Econ 424/Amath 462
Capital Asset Pricing Model

Eric Zivot

August 15, 2013

SI Model and Eﬃcient Portfolios
 assets with returns  ∼ iid ( 2)
⎛⎞

⎛

⎞

1
1
⎜ . ⎟. ⎟R=⎜
⎝ . ⎠ = ⎝ . ⎠
⎛

12.
Σ=⎜
⎝ .

⎞

· · · 1.. ⎟...
⎠2
1 · · · 

Assume risk-free asset with return Compute tangency portfolio allowing short sales
maxtan − tantan = t0

t2
tan= t0Σt

1 = t01
For each asset compute “beta” with respect to tangency portfolio using SImodel =  + tantan + tan = t0Rcov( tan)tan =var(tan)Result: For any asset  =  + tan(tan −  )That is, there is an exact linear relationship between  and tan
Recall, for an eﬃcient portfolio that is a combination of T-Bills and the tangency
portfolio that has the same expected return as asset  =  =  + tan(tan −  )tan +  = 1
Thereforetan = tan
1 − tan = 

Interpretation: The eﬃcient portfolio with• tan invested in tangency portfolio• 1 − tan invested in T-Bills
has the same expected return as an investment in asset , but has lower SD(risk).

Verifying the Proposition with Data

1. Collect data on  assets for sample  = 1     

2. Compute tangency portfolio from  assets assuming value for 

3. Estimate tan for  = 1      assets using linear regression̂ = ̂ + ̂tantan4. Compute average returns for  = 1     
1 X̂ = =1
1 X̂tan =tan =1

5. Plot ̂ vs. ̂tan
intercept = slope = (̂tan −  )

6a. Estimate the linear regression
̂ = 0 + 1̂tan + 
and we should see
̂0 = ̂1 = (̂tan −  )
2 = 16b. Alternatively, estimate the linear regression

and we should see̂ −  = 0 + 1̂tan + 
̂0 = 0̂1 = (̂tan −  )
2 = 1

Capital Asset Pricing Model (CAPM)
Assumptions

1. Many identical investors who are price takers

2. All investors plan to invest over the same time horizon

3. No taxes or transaction costs

4. Can borrow and lend at risk-free rate, 5. Investors only care about portfolio expected return and variance(a) like high  but low 2
6. Market consists of all publicly traded assets(a) market portfolio of assets = value weighted index of all publicly tradedassets = P  
=1  
 = price of asset  = total shares outstandingCAPM Conclusions
1. All investors use the Markowitz algorithm and hold 2 portfolios• T-bills• Tangency portfolio
2. Risk averse investors hold mostly T-Bills (lend at  ), risk tolerant investors hold mostly tangency portfolio (borrow at  to leverage tangency
portfolio).
3. In capital market equilibrium
demand = supply ⇒total borrowing = total lending

4. In equilibrium
demand for risky assets = tangency portfolio
supply of assets = market portfolio
which implies
tangency portfolio = market portfolio⇒ market portfolio is eﬃcient
All eﬃcient portfolios are combinations of T-Bills and the market portfolio

5. In equilibrium[] −  = ( −  )  0

6. Security Market Line (SML) pricing relationship hold for all assets
[] =  + ([] −  )cov( ) =var()or =  + ( −  )Since ( −  )  0high  ⇒ high low  ⇒ low 
7. Alternative representation of SML
 −  = ( −  )

Relationship between SI Model and CAPMSI model =  +  +  = return on market index
Steps to convert SI model to CAPM regression

1. Set  = true market portfolio

2. Subtract  from both sides of SI model −  =  −  +  + 3. Add and subtract  from right-hand-side sides of SI modelwhere −  =  −  +  −  +  + =  −  (1 − ) + ( −  ) + = ∗ + ( −  ) + ∗ =  −  (1 − )

4. CAPM security market line (SML) relationship

implies that[] −  = ([] −  )∗ = 0for every asset Regression Test of the CAPM
Use linear regression to estimate the excess returns SI model −  = ∗ + ( −  ) +  = 1      assets

Test the hypotheses
0 : ∗ = 0 (CAPM holds)
1 : ∗ 6= 0 (CAPM does not hold)for all assets  = 1      assets

Q: What happens if you reject 0 : ∗ = 0?
Suppose ∗  0 (positive “alpha”) Then³´∗ = [] −  − ([] −  )  0

so that expected excess return on asset  is greater than what CAPM predicts.
• Asset is underpriced relative to CAPM (expected return too high ⇒ current
price too low)
• If CAPM is true, then expected return should fall soon which implies thatcurrent price should rise soon.
• ∗  0 ⇒ buy asset today; ∗  0 ⇒ sell asset today

