# Part II. Portfolio Theory and Asset Pricing Models by S1av5moR

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```									Part II. Portfolio Theory and
Asset Pricing Models

1
Rates of Return: Single Period

HPR  P1  P0  D1
P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one

2
Expected Return and Standard Deviation

Expected returns

E (r )   p( s)r ( s)
s
p(s) = probability of a state
r(s) = return if a state occurs
s = state

3
Scenario Returns: Example

State        Prob. of State        r in State
1                .1                 -.05
2                .2                  .05
3                .4                  .15
4                .2                  .25
5                .1                  .35

E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35)
E(r) = .15

4
Variance or Dispersion of Returns

Variance:

   p( s )  r ( s)  E (r ) 
2                            2

s
Standard deviation = [variance]1/2
Using Our Example:
Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2…+ .1(.35-.15)2]
Var= .01199
S.D.= [ .01199] 1/2 = .1095

5
EXHIBIT 3.14 Geometric Mean rates of Return and
Standard Deviation for Sotheby's Indexes, S&P 500,
Bond Market Series, One-Year Bonds, and Inflation

Chinese Mod Paint
Amer Paint                Cont Art
Ceramic
Imp Paint
Eng Furn     FW Index      S&P500
UW Index      VW Index         Old Master
Fr+Cont Furn         Cont     19C Euro
Amer Fum               Ceramic
LBGC      Eng Silver
Cont Silver
1-Year Bond

CPI

Standard
Deviation
EXHIBIT 3.17 Alternative Investment Risk
and Return Characteristics

Futures
Art and Antiques
Coins and Stamps         Warrants and Options
US Common Stocks          Commercial Real Estate
Foreign Common Stock
Foreign Corporate Bonds     Real Estate (Personal Home)
US Corporate Bonds
Foreign Government Bonds
US Government Bonds
T-Bills
Covariance of two random
variables
• Covariance is defined as:

n
Cov(r1 , r2 )   p( s )[ r1 ( s )  E (r1 )][ r2 ( s )  E (r2 )]
i 1

8
Figure 5.4 The Normal Distribution

9
Figure 5.5A Normal and Skewed Distributions
(mean = 6% SD = 17%)

10
Figure 5.5B Normal and Fat-Tailed Distributions
(mean = .1, SD =.2)

11
Other measures of risks
• Value at Risk(Var) is another name for the quantile of
a distribution. The quantile (q) of a distribution is the
value below which lie q% of the value
• Conditional Tail Expectation (CTE) provides the
answer to the question, “Assuming the terminal
value of the portfolio falls in the bottom 5% of
possible outcomes, what is the expected value?”
• Lower Partial Standard Deviation (LPSD) is the
standard deviation computed solely from values
below the expected return

12
Table 5.5 Risk Measures for Non-Normal
Distributions

13
Figure 5.6 Frequency Distributions of Rates of
Return for 1926-2005

14
Table 5.3 History of Rates of Returns of Asset
Classes for Generations, 1926- 2005

15
Arithmetic and Geometric means

• Arithmetic mean
_          n
1
r   rt
n i 1

• Geometric mean
_
r   n   Rn  1

Rn  (1  r1 )(1  r2 )...( 1  rn )
16
Risk Aversion measured with Utility Function
1
U  E (r )  A 2

2
Where
U = utility
E ( r ) = expected return on the asset or
portfolio
A = coefficient of risk aversion
2 = variance of returns

17
Figure 6.2 The Indifference Curve: a curve
indicating the same utility level

18
Table 6.3 Utility Values of Possible Portfolios
for an Investor with Risk Aversion, A = 4

19
Indifference curve
• The slope of indifference curve indicates the
riskiness of the investor; more steeper means
more risk averse
• When indifference curve moves to the north-
west, it implies higher utility for most
investors

20
The Risk-Free Asset

• Only the government can issue default-free
bonds
– Guaranteed real rate only if the duration of the
bond is identical to the investor’s desire
holding period
• T-bills viewed as the risk-free asset
– Less sensitive to interest rate fluctuations

21
CD and T-bill Rates

22
2. Portfolio Theory: Portfolios of One Risky Asset
and a Risk-Free Asset

• It’s possible to split investment funds between
safe and risky assets.
• Risk free asset: proxy; T-bills
• Risky asset: stock (or a portfolio)

