4-1
REVIEW OF COVARIANCE
Consider an asset A, whose expected
return is ka.
ka is an unknown variable, BUT with
a known probability distribution.
e.g. Treasury Bill
Amazon.com
4-2
Probability distribution
Amazon
T-Bill
-70
2
15
100
Expected Rate of Return
Rate of
return (%)
4-3
What’s the formula for the standard
deviation and variance?
 = Standard deviation
 =
Variance
n
=
=
2

 (ki  k) Pi .
i=1

2
4-4
What’s the formula for the Covariance?
 = Covariance
(kai - ka)(kbi - kb) Pi
4-5
COVARIANCE
If A and B move up and down
together,
the covariance will be positive.
If A and B move counter to one
another,
the covariance will be negative
If A and B move with no relation,
the covariance will be small.
4-6
COVARIANCE
If the return on either stock is highly
uncertain,
cov will tend to be large, but a
random relation may cancel this.
If either stock has zero STD DEV,
COV will be zero.
4-7
COVARIANCE SUMMARY
COV(A,B) will be large and positive
if: returns on assets have large
standard deviations and move
together.
COV(A,B) will be large and negative
if: returns on assets have large
standard deviations and move
counter to one another.
4-8
COVARIANCE SUMMARY
COV(A,B) will be tend to be small if:
returns move randomly, rather than
up and down with one another;
and/or either has a small standard
deviation.
4-9
How do you know if
the covariance is
large?
4 - 10
CORRELATION COEFFICIENT
rab =

COV(A,B)
StdDev(A)xStdDev(B)

COV(A,B)=rab [StdDev(A)xStdDev(B)]
SCATTER DIAGRAMS
4 - 11
ka
kb
4 - 12
ka
kb
4 - 13
ka
kb
4 - 14
PORTFOLIO VARIANCE: TWO ASSET
CASE
VAR(P) = (x2)Var(A) + (1-x)2Var(B) +
(2)(x)(1-x) COV(A,B)
where
x=proportion of portfolio in Asset A
4 - 15
PORTFOLIO VARIANCE: THREE
ASSET CASE
VAR(P) = (xA)2 Var(A) + (xB)2Var(B) +
(xC)2Var(C) + 2 xAxB cov(A,B)+
2 xAxC cov(A,C) + 2 xBxC cov(B,C)
where
xA = proportion in asset A
xB = proportion in asset B
xC = proportion in asset C
(xA + xB+ XC) =1
4 - 16
CHAPTER 4
Risk and Return: The Basics
 Strange event in intellectual
history
 Basic return concepts
 Basic risk concepts
 Stand-alone risk
 Portfolio (market) risk
 Risk and return: CAPM/SML
4 - 17
What are investment returns?
Investment returns measure the
financial results of an investment.
Returns may be historical or
prospective (anticipated).
Returns can be expressed in:
Dollar terms.
Percentage terms.
4 - 18
What is the return on an investment
that costs $1,000 and is sold
after 1 year for $1,100?
Dollar return:
$ Received - $ Invested
$1,100
$1,000
= $100.
Percentage return:
$ Return/$ Invested
$100/$1,000
= 0.10 = 10%.
4 - 19
What is investment risk?
 Typically, investment returns are not
known with certainty.
 Investment risk pertains to the probability
of earning a return less than that
expected.
 The greater the chance of a return far
below the expected return, the greater the
risk.
 Since investors are risk averse, an asset
with a larger std.dev. Implies a greater
investment risk.
4 - 20
Probability distribution
Stock X
Stock Y
-20
0
15
50
Rate of
return (%)
 Which stock is riskier? Why?
4 - 21
Minicase 4
p.168
Simple?
4 - 22
Assume the Following
Investment Alternatives
Economy
Prob. T-Bill
Alta
Repo
Am F.
MP
Recession
0.10
8.0% -22.0%
28.0%
10.0% -13.0%
Below avg.
0.20
8.0
-2.0
14.7
-10.0
1.0
Average
0.40
8.0
20.0
0.0
7.0
15.0
Above avg.
0.20
8.0
35.0
-10.0
45.0
29.0
Boom
0.10
8.0
50.0
-20.0
30.0
43.0
1.00
4 - 23
What is unique about
the T-bill return?
The T-bill will return 8% regardless
of the state of the economy.
Is the T-bill riskless? Explain.
4 - 24
Do the returns of Alta Inds. and Repo
Men move with or counter to the
economy?
 Alta Inds. moves with the economy, so it is
positively correlated with the economy. This is
the typical situation.
 Repo Men moves counter to the economy.
Such negative correlation is unusual.
 Market portfolio: ?
 American Foam: ? See graph AM F
4 - 25
Calculate the expected rate of return
on each alternative.
^
r = expected rate of return.

