5-1
CHAPTER 5
Risk and Return: Portfolio Theory and
Asset Pricing Models
 Portfolio Theory
 Capital Asset Pricing Model (CAPM)
 Efficient frontier
 Capital Market Line (CML)
 Security Market Line (SML)
 Beta calculation
 Arbitrage pricing theory
 Fama-French 3-factor model
5-2
Portfolio Theory
Ch 5, in-class, portfolio theory exercise
 Suppose Asset A has an expected return
of 10 percent and a standard deviation of
20 percent. Asset B has an expected
return of 16 percent and a standard
deviation of 40 percent. If the correlation
between A and B is 0.4*, what are the
expected return and standard deviation for
a portfolio comprised of 30 percent Asset
A and 70 percent Asset B?
n.b. this is .4, not .6, as in binder
5-3
Portfolio Expected Return
rˆP  w A rˆA  (1  w A ) rˆB
 0.3(0.1)  0.7(0.16)
 0.142 14.2%.
5-4
Portfolio Standard Deviation
 p  WA2 A2  (1  WA ) 2  B2  2WA (1  WA )  AB  A  B
 0.32 ( 0.22 )  0.7 2 ( 0.4 2 )  2( 0.3)(0.7 )(0.4)(0.2)(0.4)
 0.309
5-5
Attainable Portfolios: AB = 0.4
 AB = +0.4: Attainable Set of
Risk/Return Combinations
Expected return
20%
15%
10%
5%
0%
0%
10%
20%
Risk,  p
30%
40%
5-6
Attainable Portfolios: AB = +1
 AB = +1.0: Attainable Set of Risk/Return
Combinations
Expected return
20%
15%
10%
5%
0%
0%
10%
20%
Risk,  p
30%
40%
5-7
Attainable Portfolios: AB = -1
 AB = -1.0: Attainable Set of Risk/Return
Combinations
Expected return
20%
15%
10%
5%
0%
0%
10%
20%
Risk,  p
30%
40%
5-8
Attainable Portfolios with Risk-Free
Asset (Expected risk-free return = 5%)
Attainable Set of Risk/Return
Combinations with Risk-Free Asset
Expected return
15%
10%
5%
0%
0%
5%
10%
Risk, p
15%
20%
Expected
Portfolio
Return, rp
5-9
Efficient Set
Feasible Set
Risk, p
Feasible and Efficient Portfolios
5 - 10
The feasible set of portfolios represents
all portfolios that can be constructed
from a given set of stocks.
An efficient portfolio is one that offers:
the most return for a given amount of risk,
or
the least risk for a give amount of return.
The collection of efficient portfolios is
called the efficient set or efficient
frontier.
5 - 11
Investor’s preferences
Add to this diagram information
about the risk aversion of investors
Expected
Return, rp
5 - 12
IB2 I
B1
IA2
IA1
Optimal Portfolio
Investor B
Optimal Portfolio
Investor A
Optimal Portfolios
Risk p
5 - 13
Indifference curves reflect an
investor’s attitude toward risk as
reflected in his or her risk/return
tradeoff function. They differ
among investors because of
differences in risk aversion.
An investor’s optimal portfolio is
defined by the tangency point
between the efficient set and the
investor’s indifference curve.
Different indifference curves
reflect different investor’s risk
aversion
5 - 14
WHAT is CAPM?
An equilibrium model specifying the
relationship between risk and required
return on assets held in diversified
portfolios.
RISK RETURN
(Systematic) RISK
RETURN
If
, then
Only one factor affects risk, and it is?
5 - 15
β
5 - 16
What are the assumptions
of the CAPM?
 Investors all think in terms of
a single holding period.
 All investors have identical expectations.
 Investors can borrow or lend unlimited
amounts at the risk-free rate.
(More...)
5 - 17
 All assets are perfectly divisible.
 There are no taxes and no transactions
costs.
 All investors are price takers, that is,
investors’ buying and selling won’t
influence stock prices.
 Quantities of all assets are given and
fixed.
5 - 18
What impact does rRF have on
the efficient frontier?
 When a risk-free asset is added to the
feasible set, investors can create
portfolios that combine this asset with a
portfolio of risky assets.
 The straight line connecting rRF with M, the
tangency point between the line and the
old efficient set, becomes the new efficient
frontier.
