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					Pojednostavljenje
portfolio selekcije
             FAKTORSKI MODELI

Markowitz model
Uvođenje faktorskih modela
Karakteristike jednofaktorskog modela
Karakteristična linija
Očekivana stopa prinosa u jednovaktorskom modelu
Pojednostavljena jednačina za varijansu portfolia u
jednofaktorskom modelu
Objašnjena u odnosu na neobjašnjenu varijansu
Multi faktorski modeli
Modeli za ocenjivanje očekivanog prinosa
                               Markowitz Model
                             M   M
               σ 2 (rp )     x x C ov(r , r )
                             j1 k 1
                                        j k    j k


               σ 2 (r )      C ov(r , r2 ) . . . C ov(r , rM ) 
                      1            1                    1      
               C ov(r , r )    σ 2 (r2 ) . . . C ov(r , rM )
                        2 1                             2      
                    .            .       .            .        
                                                               
                    .            .              .     .        
                                                               
               C ov(r , r1 )
                         M        ...      ...     σ 2 (rM )    
                                                               
Problem: Veliki broj potrebnih podataka.
      Broj potrebnih varijansi HoV = M.
      Broj potrebnih kovarijansi = (M2 - M)/2
      Ukupno = M + (M2 - M)/2
  Primer: (100 HoV)
      100 + (10,000 - 100)/2 = 5,050
  Dakle, da bi moderna portfolio teorija bila upotrebljiva za veliki
  broj HoV, proces je trebalo pojednostaviti. (Pre nekoliko
  decenija kapaciteta kompjutera su bili ograničeni)
     Uvođenje faktorskih modela
Da bismo generisali efikasni set, potrebne su nam
procene očekivanih prinosa i kovarijansi između HoV u
raspoloživom skupu. Faktorski modeli mogu se koristiti
za ovu svrhu.
Faktori rizika – (stopa inflacije rate of inflation, growth
in industrial production, and other variables that induce
stock prices to go up and down.)
   May be used to evaluate covariances of return between
   securities.
Expected Return Factors – (firm size, liquidity, etc.)
   May be used to evaluate expected returns of the securities.
In the discussion that follows, we first focus on risk
factor models. Then the discussion shifts to factors
affecting expected security returns.
      Essence of the Single-Factor Model
Fluctuations in the return of a security relative to that of
another (i.e., the correlation between the two) do not depend
upon the individual characteristics of the two securities.
Instead, relationships (covariances) between securities occur
because of their individual relationships with the overall
market (i.e., covariance with the market).
If Stock (A) is positively correlated with the market, and if
Stock (B) is positively correlated with the market, then Stocks
(A) and (B) will be positively correlated with each other.
Given the assumption that covariances between securities
can be accounted for by the pull of a single common factor
(the market), the covariance between any two stocks can be
written as:
                 Cov(r , rk )  β jβk σ 2 (rM )
                         j
     The Characteristic Line
     (See Chapter 3 for a Review of the Statistics)

Relationship between the returns on an individual
security and the returns on the market portfolio:
    rj,t  A j  β jrM, t  ε j,t
    rj,t is therateof re turnon stock(j) duringpe riod(t)
Aj = intercept of the characteristic line (the expected
rate of return on stock (j) should the market happen
to produce a zero rate of return in any given period).

j = beta of stock (j); the slope of the characteristic
line.

j,t = residual of stock (j) during period (t); the vertical
distance from the characteristic line.
Graphical Display of the Characteristic
Line

           rj,t
     0.4


     0.3


     0.2                      = j

     0.1
       Aj
      0                                rM,t
            0     0.2   0.4      0.6
    The Characteristic Line (Continued)
Note: A stock’s return can be broken down into two
parts:
   Movement along the characteristic line (changes
   in the stock’s returns caused by changes in the
   market’s returns).
   Deviations from the characteristic line (changes in
   the stock’s returns caused by events unique to the
   individual stock).
              rj,t  A j  β jrM, t  ε j,t

