Risk Return and Portfolio Theory by MikeJenny

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									Money, Banking & Finance
        Lecture 3
 Risk, Return and Portfolio Theory
                  Aims
• Explain the principles of portfolio diversification
• Demonstrate the construction of the efficient
  frontier
• Show the trade-off between risk and return
• Derive the Capital Market Line (CML)
• Show the calculation of the optimal portfolio
  choice based on the mean and variance of
  portfolio returns.
            Overview
• Investors choose a set of risky assets
  (stocks) plus a risk-free asset.
• The risk-free asset is a term deposit or
  government Treasury bill.
• Investors can borrow or lend as much as
  they like at the risk-free rate of interest.
• Investors like return but dislike risk (risk
  averse).
Preferences of Expected return
           and risk
• We have seen how expected return is defined in Lecture 2.
• The investor faces a number of stocks with different
  expected returns and differ from each other in terms of
  risk.
• The expected return on the portfolio is the weighted mean
  return of all stocks. First moment.
• Risk is measured in terms of the variance of returns or
  standard deviation. Second moment.
• Investor preferences are in terms of the first and second
  moments of the distribution of returns.
Investor Utility function
 U  U E ( R p ), p 
    U                  U
              U1  0;         
                              U2  0
  E ( R p )             p
            dU            dU
  dU  U1              
                      U2      0
           dE ( R p )     d p
  dU    dU          dE ( R p )    U1
      /                            0
  d p dE ( R p )    d p          
                                  U2
Preference Function

   E(Rp) Expected return   U2




                           U0




                            σp Risk
Expected return

        n
R p   i Ri
       i 1

R p  1R1  2 R2
E R p   1E ( R1 )  2 E ( R2 )
                                  Risk

         E R
           p  E(Rp     )  E 1  R1  E ( R1 )   2 R 2  E ( R2 ) 2
  2                       2
  p

 12 12  2  2  212 E R1  E ( R1 ) R2  E ( R2 ) 
                2 2


E R1  E ( R1 ) R2  E ( R2 )   CovR1 , R 2 
           CovR1 , R2 
 
  1, 2
             Var( R1 )Var( R2 )
 p  12 12  2  2  212 1, 2 1 2
  2              2 2
          Return and risk
• How do return and risk vary relative to each other
  as the investor alters the proportion of each of the
  assets in the portfolio?
• Assume that returns, risk and the covariance are
  fixed and simply vary the weights in the portfolio.
• Let E(R1)=8.75% and E(R2)=21.25
• Let w1=0.75 and w2=0.25
• E(Rp)=.75x8.75+.25x21.25=11.88
• σ1=10.83, σ2=19.80, ρ1,2=-.9549
         Portfolio Risk
• σ2p=(0.75)2x(10.83)2+(0.25)2x(19.80)2+2x(0
  .75)x(0.25)x(-0.95)x(10.83)x(19.80)
• =13.7
• σp=√13.7=3.7
• Calculate risk and return for different
  weights
    Portfolio risk and return
        Equity 1 Equity 2 E(Rp)    Risk
State   w1       w2
1       1        0        8.75%    10.83%
2       0.75     0.25     11.88%   3.70%
3       0.5      0.5      15%      5%
4       0        1        21.25    19.8%
Locus of risk-return points

      Expected
      return                      (0,1)

    (.5,.5)


              (.75,.25)




                          (1,0)           Risk=standard
                                          deviation
       Risk – return locus
• Can see that the locus of risk and returns vary
  according to the proportions of the equity held in
  the portfolio.
• The proportion (0.75,0.25) is the lowest risk point
  with highest return.
• The other points are either higher risk and higher
  return or low return and high risk.
• The locus of points vary with the correlation
  coefficient and is called the efficient frontier
         Choice of weights
• How does the portfolio manager choose the weights?
• That will depend on preferences of the investor.
• What happens if the number of assets grows to a large
  number.
• If n is the number of assets then will need n(n-1)/2
  covariances - becomes intractable
• A short-cut is the Single Index Model (SIM) where each
  asset return is assumed to vary only with the return of the
  whole market (FTSE100, DJ, etc).
• For ‘n’ assets the efficient frontier defines a ‘bundle’ of
  risky assets.
      ‘n’ asset case


E R p    i E Ri 
              n


             i 1
         n    n
   i j i j  ij
  2
  p
        i 1 j 1
  How is the efficient frontier
           derived?
• The shape of the efficient frontier will depend on
  the correlation between the asset returns of the two
  assets.
• If the correlation is ρ = +1 then the portfolio risk is
  the weighted average of the risk of the portfolio
  components.
• If the correlation is ρ = -1 then the portfolio risk
  can be diversified away to zero
• When ρ < +1 then not all the total risk of each
  investment is non-diversifiable. Some of it can be
  diversified away
          Correlation of +1

 p   2 12  (1   ) 2  2  2 1, 2 (1   ) 1 2 
                              2


1, 2  1
 p   2 12  (1   ) 2  2  2 (1   ) 1 2 
                              2



     ( 1  (1   ) 2 ) 2
                                     1  (1   ) 2
       Correlation of -1
 p   2 12  (1   ) 2  2  2 (1   ) 1 2 
                              2



p      1  (1   ) 2 2


min risk
 p   1  (1   ) 2  0
  1   2   2  0
     2

   1   2
           Check

     2              2 
p 
       1  1 
                         2
     1  2           1   2 

