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					The Capital Asset Pricing Model


  Global Financial Management




                                  1
           Overview

Utility and risk aversion
 » Choosing efficient portfolios
Investing with a risk-free asset
 » Borrowing and lending
 » The markt portfolio
 » The Capital Market Line (CML)
The Capital Asset Pricing Model (CAPM)
 » The Security Market Line (SML)
 » Beta
 » Project analysis



                                         2
Efficient Portfolios with Multiple Assets

             Investors
                                            Efficient
  E[r]         prefer                       Frontier


                                                 Portfolios
                               Asset 1           of other
                              Portfolios of      assets
                    Asset   2 Asset 1 and Asset 2


         Minimum-Variance
         Portfolio

         0                                              s     3
           Utility in Risk-Return Space
                     Indifference curves
         25.00%
                                  tau=0.5, Ubar=6%
                                  tau=0.5, Ubar=8%                 Investors
         20.00%                   tau=0.5, Ubar=10%                  prefer
                                  tau=0.25, Ubar=6%
                                  tau=0.25, Ubar=8%
         15.00%                   tau=0.5, Ubar=10%
Return




         10.00%



         5.00%



         0.00%
                  0.00%


                          2.00%


                                  4.00%


                                          6.00%


                                                  8.00%


                                                          10.00%


                                                                      12.00%


                                                                               14.00%


                                                                                        16.00%


                                                                                                 18.00%


                                                                                                          20.00%


                                                                                                                   22.00%


                                                                                                                            24.00%
                                                               Risk



                                                                                                                                     4
                   Individual Asset Allocations

14.00%
                                                                                                                                                       Point x is the optimal
                                                                                                                                                       portfolio for the less risk
                 Return




12.00%                                                                                                                                                 averse investor (red line)
                                                                                                                                                       Point y is the optimal
10.00%                                                                                                    x                                            portfolio for the more risk
8.00%                                                                          y                                                                       averse investor (black
                                                                                                                                                       line)
6.00%


4.00%


2.00%

                                                                     Risk
0.00%
         5.00%
                   6.00%
                           7.00%
                                   8.00%
                                           9.00%
                                                   10.00%
                                                            11.00%
                                                                      12.00%
                                                                               13.00%
                                                                                        14.00%
                                                                                                 15.00%
                                                                                                          16.00%
                                                                                                                   17.00%
                                                                                                                            18.00%
                                                                                                                                     19.00%
                                                                                                                                              20.00%




                                                                                                                                                                                     5
       Introducing a Riskfree Asset
Suppose we introduce the opportunity to invest in a riskfree
asset.
 » How does this alter investors’ portfolio choices?
The riskfree asset has a zero variance, and zero covariance with
every other asset (or portfolio).
 » var(rf) = 0.
 » cov(rf, rj) = 0 for all j.
What is the expected return and variance of a portfolio consisting
of a fraction (1-a) of the riskfree asset and a of the risky asset (or
portfolio)?




                                                                         6
Risk and Return with a Riskfree asset

 Expected Return
              
 ErP   aE rj  1  a )r f

 Variance and Standard Deviation

 VarrP   s 2  a 2 s 2  s P  as j
              P         j

 Hence, the risk-return tradeoff is:
                 sP
 E rP   r f 
                 sj
                    
                    Erj   rj   )


                                         7
      Risk and Return with a Riskfree asset
Expected
                                               The line represents
Return
                                               all portfolios
                                               depending on a
                      Asset j (a=1)


    E(rj)




      rf
            Riskfree asset
            (a=0)


      0                        sj     Standard Deviation
                                                                     8
Investing with Borrowing and Lending

Expected                     a =2
Return

               a = 0.5   M
  E[rM ]

     rf                        a =1


               a=0
               Lending         Borrowing

           0             sM                Standard
                                           Deviation
                                                       9
          Optimal Investing With Borrowing
                    and Lending
25.00%
                                                                                      Y = optimal risk-
           Return




                                                                                      return tradeoff
                     tau=0.5, Ubar=8%                                                 for risk-averse
20.00%
                     tau=0.25, Ubar=6%
                                                                                      investor
                     Portfolio
                                                                                      X = optimal risk-
15.00%
                                                                                      return tradeoff
                                                                                      for risk-tolerant
                                                                                      investor
10.00%
                                                              X

                                   Y
5.00%

           rf=4%                                                            Risk
0.00%
         1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
                                                                                                          10
      The Capital Market Line