23
A portfolio of two stocks
rp  w1r1  w2 r2
w1  w2  1
E (rp )  w1 E (r1 )  w2 E (r2 )
 p  w1  12  w2 2  2 w1w2Cov(r1 , r2 )
2      2      2 2

24
Example Using Chapter 6.4 Numbers

rf = 7%         rf = 0%

E(rp) = 15%     p = 22%

y = % in p      (1-y) = % in rf

25
Expected Returns for Combinations

E (rc )  yE (rp )  (1  y)rf

rc = complete or combined portfolio

For example, y = .75
E(rc) = .75(.15) + .25(.07)
= .13 or 13%

26
Combinations Without Leverage

If y = .75, then
c      = .75(.22) = .165 or 16.5%
If y = 1
   c   = 1(.22) = .22 or 22%
If y = 0
c      = (.22) = .00 or 0%

27
Capital Allocation Line with Leverage

Borrow at the Risk-Free Rate and invest in stock.
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = .19

c = (1.5) (.22) = .33

28
Figure 6.4 The Investment Opportunity Set with a
Risky Asset and a Risk-free Asset in the Expected
Return-Standard Deviation Plane

29
Risk Tolerance and Asset Allocation
• The investor must choose one optimal portfolio,
C, from the set of feasible choices
– Trade-off between risk and return
– Expected return of the complete portfolio is
given by:
E (rc )  rf  y  E (rP )  rf 
              
– Variance is:
 y
2
C
2   2
P

30
Table 6.5 Utility Levels for Various Positions in Risky
Assets (y) for an Investor with Risk Aversion A = 4

31
Figure 6.6 Utility as a Function of Allocation to the
Risky Asset, y

32
Figure 6.8 Finding the Optimal Complete Portfolio
Using Indifference Curves

33
Mathematically, maximize utility function of
investing to get optimum portfolio weights:
1
Maximize U  E (rc )  A c2
2

Subject to:
E ( rc )  r f  y ( E ( rp )  r f )
and
 c  y     p

34
The solution is:        E (rp )  rf
y
A p
2

-if risk premium of investing in portfolio p is
higher, y is higher
-if the investor is more risk averse, less y
-if the stock portfolio p is more risky, less y

35
Figure 6.5 The Opportunity Set with Differential
Borrowing and Lending Rates

36
3. Portfolio Theory: Portfolio with Two Risky
Securities
rp      wr
D   D
 wEr E
rP    Portfolio Return
wD  Bond Weight
rD    Bond Return
wE  Equity Weight
rE    Equity Return

E (rp )  wD E (rD )  wE E (rE )
37
Two-Security Portfolio: Risk

  w   w   2wD wE Cov(rD , rE )
2
P
2
D
2
D
2
E
2
E

 D = Variance of Security D
2

   2
E       = Variance of Security E

Cov(rD , rE ) = Covariance of returns for
Security D and Security E
38
Covariance

Cov(rD,rE) = DEDE

D,E = Correlation coefficient of
returns
D = Standard deviation of
returns for Security D
E = Standard deviation of
returns for Security E
Correlation Coefficients: Possible Values

Range of values for 1,2
+ 1.0 >     > -1.0
If  = 1.0, the securities would be perfectly
positively correlated
If  = - 1.0, the securities would be
perfectly negatively correlated
 p  wE E  wD D  2 wE wD  DE E D
2    2  2    2  2

if DE  1, then
 P  ( wE E  wD D ) 2
2

and
   ( wE E  wD D )

41
Table 7.1 Descriptive Statistics for Two
Mutual Funds
Three-Security Portfolio

E (rp )  w1E (r1 )  w2 E (r2 )  w3 E (r3 )

2p = w1212 + w2212 + w3232
+ 2w1w2       Cov(r1,r2)
+ 2w1w3 Cov(r1,r3)
+ 2w2w3 Cov(r2,r3)
Table 7.2 Computation of Portfolio Variance
From the Covariance Matrix
Table 7.3 Expected Return and Standard
Deviation with Various Correlation
Coefficients
Figure 7.3 Portfolio Expected Return as a
Function of Investment Proportions
Figure 7.4 Portfolio Standard Deviation as a
Function of Investment Proportions
Minimum Variance Portfolio as Depicted
in Figure 7.4
• Standard deviation is smaller than that of either
of the individual component assets
• Figure 7.3 and 7.4 combined demonstrate the
relationship between portfolio risk
Figure 7.5 Portfolio Expected Return as a
Function of Standard Deviation
Correlation Effects

