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Chapter 8
CAPITAL ASSET PRICING ANDARBITRAGE PRICING THEORY
The Risk Reward Relationship
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Outline
Key Issues
Basic Assumptions
Capital Market Line
Security Market Line
Inputs Required for CAPM
Calculation of Beta
Empirical Evidence on CAPM
Arbitrage Pricing Theory
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Key Issues
Essentially, the capital asset pricing model (CAPM) is concernedwith two questions:
What is the relationship between risk and return for anefficient portfolio?
What is the relationship between risk and return for an
individual security?
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Basic Assumptions
Riskaversion
Maximisation . . expected utility
Homogeneous expectation
Perfect markets
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Capital Market LineExpected
Return, E(Rp) Z
L
M
K
Rf
Standard Deviation, pE(Rj) = Rf+ j
E(RM) - Rf
=M
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E(RM) -Rf
E(Ri ) =Rf + CiM
M
Security Market Line
iMi =
ME (R i ) = R f + [ E (R M)- R f]i
PReturn SML
14%
8% 0
Alpha = Expected - Fair
Return Return
1.0 i
R l i hi B SML A d CML
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Relationship Between SML And CML
SML
E(RM ) - Rf
E(Ri) = Rf + iMM
2
Since iM = iM iM
E(RM ) - RfE(Ri) = Rf + iM i
M
IF i and M are perfectly correlated iM =1. SOE(RM ) - Rf
E(Ri) = Rf + iM
thus cml is a special case of sml
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Inputs Required For Applying CAPM
Risk-Free Return
Rate on a short-term govt security
Rate on a long term govt bond
Market Risk Premium Historical
difference between the average return on stocks and the
average risk - free return
Period : As long as possible
Average : A.M VS. G.M.
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Determinants of Risk Premium
Variance in the underlying economy
Political risk
Market structure
FINANCIAL MARKET EXAMPLES PREMIUM OVER
THE
CHARACTERISTICS GOVT BOND RATE (%)
EMERGING MARKET, WITH SOUTH AMERICAN MARKETS, 7.5 - 9.5
POLITICAL RISK CHINA, RUSSIA
EMERGING MARKETS WITH SINGAPORE, MALAYSIA, 7.5
LIMITED POLITICAL RISK THAILAND, INDIA, SOME EASTEUROPEAN MARKETS
DEVELOPED MARKETS WITH UNITED STATES, JAPAN, U.K., 5.5
WIDE STOCK LISTINGS FRANCE, ITALY
DEVELOPED MARKETS WITH GERMANY, SWITZERLAND 3.5 - 4.5
LIMITED LISTINGS AND
STABLE ECONOMIES
* Source : Aswath Damodaran Corporate Finance Theory and Practice, John Wiley.
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Triumph Of Optimists
Elroy Dimson, Paul March, and Michael Stanton triumph of theOptimists, (2001)
Equity returns 16 rich countries data 1900
Global historical risk premium 20TH century .. 4.6%
Best estimate of equity premium worldwide in future is 4 to 5
percent
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Calculation Of Beta
Rit = i + i RMt + eit
iMi =M 2
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Calculation Of Beta
Period
Return on
stockA, RA
Return on
marketportfolio, RM
Deviation of
return on stockA
from its mean
(RA - RA)
Deviation of
return on marketportfolio from its
mean (RM - RM)
Product of the
deviation,(RA - RA)
(RM - RM)
Square of the
deviation of
return on marketportfolio from its
mean
(RM - RM)2
1 10 12 0 3 0 9
2 15 14 5 5 25 25
3 18 13 8 4 32 16
4 14 10 4 1 4 15 16 9 6 0 0 0
6 16 13 6 4 24 16
7 18 14 8 5 40 25
8 4 7 -6 -2 12 4
9 - 9 1 -19 -8 152 64
10 14 12 4 3 12 9
11 15 -11 5 -20 -100 40012 14 16 4 7 28 49
13 6 8 -4 -1 4 1
14 7 7 -3 -2 6 4
15 - 8 10 -18 1 -18 1
RA = 150 RM = 135 (RA - RA) (RM - RM) 2RA =10 RM = 9 (RM - RM) = 221 = 624
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Estimation Issues
Estimation Period
A longer estimation period provides more data butthe risk profile .. firm may change
5 years
Return interval daily, weekly, monthly
Market Index
Standard Practice
Adjusting Historical Beta
Historical alignment chance factor
A companys beta may change over time
Merill lynch 0.66 Historical beta
O.34 Market beta
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Betas Based On Fundamental Information
Key factors employed are
Industry Affiliation
Corporate Growth
Earnings Variability
Financial Leverage
Size
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Betas Based On Accounting Earnings
Regress the changes in company earnings (on a quarterly or annual
basis) against changes in the aggregate earnings of all the companies
included in a market index.
