Fin 2802: Investments
Spring, 2010
Dragon Tang
Lecture 17
Asset Pricing Theories
March 25, 2010
Readings: Chapter 9
Practice Problem Sets: 1,2,3,6-20
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Chapter 9: Asset Pricing Theories
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Asset Pricing Theories
Objectives:
• Use the implications of capital market theory to
computer security risk premium
• Construct security market line
• Take advantage of an arbitrage opportunity with a
portfolio that includes mispriced securities
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Chapter 9: Asset Pricing Theories
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Asset Pricing
• Absolute pricing vs relative pricing
• Positive view vs normative view
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Capital Asset Pricing Model (CAPM)
• Equilibrium model that underlies all modern
financial theory
• Derived using principles of diversification with
simplified assumptions
• Markowitz and Sharpe won Nobel prizes for this
development
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CAPM Assumptions
•
•
•
•
•
Individual investors are price takers
Single-period investment horizon
Investments are limited to traded financial assets
No taxes, and transaction costs
Information is costless and available to all
investors
• Investors are rational mean-variance optimizers
• Homogeneous expectations
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CAPM (Equilibrium) Results
• 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.
• Risk premium on the 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.
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Capital Market Line (CML)
E(R p )
Rm
Rf
M
Sharpe Ratio
σm
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Efficient
Frontier
Chapter 9: Asset Pricing Theories
p
7
The CML and the Separation Theorem
• The CML leads all investors to invest in the M
portfolio. The only difference is the location on
the CML depending on risk preferences
– Risk averse investors will lend part of the portfolio at
the risk-free rate and invest the remainder in the market
portfolio
– Investors preferring more risk might borrow funds at
Rf and invest everything in the market portfolio
– Two-fund separation theorem or “mutual fund
theorem”
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The Market Portfolio
• Because Portfolio M lies at the point of tangency, it has
the highest portfolio possibility line
• Everybody will want to invest in Portfolio M and
borrow or lend to be somewhere on the CML
• Therefore this portfolio must include ALL RISKY
ASSETS
• Because the market is in equilibrium, all assets are
included in this portfolio in proportion to their market
value
• Therefore, Portfolio M must be the market portfolio
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The Relevant Risk Measure for A Risky Asset
• Its covariance with the market portfolio M:
– Consider the stock of GM. Its inclusion in the market
portfolio is going to increase the variance of the market
portfolio by wGMCOV(rGM, rM)
– Marginally, for each unit of GM stock included, the amount
of risk it brings to the market portfolio is COV(rGM, rM)
– The risk premium each unit of GM stock provides is
measured by rGM-r.
– The risk-return tradeoff can be measured by
» (rGM-r)/ COV(rGM, rM)
» Also termed “market price of risk”
» In equilibrium, market prices of risk for all traded asset
should be the same!
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Risk and Expected Return for A Risky Asset
Expected Return = Rf + a*Risk
E[ Ri ]  Rf  a  COV ( Ri , RM )
If i happensto be themarketportfolioM, then
E[ RM ]  Rf  a   M2  a  E[ RM ]  Rf  /  M2
T herefore,E[ Ri ]  Rf 
COV ( Ri , RM )