r=
n
 rP .
i i
i=1
^
rAlta = 0.10(-22%) + 0.20(-2%)
+ 0.40(20%) + 0.20(35%)
+ 0.10(50%) = 17.4%.
See minicase 4
4 - 26
Alta
Market
Am. Foam
T-bill
Repo Men
^r
17.4%
15.0
13.8
8.0
1.7
 Alta has the highest rate of return.
 Does that make it best?
Sumproduct function
4 - 27
What is the standard deviation
of returns for each alternative?
  Standarddeviation
  Variance  
 2


   ri  r  Pi .

i 1 
n
2
4 - 28
 2


    ri  r  Pi .

i 1 
n
Alta Inds:
 = ((-22 - 17.4)20.10 + (-2 - 17.4)20.20
+ (20 - 17.4)20.40 + (35 - 17.4)20.20
+ (50 - 17.4)20.10)1/2 = 20.0%.
T-bills = 0.0%.
Alta = 20.0%.
Repo = 13.4%.
Am Foam= 18.8%.
Market = 15.3%.
4 - 29
Prob.
T-bill
Am. F.
Alta
0
8
13.8
17.4
Rate of Return (%)
4 - 30
Standard deviation measures the
stand-alone risk of an investment.
The larger the standard deviation,
the higher the probability that
returns will be far below the
expected return.
Coefficient of variation is an
alternative measure of stand-alone
risk.
4 - 31
Expected Return versus Risk
Security
Alta Inds.
Market
Am. Foam
T-bills
Repo Men
Expected
return
17.4%
15.0
13.8
8.0
1.7
Risk, 
20.0%
15.3
18.8
0.0
13.4
Can any of these be excluded?
See Spreadsheet Chart: Northwest Rule in GraphInvestments
4 - 32
It is tempting to say that T-Bills are
least risky and HT is most risky; but
Before reaching a conclusion, we
must consider:
magnitudes of expected returns
(thus C.V)
skewness of distributions
our confidence in the prob.
distributions
relationship between each asset
and other assets that might be
4 - 33
Coefficient of Variation (CV)
Standardized measure of dispersion
about the expected value:
Std dev

CV =
=
.
^
Mean
k
Shows risk per unit of return.
Example illustrating C.V.
Consider two Assets: X & Y
Which has more risk?
 Asset X
4 - 34
 Asset Y
KX = 30%
KY = 10%
StdDev(X)= 10%
StdDev(Y)= 5%
Example illustrating C.V.
Consider two Assets: X & Y
Which has more risk?
 Asset X
4 - 35
 Asset Y
KX = 30%
KY = 10%
StdDev(X)= 10%
StdDev(Y)= 5%
 What is the
probability that
each asset will
have a return
< 10%?
Example illustrating C.V.
Consider two Assets: X & Y
Which has more risk?
 Asset X
4 - 36
 Asset Y
KX = 30%
KY = 10%
StdDev(X)= 10%
StdDev(Y)= 5%
 CV(X)=.10/.30=.33
 CV(Y)=.05/.10=.5
4 - 37
X
Y
0
X > Y , but Y is riskier. Or Alternative
Graph

:
CV
>
CV
.
Y
X
^
k
4 - 38
30%
10%
0%
4 - 39
Coefficient of Variation:
CV = Standard deviation/expected return
CVT-BILLS
= 0.0%/8.0%
= 0.0.
CVAlta Inds
= 20.0%/17.4%
= 1.1.
CVRepo Men
= 13.4%/1.7%
= 7.9.
CVAm. Foam
= 18.8%/13.8%
= 1.4.
CVM
= 15.3%/15.0%
= 1.0.
4 - 40
Expected Return versus Coefficient of
Variation
Security
Alta Inds
Market
Am. Foam
T-bills
Repo Men
Expected
return
17.4%
15.0
13.8
8.0
1.7
Risk:

