Asset pricing models

MGB Portfolio Management I
ASSET PRICING MODELS

PoA
A. Review and solve problems using the CAL, MPT, and
the Single Index model
B. Understand the implications of capital asset pricing
theory and the CAPM to compute security risk
premiums
C. Understand the arbitrage pricing theory and how it
works

Implications of Capital Market Theory and CAPM
• What have we done this far?
– We have been concerned with how an individual or
institution would select an optimum portfolio.
• If investors act as we think, we should be able to
determine how investors will behave, and how prices at
which markets will clear are set
– This market clearing of prices and returns has resulted in the
development of so-called general equilibrium models
• These models allow us to determine the risk for any asset
and the relationship between expected return and risk
for any asset when the markets are in equilibrium, i.e.
balance or constant state

Capital Asset Pricing Theory
• What is capital asset pricing theory?
– It is the theory behind the pricing of assets which takes into
account the risk and return characteristics of the asset and
the market
• What is the Capital Asset Pricing Model?
– It is an equilibrium model (i.e., a constant state model) that
underlies all modern financial theory
• It provides a precise prediction between the relationship
between the risk of an asset and its expected return
when the market is in equilibrium
• With this model, we can identify mis-pricing of securities
(in the long-run)

CAPM (continued)
• Why is it important?
– It provides a benchmark rate of return for evaluating
possible investments, and identifying potential mispricing of
investments
• For example, an analyst might want to know whether the
expected return she forecast is more or less than its
“fair” market return.
– It helps us make an “educated” guess as to the expected
return on assets that have not yet been traded in the
marketplace
• For example, how do we price an initial public offering?

CAPM (continued)
• How was it derived?
– Derived using principles of diversification with very simplified (i.e.
somewhat unrealistic) assumptions
• Does it work, i.e. withstand empirical tests in real life?
– Not totally
• But it does offer insights that are important and its accuracy may
be sufficient for some applications
• Do we use it?
– Yes, but with knowledge of its limitations

Premise of the CAPM
• The Capital Asset Pricing Model (CAPM) is a model to explain why capital
assets are priced the way they are.
• The CAPM was based on the supposition that all investors employ
Markowitz Portfolio Theory to find the portfolios in the efficient set.
Then, based on individual risk aversion, each of them invests in one of
the portfolios in the efficient set.
• Note, that if this supposition is correct, the Market Portfolio would be
efficient because it is the aggregate of all portfolios. Recall Property I - If
we combine two or more portfolios on the minimum variance set, we get
another portfolio on the minimum variance set.

CAPM Assumptions
• What does the model assume (some are unrealistic)?
– Individual investors are price takers (cannot affect prices)
– Single-period investment horizon (an its identical for all)
– Investments are limited to traded financial assets
– No taxes, and no transaction costs (costless trading)
– Information is costless and available to all investors
– Investors are rational mean-variance optimizers
– Investors analyze information in the same way, and have the
same view, i.e., homogeneous expectations
– Note: Many of the assumptions are obviously unrealistic. Later,
we will evaluate the consequences of relaxing some of these
assumptions. The assumptions are made in order to generate a
model that examines the relationship between risk and expected
return holding many other factors constant.

Resulting Equilibrium Conditions
• Based on the previous assumptions:
– All investors will hold the same portfolio for risky assets – the
market portfolio (M)
– The market portfolio (M) contains all securities and the
proportion of each security is its market value as a percentage
of total market value
– The risk premium on the market depends on the average risk
aversion of all market participants
– The risk premium on an individual security is a function of its
covariance (correlation and ss sm) with the market

E(r)
E(rM)
rf
Capital Market Line
M
CML
sm
s
M = Market portfolio rf = Risk free rate
E(rM) - rf = Market risk premium
[E(rM) - rf]/sM= Market price of risk
The efficient frontier without lending or
borrowing

Expected Return and Risk of Individual
Securities
• What does this imply?
– The risk premium on individual securities is a function
of the individual security’s contribution to the risk of
the market portfolio
– Individual security’s risk premium is a function of the
covariance of returns with the assets that make up the
market portfolio

CAPM Key Thoughts
• Key statements:
– Portfolio risk is what matters to investors, and portfolio risk is
what governs the risk premiums they demand
– Non-systematic, or diversifiable risk can be reduced through
diversification.
– Investors need to be compensated for bearing only non-systematic
risk (risk that cannot be diversified away)
– The contribution of a security to the risk of a portfolio
depends only on its systematic risk, as measured by beta. So
the risk premium of the asset is proportional to its beta.
(ß = [COV(ri,rm)] / sm
2)