2
M
E[ RM ]  Rf 
Capital Asset Pricing Model (CAPM)
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Capital Asset Pricing Model
• Relates expected return of an asset to its exposure to the
systematic risk as represented by b
E[ Ri ]  RFR   i E[ RM ]  RFR, where  i 
COV ( Ri , RM )
 M2
and E[ RM ]  RFR is themarketrisk premium
Note that  RF  0, and  M  1.
– The expected rate of return of a risky asset is determined by the
RFR plus a risk premium for the asset
– The risk premium, ERi  RFR, which can be negative (why?), is
determined by the systematic risk exposure of the asset (β) and
the prevailing market risk premium (RM-RFR)
• Security Market Line (ERi  vs  i )
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Sample Calculations for SML
If E(rm) - rf = .08 and
rf = .03
x = 1.25 then
E(rx) = .03 + 1.25(.08) = .13 or 13%
y = .6 then
E(ry) = .03 + .6(.08) = .078 or 7.8%
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Graph of Sample Calculations
E(r)
SML
.08
Rx=13%
Rm=11%
Ry=7.8%
3%
.6 1.0 1.25
ßy ßm ßx
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ß
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Determining the Expected Rate of Return
for a Risky Asset
• In equilibrium, all assets and all portfolios of assets
should plot on the SML
E(R i )  Rf   i   i (R M - Rf)
 i  0 on SML.
• Any security with an estimated return that plots above
the SML is underpriced (w.r.t. CAPM) ( i  0 )
• Any security with an estimated return that plots below
the SML is overpriced (w.r.t. CAPM) (  i  0)
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The Security Market Line and Positive Alpha Stock
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Practice
Two investment advisers are comparing performance. One
average a 19% return and the other a 16%. However, the beta
of the first adviser was 1.5, while that of the second was 1.0.
a. Can you tell which adviser was better?
b. If the T-bill rate were 6%, and market return during the period
were 14%, which would be better?
c. What if T-bill rate were 3% and market return 15%?
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 for Portfolios
• Expected return of a portfolio
n
n
i 1
i 1
E ( R p )   wi E ( Ri )   wi Rf   i RM  Rf 
 n

 Rf    wi  i RM  Rf 
 i 1

n
  p   wi  i
i 1
• Beta is additive: Portfolio beta equals to portfolio weighted
betas
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Estimating the Index Model
• Using historical data on T-bills (proxy for risk free rate), S&P
500 (proxy for market portfolio) and individual securities
• Regress risk premiums for individual stocks against the risk
premiums for the S&P 500
• Slope is the beta for the individual stock
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Monthly Return Statistics for T-bills,
S&P 500 and General Motors
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Cumulative Returns for
T-bills, S&P 500 and GM Stock
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Characteristic Line for GM
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Security Characteristic
Line for GM: Summary Output
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The CAPM and Reality
• Is the condition of zero alphas for all stocks as
implied by the CAPM met?
– Not perfect but one of the best available
• Is the CAPM testable?
– Proxies must be used for the market portfolio
• CAPM is still considered the best available
description of security pricing and is widely
accepted
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Empirical Tests of the CAPM
• Typical Tests
– Time-series test (Black-Jensen-Scholes)
Rit  R ft   i   i ( RMt  R ft )   it
T est tosee if  i  0.
– Cross-sectional tests (Fama-MacBeth)
~
~
Rit   'i   i RMt  ~it
R i   0   1 ˆi   2CHARi   i , ( i residual)
T est tosee if  0  0,  1  RM  RFR,  2  0.
– Most tests are done in portfolios
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Figure 9.4 Frequency Distribution of Alphas
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Empirical Tests of the CAPM
• High beta portfolios do not necessarily generate
high returns
– Controversial results
• Size and book-to-market value ratio seem to have
explanatory power for returns
– Fama and French
• Momentum in returns
– Relative strength
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Why still the CAPM?
• Is the CAPM wrong?
– Problems with the proxy for market portfolio
– Possible missing risk factors -> Multi-factor
models
– Relaxing assumptions
• Important intuitions from the CAPM
– Diversification
– Only covariance with systematic risks matters
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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
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CAPM & Liquidity
• Liquidity
• Illiquidity Premium
• Research supports a premium for illiquidity.
– Amihud and Mendelson
– Acharya and Pedersen
• CAPM with liquidity


E (ri )  rf   i E (ri )  rf  f (ci )
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Figure 9.5 The Relationship Between Illiquidity and Average Returns
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Other Asset Pricing Models
• CAPM is limited (true market portfolio is unobservable),
nice idea though!
• Other factors also matter, e.g., Fama-French book-to-market
and size factors
• Arbitrage Pricing Theory (APT): no free lunch (for
diversified portfolio)!
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Puzzles (for any pricing model)
•
•
•
•
•
Stocks are excessively volatile
Stocks are too rewarding
Risk free rates are too low
A lot people do not hold any stocks
Even when people buy stocks, they buy too
little
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Summary
• Assumptions for CAPM
• Market portfolio and individual covariance
with the market portfolio
• The Security Market Line
• CAPM and the Market Model
• Multifactors and Arbitrage Pricing
• Next Class: Optimal Portfolio
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