Understanding Investment Risk and Portfolio Diversification

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Copyright © 2001 by Harcourt, Inc. All rights reserved.
CHAPTER 6
Risk and Rates of Return
Stand-alone risk
Portfolio risk
Risk & return: CAPM/SML

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What is investment risk?
Investment risk pertains to the
probability of actually earning a
low or negative return.
The greater the chance of low or
negative returns, the riskier the
investment.

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RISK
Risk is defined in Webster’s as “a hazard; a peril;
exposure to loss or injury.”
Thus, risk refers to the chance that some
unfavorable event will occur.
If you go skydiving, you are taking a chance with
your life—skydiving is risky. If you bet on horse
races, you are risking your money.

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STAND-ALONE RISK
An asset’s risk can be analyzed in two ways:
(1) on a stand-alone basis, where the asset is
considered in isolation, and
(2) on a portfolio basis, where the asset is held as
one of a number of assets in a portfolio.
Thus, an asset’s stand-alone risk is the risk an
investor would face if she held only this one asset.
Obviously, most assets are held in portfolios, but it is
necessary to understand stand-alone risk in order to
understand risk in a portfolio context.

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Probability Distributions
An event’s probability is defined as the chance that
the event will occur. For example, a weather
forecaster might state: “There is a 40% chance of rain
today and a 60% chance that it will not rain.”
If all possible events, or outcomes, are listed, and if a
probability is assigned to each event, then the listing is
called a probability distribution.
Keep in mind that the probabilities must sum to 1.0, or
100%.

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Probability distribution
Expected Rate of Return
Rate of
return (%)100150-70
Firm X
Firm Y

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Investment Alternatives
(Given in the problem)
Economy Prob. T-Bill HT Coll USR MP
Recession 0.1 8.0% -22.0% 28.0% 10.0% -13.0%
Below avg. 0.2 8.0 -2.0 14.7 -10.0 1.0
Average 0.4 8.0 20.0 0.0 7.0 15.0
Above avg. 0.2 8.0 35.0 -10.0 45.0 29.0
Boom 0.1 8.0 50.0 -20.0 30.0 43.0
1.0

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Annual Total Returns,1926-1998
Average Standard
Return Deviation Distribution
Small-company
stocks 17.4% 33.8%
Large-company
stocks 13.2 20.3
Long-term
corporate bonds 6.1 8.6
Long-term
government 5.7 9.2
Intermediate-term
government 5.5 5.7
U.S. Treasury
bills 3.8 3.2
Inflation 3.2 4.5
0 17.4%
0 13.2%
0 6.1%
0 5.7%
05.5%
03.8%
03.2%

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Why is the T-bill return independent
of the economy?
Will return the promised 8%
regardless of the economy.

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Do T-bills promise a completely
risk-free return?
No, T-bills are still exposed to the
risk of inflation.
However, not much unexpected
inflation is likely to occur over a
relatively short period.

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Do the returns of HT and Coll. move
with or counter to the economy?
HT: Moves with the economy, and
has a positive correlation. This is
typical.
Coll: Is countercyclical of the
economy, and has a negative
correlation. This is unusual.

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Expected Rate of Return
If we multiply each possible outcome by its
probability of occurrence and then sum these
products, the result is a weighted average of
outcomes. The weights are the probabilities,
and the weighted average is the expected rate
of return, r^ , called “r-hat.”
 The expected rates of return for both Sale.com
and Basic Foods are shown in Figure 6-2 to be
15%. This type of table is known as a payoff
matrix.

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The range of probable returns for Sale.com is from -60% to +90%, with an expected
return of 15%.
The expected return for Basic Foods is also 15%, but its range is much narrower.

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Measuring Stand-Alone Risk:
The Standard Deviation
A common definition that is satisfactory for many
purposes is stated in terms of probability distributions
such as those presented in previous slides
The tighter the probability distribution of expected
future returns, the smaller the risk of a given
investment.
According to this definition, Basic Foods is less risky
than Sale.com because there is a smaller chance that
its actual return will end up far below its expected
return.

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Calculations
Standard Deviation

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Using Historical Data to Measure Risk

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Coefficient of Variation (CV)
Standardized measure of dispersion
about the expected value:
Shows risk per unit of return.
CV = = .
Std dev σ
^rMean

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0
A B
σA = σB , but A is riskier because larger
probability of losses.
= CVA > CVB.
σ
^r

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Portfolio Risk and Return
Assume a two-stock portfolio with
$50,000 in HT and $50,000 in
Collections.
Calculate rp and σp.^

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Portfolio Return, rp
rp is a weighted average:
rp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.
rp is between rHT and rCOLL.
^
^
^
^
^ ^
^ ^
rp = Σ wiri.
n
i = 1

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CVp = = 0.34.3.3%
9.6%
σp = = 3.3%.
1 2/






















(3.0 – 9.6)2
0.10
+ (6.4 – 9.6)2
0.20
+ (10.0 – 9.6)20.40
+ (12.5 – 9.6)2
0.20
+ (15.0 – 9.6)2
0.10

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σp = 3.3% is much lower than that of either
stock (20% and 13.4%).
σp = 3.3% is lower than average of HT and
Coll = 16.7%.
∴ Portfolio provides average r but lower
risk.
Reason: negative correlation.
^

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General statements about risk
Most stocks are positively correlated.
rr,m ≈ 0.65.
σ ≈ 35% for an average stock.
Combining stocks generally lowers risk.

