CH 04 - Risk & Return Basics

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CHAPTER 4
Risk and Return: The Basics

 Basic return concepts
 Basic risk concepts
 Stand-alone risk
 Portfolio (market) risk
 Risk and return: CAPM/SML

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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.

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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%.

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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.

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Focus on Ethics
If It Sounds Too Good To Be True...
For many years, investors around the world
clamored to invest with Bernard Madoff.
Madoff generated high returns year after year,
seemingly with very little risk.
On December 11, 2008, the U.S. Securities and
Exchange Commission (SEC) charged Madoff with
securities fraud. Madoff’s hedge fund, Ascot
Partners, turned out to be a giant Ponzi scheme.
What are some hazards of allowing investors to
pursue claims based their most recent accounts
statements?

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Risk and Return Fundamentals:
Risk Preferences
Economists use three categories to describe how
investors respond to risk.
Risk averse is the attitude toward risk in which
investors would require an increased return as
compensation for an increase in risk.
Risk-neutral is the attitude toward risk in which
investors choose the investment with the higher
return regardless of its risk.
Risk-seeking is the attitude toward risk in which
investors prefer investments with greater risk even
if they have lower expected returns.

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Probability distribution

Stock X

Stock Y

Rate of
-20 0 15 50 return (%)
 Which stock is riskier? Why?

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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

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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.

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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.

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Calculate the expected rate of return
on each alternative.
^
r = expected rate of return.
∧ n
r = ∑ rP .
i=1
i i

^ = 0.10(-22%) + 0.20(-2%)
rAlta
+ 0.40(20%) + 0.20(35%)
+ 0.10(50%) = 17.4%.

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^
r
Alta 17.4%
Market 15.0
Am. Foam 13.8
T-bill 8.0
Repo Men 1.7

 Alta has the highest rate of return.
 Does that make it best?

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What is the standard deviation
of returns for each alternative?

σ = Standard deviation

σ = Variance = σ
2

n ∧ 2
 
= ∑  ri − r  Pi .
i =1  

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n ∧ 2
 
σ = ∑  ri − r  Pi .
i =1  
Alta Inds:
σ = ((-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%.
σ T-bills = 0.0%. σ Repo = 13.4%.
σ Alta = 20.0%. σ Am Foam = 18.8%.
σ Market = 15.3%.

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Prob.
T-bill

Am. F.
Alta

0 8 13.8 17.4
Rate of Return (%)

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 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.

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Expected Return versus Risk

Expected
Security return Risk, σ
Alta Inds. 17.4% 20.0%
Market 15.0 15.3
Am. Foam 13.8 18.8
T-bills 8.0 0.0
Repo Men 1.7 13.4

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Return vs. Risk (Std. Dev.):
Which investment is best?

20.0%
18.0% Alta
16.0%
Mkt
14.0% USR
Return

12.0%
10.0%
8.0% T-bills
6.0%
4.0%
2.0% Coll.
0.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Risk (Std. Dev.)

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Risk of a Single Asset:
Coefficient of Variation
 The coefficient of variation, CV, is a measure of relative
dispersion. CV is a better measure for evaluating risk
in situations where investments have substantially
different expected returns.

 A higher coefficient of variation means that an
investment has more volatility relative to its expected
return.

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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.

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Expected Return versus Coefficient of
Variation
Expecte Risk: Risk:
d
Security return σ CV
Alta Inds 17.4% 20.0% 1.1
Market 15.0 15.3 1.0
Am. Foam 13.8 18.8 1.4
T-bills 8.0 0.0 0.0
Repo Men 1.7 13.4 7.9

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Risk of a Portfolio

 In real-world situations, the risk of any
single investment would not be viewed
independently of other assets.
 New investments must be considered in
light of their impact on the risk and
return of an investor’s portfolio of assets.
 The financial manager’s goal is to create
an efficient portfolio, a portfolio that
maximum return for a given level of risk.

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Portfolio Risk and Return

Assume a two-stock portfolio with
$50,000 in Alta Inds. and $50,000 in
Repo Men.

^ and σ .
Calculate rp p

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Portfolio Return, ^p
r

^ is a weighted average:
rp
n
^ ^
rp = Σ wiri.
i=1

^
rp = 0.5(17.4%) + 0.5(1.7%) = 9.6%.
^ ^ ^
rp is between rAlta and rRepo.

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Alternative Method
Estimated Return
Economy Prob. Alta Repo Port.
Recession 0.10 -22.0% 28.0% 3.0%
Below avg. 0.20 -2.0 14.7 6.4
Average 0.40 20.0 0.0 10.0
Above avg. 0.20 35.0 -10.0 12.5
Boom 0.10 50.0 -20.0 15.0
^ = (3.0%)0.10 + (6.4%)0.20 + (10.0%)0.40
rp
+ (12.5%)0.20 + (15.0%)0.10 = 9.6%.
(More...)

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 σ 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%.
 σ 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. The key here is
negative correlation.

