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Part II. Portfolio Theory and Asset Pricing Models

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					Part II. Portfolio Theory and
   Asset Pricing Models




                                1
    1. Risk and Risk Premiums
Rates of Return: Single Period

    HPR  P1  P0  D1
               P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one

                                  2
Expected Return and Standard Deviation

     Expected returns


     E (r )   p( s)r ( s)
                 s
 p(s) = probability of a state
 r(s) = return if a state occurs
 s = state



                                     3
        Scenario Returns: Example

State        Prob. of State        r in State
  1                .1                 -.05
  2                .2                  .05
  3                .4                  .15
  4                .2                  .25
  5                .1                  .35

E(r) = (.1)(-.05) + (.2)(.05)… + (.1)(.35)
E(r) = .15

                                                4
    Variance or Dispersion of Returns

Variance:

               p( s )  r ( s)  E (r ) 
              2                            2

                    s
    Standard deviation = [variance]1/2
     Using Our Example:
 Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2…+ .1(.35-.15)2]
 Var= .01199
 S.D.= [ .01199] 1/2 = .1095

                                                   5
EXHIBIT 3.14 Geometric Mean rates of Return and
Standard Deviation for Sotheby's Indexes, S&P 500,
Bond Market Series, One-Year Bonds, and Inflation

                                               Chinese Mod Paint
                                  Amer Paint                Cont Art
                                               Ceramic
                                                         Imp Paint
                       Eng Furn     FW Index      S&P500
                     UW Index      VW Index         Old Master
                 Fr+Cont Furn         Cont     19C Euro
               Amer Fum               Ceramic
                              LBGC      Eng Silver
                                   Cont Silver
     1-Year Bond

             CPI

                                                      Standard
                                                      Deviation
EXHIBIT 3.17 Alternative Investment Risk
      and Return Characteristics


                                        Futures
                                                  Art and Antiques
                     Coins and Stamps         Warrants and Options
             US Common Stocks          Commercial Real Estate
                                  Foreign Common Stock
   Foreign Corporate Bonds     Real Estate (Personal Home)
                         US Corporate Bonds
                    Foreign Government Bonds
               US Government Bonds
            T-Bills
      Covariance of two random
              variables
• Covariance is defined as:

                         n
      Cov(r1 , r2 )   p( s )[ r1 ( s )  E (r1 )][ r2 ( s )  E (r2 )]
                        i 1




                                                                           8
Figure 5.4 The Normal Distribution




                                     9
Figure 5.5A Normal and Skewed Distributions
           (mean = 6% SD = 17%)




                                          10
Figure 5.5B Normal and Fat-Tailed Distributions
              (mean = .1, SD =.2)




                                             11
         Other measures of risks
• Value at Risk(Var) is another name for the quantile of
  a distribution. The quantile (q) of a distribution is the
  value below which lie q% of the value
• Conditional Tail Expectation (CTE) provides the
  answer to the question, “Assuming the terminal
  value of the portfolio falls in the bottom 5% of
  possible outcomes, what is the expected value?”
• Lower Partial Standard Deviation (LPSD) is the
  standard deviation computed solely from values
  below the expected return

                                                          12
Table 5.5 Risk Measures for Non-Normal
              Distributions




                                         13
Figure 5.6 Frequency Distributions of Rates of
             Return for 1926-2005




                                             14
Table 5.3 History of Rates of Returns of Asset
    Classes for Generations, 1926- 2005




                                             15
Arithmetic and Geometric means

• Arithmetic mean
                 _            n
                   1
                r   rt
                   n i 1

• Geometric mean
                     _
                     r   n   Rn


 Rn  (1  r1 )(1  r2 )...( 1  rn )
                                        16
Figure 5.7 Nominal and Real Equity Returns
       Around the World, 1900-2000