Movement along the line: Aj + jrM,t
Deviation from the line: j,t
       Major Assumption of the Single-Factor Model
There is no relationship between the residuals of one
stock and the residuals of another stock (i.e., the
covariance between the residuals of every pair of
stocks is zero).
                   Cov(ε j , εk )  0
            Stock j’s Residuals (%)
                10




                 0                         Stock k’s Residuals (%)
          -10         0    10         20




                -10
   Expected Return in the Single-Factor Model
Actual Returns: rj,t  A j  β jrM, t  ε j,t
Expected Residual:
   Given the characteristic line is truly the line of best
   fit, the sum of the residuals would be equal to zero
                         n

                       ε
                        t 1
                               j, t  0

   Therefore, the expected value of the residual for
   any given period would also be equal to zero:
                      E(ε j )  0
Expected Returns:
  Given the characteristic line, and an expected
  residual of zero, the expected return of a security
  according to the single-factor model would be:
              E(rj )  A j  β jE(rM )
        Single-Factor Model’s Simplified Formula for
                     Portfolio Variance
 Variance of an Individual Security:
                                        n
                         σ 2 (rj )     h [r
                                       i 1
                                              i   j, i  E(r j )]
                                                                 2



 Given:              rj, i  A j  β jrM, i  ε j, i
                     E(r j )  A j  β jE(rM )
 It Follows That:
               n
σ 2 (rj )    
              i 1
                     h i ([Aj  β jrM, i  ε j, i ]  [ A j  β jE(rM )])2

                n
         =    
              i =1
                     h i (β j[rM, i  E(rM )]  ε j, i ) 2

                n
         =    
              i =1
                     h i (β 2 [rM, i  E(rM )]2  2 β j[rM, i  E(rM )]ε j, i  ε 2 i )
                            j                                                     j,

                     n                                         n                                     n
         = β2
            j      
                   i =1
                          h i [rM, i  E(rM )]2  2 β j      i 1
                                                                     h i [rM, i  E(rM )]ε j, i    
                                                                                                    i 1
                                                                                                           hiε 2 i
                                                                                                               j,
Note:      n
   β2
    j    
         i =1
                h i [rM, i  E(rM )]2  β 2σ 2 (rM )
                                          j

            n                                        n
   2βj      h [r
           i 1
                   i M, i  E(rM )]ε j, i  2 β j    h [r
                                                    i 1
                                                           i M, i  E(rM )][ε j, i  E( ε j )]

                                                             S i n ceE( ε j )  0
                                      = 2 β jC ov(r , ε j )
                                                  M
                                      =0
                      S i n ceC ov(r , ε j ) i s e qu alto z e rofor th e
                                   M
                                                        re        .
                          be stfi t l i n ei n si m pl e gre ssi on
     n

   i 1
           h i ε 2 i  σ 2 (ε j )  Re si du al
                 j,                                    (S
                                              Vari an ce e eC h apte r3)

Therefore:
                       σ 2 (rj )  β 2σ 2 (rM )  σ 2 (ε j )
                                     j
    Variance of a Portfolio
Same equation as the one for individual security
variance:    2       2 2       2
              σ (rp )  βpσ (rM )  σ (εp )

Relationship between security betas & portfolio betas
                           m
                   βp    x β
                          j1
                                    j j


Relationship between residual variances of stocks,
and the residual variance of a portfolio, given the
index model assumption.
                                m
                  σ 2 (εp )    
                                j1
                                      x 2σ 2 (ε j )
                                        j

The residual variance of a portfolio is a weighted
average of the residual variances of the stocks in the
portfolio with the weights squared.
        Explained Vs. Unexplained Variance
        (Systematic Vs. Unsystematic Risk)

Total Risk = Systematic Risk + Unsystematic
Risk
          σ 2 (rj )  β 2σ 2 (rM )  σ 2 (ε j )
                        j