Expected Return – Beta Relationship
Expected return - beta relationship of CAPM:
E(rM) - rf = E(rs) - rf
1.0 bs
In other words, the expected rate of return of an asset exceeds the
risk-free rate by a risk premium equal to the asset’s systematic risk
(its beta) times the risk premium of the market portfolio. This
leads to the familiar re-arrangement of terms to give (memorize
this):
E(rs) = rf + bs [E(rM) - rf ]

E(r)
E(rM)
rf
The Security Market Line
• Notice that instead of using standard deviation, the
SML
Security Market Line uses Beta
• SML Relationships
ß = [COV(ri,rm)] / σm
ß ß M
= 1.0
2
Slope SML = E(rm) – rf = market risk premium
SML = rf + ß[E(rm) - rf]

Differences Between the SML and CML
• What are the differences?
– The CML graphs risk premiums of efficient portfolios , i.e.
complete portfolios made up of the risk portfolio and risk-free
asset, as a function of standard deviation
– The SML graphs individual asset risk premiums as a
function of asset risk.
• The relevant measure of risk for individual assets is not
standard deviation; rather, it is beta
• The SML is also valid for portfolios

Capital Market Line (CML)
Vs.
Security Market Line (SML)
• Given a population of securities, there will be a simple
linear relationship between the beta factors of different
securities and their expected (or average) returns if and
only if the betas are computed using a minimum variance
market index portfolio.
• Therefore:
Given the CML, we can determine the SML (relationship
between beta & expected return)

CML Versus SML
E(r) E(r)
0.3
0.2
E(rM) E(rM)
0.1
0
0 1 2
0.3
0.2
0.1
0
CML
s(r)
0 0.48
SML
b
M C
B
A
C
M
B
A
rF rF
s(rM)

Example: SML Calculations
• Put the following data on the SML. Are they in
equilibrium?
Market data: E(rm) - rf = .08 rf = .03
Asset data: bx = 1.25 by = .60
– Calculations:
bx = 1.25 so E(r) on x =
E(rx) = .03 + 1.25(.08) = .13 or 13%
by = .60 so E(r) on y =
E(ry) = .03 + .6(.08) = .078 or 7.8%

E(r)
Rx=13%
SML
1.0
m
ß
ß
Rm=11%
Ry=7.8%
3%
1.25
x ß
.6
ßy
.08
Graph of Sample Calculations
They are in equilibrium

Disequilibrium Example
• Suppose a security with a beta of 1.25 is
offering expected return of 15%
– According to SML, it should be 13%
– Under priced: offering too high of a rate of return
for its level of risk. Investors therefore would:
• Buy the security, which would increase demand, which
would increase the price, which would decrease the
return until it came back into line.

Disequilibrium Example
E(r)
15%
SML
ß
The return is above the SML, so you
would buy it
1.0
Rm=11%
rf=3%
1.25
As more people bought the
security, it would push the
price up, which would bring
the return down to the line.

CAPM and Index Models
• CAPM Problems
– It relies on a theoretical market portfolio which includes all
assets
– It deals with expected returns
• To get away from these problems and make it testable, we
change it and use an Index model which:
– Uses an actual index, i.e. the S&P 500 for measurement
– Uses realized, not expected returns
• Now the Index model is testable

The Index Model
• With the Index model, we can:
– Specify a way to measure the factor that affects returns (the
return of the Index)
– Separate the rate of return on a security into its macro
(systematic) and micro (firm-specific) components
• Components
ά = excess return if market factor is zero
ßiRm = component of returns due to movements in the overall
market
ei = component attributable to company specific events
Ri = a i + ßiRm + ei
• (Notice the similarity to the Single Index model discussed earlier)

Security Characteristic Line
Excess Returns (i)
SCL
.
..
.
.
function of the excess return of the market
.
. .
. ..
.
. .
. .
Plot of a company’s excess return as a
. ..
.
.
.
. .
. ..
. .
.
. .
. .
.
. .
.
. .
.
.. . . .
. . . .
Excess returns
on market index
Ri = a i + ßiRm + ei

Does the CAPM hold?
• There is much evidence that supports the
CAPM
– There is also evidence that does not support the CAPM
• Is the CAPM useful?
– Yes. Return and risk are linearly related for securities and
portfolios over long periods of time
– Yes. Investors are compensated for taking on added market
risk, but not diversifiable risk
• Perhaps instead of determining whether the CAPM is true or not,
we might ask: Are there better models?