• The relationship depends on the correlation
coefficient
• -1.0 <  < +1.0
• The smaller the correlation, the greater the risk
reduction potential
• If  = +1.0, no risk reduction is possible
Figure 7.10 The Minimum-Variance Frontier of
Risky Assets

51
Figure 7.6 The Opportunity Set of the Debt
and Equity Funds and Two Feasible CALs
The Sharpe Ratio

• Maximize the slope of the CAL for any possible
portfolio, p
• The objective function is the slope:

E (rP )  rf
SP 
P

53
Mathematically, we solve for the tangent
portfolio weights from maximization the
sharpe ratio:
E ( rp )  r f
Sp 
p
subjectto :
E ( rp )  w1 E ( r )  w2 E ( r2 )
1

p      w1  1  w2  2  2 12 w1 w2 1 2
2   2    2   2

The solutions are:
E ( RD ) E  E ( RE )Cov( RD , RE )
2
wD 
E ( RD ) E  E ( RE ) D  [ E ( RD )  E ( RE )]Cov( RD , RE )
2             2

wE  1  wD
54
Figure 7.7 The Opportunity Set of the Debt
and Equity Funds with the Optimal CAL and
the Optimal Risky Portfolio
Figure 7.8 Determination of the Optimal
Overall Portfolio
Efficient Frontier with
Lending & Borrowing
CAL
E(r)
B
Q
M

A

rf          F

57
4. Asset Pricing Models: Capital Asset Pricing Model
(CAPM) and Arbitrage Pricing Theory (APT)

• It is the equilibrium model that underlies all
modern financial theory.
• Derived using principles of diversification with
simplified assumptions.

58
Assumptions
• Individual investors are price takers.
• Single-period investment horizon.
• Investments are limited to traded financial
assets.
• No taxes and transaction costs.

59
Assumptions (cont’d)
• Information is costless and available to all
investors.
• Investors are rational mean-variance
optimizers.
• There are homogeneous expectations.

60
Resulting Equilibrium Conditions
• All investors will hold the same portfolio for
risky assets – market portfolio.
• Market portfolio contains all securities and the
proportion of each security is its market value
as a percentage of total market value.

61
Resulting Equilibrium Conditions
(cont’d)
• Risk premium on the market depends on the
average risk aversion of all market participants.
• Risk premium on an individual security is a
function of its covariance with the market.


E (ri )  rf  i E (rM )  rf   

62
Using GE Text Example
• Reward-to-risk ratio for investment in market
portfolio:
Market risk premium E (rM )  rf

Market variance        M2

• In equilibrium, reward-to-risk ratios of GE and
the market portfolio should be equal to each
other:        E (r )  r E (r )  r

GE      f        M       f

Cov(rGE , rM )          2
M

• And the risk premium for GE:
Cov(rGE , rM )
E (rGE )  rf                         E (rM )  rf 
              
   2
M
Expected Return-Beta Relationship

• CAPM holds for the overall portfolio because:
E (rP )   wk E (rk ) and
k

 P   wk  k
k
• This also holds for the market portfolio:

E (rM )  rf   M  E (rM )  rf 
              
Cov(rM , rM )  M
So  M  1 or alternatively
2
M                2 1
M 2
M
Figure 9.2 The Security Market Line
Security Market Line (SML)
i = [COV(ri,rm)] / m2
Slope SML =      E(rm) - rf
SML = rf + i [E(rm) - rf]
Betam = [Cov (ri,rm)] / m2
= m2 / m2 = 1

66
Sample Calculations for SML
E(rm) - rf = .08   rf = .03

x = 1.25
E(rx) = .03 + 1.25(.08) = .13 or 13%

y = .6
E(ry) = .03 + .6(.08) = .078 or 7.8%

67
Graph of Sample Calculations
E(r)
SML

Rx=13%                                   .08
Rm=11%
Ry=7.8%

3%

.6   1.0   1.25
y         x
By
68
Disequilibrium Example
E(r)

SML
15%

Rm=11%

rf=3%


1.0   1.25

69
Disequilibrium Example
• Suppose a security with a  of 1.25 is offering
expected return of 15%.
• According to SML, it should be 13%.
• Under-priced: offering higher rate of return
for its level of risk.