5 - 19
Portfolios Combining the Risk Free
Asset and the Market Portfolio
Expected Portfolio Return
M
^
rM
rRF
A
.
.
B
The Capital Market
Line (CML)
.
M
Slope =?
Z
Risk, p
5 - 20
Portfolio M must contain every asset
in exact proportion to that asset’s
fraction of total market value of all
securities
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What is the Capital Market Line?
The Capital Market Line (CML) is all
linear combinations of the risk-free
asset and Portfolio M.
Portfolios below the CML are inferior.
The CML defines the new efficient set.
All investors will choose a portfolio on
the CML.
5 - 22
The CML Equation
^
rp =
rRF +
Intercept
^
rM - rRF
M
Slope
 p.
Risk
measure
5 - 23
What does the CML tell us?
The expected rate of return on any
efficient portfolio is equal to the
risk-free rate plus a risk premium.
The optimal portfolio for any
investor is the point of tangency
between the CML and the
investor’s indifference curves.
5 - 24
Expected
Return, rp
CML
I2
^
rM
^r
R
I1
.
.
M
R
R = Optimal
Portfolio
rRF
R0=original
portfolio
R M
Risk, p
5 - 25
What is the Security Market Line (SML)?
The CML gives the risk/return
relationship for efficient portfolios.
The Security Market Line (SML), also
part of the CAPM, gives the risk/return
relationship for individual stocks.
5 - 26
The SML Equation
The measure of risk used in the SML
is the beta coefficient of company i,
βi.
The SML equation:
ri = rRF + (RPM) βi
5 - 27
Recall how are betas calculated?
Run a regression line of past
returns on Stock i versus returns
on the market.
The regression line is called the
characteristic line.
The slope coefficient of the
characteristic line is defined as the
beta coefficient.
5 - 28
Illustration of beta calculation
_
ri
.
.
20
15
10
Year rM
1
15%
2
-5
3
12
ri
18%
-10
16
5
-5
0
-5
.
-10
5
10
15
20
^
ri = -2.59 + 1.44 k^M
_
rM
5 - 29
Method of Calculation
Analysts use a computer with
statistical or spreadsheet software to
perform the regression.
At least 3 year’s of monthly returns or 1
year’s of weekly returns are used.
Many analysts use 5 years of monthly
returns.
(More...)
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If beta = 1.0, stock is average risk.
If beta > 1.0, stock is riskier than
average.
If beta < 1.0, stock is less risky than
average.
Most stocks have betas in the range
of 0.5 to 1.5.
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Interpreting Regression Results
The R2 measures the percent of a
stock’s variance that is explained by
the market. The typical R2 is:
0.3 for an individual stock
over 0.9 for a well diversified portfolio
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Interpreting Regression Results
(Continued)
The 95% confidence interval shows
the range in which we are 95% sure
that the true value of beta lies. The
typical range is:
from about 0.5 to 1.5 for an individual
stock
from about .92 to 1.08 for a well
diversified portfolio
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What is the relationship between standalone, market, and diversifiable risk.
2j = b2j M2 + e2j .
2j = variance
= stand-alone risk of Stock j.
2 = market risk of Stock j.
b2j M
e2j = variance of error term
= diversifiable risk of Stock j.
5 - 34
What are two potential tests that can be
conducted to verify the CAPM?
Beta stability tests
Tests based on the slope
of the SML
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Tests of the SML indicate:
A more-or-less linear relationship
between realized returns and market
risk.
Slope is less than predicted.
Irrelevance of diversifiable risk
specified in the CAPM model can be
questioned.
(More...)
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Betas of individual securities are not
good estimators of future risk.
Betas of portfolios of 10 or more
randomly selected stocks are
reasonably stable.
Past portfolio betas are good
estimates of future portfolio volatility.
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Are there problems with the
CAPM tests?
Yes.
Richard Roll questioned whether it was
even conceptually possible to test the
CAPM.
Roll showed that it is virtually
impossible to prove investors behave
in accordance with CAPM theory.
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What are our conclusions
regarding the CAPM?
It is impossible to verify.
Recent studies have questioned its
validity.
Investors seem to be concerned with
both market risk and stand-alone
risk. Therefore, the SML may not
produce a correct estimate of ri. (More...)
5 - 39
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.