20.0%
15.3
18.8
0.0
13.4
Risk:
CV
1.1
1.0
1.4
0.0
7.9
4 - 41
Return
Return vs. Risk (Std. Dev.):
Which investment is best?
20.0%
18.0%
16.0%
14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
Alta
Mkt
USR
T-bills
0.0%
Coll.
5.0%
10.0%
15.0%
Risk (Std. Dev.)
20.0%
25.0%
4 - 42
Portfolio Risk and Return
Assume a two-stock portfolio with
$50,000 in Alta Inds. and $50,000 in
Repo Men.
^
Calculate rp and p.
4 - 43
^
Portfolio Return, rp
^
rp is a weighted average:
n
^
^
rp =  wiri
i=1
^
rp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.
^
^
^
rp is between rAlta and rRepo.
4 - 44
Alternative Method
Estimated Return
Economy
Recession
Below avg.
Average
Above avg.
Boom
Prob.
0.10
0.20
0.40
0.20
0.10
Alta
-22.0%
-2.0
20.0
35.0
50.0
Repo
28.0%
14.7
0.0
-10.0
-20.0
Port.
3.0%
6.4
10.0
12.5
15.0
^
rp = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40
+ (12.5%)0.20 + (15.0%)0.10 = 9.6%.
(More...)
4 - 45
 p = ((3.0 - 9.6)20.10 + (6.4 - 9.6)20.20 +
(10.0 - 9.6)20.40 + (12.5 - 9.6)20.20
+ (15.0
- 9.6)20.10)1/2 = 3.3%.
 p is much lower than:
either stock (20% and 13.4%).
average of Alta and Repo (16.7%).
 The portfolio provides average return but
much lower risk.
 Reason: ?
 See spreadsheet for alternate calculation.
4 - 46
Two-Stock Portfolios
Two stocks can be combined to form a
riskless portfolio if r = -1.0.
Risk is not reduced at all if the two
stocks have r = +1.0.
In general, stocks have r  0.65, so
risk is lowered but not eliminated.
Investors typically hold many stocks.
What happens when r = 0?
4 - 47
General Statements About Risk
Most stocks are positively
correlated. rk,m  0.65.
35% for an average stock.
Combining stocks generally lowers
risk.
4 - 48
DIGRESSION:
CONSIDER TWO STOCKS: W & M
Stock W
Stock M
Portfolio:
(50%W,50%M)
kW = 15%
kM= 15%
KP = 15%
StdDev(W)
= 22.6%
StdDev(M)
= 22.6%
StdDev(P)
=?
4 - 49
CONSIDER TWO STOCKS: W & M
Stock W
Stock M
Portfolio:
(50%W,50%M)
kW = 15%
kM= 15%
KP = 15%
StdDev(W)
= 22.6%
StdDev(M)
= 22.6%
StdDev(P)
=?
DEPENDS on COV(A,B) or
rab
4 - 50
W&M: rWM = -1.0
Year
1986
1987
1988
1989
1990
AvgRet.
StdDev.
Stock W
40%
(10%)
35
(5)
15
15%
22.6%
Stock M
(10%)
40%
(5)
35
15
15%
22.6%
Port.WM
15%
15
15
15
15
15%
0.0%
4 - 51
Returns Distribution for Two Perfectly
Negatively Correlated Stocks (r = -1.0) and
for Portfolio WM
Stock W
.
25 .
.
0
.
.
.
25
15
-10
Stock M
.
25
.
15
15
0
0
-10
Portfolio WM
.
.
-10
. . . . .
4 - 52
W&M: rM’M = +1.0
Year
1986
1987
1988
1989
1990
AvgRet.
StdDev.
Stock M’
(10%)
40%
(5)
35
15
15%
22.6%
Stock M
(10%)
40%
(5)
35
15
15%
22.6%
Port.WM
(10%)
40%
(5)
35
15
15%
22.6%
4 - 53
Returns Distributions for Two Perfectly
Positively Correlated Stocks (r = +1.0) and
for Portfolio MM’
Stock M’
Stock M
Portfolio MM’
25
25
25
15
15
15
0
0
0
-10
-10
-10
4 - 54
W&M: rWM =+.65
Year
1986
1987
1988
1989
1990
AvgRet.
StdDev.
Stock W
40%
(10)
35
(5)
15
15%
22.6%
Stock M
28%
20
41
(17)
3
15%
22.6%
Port.WM
34%
5
38
(11)
9
15%
20.6%
4 - 55
See p. 145
4 - 56
Does the Expected Return and Risk
depend on the percentage of the
Portfolio invested in each Stock?
MIinics4.xls
Consider 50% in High Tech and 50%
in U.S. Collections.
Consider other splits.
Chart HT&COLL
Why is there no risk when we have
40% High Tech, 60% Collections?
4 - 57
Does the Expected Return and Risk
depend on the percentage of the
Portfolio invested in each Stock?
What would the profile between
different proportions of two other
assets look like? See graph-RM&ALTA.
Coming attraction: What would the
envelope look like?
4 - 58
What would happen to the
risk of an average 1-stock
portfolio as more randomly
selected stocks were added?
p would decrease because the added
stocks would not be perfectly correlated,
but ^rp would remain relatively constant.
4 - 59
Prob.
Large
2
1
0
