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 Analysis of two or more is very similar This will allow us to delineate the preferred portfolio This will allow us to delineate the preferred portfolio THE EFFICIENT FRONTIER!!!!THE EFFICIENT FRONTIER!!!!

Combination of two risky assets  Expected Return  Investor must be fully invested Therefore weights add to one Therefore weights add to one  Standard deviation Not a simple weighted average Not 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 securities to understand this We next examine co-movement between securities to understand this

Case 1-Perfect Positive Correlation (p=+1)  C=Colonel Motors  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 Due to the square root of a negative number  This std. deviation is always smaller than p=+1 Risk is smaller when p=-1 Risk is smaller when p=-1 It is possible to find two securities with zero risk 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  The point on the Mean Variance Efficient Frontier that has the lowest variance  To find the optimal percentage in each asset, take the derivative of the risk equation with respect to X c  Then set this derivative equal to 0 and solve for X c

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  The closer 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 Only legitimate shape is a concave curve  The Efficient Frontier with No Short Sales All portfolios between global min and max return portfolios All portfolios between global min and max return portfolios  The Efficient Frontier with Short Sales No finite upper bound No finite upper bound

The Efficient Frontier with Riskless Lending and Borrowing  All combinations 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 σ Pred 2 = σ 2 + σ 2 /T where: variance σ Pred 2 = σ 2 + σ 2 /T where: σ Pred 2 is the predicted variance series σ Pred 2 is the predicted variance series σ 2 is the variance of monthly return σ 2 is the variance of monthly return T is the number of time periods T is the number of time periods  σ 2 captures inherent risk  σ 2 /T captures the uncertainty that comes from lack of knowledge about true mean return  In Bayesian analysis, σ 2 + σ 2 /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 = (σ s 2 – σ c σ s ρ cs ) (σ c 2 + σ s 2 - 2σ c σ s ρ cs )  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.