Other models are being developed
that will one day replace the CAPM,
but it still provides a good framework
for thinking about risk and return.
5 - 40
When using the CAPM,
there are choices that have
to be made:
5 - 41
CAPM: COMMENTS ON ESTIMATING
Required rates of return
Under the CAPM, required rate of
return (or cost of equity) depends on
the risk free rate, the market risk
premium, and the Beta, as shown in
the Security market Line
ki = krf + (km -krf) i
5 - 42
Risk free rate: short term or long
term?
Long term used because (1)equities are
long term instruments, and (2)there is
less volatility in long term rate.
Market risk premium: ex-post or exante?
Ex-post from Ibbotson
Ex-ante from IBES (Will vary over time).
Estimating Betas: historical or
anticipated?
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Historical from characteristic line:
assumes that future will reflect the past.
Adjusted Betas (Marshall Blume): Betas
tend to move towards 1 over time.
Math. procedure used to estimate this.
Fundamental Betas: Adjustment
process to include (1)financial leverage
(2)sales volatility, etc. Adjusted
continuously.
5 - 44
Choices in estimating Betas
Historical periods of different length:
one, two, ...five,... years?
Different holding periods: daily,
weekly, monthly, annual returns?
More frequent--more data, but also more
noise.
What is used to represent km?
S&P500, NYSE index, Wilshire index or
what?
5 - 45
TWO CALCULATIONS OF BETAS
 Merrill Lynch (&
Yahoo!Finance)
 km: S&P500
 B: Pure Historical
 Data: Monthly(60
observations)
 Value Line
 km: NYSE index
 B: Adjusted
 Data: Weekly:(260
observations)
5 - 46
CONCLUSION
One can calculate Betas in many
ways; depending on the method
used, different betas may result from
SML: ki = krf + (km -krf)B
With luck, different Betas from
different sources will be close.
5 - 47
What is the difference between the
CAPM and the Arbitrage
Pricing Theory (APT)?
The CAPM is a single factor model.
The APT proposes that the
relationship between risk and return
is more complex and may be due to
multiple factors such as GDP
growth, expected inflation, tax rate
changes, and dividend yield.
5 - 48
Required Return for Stock i
under the APT
ri = rRF + (r1 - rRF)1 + (r2 - rRF)  2
+ ... + (rj - rRF)  j.
rj = required rate of return on a portfolio
sensitive only to economic Factor j.
j
= sensitivity of Stock i to economic
Factor j.
5 - 49
What is the status of the APT?
The APT is being used for some real
world applications.
Its acceptance has been slow because
the model does not specify what
factors influence stock returns.
More research on risk and return
models is needed to find a model that
is theoretically sound, empirically
verified, and easy to use.
5 - 50
Fama-French 3-Factor Model
Fama and French propose three
factors:
The excess market return, rM-rRF.
the return on, S, a portfolio of small
firms (where size is based on the market
value of equity) minus the return on B, a
portfolio of big firms. This return is
called rSMB, for S minus B.
5 - 51
Fama-French 3-Factor Model
(Continued)
the return on, H, a portfolio of firms with
high book-to-market ratios (using
market equity and book equity) minus
the return on L, a portfolio of firms with
low book-to-market ratios. This return
is called rHML, for H minus L.
5 - 52
Required Return for Stock i
under the Fama-French 3-Factor Model
ri = rRF + (rM - rRF)bi + (rSMB)ci + (rHMB)di
bi = sensitivity of Stock i to the market
return.
cj = sensitivity of Stock i to the size
factor.
dj = sensitivity of Stock i to the bookto-market factor.
5 - 53
Required Return for Stock i: bi=0.9,
rRF=6.8%, the market risk premium is
6.3%, ci=-0.5, the expected value for the
size factor is 4%, di=-0.3, and the
expected value for the book-to-market
factor is 5%.
ri = rRF + (rM - rRF)bi + (rSMB)ci + (rHMB)di
ri = 6.8% + (6.3%)(0.9) + (4%)(-0.5) +
(5%)(-0.3)
= 8.97%
5 - 54
CAPM Required Return for Stock i
CAPM:
ri = rRF + (rM - rRF)bi
ri = 6.8% + (6.3%)(0.9)
= 12.47%
Fama-French (previous slide):
ri = 8.97%
5 - 55
THE END, thank god.