Probabilistic Cash Flow Analysis Lecture No 39 Chapter 12 Contemporary Engineering Economics Copyright, © 2016 Contemporary Engineering Economics, 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, 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 Contemporary Engineering Economics, th edition Park • Cumulative probability distribution o Discrete F ( x ) P ( X �x ) o Continuous j �p j 1 f(x)dx Copyright © 2016 by Pearson Education, Inc All Rights Reserved j Useful Continuous Probability Distributions in Cash Flow Analysis (b) Uniform Distribution (a) Triangular Distribution L: minimum value Mo: mode (most-likely) H: maximum value Contemporary Engineering Economics, 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) 1,600 0.20 $48 0.30 2,000 0.60 50 0.50 2,400 0.20 53 0.20 Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Cumulative Probability Distribution for X Unit Demand (x) 1,600 2,000 2,400 Probability P(X = x) 0.2 0.6 0.2 F (x) P( X �x) 0.2, x �1,600 0.8, x �2,000 1.0, x �2,400 Contemporary Engineering Economics, 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) Contemporary Engineering Economics, th edition Park Unit Price (Y) Copyright © 2016 by Pearson Education, Inc All Rights Reserved Measure of Expectation • Discrete case j E [ X ] �( p j ) x j j 1 • Continuous case E[X] = xf(x)dx Contemporary Engineering Economics, th edition Park Event Return (%) 6% 9% 18% Probability 0.40 0.30 0.30 Weighted 2.4% 2.7% 5.4% Expected Return (μ) 10.5% Copyright © 2016 by Pearson Education, Inc All Rights Reserved Measure of Variation Formula Variance Calculation μ = 10.5% �j (xj )2 (pj ), discrete case �� �j1 Var X X2 � H �(x )2 f (x)dx, continuous case � � �L or Var X E � X2 � (E X )2 � � Event Probability Deviation Squared 0.40 (6 − 10.5)2 8.10 0.30 (9 − 10.5)2 0.68 0.30 (18 − 10.5)2 16.88 Variance (σ2) = 25.66 σ= 5.07% x Var X Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Weighted Deviation Example 12.5: Calculation of Mean and Variance Xj Pj Xj(Pj) (Xj-E[X]) (Xj-E[X])2 (Pj) 1,600 0.20 320 (-400)2 32,000 2,000 0.60 1,200 0 2,400 0.20 480 (400)2 32,000 E[X] = 2,000 Var[X] = 64,000 σ = 252.98 Yj Pj Yj(Pj) [Yj-E[Y]]2 (Yj-E[Y])2 (Pj) $48 0.30 $14.40 (-2)2 1.20 50 0.50 25.00 (0) 53 0.20 10.60 E[Y] = 50.00 (3)2 1.80 Var[Y] = 3.00 σ = $1.73 Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Case 1: Independent Random Cash Flows E [P W (i ) ] N � n V a r[P W (i ) ] E (An ) (1 i ) n N � n V a r(A n ) (1 i ) n where i = a risk-free discount rate, An = net cash flows in period n, E[An ] = expected net cash flows in period n, Var[An ] = variance of the net cash flows in period n E[PW(i)] = expected net present worth of the project, and Var[PW(i)] = variance of the net present worth of the project Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Case 2: Dependent Cash Flows Contemporary Engineering Economics, 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, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Solution: NPW Distribution Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Case 1: Independent Cash Flows Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Case 2: Dependent Cash Flows Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Normal Distribution Assumption Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved NPW Distribution with ±3σ Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Expected Return/Risk Trade-of Probability (%) Investment A Investment B -30 -20 -10 10 20 30 40 50 Return (%) Contemporary Engineering Economics, 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 diferent NPW distributions at MARR = 10% Find: The best model to market Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved Comparison Rule o If EA > EB and VA ≤ VB, select A o If EA = EB and VA ≤ VB, select A o If EA < EB and VA ≥ VB, select B o If EA > EB and VA > VB, Not clear Contemporary Engineering Economics, th edition Park Model Type E (NPW) Var (NPW) Model $1,950 747,500 Model 2,100 915,000 Model 2,100 1,190,000 Model 2,000 1,000,000 Model vs Model Model vs Model Model vs Model Model >>> Model Model >>> Model Can’t decide Copyright © 2016 by Pearson Education, Inc All Rights Reserved Mean-Variance Chart Showing Project Dominance Contemporary Engineering Economics, 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, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved o o 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 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 The end goal of a probabilistic analysis of project variables is to produce an NPW distribution Contemporary Engineering Economics, 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 bestand 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) Contemporary Engineering Economics, th edition Park Copyright © 2016 by Pearson Education, Inc All Rights Reserved ... +245 -9X -6,000 0.6X(Y-15) +33,617 Cash inflow: Net salvage X(1-0.4)Y 0.4 (dep) Cash outflow: Investment -125,000 -X(1-0.4)($15) -(1-0.4)($10,000) Net Cash Flow -125,000 Contemporary Engineering... Function o Cash inflow: o PW(15%) = 0.6 XY (P/A, 15%, 5) + $44,490 = 2.0113XY + $44,490 o Cash outflow: o PW(15%) = $125,000 + (9 X + $6,000)(P/A, 15%, 5) = 30.1694X + $145,113 o Net cash flows:... Independent Random Cash Flows E [P W (i ) ] N � n V a r[P W (i ) ] E (An ) (1 i ) n N � n V a r(A n ) (1 i ) n where i = a risk-free discount rate, An = net cash flows in period n,