Expected
Return
                            M
    E [ rm ]
   E [ rIBM ]       A                    IBM


       rf




            Systematic   Diversifiable         Standard
            Risk         Risk                  Deviation
                                                           11
             The Capital Market Line
The CML gives the tradeoff between risk and return for portfolios
consisting of the riskfree asset and the tangency portfolio M.
 » Portfolio M is the market portfolio.
The equation of the CML is:

                       E (rM )  rf
  E (rp )  rf  s p
                           sM

The expected rate of return on a risky asset can be thought of as
composed of two terms.
 » The return on a riskfree security, like U.S. Treasury bills;
   compensating investors for the time value of money.
 » A risk premium to compensate investors for bearing risk.
       E(r) = rf + Risk x [Market Price of Risk]
                                                                    12
     Everybody holds the Market

Everybody holds the tangency portfolio M
 » If all hold the same portfolio, it must be the market!
Nobody can do better than holding the market
 » If another asset existed which offers a better return for the
   same risk, buy that!
        Can’t be an equilibrium
Write the weight of asset j in the market portfolio as wj. Then we
have:
                                )         )
          E rM )   j 1 w j E rj  rf  rf
                      j N



      Var rM )  i 1     j 1 wi w j Covrj , ri )
                    i N      j N



 » Simply use expressions for multi-asset case

                                                                     13
All Risk-Return Tradeoffs are Equal
Hence, if you increase the weight of asset j in your portfolio
(relative to the market),
 » Then expected returns increase by:
   E  rj )  rf
 » Then the riskiness of the portfolio increases by:

    i 1 wi Covrj , ri )  Covrj , rM )
       N



 » Hence, the return/risk gain is:
     E  rj )  rf
    Cov rj , rM )

 » This must be the same for all assets
    – Why?
                                                                 14
              All Assets are Equal

Suppose that for two assets A and B:

    E rA )  rf             E rB )  rf
                     
    CovrA , rM )        CovrB , rM )

» Asset A offers a better return/risk ratio than asset B
    – Buy A, sell B
    – What if everybody does this?
» Hence, in equilibrium, all return/risk ratios must be equal for
  all assets
      E rA )  rf            E rB )  rf
                         
     CovrA , rM )           CovrB , rM )



                                                                    15
 The Capital Asset Pricing Model
If the risk-return tradeoff is the same for all assets, than it is the
one of the market:
      E rA )  rf       E rB )  rf           E rM )  rf
                                           
     CovrA , rM )       CovrB , rM )           Var rM )

This gives the relationship between risk and expected return for
individual stocks and portfolios.
 » This is called the Security Market Line.
                      Cov rA , rM )
      E  rA )  rf 
                       Var  rM )
                                                        )     
                                     E  rM )  rf  rf   A E  rM )  rf   )

                            Cov rA , rM )
                     A 
     where                   Var  rM )
                                                                                  16
            Capital Asset Pricing Model
                     A Graphical Illustration


Expected
 Return


Expected
 Market
 Return
                                               Expected
                                               market risk
                                               premium

Risk free
   rate



            0              0.5          1.0                               Beta
      Expected        Risk free        Beta                  Expected market
                 =                +               x
       return            rate         factor                  risk premium
                                                                                 17
     The Intuitive Argument For the
                 CAPM

  Everybody holds the same portfolio, hence the market.
  Portfolio-risk cannot be diversified.
  Investors demand a premium on non-diversifiable risk only,
  hence portfolio or market risk.
  Beta measures the market risk, hence it is the correct measure
  for non-diversifiable risk.

Conclusion:
  In a market where investors can diversify by holding many
  assets in their portfolio, they demand a risk premium
  proportional to beta.


                                                                   18
   The SML and mispriced stocks
Suppose for a particular stock:
                        ) Er )  r
               Cov rj , rM
   )
E rj  r f 
                VarrM )
                           M f
Remember the definition of expected returns:

                      )
         E Pj1  D1  Pj0
  )
E rj 
                  j

                 Pj0


Then P0 falls, so that E(rj) increases until disequilibrium
vanishes and the equation holds!



                                                              19
           The SML and mispriced stocks
        Expected
        Return                 Stock j is overvalued at X:
                                » price drops,
E(rM)                           » expected return rises.
                    Y          At Y, stock j would be
                               undervalued!
                                » expected return falls
                    X           » price increases

  rf




                   j   =1
                                                         20
                     The CML and SML


  E(r)                      CML       E(r)

                                                          SML

                M                                   M
E(rM)
E(rIBM)
                       IBM
    rf                                  rf




    sIBM,M/sM   sM   sIBM         s          IBM   1.0    



                                                                21
    The Capital Asset Pricing Model
The appropriate measure of risk for an individual stock is its beta.
 Beta measures the stock’s sensitivity to market risk factors.
 » The higher the beta, the more sensitive the stock is to market
   movements.
The average stock has a beta of 1.0.
Portfolio betas are weighted averages of the betas for the individual
stocks in the portfolio.
The market price of risk is [E(rM)-rf].