CAPM Problem
• Suppose the risk premium on the market portfolio is
9%, and we estimate the beta of Dell as bs = 1.3. The
risk premium predicted for the stock is therefore 1.3
times the market risk premium of 9% or 11.7%. The
expected return on Dell is the risk-free rate plus the
risk premium. For example, if the T-bill rate were 5%,
the expected return of Dell would be 5% +(1.3 * 9%) =
16.7%.
a. If the estimate of the beta of Dell were only 1.2,
what would be Dells required risk premium?
b. If the market risk premium were only 8% and Dell’s
beta was 1.3, what would be Dell’s risk premium?

Answer
• a. If Dell’s beta was 1.2 the required risk premium
would be (remember the risk premium is the
expected return less the risk-free rate):
E(rs) = rf + bs [E(rM) - rf ] or the expected return on
Dell = 5% + 1.2 (9%) = 15.8%
Dell’s risk premium (over the risk free rate) =
15.8% - 5% = 10.8%
• b. If the market risk premium was 8%:
E(rs) = rf + bs [E(rM) - rf ]
E(r) of Dell = 5% + 1.3 (8%) = 15.4%
Dell’s new risk premium is 15.4 – 5% = 10.4%

ASSET PRICING MODELS – RELAXING THE ASSUMPTIONS

Modeling Risk & Return
Part Two:
• Extensions,
• Testing, and
• The Arbitrage Pricing Theory (APT)

Relaxing the Assumptions of the CAPM
• CAPM assumption: all investors can borrow or lend at
the risk-free rate – unrealistic
• Two possible alternatives:
1. Differential borrowing and lending rates
• Unlimited lending at risk-free rate
• Borrowing at higher rate
• Leads to “bent” Capital Market Line
2. Zero-Beta CAPM
• Eliminates theoretical need for risk-free asset
• Leads to same form for SML but with a shallower slope

Differential Borrowing and Lending Rates
(Cost of Borrowing higher than Cost of Lending)
E(R)
Rb
RFR
Risk (standard deviation s)
F
G
K

Zero-Beta CAPM
• Zero-beta portfolio: create a portfolio that is
uncorrelated to the market (beta 0)
– The return of the zero-beta portfolio may differ from the risk-free
rate
• Any combination of portfolios on the efficient frontier
will be on the frontier
• Any efficient portfolio will have associated with it a
zero-beta portfolio

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(r,r ) Cov(r ,r )
Cov(r ,r )
E(r ) E(r ) E(r ) E(r )
P Q
2
P
i P P Q
i Q P Q
s 

=  

Efficient Portfolios and Zero
Companions
Q
P
Z(Q)
Z(P)
E(r)
E[rz (Q)]
E[rz (P)]
s

Zero Beta Market Model
  2
i M
M
i Z(M) M Z(M)
Cov(r ,r )
E(r ) E(r ) E(r ) E(r )
s
=  
CAPM with E(rz (M)) replacing rf

Implications of
Black’s Zero-beta model
• The expected return of any security can be expressed as a linear
relationship of any two efficient portfolios
E(Ri) = E(Rz) + bi[E(Rm) - E(Rz)]
• If original CAPM defines the relationship between risk and
return, then the return on the zero-beta portfolio should equal
RF
– Typically, in real world, RFR < E(RZ), so the zero-beta SML would be less
steep than the original SML
– Consistent with empirical results of tests of original CAPM
• To test directly - identify a market portfolio and solve for the
return of a zero-beta portfolio
– Leads to less consistent results

Security Market Line
With A Zero-Beta Portfolio
E(R)
E(Rm)
bi
SML
M
0.0 1.0
E(Rz)
E(Rm) - E(Rz)

• Another assumption of CAPM – zero transactions costs
• Existence of transaction costs:
– affect mispricing corrections
– affect diversification
– Leads to a “security market ‘band’” in place of the security
market line

Security Market Line With Transaction Costs
E(R)
E(Rm)
bi
SML
0.0 1.0
E(RFR) or
E(Rz)

• Heterogenous expectations
– If all investors have different expectations about risk and return, each
investor would have a different idea about the position and composition of
the efficient frontier, hence would have a different idea about the location
and composition of the tangency portfolio, M
– Hence, each would have a unique CML and/or SML, and the composite
graph would be a band of lines with a breadth determined by the
divergence of expectations
– Since each investor would have a different idea about where the SML lies,
each would also have unique conclusions about which securities are under-and
which are over-valued
– Also note that small differences in initial expectations can lead to vastly
different conclusions in this regard!

