Citibank: basic of corporate finance
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6-8 CORPORATE VALUATION – RISK
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The analysts think that there is a 15% chance that the project will
generate $31,000 in cash flow, a 15% chance it will generate
$32,000, and so forth. The calculation is done the same way as the
calculation for the expected rate of return — by multiplying each
possible cash flow by the probability of its occurrence (Column 2
and Column 3). The generic formula for calculating the expected
cash flow E(CF) of an investment or project is the same one we use
for calculating an expected return.
n
Expected Cash Flow = E(CF) = S = P
i
CF
i
i = 1
Suppose there is another project for which analysts have computed
the expected cash flow. Let's call it Project B. The payoff matrix for
Project B is shown in Figure 6.5.
(1) (2) (3) (4)
Estimate
Number Probability
Possible
Cash
Flow
Weighted
Cash Flow
(2) x (3)
1
2
3
4
5
0.10
0.25
0.50
0.10
0.05
$10,000
30,000
35,000
45,000
50,000
$1,000
7,500
17,500
4,500
2,500
E(CF) = $33,000
Figure 6.5: Project B — Payoff Matrix
You can see that the cash flows for the projects are different — but
the expected cash flow for both projects is exactly the same.
Before we compare the merits of Project A and Project B, practice what you know about
calculating the expected cash flow by completing the exercise on the next page.
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CORPORATE VALUATION – RISK 6-9
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PRACTICE EXERCISE 6.1
Directions: For Projects X and Y, calculate the weighted cash flows in Column 4 and
the expected cash flow E(CF). Compare your calculations to the correct
ones on the next page.
Project X
(1) (2) (3) (4)
Estimate
Number Probability
Possible
Cash
Flow
Weighted
Cash Flow
(2) x (3)
1
2
3
0.20
0.50
0.30
$16,000
22,000
26,000
E(CF) =
$_______
_______
_______
$_______
____________________________________________________________________
Project Y
(1) (2) (3) (4)
Estimate
Number Probability
Possible
Cash
Flow
Weighted
Cash Flow
(2) x (3)
1
2
3
0.20
0.50
0.30
$15,000
22,000
40,000
E(CF) =
$_______
_______
_______
$_______
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6-10 CORPORATE VALUATION – RISK
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ANSWER KEY
Project X – Payoff Matrix
(1) (2) (3) (4)
Estimate
Number Probability
Possible
Cash
Flow
Weighted
Cash Flow
(2) x (3)
1
2
3
0.20
0.50
0.30
$16,000
22,000
26,000
E(CF) =
$3,200
11,000
7,800
$22,000
_______________________________________________________________________
Project Y – Payoff Matrix
(1) (2) (3) (4)
Estimate
Number Probability
Possible
Cash
Flow
Weighted
Cash Flow
(2) x (3)
1
2
3
0.20
0.50
0.30
$15,000
22,000
40,000
E(CF) =
$3,000
11,000
12,000
$26,000
If you did not get the correct solutions, we suggest that you review the appropriate text. If
you solved the problems correctly, go on to the next section on "Measuring Risk."
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CORPORATE VALUATION – RISK 6-11
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MEASURING RISK
The payoff matrices for Project A and Project B on pages 6-7 and
6-8 show that the expected cash flow for each project is $33,000.
If you could only invest in one project, which would you choose –
Project A or Project B?
Payoff matrix
analysis
Before you answer that question, let's analyze the payoff matrices
for both projects. In Figures 6.6 and 6.7, you can see graphs of the
possible cash flows from the payoff matrices of both projects.
Along the vertical axis are the probabilities of each possible cash
flow. Along the horizontal axis are the calculated amounts of each
possible cash flow. By plotting each possible cash flow along the
two axes, the differences between them are clearer.
Project A
Figure 6.6: Project A – Possible Cash Flows
Project A:
1) The possible cash flows are concentrated into a narrow range.
2) In the worst case scenario, the project will generate a possible
cash flow of $31,000.
3) In the best case scenario, the project will generate a possible
cash flow of $35,000.
4) The expected cash flow is $33,000.
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Project B
Figure 6.7: Project B – Possible Cash Flows
Project B:
1) The range of possible cash flows is broad.
2) In the worst case scenario, the project will generate a possible
cash flow of $10,000.
3) In the best case scenario, the project will generate a possible
cash flow of $50,000.
4) The expected cash flow is $33,000.
Indicator of
risk
This "spread" between possible cash flows is one indicator of risk for
a potential investment. Project B has a greater spread between
possible cash flows than Project A and, therefore, is considered the
more risky investment.
Project A is probably a better investment because the expected cash
flow is equal to that of Project B, but with much less risk.
We can quantify the risk associated with expected cash flows by using
two statistical computations: variance and standard deviation.
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CORPORATE VALUATION – RISK 6-13
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Variance
Variance for
Project A
The variance is an average of the squared deviations from the possible
cash flows that have been weighted by the probability of each
deviation's occurrence. The deviation is the difference between each
possible cash flow and the expected cash flow. We can modify the
payoff matrix to illustrate the calculation of the variance for Project
A (Figure 6.8).
(1) (2) (3) (4) (5) (6)
Estimate
Number
Probability
P
1
Possible
Cash Flow
CF
1
Deviation
CF
1
–E(CF)
Deviation
Squared
[CF
1
–E(CF)]
2
(2) x (5)
Weighted
Deviation
1
2
3
4
5
0.15
0.15
0.40
0.15
0.15
$31,000
32,000
33,000
34,000
35,000
-2,000
-1,000
0
1,000
2,000
4,000,000
1,000,000
0
1,000,000
4,000,000
Variance =
600,000
150,000
0
150,000
600,000
________
1,500,000
Figure 6.8: Variance for Project A
Calculation of
variance
So that you can fully understand the variance calculation, let's look at
each column of the Project A pay off matrix.