                                                                        22
Using Regression Analysis
    to Measure Betas
        Rate of Return
        on Stock A                               Slope = Beta

                                     x       x
                                                     x
                         x                                 x
    x                                    x       x
                x
                                 x                       Rate of Return
            x       x
                                                         on the Market
x                            x
        x
                              Jan 1995



                                                                          23
                 Calculating the beta of BA
Return on B A

                                                                                                  40




                                                                                                  30




                                                                                                  20




                                                                                                  10




                                                                                                  0




                                                                                                  -10




                                                                                                  -20

                                               B eta
                                                                                                  -30




                                                                                                  -40
       -30              -20              -10                0         10                     20



       Beta is the slope of a regression line which best fits   Return on the market index
       the scatter of monthly returns on the share and on
       the market index.
                                                                                                        24
                  Betas of Selected
                  Common Stocks
        Stock           Beta             Stock            Beta
AT&T                    0.96     Ford Motor               1.03
Boston Ed.              0.49     Home Depot               1.34
BM Squibb               0.92     McDonalds                1.06
Delta Airlines          1.31     Microsoft                1.20
Digital Equip.          1.23     Nynex                    0.77
Dow Chem.               1.05     Polaroid                 0.96
Exxon                   0.46     Tandem                   1.73
Merck                   1.11     UAL                      1.84
Betas based on 5 years of monthly returns through mid-1993.


                                                                 25
 Beta and Standard Deviation

  Risk of a               Market risk                   Specific risk
                   =                             +
Share (Variance)          of the share                   of the share



                        Beta of        Risk of           This is the major
                         share     x   market         element of a share's risk




  Risk of a             Market risk of                Specific risk of
  portfolio        =     the portfolio           +      the portfolio



                        Beta of        Risk of            This is negligible
                       Portfolio   x   market        for a diversified portfolio



                                                                                   26
          Testing the CAPM
          Black, Jensen and Scholes




Average
Monthly
Return
                             Theoretical
                                Line            •

                             •
                                    •      Fitted Line
                     •
                         •
           •
               •
      •

•

                                                         Beta



                                                                27
    Estimating the Expected Rate of
           Return on Equity
The SML gives us a way to estimate the expected (or required) rate of
return on equity.

      )            
   E rj  r f   j E rM )  r f   
We need estimates of three things:
» Riskfree interest rate, rf.
» Market price of risk, [E(rM)-rf].
» Beta for the stock,j.




                                                                        28
      Estimating the Expected Rate of
             Return on Equity
The riskfree rate can be estimated by the current yield on one-year
Treasury bills.
 » As of early 1997, one-year Treasury bills were yielding about 5.0%.
The market price of risk can be estimated by looking at the historical
difference between the return on stocks and the return on Treasury
bills.
 » This difference has averaged about 8.6% since 1926.
The betas are estimated by regression analysis.




                                                                         29
Estimating the Expected Rate of
       Return on Equity

          E(r) = 5.0% + (8.6%)

     Stock        E(r)        Stock    E(r)
AT&T             13.3%   Ford Motor   13.9%
Boston Ed.        9.2%   Home Depot   16.5%
BM Squibb        12.9%   McDonalds    14.1%
Delta Airlines   16.3%   Microsoft    15.3%
Digital Equip.   15.6%   Nynex        11.6%
Dow Chem.        14.0%   Polaroid     13.3%
Exxon             9.0%   Tandem       19.9%
Merck            14.5%   UAL          20.8%
                                              30
       Example of Portfolio Betas and
            Expected Returns
What is the beta and expected rate of return of an equally-weighted
portfolio consisting of Exxon and Polaroid?
Portfolio Beta

 p  (1 / 2)(.46)  (1 / 2)(.96)
 p  0.71

Expected Rate of Return
 E (rp )  5.0%  (8.6%)(0.71)  111%
                                   .

How would you construct a portfolio with the same beta and expected
return, but with the lowest possible standard deviation?
Use the figure on the following page to locate the equally-weighted
portfolio of Exxon and Polaroid. Also locate the minimum variance
portfolio with the same expected return.
                                                                      31
            Graphical Illustration

 E(r)                       E(r)
                  CML                           SML



        M                                 M
13.6%

11.1%



5.0%                        5.0%




        sM              s          0.71   1.0        32
                       Example
The S&P500 Index has a standard deviation of about 12%
per year.
Gold mining stocks have a standard deviation of about 24%
per year and a correlation with the S&P500 of about r = 0.15.
If the yield on U.S. Treasury bills is 6% and the market risk
premium is [E(rM)-rf] = 7.0%, what is the expected rate of
return on gold mining stocks?