                                             17
Figure 5.8 Standard Deviations of Real Equity and
   Bond Returns Around the World, 1900-2000




                                               18
Risk Aversion measured with Utility Function
                           1
               U  E (r )  A 2

                           2
      Where
      U = utility
      E ( r ) = expected return on the asset or
         portfolio
      A = coefficient of risk aversion
      2 = variance of returns


                                                  19
Figure 6.2 The Indifference Curve: a curve
     indicating the same utility level




                                        20
Table 6.3 Utility Values of Possible Portfolios
  for an Investor with Risk Aversion, A = 4




                                            21
           Indifference curve
• The curvature of indifference curve indicates
  the riskiness of the investor; more steeper
  means more risk averse
• When indifference curve moves to the north-
  west, it implies higher utility for most
  investors



                                                  22
              The Risk-Free Asset

• Only the government can issue default-free
  bonds
   – Guaranteed real rate only if the duration of the
     bond is identical to the investor’s desire
     holding period
• T-bills viewed as the risk-free asset
   – Less sensitive to interest rate fluctuations



                                                    23
Figure 6.3 Spread Between 3-Month
         CD and T-bill Rates




                                    24
2. Portfolio Theory: Portfolios of One Risky Asset
               and a Risk-Free Asset

 • It’s possible to split investment funds between
   safe and risky assets.
 • Risk free asset: proxy; T-bills
 • Risky asset: stock (or a portfolio)




                                                     25
Example Using Chapter 6.4 Numbers


 rf = 7%         rf = 0%

 E(rp) = 15%     p = 22%

 y = % in p      (1-y) = % in rf


                                    26
Expected Returns for Combinations

   E (rc )  yE (rp )  (1  y)rf

rc = complete or combined portfolio

     For example, y = .75
     E(rc) = .75(.15) + .25(.07)
           = .13 or 13%

                                      27
Combinations Without Leverage

If y = .75, then
c      = .75(.22) = .165 or 16.5%
If y = 1
   c   = 1(.22) = .22 or 22%
If y = 0
c      = (.22) = .00 or 0%

                                     28
 Capital Allocation Line with Leverage

Borrow at the Risk-Free Rate and invest in stock.
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = .19

c = (1.5) (.22) = .33




                                                    29
Figure 6.4 The Investment Opportunity Set with a
Risky Asset and a Risk-free Asset in the Expected
        Return-Standard Deviation Plane




                                               30
   Risk Tolerance and Asset Allocation
• The investor must choose one optimal portfolio,
  C, from the set of feasible choices
   – Trade-off between risk and return
   – Expected return of the complete portfolio is
     given by:
         E (rc )  rf  y  E (rP )  rf 
                                        
  – Variance is:
                 y
                   2
                   C
                           2   2
                               P


                                                    31
Table 6.5 Utility Levels for Various Positions in Risky
 Assets (y) for an Investor with Risk Aversion A = 4




                                                   32
Figure 6.6 Utility as a Function of Allocation to the
                    Risky Asset, y




                                                 33
Figure 6.8 Finding the Optimal Complete Portfolio
             Using Indifference Curves




                                              34
Mathematically, maximize utility function of
 investing to get optimum portfolio weights:
                          1
    Maximize U  E (rc )  A c2
                               2


     Subject to:
       E ( rc )  r f  y ( E ( rp )  r f )
       and
       c  y     p




                                               35
The solution is:        E (rp )  rf
                   y
                           A p
                              2




  -if risk premium of investing in portfolio p is
  higher, y is higher
  -if the investor is more risk averse, less y
  -if the stock portfolio p is more risky, less y

                                                    36
Figure 6.5 The Opportunity Set with Differential
         Borrowing and Lending Rates




                                              37
3. Portfolio Theory: Portfolio with Two Risky
                  Securities
    rp      wr
              D   D
                       wEr E
    rP    Portfolio Return
    wD  Bond Weight
    rD    Bond Return
    wE  Equity Weight
    rE    Equity Return


