Transcript Lecture 8

Updated: 24 April 2007
EMU
FINA 522: Project Finance and Risk
Management
Lecture Eight
0
FOUNDATIONS OF RISK AND
UNCERTAINTY:
APPLIED STATISTICS FOR
PROJECT APPRAISAL
AND RISK ANALYSIS
1
Why do we need statistics in project appraisal?
 Deterministic Analysis versus Probabilistic (or Stochastic)
Analysis
 What is the value of going beyond deterministic analysis?
Table of Parameters
Price
10
Quantity
200
Yr
Revenues
0
1
2,000
How do we know that the price will be $10 per unit?
How do we know that the quantity sold will be 200 units?
2
• If there is no certainty about the price and the quantity, then
there is no certainty about the outcome, i.e. revenue
• Assumptions about Expected Price and Expected Quantity will
lead to Expected Revenues
• Two-way table for the revenues with price and quantity. The
price is expected to range in discrete values from $8 per unit to
$12 per unit. The quantity is expected to range in discrete
values from 180 units to 220 units
Two-way table for
Revenues with Price and Quantity
Price
8
9
10
11
12
180
1,440
1,620
1,800
1,980
2,160
Quantity
190
200
1,520
1,600
1,710
1,800
1,900
2,000
2,090
2,200
2,280
2,400
210
1,680
1,890
2,100
2,310
2,520
220
1,760
1,980
2,200
2,420
2,640
3
• With the assumptions about the discrete values
for price and quantity, the revenues will range
from $1,440 to $2,640
• With the assignment of discrete values to
prices, what is the implicit assumption about
the probabilities we are giving to the occurrence
of the prices?
• One possible solution is to assign equal
probabilities to the possible discrete prices
4
Case 1: Discrete Uniform Distribution
Price 8
Prob. 20%
9
20%
10
20%
11
20%
12
20%
Uniform, Discrete Probability Distribution
35.0%
Probability
30.0%
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
8
9
10
11
12
Discrete Values for Price
5
Other Possible Allocation
of Probabilities to the Price
Case 2: Symmetric
Price 8
9
10
11
12
Prob. 10% 25% 30% 25% 10%
Probability
Symmetric Discrete Probability Distribution
35.0%
30.0%
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
8
9
10
11
Discrete Values for Price
12
6
Case 3: Non-symmetric
Price
Prob.
8
5%
9
25%
10
30%
11
25%
12
15%
Non-symmetric discrete probability distribution
35.0%
30.0%
Probability
25.0%
20.0%
15.0%
10.0%
5.0%
0.0%
8
9
10
Discrete Values for Price
11
12
7
Measures of Central Tendency:
Mean and Expected Value
MEAN or Average
n
=
X
i 1
n
i
Sum all values
Number of observations
8
Expected or Average Value
n
u  (Prob X )(X )
E( ) 
i
i 1
i
Sum all probabilities times values
9
Calculating Expected Values
For simplicity, suppose that the price of an input
only assumes discrete values as in the following:
Probability of Price
Price of Input
1/4
12
1/2
15
1/4
18
Expected Value or Weighted Mean Price
= (12*1/4 + 15*1/2 + 18*1/4) = 15
10
Price
Prob.
8
5%
9
25%
10
30%
11
25%
12
15%
• For a quantity of 200, what are the expected sales if
the probability distribution is as given ?
• Mean Price = 8*0.05 + 9*0.25+10*0.30 + 11*0.25
+ 12*0.15 = 10.20
• Mean Sales = 10.20 * 200 = 2040
• How do you choose the appropriate probability
distribution for each price? Historical data provides
a good background.
11
Expected Value (Summary)
The expected value or mean is a measure of the central
tendency.
 =
 (Prob of the Value)*(Value)
If all values have the same probability of occurring, the
expected value is simply the mean of all the values.

=
( Pi) / N
where Pi is the value of the ith price and N is the number
of values for the prices.
12
VARIANCE
n
2
(
X


)
 i
i 1
n
The sum of the squares of the difference
between each
observation and the mean
Number of Observations
13
Variance and Standard Deviation:
These are the measures of Dispersion or Variability. Often,
we are interested not only in the price of the input, but also in
the variability of the price around the mean. In other words,
we want to know the level of dispersion of the price near the
mean.
Variance, s2 :
= (Sum of the deviations of the mean, squared)
Number of Prices
= (Xi - )2
n
14
STANDARD DEVIATION
n
2
(
X