Systematic: That part of total variance which
is explained by the variance in the market’s
returns.
Unsystematic: The unexplained variance, or
that part of total variance which is due to the
stock’s unique characteristics.
Note:          C ov(r , rM )
                     j             ρ j,M σ(rj ) σ(rM )       ρ j,Mσ(rj )
        βj                                             
                  2
                 σ (rM )                 2
                                        σ (rM )               σ(rM )
        The re fore β jσ(rM )  ρ j,M σ(rj )
                   :

                       β 2σ 2 (rM )  ρ 2 M σ 2 (rj )
                         j              j,
[i.e., j22(rM) is equal to the coefficient of
determination (the % of the variance in the security’s
returns explained by the variance in the market’s
returns) times the security’s total variance]
Total Variance = Explained + Unexplained
                            2
               σ 2 (rp )  βpσ 2 (rM )  σ 2 (εp )
                          2
                       = ρp, M σ 2 (rp )  σ 2 (εp )
As the number of stocks in a portfolio increases, the
residual variance becomes smaller, and the
coefficient of determination becomes larger.
             Explained Vs. Unexplained Variance
                    (A Graphical Display)

Residual Variance              Coefficient of Determination
 10                                 1.2

                                      1
  8

                                    0.8
  6
                                    0.6
  4
                                    0.4

  2
                                    0.2

  0                                   0
      1     5    9   13   17              1     5   9   13    17
          Number of Stocks                    Number of Stocks
          Explained Vs. Unexplained Variance
            (A Two Stock Portfolio Example)
      β 2 σ 2 (r )     β Aβ Bσ 2 (rM )       σ 2 (ε )    C ov(ε A , ε B )
      A         M                                  A                     
     β β σ 2 (r )        β Bσ 2 (rM ) 
                            2                 C ov(ε , ε )   σ 2 (ε B ) 
      B A        M                                B A                    
  Covariance Matrix for                    Covariance Matrix for
   Explained Variance                      Unexplained Variance
                                  2 2
σ 2 (rp )  x 2 β 2 σ 2 (rM )  x Bβ Bσ 2 (rM )  2 x A x Bβ Aβ Bσ 2 (rM )
              A A
                               Explaine dVariance
                               2
          + x 2 σ 2 (ε A )  x Bσ 2 (ε B )
              A
                       d
             Une xplaine Variance
               mingC ov(ε A , ε B )  0
            Assu
       Explained Vs. Unexplained Variance (A Two
          Stock Portfolio Example) Continued
                                              m
σ 2 (rp )  (xAβ A  x Bβ B ) 2 σ 2 (rM )    
                                              j1
                                                    x 2 σ 2 (ε j )
                                                      j


                  Explain e d                          e
                                            Un e xplaind
                        2
          m                         m
         
        = 
          j =1
                       
                x jβ j  σ 2 (rM ) 
                       
                                         x 2 σ 2 (ε j )
                                            j
                                   j =1
           2
        = βpσ 2 (rM )  σ 2 (ε p )
If C ov(ε A , ε B )  0, re si du al                            d
                                   vari an cewillbe u n de rstate .
If C ov(ε A , ε B )  0, re si du al                           .
                                   vari an cewillbe ove rstate d
       A Note on Residual Variance
The Single-Factor Model assumes zero correlation
between residuals:
                   Cov(ε j , εk )  0
In this case, portfolio residual variance is expressed
as:                         m
                          σ 2 (εp )    
                                        j1
                                              x 2σ 2 (ε j )
                                                j