         E (rp )  wD E (rD )  wE E (rE )
                                                38
              Two-Security Portfolio: Risk

       w   w   2wD wE Cov(rD , rE )
         2
         P
                2
                D
                    2
                    D
                        2
                        E
                            2
                            E



   D = Variance of Security D
    2



    2
     E       = Variance of Security E

Cov(rD , rE ) = Covariance of returns for
                 Security D and Security E
                                             39
              Covariance

Cov(rD,rE) = DEDE

D,E = Correlation coefficient of
       returns
 D = Standard deviation of
      returns for Security D
 E = Standard deviation of
      returns for Security E
Correlation Coefficients: Possible Values


Range of values for 1,2
  + 1.0 >     > -1.0
If  = 1.0, the securities would be perfectly
positively correlated
If  = - 1.0, the securities would be
perfectly negatively correlated
 p  wE E  wD D  2 wE wD  DE E D
  2    2  2    2  2




if DE  1, then
 P  ( wE E  wD D ) 2
  2


and
   ( wE E  wD D )




                                           42
Table 7.1 Descriptive Statistics for Two
            Mutual Funds
       Three-Security Portfolio

   E (rp )  w1E (r1 )  w2 E (r2 )  w3 E (r3 )

2p = w1212 + w2212 + w3232
                       + 2w1w2       Cov(r1,r2)
                       + 2w1w3 Cov(r1,r3)
                        + 2w2w3 Cov(r2,r3)
Table 7.2 Computation of Portfolio Variance
        From the Covariance Matrix
Table 7.3 Expected Return and Standard
  Deviation with Various Correlation
              Coefficients
Figure 7.3 Portfolio Expected Return as a
   Function of Investment Proportions
Figure 7.4 Portfolio Standard Deviation as a
    Function of Investment Proportions
Minimum Variance Portfolio as Depicted
            in Figure 7.4
• Standard deviation is smaller than that of either
  of the individual component assets
• Figure 7.3 and 7.4 combined demonstrate the
  relationship between portfolio risk
Figure 7.5 Portfolio Expected Return as a
     Function of Standard Deviation
              Correlation Effects

• The relationship depends on the correlation
  coefficient
• -1.0 <  < +1.0
• The smaller the correlation, the greater the risk
  reduction potential
• If  = +1.0, no risk reduction is possible
Figure 7.10 The Minimum-Variance Frontier of
                 Risky Assets




                                           52
Figure 7.6 The Opportunity Set of the Debt
 and Equity Funds and Two Feasible CALs
               The Sharpe Ratio

• Maximize the slope of the CAL for any possible
  portfolio, p
• The objective function is the slope:

                    E (rP )  rf
             SP 
                        P


                                                   54
Mathematically, we solve for the tangent
 portfolio weights from maximization the
 sharpe ratio:
           E ( rp )  r f
    Sp 
               p
    subjectto :
    E ( rp )  w1 E ( r )  w2 E ( r2 )
                       1

    p      w1  1  w2  2  2 12 w1 w2 1 2
              2   2    2   2




The solutions are:
                                    E ( RD ) E  E ( RE )Cov( RD , RE )
                                              2
                wD 
                       E ( RD ) E  E ( RE ) D  [ E ( RD )  E ( RE )]Cov( RD , RE )
                                 2             2




                       wE  1  wD
                                                                                          55
Figure 7.7 The Opportunity Set of the Debt
and Equity Funds with the Optimal CAL and
        the Optimal Risky Portfolio
Figure 7.8 Determination of the Optimal
            Overall Portfolio
                Efficient Frontier with
                Lending & Borrowing
                                      CAL
     E(r)
                                  B
                              Q
                        M


                  A

rf          F



                                            58
  4. Asset Pricing Models: Capital Asset Pricing Model
       (CAPM) and Arbitrage Pricing Theory (APT)


• It is the equilibrium model that underlies all
  modern financial theory.
• Derived using principles of diversification with
  simplified assumptions.




