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					Ch. 7 Introduction to Risk, Return, and the Opportunity Cost of Capital

━ three issues on risk:


  1) How risk is defined (ch. 7)
  2) What the links are between risk and the OCC (ch. 8)
  3) How the financial manager can cope with risk in practical situations (ch. 9)


  notes: 1) two major modern developments in finance: risk --> r and OPM
         2) CAPM, APT, and three-factor model --> cross-sectional variations in stock returns

━ some concepts on risk:


  The risk in investment means that future returns are unpredictable. The spread of possible
  outcomes is usually measured by the standard deviation of return.


  However, we will explain that the effective risk of any security can not be judged by an
  examination of that security alone. Risk is best judged in a portfolio context.


  We will introduce the concept of beta, the standard risk measure for individual securities. It is a
  security‟s contribution to the risk of a well-diversified portfolio.

━ main themes in the chapter: diversification and portfolio risk


 notes: 1) implicit assumption: Investors are diversified investors. Why?
        2) Diversification practice in Philippines? STD as risk measure?
        3) diversified investors => diversification => portfolio risk => beta


7-1 Over A Century of Capital Market History in One Easy Lesson

━ CRSP (University of Chicago‟s Center for Research in Security Prices) database


  a note: Other well-known databases: a) bond, stock, option data - Datastream and Bloomberg
                                      b) financial statement data - Compustat

━ a study by Dimson, Marsh, and Staunton (2002) on the historical performance of 3 portfolios
  of securities for the period 1900 to 2003 (updated by Brealey and Myers):


  3 portfolios: 1) Treasury bills (some uncertainty of inflation)
                2) long-term government bonds (interest-rate risk)
                                                 1
                         [3) long-term corporate bonds (default risk)]
                         4) common stocks of large firms (S&P 500) (business risk)
                         [5) common stocks of small firms (greater business risk and some „size‟ risk)]


  the performance: Figures 7.1 and 7.2, Table 7.1


  a note: an earlier study by Ibbotson Associates on the historical performance of 5 portfolios of
          securities for the period 1926 to 2000:


                                             Average Annual Rate of Return
             Portfolio               Nomial                         Real          Average Risk Premium
             Treasury Bills            3.9                          0.8                     0
             Government Bonds          5.7                          2.7                    1.8
             Corporate Bonds           6.0                          3.0                    2.1
             Common Stocks (S&P 500) 13.0                           9.7                    9.1
             Small-firm Common Stocks 17.3                         13.8                   13.4


  a lesson: Portfolio performance (or return) coincides with our intuitive risk rankings among the
             five portfolios.

━ arithmetic averages and compound annual returns (i.e., geometric averages)
              T
 ● AM    =   r /T
             t 1
                     t


                    T
 ●   GM = [ (1  rt )]1 / T  1  ( AT / A0 )1 / T  1 ,    where A0 = the initial wealth, AT = the
                    t 1

                                                             ending-period wealth

     Notice that the GM is the rate r such that A0 (1  r ) T  AT , which indicates that the GM is a

     compound return.

 ● comparison:



     AM is biased upward if you are attempting to measure an asset‟s long-term performance. GM
     is considered a superior measure of the long-term mean rate of return because it indicates the
     compound annual rate of return.


     footnote 5: 1) For S&P 500 in the period 1900-2003, GM = (15,579)1/75 – 1 = 0.097 while AM
                    = 0.117.
                 2) For lognormally distributed returns, GM = AM – 0.5 ╳ variance


                                                         2
 ● moral:     If the cost of capital is estimated from historical returns or risk premiums over a very
            long period, use arithmetic averages, not compound annual rates of return. [Notice that
            in calculating PV, both the cash flows and the discount rate are expectations.]

━ using historical evidence to evaluate today‟s cost of capital


 ● case:   An asset has the same risk as the market.


 ● marketportfolio: a portfolio including all risky assets  S&P Composite Index
   the expected rate of return on the market portfolio: market return, rm


 ● estimate rm    in 2004:


  the fact:     rf varies over time. In 1981, it is 15%   => rm is not likely to be stable over time.


  a more sensible procedure:


     rm (2004) = rf (2004) + normal risk premium,         where rf = the interest rate on Treasury bill
                  = 0.01 + 0.076 = 0.086


   the crucial assumption here: There is a normal, stable risk premium on the market portfolio,
   so that the expected future risk premium can be measured by the average past risk premium.