In reality, firms’ residuals may be correlated with
each other. That is, extra-market events may impact
on many firms, and: Cov(ε j , εk )  0
In this case, portfolio residual variance would be:
                  m                           m      m
    σ 2 (εp )    
                  j1
                        x 2σ 2 (ε j )  2
                          j                 x x Cov(ε , ε )
                                            j1 k  j1
                                                              j k   j k
  Markowitz Model Versus the Single-Factor Model
      (A Summary of the Data Requirements)
Markowitz Model
   Number of security variances = m
   Number of covariances = (m2 - m)/2
   Total = m + (m2 - m)/2
   Example - 100 securities:
    100 + (10,000 - 100)/2 = 5,050
Single-Factor Model
   Number of betas = m
   Number of residual variances = m
   Plus one estimate of 2(rM)
   Total = 2m + 1
   Example - 100 securities:
    2(100) + 1 = 201
             Multi-Factor Models
Recall the Single-Factor Model’s formula for portfolio
variance:               2
           σ 2 (rp )  βpσ 2 (rM )  σ 2 (εp )
                     2
                  = ρp, M σ 2 (rp )  σ 2 (εp )
If there is positive covariance between the residuals
of stocks, residual variance would be high and the
coefficient of determination would be low. In this
case, a multi-factor model may be necessary in order
to reduce residual variance.
A Two Factor Model Example
        rj,t  A j  βg, jrg,t  βI, jrI,t  ε j,t
where: rg = growth rate in industrial production
       rI = % change in an inflation index
      Two Factor Model Example - Continued
Once again, it is assumed that the covariance
between the residuals of the the individual stocks are
equal to zero: Cov(ε j , εk )  0
Furthermore, the following covariances are also
presumed:       C ov(r , ε j )  0
                       g
                 C ov(r , ε j )  0
                      I
                 C ov(r , rI )  0
                      g
Portfolio Variance in a Two Factor Model:

                2               2
   σ 2 (rp )  βg,pσ 2 (rg )  βI,pσ 2 (rI )  σ 2 (εp )
                          m

                          x β
where:
                β g,p              j     g, j
                          j1
                          m
                β I,p    x β
                          j1
                                    j    I, j



                                m
                σ 2 (ε p )    
                               j1
                                        x 2 σ 2 (ε j )
                                          j


                Assu m i n g ov(ε j , ε k )  0
                           C

Note that if the covariances between the residuals of
the individual securities are still significantly different
from zero, you may need to develop a different model
(perhaps a three, four, or five factor model).
             Note on the Assumption Cov(rg,rI ) = 0

   If the Cov(rg,rI) is not equal to zero, the two factor
   model becomes a bit more complex. In general,
   for a two factor model, the systematic risk of a
   portfolio can be computed using the following
   covariance matrix:
                     g,            I,p
            g,p  2 ( r )
                     p
                                 Cov(rg , rI )
                        g                    
                                   2 (r ) 
            I,p                   I 
                                             
             2
 2 (rp )  g, p 2 (rg )   2 p 2 (rI )  2g, p I, p Cov(rg , rI )
                               I,

   To simplify matters, we will assume that the
   factors in a multi-factor model are uncorrelated
   with each other.
      Models for Estimating Expected Return
One Simplistic Approach
  Use past returns to predict expected future
  returns. Perhaps useful as a starting point.
  Evidence indicates, however, that the future
  frequently differs from the past. Therefore,
  “subjective adjustments” to past patterns of returns
  are required.
Systematic Risk Models
  One Factor Systematic Risk Model:

           E(rj )  A j  β jE(rM )
  Given a firm’s estimated characteristic line and an
  estimate of the future return on the market, the
  security’s expected return can be calculated.
     Models for Estimating Expected Return
                  (Continued)