15
1 35% ; Large 20%.
Return
4 - 60
p (%)
Company Specific
(Diversifiable) Risk
35
Stand-Alone Risk, p
20
Market Risk
0
10
20
30
40
2,000+
# Stocks in Portfolio
4 - 61
Stand-alone Market
Diversifiable
= risk
+
.
risk
risk
Market risk is that part of a security’s
stand-alone risk that cannot be
eliminated by diversification.
Firm-specific, or diversifiable, risk is
that part of a security’s stand-alone risk
that can be eliminated by
diversification.
4 - 62
Standard deviation (i) measures
the total or stand-alone risk.
The larger the i , the higher the
probability that actual returns will
be far below the expected return.
4 - 63
Diversifiable risk is caused by
company-specific events such as
lawsuits
strikes
winning and losing major contracts
successful or unsuccessful mktg.
The impact of these events can be
eliminated by diversification: bad
events in one firm or industry will be
offset by good events in another.
4 - 64
Market risk stems from unavoidable
events such as war, inflation,
recession, and high interest rates
which have an impact on all firms-cannot be eliminated by
diversification.
Called systematic risk, because it
shows the degree to which a stock
moves systematically with other
stocks. Measured by?
4 - 65
Conclusions
As more stocks are added, each new
stock has a smaller risk-reducing
impact on the portfolio.
p falls very slowly after about 40
stocks are included. The lower limit
for p is about 20% = M .
By forming well-diversified portfolios,
investors can eliminate about half the
riskiness of owning a single stock.
4 - 66
Can an investor holding one stock earn
a return commensurate with its risk?
No. Rational investors will minimize
risk by holding portfolios.
They bear only market risk, so prices
and returns reflect this lower risk.
The one-stock investor bears higher
(stand-alone) risk, so the return is less
than that required by the risk.
4 - 67
G. 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?
4 - 68
NO!
Stand-alone risk as measured by a
stock’s  or CV is not important to a
well-diversified investor.
Rational, risk averse investors are
concerned with portfolio risk, and
here the relevant risk of an
individual stock is its contribution
to the riskiness of a portfolio.
4 - 69
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.
A little cryptic, but consider the
following:
4 - 70
If you hold a one-stock portfolio, you
will be exposed to a high degree of
risk and won’t be compensated for it.
REASONING: IF you were
compensated for TOTAL risk,
(including both undiversifiable and
unavoidable risk), you would have
to earn a high rate of return.
4 - 71
BUT IF this were true, and this high
rate of return were available to
diversified investors (for whom the
risk associated with this asset is
lower), they would have bought
this asset, driving up its price and
thereby driving down its return.
CONCLUSION: the high rate of
return would NOT have been
available to you, as a non-
4 - 72
FURTHER, the lower rate of return
which you do earn would NOT
compensate you for the risk of
holding a single stock portfolio.
4 - 73
How is market risk measured for
individual securities?
Market risk, which is relevant for stocks
held in well-diversified portfolios, is
defined as the contribution of a security
to the overall riskiness of the portfolio.
It is measured by a stock’s beta
coefficient. For stock i, its beta is:
bi = (riM i) / M
4 - 74
How are betas calculated?
In addition to measuring a stock’s
contribution of risk to a portfolio,
beta also which measures the stock’s
volatility relative to the market.
4 - 75
Using a Regression to Estimate Beta
Run a regression with returns on
the stock in question plotted on the
Y axis and returns on the market
portfolio plotted on the X axis.