70
Active investment with beta
• Those who think they are able to time the
market can do the followings.
– Buy high-beta stocks when they think the market
is up
– Switch to low-beta stock when they fear the
market is down

71
Extensions of the CAPM
• Zero-Beta Model
– Helps to explain positive alphas on low beta
stocks and negative alphas on high beta stocks
• Consideration of labor income and non-traded
assets
• Merton’s Multiperiod Model and hedge portfolios
– Incorporation of the effects of changes in the
real rate of interest and inflation
Black’s Zero Beta Model

• Absence of a risk-free asset
• Combinations of portfolios on the efficient
frontier are efficient.
• All frontier portfolios have companion
portfolios that are uncorrelated.
• Returns on individual assets can be
expressed as linear combinations of
efficient portfolios.
Black’s Zero Beta Model
Formulation

Cov(ri , rP )  Cov(rP , rQ )

E (ri )  E (rQ )  E (rP )  E (rQ )          P  Cov(rP , rQ )
2
Efficient Portfolios and Zero
Companions
E(r)

Q
P
E[rz (Q)]           Z(Q)
E[rz (P)]                 Z(P)


Zero Beta Market Model


E (ri )  E (rZ ( M ) )  E (rM )  E (rZ ( M ) )      Cov(ri , rM )
   2
M

CAPM with E(rz (m)) replacing rf
-To the extent that risk characteristics of private enterprises
that best hedges the risk of typical private business would
enjoy excess demand from the population of private business
owners. The price of assets in this portfolio will bid up relative
to the CAPM considerations, and the expected returns on
these securities will be lower in relation to their systematic
risk. Conversely, securities highly correlated with such risk will
have high equilibrium risk premiums and may appear exhibit
positive alphas relative to the conventional SML
CAPM with Labor Income
• An individual seeking diversification should
avoid investing in his employer’s stock and
limit investments in the same industry. Thus,
the demand for stocks of labor-intensive firms
may be reduced, and these stocks may require
a higher expected return than predicated by
the CAPM
• Mayer’s model (9.13)
Mayers derives the equilibrium expected return-beta
for an economy in which individuals are endowed
with labor income of varying size relative to their
nonlabor capital.
Cov( Ri , RM )  PM Cov( Ri , RH )
PH

E ( Ri )  E ( RM )
 M  P Cov( RM , RH )
2   P H
M

Where PH =value of aggregate human capital
PM =market value of traded assets (market
portolio)
RH =excess return on aggregate human capital

79
A Multiperiod Model and Hedge
Portfolios
• Merton relaxes the “single-period” myopic assumptions about investors.
He envisions individuals who optimize a lifetime consumption/investment
plan, and who continually adapt consumption/investment decisions to
current wealth and planned retirement age
• One key parameter is the future risk-free rate. If it falls in some future
period, one’s level of wealth will now support a lower stream of real
consumption. To the extent that returns on some securities are correlated
with changes in the risk-free rate, a portfolio can be formed to hedge such
risk, and investors will bid up the prices (and bid down the expected
return) of those hedge assets.
• Another key parameter is inflation risk. For example, investors may bid up
share prices of energy companies that will hedge energy price uncertainty
K
E ( Ri )   iM E ( RM )    ik E ( Rk )
k 1
Extensions of the CAPM Continued

• A consumption-based CAPM
– Models by Rubinstein, Lucas, and Breeden
• Investor must allocate current wealth between today’s
consumption and investment for the future
• As a general rule, investors will value additional income more
highly during difficult times than in affluent times. An asset will
therefore be viewed as riskier in terms of consumption if it has
positive covariance with consumption path (9.15)
E ( Ri )   iC RPC

where
RP  E ( RC )  E (rC )  rf
C
Liquidity and the CAPM

• Liquidity is prefered by investors
• Research supports a premium for illiquidity.
– Amihud and Mendelson
– Acharya and Pedersen
Figure 9.5 The Relationship Between
Illiquidity and Average Returns
Acharya and Pedersen proposed the following model:
E ( Ri )  kE(Ci )   (   L1   L 2   L3 )
Where    E (Ci )        =expected cost of illiquidity
k =adjustment for average holding period

over all securities
 =market risk premium net of average
market illiquidity cost,
 =measure of systematic market risk
 L1 ,  L 2 ,  L 3 =liquidity betas