                                                         59
               Assumptions
• Individual investors are price takers.
• Single-period investment horizon.
• Investments are limited to traded financial
  assets.
• No taxes and transaction costs.




                                                60
          Assumptions (cont’d)
• Information is costless and available to all
  investors.
• Investors are rational mean-variance
  optimizers.
• There are homogeneous expectations.




                                                 61
  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.




                                                 62
    Resulting Equilibrium Conditions
                (cont’d)
• Risk premium on 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.

                           
         E (ri )  rf  i E (ri )  rf   

                                                      63
   Using GE Text Example Continued
• Reward-to-risk ratio for investment in market
  portfolio:
         Market risk premium E (rM )  rf
                            
          Market variance        M2




• In equilibrium, reward-to-risk ratios of GE and
  the market portfolio should be equal to each
  other:          E (r )  r
                        GE
                             
                               E (r (r )
                                 f           M       f

                   Cov ( rGE , rM )             2
                                                 M




• And the risk premium for GE:
                             Cov(rGE , rM )
           E (rGE )  rf                      E (rM )  rf 
                                                            
                                        2
                                         M
    Expected Return-Beta Relationship

• CAPM holds for the overall portfolio because:
             E (rP )   wk E (rk ) and
                       k

              P   wk  k
                   k
• This also holds for the market portfolio:

           E (rM )  rf   M  E (rM )  rf 
                                            
Figure 9.2 The Security Market Line
     Security Market Line (SML)
          i = [COV(ri,rm)] / m2
Slope SML =      E(rm) - rf
            = market risk premium
      SML = rf + i [E(rm) - rf]
      Betam = [Cov (ri,rm)] / m2
            = m2 / m2 = 1



                                    67
     Sample Calculations for SML
E(rm) - rf = .08   rf = .03

x = 1.25
  E(rx) = .03 + 1.25(.08) = .13 or 13%

y = .6
  E(ry) = .03 + .6(.08) = .078 or 7.8%


                                         68
Graph of Sample Calculations
         E(r)
                                       SML

Rx=13%                                   .08
Rm=11%
Ry=7.8%

    3%
                                             
                     .6   1.0   1.25
                     y         x
                By
                                                 69
     Disequilibrium Example
          E(r)

                              SML
   15%

Rm=11%


  rf=3%

                                    
                 1.0   1.25

                                        70
        Disequilibrium Example
• Suppose a security with a  of 1.25 is offering
  expected return of 15%.
• According to SML, it should be 13%.
• Under-priced: offering higher rate of return
  for its level of risk.




                                                    71
     Active investment with beta
• Those who think they are able to time the
  market can do the followings.
  – Buy high-beta stocks when they think the market
    is up
  – Switch to low-beta stock when they fear the
    market is down




                                                      72
          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
      Black’s Zero Beta Model

• Absence of a risk-free asset
• Combinations of portfolios on the efficient
  frontier are efficient.
• All frontier portfolios have companion
  portfolios that are uncorrelated.
• Returns on individual assets can be
  expressed as linear combinations of
  efficient portfolios.
          Black’s Zero Beta Model
                Formulation


                                            Cov(ri , rP )  Cov(rP , rQ )
                   
E (ri )  E (rQ )  E (rP )  E (rQ )          P  Cov(rP , rQ )
                                                 2
      Efficient Portfolios and Zero
              Companions
             E(r)




                           Q
                    P
E[rz (Q)]           Z(Q)
 E[rz (P)]                 Z(P)



                                  
   Zero Beta Market Model



                         
E (ri )  E (rZ ( M ) )  E (rM )  E (rZ ( M ) )      Cov(ri , rM )
                                                               2
                                                                M