Limitations
Accounting earnings .. generally smoothed out ..relative .. value of the company
Accounting earnings influenced by non - operating
factors
Less frequent measurement
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Betas from CrossSectional Regressions
1. Estimate a cross - sectional regressionRelationship for publicly traded firms:
Beta = 0.6507 + 0.27 coefficient of variation
In operating income + 0.09 D/E + 0.54
Earnings - .00009 total assets(million $)
2. Plug the characteristics of the project, division, or
unlisted company in the regrn reln to arrive at an
estimate of beta
Beta = 0.6507 + 0.27 (1.85) + 0.09 (0.90) + 0.54 (0.12) -
0.00009 (150) = 1.2095
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Empirical Evidence On CAPM
1. Set up the sample data
Rit , RMt , Rft
2. Estimate the security characteristic linesRit - Rft = ai +bi (RMt -Rft) + eit
3. Estimate the security market line
Ri =0 +1 bi + ei , i= 1, 75
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General Findings
The relation appears .. linear
0 > Rf 1 < RM -Rf In addition to beta, some other factors, such as standard
deviation of returns and company size, too have a bearing
on return
Beta does not explain a very high percentage of thevariance in return
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Conclusions
Problems
Studies use historical returns as proxies for expectations
Studies use a market index as a proxy
Popularity
Some objective estimate of risk premium .. better than a
completely subjective estimate
Basic message .. accepted by all
No consensus on alternative
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ArbitragePricing Theory
Return generating process
Ri = ai +bi 1 I1+ bi2 I2 +bij Ij+ ei
Equilibrium riskreturn relationshipE(Ri) = 0 + bi1 1 + bi2 2 + bijjj = Risk premium for the type ofRisk associated with factor j
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Comparison of CAPM and APT
CAPM APT
Nature of relation Linear Linear
Number of risk factors 1 k
Factor risk premium [E(RM)Rf] ljFactor risk sensitivity bi bij
Zero-beta return Rf l0
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Multifactor Models
Given the practical difficulties in using the above approach,researchers have followed a different approach that captures the
essence of the APT. In this approach, the researcher chooses a priorithe exact number and identify of risk factors and specifies the
multifactor model of the following kind.Rit = ai +[bit F1t + bi2 F2t+.. + bik Fkt] + eit
where Rit is the return on security i in period t, and Fjt is the returnassociated with thej th risk factor in period t.
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The advantage of a factor model like this is that the researcher
can specify the risk factors; the disadvantage of such a model is that
there is very little theory to guide it. Hence, developing a useful
factor model is as much an art as science.
The variety of multifactor models employed in practice may be
divided into two broad categories: macro-economic based risk factor
models and micro-economic based risk factor models.
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Macroeconomic Based Risk Factor Models
These models consider risk factors that are macroeconomic in
nature. Typical of this approach is the following model proposed by
Chen, Roll, and Ross in their classic paper, "Economic Forces and
the Stock Market," published in the April 1986 issue ofJ ournal ofBusiness.
Rit = ai + bi1 Rmt + bi2 MPt + bi3DEI t + bi4UI t + b5UPRt + bi6 UTSt + eit
Where Rm is the return on a value weighted index of NYSE listed stocks, MP is the monthly growth rate in the US industrial
production, DEI is the change in inflation, measured by the USconsumer price index, UI is the difference between actual and
expected levels of inflation, UPR is the unanticipated change in the
bond credit spread (Baa yieldRFR), and UTS is the unanticipated
term structure shift (long term RFRshort term RFR).
Mi i B d Ri k F M d l
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Microeconomic Based Risk Factor Models
Instead of specifying risk in macroeconomic terms, you can delineate
risk in microeconomic terms. Typical of this approach is the
following model proposed by Fama and French in their celebrated
paper "Common Risk Factors in the Returns on Stocks and Bonds," published in the January 1993 issue of the J ournal of FinancialEconomics:
(RitRFRt) = i + bi1 (RmtRFRt) + bi2SMBt + bi3HMLt + eitIn this model, in addition to (R
mtRFR
t), the excess return on a
stock market portfolio, there are two other microeconomic riskfactors: SMB
t and HML
t. SMB
t(i.e.,
contd
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St k M k t C l
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Stock Market as a Complex
Adaptive System
To understand what a complex adaptive system is let us begin with asimple situation where two people are put in a room and asked totrade a commodity. What happens? Hardly anything. If a few more
people are added, the activity picks up, but the interactions remain
somewhat subdued. The system remains static and lifeless comparedto what we see in the capital markets. As more and more people are
added to the system, something remarkable happens: it acquires
lifelike characteristics.As Mauboussin put it: In a tangible way, the
system becomes more complex than the pieces that it comprises.
