Lecture no39 probabilistic cash flow analysis

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

Probabilistic Cash Flow Analysis

Lecture No 39 Chapter 12 Contemporary Engineering Economics

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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

Types of Probability Distribution

Useful Continuous Probability Distributions

in Cash Flow Analysis

(a) Triangular Distribution (b) Uniform Distribution

L: minimum value

Mo: mode (most-likely)

H: maximum value

Discrete Distribution: Probability

Distributions for Unit Demand (X) and Unit

Price (Y) for BMC’s Project

Product Demand (X) Unit Sale Price (Y)

Cumulative Probability Distribution for X

x x

�

�

Probability and Cumulative Probability

Distributions for Random Variable X and Y

9%

18%

0.40 0.30 0.30

2.4% 2.7% 5.4% Expected Return (μ) 10.5%



2

( ) ( ), discrete case Var

( ) ( ) , continuous case or

Example 12.5: Calculation of Mean and Variance

Joint and Conditional Probabilities



 ( , ) (1,600,$48)

( , ) ( ) ( )

P x y P X x Y y P Y y

Assessments of Conditional and Joint

Marginal Distribution for X

X j

1,600 P(1,600, $48) + P(1,600, $50) + P(1,600, $53) = 0.18 2,000 P(2,000, $48) + P(2,000, $50) + P(2,000, $53) = 0.52 2,400 P(2,400, $48) + P(2,400, $50) + P(2,400, $53) = 0.30

y

P x  � P x y

Covariance and Coefficient of Correlation

Calculating the Correlation Coefficient

between X and Y

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.

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

Step 1: Express After-Tax Cash Flow as a Function of

Unknown Unit Demand (X) and Unit Price (Y)

Step 2: Develop an NPW Function

Step 3: Calculate the NPW for Each Event

Event No. x y P[ x,y ]

Cumulative Joint Probability

Step 4: Plot the NPW Distribution

Step 5: Calculate the Mean

Step 6: Calculate the Variance of NPW

Event

No. x y P[x,y] NPW (NPW- E[NPW])2

Weighted (NPW- E[NPW])

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

Case 1: Independent Random Cash Flows

where = a risk-free discount rate, = net cash flows in period , E[ ] = expected net cash flows in period , Var[ ] = variance of the net cash flows in perio

n n n

(1 )

N

n n n

A i

i

Case 2: Dependent Cash Flows

Example 12.7: Aggregation of Risk Over

Time

Solution: NPW Distribution

Case 1: Independent Cash Flows

Case 2: Dependent Cash Flows

Normal Distribution Assumption

NPW Distribution with ±3σ

Expected Return/Risk Trade-of

0 10 20 30 40 50 -10

-20 -30

Investment A

Investment B

Probability (%)

Example 12.8: Comparing Risky Mutually

Exclusive Projects

Green engineering has

developed a prototype

conversion unit that allows a

motorist to switch from

Model 2 vs Model 3 Model 2 >>> Model 3 Model 2 vs Model 4 Model 2 >>> Model 4

Mean-Variance Chart Showing Project Dominance

will not meet our minimum return requirements for

acceptability.

don’t exist in real life The challenge is to decide what

level of risk we are willing to assume and then, having

implications of that choice.

are (1) sensitivity analysis, (2) breakeven analysis, and (3)

o Sensitivity, breakeven, and scenario analyses are

reasonably simple to apply, but also somewhat simplistic and imprecise in cases where we must

analysis of project risk by assigning numerical

values to the likelihood that project variables will have certain values.

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).