 ● the   assumption above and the argument about the market risk premium:


  Many financial managers and economists believe that the long-run historical risk premium
  (7.6%) is the best measure available. Others argue that investors don‟t need such a large risk
  premium to persuade them to hold common stocks. The argument is based on two reasons.

 ● an   alternative measure of the risk premium:      r = DIV1/P0 + g


   Since 1900, DIV1/P0 = 4.7% and g = 4.7%. => rm = 9.4% or about 5.3% above the
   risk-free interest rate. This is 2.3% lower than the realized risk premium reported in Table 7.1.


 ● Why     have we talked so much about the market risk premium? Why is it so important?


  The CAPM, the theory linking risk to the OCC, hinges on the market risk premium:

                       ri  rf   i  market risk premium



                                                     3
7-2 Measuring Portfolio Risk

━ two benchmarks we have obtained: the discount rate for safe projects and the estimate of the
  rate for average-risk projects


  To estimate discount rates for assets that do not fit these simple cases, however, we have to
  learn (1) how to measure risk and (2) the relationship between risks borne and risk premiums
  demanded.

━ Figures 7.4 and 7.5: remarkably wide fluctuations in annual rates or market return rm


━ the standard statistical measures of spread: variance and standard deviation


  example: the market portfolio

     ~ = the actual market return (random return)
     rm
     rm = the expected market return (mean)

    variance of ~m =  m =
                r      2
                                          E[( ~m - rm )2]
                                              r


    standard deviation of ~m =  m
                          r

    When variance is estimated from a sample of observed returns, the formula of variance is
        N

      (~
        r
     t 1
            mt    rm ) 2 /( N  1) ,   where N is the number of observations and ~ is the market
                                                                                  rmt

    return in period t.


  example: a game of flipping two coins (Table 7.2)


  notes:
   1) footnote 18: Since STD is in the same units as the rate of return, it is generally more
     convenient to use STD. However, when we are talking about the proportion of risk that is
     due to some factor, it is usually less confusing to work in terms of the VAR.
   2) rate of return in percentage => the STD in percentage (i.e., the spread in percentage)
     The measure of risk in terms of the STD in value (i.e., the spread in value) will be easily
     interpreted to investors. How do we convert the STD in percentage into the STD in value?
   3) value at risk (VaR)

━ observing past variability


● the   annual STDs and VARs observed for the 3 portfolios over the period 1900-2003:
                                                            4
            Portfolio                          STD(%)          VAR(%)
            Treasury bills                        2.8            7.9
            Government bonds                      8.2           68.0
            Common stocks (S&P 500)              20.1          402.6


    a note: the annual STDs and VARs observed for the 5 portfolios over the period 1926-2000:


            Portfolio                          STD(%)          VAR(%)
            Treasury bills                       3.2             10.1
            Government bonds                     9.4             88.7
            Corporate bonds                       8.7            75.5
            Common stocks (S&P 500)              20.2           406.9
            Small-firm common stocks             33.4           1118.4


 ● the   annual STDs of the market returns for successive periods:


            Period              Market STD (%)
            1926-1930               21.7
            1931-1940                37.8
            1941-1950                14.0
            1951-1960                12.1
            1961-1970                13.0
            1971-1980                15.8
            1981-1990                16.5
            1991-2003                14.8


    footnote 21: VAR is proportional to the length of time interval over which a security or
                 portfolio return is measured, STD is proportional to the square root of the interval.

 ● Figure   7.6: the STDs of stock market returns in 16 countries over the period 1900-2003


   Most of the countries cluster together with percentage STDs in the low 20s.

━ How diversification reduces risk.


 ● comparison    and question raised:


  individual stocks: Tables 7.3 and 7.4
  market portfolio: Tables above and 7.4


                                                   5
  => Q:  m < average  of individual stocks
             The market portfolio is made up of individual stocks, so why doesn‟t its variability
             reflect the average variability of its components?