Importantly, the transitionoften called self-organised criticalityoccurs without design or help from outside agent. Rather, it is a
direct function of the dynamic interactions among the agents in the
system.
P ti f C l Ad ti S t
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Properties of a Complex Adaptive System
Aggregation The collective interactions of many less-complex agents
produces complex, large-scale behaviour.
Adaptive Decision Rules Agents in the system take information from
the environment and develop decision rules. The competition
between various decision rules ensures that eventually the most
effective decision rules survive.
Non-Linearity Unlike a linear system, wherein the value of the whole
is equal to the sum of its parts, a non-linear system is one wherein
the aggregate behaviour is very complex because of interaction
effects.
contd...
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contd
Feedback Loops In a system that has feedback loops the output ofone interaction becomes the input of the next. A positive feedback
can magnify an effect, whereas a negative feedback can dampen an
effect.
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How Does the New Model Compare with
Classical Market Theory
The complex adaptive expectations model seems to conform to
reality better than the classical capital market theory. The
following evidence bears this out:
1. The high kurtosis (fattails) in return distribution suggeststhat periods of stability are interspersed by rapid change.
2. The price behaviour in a complex adaptive system would not
be very different from a classic random walk. However, the
new model explains better the observed persistence in
returns, to the extent that the same exists.
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3. Under most circumstances, the aggregation of theheterogeneous expectations of investors would yield prices
that are similar to intrinsic values. However, if certain
decision rules become pervasive, the resulting homogeneity ofviews may lead to self-reinforcing trends, leading to booms
and crashes.
4. The poor performance of active portfolio managers is
consistent with the classical market theory as well as thecomplex adaptive model. Still, it is possible that some
investors would do well. As Mauboussin put it: That point
made, it remains possible under theory that certain investors
Warren Buffett and Bill Miller, e.g. may be hard-wired tobe successful investors. In this sense, hard-wired suggest
innate mental processes, fortified with practice, that allow for
systematically superior security selection.
I li ti f th N M d l
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Implications of the New ModelThe important implications of the new model for investors and
corporate practitioners are as follows:
1. While the CAPM is still probably the best available estimate
of risk for most corporate investment decision, managers
must recognise that their stock price may fluctuate more than
what the standard theory suggests.
2. The market is usually smarter than the individual. Hence
managers should weight the evidence of the market over the
evidence of experts.
3. Markets function well when participants pursue diverse
decision rules and their errors are independent. Markets,
however, can become very fragile when participants display
herd-like behaviour, imitating one another.
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4. It may be futile to identify the cause of a crash or boom
because in a non- linear system small things can cause large-
scale changes.
5. The discounted cash flow model provides an excellent
framework for valuation. Indeed, it is the best model for
figuring out the expectations embedded in stock prices.
Mauboussin summed up the implications of the new model as
follows: From a practical standpoint, managers who
subscribe to standard capital market theory and operate on
the premise of stock market efficiency will probably not gotoo far astray. However, complex adaptive systems may
provide a useful perspective in areas like risk management
and investor communication.
S mming Up
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Summing Up
The relationship between risk and expected return for
efficient portfolios, as given by the capital market line, is:
E (Ri) = Rf+ i The relationship between risk and expected return for an
inefficient portfolio or a single security as given by the
security market line is:
E (Ri) = Rf+ [E (RM)Rf] x i The beta of a security is the slope of the following
regression relationship:
Rit = i + iRMt + eit
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The commonly followed procedure for testing CAPM involves
two steps. In the first step, the security betas are estimated. In
the second step, the relationship between security beta and
return is examined.
Empirical evidence is favour of CAPM is mixed.
Notwithstanding this, the CAPM is the most widely used risk-
return model because it is simple and intuitively appealing
and its basic message that diversifiable risk does not matter is
generally accepted.
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The APT is much more general in that asset prices can be
influenced by factors beyond means and variances. The APT
assumes that the return on any security is linearly related to a
set of systematic factors.