• Planning periods
– CAPM is a one period model, and the period employed should
be the planning period for the individual investor, which will
vary by individual, affecting both the CML and the SML
• Taxes
– Tax rates affect returns
– Tax rates differ between individuals and institutions

Empirical Testing of CAPM
Key questions asked:
• How stable is the measure of systematic risk (beta)?
• Is there a positive linear relationship as hypothesized
between beta and the rate of return on risky assets?
• How well do returns conform to the SML equation?

• Beta is not stable for individual stocks over short periods of
time (52 weeks or less)
– Need to estimate over 3 or more years (5 typically used)
• Stability increases significantly for portfolios
• The larger the portfolio and the longer the period, the more
stable the beta of the portfolio
• Betas tend to regress toward the mean ( = 1.0)

• In general, the empirical evidence regarding CAPM
has been mixed.
• Empirically, the most serious challenge to CAPM
was provided by Fama and French (discussed in the
Introductory lecture)
• Conceptually, the most serious challenge is provided
by Roll’s Critique

The Market Portfolio: Theory Versus Practice
• Impossible to test full market
• Portfolio used as market proxy may be correlated to
true market portfolio
• Benchmark error – 2 possible effects:
– Beta will be wrong
– SML will be wrong

Criticism of CAPM by Richard Roll
• Key limit on potential tests of CAPM:
– Ultimately, the only testable implication from CAPM is
whether the market portfolio is efficient (i.e., whether it
lies on the efficient frontier)
• Range of SML’s - infinite number of possible SML’s,
each of which produces a unique estimate of beta
• Market efficiency effects - substituting a proxy, such as
the S&P 500, creates two problems
– Proxy does not represent the true market portfolio
– Even if the proxy is not efficient, the market portfolio might be
(or vice versa)

Criticism of CAPM by Richard Roll
• Conflicts between proxies - different substitutes may be
highly correlated even though some may be efficient and
others are not, which can lead to different conclusions
regarding beta risk/return relationships
• So, ultimately, CAPM is not testable and cannot be
verified, so it must be used with great caution
• Stephen Ross devised an alternative way to look at asset
pricing that uses fewer assumptions – the Arbitrage
Pricing Theory, or APT

Understand Arbitrage Pricing Theory (APT)
and How it Works
• What is arbitrage?
– The exploitation of security mis-pricing to earn risk-free
economic profits
• It rises if an investor can construct a zero investment
portfolio (with a zero net investment position netting out
buys and sells) with a sure profit
• Should arbitrage exist?
– In efficient markets (and in CAPM theory), profitable
arbitrage opportunities will quickly disappear as more
investors try to take advantage of them

Arbitrage Pricing Theory (APT) (continued)
• What is APT based on?
– It is a variant of the CAPM, and is an attempt to move away
from the mean-variance efficient portfolios (the calculation
problem)
– Ross instead calculated relationships among expected returns
that would rule out riskless profits by any investor in a well-functioning
capital market
• What is it?
– It is a another theory of risk and return similar to the CAPM.
– It is based on the law of one price: two items that are the
same can’t sell at different prices

APT (continued)
• In its simplest form, it is:
Ri = a i + ßiRm + ei the same as CAPM
The only value for a which rules out arbitrage
opportunities is zero. So set a to zero and you get:
Ri = ßiRm Subtract the risk-free rate and you get the
well-known equation:
E(rs) = rf + bs [E(rM) - rf ] from CAPM

APT and CAPM Compared
• Differences:
– APT applies to well diversified portfolios and not
necessarily to individual stocks
– With APT it is possible for some individual stocks to
be mispriced – to 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, such
as:
Ri = a i + ß1R1 + ß2R2 + ß3R3 + ßnRn + ei

APT and Investment Decisions
APT offers an approach to strategic portfolio planning
– Investors need to recognize that a few systematic
factors affect long-term average returns
• Investors should understand those factors and
set up their portfolios to take those factors into
account
– Key Factors:
• Changes in expected inflation
• Unanticipated changes in inflation
• Unanticipated changes in industrial production
• Unanticipated changes in default-risk premium
• Unanticipated changes in the term structure of
interest rates

Problem
• Suppose two factors are identified for the U.S.
economy: the growth rate of industrial production
(IP) and the inflation rate (IR). IP is expected to be
4% and IR 6% this year. A stock with a beta of 1.0 on
IP and 0.4 on IR currently is expected to provide a
rate of return of 14%. If industrial production
actually grows by 5% while the inflation rate turns
out to be 7%, what is your best guess on the rate of
return on the stock?

Answer
• The revised estimate on the rate of return on
the stock would be:
– Before
• 14% = a +[4%*1] + [6%*.4]
a = 7.6%
– With the changes:
• 7.6% + [5%*1] + [7%*.4]
The new rate of return is 15.4%

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