Download Probabilistic Cash Flow Analysis

Probabilistic Cash Flow
Analysis
Lecture No. 47
Chapter 12
Contemporary Engineering Economics
Copyright, © 2006
Contemporary Engineering Economics, 4th
edition, © 2007
Probability Concepts for Investment
Decisions




Random variable: variable that
can have more than one
possible value
Discrete random variables:
random variables that take on
only isolated (countable) values
Continuous random variables:
random variables that can have
any value in a certain interval
Probability distribution: the
assessment of probability for
each random event
Contemporary Engineering Economics, 4th
edition, © 2007
Types of Probability Distribution

Continuous Probability Distribution




Triangular distribution
Uniform distribution
Normal distribution
Discrete Probability Distribution
Contemporary Engineering Economics, 4th
edition, © 2007
Cumulative Probability Distribution
j
F( x )  P( X  x )   p j
(for a discrete
random variable)
j 1
 f(x)dx
Contemporary Engineering Economics, 4th
edition, © 2007
(for a continuous
random variable)
Useful Continuous Probability
Distributions in Cash Flow Analysis
(a) Triangular Distribution
(b) Uniform Distribution
L: minimum value
Mo: mode (most-likely)
H: maximum value
Contemporary Engineering Economics, 4th
edition, © 2007
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, 4th
edition, © 2007
Cumulative Probability Distribution for
X
Unit Demand
(x)
Probability
P(X = x)
1,600
0.2
2,000
0.6
2,400
0.2
F ( x)  P( X  x)  0.2,
x  1,600
0.8,
10
. ,
x  2,000
x  2,400
Contemporary Engineering Economics, 4th
edition, © 2007
Probability and Cumulative Probability
Distributions for Random Variable X
Contemporary Engineering Economics, 4th
edition, © 2007
Probability and Cumulative Probability
Distributions for Random Variable Y
Contemporary Engineering Economics, 4th
edition, © 2007
Measure of Expectation
j
E[ X ]     ( p j ) x j
(discrete case)
j 1
 xf(x)dx
Contemporary Engineering Economics, 4th
edition, © 2007
(continuous case)
Expected Return Calculation
Event
1
2
3
Return
(%)
6%
9%
18%
Probability
Weighted
0.40
0.30
0.30
2.4%
2.7%
5.4%
Expected Return (μ)
10.5%
Contemporary Engineering Economics, 4th
edition, © 2007
Measure of Variation
j
Var  X      ( x j   ) ( p j )
2
x
x 
2
j 1
Var X
Var X   p x  ( p j x j )
2
j j
E X
2
 (E X )
Contemporary Engineering Economics, 4th
edition, © 2007
2
2
Variance Calculation
Event
Deviations
Weighted Deviations
1
(6% - 10.5%)2
0.40(6% - 10.5%)2
2
(9% - 10.5%)2
0.30(9% - 10.5%)2
3
(18% - 10.5%)2
0.30(18% - 10.5%)2
( 2) = 25.65
σ = 5.06%
Contemporary Engineering Economics, 4th
edition, © 2007
Example 12.5 Calculation of Mean & 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
  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
(3)2
1.80
E[Y] = 50.00
Var[Y] = 3.00
  1.73
Contemporary Engineering Economics, 4th
edition, © 2007
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)
 (010
. )(0.30)
 0.03
Contemporary Engineering Economics, 4th
edition, © 2007
Assessments of Conditional and Joint
Probabilities
Unit Price Y
$48
50
53
Marginal
Probability
0.30
0.50
0.20
Conditional
Unit Sales X
Conditional
Probability
Joint
Probability
1,600
0.10
0.03
2,000
0.40
0.12
2,400
0.50
0.15
1,600
0.10
0.05
2,000
0.64
0.32
2,400
0.26
0.13
1,600
0.50
0.10
2,000
0.40
0.08
2,400
0.10
0.02
Contemporary Engineering Economics, 4th
edition, © 2007
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
Contemporary Engineering Economics, 4th
edition, © 2007
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
 xy 
Cov( X , Y )
 x y
Contemporary Engineering Economics, 4th
edition, © 2007
Calculating the Correlation Coefficient
between X and Y
Contemporary Engineering Economics, 4th
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Meaning of Coefficient of Correlation

Case 1:

Case 2:

Case 3:
Contemporary Engineering Economics, 4th
edition, © 2007