84
Three Elements of Liquidity

• Sensitivity of security’s illiquidity to market
illiquidity:   Cov(Ci , CM )
Var ( RM  CM )
L1

Cov( Ri , CM )
L2 
Var ( RM  CM )
• Sensitivity of the security illiquidity to the market
rate of return:
Cov(Ci , RM )
 L3 
Var ( RM  CM )
How to estimate beta: Single Index Model

rit   i  i rmt  eit
factor
m = a broad market index like the S&P 500 is the
common factor
ei = uncertainty about the firm
Where Cov(ei , rm )  0
Cov(ei , e j )  0

86
Single-Index Model

• Alternatively, we run regression equation:
ri  r f   i   i ( rm  r f )  ei
Ri  ri  r f
Rm  rm  r f
• A Portfolio
n                n
R p   wi Ri   wi [ i   i Rm  ei ]
i 1             i 1
n                n              M
  wi i  ( wi  i ) Rm  ( wi ei )   p   p Rm  e p
i 1             i 1            i 1

87
Single-Index Model Continued

• Systematic and unsystematic risk
– Total risk = Systematic risk + Firm-specific risk (or
unsystematic risk):
 i2  i2 M   2 (ei )
2

88
Index Model and Diversification
• Portfolio’s variance:

      (eP )
2
P
2
P
2
M
2

• Variance of the equally weighted portfolio of
firm-specific components:
2
n
1 2     1 2
 (eP )      (ei )   (e)
2

i 1  n      n

• When n gets large,  2 (eP ) becomes negligible

89
Figure 8.1 The Variance of an Equally
Weighted Portfolio with Risk Coefficient βp in
the Single-Factor Economy

90
Figure 8.2 Excess Returns on HP and S&P 500
April 2001 – March 2006

91
Figure 8.3 Scatter Diagram of HP, the S&P
500, and the Security Characteristic Line
(SCL) for HP

92
Table 8.1 Excel Output: Regression Statistics
for the SCL of Hewlett-Packard

93
Alpha and Security Analysis

• Macroeconomic analysis is used to estimate the
risk premium and risk of the market index
• Statistical analysis is used to estimate the beta
coefficients of all securities and their residual
variances
• According to CAPM,  P  0 . A portfolio manager
who beats the market implies alpha is greater
than zero

94
Optimal Risky Portfolio of the Single-Index
Model
• Maximize the Sharpe ratio by choosing weights
– Expected return, SD, and Sharpe ratio:
n 1              n 1
E ( RP )   P  E ( RM )  P   wi i  E ( RM ) wi i
i 1              i 1
1
1  2  n 1       2
 n 1 2 2          2
 P    P M   (eP )    M   wi  i    wi  (ei ) 

2 2     2

2

  i 1
                i 1           

E ( RP )
SP 
P

95
The Information Ratio

• The Sharpe ratio of an optimally constructed risky
portfolio will exceed that of the index portfolio
(the passive strategy):
2
 A 
 sM  
2        2
sP                   
  (e A ) 

96
Treynor-Black Allocation
CAL
E(r)                  CML

P
A

M

Rf


Arbitrage Pricing Theory and Multifactor
Models of Risk and Return
Single Factor Model Equation
ri  E (ri )  i F  ei

Or alternatively, ri   i   i F1  ei
ri = Return for security i
F = Surprise in macro-economic factor
(F could be positive, negative or zero)
ei = Firm specific events
Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero
investment portfolio with a sure profit
• Since no investment is required, an investor can
create large positions to secure large levels of
profit
• In efficient markets, profitable arbitrage
opportunities will quickly disappear
Arbitrage Pricing Theory (APT)
• Assume factor model such as
ri   i   i F1  ei

• And no arbitrage opportunities exist in
equilibrium
• Then, we have
E (ri )  0  1 i

