Transcript Chapter 9
FIN 2802, Spring 10 - Tang
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
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
FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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Asset Pricing
• Absolute pricing vs relative pricing • Positive view vs normative view FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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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 FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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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 Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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.
Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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E(R p )
Capital Market Line (CML)
R m M R f FIN 2802, Spring 10 - Tang Sharpe Ratio σ m Chapter 9: Asset Pricing Theories Efficient Frontier p
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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” Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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
Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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 w GM COV(r GM , r M ) – Marginally, for each unit of GM stock included, the amount of risk it brings to the market portfolio is COV(r GM , r M ) – The risk premium each unit of GM stock provides is measured by r GM -r.
– The risk-return tradeoff can be measured by » (r GM -r)/ COV(r GM , r M ) » Also termed “market price of risk” » In equilibrium, market prices of risk for all traded asset should be the same!
Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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Risk and Expected Return for A Risky Asset
Expected Return = Rf + a*Risk
E
[
R i
]
Rf
a
COV
(
R i
,
R M
) If
i
happens
E
[
R M
]
Rf
to be the market
a
2
M
a
portfolio
E
[
R M
]
M, then
Rf
/
2
M
Therefore,
E
[
R i
]
Rf
COV
(
R i
2
M
,
R M
)
E
[
R M
]
Rf
Capital Asset Pricing Model (CAPM)
Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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
[
R i
]
RFR
i
E
[
R M
]
RFR
, where
i
and
E
[
R M
] Note that
RF RFR
is the market risk premium 0 , and
M
1 .
COV
(
R i
2
M
,
R M
) – The expected rate of return of a risky asset is determined by the RFR plus a risk premium for the asset determined by the systematic risk exposure of the asset (β) and the prevailing market risk premium (
R M -RFR
)
E
vs
Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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Sample Calculations for SML
If E(r m ) - r f = .08 and r f = .03
x = 1.25 then E(r x ) = .03 + 1.25(.08) = .13 or 13%
y = .6 then E(r y ) = .03 + .6(.08) = .078 or 7.8%
FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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R x =13% R m =11 % R y =7.8
% 3% Graph of Sample Calculations E(r) SML .08
ß .6
ß y 1.0
ß m 1.25
ß x
Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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 ) Rf
i
i
(R Rf)
i
0 on SML.
i M • Any security with an estimated return that plots above the SML is
underpriced
0 • Any security with an estimated return that plots below the SML is
overpriced
i
0 FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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The Security Market Line and Positive Alpha Stock FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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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%?
FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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for Portfolios
• • Expected return of a portfolio
E
(
R p
)
i n
1
w i E
(
R i
)
i n
1
w i
Rf
i
R M
Rf
Rf
i n
1
w i
i
R M
Rf
p
i n
1
w i
i
Beta is additive:
Portfolio beta equals to portfolio weighted betas Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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 FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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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 FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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Empirical Tests of the CAPM
•
Typical Tests
–
Time-series test (Black-Jensen-Scholes)
R it
R ft
i
Test to see if
i
i
(
R Mt
0 .
R ft
)
it
–
Cross-sectional tests (Fama-MacBeth)
~
R it
'
i
i
~
R Mt
~
it
–
R i
0 1
ˆ
i
Test to see if
0 2
CHAR i
0 ,
1
Most tests are done in portfolios
R M
i
, (
i
residual)
RFR
,
2
0 .
Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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 Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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 Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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 FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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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
(
r i
)
r f
i
E
(
r i
)
r f
f
(
c i
) Chapter 9: Asset Pricing Theories FIN 2802, Spring 10 - Tang
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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 FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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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 FIN 2802, Spring 10 - Tang Chapter 9: Asset Pricing Theories
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