)
 i
i 1
Variance
n
15
Combining Trend with Random
Variation of Variable
• Oil prices may be low now but we expect that there will
be a positive trend e.g. mean of 1% a year and standard
deviation of 0.5%.
– We first use the Monte Carlo analysis trend to project the mean
value for each year.
– We then do a Monte Carlo analysis to project the value of the
variable given it first between around the mean for the period.
16
Cyclical Effects
• Some of the variation around the trend can be explained
by changes in other variables such as real income.
• We may wish to take out this effect econometrically
before defining the distribution of the pure random
movement of the variable.
17
Application of Risk Distribution a year at a time
versus applying average to entire period of project
• Suppose we calculate from annual data that the mean of a variable
X with a normal distribution is X and its standard deviation is sx.
• Therefore, if we undertake an analysis where we do the risk analysis
or annual basis then we can use the distribution (
X ,sx)
• If we do a risk analysis where the distribution is applied to n years
of a project. Then the value of the mean of X is still
X
, but its
standard deviation is s x
N
18
Example
•
Example from Annual observation of X, we find that
Mean of X, = X = 80, and S.D. of X is = sx = 60
1.
If applied in risk analysis each year independent of the other than
for each year distribution of X has mean of
X = 80 and S.D. of sx = 60
2.
If applied to 5 years of a project all at the same time then,
X
3.
= 80 and S.D. of sx5 =
sx
60
60


 26.79
N
5 2.24
If applied to 20 years of a project all at the same time then,
X
= 80 and S.D. of sx20 =
sx
60
60


 13.42
N
20 4.47
19
Example (cont’d)
Case 1: Independent Annual Input Distribution
•
Variable X is modeled into risk analysis as an input distribution,
applied independently every year, with:
X = 80
•
and
S.D. of sx = 60
The resulting output distribution of NPV is:
NPV = $2.5 million and sNPV = $1.1million
Case 2: Input Distribution Applied Once at a Time
•
Variable X is modeled into risk analysis of a 5-year project as an
input distribution, applied once at a time, with:
X = 80
•
and
S.D. of sx = 60 (S.D. is not adjusted)
The resulting output distribution of NPV is:
NPV = $2.5 million and sNPV = $2.3 million
20
Example (cont’d)
Case 3: Input Distribution Applied Once at a Time, Standard
Deviation is Adjusted
•
Variable X is modeled into risk analysis of a 5-year project as an
input distribution, applied once at a time, with:
X = 80
•
and
S.D. of sx = 26.79 (S.D. is adjusted)
The resulting output distribution of NPV is:
NPV = $2.5 million and sNPV = $1.1 million
21
COEFFICIENT OF VARIATION
s
X
Standard Deviation
Mean
Relative variation in
relation to the mean
Not a good indicator of risk when considering the variability of NPV for
the project, i.e. normal to have mean NPV = 0, therefore coefficient of
variation = ∞
22
STANDARD DEVIATION OF NPV OVER
THE VALUE OF THE PROJECT
s
Standard Deviation of NPV
X  PVInv. Costs
Mean of NPV + PV of Investment Costs
Relative variation NPV in
relation to the Value of the Project
23
Comparison of Different
Profiles: the Mean-Variance Rule
• The expected return is an indicator of a project’s
profitability
• The variance is an index of its risk
• If we assume that returns are normally distributed so that
the mean and variance provide us with all of the relevant
information about their distributions, all risk averters will
use the Mean-Variance Rule. They can maximize their
expected utility simply by selecting assets that yield the
‘best’ combinations of mean and variance of returns.
24
Case 1:
E(A)
Var(A)
>=
<
E(B) and
Var(B)
Case 2:
E(A)
Var(A)
>
<=
E(B) and
Var(B)
In both cases, a risk averse investors will prefer
project A to B. The outcome is not so clear if
E(A)
Var(A)
>
>
E(B), and
Var(B)
25
Relationship between variables
Covariance: A measure of the strength of the
relationship between two variables.
n
Cov (X, Y) =
1
(x j  μ X )(y j  μ Y )

