Probabilistic Risk Analysis

1. Probabilistic Risk Analysis Part 1

2. Topics to be discussed • The definition of a random variable • The basic characteristics of probability distributions • Evaluation of projects with discrete random variables • Evaluation of projects with continuous random variables

3. Risk • The chance that the cash flow will fall short or exceed the estimate – chance of loss. • Sensitivity analysis questions the effect of cash flow deviations and the cost of capital when risk is considered. • Risk analysis is appropriate when significant outcome variations are likely for different future states and meaningful probabilities can be assigned to those states.

4. Decision Making under Uncertainty vs. Risk • Under uncertainty • There are only two or more observable values; • However, it is most difficult to assign the probability of occurrence of the possible outcomes; • At times, no one is even willing to try to assign probabilities to the possible outcomes.

5. Decision Making under Uncertainty vs. Risk • Risk • The process of incorporating explicitly random variation in the estimates of measure of merit for an investment proposal • Initial investments, Operating expenses, Revenues, Useful life, and other economic factors are seen as random variables

6. Risk Analysis • Risk is associated with knowing the following about a parameter: • The number of observable values and, • The probability of each value occurring. • The “state of nature” of the process is known at hand.

7. Risk Analysis • Approaches: • Expected value analysis • Discrete or continuous? • Must assign or assume probabilities/probability distributions. • Simulation Analysis • Assign relevant probability distributions: • Generate simulated data by applying sampling techniques from the assumed distributions

8. Before a Study Is Started • Must decide the following: • Analysis under certainty (point estimates); • Analysis under risk: • Assign probability values or distributions to the specified parameters; • Account for variances; • Which of the parameters are to be probabilistic and which are to be treated as “certain” to occur?

9. Basic Probability and Statistics • RANDOM VARIABLE • A rule that assigns a numerical outcome to a sample space. • Describes a parameter that can assume any one of several values over some range. • Random Variables (RV’s) can be: • Discrete or, • Continuous.

10. Discrete Probability Distributions

11. Continuous Distribution

12. Probability Distributions • Individual probability values are stated as: P(Xi) = probability that X = Xi • “X” represents the random variable or rule (math function, for example) • Xi represents a specific value generated from the random variable, X. • Remember, the random variable, X, is most likely a rule or function that assigns probabilities.

13. Probability Distributions • Probabilities are developed two ways: 1. Listing the outcome and the associated probability: 2. From a mathematical function that is a proper probability function.

14. Probability Concepts • Expected value or mean • Discrete: • Continuous:

15. Probability Concepts • Variance: the sum of squared deviations about the population mean. • Var(X)= E{[X-E(X)]2}= E(X2)-[E(X)]2 • Discrete: • Continuous:

16. Probability Concepts • Standard deviation: • Multiplication of a random variable by a constant • Expected value • Variance

17. Probability Concepts • Addition of two independent random variables • Expected value • Variance

18. Probability Concepts • Linear combination of two or more independent variables. • Expected value • Variance

19. Probability Concepts • Multiplication of two independent random variables • Expected value • Variance

20. Evaluation of Projects with Discrete Random Variables • Factors are modeled as discrete random variables. • Expected value and variance of equivalent measure of merit are investigated and used in the evaluation.

21. Example 1 • A drainage channel in a community where flash floods are experienced has a capacity sufficient to carry 700 cubic feet per second. Engineering studies produce the following data regarding the probability that a given water flow in any one year will be exceeded and the cost of enlarging the channel

22. Example 1 (cont.) The average property damage amounts to $20,000 when serious overflow occurs. Reconstruction of the channel would be financed by 40-year bonds bearing 8% interest per year. Solution: If nothing is done and the drainage’s maximum capacity stays at 700 ft3/sec, Capital Recovery Amount = 0 Expected AW(Property damage cost) = $20,000(0.2)+0(0.8) = $4,000 Total expected AW of this “do nothing” choice would be $4,000

23. Example 1 (cont.) If decide to enlarge the drainage’s maximum capacity to 1,000 ft3/sec, Capital Recovery Amount = $20,000(A/P,8%,40) = $20,000(0.0839) = $1,678 Expected AW(Property damage cost) = $20,000(0.10)+0(0.9) = $2,000 Total expected AW of this “do nothing” choice would be $3,678

24. Example 1 (cont.)

25. Example 1 (cont.)

26. Example 2 • The Heating, Ventilating, and Air-Conditioning (HVAC) system in a commercial building has become unreliable and inefficient. Rental income is being hurt, and the annual expenses of the system continue to increase. Your engineering firm has been hired by the owners to • Perform a technical analysis of the system, • Develop a preliminary design for rebuilding the system, and • Accomplish an engineering economic analysis to assist the owners in making decision. The estimated capital investment cost and annual savings in O&M expenses, based on the preliminary design, are shown in the following table.

27. Example 2 (cont.) The estimated annual increase in rental income with a modern HVAC system has been developed by the owner’s marketing staff and is also provided in the following table. These estimates are considered reliable because of the extensive information available. The useful life of the rebuilt system, however, is quite uncertain. The estimated probabilities of various useful lives are provided. Assume that the MARR = 12% per year and the estimated market value of the rebuilt system at the end of its useful life is zero. Based on this information, what is the E(PW), V(PW), and SD(PW) of the project’s cash flows? Also, what is the probability of the PW >= 0? What decision would you make regarding the project, and how would you justify your decision using the available information?

28. Example 2 (cont.)

29. Example 2 (cont.) • Solution: The PW of the project’s cash flows, as a function of project life (N), is PW(N) = - $521,000 + $79,600(P/A,12%,N)

30. Example 2 (cont.)

31. Example 2 (cont.) E(PW) = $9,984 V(PW) = 477,847x106 SD(PW) = $21,859 Pr{PW >= 0} = 0.3+0.2+0.1+0.05+0.05 = 0.7

32. Evaluation of Projects with Continuous Random Variables • Expected values and the variances of probabilistic factors are computed mathematically. • Simplifying assumptions are made about the distribution of the random variable and the statistical relationship among the values it takes on, for example, • Cash flow amounts are normal distribution. • Cash flow amounts are statistically independent. • When the situation is more complicated, Monte Carlo simulation is normally used.

33. Example 3 • For the following annual cash flow estimates, find the E(PW), V(PW), and SD(PW) of the project. Assume that the annual net cash flow amounts are normally distributed with the expected values and standard deviations as given and statistically independent, and that the MARR = 15% per year.

34. Example 3 (cont.) • Solution: • Assume independency among cash flow amounts. • Expected PW are then linear combination of cash flow amounts.

35. Example 3 (cont.)

36. The probability that the IRR is less than the MARR is the same as the probability that PW is less than zero Example 4 • Refer to example 3. For this problem, what is the probability that the IRR of the cash flow estimates is less than the MARR, Pr{IRR<MARR}? Assume that the PW of the project is a normally distributed random variable, with its mean and variance equal to the values calculated in example 3.