ISE 5434
Economic Evaluation of
(Industrial) Projects and
Processes
Unit #11
Risk Analysis
 Part 2 
Analysis of Risk
•
Risk analysis provides
–
behavior of the measure of economic effectiveness* due
to
random
events
–
potential for reversal in preferences for investment
alternatives
•
Risk analysis measures the “sensitivity” of a
decision to random events
with known distribution
•
Risk analysis is generally performed via Monte
Carlo simulation
* aka measure of merit
Monte Carlo
•
Given Model
–
independent variables(s):
x
i
–
dependent variable(s):
y = f(x)
•
Generate
variate
for each x
–
distributions known
•
Determine Cholesky Factor for Covariance Matrix
–
C = L
T
L;
L ~ Cholesky Factor, C ~ covariance matrix
–
x = Lg + m;
g ~ variate of x w/ mean = 0 and std dev = 1
•
Compute
y
•
Repeat
Random Event
•
A random event is a phenomenon described by a
random variable
•
A random variable is a
function
that assigns real
numbers to elements of a sample space, i.e., the
set of all possible events
–
coin flip sample space: S = {head, tail} ~ {h,t}
–
random variable, X: S
→
reals, e.g., X(h) = 1, X(t) = 0
•
A
random event
is a subset of a sample space, to
which a probability is assigned (by a random
variable)
Probability Distribution
•
Associated with every random variable, X, is a
cumulative
probability distribution
, F
∑
≤
=
≤
=
x
x
i
i
)
x
(
p
}
x
X
Pr{
)
x
(
F
∫
∞

=
≤
=
x
dt
)
t
(
f
}
x
X
Pr{
)
x
(
F
discrete
continuous
probability density function ~ frequency
probability mass function ~ frequency
Probability Mass Function
Sample space is countable: S= {x
1
, x
2
,…}
1
)
x
(
p
S
x
=
∑
∈
]
1
,
0
[
S
:
p
→
=
=
0
)
x
X
Pr(
)
x
(
p
x
∈
S
x
∉
S
1
2
3
4
5
6
1/6
1/6
1/6
1/6
1/6
1/6
x
p(x)
pmf for a fair die
Probability Density Function
Sample space is uncountable (continuous)
]
1
,
0
[
S
:
f
→
1
dt
)
t
(
f
=
∫
∞
∞

f(t)
t
F(x)
t = x
Useful Excel Functions
•
RAND()
–
U(a,b) ~ RAND()*(ba)+a
–
0
≤
RAND()
≤
1
•
NORMSDIST(z)
•
NORMDIST(x,mean,standard_dev,cumulative)
–
cumulative = 1: f(x)
–
cumulative = 0: F(x)
•
NORMSINV(probability)
•
NORMINV(probability,mean,standard_dev)
F
1
Expectation Values
•
The
expected value
of a random variable X, E[X], is
the sum of X’s values weighted by X’s distribution
discrete
continuous
∫
∞
∞

=
=
=
μ
dx
)
x
(
xf
X
]
X
[
E
∑
=
=
=
=
μ
n
1
i
i
i
)
x
(
p
x
X
]
X
[
E
Expectation Values
]
y
[
E
]
x
[
E
]
y
x
[
E
+
=
+
b
]
x
[
aE
]
b
ax
[
E
+
=
+
P1
P2
Proofs: P1
=
=
0
)
x
X
Pr(
)
x
(
p
p(ax)
= p(x)
p(x+b) = p(x)
multiplying X by a constant does not change Pr(X=x)
adding to X a constant does not change Pr(X=x)
b
]
x
[
aE
)
x
(
p
)
b
ax
(
)
b
ax
(
p
)
b
ax
(
]
b
ax
[
E
N
1
i
i
i
N
1
i
i
i
+
=
+
=
+
+
=
+
∑
∑
=
=
Proofs: P2
)
y
Y
,
x
X
Pr(
)
y
x
(
]
y
x
[
E
j
n
1
i
n
1
j
i
j
i
=
=
+
=
+
∑∑
=
=
)
y
Y
,
x
X
Pr(
y
)
y
Y
,
x
X
Pr(
x
j
n
1
i
n
1
j
i
j
j
n
1
i
n
1
j
i
i
=
=
+
=
=
=
∑∑
∑∑
=
=
=
=
Proofs: P2
)
x
X
Pr(
)
y
Y
,
x
X
Pr(
i
n
1
j
j
i
=
=
=
=
∑
=
)
y
Y
Pr(
)
y
Y
,
x
X
Pr(
j
n
1
i
j
i
=
=
=
=
∑
=
∑
∑
=
=
=
+
=
=
+
n
1
j
j
j
n
1
i
i
i
)
y
Y
Pr(
y
)
x
X
Pr(
x
]
y
x
[
E
]
y
[
E
]
x
[
E
]
y
x
[
E
+
=
+
Variance
]
])
x
[
E
x
[(
E
)
x
(
VAR
2

=
2
2
2
]
x
[
E
]
x
[
E
σ
=

]
]
x
[
E
]
x
[
xE
2
x
[
E
]
])
x
[
E
x
[(
E
2
2
2
+

=

2
2
2
]
x
[
E
]
x
[
E
]
])
x
[
E
x
[(
E

=

Variance
)
x
(
VAR
a
)
b
ax
(
VAR
2
=
+
vanishes when independent
correlation = 0
)
y
,
x
(
COV
2
)
y
(
VAR
)
x
(
VAR
)
y
x
(
VAR
+
+
=
+
P3
P4
Proof: P3
]
)
b
]
x
[
aE
b
ax
[(
E
)
b
You've reached the end of your free preview.
Want to read all 67 pages?
 Winter '14
 Normal Distribution, Probability theory, probability density function, Cumulative distribution function