Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Probabilistic Cash Flow Analysis Lecture No. 39 Chapter 12 Contemporary Engineering Economics Copyright, © 2016

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Probability Concepts for Investment Decisions o Random variable: A variable that can have more than one possible value o Discrete random variables: Random variables that take on only isolated (countable) values o Continuous random variables: Random variables that can have any value in a certain interval o Probability distribution: The assessment of probability for each random event

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Types of Probability Distribution Continuous probability distribution o Triangular distribution o Uniform distribution o Normal distribution Discrete probability distribution Cumulative probability distribution o Discrete o Continuous f(x)dx

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Useful Continuous Probability Distributions in Cash Flow Analysis (a) Triangular Distribution (b) Uniform Distribution L: minimum value M o : mode (most-likely) H: maximum value

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Discrete Distribution: Probability Distributions for Unit Demand (X) and Unit Price (Y) for BMC’s Project Product Demand (X)Unit Sale Price (Y) Units (x)P(X = x)Unit price (y)P(Y = y) 1, $ , ,

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Cumulative Probability Distribution for X Unit Demand (x) Probability P(X = x) 1, , ,4000.2

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Probability and Cumulative Probability Distributions for Random Variable X and Y Unit Demand (X)Unit Price (Y)

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Measure of Expectation Discrete case Continuous case E[X] = xf(x)dx EventReturn (%) ProbabilityWeighted % 9% 18% % 2.7% 5.4% Expected Return (μ)10.5%

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Measure of Variation  Formula EventProbabilityDeviation SquaredWeighted Deviation 10.40(6 − 10.5) (9 − 10.5) (18 − 10.5) Variance (σ 2 ) =25.66 σ =5.07%  Variance Calculation μ = 10.5%

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Example 12.5: Calculation of Mean and Variance XjXj PjPj Xj(Pj)Xj(Pj)(X j -E[X])(X j -E[X]) 2 (P j ) 1, (-400) 2 32,000 2, , , (400) 2 32,000 E[X] = 2,000Var[X] = 64,000 σ = YjYj PjPj Yj(Pj)Yj(Pj)[Y j -E[Y]] 2 (Y j -E[Y]) 2 (P j ) $480.30$14.40(-2) (0) (3) E[Y] = 50.00Var[Y] = 3.00 σ = $1.73

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Joint and Conditional Probabilities

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Assessments of Conditional and Joint Probabilities Unit Price Y Marginal Probability Conditional Unit Sales X Conditional Probability Joint Probability $ , , , , , , , , ,

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Marginal Distribution for X XjXj 1,600P(1,600, $48) + P(1,600, $50) + P(1,600, $53) = ,000P(2,000, $48) + P(2,000, $50) + P(2,000, $53) = ,400P(2,400, $48) + P(2,400, $50) + P(2,400, $53) = 0.30

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Covariance and Coefficient of Correlation

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Calculating the Correlation Coefficient between X and Y

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Meanings of Coefficient of Correlation Case 1: 0 < ρ XY < 1 – Positively correlated. When X increases in value, there is a tendency that Y also increases in value. When ρ XY = 1, it is known as a perfect positive correlation. Case 2: ρ XY = 0 – No correlation between X and Y. If X and Y are statistically independent each other, ρ XY = 0. Case 3: -1 < ρ XY < 0 – Negatively correlated. When X increases in value, there is a tendency that Y will decrease in value. When ρ XY = −1, it is known as a perfect negative correlation.

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Estimating the Amount of Risk Involved in an Investment Project o How to develop a probability distribution of NPW o How to calculate the mean and variance of NPW o How to aggregate risks over time o How to compare mutually exclusive risky alternatives

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Step 1: Express After-Tax Cash Flow as a Function of Unknown Unit Demand (X) and Unit Price (Y). Item Cash inflow: Net salvage X(1-0.4)Y 0.6XY 0.4 (dep) 7,14512,2458,7456,2452,230 Cash outflow: Investment-125,000 -X(1-0.4)($15) -9X -(1-0.4)($10,000) -6,000 Net Cash Flow -125,0000.6X(Y-15) +1, X(Y-15) +6, X(Y-15) +2, X(Y-15) X(Y-15) +33,617

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Step 2: Develop an NPW Function o Cash inflow: o PW(15%) = 0.6 XY (P/A, 15%, 5) + $44,490 = XY + $44,490 o Cash outflow: o PW(15%) = $125,000 + (9 X + $6,000)(P/A, 15%, 5) = X + $145,113. o Net cash flows: o PW(15%) = X(Y − $15) − $100,623

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Step 3: Calculate the NPW for Each Event

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Step 4: Plot the NPW Distribution

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Step 5: Calculate the Mean

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Step 6: Calculate the Variance of NPW

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Aggregating Risk Over Time Approach: Determine the mean and variance of cash flows in each period, and then aggregate the risk over the project life in terms of NPW NPW E[NPW] Var[NPW]

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Case 1: Independent Random Cash Flows

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Case 2: Dependent Cash Flows

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Example 12.7: Aggregation of Risk Over Time  Given: Generalized project cash flows  Find: Mean and variance of NPW

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Solution: NPW Distribution

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Case 1: Independent Cash Flows

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Case 2: Dependent Cash Flows

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Normal Distribution Assumption

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved NPW Distribution with ±3σ

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Expected Return/Risk Trade-off

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Example 12.8: Comparing Risky Mutually Exclusive Projects Green engineering has developed a prototype conversion unit that allows a motorist to switch from gasoline to compressed natural gas.  Given: Four models with different NPW distributions at MARR = 10%.  Find: The best model to market.

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Comparison Rule o If E A > E B and V A ≤ V B, select A. o If E A = E B and V A ≤ V B, select A. o If E A < E B and V A ≥ V B, select B. o If E A > E B and V A > V B, Not clear. Model TypeE (NPW)Var (NPW) Model 1$1,950747,500 Model 22,100915,000 Model 32,1001,190,000 Model 42,0001,000,000 Model 2 vs. Model 3 Model 2 >>> Model 3 Model 2 vs. Model 4 Model 2 >>> Model 4 Model 2 vs. Model 1 Can’t decide

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Mean-Variance Chart Showing Project Dominance

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved Summary o Project risk: the possibility that an investment project will not meet our minimum return requirements for acceptability. o Our real task is not to try to find “risk-free” projects; they don’t exist in real life. The challenge is to decide what level of risk we are willing to assume and then, having decided on your risk tolerance, to understand the implications of that choice. o Three of the most basic tools for assessing project risk are (1) sensitivity analysis, (2) breakeven analysis, and (3) scenario analysis.

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved o Sensitivity, breakeven, and scenario analyses are reasonably simple to apply, but also somewhat simplistic and imprecise in cases where we must deal with multifaceted project uncertainty. o Probability concepts allow us to further refine the analysis of project risk by assigning numerical values to the likelihood that project variables will have certain values. o The end goal of a probabilistic analysis of project variables is to produce an NPW distribution.

Contemporary Engineering Economics, 6 th edition Park Copyright © 2016 by Pearson Education, Inc. All Rights Reserved o From the NPW distribution, we can extract such useful information as the expected NPW value, the extent to which other NPW values vary from, or are clustered around the expected value, (variance), and the best- and worst-case NPWs. o All other things being equal, if the expected returns are approximately the same, choose the portfolio with the lowest expected risk (variance).