  CAPM with E(rz (m)) replacing rf
     CAPM with Nontraded Assets
-To the extent that risk characteristics of private enterprises
   differ from those of traded assets, a portfolio of traded assets
   that best hedges the risk of typical private business would
   enjoy excess demand from the population of private business
   owners. The price of assets in this portfolio will bid up relative
   to the CAPM considerations, and the expected returns on
   these securities will be lower in relation to their systematic
   risk. Conversely, securities highly correlated with such risk will
   have high equilibrium risk premiums and may appear exhibit
   positive alphas relative to the conventional SML
      CAPM with Labor Income
• An individual seeking diversification should
  avoid investing in his employer’s stock and
  limit investments in the same industry. Thus,
  the demand for stocks of labor-intensive firms
  may be reduced, and these stocks may require
  a higher expected return than predicated by
  the CAPM
• Mayer’s model (9.13)
Mayers derives the equilibrium expected return-beta
 for an economy in which individuals are endowed
 with labor income of varying size relative to their
 nonlabor capital.
                                    Cov( Ri , RM )  PM Cov( Ri , RH )
                                                     PH

              E ( Ri )  E ( RM )
                                         M  P Cov( RM , RH )
                                          2   P H
                                                M


Where PH =value of aggregate human capital
        PM =market value of traded assets (market
 portolio)
       RH =excess return on aggregate human capital


                                                                         80
      A Multiperiod Model and Hedge
                 Portfolios
• Merton relaxes the “single-period” myopic assumptions about investors.
  He envisions individuals who optimize a lifetime consumption/investment
  plan, and who continually adapt consumption/investment decisions to
  current wealth and planned retirement age
• One key parameter is the future risk-free rate. If it falls in some future
  period, one’s level of wealth will now support a lower stream of real
  consumption. To the extent that returns on some securities are correlated
  with changes in the risk-free rate, a portfolio can be formed to hedge such
  risk, and investors will bid up the prices (and bid down the expected
  return) of those hedge assets.
• Another key parameter is inflation risk. For example, investors may bid up
  share prices of energy companies that will hedge energy price uncertainty
                                             K
                  E ( Ri )   iM E ( RM )    ik E ( Rk )
                                            k 1
      Extensions of the CAPM Continued

• A consumption-based CAPM
   – Models by Rubinstein, Lucas, and Breeden
       • Investor must allocate current wealth between today’s
         consumption and investment for the future
       • As a general rule, investors will value additional income more
         highly during difficult times than in affluent times. An asset will
         therefore be viewed as riskier in terms of consumption if it has
         positive covariance with consumption path (9.15)
                  E ( Ri )   iC RPC

                  where
                  RP  E ( RC )  E (rC )  rf
                    C
           Liquidity and the CAPM

• Liquidity is prefered by investors
• Illiquidity Premium
• Research supports a premium for illiquidity.
   – Amihud and Mendelson
   – Acharya and Pedersen
Figure 9.5 The Relationship Between
   Illiquidity and Average Returns
Acharya and Pedersen proposed the following model:
      E ( Ri )  kE(Ci )   (   L1   L 2   L3 )
Where    E (Ci )        =expected cost of illiquidity
                   k =adjustment for average holding period

  over all securities
                    =market risk premium net of average
  market illiquidity cost,
                   =measure of systematic market risk
    L1 ,  L 2 ,  L 3 =liquidity betas

                                                              85
         Three Elements of Liquidity

• Sensitivity of security’s illiquidity to market
  illiquidity:   Cov(Ci , CM )
                        Var ( RM  CM )
                 L1



• Sensitivity of stock’s return to market illiquidity:
                        Cov( Ri , CM )
               L2 
                       Var ( RM  CM )
• Sensitivity of the security illiquidity to the market
  rate of return:
                         Cov(Ci , RM )
                L3 
                        Var ( RM  CM )
How to estimate beta: Single Index Model

                 rit   i  i rmt  eit
 ßi = index of a securities’ particular return to the
 factor
 m = a broad market index like the S&P 500 is the
 common factor
 ei = uncertainty about the firm
Where Cov(ei , rm )  0
         Cov(ei , e j )  0


