The work presents an analytical model that establishes criteria for quantifying the combined effects of the inaccuracy of the investment estimates and operational cost in relation to the value of the project, discussing its relative importance.

1. 1
The CAPEX and OPEX precision trade-off
Tito Cardoso | Associated Director
Berkeley Research Group
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March, 12. 2017
Project delivery systems often specify precision requirements for capital expenditures estimates (CAPEX), but in
many cases, no similar requirement is reported for other estimates, such for the operational expenditure (OPEX).
This question is also relevant to the selection among alternatives for defining the project scope. On one occasion, a
team was meeting to decide on a trade-off of the project and one of the participants suggested: "If the total cost of
the two alternatives is similar, choose the one of smaller OPEX, because the CAPEX precision is always greater". In
general, the observation is true and the recommendation, in principle, is right. Mathematically, however, even if the i
OPEX estimate precision is smaller, its effect will depend on the relative magnitude of OPEX in relation of the CAPEX.
Modeling
Consider a project with a cash flow of N = 30 years, annual cost of capital rate, i = 12% and Investment Present Value,
IPV, approximately equal to CAPEX, that is, has a reduced time frame for the project implementation. Let us assume
that OPEX can be represented by the function: OPEX = O1 (1+b.tc), where O1 is the amount of OPEX in the 1st year of
operation, "b", Is the OPEX annual growth rate (b = 0.1 means a 10% increase in OPEX each year). If b = 0, OPEX is
constant and equal to O1 in each year. "C" is the OPEX annual growth acceleration parameter (if C > 1, the growth is
accelerated, C = 1 the change is non-accelerated and, when C < 1, the change is damped). The following graph
illustrate the OPEX curve when O1 = US$ 100 MM/year, b = 0.1 and C = 0.6.
0
50
100
150
200
250
300
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ano
OPEXanual(US$/ano)
Considering only investment and operating cost, the total cost of the project can be described as the sum of the
Capex and Opex Present Value: Total Cost = IPV + PV (OPEX). Therefore, the function IPV - PV (OPEX) = 0 describes
the curve where the present values of CAPEX and OPEX are equivalent. Replacing the above definitions:

2. 2
Let's admit that β is the CAPEX uncertainty and γ Is the OPEX uncertainty, both constants in time. Inserting the
uncertainties and dividing all terms by CAPEX, we have an expression of the combined effect of both uncertainties at
Present Value:
By plotting this function on a graph β versus γ, with a fixed ratio of O1 / CAPEX, we obtain a curve that defines the
points where the effect of the CAPEX and OPEX uncertainties is equal. Below this curve, the effect of the OPEX
uncertainty on the project Net Present Value, NPV, is greater than that of the CAPEX uncertainty. Above this curve
we have the opposite: the uncertainty of CAPEX is dominant. The following chart presents several of these curves
each with a value different from the OPEX / CAPEX ratio for b = 0, that is, OPEX constant in all years and equal to the
O1.
Combined uncertainty curves in function of the OPEX / CAPEX ratio (for b = 0)Curvas de incerteza combinada em função da razão OPEX / CAPEX
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 10% 20% 30% 40% 50% 60% 70% 80%
Incerteza de OPEX
IncertezadeCAPEX
opex(1) / capex = 0,08
opex(1) / capex = 0,10
opex(1) / capex = 0,12
opex(1) / capex = 0,15
opex(1) / capex = 0,18
opex(1) / capex = 0,20
Results
For OPEX constant in all years, when annual OPEX is very small in the presence of CAPEX (8% of CAPEX), only
excessively large uncertainties in the OPEX estimate (> ± 55%) may in some way influence the uncertainty of the
project NPV. Below, the only influence is uncertainty in CAPEX. This situation changes rapidly with the growth of
OPEX / CAPEX ratio. For OPEX = 10% of CAPEX, uncertainties in OPEX above ± 25% are already important. When OPEX
= 15% of the CAPEX its uncertainty predominates whenever the CAPEX uncertainty is below ± 21%. And when OPEX =
18% of CAPEX the uncertainty of OPEX prevails whenever CAPEX uncertainty is below ± 45%.
OPEX uncertainty
CAPEXuncertainty

3. 3
Regarding the parameters of the OPEX function, we observe indifference with respect to the acceleration, "C", in the
range between 0.3 and 1.3. The effect of the OPEX annual growth parameter, "b", is radical, as shown below:
Combined uncertainty curves in function of the 1st year OPEX / CAPEX ratio (for b = 0.1)
Curvas de incerteza combinada em função da razão OPEX no 1o ano / CAPEX (b=0,1)
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 10% 20% 30% 40% 50% 60% 70% 80%
Incerteza de OPEX
IncertezadeCAPEX
opex(1) / capex = 0,08
opex(1) / capex = 0,10
opex(1) / capex = 0,12
opex(1) / capex = 0,15
opex(1) / capex = 0,18
opex(1) / capex = 0,20
Conclusion
1- The OPEX estimate uncertainty is important for annual OPEX values > 12% of CAPEX
2 - As CAPEX uncertainties lesser than ± 20% are commonly obtained in projects with basic engineering completed, it
can be stated that whenever OPEX annual> 15% of CAPEX, the effect of OPEX uncertainty should always be a decision
factor among alternatives of similar total cost because its influence predominates in the project NPV composite
uncertainty.
3- The OPEX annual growth rate is a key factor, changing the conclusion (1), even when OPEX in the 1st year is small
in the presence of CAPEX.
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OPEX uncertainty
CAPEXuncertainty