Capital Asset Pricing Model and Arbitrage Theory Capital by rjt20314

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									Capital Asset Pricing Model
  and Arbitrage Theory
       Riccardo Colacito
  Capital Asset Pricing Model (CAPM)

• Equilibrium model that underlies all
  modern financial theory
• Derived using principles of diversification
  with simplified assumptions
• Markowitz, Sharpe, Lintner and Mossin
  are researchers credited with its
  development


Foundations of Financial Markets                2
                    Link to Factor Models


• What risk should be priced?
        – Idiosyncratic risk: no
        – Aggregate risk: yes


• Only aggregate/macro risk commands a
  premium


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                                   Why?

Because:

1. idiosyncratic risk can be diversified away
2. Macro risk affects all assets and cannot
   be diversified

Remember our example using factor models?

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                      Example: two assets

      Two assets:
      R1  1  1 Rm  e1 and R2   2   2 Rm  e2
      Assume:
      1   2  1
     Var(e1 )  Var(e2 )  1
      cov(e1 , e2 )  0
      Invest equal portfolio shares in the two assets
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     What is the portfolio variance?




             Var ( w1 R1  w2 R2 )  Var ( Rm )  1 / 2
                                      Systematic   Idiosyncratic
                                         Risk          Risk




Foundations of Financial Markets                                   6
                  Example: three assets

    Two assets :
     Ri   i   i Rm  ei , i  1,2,3
    Assume :
    1   2   3  1
    Var(e1 )  Var(e2 )  Var(e3 )  1
    cov(e1 , e2 )  cov(e1 , e3 )  cov(e2 , e3 )  0
    Invest equal portfolio shares in the three assets
Foundations of Financial Markets                        7
     What is the portfolio variance?


  Var ( w1 R1  w2 R2  w3 R3 )  Var ( Rm )  1 / 3
                                      Systematic   Idiosyncratic
                                         Risk          Risk

    •Systematic risk: unchanged
    •Idiosyncratic risk: decreased
    •Can you guess what would happen if we had an infinite
    number of assets?


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                                   Infinite assets

• For a well diversified portfolio
                                              2
                                           
                 Var   wi Ri      i wi  Var Rm ,
                      i         i          
                 or  R p    p m

• That is: we got rid of any idiosyncratic
  shock and we are left only with systematic
  risk
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         What lesson did we learn?

• The only source of risk that we are entitled
  to ask a compensation for is aggregate
  risk.

• Idiosyncratic risk does not entitle to any
  compensation because it can be
  diversified away.


Foundations of Financial Markets               10
                         Risk compensation

                                         on
        Any asset entitles to a compensati proportional
        to its contribution to aggregate risk :
         E ( R1  rf )                 E ( R2  rf )             E ( Rm  rf )
                                                       ... 
               1 m                       2 m                     m
        or, equivalently
        E ( R1 )  rf  1 E ( Rm  rf )
        E ( R2 )  rf   2 E ( Rm  rf )
        ...
Foundations of Financial Markets                                                 11
   The fundamental equation of the
  Capital Asset Pricing Model (CAPM)

  Any asset entitles to a risk premium that is
  proportional to the risk premium of the market
  portfolio. The  of the asset is the coefficient of
  proportionality.


     E ( Ri )  rf   i E ( Rm  rf ),   i  1,2,...


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                                   Questions

• What assumptions need to be satisfied for
  the CAPM to hold?
• Why the market portfolio? What is it
  anyway?
• Why b? What is b anyway?
• What are the empirical predictions of the
  CAPM?
• Can we measure it from the data?
• If so, is it empirically accepted?
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                                   Assumptions

• Individual investors are price takers
• Single-period investment horizon
• Investments are limited to traded financial assets
• No taxes, and transaction costs
• Lending and borrowing can be done at same
  rate
• Information is costless and available to all
  investors
• Investors are rational mean-variance optimizers
• Homogeneous expectations

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    Assumptions… in other words!

• Everybody knows how to solve the
  problem that we discussed during the last
  three classes
• Everybody is forecasting returns,
  variances and correlations in the same
  way
• … if this is the case we are all going to
  end up with same optimal risky portfolio!

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        The Efficient Frontier and the
            Capital Market Line




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                               Some re-labeling

• We call the optimal CAL, the Capital
  Market Line

• We call the optimal risky portfolio, the
  Market Portfolio

• Remember the separation property?

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                        Separation property

• Aka mutual fund theorem

• All investors desire the same portfolio of
  risky assets: the optimal risky portfolio or
  market portfolio.




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      This answers: Why the market
               portfolio?
• If everybody is holding this portfolio, then it
  is an excellent candidate to proxy for
  aggregate market risk.

• We are now left with the question of why 
  is the coefficient of proportionality!



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      First things first: what is ?

• Remember factor models

                          Ri  r f   i   i Rm  r f   ei

• Remember how we computed 

                                     covRm  rf , Ri  rf 
                              i 
                                         VarRm  rf 

Foundations of Financial Markets                                  20
                                    in words

• The more correlated the asset is with
  market the larger  is.

• What does it mean in terms of the CAPM?