                                                        87
                    Single-Index Model

• Regression Equation:

                  ri  rf   i   i (rm  rf )  ei
• Portfolio
              n                n
    R p   wi Ri   wi [ i   i Rm ]
             i 1             i 1
       n                n
      wi i  ( wi  i ) Rm   p   p Rm
      i 1             i 1



                                                        88
      Single-Index Model Continued

• Systematic and unsystematic risk
  – Total risk = Systematic risk + Firm-specific risk (or
    unsystematic risk):
             i2  i2 M   2 (ei )
                        2




                                                            89
     Index Model and Diversification
• Portfolio’s variance:

                   (eP )
              2
              P
                      2
                      P
                          2
                          M
                                   2


• Variance of the equally weighted portfolio of
  firm-specific components:
                      2
                  n
                      1 2     1 2
        (eP )      (ei )   (e)
         2

                 i 1  n      n

• When n gets large,  2 (eP ) becomes negligible

                                                    90
   Figure 8.1 The Variance of an Equally
Weighted Portfolio with Risk Coefficient βp in
        the Single-Factor Economy




                                           91
Figure 8.2 Excess Returns on HP and S&P 500
           April 2001 – March 2006




                                        92
Figure 8.3 Scatter Diagram of HP, the S&P
500, and the Security Characteristic Line
               (SCL) for HP




                                        93
Table 8.1 Excel Output: Regression Statistics
       for the SCL of Hewlett-Packard




                                          94
        Alpha and Security Analysis

• Macroeconomic analysis is used to estimate the
  risk premium and risk of the market index
• Statistical analysis is used to estimate the beta
  coefficients of all securities and their residual
  variances




                                                      95
Optimal Risky Portfolio of the Single-Index
                 Model
• Maximize the Sharpe ratio by choosing weights
  – Expected return, SD, and Sharpe ratio:
                                      n 1              n 1
       E ( RP )   P  E ( RM )  P   wi i  E ( RM ) wi i
                                      i 1              i 1
                                                                         1
                                  1  2  n 1       2
                                                     n 1 2 2          2
        P    P M   (eP )    M   wi  i    wi  (ei ) 
             
                 2 2     2
                                
                                  2

                                      i 1
                                                    i 1           
                                                                     
             E ( RP )
       SP 
              P


                                                                     96
              The Information Ratio

• The Sharpe ratio of an optimally constructed risky
  portfolio will exceed that of the index portfolio
  (the passive strategy):
                                        2
                  A 
           sM  
          2        2
       sP                   
                   (e A ) 

                                                   97
     Treynor-Black Allocation
                       CAL
 E(r)                  CML

             P
                  A

             M

Rf

                          
Arbitrage Pricing Theory and Multifactor
       Models of Risk and Return
         Single Factor Model Equation
    ri  E (ri )  i F  ei

 Or alternatively, ri   i   i F1  ei
 ri = Return for security i
 i = Factor sensitivity or factor loading or factor beta
 F = Surprise in macro-economic factor
     (F could be positive, negative or zero)
 ei = Firm specific events
           Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero
  investment portfolio with a sure profit
• Since no investment is required, an investor can
  create large positions to secure large levels of
  profit
• In efficient markets, profitable arbitrage
  opportunities will quickly disappear
   Arbitrage Pricing Theory (APT)
• Assume factor model such as
                  ri   i   i F1  ei

• And no arbitrage opportunities exist in
  equilibrium
• Then, we have
         E (ri )  0  1 i



                                            102
         APT and CAPM Compared

• APT applies to almost all individual securities
• With APT it is possible for some individual stocks
  to be mispriced - 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
               Multifactor APT

• Use of more than a single factor
• Requires formation of factor portfolios
• What factors?
   – Factors that are important to performance of
     the general economy
   – Fama-French Three Factor Model
              Two-Factor Model

      ri  E (ri )  i1F1  i 2 F2  ei
• The multifactor APR is similar to the one-
  factor case
  – But need to think in terms of a factor portfolio
  Example of the Multifactor Approach