Columns
1, 2, and 3 =
The estimate number, probability, and amount
of possible cash flow – data that is taken
straight from the original payoff matrix.
Column 4 = The deviation, which is computed by subtracting
the expected cash flow for the project ($33,000)
from each possible cash flow (Column 3). It
represents the difference between the possible
cash flow and the expected cash flow.
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Column 5 = Deviation squared (raised to the second power).
The deviations in Column 4 are squared so that
the negative deviations will not cancel out the
positive deviations when a sum is calculated.
Column 6 = The probability (Column 2) multiplied by the
squared deviation (Column 5). This weights
each squared deviation according to the
probability of its occurrence.
Variance = Sum of the weighted deviations.
Statistical
formula for
variance
The statistical formula used to calculate variance is shorthand for
the calculations in Figure 6.8.
n
Variance = s
2
= S [CF
i
– E(CF)]
2
P
i
i = 1
Where:
s
2
= Variance
S = Sum of the series of all possible occurrences
i = 1 = Series begins at the first possible occurrence
n = Number of possible occurrences
[CF
i
– E(CF)]
2
= Each possible cash flow minus the expected
cash flow, squared
P
i
= Probability of occurrence
Let's summarize the steps for calculating the variance:
1) Calculate the expected cash flow.
2) Subtract the expected cash flow from each possible cash flow
to find the deviations.
3) Square each of these deviations.
4) Multiply the squared deviations by their respective probabilities
(to weight them).
5) Total the weighted deviations to arrive at the variance.
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CORPORATE VALUATION – RISK 6-15
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Variance for The variance calculation for Project B is shown in Figure 6.9.
Project B
(1) (2) (3) (4) (5) (6)
Estimate
Number
Probability
P
1
Possible
Cash Flow
CF
1
Deviation
CF
1
–E(CF)
Deviation
Squared
[CF
1
–E(CF)]
2
(2) x (5)
Weighted
Deviation
1
2
3
4
5
0.10
0.25
0.50
0.10
0.05
$10,000
30,000
35,000
45,000
50,000
-23,000
-3,000
2,000
12,000
17,000
529,000,000
9,000,000
4,000,000
144,000,000
289,000,000
Variance =
52,900,000
2,250,000
2,000,000
14,400,000
14,450,000
__________
86,000,000
Figure 6.9: Variance for Project B
Greater
variance
indicates
riskier project
Now we can compare the variance for Project A (1,500,000) with the
variance for Project B (86,000,000). As you can see, the variance of
Project B's possible cash flows is greater than the variance of Project
A's possible cash flows. Therefore, we consider Project B to be
riskier than Project A.
Variance of
possible
returns
Let's return to the XYZ Corporation example (page 6-5) to
demonstrate how to calculate the variance of a set of possible
returns. Remember that the payoff matrix of XYZ Corporation
looked like this:
(1)
State of
Economy
(2)
Probability
(3)
Possible
Return
(4)
Weighted Possible
Return = (2) x (3)
Boom 0.30 100% 30%
Normal 0.40 15% 6%
Recession 0.30 -70% -21%
Expected Return E(k) = 15%
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Let's continue to build the table to demonstrate how to calculate the
variance of XYZ Corporation. In Figure 6.10, the calculations for
Columns 4, 5, and 6 have been added to the first three columns of
the payoff matrix. The deviation is the difference between the
possible return (k
1
) and the expected return [E(k)].
(1) (2) (3) (4) (5) (6)
State of
Economy Probability
Possible
Return
Deviation
k
1
–E(k)
Deviation
Squared
[CF
1
–E(CF)]
2
(2) x (5)
Weighted
Deviation
Boom
Normal
Recession
0.30
0.40
0.30
100%
15%
-70%
85
0
-85
7,225
0
7,225
Variance =
2,167.5
0.0
2,167.5
________
4,335.0
Figure 6.10: Variance Calculation for XYZ Corporation
The variance of the possible stock returns for the XYZ Corporation is
4,335.0. You can see that the variance method used for calculating
possible returns is identical to the variance used for possible cash
flows.
Standard Deviation
Because variance is expressed in terms of squared units, it is difficult
to interpret. Analysts often will convert variance to an expression of
standard deviation as a more convenient measure of risk.
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CORPORATE VALUATION – RISK 6-17
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Square root
of the
variance
Standard deviation is the square root of the variance. The units of
measure for standard deviation are the same as the units of measure
for the data that is being analyzed (e.g. dollars, percentages). The
mathematical formula for the standard deviation of expected cash
flows is:
Where:
s = The square root of the variance, or the variance
raised to the ½ power
S = Sum of the series of all possible occurrences
n = Number of possible occurrences
i = 1 = Series begins at the first possible occurrence
[CF
i
–E(CF)]
2
P
i
= Each possible cash flow minus the expected cash
flow, squared, and then multiplied by each
probability of occurrence
P
i
= Probability of occurrence
To calculate the standard deviation, first find the variance and then take
the square root of the variance. We interpret the standard deviation by
thinking in terms of the average deviation from the expected cash flow
(or return).
Standard
deviations of
Project A and
B cash flows
Let's compare the standard deviations of Projects A and B. Notice
that taking the square root of a number and raising that number to the
½ power are identical calculations.
Project A
• Variance = 1,500,000
• Standard deviation = (1,500,000)
½
or $1,224.74
Project B
• Variance = 86,000,000
• Standard deviation = (86,000,000)
½
or $9,273.61
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