 ● answer:     Diversification reduces variability. (Figure 7.7)


  Diversification works because prices of different stocks do not move exactly together, i.e.,
  stock price changes are less than perfectly positively correlated. On many occasions, a decline
  in the value of one stock was offset by a rise in the price of another. Therefore there was an
  opportunity to reduce our risk by diversification. (Figure 7.8 and footnotes 24 and 25)


  footnote 23:


 ● unique   risk and market risk:


   unique risk (diversifiable, unsystematic, specific, or residual risk): the perils that is peculiar to
      an individual company


   market risk (systematic or undiversifiable risk): the perils that is economywide, threatening
      all businesses


   Figure 7.9


7-3 Calculating Portfolio Risk

━ point: To understand fully the effect of diversification, we need to know how the risk of a
       portfolio depends on the risk of the individual stocks.

━ covariance of two random variables X and Y:

                                                      T
  Var ( X )   X  E[( X   X ) 2 ] or s X  [ ( xt  X ) 2 ] /(T  1) ,  X , s X  0
                2                          2                                  2     2

                                                      t 1
                                                T
  Var (Y )   Y  E[(Y  Y ) 2 ] or sY  [ ( y t  Y ) 2 ] /(T  1) ,  Y , sY  0
               2                       2                                   2    2

                                               t 1
                                                                              T
  Cov( X , Y )   XY  E[( X   X )(Y  Y )]                  or   s XY  [ ( xt  X )( y t  Y )] /(T  1)   ,
                                                                              t 1

                                                                                  XY , s XY  


  figure for S XY > 0:
  figure for S XY < 0:
                                                             6
  a measure free from units: the coefficient of correlation

            XY   XY /  X  Y or rXY  s XY / s X sY ,  1   XY , rXY  1

  figure for rXY = +1 (perfectly positive correlation):


  figure for 0 < rXY < +1 (positive correlation):


  figure for rXY = -1 (perfectly negative correlation):


  figure for -1 < rXY < 0 (negative correlation):


  figure for rXY = 0 (no correlation):


━ the expected return and risk for a portfolio consisting of two stocks:

  ~ = stock i‟s actual return, r = stock i‟s expected return, i = 1, 2
  ri                            i

  xi = proportion invested in stock i, i = 1, 2
  ~ = portfolio‟s actual return = ~  x ~  x ~
  rp                              rp   1r1   2 r2


● portfolio‟s    expected return:

  rp  E ( ~p )  E ( x1~  x 2 ~ )  x1 E ( ~ )  x 2 E ( ~ )  x1r1  x 2 r2 , the weighted average of the
           r            r1      r2           r1            r2

                                                                       expected return on the two stocks
 ● portfolio‟s   variance:

   p  V a (~p )  x12V a (~ )  x2V a (~ )  2 x1 x2C o (~, ~ )
    2
            rr             rr1     2
                                        rr2               vr1 r2

      = x12 12  x2  2  2 x1 x2 12  x12 12  x2  2  2 x1 x2 12 1 2 ,
                   2 2                              2 2
                                                                                  where 12   12 /  1 2


   See the box in Figure 7.10.

 ● example   for calculating  p :  1 = 18.2%,  2 = 27.3%, x1 = 0.60, x 2 = 0.40
                               2



                                        r1 = 10%, r2 = 15%

  1) 12 = +1.0          =>      p = 477%,  p = 21.8% = 0.60 ╳ 18.2% + 0.40 ╳ 27.3%, the
                                  2



                                                                          weighted average of the two STDs

      Notice that  p  x1 1  x 2 2 when 12 = +1.0.


  2) 12 = +0.4       =>       p = 333.9%,  p = 18.3%
                                2



                                                         7
  3) 12 = -1.0 =>                  p = 0%
                                     2




  notes: 1) an intuitive way to calculate the risk of the portfolios: the weighted average of the
           STDs on the two stocks = 0.60╳18.2% + 0.40╳27.3% = 21.8%

             However, we see  p = weighted average if 12 = +1.0

                                           p < weighted average if 12 < +1.0

             Hence, diversification would reduce risk if individual stocks are not perfectly
             positively correlated.

            2) For any 12 , there is some x1* such that  p is minimized. If 12 = -1.0, we can


                   easily find the optimal x1* such that  p = 0. In the example, x1* = 0.60.