• Work of Chen, Roll, and Ross
  – Chose a set of factors based on the ability of
    the factors to paint a broad picture of the
    macro-economy
  – GDP factor, inflation factor, and interest rate
    factor
               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
     rit  i  iM RMt  iSMB SMBt  iHML HMLt  eit
•   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
    The Multifactor CAPM and the APT

• A multi-index CAPM will inherit its risk factors
  from sources of risk that a broad group of
  investors deem important enough to hedge
• The APT is largely silent on where to look for
  priced sources of risk
Empirical tests of asset pricing
            models
         Overview of Investigation

• Tests of the single factor CAPM or APT Model
• Tests of the Multifactor APT Model
• Studies on volatility of returns over time
The Index Model and the Single-Factor APT

• Test the linear expected return-beta relationship
          E (ri )  rf  i  E (rM  rf 
                                        
              Tests of the CAPM
Tests of the expected return beta relationship:
• First Pass Regression
   – Estimate beta, average risk premiums and
     unsystematic risk
• Second Pass: Using estimates from the first pass
  to determine if model is supported by the data
• Most tests do not generally support the single
  factor model
     Single Factor Test Results


Return %
                            Predicted

                             Actual




                                  Beta
              Roll’s Criticism
• The only testable hypothesis is on the efficiency
  of the market portfolio
• CAPM is not testable unless we know the exact
  composition of the true market portfolio and use
  it in the tests
• Benchmark error
        Measurement Error in Beta

• Statistical property
• If beta is measured with error in the first stage,
  second stage results will be biased in the
  direction the tests have supported
• Test results could result from measurement error
        Jaganathan and Wang Study

• Included factors for cyclical behavior of betas and
  human capital
• When these factors were included the results
  showed returns were a function of beta
• Size is not an important factor when cyclical
  behavior and human capital are included
Table 13.2 Evaluation of Various CAPM
            Specifications
Table 13.4 Determinants of Stockholdings
    Tests of the Multifactor Model
• Chen, Roll and Ross 1986 Study
Factors
Growth rate in industrial production
Changes in expected inflation
Unexpected inflation
Unexpected Changes in risk premiums on bonds
Unexpected changes in term premium on bonds
          Study Structure & Results

• Method: Two -stage regression with portfolios
  constructed by size based on market value of
  equity Fidings
• Significant factors: industrial production, risk
  premium on bonds and unanticipated inflation
• Market index returns were not statistically
  significant in the multifactor model
Table 13.5 Economic Variables and Pricing
  (Percent per Month x 10), Multivariate
                Approach
    Fama-French Three Factor Model

• Size and book-to-market ratios explain returns on
  securities
• Smaller firms experience higher returns
• High book to market firms experience higher
  returns
• Returns are explained by size, book to market and
  by beta
 Table 13.6 Three Factor Regressions for
Portfolios Formed from Sorts on Size and
      Book-to-Market Ratios (B/M)
  Interpretation of Three-Factor Model

• Size is a proxy for risk that is not captured in
  CAPM Beta
• Premiums are due to investor irrationality or
  behavioral biases
        Risk-Based Interpretations

• In figure 13.1, it shows that returns on style
  portfolios (HML or SMB) seem to predict GDP
  growth, and thus may in fact capture some
  aspects of business cycle risks
• In figure 13.2, it shows the beta of the HML
  portfolio is lower in good economies while
  becomes higher in recessions, suggesting also
  that HML captures some aspects of business cycle
  risks
Zhang (2005) finds that value firms (with high
  book-to-market ratios) on average have
  greater amount of tangible capital. Investment
  irreversibility puts such firms more at risk for
  economic downturns. In contrast, growth
  firms are better able to deal with a downturn
  by deferring investment plans. The greater
  exposure of high book-to-market firms to
  recessions will result in higher down-market
  betas.                                        126
To quantify this, Petkova and Zhang fit the following model:
      rHML    rMt  ei
         [b0  b1DIVt  b2 DEFLT  b3TERMt  b4TBt ]  et
                                   t

