Citibank: basic of corporate finance
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6-18 CORPORATE VALUATION – RISK
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As you can see, the standard deviation of Project A's possible cash
flows is considerably less than the standard deviation of Project B's. In
other words, Project A's possible cash flows, on average, deviate less
from the expected cash flow.
Lower standard
deviation means
less volatility
Project A has the same expected cash flow as Project B, but with
considerably less volatility (or spread) among the possible cash flows.
Project A is considered less risky than Project B and, therefore, the
better investment.
Standard
deviation of
expected
returns
The method for calculating the standard deviation of a set of possible
returns is to take the square root of the variance of that set of returns.
For example, the standard deviation for the possible stock returns of
XYZ Corporation would be s = (4,335)
½
= 65.84%.
The formula for the standard deviation of expected returns is:
Summary
In this section, we discussed the use of the statistical measures of
variance and standard deviation to measure the risk associated with
a project or investment. The variance is the sum of the squared
deviations from the possible cash flows or expected returns that have
been weighted by the probability of each deviation's occurrence.
Standard deviation (the square root of the variance) is preferred by
analysts because it is easier to use as a measure of risk. A lower
standard deviation indicates less volatility among possible cash flows
or returns and, therefore, a lower amount of risk.
Before continuing to the section on "Portfolio Risk," please complete the Practice
Exercise which follows.
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PRACTICE EXERCISE 6.2
Directions: For Project X and Project Y, use the information provided in the payoff
matrices to calculate the variation and the standard deviation. After you
have completed every exercise on this page, compare your answers to the
correct ones on the next page.
1. For Project X, the expected cash flow is $22,000.
(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
0.20
0.50
0.30
$16,000
22,000
26,000
________
________
________
________
________
________
Variance =
________
________
________
_____________
For Project X the standard deviation is ________________________
2. For Project Y, the expected cash flow is $26,000.
(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
0.20
0.50
0.30
$15,000
22,000
40,000
_______
_______
_______
_______
_______
_______
Variance =
_______
_______
_______
_____________
For Project Y the standard deviation is ________________________
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ANSWER KEY
1. For Project X, the expected cash flow is $22,000.
(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
0.20
0.50
0.30
$16,000
22,000
26,000
-6,000
0
4,000
36,000,000
0
16,000,000
Variance =
7,200,000
0
4,800,000
12,000,000
For Project X the standard deviation = (variance)
½
= (12,000,000)
½
= 3,464.10
2. For Project Y, the expected cash flow is $26,000.
(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
0.20
0.50
0.30
$15,000
22,000
40,000
-11,000
-4,000
14,000
121,000,000
16,000,000
196,000,000
Variance =
24,200,000
8,000,000
58,800,000
91,000,000
For Project Y the standard deviation = (variance)
½
= (91,000,000)
½
= 9,539.4
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PORTFOLIO RISK
Up to now, the discussion has focused on the methods used to identify
and measure risk associated with individual investment securities. Large
investors such as mutual funds, pension funds, insurance companies, and
other financial institutions own stocks of many different companies —
government regulations require that they do so. Individual investors also
purchase the stock
of several different firms rather than invest with one company. In this
section, we explain the reasons why investors prefer to vary their stock
holdings.
Portfolio Diversification
The collection of investment securities held by an investor is
referred to as a portfolio. Diversification is the strategy of investing
with several different industries. If an investor maintains a portfolio
that holds Ford Motor Company, Exxon, and IBM stocks, that
investor is said to have a portfolio diversification.
Focus on
expected return
of a portfolio
An investor with a diversified portfolio is not overly concerned when
an individual security in the portfolio moves up or down. The main
concern is the riskiness and overall return of the portfolio.
Therefore, the risk and return of each security needs to be analyzed in
terms of how that security affects the risk and return of the entire
portfolio.
The method for calculating the expected rate of return of a portfolio
of stocks can be illustrated with an example.