Prediction Test of CAPM
Security Market Line (SML) says
 −  = ( −  ) −   0
Implication:• High  stocks should have high average returns • Low  stocks should have low average returns Simple prediction test
1. Compute estimates ̂ for a bunch of assets using excess returns SI model
2. Compute average excess returns ̂ −  using sample means
3. Plot ̂ −  against ̂
4. Estimate SML using regression

and test hypotheses

̂ −  = 0 + 1̂ + 

0 : 0 = 0 and 1 =  − 
1 : 0 6= 0 or 1 6=  −  or bothPrediction Test IIIdea: A true prediction test would use  0 estimated during one period topredict average returns in another period.
Example: 10 years of monthly data
1. Split sample into 2 non-overlapping 5 year sub-samples
2. Estimate  over first 5 years
3. Estimate  −  over second 5 years
4. Perform prediction test as described above

Survey of Classical Papers that Test CAPM
1. Litner (1965), “Security Prices, Risk and Maximal Gains from Diversification,” Journal of Finance.

• Uses annual data on 631 NYSE stocks for 10 years: 1954 - 1963
• Uses In-Sample Prediction Test (2 step process)— Compute ̂1     ̂631 and ̂1 −       ̂631 −  using full 10 yearsample— Estimate SML̂ −  = 0 + 1̂ + 

— Test0 : 0 = 0 and 1 =  − 
1 : 0 6= 0 or 1 6=  −  or bothResults:̂ −  = 0124 + 0042 · ̂(0006)(0006)̂ −  = 0165
0124
0=0 == 21160006
0042 − 0165
1=0165 == −2050006
Conclusion: Estimated SML is too flat!
Problem: Measurement error in ̂ causes downward bias (toward zero) in ̂1

2. Black, Jensen and Scholes, 1972. “The Capital Asset Pricing Model: Some
Empirical Tests,” in Studies in the Theory of Capital Markets.
3. Fama and MacBeth, 1973“Risk, Return and Equilibrium: Empirical Tests,”
Journal of Political Economy.
Contributions: Developed method to get around measurement error in ̂
Main idea: Estimate  for portfolios• Diversification ⇒ higher % market risk ⇒ more precise ̂• Construct portfolios with broad range of  values (low to high) and estimate SML using portfolios

Three step technique: perform Prediction Test II using 3 non-overlapping subsamples:
1. 1st sample - estimate  for individual assets. Sort assets into 10 portfolios
based on 
2. 2nd sample - estimate  for 10 portfolios
3. 3rd sample - estimate  −  for 10 portfolios, estimate SML
Result: Measurement error in  for 10 portfolios is small• — measurement errors for portfolios are independent from measurement
error for individual assets

Black et. al. results

̂ −  = 00036 + 00108 · ̂(00006)(000052)

̂ −  = 00142
00036
0=0 =
=600006
00108 − 00142
1=00142 == 654
0006
Conclusion: Estimated SML is still too flat, but results look better.

Problems with Prediction Tests• SML is a relationship between expected returns, [] and true  Bothvalues cannot be observed without error— Test uses estimates ̂ and ̂ :̂ =  + 
̂ =  + 
[] = 0since ̂ and ̂ are unbiased estimates•  is not estimated very precisely for individual assets— measurement error in ̂ is large• Measurement error in ̂ creates bias in estimates of 0 and 1•  and  may change over timeMeasurement Error in Regression
Simple regression
 =  + cov( ) = 0Thencov( )cov( +  )=var()var()cov( ) + cov( )=var()() + 0==var()Least square estimation: as  gets large

so thatd  )cov(cov( )
̂ =→=
dvar()var()̂ is consistent for 

Measurement Error in Suppose  is measured with error∗ =  + = mis-measured  = measurement error
cov( ) = 0
Regression model with mis-measured ∗∗ −  =  + Add  to both sides
∗ =  + ( + )=  + Result: As long as cov( ) = 0 measurement error in ∗ does not cause
any bias in least squares estimate of  :cov( ) = cov(  + )= cov( ) + cov( )=0Measurement Error in Suppose  is measured with error∗ =  + = mis-measured  = measurement error
cov(∗  ) = 0Regression model with mis-measured ∗ = (∗ − ) + = ∗  +  − = ∗  + ∗∗ =  − Note:cov(∗  ∗ ) = cov( +   − )= cov(− )= −()Consequently
cov( ∗ )cov(∗ + ∗  ∗ )=∗var( )var(∗ )cov(∗  ∗ ) + cov(∗  ∗ )=var(∗ )(∗ ) −()=+∗var( )var(∗ )var()=−·6= ∗var( )Result: If  is measured with error then as  gets large

so thatd  )cov(var()̂ =→−·6= ∗dvar()var( )̂ is not consistent for Note thatÃvar()var()−·= 1−
var(∗ )var(∗ )var()1∗var( )so that ̂ is downward biased (toward zero)!