Lecture no39 probabilistic cash flow analysis

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

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Types of Probability Distribution
• Continuous probability
distribution
o Triangular distribution
o Uniform distribution
o Normal distribution

• Discrete probability
distribution

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• Cumulative probability
distribution
o Discrete
F ( x )  P ( X �x ) 

o Continuous

j

�p
j 1

f(x)dx

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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

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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

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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
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Probability and Cumulative Probability
Distributions for Random Variable X and Y
Unit Demand (X)

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Unit Price (Y)

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Measure of Expectation
• Discrete case
j

E [ X ]    �( p j ) x j
j 1

• Continuous case

E[X] =  xf(x)dx
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Event

Return
(%)

1
2
3

6%
9%
18%

Probability

0.40
0.30
0.30

Weighted

2.4%
2.7%
5.4%

Expected Return (μ) 10.5%

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Measure of Variation
Formula

Variance Calculation
μ = 10.5%

�j
(xj  )2 (pj ), discrete case
��
�j1
Var X    X2  �
H
�(x  )2 f (x)dx, continuous case
�
�
�L
or
Var X   E �
X2 �
 (E  X  )2
�
�

Event

Probability

Deviation Squared

1

0.40

(6 − 10.5)2

8.10

2

0.30

(9 − 10.5)2

0.68

3

0.30

(18 − 10.5)2

16.88

Variance (σ2) =

25.66

σ=

5.07%

 x  Var  X 

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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

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)

0

53

0.20

10.60
E[Y] = 50.00

(3)2

1.80
Var[Y] = 3.00
σ = $1.73

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Joint and Conditional Probabilities
P(x, y)  P(X  x Y  y)P(Y  y)

P ( x , y )  P ( x )P ( y )
P ( x , y )  P (1,600,$48)

 P ( x  1,600 y  $48)P(y  $48)
 (0.10)(0.30)
 0.03

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Assessments of Conditional and Joint
Probabilities
Unit Price Y

Marginal
Probability

$48

0.30

50

0.50

53

0.20

Conditional
Unit Sales X
1,600
2,000
2,400
1,600
2,000
2,400

1,600
2,000
2,400

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Conditional
Probability
0.10
0.40
0.50
0.10
0.64
0.26
0.50
0.40
0.10

Joint
Probability
0.03
0.12
0.15
0.05
0.32
0.13
0.10
0.08
0.02

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Marginal Distribution for X
Xj

P(x)  �P(x , y)
y

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

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Covariance and Coefficient of Correlation
Cov( X ,Y )   xy
 E  ( X  E[ X ])(Y  E[Y ])
 E ( XY )  E ( X )E (Y )
  xy x y
Cov( X ,Y )
 xy 
 x y
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Calculating the Correlation Coefficient
between X and Y

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Meanings of Coefficient of Correlation
• Case 1: 0 < ρXY < 1
– Positively correlated. When X increases in value, there is a
tendency that Y also increases in value. When ρXY = 1, it is
known as a perfect positive correlation.

• Case 2: ρXY = 0
– No correlation between X and Y. If X and Y are statistically
independent each other, ρXY = 0.

• Case 3: -1 < ρXY < 0
– Negatively correlated. When X increases in value, there is a
tendency that Y will decrease in value. When ρXY = −1, it is
known as a perfect negative correlation.
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Estimating the Amount of Risk Involved in
an Investment Project
o
o
o
o

How to develop a probability distribution of NPW
How to calculate the mean and variance of NPW
How to aggregate risks over time
How to compare mutually exclusive risky
alternatives

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Step 1: Express After-Tax Cash Flow as a Function of
Unknown Unit Demand (X) and Unit Price (Y).
Item

0

1

2

3

4

5

0.6XY
7,145

0.6XY
12,245

0.6XY
8,745

0.6XY
6,245

0.6XY
2,230

-9X
-6,000
0.6X(Y-15)
+1,145

-9X
-6,000
0.6X(Y-15)
+6,245

-9X
-6,000
0.6X(Y-15)
+2,745

-9X
-6,000
0.6X(Y-15)
+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

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Step 2: Develop an NPW 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:
o PW(15%) = 2.0113 X(Y − $15) − $100,623

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Step 3: Calculate the NPW for Each Event
x

y

P[ x,y ]

Cumulative
Joint
Probability

1

1,600

$48.00

0.06

0.06

$5,574

2

1,600

50.00

0.10

0.16

12,010

3

1,600

53.00

0.04

0.20

21,664

4

2,000

48.00

0.18

0.38

32,123

5

2,000

50.00

0.30

0.68

40,168

6

2,000

53.00

0.12

0.80

52,236

7

2,400

48.00

0.06

0.86

58,672

8

2,400

50.00

0.10

0.96

68,326

9

2,400

53.00

0.04

1.00

82,808

Event No.

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NPW

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Step 4: Plot the NPW Distribution

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Step 5: Calculate the Mean

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Step 6: Calculate the Variance of NPW
Event
No.

Weighted
(NPW- E[NPW])

x

y

P[x,y]

NPW

1

1,600

$48.00

0.06

$5,574

1,196,769,744

$71,806,185

2

1,600

50.00

0.10

12,010

792,884,227

79,228,423

3

1,600

53.00

0.04

21,664

342,396,536

13,695,861

4

2,000

48.00

0.18

32,123

64,725,243

11,650,544

5

2,000

50.00

0.30

40,168

0

0

6

2,000

53.00

0.12

52,236

145,631,797

17,475,816

7

2,400

48.00

0.06

58,672

342,396,536

20,543,792

8

2,400

50.00

0.10

68,326

792,884,227

79,288,423

9

2,400

53.00

0.04

82,808

1,818,132,077

72,725,283

Var[NPW] =

366,474,326

(NPW- E[NPW])

2

σ = $19,144

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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.

0

1

2

3

4

E[NPW]
Var[NPW]
NPW

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5

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Case 1: Independent Random Cash Flows
E [P W (i ) ] 

N

�
n  0

V a r[P W (i ) ] 

E (An )
(1  i ) n

N

�
n 0

V a r(A n )
(1  i ) 2 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.

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