            3) a helpful diagram:



━ general formula for the portfolio consisting of N stocks:
             ~  x ~  x ~  ...  x ~
             rp   1r1   2 r2        N rN



 ● expected    return: rp  x1r1  x 2 r2  ...  x N rN , a weighted average

                               N    N

 ● variance:        x i x j ij
                   2
                   P
                           i 1 j 1

                           N                N      N                      N         N         N

                            xi  i               x i x j ij or        xi  i  2       x x 
                              2 2                                            2 2
                       =                                                                          i   j   ij
                           i 1              i     j                     i 1      i        j




   point: The variability of a portfolio reflects mainly the covariances. See Figure 7.11.


 ● limits   to diversification:
                                                        1
  assumption: equal investment ( xi                      )
                                                        N
                                         1                                      1
  portfolio variance = N                  2
                                              average variance + ( N 2  N )  2  average covariance
                                        N                                      N
                                   1                          1
                           =          average variance + (1  )  average covariance
                                   N                          N


  Thus, as N   , portfolio variance  average covariance.
                                                                      8
  point: Most of common stocks do move together, not independently
        => positive covariances
        => average covariance > 0


        Therefore, positive covariances between most of common stocks set the limit to the
        benefits of diversification.


  conclusion: market risk = the average covariance remaining after diversification, i.e., the
                            bedrock of risk remaining after diversification has done its work


7-4 How Individual Securities Affect Portfolio Risk

━ diversified investors‟ concern: the effect that each security will have on the risk of their
 portfolio, i.e., the (marginal) contribution of an individual security to the risk of the portfolio

━ the risk of any security = market risk + unique risk
  unique risk: the risk that is peculiar to the stock (the firm)
  market risk: the risk that is associated with market-wide variations


  a note: Recall that when we are talking about the proportion or component of risk that is due to
          some factor, it usually works in terms of variance.


  => The risk of a well-diversified portfolio (i.e., a portfolio without unique risk) depends on the
     market risk of the securities included in the portfolio.


      notes: 1) a helpful diagram for the statement above:
             2) There are 3 stars ☆ in the following diagram, indicating the 3 stages about the
                  issues discussed here:

              portfolio risk  p     <---------- 1 ----------     systematic risk of a security i

                        ↑                                                         │

                      p │3                                                       │2

                         | <-------------------   i       <----------------------- |
              Now we have gone by the first stage (☆1).


━ Market risk (for a security) is measured by beta.


 ● concept:   The market risk of a security indicates that how sensitive its return is to market
                  movements (☆1). This sensitivity is called beta.
                                                       9
 ● mathematical     expression of beta:


  market portfolio  portfolio of S&P 500
  actual return of market portfolio:
                                                  N
           rm   1r1   2r2         N rN  j rj , where ~j = actual return for stock j and
           ~  x ~  x ~  ...  x ~  x ~
                                                 j 1
                                                      r


                                                              x j = stock j‟s weight

  security i‟s beta:
                                   Cov( ~ , ~ )  im
                                        ri rm
                            i                 2
                                    Var ( ~ )
                                          rm    m

            where  im  Cov( ~, x1~  x2 ~  ...  x N ~ )
                              ri r1       r2            rN
                                                                                       N
                           = x1Cov( ~, ~)  x2Cov( ~, ~ )  ...  xN Cov( ~, ~ ) =
                                    ri r1          ri r2                  ri rN        x 
                                                                                       j 1
                                                                                              j   ij   ,

                       the weighted average of covariances among stock i and others in the
                       market (i.e., the market risk of security i)


 ● interpretation   to the formula:

  1) The formula can be rewritten as  im   i   m :
                                                    2
                                                                 The security beta measures how the

     stock‟s return is liable to be affected by general market movements, i.e., it measures the
     degree of sensitivity of the stock‟s return to market movements.
                                                         
  2) Think of the market portfolio as a security:  m  mm  1 . Thus, the “average” stock has a
                                                          m2


      beta of 1. =>   i  1 : unusually sensitive to market movements
                      i  1 : unusually insensitive to market movements
  3)  i = 1.77 means that on average when the market (index) rises an extra 1.0% of return,
      stock i‟s return will rise by an extra of 1.77%.         A diagram illustrating the empirical use
      may be helpful to explain (Figure 7.12):


● Tables   7.5 and 7.6:    High STDs do not always have high betas.