Example
An investor has invested $10,000 in a portfolio of stocks. The
amount invested and the expected rate of return for each stock in that
portfolio is as follows:
• $2,000 invested in General Motors, which is expected to earn a
14% rate of return
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• $5,000 invested in Ford Motor Company, which is expected to
generate a 17% return
• $3,000 invested in Chrysler Motors, which is expected to earn
an 18% return
Now, let's arrange the portfolio in a payoff matrix.
Stock Amount
(W
i
)
Weight
(k
i
)
Expected
Return
(W
i
x k
i
)
Weighted
Return
GM
Ford
Chrysler
$2,000
5,000
3,000
0.20
0.50
0.30
14%
17%
18%
Expected
Return E (k
p
) =
2.80
8.50
5.40
16.70%
Figure 6.11: Payoff Matrix for Stock Portfolio
The expected return of the portfolio is calculated using almost the
same method as we use to calculate the expected return for a range of
possible returns of an individual stock. The only difference is that the
weights in a portfolio are derived from the percentage of an
individual investment relative to the total amount invested in the
portfolio. In this example, the total portfolio investment is $10,000;
the investment in General Motors stock is $2,000 or 20% of the
portfolio. Therefore, the portfolio weight of GM stock in the
payoff matrix is .20. The weights always total 1.0.
Calculating risk
for portfolio of
stocks
Unlike the portfolio's expected return, the riskiness of the portfolio
is not a weighted average of the standard deviations of the individual
stocks in the portfolio. The portfolio's risk will be smaller than the
weighted average of the standard deviations because the variations in
the individual stocks will be offsetting to some degree.
Example
Let's consider an example to illustrate the concept of offsetting
variations. The rates of return over a five year period for two stocks
(W and M) are listed in Figure 6.12.
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Stock W
k
w
(%)
Stock M
k
M
(%)
1988
1989
1990
1991
1992
Average Return
Standard Deviation
40
-10
35
-5
15
15
22.6
-10
40
-5
35
15
15
22.6
Figure 6.12: Rates of Return for Stocks W and M over Five Years
In Figure 6.13 you can see the rates of return for both stocks plotted
on graphs. Notice the patterns of rate variation.
Figure 6.13: Rates of Return for Stock W and Stock M
Offsetting
variations
If held individually, both stock W and stock M would have a high level
of risk. As you can see from the graphs, a rise in the rate of return in
stock W is offset by an equal fall in the rate of return of stock M.
Imagine the profile of a portfolio where each of these stocks r
epresents 50% of the total investment.
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The rate of return distribution for portfolio WM is shown in
Figure 6.14.
Figure 6.14: Rate of Return Distribution for Portfolio WM
Risk-free
portfolio
Portfolio WM shows a 15% rate of return for each year, with no
risk at all. The reason that stock W and stock M can be combined to
form a risk-free portfolio is that their returns move in opposition to
each other.
Correlation
coefficient
In statistics, the tendency of two variables to move together is
called correlation. The correlation coefficient (r) is the measure
of this tendency. To learn how to calculate the correlation
coefficient, consult any beginning statistics text. All of the
examples in this workbook will have the correlation coefficient
already calculated. You should become familiar with what the
correlation coefficient represents and how it is used.
Value of r
The correlation coefficient (r) ranges between +1.0 to -1.0. Let's
see what the values represent.
• If r = +1.0 (perfectly positively correlated), the two variables
move up and down in perfect synchronization.
• If r = -1.0 (perfectly negatively correlated), the two variables
always move in exact opposite directions.
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• If r falls between +1.0 and -1.0, correlation occurs in varying
degrees.
• If r = 0.0, the two variables are not related to each other –
changes in one variable are independent of changes in the other.
In the example, stock W and stock M have a correlation coefficient of
-1.0 (perfectly negatively correlated). Diversification with these two
stocks creates a risk-free portfolio.