                            E ( Ri )  rf   i E ( Rm  rf )


Foundations of Financial Markets                                21
                  and the risk premium

• It means that the higher the correlation
  with the market, the higher the risk
  premium that an asset commands.

• Why?




Foundations of Financial Markets             22
  and the risk premium: intuition

Which asset is more appealing:

        1. An asset whose return goes up when the
           market goes up
        2. An asset whose return goes down when the
           market goes up




Foundations of Financial Markets                      23
                                   Answer

• Everything else equal you prefer an asset that
  co-varies negatively with the market, because it
  gives you an insurance against bad states of the
  world.
• If an asset is perceived as a good asset because
  it provides an hedge against bad states, its
  demand will go up, increasing the price and
  decreasing the expected return.
• How about an example?

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                                   Example

• Two assets, the market portfolio, a risk free rate.
  Two states of the world.

                                    R1   R2   Rm    rf
  Boom                              -2    4    4   .5
  Recession                          4   -2   -2   .5


 • Let’s compute ’s and required risk premia!

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                                       Results

• Some computations:

                            Var(r1 )  Var(r2 )  Var(rm )  9
                            cov(r1 , rm )  9, cov(r2 , rm )  9
                             1  1,  2  1, E (rm )  1

• Hence
                       E r1   rf  1 E rm   rf   .5  1 .5  0
                       E r2   rf   2 E rm   rf   .5  1 .5  1

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                          What did we learn?

• Asset 2 doesn’t do too much to protect us
  against market fluctuations: therefore it
  must `promise’ a higher return to convince
  us to buy it
• Asset 1 can protect us against market
  fluctuations and therefore we are willing to
  buy it even if its return is low in
  expectation.

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 What happens to the price of the
         first security?
• The CAPM says that the expected return of asset 1
  should be 0
• However at the current prices, asset 1 has an expected
  return of 1
• Looks like a great deal!
• Investors will want to buy more of asset 1
• But if its current price goes up, its return will go down in
  expectation
• When does the price increase stop?
        – When the expected return at the current prices equals the CAPM
          expected return
• What happens to the current price of security 2?
        – Nothing! This security is price correctly!
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    Empirical predictions of the CAPM

• Remember the one factor model:

                Ri   i   i Rm  ei
                            E ( Ri )   i E ( Rm )



• The intercept should be equal to zero!


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        The Security Market Line and
            Positive Alpha Stock




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                                   Alpha 

• The abnormal rate of return on a security
  in excess of what would be predicted by
  an equilibrium model such as the CAPM

• Can we test this prediction?




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       Estimating the Index Model

• Using historical data on T-bills, S&P 500
  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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         Characteristic Line for GM




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                          Statistical analysis




    •In this example we cannot reject that 
    is equal to zero
    •However in many cases a is significantly
    different from zero: what does it mean?

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                     Rejecting the CAPM?

• Not necessarily!

• We may have chosen an imprecise proxy
  for the market risk: after all the Market
  Portfolio is not directly observable.

• We may be omitting some risk factors:
  research shows that this is possible.
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                Fama-French Research

• Returns are related to factors other than
  market returns
• Size
• Book value relative to market value
• Three factor model better describes
  returns



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  Why are these factors supposed
             to help?
• Firms with high ratios of book to market
  value are more likely to be in financial
  distress
• Small firms are more sensitive to changes
  in business conditions




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                     Fama-French applied

• A Fama-French three factors regression for GM
    rGM  rf   GM   M rm  rf    HMLrHML   SMB rSMB  eGM

• where

   rHML  differencein returns between small and large firms
   rSMB  differencein returns between firms with a high vs low B/M




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 Regression Statistics for the Single-
  index and FF Three-factor Model




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Resulting Equilibrium Conditions

• 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 an individual security is a
  function of its covariance with the market

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     Arbitrage Pricing Theory (APT)

Arbitrage - arises if an investor can construct
  a zero beta investment portfolio with a
  return greater than the risk-free rate
• If two portfolios are mispriced, the investor
  could buy the low-priced portfolio and sell
  the high-priced portfolio
• In efficient markets, profitable arbitrage
  opportunities will quickly disappear

Foundations of Financial Markets              41
                   APT and well diversified
                         portfolios
• A well diversified portfolio has no exposure to
  idiosyncratic risk

                      RP  rf   P   P Rm  rf 

• Claim: P must equal zero. Why?




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                      P=0 to avoid arbitrage
                          opportunities
• Take two well-diversified portfolios

            Rv  rf   v   v Rm  rf , Ru  rf   u   u Rm  rf 
• Pick the following shares of investment

                                            u            v
                                   wv            , wu 
                                          v  u        v  u
• The resulting portfolio is not subject to any risk and provides a non-
  zero return
                                 u          v
              wv Rv  wu Ru          v           u  0
                              v  u       v  u
• Hence ’s must be equal to zero

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                                   Conclusion

• The equilibrium condition is the same as the
  CAPM

                   E ( R p  rf )   p   p E ( Rm )  r f 
• Only assumption needed is the absence of
  arbitrage opportunities
• Can be extended beyond well diversified
  portfolios


Foundations of Financial Markets                                  44

								
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