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Returns Distribution for Two Perfectly
Negatively Correlated Stocks (r = -1.0) and
for Portfolio WM
25
15
0
-10 -10 -10
0 0
15 15
25 25
Stock W Stock M Portfolio WM
.
. .
. .
.
.
..
.
. . . . .

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The reason Stocks W and M can be combined to
form a riskless portfolio is that their returns move
countercyclically to each other—when W’s returns
fall, those of M rise, and vice versa. The tendency of
two variables to move together is called correlation,
and the correlation coefficient measures this
tendency. In statistical terms, we say that the returns
on Stocks W and M are perfectly negatively
correlated, with ρ = −1.0.

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What would happen to the
riskiness of an average 1-stock
portfolio as more randomly
selected stocks were added?
σp would decrease because the
added stocks would not be
perfectly correlated but rp would
remain relatively constant.
^

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The tendency of two variables to move
together is called correlation, and the
correlation coefficient measures this
tendency.

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Large
0 15
Prob.
2
1
Even with large N, σp ≈ 20%

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# Stocks in Portfolio
10 20 30 40 2,000+
Company Specific Risk
Market Risk
20
0
Stand-Alone Risk, σp
σp (%)
35

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Diversifiable risk
Diversifiable risk is caused by such random events
as lawsuits, strikes, successful and unsuccessful
marketing programs, winning or losing a major
contract, and other events that are unique to a
particular firm.
Because these events are random, their effects on a
portfolio can be eliminated by diversification—bad
events in one firm will be offset by good events in
another

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Market risk
Market risk, on the other hand, stems from
factors that systematically affect most firms:
war, inflation, recessions, and high interest
rates. Because most stocks are negatively
affected by these factors, market risk cannot
be eliminated by diversification.

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As more stocks are added, each
new stock has a smaller risk-
reducing impact.
σp falls very slowly after about 10
stocks are included, and after 40
stocks, there is little, if any, effect.
The lower limit for σp is about
20% = σM .

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Stand-alone Market Firm-specific
Market risk is that part of a security’s
stand-alone risk that cannot be eliminated
by diversification, and is measured by
beta.
Firm-specific risk is that part of a security’s
stand-alone risk that can be eliminated by
proper diversification.
risk risk risk= +

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By forming portfolios, we can
eliminate about half the riskiness of
individual stocks (35% vs. 20%).
If you chose to hold a one-stock
portfolio and thus are exposed to
more risk than diversified investors,
would you be compensated for all the
risk you bear?

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NO!
Stand-alone risk as measured by a
stock’s σor CV is not important to a
well-diversified investor.
Rational, risk averse investors are
concerned with σp , which is based on
market risk.

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There can only be one price, hence
market return, for a given security.
Therefore, no compensation can be
earned for the additional risk of a one-
stock portfolio.

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Individual Stocks’ Betas
The tendency of a stock to move up and down with
the market is reflected in its beta coefficient.
An average-risk stock is defined as one with a beta
equal to 1 (b = 1.0).
Beta measures a stock’s market risk.
It shows a stock’s volatility relative to the market.
Beta shows how risky a stock is if the stock is held
in a well-diversified portfolio.

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A portfolio of such b = 1.0 stocks will move up and down
with the broad market indexes, and it will be just as risky
as the market.
A portfolio of b = 0.5 stocks tends to move in the same
direction as the market, but to a lesser degree.
On the other hand, a portfolio of b = 2.0 stocks also tends
to move with the market, but it will have even bigger
swings than the market.

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How are betas calculated?
Run a regression of past returns on
Stock i versus returns on the market.
Returns = D/P + g.
The slope of the regression line is
defined as the beta coefficient.

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Year rM ri
1 15% 18%
2 -5 -10
3 12 16
.
.
.
ri
_
rM
_-5 0 5 10 15 20
20
15
10
5
-5
-10
Illustration of beta calculation:
Regression line:
ri = -2.59 + 1.44 rM
^ ^

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Portfolio Betas
 Here bp is the beta of the portfolio, which shows its
tendency to move with the market; wi is the fraction of
the portfolio invested in Stock i; and bi is the beta
coefficient of Stock i.
 For example, if an investor holds a $100,000 portfolio
consisting of $33,333.333 invested in each of three
stocks, and if each of the stocks has a beta of 0.70, then
the portfolio’s beta will be
bp = 0.70: bp = 0.3333(0.70) + 0.3333(0.70) + 0.3333(0.70) = 0.70
 An important aspect of the CAPM is that the beta of a
portfolio is a weighted average of its individual
securities’ betas:

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If beta = 1.0, average stock.
If beta > 1.0, stock riskier than
average.
If beta < 1.0, stock less risky than
average.
Most stocks have betas in the range
of 0.5 to 1.5.

