Portfolio Management 3-228-07 Albert Lee Chun

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Portfolio Management3-228-07 Albert Lee Chun
Multifactor Equity Pricing Models

• Lecture 7

6 Nov 2008
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Todays Lecture

• Single Factor Model
• Multifactor Models
• Fama-French
• APT

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Alpha
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Alpha

• Suppose a security with a particular ? is
  offering expected return of 17 , yet according
  to the CAPM, it should be 14.8.
• Its under-priced offering too high of a rate
  of return for its level of risk
• Its alpha is 17-14.8 2.2
• According to CAPM alpha should be equal to 0.

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Frequency Distribution of Alphas
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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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Single Factor Model

• Returns on a security come from two sources
• Common macro-economic factor
• Firm specific events
• Possible common macro-economic factors
• Gross Domestic Product Growth
• Interest Rates

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Single Factor Model

• ßi index of a securitys particular return to
  the factor
• F some macro factor in this case F is
  unanticipated movement F is commonly related to
  security returns

• Assumption a broad market index like
• the SP/TSX Composite is the common factor

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Regression Equation Single Index Model
ai alpha bi(rM-ri) the component of return
due to market movements (systematic risk) ei
the component of return due to unexpected
firm-specific events (non-systematic risk)
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Risk Premium Format
The above equation regression is the single-index
model using excess returns.
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Measuring Components of Risk

• ?i2 total variance
• ?i2 ?m2 systematic variance
• ?2(ei) unsystematic variance

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Index Models and Diversification
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The Variance of a Portfolio
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Security Characteristic Line for X
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Multi Factor Models
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More than 1 factor?

• CAPM is a one factor model The only determinant
  of expected returns is the systematic risk of the
  market. This is the only factor.
• What if there are multiple factors that determine
  returns?
• Multifactor Models Allow for multiple sources of
  risk, that is multiple risk factors.

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Multifactor Models

• Use other factors in addition to market returns
• Examples include industrial production, expected
  inflation etc.
• Estimate a beta or factor loading for each factor
  using multiple regression

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Example Multifactor Model Equation

• Ri E(ri) BetaGDP (GDP) BetaIR (IR) ei
• Ri Return for security i
• BetaGDP Factor sensitivity for GDP
• BetaIR Factor sensitivity for Interest Rate
• ei Firm specific events

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Multifactor SML

• E(r) rf BGDPRPGDP BIRRPIR
• BGDP Factor sensitivity for GDP
• RPGDP Risk premium for GDP
• BIR Factor sensitivity for Interest Rates
• RPIR Risk premium for GDP

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Multifactor Models

• CAPM say that a single factor, Beta, determines
  the relative excess return between a portfolio
  and the market as a whole.
• Suppose however there are other factors that are
  important for determining portfolio returns.
• The inclusion of additional factors would allow
  the model to improve the models fit of the data.
• The best known approach is the three factor
  model developed by Gene Fama and Ken French.

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Fama French 3-Factor Model
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The Fama-French 3 Factor Model

• Fama and French observed that two classes of
  stocks tended to outperform the market as a
  whole
• (i) small caps
• (ii) high book-to-market ratio

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Small Value Stocks Outperform
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(No Transcript)
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Fama-French 3-Factor Model

• They added these two factors to a standard CAPM
• SMB small market capitalization minus big
  
• "Size" This is the return of small stocks minus
  that of large stocks. When small stocks do well
  relative to large stocks this will be positive,
  and when they do worse than large stocks, this
  will be negative.
• HML high book/price minus low
• "Value" This is the return of value stocks minus
  growth stocks, which can likewise be positive or
  negative.

The Fama-French Three Factor model explains over
90 of stock returns.
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Arbitrage Pricing Theory (APT)
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APT

• Ross (1976) intuitive model, only a few
  assumptions, considers many sources of risk
• Assumptions
• There are sufficient number of securities to
  diversify away idiosyncratic risk
• The return on securities is a function of K
  different risk factors.
• No arbitrage opportunities

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APT

• APT does not require the following CAPM
  assumptions
• Investors are mean-variance optimizers in the
  sense of Markowitz.
• Returns are normally distributed.
• The market portfolio contains all the risky
  securities and it is efficient in the
  mean-variance sense.

