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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
 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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