chapter 10 monte carlo analysis

Chapter 10 Monte Carlo Simulation and the

Evaluation of Risk

Chemical Engineering Department

West Virginia University

Copyright - R.Turton and J. Shaeiwitz 2012 1

Outline


Causes of uncertainty in profitability calculations

Forecasting

Quantification of risk

Best-case - worst-case

Monte-Carlo method and probability distributions

Using CAPCOST

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


Forecasting – Prediction of Future Trends


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

Historical Data


• Variation around trend line = ± 35c/gal

• Build Plant in 1998 or 2005!

Difficulty in Forecasting


According to Yogi Berra

“It’s tough to make predictions, especially about the future”

Quantifying Risk


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%?

Quantifying Risk


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!

Quantifying Risk


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

Quantifying Risk


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%

Quantifying Risk


Assign Probabilities to values using probability distributions leads to the Monte Carlo Method (MC)

We use an 8-step method to describe MC

Quantifying Risk


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.

Probability Distributions


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)



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)



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

Monte Carlo Method


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

Monte Carlo Method


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

Monte Carlo Method


b = 75

First part of curve – Eqn (10.10) x<b

0.3501

x = 69.92

0.80

Monte Carlo Method


• 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

Monte Carlo Method


Monte Carlo Method


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

Monte Carlo Method

Copyright - R.Turton and J. Shaeiwitz 2012

23

Results using Capcost for Monte Carlo Simulations

Summary


• 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