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APT Well-Diversified Portfolios

• F is some macroeconomic factor
• For a well-diversified portfolio eP approaches
  zero

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Returns as a Function of the Systematic Factor
Well-diversified portfolio
Single Stocks
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Returns as a Function of the Systematic Factor
An Arbitrage Opportunity
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Example An Arbitrage Opportunity
Risk premiums must be proportional to Betas!
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Disequilibrium Example

• Short Portfolio C, with Beta .5
• One can construct a portfolio with equivalent
  risk and higher return Portfolio D
• D .5x A .5 x Risk-Free Asset
• D has Beta .5
• Arbitrage opportunity riskless profit of 1

Risk premiums must be proportional to Betas!
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APT Security Market Line
This is CAPM!
Risk premiums must be proportional to Betas!
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APT and CAPM Compared

• APT applies to well diversified portfolios and
  not necessarily to individual stocks
• With APT it is possible for some individual
  stocks to be mispriced that is 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

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A Multifactor APT

• A factor portfolio is a portfolio constructed so
  that it would have a beta equal to one on a given
  factor and zero on any other factor
• These factor portfolios are the building blocks
  for a multifactor security market line for an
  economy with multiple sources of risk

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Where Should we Look for Factors?

• The multifactor APT gives no guidance on where to
  look for factors
• Chen, Roll and Ross
• Returns a function of several macroeconomic and
  bond market variables instead of market returns
• Fama and French
• Returns a function of size and book-to-market
  value as well as market returns

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Generalized Factor Model

• In theory, the APT supposes a stochastic process
  that generates returns and that may be
  represented by a model of K factors, such that
• where
• Ri One period realized return on security i,
  i 1,2,3,n
• E(Ri) expected return of security i
• Sensitivity of the reutrn of the ith
  stock to the jth risk factor
• j-th risk factor
• captures the unique risk
  associated with security i
• Similar to CAPM, the APT assumes that the
  idiosyncratic effects can be diversified away in
  a large portfolio.

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Multifactor APT
APT Model
The expected return on a secutity depends on
the product of the risk premiums and the factor
betas (or factor loadings) E(Ri) rf is the
risk premium on the ith factor portfolio.
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Sample APT Problem

• Suppose that the equity market in a large economy
  can be described by 3 sources of risk A, B and
  C.
• Factor Risk Premium
• A .06
• B .04
• C .02

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Example APT Problem

• Suppose that the return on Maggies Mushroom
  Factory is given by the following equation, with
  an expected return of 17.
• r(t) .17 1.0 x A .75 x B .05 x C
  error(t)

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Sample APT problem

• The risk free rate is given by 6
• 1. Find the expected rate of return of the
  mushroom factory under the APT model.
• 2. Is the stock-under or over-valued? Why?

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Sample APT Problem

• Factor Risk Premium
• A .06
• B .04
• C .02
• Risk-Free Rate 6
• Return(t) .17 1.0A 0.75B .05C
  e(t)
• The factor loadings are in green.

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Sample APT Problem

• Factor Risk Premium
• A .06
• B .04
• C .02
• Risk-Free Rate 6
• Return .17 1.0A 0.75B .05C e
• So plug in risk-premia into the APT formula
• ERi .06 1.00.060.750.040.50.02
  .16
• 16 lt 17 gt Undervalued!

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Quick Review of Underpricing

• Undervalued Underpriced Return Too High
• Overvalued Overpriced Return Too Low
• P(t) P(t1)/ 1 r
• r P(t1)/P(t) 1
• where r is the return for a risky payoff
  P(t1).
• This is easy to remember if you think about the
  inverse relationship between price (value) today
  and return.

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Examples 9.3 and 9.4
Factor portfolio 1 E(R1) 10 Factor Portfolio
2 E(R2) 12 Rf 4 Portfolio A with B1
.5 and B2 .75 Construct aPortfolio Q using
weights of B1 .5 on factor portfolio 1 and a
weight of B2 .75 on factor portfolio 2 and a
weight of 1- B1 B2 -.25 on the risk free
rate. E(Rq) B1E(R1) B2 E(R2) (1-B1-B2)
Rf rf B1(E(R1) rf ) B2(E(R2) rf) 13
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Example 9.4
Suppose that E(RA) 12 lt 13 Portfolio
Q Ponderation B1 .5 facteur portefeuille
1 Ponderation B2 .75 facteur portefeuille
2 Ponderation 1- B1 B2 -.25 rf E(Rq )
12 1 x E(Rq) - 1x E(RA)1
There is a riskless arbitrage opportunity of 1!
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Next Week

• We will continue our lecture with Chapter 12
• Market Efficiency (Chapter 10 Section 11.1)