If two stocks in a portfolio are perfectly positively correlated (r =
+1.0), and if the standard deviations for the stocks are not equal,
diversification does not eliminate any risk. The portfolio of the two
stocks will have the same risk as holding either stock on its own.
Impossible to
eliminate all
risk
Portfolios with two perfectly negatively correlated stocks (r =
-1.0) or two perfectly positively correlated stocks (r = +1.0) are
extremely rare. In fact, most pairs of stocks are positively
correlated, but not perfectly. On average, the correlation coefficient
for the returns of two randomly selected stocks would be about
+0.6. For most pairs of stocks, the correlation coefficient will be
between +0.5 and +0.7. For this reason, it is impossible to eliminate
all risk by combining stocks into a portfolio.
Example
We used portfolio WM to illustrate a point. Let's look at a more
realistic example. Consider stock W and stock Y. The rates of return
for the two stocks and for a portfolio equally invested in the two
stocks are listed in Figure 6.15. The standard deviations for each set
of returns, and the portfolio as a whole, have already been
calculated. The correlation coefficient is r = +0.65.
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Stock W
k
w
(%)
Stock Y
k
y
(%)
Portfolio WY
k
p
(%)
1988
1989
1990
1991
1992
Average Return
Standard Deviation
40
-10
35
-5
15
15
22.6
28
20
41
-17
3
15
22.6
34
5
38
-11
9
15
20.6
Figure 6.15: Return Rates and Standard Deviation for Stocks W
and Y and Portfolio WY
You can see that stock W, stock Y, and portfolio WY each have an
average return of 15%. The average return on the portfolio is the same
as the average return on either stock. The standard deviation is 22.6 for
each stock and 20.6 for the portfolio. Even though the two stocks are
positively correlated (+0.65), the risk is reduced by combining the
stocks into a portfolio. A stock analyst might say that combining the
two stocks "diversifies away some of the risk."
Diversification
eliminates some
risk
This example illustrates that the risk of the portfolio as measured by
the standard deviation is not the average of the risk of individual
stocks in the portfolio. Typically, if the correlations among individual
stocks are positive, but less than +1.0, some of the risk may be
eliminated.
Positive
correlation
between same-
industry
companies
It is important to recognize that companies in the same industry
generally have a more positive correlation coefficient for expected
returns than companies in different industries. For example, you
would expect the correlation between the returns of Ford Motor
Company and the returns of General Motors to be more positive than
the correlation between the returns of Ford Motor Company and
IBM. Thus, a two-stock portfolio of companies within the same
industry is probably riskier than a two-stock portfolio of companies
operating in different industries. Portfolios should be diversified
between industries as well as companies to eliminate the maximum
amount of risk.
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Systematic and Unsystematic Risk
Expected vs.
uncertain return
To evaluate the risk associated with an individual security or a
portfolio of securities, the realized return to investors must be
broken into two parts:
1) Expected return
2) Uncertain return
Expected return
based on
current market
factors
For a stock (equity security), the expected return is the part that
investors and analysts can predict. These predictions are based on
information about current market factors that will influence the stock
in the future. We discussed this process in Unit Two in our
discussion on interest rates. By studying external and internal factors
that affect a company's operations, analysts can assess how each
factor may affect the company. The stock price of a company is
determined by expectations of a company's future prospects based on
the influence of all factors.
Uncertain
return
influenced by
unexpected
factors
The other part of the return of a stock is the uncertain or risky part
that comes from unexpected factors arising during the period. For
example, an unexpected research breakthrough may cause a
company's stock to rise; a sudden, unexpected rise in inflation rates
may cause a company's stock to fall.
Company
specific,
diversifiable
risk
These two parts of the realized return translate into two types of risk:
(1) risk specific to a single company or a small group of companies,
and (2) risk that affects all companies in the market.
In the stock W and stock Y example, when we created a diversified
portfolio to eliminate some of the risk associated with each stock,
we eliminated the company-specific risk. This type of company-
specific, diversifiable risk is called unsystematic risk.
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