101
APT and CAPM Compared

• APT applies to almost all individual securities
• With APT it is possible for some individual stocks
to be mispriced - not lie on the SML
• APT is more general in that it gets to an expected
return and beta relationship without the
assumption of the market portfolio
• APT can be extended to multifactor models
Multifactor APT

• Use of more than a single factor
• Requires formation of factor portfolios
• What factors?
– Factors that are important to performance of
the general economy
– Fama-French Three Factor Model
Two-Factor Model

ri  E (ri )  i1F1  i 2 F2  ei
• The multifactor APT is similar to the one-
factor case
– But need to think in terms of a factor portfolio
Example of the Multifactor Approach

• Work of Chen, Roll, and Ross
– Chose a set of factors based on the ability of
the factors to paint a broad picture of the
macro-economy
– GDP factor, inflation factor, and interest rate
factor
Another Example:
Fama-French Three-Factor Model

• The factors chosen are variables that on past
evidence seem to predict average returns well
and may capture the risk premiums
rit  i  iM RMt  iSMB SMBt  iHML HMLt  eit
•   Where:
– SMB = Small Minus Big, i.e., the return of a portfolio of small stocks in excess
of the return on a portfolio of large stocks
– HML = High Minus Low, i.e., the return of a portfolio of stocks with a high book
to-market ratio in excess of the return on a portfolio of stocks with a low book-
to-market ratio
The Multifactor CAPM and the APT

• A multi-index CAPM will inherit its risk factors
from sources of risk that a broad group of
investors deem important enough to hedge
• The APT is largely silent on where to look for
priced sources of risk
Empirical tests of asset pricing
models
Overview of Investigation

• Tests of the single factor CAPM or APT Model
• Tests of the Multifactor APT Model
• Studies on volatility of returns over time
The Index Model and the Single-Factor APT

• Test the linear expected return-beta relationship

E (ri )  rf   i [ E (rM )  rf ]
Tests of the CAPM
Tests of the expected return beta relationship:
• First Pass Regression
– Estimate beta, average risk premiums and
unsystematic risk
• Second Pass: Using estimates from the first pass
to determine if model is supported by the data
• Most tests do not generally support the single
factor model
Single Factor Test Results

Return %
Predicted

Actual

Beta
Roll’s Criticism
• The only testable hypothesis is on the efficiency
of the market portfolio
• CAPM is not testable unless we know the exact
composition of the true market portfolio and use
it in the tests
• Benchmark error
Measurement Error in Beta

• Statistical property
• If beta is measured with error in the first stage,
second stage results will be biased in the
direction the tests have supported
• Test results could result from measurement error
Jaganathan and Wang Study

• Included factors for cyclical behavior of betas and
human capital
• When these factors were included the results
showed returns were a function of beta
• Size is not an important factor when cyclical
behavior and human capital are included
Table 13.2 Evaluation of Various CAPM
Specifications
Table 13.4 Determinants of Stockholdings
• We expect that private-business owners will
reduce demand for traded securities that are
positively related with their specific
entrepreneurial income. The empirical
evidence form Table 13.4 is consistent with
that reasoning

118
Tests of the Multifactor Model
• Chen, Roll and Ross 1986 Study
Factors
Growth rate in industrial production
Changes in expected inflation
Unexpected inflation
Unexpected Changes in risk premiums on bonds
Unexpected changes in term premium on bonds
Study Structure & Results

• Method: Two -stage regression with portfolios
constructed by size based on market value of
equity Fidings
• Significant factors: industrial production, risk
premium on bonds and unanticipated inflation
• Market index returns were not statistically
significant in the multifactor model
Table 13.5 Economic Variables and Pricing
(Percent per Month x 10), Multivariate
Approach
Fama-French Three Factor Model

• Size and book-to-market ratios explain returns on
securities
• Smaller firms experience higher returns
• High book to market firms experience higher
returns
• Returns are explained by size, book to market and
by beta
Table 13.6 Three Factor Regressions for
Portfolios Formed from Sorts on Size and
Book-to-Market Ratios (B/M)
Interpretation of Three-Factor Model

• Size is a proxy for risk that is not captured in
CAPM Beta
• Premiums are due to investor irrationality or
behavioral biases
Risk-Based Interpretations