       1 2    1 2
p                   0
     1   2 1   2
     Correlation < +1



 p   1  (1   ) 2
        Efficient frontier
E(Rp)




         Ρ = -1



                  -1 < Ρ < +1   Ρ = +1




                                         σp
The general case – applied to
        two assets
  p   2 12  (1   ) 2  2  2 1, 2 (1   ) 1 2 
                               2


  p   2 12  (1   ) 2  2  2 1, 2 (1   ) 1 2 
   2                           2


  d p
     2

          2 12  2(1   ) 2  2 1, 2 1 2  4 1, 2 1 2  0
                               2

 d
   12  (1   ) 2  1, 2 1 2  2 1, 2 1 2  0
                      2


   ( 12   2 )   2  2 1, 2 1 2   1, 2 1 2
                2       2


   ( 12   2  2 1, 2 1 2 )   2  1, 2 1 2
                2                        2


         2 ( 2  1, 2 1 )
  2
   [ 1   2  2 1, 2 1 2 ]
             2
  Efficient Frontier

E(Rp)




          X


         Y


                       σp
          Risk-free asset
• Lets introduce a risk-free asset that pays a
  rate of interest Rf.
• The rate Rf is known with certainty and has
  zero variance and therefore no covariance
  with the portfolio.
• Such a rate could be a short-term
  government bill or commercial bank
  deposit.
  One bundle of risky assets
• Take one bundle of risky assets and allow the investor to
  lend or borrow at the safe rate of interest. The investor can;
• Invest all his wealth in the risky bundle and undertake no
  lending or borrowing.
• Invest less than his total wealth in the single risky bundle
  and the rest in the risk-free asset.
• Invest more than his total wealth in the risky bundle by
  borrowing at the risk-free rate and hold a levered portfolio.
• These choices are shown by the transformation line that
  relates the return on the portfolio with one risk-free asset
  and risk.
        Transformation line

E R p   R f  (1   ) RN
 p   2 2  (1   ) 2  N   (1   ) f  N  fn
  2
           f
                            2


  (1   ) 
  2
  p
               2    2
                    N

 p  (1   ) N
    Linear Opportunity set
• Let the risk-free rate Rf = 10% and the return on the
  bundle of assets RN = 22.5%.
• The standard deviation of the returns on the bundle
  σN = 24.87%.
• The weights on the risky bundle and the risk-free
  asset can be varied to produce a range of new
  portfolio returns.
    Portfolio Risk and Return
State    T-bill   Equity   E(Rp)    σp
         (1-φ)    φ
1        1        0        10%      0%
2        0.5      0.5      16.25%   12.44%
3        0        1        22.5%    24.87%
4        -0.5     1.5      28.75%   37.31%
       Transformation line
• The transformation line describes the linear risk-
  return relationship for any portfolio consisting of a
  combination of investment in one safe asset and
  one ‘bundle’ of risky assets.
• At every point on a given transformation line the
  investor holds the risky assets in the same fixed
  proportions of the risky portfolio ωi.
     Transformation line

     E(Rp)

                                              -0.5 borrowing + 1.5
         0.5 lending + 0.5                    in risky bundle
         in risky bundle
                             No lending all
                             investment in
                             bundle



Rf
         All lending
                                                   σp
   A riskless asset and a risky
            portfolio
• An investor faces many bundles of risky assets (eg
  from the London Stock Exchange).
• The efficient frontier defines the boundary of
  efficient portfolios.
• The single risky asset is replaced by a risky
  portfolio.
• We can find a dominant portfolio with the riskless
  asset that will be superior to all other
  combinations.
Combining risk-free and risky
        portfolios
     E(Rp)



                     C
                 B
             A
Rf



                          σp
   Borrowing and Lending
• The investor can lend or borrow at the risk-
  free rate of interest rate.
• The risk-free rate of interest Rf represents
  the rate on Treasury Bills or some other
  risk-free asset.
• The efficiency boundary is redefined to
  include borrowing.
Borrowing and lending frontier
                         C
     E(Rp)



                 B

Rf


             A
                             σp
   Combined borrowing and
  lending at different rates of
            interest
• The investor can borrow at the rate of
  interest Rb
• Lend at the rate of interest Rf
• The borrowing rate is greater than the risk-
  free rate. Rb > Rf
• Preferences determine the proportions of
  lending or borrowing,
     Combining borrowing and
             lending
                                  D
      E(Rp)
                              Q

                          C
                      B
Rb
              P

 Rf               A

                                      σp
      Separation Principle
• Investor makes 2 separate decisions
• Given knowledge of expected returns, variances
  and covariances the investor determines the
  efficient frontier. The point M is located with
  reference to Rf.
• The investor determines the combination of the
  risky portfolio and the safe asset (lending) or a
  leveraged portfolio (borrowing).
                Market portfolio and risk
                       reduction
    Portfolio
    risk




Diversifiable
risk




Non-                                        Number of
diversifiable                               securities
risk
                           20
            Summary
• We have examine the theory of portfolio
  diversification
• We have seen how the efficient frontier is
  constructed.
• We have seen that portfolio diversification
  reduces risk to the non-diversifiable
  component.

								
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