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Chapter 3Delineating Efficient

PortfoliosJordan Eimer

Danielle Ko

Raegen RichardJon Greenwald

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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!!!!

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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

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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

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Case2-Perfect Negative Correlation

(p=-1)

This examination yields two straight lines

Due to the square root of a negative number

This std. deviation is always smaller thanp=+1

Risk is smaller when p=-1

It is possible to find two securities with zero

risk

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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 forthe standard deviation of a 2 asset

portfolio

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Minimum Variance Portfolio

The point on the Mean Variance EfficientFrontier that has the lowest variance

To find the optimal percentage in eachasset, take the derivative of the riskequation with respect to Xc

Then set this derivative equal to 0 andsolve for Xc

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Intermediate Risk ( = .5)

A more practical example

There may be a combination of assets that

results in a lower overall variance with ahigher expected return when 0 < < 1

Note: Depending on the correlation betweenthe assets, the minimum risk portfolio may

only contain one asset

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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

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The Efficient Frontier with Riskless

Lending and Borrowing

All combinations of riskless lending and

borrowing lie on a straight line

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Input Estimation Uncertainty

Reliable inputs are crucial to the proper use of

mean-variance optimization in the asset

allocation decision

Assuming stationary expected returns andreturns 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)

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Input Estimation Uncertainty

Predicted returns with have mean R and

variance Pred2= 2+ 2/T where:

Pred2is the predicted variance series

2is the variance of monthly return

T is the number of time periods

2 captures inherent risk

2/T captures the uncertainty that comes from lack ofknowledge about true mean return

In Bayesian analysis, 2+ 2/T is known as thepredictive distribution of returns

Uncertainty: predicted variance > historical variance

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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 whena relatively new asset class is added to the

mix.

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Short Horizon Inputs and Long

Horizon Portfolio Choice

Important consideration in estimate inputs: Timehorizon 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 overshort intervals but increases on a percentagebasis as time period increases

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Example

Solving for Xc yields for the minimumvariance portfolio:

Xc = (s2cscs)

(c2+ s2- 2cscs)

In a portfolio of assets, adding bonds tocombination of S&P and international

portfolio does not lead to muchimprovement in the efficient frontier withriskless lending and borrowing.