Chapter 5 - Portfolio Theory and Asset Pricing Models.pptx

Chapter 5
Portfolio Theory and Asset Pricing Models
By:
Prof. Saad Abdel-Hamid Abdel-Hamid Metawa
(
Full Professor of Investment & Financet
)
(Faculty of Commerce - Mansoura University))
2021

• 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.

Portfolio Theory:
• 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 0.6, what are
the expected return and standard deviation for
a portfolio comprised of 30 percent Asset A
and 70 percent Asset B?

Portfolio Expected Return:
%.
2
.
14
142
.
0
)
16
.
0
(
7
.
0
)
1
.
0
(
3
.
0
r̂
)
w
1
(
r̂
w
r̂ B
A
A
A
P








Portfolio Standard Deviation:
320
.
0
)
4
.
0
)(
2
.
0
)(
6
.
0
)(
7
.
0
)(
3
.
0
(
2
)
4
.
0
(
7
.
0
)
2
.
0
(
3
.
0
)
W
1
(
W
2
)
W
1
(
W
2
2
2
2
B
A
AB
A
A
2
B
2
A
2
A
2
A
p
















Attainable Portfolios: rAB = 0.4
rAB = +0.4: Attainable Set of
Risk/Return Combinations
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Risk, sp
Expected
return

Attainable Portfolios: rAB = +1
AB = +1.0: Attainable Set of Risk/Return
Combinations
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Risk, p
Expected
return

Attainable Portfolios: rAB = -1
AB = -1.0: Attainable Set of Risk/Return
Combinations
0%
5%
10%
15%
20%
0% 10% 20% 30% 40%
Risk, p
Expected
return

Attainable Portfolios with Risk-Free Asset
(Expected risk-free return = 5%)
Attainable Set of Risk/Return
Combinations with Risk-Free Asset
0%
5%
10%
15%
0% 5% 10% 15% 20%
Risk, p
Expected
return

Expected
Portfolio
Return, rp
Risk, p
Efficient Set
Feasible Set
Feasible and Efficient Portfolios

• 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.

IB2 IB1
IA2
IA1
Optimal Portfolio
Investor A
Optimal
Portfolio
Investor B
Risk p
Expected
Return, rp
Optimal Portfolios

• 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.

What is the CAPM?
• The CAPM is an equilibrium model that
specifies the relationship between risk and
required rate of return for assets held in well-
diversified portfolios.
• It is based on the premise that only one factor
affects risk.
• What is that factor?

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.

• 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.

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.

Efficient Set with a Risk-Free Asset
M
Z
.
A
rRF
M Risk, p
The Capital Market
Line (CML):
New Efficient Set
.
.B
rM
^
Expected
Return, rp

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.

The CML Equation
rp = rRF +
Slope
Intercept
^ p.
rM - rRF
^

M
Risk
measure

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.

rRF
M
Risk, p
I1
I2
CML
R = Optimal
Portfolio
.
R
.
M
rR
rM
R
^
^
Expected
Return, rp

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.

The SML Equation
• The measure of risk used in the SML is the
beta coefficient of company i, bi.
• The SML equation:
ri = rRF + (RPM) bi

rRF
M
Risk, Risk, p
I1
I2
CML
R = Optimal
Portfolio
.
R
.
M
rR
rM
R
^
^
Expected
Return, rp

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.

Illustration of beta calculation
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
.
.
.
ri = -2.59 + 1.44 kM
^ ^

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.
• 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

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
• 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

What is the relationship between stand-alone,
market, and diversifiable risk.
2
= b2
2
+ e
2
.
2
= variance
= stand-alone risk of Stock j.
b2
2
= market risk of Stock j.
e
2 = variance of error term
= diversifiable risk of Stock j.

What are two potential tests that can be
conducted to verify the CAPM?
Beta stability tests
Tests based on the slope of the SML

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.
• 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.

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.

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.
• 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.

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.

Required Return for Stock i
under the APT
ri = rRF + (r1 - rRF)b1 + (r2 - rRF)b2
+ ... + (rj - rRF)bj.
bj = sensitivity of Stock i to economic
Factor j.

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.

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.
• 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.

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.
ci = sensitivity of Stock i to the size factor.
di = sensitivity of Stock i to the book-to-market
factor.

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%

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%

Chapter 5 - Portfolio Theory and Asset Pricing Models.pptx

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