n j1
26
The correlation coefficient is a measure of the linear
association between two variables
Correlation Coefficient
r xy =
Cov(X,Y)
s s
x
y
By definition, the correlation coefficient ranges from + 1 to -1
-1 < rxy < 1
If two variables are perfectly positively correlated, then the correlation
coefficient is +1
If two variables are perfectly negatively correlated, the correlation
coefficient is -1
If two variables are independent, then the correlation coefficient is
zero
27
Positive Correlation
40.00
35.00
30.00
25.00
20.00
15.00
10.00
8
10
12
14
16
18
20
22
24
26
Negative Correlation
40.00
35.00
30.00
25.00
20.00
15.00
10.00
8
10
12
14
16
18
20
22
24
26
28
What to Look for in Published
Statistics...
• Does the chosen functional form fit the
data?
• How would you select the distribution, and
what does it say about the variable being
summarized?
• What is the range of the data for the
variable?
• What is the variability of the variable?
29
USING HISTORICAL DATA...
Price 1
60
Price
58
56
54
52
50
48
0
2
4
6
8
10
12
14
Years
30
Forecasting the Outcome of a Future Event
From a Frequency to a Probability Distribution
Variable
Value
Frequency
Observations
x
x x x
x
x
x
x
x
Maximum
5
x
3
Minimum
1
Minimum
Now
Time
1
Maximum
Variable Value
Probability
0.5
0.3
0.1
Minimum
0.1
Maximum
Variable Value
31
The Probability Distribution versus
The Cumulative Probability Distribution
The Cumulative Probability Distribution allows us to estimate the
probability for a range of prices
Cumulative Probability Distribution in case of the Non-symmetric
Distribution:
Price
Prob.
Cum.Prob.
8
5%
5%
9
25%
30%
10
30%
60%
11
25%
85%
12
15%
100%
What is the probability of getting a price equal to or less than 9?
What is the probability of getting a price greater than 9?
32
Frequency to a Cumulative Distribution
Cumulative Distribution
Frequency Distribution
Cumulative
Probability
Probability
30%
25%
100%
85%
15%
60%
30%
5%
5%
8
9
10
11
12
X
X
8
9
10
11
12
33
Discrete versus Continuous Distributions
What are the shortcomings of the discrete probability
distribution?
The number of possible values for the price may be greater
than the specified discrete values. For example, the price
may take on a value between 8 and 9. Or the price may be
beyond the specified range from 8 to 12.
Continuous Distributions:
(a)
Uniform
(b)
Triangular
(c)
Step/Custom
(d)
Normal
34
In continuous distributions we have to refer to a
range of values rather than to single values
Uniform Probability Distribution
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
30
29
28
27
26
25
24
23
22
Series1
21
20
Probability
Uniform Probability Distribution, with a range from 20 to
30.
Price of Input
What is the probability of obtaining any single price? What is the
probability of obtaining a price less than 24?
For a uniform distribution, how do we determine the probability for a
certain range of values? What would be the uniform probability if the range
of the price was from 20 to 25?
35
Cumulative Uniform Probability
Distribution
Cumulative Probability
Cumulative Uniform Probability Distribution
100%
80%
60%
Series1
40%
20%
0%
20
21
22
23
24
25
26
27
28
29
30
Price of Input
What is the probability of obtaining a value
greater than 25?
36
Triangular Distribution (Symmetric)
Triangular Probability Distribution, with a range from 20 to 30.
Probability
20.0%
15.0%
Series A
10.0%
5.0%
30
29
28
27
26
25
24
23
22
21
20
0.0%
Price of Input
What is the probability of obtaining a
value less than 23? How would you
compare the triangular distribution with
the uniform distribution?
37
100.0%
80.0%
60.0%
Series A
40.0%
20.0%
30
29
28
27
26
25
24
23
22
21
0.0%
20
Cumulative Probability
Cumulative Triangular Probability Distribution.
Price of Input
Estimate the probability of obtaining a
price less than 27.
38
Step or Custom Distribution
S te p
50%
4 5%
45%
P ro b a b ilit y
40%
35%
30%
30%
25%
20%
20%
15%
10%
5%
5%
0%
20
22.5
25
27.5
30
Ra n g e
What is the probability of obtaining a
value less than 25?
39
Cumulative Step Probability Distribution
Cumulative Probability
Prob.
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
20
22.5
25
27.5
30
Price Range
Estimate the probability of obtaining a
price less than 27.5.
40
Normal (or Gaussian) Distribution
Normal Probability Distribution, with a mean of 52 and a
standard deviation of 4.
Probability
0.200
0.150
0.100
0.050
66
64
62
60
58
56
54
52
50
48
46
44
42
40
38
0.000
Price of Input
The shape of the normal probability distribution is fully
determined by the mean and the standard deviation.
Within one standard deviation of the mean, the total
probability is approximately 68%.
41
• What is the probability that price will fall between 48 and
56? What is the probability that the price will be greater
than 56?
• Within two standard deviations of the mean, the total
probability is 95%.
• What is the probability that price will be between 44 and
60? What is the probability that the price will be greater
than 60?
• Within three standard deviations of the mean, the total
probability is 99%.
• What is the probability that price will be greater than 64%?
What is the probability that the price will be between 56 and
60?
42
Cumulative Normal Probability Distribution.
100.0%
90.0%
80.0%
70.0%
60.0%
40.0%
30.0%
20.0%
10.0%
66
64
62
60
58
56
54
52
50
48
46
44
42
40
0.0%
38
Cumulative Probability
50.0%
Price of Input
What is the probability that the price will be less than 50?
What is the probability that the price will be greater than 50?
43
Normal Probability Distribution, with a mean of 2 and a standard
deviation of 4
0.200
Probability
0.150
0.100
0.050
0.000
-12
-8
-4
0
4
8
12
16
NPV of Project
What is the probability that the NPV is negative?
44
Cumulative Normal Probability Distribution,
100.0%
Cumulative Probability
90.0%
80.0%
70.0%
60.0%
50.0%
40.0%
30.0%
20.0%
10.0%
0.0%
-12
-8
-4
0
4
8
12
16
NPV of Project
45