Where:
 DIV =market dividend yield
 Default=default spread on corporate bonds (Baa-Ass rates)
 Term=term structure spread (10-year – 1-year Treasury rates)
 TB=1-month T-bill rate




                                                               127
Figure 13.1 Difference in Return to Factor
Portfolios in Year Prior to Above-Average
          versus Below-Average
               GDP Growth
Figure 13.2 HML Beta in Different Economic
                 States
          Behavioral Explanations
• Market participants are overly optimistic
   – Analysts extrapolate recent performance too
     far into the future
   – Prices on these glamour stocks are overly
     optimistic
   – Lower book-to-market on these glamour firms
     leads to underperformance compared to value
     stocks
• Chan, Karceski and Lakonishok find that B/M
  ration reflects the past earning growth but not
  the future, as in Fig. 13.3
La Porta, Lakonishok, Shleifer, and Vishny (2007)
  demonstrate that growth stocks
  underperforms value stocks surrounding
  earnings announcements, suggesting that
  when news of actual earnings is released to
  the public, the market is relatively
  disappointed in stocks is has been predicting
  as growth firms.

                                                132
         Liquidity and Asset Pricing

• Acharya and Pedersen
   – Premiums observed in the three-factor model
     may be illiquidity premiums, as shown in the
     first three row of results in Table 13.7
   – In table 13.8, it shows that despite that the
     liquidity adjustments to the market beat are
     relatively small, accounting for portfolio
     liquidity materially improves the fit of the
     model
Table 13.7 Properties of Liquidity Portfolios
Table 13.8 Estimates of the CAPM With and
         Without Liquidity Factors
Consumption-based Asset Pricing
      Model (CCAPM)
Each individual’s plan is to maximize a utility function of
  lifetime consumption, and consumption/investment
  in each period is based on age and current wealth, as
  well as risk-free rate and the market portfolio’s risk
  and risk premium
                E (rM )  rf  ACov(rM , rC )

Table 13.10 and Figure 13.7 indicate that Fama-French
  factors for average returns may in fact reflect the
  differing consumption risk of those portfolios

                                                        137
138
           Equity Premium Puzzle

• Rewards for bearing risk appear to be excessive
• Possible Causes
   – Predicting returns from realized returns;
     people underestimated the realized returns in
     post-war America
• Survivorship bias also creates the appearance of
  abnormal returns in market efficiency studies;
  see Figure 13.8
            Equity Premium Puzzle
Period       Risk-Free Rate   S&P 500 return   Equity Premium

1872-1999    4.87             10.97            6.10
1972-1949    4.05             8.67             4.62
1950-1999    6.15             14.56            8.41
   Extensions to the CAPM may
resolve the equity premium puzzle
Constantinides (2008) argues that the standard
  CAPM can be extended to including habit
  formation, incomplete markets, the life cycle,
  borrowing constraints, and other forces of
  limited stock market participants, to help
  explain equity premium puzzle




                                               143
   Behavioral Explanations of the
      Equity Premium Puzzle
Barberis and Huang (2008) incorporate loss
  aversion and narrow framing to explain the
  puzzle. Narrow framing is the idea that
  investors evaluate every risk they face in
  isolation. Thus, investors will ignore low
  correlation of the risk of stock portfolio with
  other components of wealth, and therefore
  require a higher risk premium than rational
  models would predict. Loss aversion also
  generate larger risk premium                      144
            Time-Varying Volatility

• Stock prices change primarily in reaction to
  information
• New information arrival is time varying
• Volatility is therefore not constant through time
 Stock Volatility Studies and Techniques

• Volatility is not constant through time
• Improved modeling techniques should improve
  results of tests of the risk-return relationship
• ARCH and GARCH models incorporate time
  varying volatility
Figure 13.5 Estimates of the Monthly Stock
       Return Variance 1835 - 1987

				
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