• In figure 13.1, it shows that returns on style
portfolios (HML or SMB) seem to predict GDP
growth, and thus may in fact capture some
• In figure 13.2, it shows the beta of the HML
portfolio is lower in good economies while
becomes higher in recessions, suggesting also
that HML captures some aspects of business cycle
risks
Portfolios in Year Prior to Above-Average
versus Below-Average
GDP Growth
Figure 13.2 HML Beta in Different Economic
States
Zhang (2005) finds that value firms (with high
book-to-market ratios) on average have
greater amount of tangible capital. Investment
irreversibility puts such firms more at risk for
economic downturns. In contrast, growth
firms are better able to deal with a downturn
by deferring investment plans. The greater
exposure of high book-to-market firms to
recessions will result in higher down-market
betas.                                        128
To quantify this, Petkova and Zhang fit the following model:
rHML    rMt  ei
   [b0  b1DIVt  b2 DEFLT  b3TERMt  b4TBt ]rM  et
t

Where:
DIV =market dividend yield
Default=default spread on corporate bonds (Baa-Ass rates)
Term=term structure spread (10-year – 1-year Treasury rates)
TB=1-month T-bill rate

129
Behavioral Explanations
• Market participants are overly optimistic
– Analysts extrapolate recent performance too
far into the future
– Prices on these glamour stocks are overly
optimistic
– Lower book-to-market on these glamour firms
leads to underperformance compared to value
stocks
• Chan, Karceski and Lakonishok find that B/M ratio
reflects the past earning growth but not the
future, as in Fig. 13.3
La Porta, Lakonishok, Shleifer, and Vishny (2007)
demonstrate that growth stocks
underperforms value stocks surrounding
earnings announcements, suggesting that
when news of actual earnings is released to
the public, the market is relatively
disappointed in stocks it has been predicting
as growth firms.

132
Liquidity and Asset Pricing

• Acharya and Pedersen
– Premiums observed in the three-factor model
may be illiquidity premiums, as shown in the
first three row of results in Table 13.7
– In table 13.8, it shows that despite that the
liquidity adjustments to the market beat are
relatively small, accounting for portfolio
liquidity materially improves the fit of the
model
Table 13.7 Properties of Liquidity Portfolios
Table 13.8 Estimates of the CAPM With and
Without Liquidity Factors
Consumption-based Asset Pricing
Model (CCAPM)
Each individual’s plan is to maximize a utility function of
in each period is based on age and current wealth, as
well as risk-free rate and the market portfolio’s risk
E (rM )  rf  ACov(rM , rC )

Table 13.10 and Figure 13.7 indicate that Fama-French
factors for average returns may in fact reflect the
differing consumption risk of those portfolios

137
138

• Rewards for bearing risk appear to be excessive
• Possible Causes
– Predicting returns from realized returns;
people underestimated the realized returns in
post-war America
• Survivorship bias also creates the appearance of
abnormal returns in market efficiency studies;
see Figure 13.8
Period       Risk-Free Rate   S&P 500 return   Equity Premium

1872-1999    4.87             10.97            6.10
1972-1949    4.05             8.67             4.62
1950-1999    6.15             14.56            8.41
Extensions to the CAPM may
Constantinides (2008) argues that the standard
CAPM can be extended to including habit
formation, incomplete markets, the life cycle,
borrowing constraints, and other forces of
limited stock market participants, to help

143
Behavioral Explanations of the
Barberis and Huang (2008) incorporate loss
aversion and narrow framing to explain the
puzzle. Narrow framing is the idea that
investors evaluate every risk they face in
isolation. Thus, investors will ignore low
correlation of the risk of stock portfolio with
other components of wealth, and therefore
require a higher risk premium than rational
models would predict. Loss aversion also
Time-Varying Volatility

• Stock prices change primarily in reaction to
information
• New information arrival is time varying
• Volatility is therefore not constant through time
Stock Volatility Studies and Techniques

• Volatility is not constant through time
• Improved modeling techniques should improve
results of tests of the risk-return relationship
• ARCH and GARCH models incorporate time
varying volatility
Figure 13.5 Estimates of the Monthly Stock
Return Variance 1835 - 1987

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