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Chapter 5

Finding the Efficient Set

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Attainable Portfolios

In the last chapter we identified the risk return relationships between different portfolios. This chapter is designed to determine which portfolio is best

Attainable portfolios (all possible combinations) Fig 5.1 pg 93 entire shaded area & line There are no portfolios that can be created with

risky assets that have a level of risk and return outside the bullet

• (IE. You cannot create a portfolio that has a E(r) of 20% and a standard deviation of 5%)

Minimum variance set - bullet shaped curve (only the line)

Given a particular level of return, has lowest standard deviation possible

There are two important components• 1) MVP

• global minimum variance portfolio• 2) efficient set

• given level of standard deviation, portfolios with highest return

• Top half of the bullet These are the most desirable portfolios

Efficient portfolio (on the efficient set)A) minimize risk for given returnB) maximize return for given risk

Efficient set

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Finding Efficient Set (with Short Selling)

Attainable set w 3 stocks Start with

covariance matrix Expected returns Standard deviation (get from

covariance matrix)

If you look at Fig 5.2 pg 95, shows the 3 stocks in the attainable set The line between points are long

positions in each security

The rest of the shaded area represents combinations of all three

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Portfolio Weights Computer plugs in different weights for each

security Fig 5.3 pg 96 (Weights of portfolios)

Pt R = 100% in B (Brown) Pt T = 100% in A (Acme) Pt S = 100% in C (Consolidated)

Inside triangle = + amounts of each stock PT L

On perimeter = + amounts in 2 & zero in third • PT Q + amount s in B & C nothing invested in A

Outside perimeter = short selling is taking place• Above line XY’ (northeast) -C• West of vertical -A• South of horizontal -B

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Expected return Plane

B) Expected return plane Solve the portfolio return line for the weight of 1 of the assets

Given that we know the returns of the individual assets we can further simplify this equation

Assume that we want a certain return; we place that number in the numerator (Rp) Assume then that we choose to invest 90% in Asset A (XA) The formula will then solve for the amount we must invest in Asset B Since we know the weights for A & B we can solve for C

If we repeat this for another weight of XA; we will have 2 portfolios with equal returns

CCBBAAp R*XR*XR*X)R(E

CBABBAAp R*)XX1(R*XR*X)R(E

CBCACBBAAp R*XR*XRR*XR*X)R(E

ABC

CA

BC

pCB X*

RR

RR

RR

RRX

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Returns Variance Standard deviationAmerican express .04% .0071 .0852Anheuser Busch .54% .0035 .0593Apple computer .26% .0167 .1293

We want a portfolio that Rp = .70

Assume we invest 90 % of our funds in Asset A (Xa = .90)

• Xa = .90 Xb = 2.277 Xc = 1 - .90 - 2.277 = (-2.177)

Assume want to invest 150% of our money in Asset A

• Xa = 1.50 Xb = 2.748 Xc = 1 - 1.50 - 2.748 = (-3.248)

We would now repeat this but for a different return level and we would get another Iso- return line

AAB X*785.571.1X*54.26.

26.04.

54.26.

70.26.X

277.290.*785.571.1XB

748.250.1*785.571.1XB

7 of 59Iso-Return - Variance

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Iso-Return Lines

We would repeat this process for many different returns (thank goodness for computers)

We would then graph the lines and place it over the portfolio weight graph

Shaded areas – long positions in all 3 assets

all points on a line equal returns

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Iso-Variance Elipses

Lets assume we want a portfolio with a variance of .02

First invest 90% of our assets in Asset A

Multiply and rearrange terms

This is a quadratic equation, with some arranging of terms we can solve for the value of XB

quadratic formula has the following form:

In a quadratic formula X has two possible values use the following formula to find them

)X10(.XX1X XX 90.X bbacbba

bccbacca

abba2cc

22b

2b

2a

2a

2p

COV*X*X*2COV*X*X*2

COV*X*X*2*X*X*X

2bb X*0138.X*00352.006169.02.

0cbXaX2

a2

ac4bbX,X

2

21

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When we solve for quadratic formula it will provide 2 points with equal risk on an ellipse.