The slope of the regression line,
which measures relative volatility,
is defined as the stock’s beta
coefficient, or β.
4 - 76
Use the historical stock returns to
calculate the beta for PQU.
Year
1
2
3
4
5
6
7
8
9
10
Market
25.7%
8.0%
-11.0%
15.0%
32.5%
13.7%
40.0%
10.0%
-10.8%
-13.1%
PQU
40.0%
-15.0%
-15.0%
35.0%
10.0%
30.0%
42.0%
-10.0%
-25.0%
25.0%
4 - 77
Calculating Beta for PQU
40%
r KWE
20%
rM
0%
-40%
-20%
0%
20%
40%
-20%
-40%
r PQU = 0.83r M + 0.03
2
R = 0.36
4 - 78
What is beta for PQU?
The regression line, and hence
beta, can be found using a
calculator with a regression
function or a spreadsheet program.
In this example, β = 0.83.
4 - 79
Calculating Beta in Practice
Many analysts use the S&P 500 to
find the market return.
Analysts typically use four or five
years’ of monthly returns to
establish the regression line.
 Some analysts use 52 weeks of
weekly returns.
4 - 80
How is beta interpreted?
 If β = 1.0, stock has average risk.
 If β > 1.0, stock is riskier than average.
 If β < 1.0, stock is less risky than average.
 Most stocks have betas in the range of 0.5
to 1.5.
 See notebook for examples
 Can a stock have a negative beta?
4 - 81
Finding Beta Estimates on the Web
Go to www.thomsonfn.com.
(or www.finance.yahoo.com)
Enter the ticker symbol for a
“Stock Quote”, such as IBM
or Dell, then click GO.
When the quote comes up,
select Company Earnings,
then GO.
4 - 82
Expected Return versus Market Risk
Security
Alta
Market
Am. Foam
T-bills
Repo Men
Expected
return
17.4%
15.0
13.8
8.0
1.7
Risk, β
1.29
1.00
0.68
0.00
-0.86
 Which of the alternatives is best?
4 - 83
I. Use the SML to calculate the required
returns.
SML: ki = kRF + (kM - kRF) i .
intercept
slope
Assume kRF = 8%.
^
Note that kM = kM is 15%. (From
market portfolio.)
RPM = kM - kRF = 15% - 8% = 7%.
4 - 84
Required Rates of Return
rAlta = 8.0% + (7%)(1.29)
= 8.0% + 9.0%
rM
=
15.0%.
rAm. F. =
12.8%.
rT-bill =
rRepo =
= 17.0%.
8.0% + (7%)(1.00)
=
8.0% + (7%)(0.68)
=
8.0% + (7%)(0.00) =
8.0% + (7%)(-0.86) =
8.0%.
2.0%.
4 - 85
Expected versus Required Returns
Alta
r^
17.4%
r
17.0% Undervalued
Market 15.0
15.0
Fairly valued
Am. F.
13.8
12.8
Undervalued
T-bills
8.0
8.0
Fairly valued
Repo
1.7
2.0
Overvalued
4 - 86
ri (%) SML: ri = rRF + (RPM) bi
ri = 8% + (7%) bi
.
Alta
rM = 15
rRF = 8
.
. .
. T-bills
Market
Am. Foam
Repo
-1
0
1
2
Risk, bi
SML and Investment Alternatives
4 - 87
Calculate beta for a portfolio with 50%
Alta and 50% Repo
βp
= Weighted average
= 0.5(bAlta) + 0.5(bRepo)
= 0.5(1.29) + 0.5(-0.86)
= 0.22.
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What is the required rate of return
on the Alta/Repo portfolio?
rp = Weighted average r
= 0.5(17%) + 0.5(2%) = 9.5%.
Or use SML:
rp = rRF + (RPM) bp
= 8.0% + 7%(0.22) = 9.5%.
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Impact of Inflation Change on SML
Required Rate
of Return r (%)
 I = 3%
New SML
SML2
SML1
18
15
11
8
Original situation
0
0.5
1.0
1.5
2.0
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Impact of Risk Aversion Change
Required Rate
of Return (%)
After increase
in risk aversion
SML2
rM = 18%
rM = 15%
SML1
18
 RPM =
3%
15
8
Original situation
1.0
Risk, bi
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Has the CAPM been completely confirmed
or refuted through empirical tests?
No. The statistical tests have
problems that make empirical
verification or rejection virtually
impossible.
Investors’ required returns are
based on future risk, but betas are
calculated with historical data.
and
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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 ki:
ki = kRF + (kM - kRF)i + ?
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The End!
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