Chapter 3Chapter 3Delineating Efficient Delineating Efficient

PortfoliosPortfolios

Jordan EimerJordan EimerDanielle KoDanielle Ko

Raegen RichardRaegen RichardJon GreenwaldJon Greenwald

GoalGoal

Examine attributes of combinations of two Examine attributes of combinations of two risky assetsrisky assets Analysis of two or more is very similarAnalysis of two or more is very similar This will allow us to delineate the preferred This will allow us to delineate the preferred

portfolioportfolio• THE EFFICIENT FRONTIER!!!!THE EFFICIENT FRONTIER!!!!

Combination of two risky assetsCombination of two risky assets

Expected ReturnExpected Return Investor must be fully investedInvestor must be fully invested

Therefore weights add to oneTherefore weights add to one Standard deviationStandard deviation

Not a simple weighted averageNot a simple weighted average• Weights do not, in general add to oneWeights do not, in general add to one• Cross-product terms are involvedCross-product terms are involved

We next examine co-movement between We next examine co-movement between securities to understand thissecurities to understand this

Case 1-Perfect Positive CorrelationCase 1-Perfect Positive Correlation(p=+1)(p=+1)

C=Colonel MotorsC=Colonel Motors S=Separated EdisonS=Separated Edison

Here, risk and return of the portfolio are Here, risk and return of the portfolio are linear combinations of the risk and return linear combinations of the risk and return of each securityof each security

Case2-Perfect Negative Correlation Case2-Perfect Negative Correlation (p=-1)(p=-1)

This examination yields two straight linesThis examination yields two straight lines Due to the square root of a negative numberDue to the square root of a negative number

This std. deviation is always smaller than This std. deviation is always smaller than p=+1p=+1 Risk is smaller when p=-1Risk is smaller when p=-1 It is possible to find two securities with zero It is possible to find two securities with zero

riskrisk

No Relationship between Returns No Relationship between Returns on the Assets ( = 0)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 PortfolioMinimum Variance Portfolio

The point on the Mean Variance Efficient The point on the Mean Variance Efficient Frontier that has the lowest varianceFrontier that has the lowest variance

To find the optimal percentage in each To find the optimal percentage in each asset, take the derivative of the risk asset, take the derivative of the risk equation with respect to Xequation with respect to Xcc

Then set this derivative equal to 0 and Then set this derivative equal to 0 and solve for Xsolve for Xcc

Intermediate Risk ( = .5) Intermediate Risk ( = .5)

A more practical exampleA more practical example

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

Note: Depending on the correlation between Note: Depending on the correlation between the assets, the minimum risk portfolio may the assets, the minimum risk portfolio may only contain one assetonly contain one asset

2 Asset Portfolio Conclusions2 Asset Portfolio Conclusions

The closer the correlation between the two The closer the correlation between the two assets is to -1.0, the greater the assets is to -1.0, the greater the diversification benefitsdiversification benefits

The combination of two assets can never The combination of two assets can never have more risk than their individual have more risk than their individual variancesvariances

The Shape of the Portfolio The Shape of the Portfolio Possibilities CurvePossibilities Curve

The Minimum Variance PortfolioThe Minimum Variance PortfolioOnly legitimate shape is a concave curveOnly legitimate shape is a concave curve

The Efficient Frontier with No Short SalesThe Efficient Frontier with No Short SalesAll portfolios between global min and max return All portfolios between global min and max return portfoliosportfolios

The Efficient Frontier with Short SalesThe Efficient Frontier with Short SalesNo finite upper boundNo finite upper bound

The Efficient Frontier with Riskless The Efficient Frontier with Riskless Lending and BorrowingLending and Borrowing

All combinations of riskless lending and All combinations of riskless lending and borrowing lie on a straight lineborrowing lie on a straight line

Input Estimation UncertaintyInput Estimation Uncertainty

Reliable inputs are crucial to the proper use of Reliable inputs are crucial to the proper use of mean-variance optimization in the asset mean-variance optimization in the asset allocation decisionallocation decision

Assuming stationary expected returns and Assuming stationary expected returns and returns uncorrelated through time, increasing N returns uncorrelated through time, increasing N improves expected return estimateimproves expected return estimate

All else equal, given two investments with equal All else equal, given two investments with equal return and variance, prefer investment with more return and variance, prefer investment with more data (less risky)data (less risky)

Input Estimation UncertaintyInput Estimation Uncertainty

Predicted returns with have mean R and Predicted returns with have mean R and

variance variance σσPredPred22 = = σσ22 + + σσ22/T where:/T where:

σσPredPred22 is the predicted variance series is the predicted variance series

σσ22 is the variance of monthly return is the variance of monthly return T is the number of time periodsT is the number of time periods σσ2 2 captures inherent riskcaptures inherent risk σσ22/T captures the uncertainty that comes from lack of /T captures the uncertainty that comes from lack of

knowledge about true mean returnknowledge about true mean return In Bayesian analysis, In Bayesian analysis, σσ22 + + σσ22/T is known as the /T is known as the

predictive distribution of returnspredictive distribution of returns Uncertainty: predicted variance > historical varianceUncertainty: predicted variance > historical variance

Input Estimation UncertaintyInput Estimation Uncertainty

Characteristics of security returns usually Characteristics of security returns usually change over time.change over time.

There is a tradeoff between using a longer There is a tradeoff between using a longer time frame and having inaccuracies.time frame and having inaccuracies.

Most analysts modify their estimates.Most analysts modify their estimates. Choice of time period is complicated when Choice of time period is complicated when

a relatively new asset class is added to the a relatively new asset class is added to the mix. mix.

Short Horizon Inputs and Long Short Horizon Inputs and Long Horizon Portfolio ChoiceHorizon Portfolio Choice

Important consideration in estimate inputs: Time Important consideration in estimate inputs: Time horizon affects variancehorizon affects variance

In theory, returns are uncorrelated from one In theory, returns are uncorrelated from one period to the next.period to the next.

In reality, some securities have highly correlated In reality, some securities have highly correlated returns over time.returns over time.

Treasury bill returns tend to be highly Treasury bill returns tend to be highly autocorrelated – standard deviation is low over autocorrelated – standard deviation is low over short intervals but increases on a percentage short intervals but increases on a percentage basis as time period increasesbasis as time period increases

ExampleExample

Solving for Xc yields for the minimum Solving for Xc yields for the minimum variance portfolio:variance portfolio:

Xc =Xc = ( (σσss22 – – σσccσσssρρcscs) )

((σσcc22 + + σσss

22 - 2 - 2σσccσσssρρcscs)) In a portfolio of assets, adding bonds to In a portfolio of assets, adding bonds to

combination of S&P and international combination of S&P and international portfolio does not lead to much portfolio does not lead to much improvement in the efficient frontier with improvement in the efficient frontier with riskless lending and borrowing.riskless lending and borrowing.