If we continue to change the weight of A and solve we would create the points of an ISO-variance ellipse

Now we would continue the process but this time change the desired variance

This set of ellipses would then be placed over the portfolio weight graph

Pts are all concentric about MVP (Fig 5.5 pg 100)

MVP is the bottom of a "valley“ Each point on ellipse = risk

11 of 59Portfolio Management / Iso return Variance

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Critical line

Critical line - this line shows the portfolios of the Min variance set (equivalent to the bullet shaped curve)

MVP to Northwest = efficient portfolios Superimpose the iso-return lines and the iso

variance ellipses Find pt on iso-variance ellipse tangent to iso-

return (Highest possible return given the risk ellipse)

Fig 5.7 pg 102

pt Q on border of triangle Invest positive amounts in brown and

consolidated, 0 in acme from pts Q to MVP , positive amounts in all

three to the south east of MVP not efficient

(Can find portfolio with higher return and the same risk)

To west of Q short Acme and positive amounts in other

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Finding Efficient Set (without Short Selling)

for Fig 5.8 pg 107 must be on or inside the triangle

Get a line (SQZ) Minimum variance set

Fig 5.11 pg 111 Note the two bullet curves One superior to the other Superior meaning more

efficient

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Property 1 of minimum variance set

If combine two or more portfolios from minimum variance set get another portfolio on the Minimum variance set

When discuss CAPM important because we assume that all investors hold efficient portfolios

If we combine all of them, we must then also get an efficient portfolio

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Property 2 of the Minimum Variance Set

given the population of sec., there is a linear relationship between beta factors and their expected returns, if and only if we use a minimum variance portfolio as the index portfolio

Market index used for calculation of beta is minimum variance portfolio

Beta measures responsiveness of sec returns to a market portfolio ( all risky assets)

Must get a relationship as in right side of Fig 5.13 pg 113

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Allocation to Risky Assets

Investors will avoid risk unless there is a reward.

The utility model gives the optimal allocation between a risky portfolio and a risk-free asset.

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Risk and Risk Aversion

Speculation Taking considerable risk for a commensurate

gain Parties have heterogeneous expectations

Gamble Bet or wager on an uncertain outcome for

enjoyment Parties assign the same probabilities to the

possible outcomes

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Risk Aversion and Utility Values

Investors are willing to consider: risk-free assets

speculative positions with positive risk premiums

Portfolio attractiveness increases with expected return and decreases with risk.

What happens when return increases with risk?

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Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%)

Each portfolio receives a utility score to assess the investor’s risk/return trade off

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Utility Function

U = utility – measures benefit Investors would like to maximize utility. Utility incorporates risk and return as well as individual

sensitivity to risk Certainty Equivalent rate – rate willing to accept from a

RF rate to buy it instead of the risky asset

E ( r ) = expected return

A = coefficient of risk aversion

= variance of returns

½ = a scaling factor

21( )

2U E r A

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Risky InvestmentStandard Deviation 19.70%Expected Returns 12.75%Risk Aversion 2.75Utility 0.074138

Risky InvestmentStandard Deviation 19.70%Expected Returns 12.75%Risk Aversion 3.5Utility 0.059584

Risky InvestmentStandard Deviation 19.70%Expected Returns 12.75%Risk Aversion 4.5Utility 0.04018

Risk Free AssetStandard Deviation 0.00%Expected Returns 4.00%Risk Aversion 4.5Utility 0.04

Utility

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Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion

BA rErE

BA

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What is Risk Aversion?

Risk aversion measures how sensitive a person is to changing risk characteristics of an asset. They use this sensitivity to establish a difference in preference for an asset. In the utility formula it is the variable (could be negative) that determines the change in value necessary to compensate for the changes.

Risk Averse investors require higher levels of return as risk increases.

• (A > 0) Risk neutral investors pick securities solely by their expected utility

• (A = 0) Risk lovers are willing to engage in gambling

• (A < 0) Research has shown that most investors are between A = 2 & 4

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Portfolio Dominance

What does dominance mean? Mean Variance Criterion

• Portfolio A dominates portfolio B if:

• And

BA rErE

BA

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