● summary:     The market risk of a security is measured by its beta (☆2).


━ Why security betas determine portfolio risk


 ● security   betas and portfolio risk:


                                                        10
    With more securities, portfolio risk declines until all unique risk of the portfolio is eliminated
    and only the bedrock of risk (i.e., the market risk or systematic risk) remains. The bedrock of
    portfolio risk depends on the average beta of the securities included.

    In fact, portfolio beta (  p ) equals the weighted average beta of the securities included, i.e.,

              N
     p   xi  i .         [Try to prove it by yourself.]
              i 1


 ● The      risk of a portfolio (  p ) is proportional to the portfolio beta:             p   p m .

  sketch of proof:           For a well-diversified portfolio p, we will have

                                                        pm
                                p ,m  1      =>            1         =>       pm   p m
                                                       p m

                                                          pm  p m  p
                              By definition,  p                                  =>      p   p m .
                                                         m 2
                                                                m
                                                                 2
                                                                      m

    example:  p = 1.2 The portfolio‟s standard deviation would be 1.2 times the market‟s

                     standard deviation.
                                                                                                            N
 ● summary:          Securities betas determine portfolio risk (☆3).                [  p   p   m = (  xi  i ) m ]
                                                                                                            i 1

━ conclusion:             For a diversified investor, the appropriate risk measure is beta rather than the
                         standard deviation.

━ example for calculating beta:
 ● assumption:           The market portfolio consists of the two securities as illustrated in Section 7-3.
 ● calculation:

    stock 1                  stock 2
  --------------------------------------
                                                            2
1 x12 12                    x1 x2 12 1 2        => x1  x j 1 j  x1 1m
                                                            j 1


   (. 60 ) 2 (18 .2) 2       .60  .40 

                       .4  18.2  27.3             => .60[.60  (18.2)2 + .40  .4  18.2  27.3] = .60  278.2
  ---------------------------------------
                                                             2
2 x1 x2 12 1 2            x2  2
                              2 2
                                                    => x 2  x j 2 j  x 2 2 m
                                                            j 1


  .60  .40                 (. 40 ) 2 ( 27 .3) 2

  .4  18.2  27.3                         => .40[ .60  .4  18.2  27.3 + .40  (27.3)2] = .40  417.4
  ----------------------------------------
                                                                   11
   We see  m = .60  278.2 + .40  417.4 = 333.9
            2



                                                                 2
                   = x1 1m + x2 2 m =                         x
                                                                i 1
                                                                       i   im   ,


    notes: 1)  im    x 
                         j
                                 j       ij   , the weighted average of covariances of stock i with other securities in

                                                   the market

            2)  m =
                 2
                         x i
                                     i        im    The market risk is the weighted average of covariances among

                                                    securities. Security i‟s contribution to the market risk depends on its
                                                    relative importance (xi) and its average covariance with other
                                                    securities (  im ).
            3) The proportion of stock 1‟s contribution to the market risk = 0.60  (278.2/333.9) =
                                                                            
               0.60  0.83 = 0.5. For any security i, the proportion is xi  im  xi  i .
                                                                            m


  ● any   portfolio p:


   1) stock i‟s contribution to portfolio risk = xi ip


                                                                                      ip
   2) the proportion of stock i‟s contribution = xi 
                                                                                     p

                      ip
   3) Let  ' i =         , called the beta of stock i relative to the portfolio p. Then, the proportion
                     p
       of stock i‟s contribution = xi   ' i .

   4) If p is well-diversified, then  ' i   i . The proportion of stock i‟s contribution  xi   i .


7-5 Diversification and Value Additivity

━ Q: Is a diversified firm more attractive to investors?


   If it is, the value additivity no longer holds.
   Assuming two assets or plants A and B, this means that PV(A + B) > PV(A) + PV(B).

  notes: 1) the argument: diversification => the decrease in rA B
          2) another argument: economies of scale or scope (such as synergies) derived from
           mergers


                                                                                12
━ point: If investors can diversify on their own, they will not pay any extra for firms that
           diversify.


   If capital markets are large and competitive, investors can diversify more easily than firms.
   Diversification thus does not add to a firm‟s value. => PV(A+B) = PV(A) + PV(B).

━ The concept of value additivity is very general.




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