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List of Beta Coefficients
Stock Beta
Merrill Lynch 2.00
America Online 1.70
General Electric 1.20
Microsoft Corp. 1.10
Coca-Cola 1.05
IBM 1.05
Procter & Gamble 0.85
Heinz 0.80
Energen Corp. 0.80
Empire District Electric 0.45

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Can a beta be negative?
Answer: Yes, if ri, m is negative.
Then in a “beta graph” the
regression line will slope
downward. Though, a negative
beta is highly unlikely.

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HT
T-Bills
b = 0
ri
_
rM
_
-20 0 20 40
40
20
-20
b = 1.29
Coll.
b = -0.86

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Riskier securities have higher
returns, so the rank order is OK.
HT 17.4% 1.29
Market 15.0 1.00
USR 13.8 0.68
T-bills 8.0 0.00
Coll. 1.7 -0.86
 Expected Risk
Security Return (Beta)

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Use the SML to calculate the
required returns.
Assume rRF = 8%.
Note that rM = rM is 15%. (Equil.)
RPM = rM – rRF = 15% – 8% = 7%.
SML: ri = rRF + (rM – rRF)bi .
^

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Required Rates of Return
rHT = 8.0% + (15.0% – 8.0%)(1.29)
= 8.0% + (7%)(1.29)
= 8.0% + 9.0% = 17.0%.
rM = 8.0% + (7%)(1.00)= 15.0%.
rUSR = 8.0% + (7%)(0.68)= 12.8%.
rT-bill = 8.0% + (7%)(0.00)= 8.0%.
rColl = 8.0% + (7%)(-0.86)= 2.0%.

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HT 17.4% 17.0% Undervalued:
r > r
Market 15.0 15.0 Fairly valued
USR 13.8 12.8 Undervalued:
r > r
T-bills 8.0 8.0 Fairly valued
Coll. 1.7 2.0 Overvalued:
r < r
Expected vs. Required Returns
^
^
^
^
r r

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.
.Coll.
.HT
T-bills
. USR
SML
rM = 15
rRF = 8
-1 0 1 2
.
SML: ri = 8% + (15% – 8%) bi .
ri (%)
Risr, bi

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Calculate beta for a portfolio with 50%
HT and 50% Collections
bp= Weighted average
= 0.5(bHT) + 0.5(bColl)
= 0.5(1.29) + 0.5(-0.86)
= 0.22.

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The required return on the HT/Coll.
portfolio is:
rp = Weighted average r
= 0.5(17%) + 0.5(2%) = 9.5%.
Or use SML:
rp= rRF + (rM – rRF) bp
= 8.0% + (15.0% – 8.0%)(0.22)
= 8.0% + 7%(0.22) = 9.5%.

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If investors raise inflation
expectations by 3%, what would
happen to the SML?

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SML1
Original situation
Required Rate
of Return r (%)
SML2
0 0.5 1.0 1.5 Risk, bi
18
15
11
8
New SML
∆ I = 3%

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If inflation did not change but risk
aversion increased enough to cause
the market risk premium to increase
by 3 percentage points,
what would happen to the SML?

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rM = 18%
rM = 15%
SML1
Original situation
Required Rate
of Return (%)
SML2
After increase
in risk aversion
Risk, bi
18
15
8
1.0
∆ RPM = 3%

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Has the CAPM been verified through
empirical tests?
Not completely.
Those statistical tests have
problems that make verification
almost impossible.

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Investors seem to be concerned
with both market risk and total
risk. Therefore, the SML may not
produce a correct estimate of ri:
ri = rRF + (rM – rRF)b + ?

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Also, CAPM/SML concepts are
based on expectations, yet betas
are calculated using historical
data. A company’s historical data
may not reflect investors’
expectations about future
riskiness.

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Capital Asset Pricing Model (CAPM)
 Capital Asset Pricing Model (CAPM), an important
tool used to analyze the relationship between risk and
rates of return.
 The primary conclusion of the CAPM is this: The
relevant risk of an individual stock is its contribution to
the risk of a well diversified portfolio.
 A stock might be quite risky if held by itself, but—
since about half of its risk can be eliminated by
diversification—the stock’s relevant risk is its
contribution to the portfolio’s risk, which is much
smaller than its stand-alone risk.
 The risk that remains after diversifying is called market
risk, the risk that is inherent in the market

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