                                                                33
                                Example
The beta for gold mining stocks is calculated as follows:
    s gM r gM s g s M .15(.24)
 2                          0.30
    sM        sM2
                         .12
The expected rate of return on gold mining stocks is:
 E(rg )  6.0%  ( 7.0%)(0.30)  7.1%

Question: What portfolio has the same expected return as gold mining
stocks, but the lowest possible standard deviation?
Answer: A portfolio consisting of 70% invested in U.S. Treasury bills
and 30% invested in the S&P500 Index.
  Beta  (.7)( 0)  (.3)(1.0)  0.30
  E ( rp )  6.0%  ( 7.0%)(0.30)  8.1%
 Sd ( rp )  (.7)( 0)  (.3)(12.0%)  3.6%

                                                                        34
             Using the CAPM for
              Project Evaluation

Suppose Microsoft is considering an expansion of its current
operations.
 » The expansion will cost $100 million today
 » expected to generate a net cash flow of $25 million per year
   for the next 20 years.
 » What is the appropriate risk-adjusted discount rate for the
   expansion project?
 » What is the NPV of Microsoft’s investment project?




                                                                  35
      Microsoft’s Expansion Project
  The risk-adjusted discount rate for the project, rp, can be
  estimated by using Microsoft’s beta and the CAPM.

                
  rP  r f   E rm   r f   )

  Thus, the NPV of the project is:

  rP  0.05  1.2 *  0.086)

                 $25
NPV  t 1
          20
                         $100  $53.92 million
               1153)
                 .    t




                                                                36
             Company Risk Versus
                Project Risk
The company-wide discount rate is the appropriate discount rate for
evaluating investment projects that have the same risk as the firm as
a whole.
For investment projects that have different risk from the firm’s
existing assets, the company-wide discount rate is not the appropriate
discount rate.
In these cases, we must rely on industry betas for estimates of
project risk.




                                                                         37
              Company Risk versus
                 Project Risk

Suppose Microsoft is considering investing in the development of a
new airline.
 » What is the risk of this investment?
 » What is the appropriate risk-adjusted discount rate for evaluating
   the project?
 » Suppose the project offers a 17% rate of return. Is the investment
   a good one for Microsoft?




                                                                        38
               Industry Asset Betas

       Industry         Beta           Industry        Beta
Airlines                1.80 Agriculture               1.00
Electronics             1.60 Food                      1.00
Consumer Durables       1.45 Liquor                    0.90
Producer Goods          1.30 Banks                     0.85
Chemicals               1.25 International Oils        0.85
Shipping                1.20 Tobacco                   0.80
Steel                   1.05 Telephone Utilities       0.75
Containers              1.05 Energy Utilities          0.60
Nonferrous Metals       1.00 Gold                      0.35
Source: D. Mullins, “Does the Capital Asset Pricing Model
Work?,” Havard Business Review, vol. 60, pp. 105-114.
                                                              39
             Company Risk versus
                Project Risk
The project risk is closer to the risk of other airlines than it is to the risk
of Microsoft’s software business.
The appropriate risk-adjusted discount rate for the project depends
upon the risk of the project. If the average asset beta for airlines is
1.8, then the project’s cost of capital is:

rp  rf   p  E rm   rf )

 rp  0.05  180.086)  20.5%
              .




                                                                              40
                   Company Risk versus
                      Project Risk
     Required
     Return                                           SML


Project-specific
Discount Rate

Project IRR                                  A

Company-wide
Discount Rate




                        Company Beta   Project Beta            41
          Project Evaluation: Rules
The risk of an investment project is given by the project’s beta.
 » Can be different from company’s beta
 » Can often use industry as approximation
The Security Market Line provides an estimate of an appropriate
discount rate for the project based upon the project’s beta.
 » Same company may use different discount rates for different
    projects
This discount rate is used when computing the project’s net present
value.




                                                                      42
                       Summary
Optimal investments depend on trading off risk and return
  » Investors with higher risk tolerance invest more in risky
     assets
  » Only risk that can’t be diversified counts
If investors can borrow and lend, then everybody holds a
combination of two portfolios
  » The market portfolio of all risky assets
  » The riskless asset
      – Covariance with the market portfolio counts
In equilibrium, all stocks must lie on the security market line
  » Beta measures the amount of nondiversifiable risk
  » Expected returns reflect only market risk
  » Use these as required returns in project evaluation
                                                                  43

				
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