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VII. Choices Among Risky Portfolios

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Choices Among Risky Portfolios

1. Utility Analysis

2. Safety First

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Utility Analysis

Choice among risky portfolios depends on risk returntrade off

More formally depends on maximizing value to me or utility of outcomes

Utility functions are a mathematical way of determining the value of different choices to the investor

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Properties we believe utility functions for most individual should have

1. Prefer more to less – non satiation

2. Require compensation for taking risk

Risk Aversion

Additional Qualities to Consider

3. More of less dollars at risk as wealthier

4. Larger or smaller percentage of wealth at risk as wealthier

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1). Non Satiation

( 1) ( )

( 1) - ( ) 0

( ) 0 OR ' 0

U w U w

U w U w

dU wU

dw

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2) RiskAversion

(1)

2U(1)>U(2)+U(0)

(1) - (0) (2) - (1)

" 0

1 1(2) (0)

2 2

U U

U

U U U

U U

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8

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3) What happen to willingness to take a bet (put a sum of money at risk) as wealth changes.

Investor Absolute risk Aversion Measured by

- ''( )( )

'( )

U WA W

U W

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What happens to willingness to risk a fraction of money as wealth changes

"( )( ) ( )

'( )

WU WR W WA W

U W

Relative Risk Aversion

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What do we know – most individuals exhibit

1. Non Satiation

2. Risk Aversion

3. Decreasing Absolute Risk Aversion

4. Either Constant or decreasing relative risk aversion

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-1

2

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1

2

Investing the properties of a utility fuction - one

of most populat 1n W

U(W)=1nW

U'(W)=W

"( )

"( ) ( )( )

'

'( ) Decresing absolute

( ) - ( ) 1

'( ) 0

U W W

U W WA W W

U W W

A W W

R W WA W

R W

Constant Relative

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Other Portfolios Selection Criteria

1. Safety First

2. Maximize Geometric Mean

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Safety First

Investors wont go through complex UtilityCalculations but need a simpler way to select a

portfolio.

Investors thinks in terms of bad outcomes.

Criteria

1. Telser

2. Kataoka

3. Roy

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Roy’s Criteria

Minimize Prob Rp < RL

Minimize the probability of a return lowerthan some limit – e.g minimize the Probability of the return below zero

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Tesler Criteria

Maximize expected return subject to theProbability of a lower limit is no greater than some number

e.g., Maximize expected return given that the chance of having a negative return is no greater than 10%

e.g., Maximize expected return given that the chance of not earning the actuarial rate is no greater then 5%.

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P

P

Assume .05 from the above distribution about

5% of observations fall more than 1.65 standard

deviations below the mean.

So the con

Tesler Criteria

Max R

Subject to

Prob (R )L

a

R a

P

straint is met as long as

R 1.65PLR

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Maximize the Geometric Mean Return

1. Has the highest expected value of terminal

wealth

2. Has the highest probability of exceeding given wealth level

What is geometric mean

(

1 2 3

1/3

1 2 3)

1/

1

11.0

3

1

(1 ) (1 ) (1 )

1(1 ) 1.0

(1 )(1 )(1 1.0

(1 ) 1.0

A

A t

G

T

G tt

t

R R R R

R RT

R R R R

R R

T

T

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Maximizing the geometric Mean

1. In general will not maximize expected utility

2. May select a portfolio not on the efficient frontier

3. If the portfolio is on the efficient frontier it involves a particular risk – return trade off

If returns are log normally distributed or utilityFunctions are log normal

U(W) = ln (W)

Then we can show that the portfolio which hasMaximum geometric mean return lies on the Efficient frontier.