chapter 10 monte carlo analysis
TRANSCRIPT
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Chapter 10 Monte Carlo Simulation and the
Evaluation of Risk
Chemical Engineering Department
West Virginia University
Copyright - R.Turton and J. Shaeiwitz 2012 1
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Outline
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Causes of uncertainty in profitability calculations
Forecasting
Quantification of risk
Best-case - worst-case
Monte-Carlo method and probability distributions
Using CAPCOST
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Factors Affecting Profitability
From Table 10.1 Cost of Fixed Capital Investment1 -10 to +25 Construction Time -5 to +50 Start-up Costs and Time -10 to +100 Sales Volume -50 to +150 Price of Product -50 to +20 Plant Replacement and Maintenance Costs -10 to +100 Income Tax Rate -5 to +15 Inflation Rates -10 to +100 Interest Rates -50 to + 50 Working Capital -20 to +50 Raw Material Availability and Price -25 to +50 Salvage Value -100 to +10 Profit -100 to +10
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Forecasting – Prediction of Future Trends
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demand
supply
Quantity of X demanded, Q (per year)
Demand: As P demand increases
Supply: As P more supply will become available
Market will reach equilibrium when Supply = Demand
New plant comes on line – so supply curve shifts down and Pequilib
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Historical Data
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• Variation around trend line = ± 35c/gal
• Build Plant in 1998 or 2005!
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Difficulty in Forecasting
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According to Yogi Berra
“It’s tough to make predictions, especially about the future”
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Quantifying Risk
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Example 10.1 and 10.2
R= $75 million per year
COMd = $30 million per year
FCIL = $150 million
NPV = $17.12 million
What if variation of 3 parameters is
R – 20% to +5%, COMd –10% to +10%,
FCIL +30% to –20%?
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Quantifying Risk
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Best Case – Worst Case Scenario
Worst Case (all figures in $million or $million/yr)
R = (75)(0.8) = 60
COMd = (30)(1.1) = 33
FCIL = (150)(1.3) = 195
Best Case
R = (75)(1.05) = 78.75
COMd = (30)(0.9) = 27
FCIL = (150)(0.8) = 120
NPV = -59.64
NPV = 53.62
What does this tell us? - not much!
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Quantifying Risk
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The problem with the best case –worst case scenario is that neither case is very likely!
If each variation were equally likely, i.e., the high, average, and low values could each occur with the same probability then we would have
33 = 27 equally possible outcomes
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Quantifying Risk
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Scenario R1 COMd1 FCIL
1 Probability of Occurrence
1 -20% -10% -20% (1/3)(1/3)(1/3) = 1/27
2 -20% -10% 0%
3 -20% -10% +30%
4 -20% 0% -20%
5 -20% 0% 0%
6 -20% 0% +30%
7 -20% +10% -20%
8 -20% +10% 0%
9 (worst) -20% +10% +30%
10 0% -10% -20%
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Quantifying Risk
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Assign Probabilities to values using probability distributions leads to the Monte Carlo Method (MC)
We use an 8-step method to describe MC
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Quantifying Risk
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1. All the parameters for which uncertainty is to be quantified are identified.
2. Probability distributions are assigned for all parameters in step 1 above.
3. A random number is assigned for each parameter in step 1 above.
4. Using the random number from step 3, the value of the parameter is assigned using the probability distribution (from step 2) for that parameter.
5. Once values have been assigned to all parameters, these values are used to calculate the profitability (NPV or other criterion) of the project.
6. Steps 3, 4, and 5 are repeated many times (say 1000).
7. A histogram and cumulative probability curve for the profitability criteria calculated from step 6 are created.
8. The results of step 7 are used to analyze the profitability of the project.
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Probability Distributions
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Uniform Distribution
Probability density function p(x)
a b
1
b - a
a b
1
0
Cumulative probability function P(x)
x
p(x)
x
P(x)
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Probability Distributions
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Triangular Distribution
Probability density function, p(x)
a b c
2
c - a
a b c
1
0
Cumulative probability function, P(x)
P(x)
x x
p(x)
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Probability Distributions
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Triangular Distribution – used in CAPCOST
Triangular probability density function:
(10.9)
Triangular cumulative probability function
(10.10)
2( )( ) for
( )( )
2( )( ) for
( )( )
x ap x x b
c a b a
c xp x x b
c a c b
2( )( ) for
( )( )
( ) ( )(2 )( ) for
( ) ( )( )
x aP x x b
c a b a
b a x b c x bP x x b
c a c a c b
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Monte Carlo Method
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Monte Carlo Method
1. Identify parameters = R, COMd, FCIL
2. Probability distributions assigned – use low, medium and high values for a, b, c in triangular distribution
3. and 4. As an example – look at R
a = 60, b = 75, c = 78.75 (-20% - +5%, BC = 75)
P(x = b) = (b-a)2/(c-a)(b-a) =15/18.75= 0.8
Generate a random number (RN) (0,1) = 0.3501
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Monte Carlo Method
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Monte Carlo Method
Since RN < 0.8 use first part of Eqn (10.10)
2
2
( )( ) for
( )( )
( 60)0.3501 69.92
(78.75 60)(75 60)
x aP x x b
c a b a
xx
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Monte Carlo Method
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b = 75
First part of curve – Eqn (10.10) x<b
0.3501
x = 69.92
0.80
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Monte Carlo Method
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• Using R = x = 69.92
• Choose RNs for COMd and FCIL and repeat procedure to get values for these parameters
• Calculate NPV
• Repeat many times (1000) and plot frequency (distribution) of NPV
Figure 10.15 shows NPV distribution for this problem
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Monte Carlo Method
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Monte Carlo Method
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Project B
Project A
1.00
0.50
0.00-40 -20 0 20 40 60
0.17
0.02
Net Present Value ($ Millions)
CumulativeProbability
Figure 8.16: A Comparison of the Profitability of Two Projects Showing the NPV with Respect to the Estimated Cumulative Probability from a Monte Carlo Analysis
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Monte Carlo Method
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22
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Monte Carlo Method
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Results using Capcost for Monte Carlo Simulations
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Summary
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• The quantification of risk allows a more complete interpretation of the economic potential of a new project
• The Monte-Carlo method is a convenient tool for quantifying the risk associated with factors affecting a project’s profitability
• Capcost may be used to run Monte-Carlo simulations on a process