expected investment based on probabilities + PVs of future cash expenses = 2537250 + 411989.79 = $2949239.79
Annual Rainfall
Probability of higher rainfall
Estimated investment
110
0.6
2500000
120
0.2
2550000
130
0.1
2600000
140
0.055
2650000
150
0.045
2700000