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1st case study
Dimensional Fund Advisors, 2002The question set is available onlineThe case is due on Feb 27.
Outline for today
Tracking portfolioBeta
Adjustmentsestimation
CAPM vs. single-factor index modelStructural multifactor model
Tracking Portfolio
A Tracking Portfolio (T) is designed to match the systematic component of a portfolio (P)’s return, and has as little nonsystematic risk as possible.
This procedure is called Beta Capture. Example:
R(P) = 0.04 + 1.4R(S&P500) + e(P) R(T) = 1.4R(S&P500)
A long position in P + a short position in T R(C) = R(P) – R(T) = 0.04 + 1.4R(S&P500) + e(P) -1.4R(S&P500) = 0.04 + e(P) Long-Short Strategy to achieve a Market Neutral position Alpha Transport
Beta
Adjustment Beta coefficients seem to move to 1 over time Example by Merrill Lynch
Adjusted beta=2/3 sample beta + 1/3 (1) Prediction
Simple approach Current beta = a + b (Past beta) Forecast beta= a + b (current beta)
An expanded version Current beta = a + b1 (Past beta) + b2 (Size) + b3 (debt
ratio) Variance of earnings, growth in earnings, dividend yield
etc.
CAPM vs. the Single-Factor Index Model
For the index model
For the CAPM model
What is different?
),cov(),cov( MiMiiMi ReRaRR 2),cov(),cov(),cov( MiMiMMiMi ReRRRR
2
),cov(
M
Mii
RR
2
),cov(
M
Mii
rr
CAPM vs. the Single-Factor Index Model
CAPM Single-Factor Index Model
Returns Expected Realized
Equilibrium Yes No
Residual returns Not uncorrelated
Uncorrelated
Expression fMifi rrErrE )()( iMiii eRaR
The CAPM and Reality
Is the condition of zero alphas for all stocks as implied by the CAPM metNot perfect but one of the best available
The CAPM and Reality
Is the condition of zero alphas for all stocks as implied by the CAPM metNot perfect but one of the best available
Is the CAPM testableProxies must be used for the market
portfolio CAPM is still considered the best available
description of security pricing and is widely accepted
Structural Multifactor Models
)()()( , tetbXtr ikk
kii )(tri
)(, tX ki
Excess return of stock i from time t to t+1
Exposure (factor loading) of stock i to factor k, estimated at time t.
For industry: 0/1
For other factors: standardized with mean=0 and SD=1)(tbk Factor return to factor k from time t to t+1
)(tei Stock i’s idiosyncratic return from time t to t+1
Return structure:
Structural Multifactor Models
jikjkk
k
kkkiji XFXV ,2,2,1
12,11,,
jiV ,
1,kiX
Covariance of stock i with stock j
Exposure (factor loading) of stock i to factor k1, estimated at time t.
2,1 kkF Covariance of factor k1 with factor k2
ji , idiosyncratic covariance of stock i with stock j
Risk structure:
Structural Multifactor Models
The choice of factors A priori factors Three categories
Response to external influences (macro factors) inflation oil price Exchange rates GDP Problems:
Estimation error (error in variable) Nonstationary response coefficients Poor data quality
Structural Multifactor Models
The choice of factors A priori factors Three categories
Response to external influences (macro factors) Cross-sectional comparisons of asset attributes
Compare attributes of the stocks Fundamental
Ratios (dividend yield etc) analysts forecasts
Market Volatility, return, share turnover, etc.
Problem: Error in variable Nonstationary coefficient
Structural Multifactor Models
The choice of factors A priori factors Three categories
Response to external influences (macro factors) Cross-sectional comparisons of asset attributes Internal or statistical factors
Factors produced by statistical methods: Principal component analysis Maximum likelihood analysis Expectations maximization analysis
Problems: Very difficult to interpret Spurious correlations Cannot capture changes over time
Structural Multifactor Models
The choice of factors A priori factors Three categories
Response to external influences (macro factors) Cross-sectional comparisons of asset attributes Internal or statistical factors
Criteria: Incisive: distinguish returns Intuitive: interpretable and recognizable
Recognizable investment themes: industry, size, value, growth etc.
Interesting: help to explain alpha or beta or volatility
Structural Multifactor Models
Typical factors Industry factors : 0/1 factors Risk indices
Volatility Momentum Size Liquidity Growth Value Earnings volatility Financial leverage
Another Example:Fama-French Three-Factor Model
The factors chosen are variables that on past evidence seem to predict average returns well and may capture the risk premiums
Where: SMB = Small Minus Big, i.e., the return of a portfolio of small stocks
in excess of the return on a portfolio of large stocks HML = High Minus Low, i.e., the return of a portfolio of stocks with a
high book to-market ratio in excess of the return on a portfolio of stocks with a low book-to-market ratio
it i iM Mt iSMB t iHML t itr R SMB HML e