1. Chapter 10 - Monte Carlo Simulation and the Evaluation of Risk

2. Outline Causes of uncertainty in profitability calculations
Forecasting
Quantification of risk
Best-case - worst-case
Monte-Carlo method and probability distributions
Using CAPCOST
Note the large variability in sales volume, this is probably the single most uncertain factor in the evaluation of profitability and yet it is often overlooked or ignored. All these factors introduce uncertainty into the calculation of a process’s or plant’s profitability

3. Factors Affecting Profitability
From Table 10.1
Cost of Fixed Capital Investment 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
Note the large variability in sales volume, this is probably the single most uncertain factor in the evaluation of profitability and yet it is often overlooked or ignored. All these factors introduce uncertainty into the calculation of a process’s or plant’s profitability

4. Forecasting – Prediction of Future Trends
Demand: As P demand increases
Supply: As P more supply will become available
Market will reach equilibrium when Supply = Demand
demand
supply
Price of X, $
The concept here is that as new production capacity comes on-line, so the price of the product will drop. The larger the % of market share that the capacity of the new process represents, the larger the decrease in product price.
New plant comes on line – so supply curve shifts down and Pequilib
Quantity of X demanded, Q (per year)

5. Historical Data Variation around trend line = ± 35c/gal
Build Plant in 1998!
Even an excellent prediction of future prices may not be enough to guarantee a profitable process. If the refinery were to be built in 1998 and the economics used were based on the trendline through the data then the first 2-3 years of production would be less profitable than predicted especially if we were locked into a contract price for crude oil based on the same predictions!
Figure 10.10:

6. Difficulty in Forecasting
“It’s tough to make predictions, especially about the future”

7. 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%?

8. Quantifying Risk Best Case – Worst Case Scenario NPV = -59.64
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 =
As the slide shows these numbers don’t tell us very much. We could make a lot or loose a lot. Also the likelihood of either scenario occuring is very remote.
What does this tell us?
- not much!
NPV = 53.62

9. 33 = 27 equally possible outcomes
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
Of course the number of possibilities is infinite since each variable is continuous between the high and low limits.

10. Quantifying Risk Scenario R1 COMd1 FCIL1 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% %
% -10% %

11. 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

12. Quantifying Risk
All the parameters for which uncertainty is to be quantified are identified.
Probability distributions are assigned for all parameters in step 1 above.
A random number is assigned for each parameter in step 1 above.
Using the random number from step 3, the value of the parameter is assigned using the probability distribution (from step 2) for that parameter.
Once values have been assigned to all parameters, these values are used to calculate the profitability (NPV or other criterion) of the project.
Steps 3, 4, and 5 are repeated many times (say 1000).
A histogram and cumulative probability curve for the profitability criteria calculated from step 6 are created.
The results of step 7 are used to analyze the profitability of the project.

13. Probability Distributions
Uniform Distribution
P(x)
p(x) 1 1
b - a x x
a b
a b
Probability density function p(x)
Cumulative probability function P(x)

14. Probability Distributions
Triangular Distribution
P(x)
p(x) 1 2
c - a
This is a good compromise between a normal distribution (which most likely describes the true variability of the different factors) and the limited information that will be available for the variables. All we need here is a likely high, low and average or most likely value. x x
a b c
a b c
Probability density function, p(x)
Cumulative probability function, P(x)

15. Probability Distributions
Triangular Distribution – used in CAPCOST
Triangular probability density function:
(10.9)
Triangular cumulative probability function
(10.10)

16. Monte Carlo Method Monte Carlo Method
Identify parameters = R, COMd, FCIL
Probability distributions assigned – use low, medium and high values for a, b, c in triangular distribution
and 4. As an example – look at R
Steps 3 and 4 illustrate that once a RN has been chosen, we must deterine which part of the triangular distribution the value lies on and hence which form of the equation should be used.
a = 60, b = 75, c = (-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) =

17. Monte Carlo Method Monte Carlo Method
Since RN < 0.8 use first part of Eqn (10.10)

18. Monte Carlo Method First part of curve – Eqn (10.10) x<b 0.80
0.3501
x = 69.92
b = 75

19. 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

20. Monte Carlo Method
So this figure shows that about 38% of the simulations show a loss. Also that our “average” values that give rise to a NPV of 17.2 are only likely to be exceeded ~20% of the time. The choice of “should we build or not?” depends on many things such as risk philosophy in the company and how alternative projects stack up against this one and also the risk associated with alternative investments.

21. Monte Carlo Method
Figure shows two projects A and B. Although Project B may look less favorable based on average values, when the distribution of NPV is calculated, Project B is much less likely to loose money and this may affect the decision in which project to invest.

22. Monte Carlo Method
As a follow on to Example 8.1, set up the problem in CAPCOST (already done if following the Powerpoint slides for the beginning of the chapter) and set the ranges for R, FCI, and COMd (which will be the raw materials cost for this example) to the values used in this example and then generate the above figure. Note that every MC simulation is a little different and so actual values may deviate slightly from the above. After Depreciation methods screen has been filled in, press “run Economic Analysis”

23. Monte Carlo Method Results using Capcost for Monte Carlo Simulations
Results show that only about 37% of the simulations are profitable and that the NPV using the “expected” values is exceeded in only 20% of the runs. These results are essentially the same as shown in Figure

24. 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
Results show that only about 37% of the simulations are profitable and that the NPV using the “expected” values is exceeded in only 20% of the runs. These results are essentially the same as shown in Figure