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Chapter 3 Delineating Efficient Portfolios by 4mD4ar

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									     Chapter 3
Delineating Efficient
     Portfolios
       Jordan Eimer
        Danielle Ko
      Raegen Richard
      Jon Greenwald
                       Goal
 Examine attributes of combinations of two
 risky assets
     Analysis of two or more is very similar
     This will allow us to delineate the preferred
      portfolio
       • THE EFFICIENT FRONTIER!!!!
Combination of two risky assets
 Expected  Return
 Investor must be fully invested
     Therefore weights add to one
 Standard     deviation
     Not a simple weighted average
       • Weights do not, in general add to one
       • Cross-product terms are involved
     We next examine co-movement between
      securities to understand this
Case 1-Perfect Positive Correlation
             (p=+1)
 C=ColonelMotors
 S=Separated Edison


 Here, risk and return of the portfolio are
 linear combinations of the risk and return
 of each security
Case2-Perfect Negative Correlation
              (p=-1)
 This   examination yields two straight lines
     Due to the square root of a negative number
     std. deviation is always smaller than
 This
 p=+1
     Risk is smaller when p=-1
     It is possible to find two securities with zero
      risk
No Relationship between Returns
     on the Assets (  = 0)

•The expression for return on the
portfolio remains the same
•The covariance term is eliminated from
the standard deviation
•Resulting in the following equation for
the standard deviation of a 2 asset
portfolio
  Minimum Variance Portfolio
 Thepoint on the Mean Variance Efficient
 Frontier that has the lowest variance

 Tofind the optimal percentage in each
 asset, take the derivative of the risk
 equation with respect to Xc

 Then set this derivative equal to 0 and
 solve for Xc
      Intermediate Risk (  = .5)
   A more practical example

   There may be a combination of assets that
    results in a lower overall variance with a
    higher expected return when 0 <  < 1

   Note: Depending on the correlation between
    the assets, the minimum risk portfolio may
    only contain one asset
 2 Asset Portfolio Conclusions
 Thecloser the correlation between the two
 assets is to -1.0, the greater the
 diversification benefits

 The combination of two assets can never
 have more risk than their individual
 variances
        The Shape of the Portfolio
           Possibilities Curve
   The Minimum Variance Portfolio
       Only legitimate shape is a concave curve


The      Efficient Frontier with No Short Sales
    All portfolios between global min and max return
    portfolios
The      Efficient Frontier with Short Sales
       No finite upper bound
The Efficient Frontier with Riskless
     Lending and Borrowing
 Allcombinations of riskless lending and
  borrowing lie on a straight line
    Input Estimation Uncertainty
 Reliable inputs are crucial to the proper use of
  mean-variance optimization in the asset
  allocation decision
 Assuming stationary expected returns and
  returns uncorrelated through time, increasing N
  improves expected return estimate
 All else equal, given two investments with equal
  return and variance, prefer investment with more
  data (less risky)
    Input Estimation Uncertainty
 Predicted returns with have mean R and
variance σPred2 = σ2 + σ2/T where:
    σPred2 is the predicted variance series
    σ2 is the variance of monthly return
    T is the number of time periods
 σ2 captures inherent risk
 σ2/T captures the uncertainty that comes from lack of
  knowledge about true mean return
 In Bayesian analysis, σ2 + σ2/T is known as the
  predictive distribution of returns
 Uncertainty: predicted variance > historical variance
   Input Estimation Uncertainty
 Characteristics of security returns usually
  change over time.
 There is a tradeoff between using a longer
  time frame and having inaccuracies.
 Most analysts modify their estimates.
 Choice of time period is complicated when
  a relatively new asset class is added to the
  mix.
    Short Horizon Inputs and Long
      Horizon Portfolio Choice
 Important consideration in estimate inputs: Time
  horizon affects variance
 In theory, returns are uncorrelated from one
  period to the next.
 In reality, some securities have highly correlated
  returns over time.
 Treasury bill returns tend to be highly
  autocorrelated – standard deviation is low over
  short intervals but increases on a percentage
  basis as time period increases
                 Example
 Solving  for Xc yields for the minimum
  variance portfolio:
      Xc =     (σs2 – σcσsρcs)
           (σc2 + σs2 - 2σcσsρcs)
 In a portfolio of assets, adding bonds to
  combination of S&P and international
  portfolio does not lead to much
  improvement in the efficient frontier with
  riskless lending and borrowing.

								
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