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Risk of a Portfolio: Correlation (ρ)
 Correlation is a statistical measure of the relationship
between any two series of numbers.
 Positively correlated describes two series that move in the
same direction.
 Negatively correlated describes two series that move in
opposite directions.
 The correlation coefficient is a measure of the degree of
correlation between two series.
 Perfectly positively correlated describes two positively
correlated series that have a correlation coefficient of +1.
 Perfectly negatively correlated describes two negatively
correlated series that have a correlation coefficient of –1.ρ

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Figure 8.4 Correlations

© 2012 Pearson Education 8-28

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Risk of a Portfolio: Diversification

 To reduce overall risk, it is best to diversify by
combining, or adding to the portfolio, assets that have
the lowest possible correlation.
 Combining assets that have a low correlation with each
other can reduce the overall variability of a portfolio’s
returns.
 Uncorrelated describes two series that lack any
interaction and therefore have a correlation coefficient
close to zero.


Figure 8.5 4 - 30
Diversification


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Two-Stock Portfolios

 Two stocks can be combined to form
a riskless portfolio if ρ = -1.0.
 Risk is not reduced at all if the two
stocks have ρ = +1.0.
 In general, stocks in US markets have
ρ ≈ 0.65, so risk is lowered but not
eliminated.
 Investors typically hold many stocks.
 What happens when ρ = 0?

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What would happen to the
risk of an average 1-stock
portfolio as more randomly
selected stocks were added?

 σ p would decrease because the added
stocks would not be perfectly correlated,
but rp would remain relatively constant.
^
 In the real world, it is impossible to form a
completely riskless stock portfolio.

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Prob.
Large

2

1

0 15 Return
σ 1 ≈ 35% ; σ Large ≈ 20%.

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σ p (%)
Company Specific
35
(Diversifiable) Risk
Stand-Alone Risk, σ p

20
Market Risk

0
10 20 30 40 2,000+

# Stocks in Portfolio

4 - 35

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.

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Conclusions

 As more stocks are added, each new
stock has a smaller risk-reducing
impact on the portfolio.
 σ p falls very slowly after about 40
stocks are included. The lower limit
for σ p is about 20% = σ M .
 By forming well-diversified
portfolios, investors can eliminate
about half the riskiness of owning a
single stock.

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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.

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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 = (ρ iM σ i) / σ M

4 - 39

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 - 40

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 b.

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Use the historical stock returns to
calculate the beta for PQU.
Year Market PQU
1 25.7% 40.0%
2 8.0% -15.0%
3 -11.0% -15.0%
4 15.0% 35.0%
5 32.5% 10.0%
6 13.7% 30.0%
7 40.0% 42.0%
8 10.0% -10.0%
9 -10.8% -25.0%
10 -13.1% 25.0%

4 - 42
Calculating Beta for PQU

40%
r KWE

20%

0% r M

-40% -20% 0% 20% 40%
-20%

r PQU = 0.83r M + 0.03
-40% 2
R = 0.36

4 - 43

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, b = 0.83.

4 - 44

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.

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How is beta interpreted?
 If b = 1.0, stock has average risk.
 If b > 1.0, stock is riskier than average.
 If b < 1.0, stock is less risky than
average.
 Most stocks have betas in the range of
0.5 to 1.5.
 Can a stock have a negative beta?

4 - 46

Finding Beta Estimates on the Web

Go to www.thomsonfn.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 - 47

Expected Return versus Market Risk

Expected
Security return Risk, b
Alta 17.4% 1.29
Market 15.0 1.00
Am. Foam 13.8 0.68
T-bills 8.0 0.00
Repo Men 1.7 -0.86
 Which of the alternatives is best?

4 - 48

Use the SML to calculate each
alternative’s required return.

 The Security Market Line (SML) is
part of the Capital Asset Pricing
Model (CAPM).

SML: ri = rRF + (RPM)bi .
^
 Assume rRF = 8%; rM = rM = 15%.
 RPM = (rM - rRF) = 15% - 8% = 7%.

4 - 49

Required Rates of Return

rAlta = 8.0% + (7%)(1.29)
= 8.0% + 9.0% = 17.0%.
rM = 8.0% + (7%)(1.00) = 15.0%.
rAm. F. = 8.0% + (7%)(0.68) = 12.8%.
rT-bill = 8.0% + (7%)(0.00) = 8.0%.
rRepo = 8.0% + (7%)(-0.86) = 2.0%.

4 - 50

Expected versus Required Returns

^
r r
Alta 17.4% 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 - 51

ri (%) SML: ri = rRF + (RPM) bi
ri = 8% + (7%) bi

Alta . Market
rM = 15 . .
rRF = 8 . T-bills Am. Foam

Repo
. Risk, bi
-1 0 1 2

SML and Investment Alternatives

4 - 52

Calculate beta for a portfolio with 50%
Alta and 50% Repo

bp = Weighted average
= 0.5(bAlta) + 0.5(bRepo)
= 0.5(1.29) + 0.5(-0.86)
= 0.22.

4 - 53

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%.

4 - 54

Impact of Inflation Change on SML
Required Rate
of Return r (%)
∆ I = 3%
New SML
SML2

18 SML1
15
11 Original situation
8

0 0.5 1.0 1.5 2.0

Impact of Risk Aversion Change 4 - 55

After increase
Required Rate in risk aversion
of Return (%)
SML2
rM = 18%
rM = 15%
18 SML1
15 ∆ RPM = 3%

8
Original situation
Risk, bi
1.0

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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.
Investors may be concerned about
both stand-alone and market risk.

CH 04 - Risk & Return Basics

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