   Two Factor Systematic Risk Model:
         E(rj )  A j  β1, jE(r1 )  β 2, jE(r2 )
   N Factor Systematic Risk Model:
E(r j )  A j  β1, jE(r1 )  β 2, jE(r2 )  . . . + β N, jE(rN )
Other Factors That May Be Used in Predicting
Expected Return
  Note that the author discusses numerous factors
  in the text that may affect expected return. A
  review of the literature, however, will reveal that
  this subject is indeed controversial. In essence,
  you can spend the rest of your lives trying to
  determine the “best factors” to use. The following
  summarizes “some” of the evidence.
        Other Factors That May Be Used in Predicting
                        Expected Return
Liquidity (e.g., bid-asked spread)
   Negatively related to return [e.g., Low liquidity
   stocks (high bid-asked spreads) should provide
   higher returns to compensate investors for the
   additional risk involved.]
Value Stock Versus Growth Stock
   P/E Ratios
    • Low P/E stocks (value stocks) tend to
      outperform high P/E stocks (growth stocks).
   Price/(Book Value)
    • Low Price/(Book Value) stocks (value stocks)
      tend to outperform high Price/(Book Value)
      stocks (growth stocks).
   Other Factors That May Be Used in Predicting Expected
                    Return (continued)

Technical Analysis
   Analyze past patterns of market data (e.g., price
   changes) in order to predict future patterns of
   market data. “Volumes have been written on this
   subject.
Size Effect
   Returns on small stocks (small market value) tend
   to be superior to returns on large stocks. Note:
   Small NYSE stocks tend to outperform small
   NASDAQ stocks.
January Effect
   Abnormally high returns tend to be earned
   (especially on small stocks) during the month of
   January.
 Other Factors That May Be Used in Predicting Expected
                  Return (continued)

And the List Goes On
  If you are truly interested in factors that
  affect expected return, spend time in the
  library reading articles in Financial Analysts
  Journal, Journal of Portfolio Management,
  and numerous other academic journals.
  This could be an ongoing venture the rest
  of your life.
     Building a Multi-Factor Expected Return Model:
                 One Possible Approach

Estimate the historical relationship between return
and “chosen” variables. Then use this relationship to
predict future returns.
   Historical Relationship:
 rt  a0  a1 (P/E Ratio) 1  a 2 (Firm Siz e t 1
                            t                  )
      + a 3 (Be ta)1  . . .
                  t

   Future Estimate:

    rt 1  a0  a1 (P/E Ratio)  a 2 (Firm Siz e t
                               t                  )
         + a 3 (Be ta)  . . .
                      t
        Using the Markowitz and Factor Models
          to Make Asset Allocation Decisions


Asset Allocation Decisions
  Portfolio optimization is widely
  employed to allocate money between
  the major classes of investments:
  •   Large capitalization domestic stocks
  •   Small capitalization domestic stocks
  •   Domestic bonds
  •   International stocks
  •   International bonds
  •   Real estate
     Using the Markowitz and Factor Models
 to Make Asset Allocation Decisions Continued:
   Strategic Versus Tactical Asset Allocation
Strategic Asset Allocation
   Decisions relate to relative amounts invested
   in different asset classes over the long-term.
   Rebalancing occurs periodically to reflect
   changes in assumptions regarding long-term
   risk and return, changes in the risk tolerance
   of the investors, and changes in the weights of
   the asset classes due to past realized returns.
Tactical Asset Allocation
   Short-term asset allocation decisions based on
   changes in economic and financial conditions,
   and assessments as to whether markets are
   currently underpriced or overpriced.
        Using the Markowitz and Factor Models
     to Make Asset Allocation Decisions Continued
Markowitz Full Covariance Model
  Use to allocate investments in the portfolio among the
  various classes of investments (e.g., stocks, bonds,
  cash). Note that the number of classes is usually rather
  small.
Factor Models
  Use to determine which individual securities to include
  in the various asset classes. The number of securities
  available may be quite large. Expected return factor
  models could also be employed to provide inputs
  regarding expected return into the Markowitz model.
Further Information
  Interested readers may refer to Chapter 7, Asset
  Allocation, for a more indepth discussion of this
  subject. In addition, the author has provided “hands
  on” examples of manipulating data using the
  PManager software in the process of making asset
  allocation decisions.

				
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posted:9/28/2012
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