Risky Assets

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Hal Varian

Intermediate Microeconomics

Chapter Thirteen

Risky Assets


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Mean of a Distribution

x A random variable (r.v.) wtakesvalues w

1,,w

Swith probabilities

1,..., S ( 1 + + S = 1).x The mean (expected value) of the

distribution is the av. value of the

r.v.; E[ ] .w ww s ss

S= == 1


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Variance of a Distribution

x The distributions variance is the r.v.s av.squared deviation from the mean;

x Variance measures the r.v.s variation.var[ ] ( ) .w ww s w s

s

S= =

= 2 2

1


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Standard Deviation of a

Distributionx The distributions standard deviation

is the square root of its variance;

x St. deviation also measures the r.v.svariability.

st. dev[ ] ( ) .w ww w s w ss

S

= = = =

2 2

1


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Mean and Variance

Probability

Random Variable Values

Two distributions with the samemean and different variances.


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Preferences over Risky Assets

x Higher mean return is preferred.

x Less variation in return is preferred

(less risk).x Preferences are represented by a

utility function U( , ).x U as mean return .x U as risk .


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Preferred Higher mean return is a good.

Higher risk is a bad.

Mean Return,

St. Dev. of Return,


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Mean Return,

St. Dev. of Return,


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x How is the MRS computed?


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x How is the MRS computed?

dUU

dU

d

Ud

Ud

d

d

U

U

= + =

=

=

0

/

/.


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Mean Return,

St. Dev. of Return,



d

d

U

U

=

/

/


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Budget Constraints for Risky

Assets

x Two assets.

x Risk-free assets rate-or-return is rf.

x Risky stocks rate-or-return is ms ifstate s occurs, with prob. s.

x Risky stocks mean rate-of-return is

r mm s ss

S== .1


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Assets

x A bundle containing some of therisky stock and some of the risk-freeasset is a portfolio.

xxis the fraction of wealth used tobuy the risky stock.

x

Givenx

, the portfolios av. rate-of-return is r xr x r x m f = + ( ) .1


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Assets

r xr x r x m f = + ( ) .1

x= 0 r rx f= andx= 1 r rx m= .


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Assets

r xr x r x m f = + ( ) .1

x= 0 r rx f= andx= 1 r rx m= .Since stock is risky and risk is a bad, for stockto be purchased must have r rm f> .


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Assets

r xr x r x m f = + ( ) .1

x= 0 r rx f= andx= 1 r rx m= .Since stock is risky and risk is a bad, for stockto be purchased must have r rm f> .

So portfolios expected rate-of-return rises withx

(more stock in the portfolio).


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Assets

x Portfolios rate-of-return variance is

x s f x s

s

S xm x r r 2 2

1

1= +

=

( ( ) ) .


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Assets


x s f x s

s

S xm x r r 2 2

1

1= +

=

( ( ) ) .

r xr x r x m f = + ( ) .1


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Assets


x s f x s

s

S xm x r r 2 2

1

1= +

=

( ( ) ) .

r xr x r x m f = + ( ) .1

x s f m f ss

S xm x r xr x r 2 2

1

1 1= + =

( ( ) ( ) )


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Assets


x s f x s

s

S xm x r r 2 2

1

1= +

=

( ( ) ) .

r xr x r x m f = + ( ) .1

x s f m f ss

S

s m ss

S

xm x r xr x r

xm xr

2 2

1

2

1

1 1= +

=

=

=

( ( ) ( ) )

( )


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Assets


x s f x s

s

S xm x r r 2 2

1

1= +

=

( ( ) ) .

r xr x r x m f = + ( ) .1

x s f m f ss

S

s m ss

Ss m s

s

S

xm x r xr x r

xm xr x m r

2 2

1

2

1

2 2

1

1 1= +

= =

=

=

=

( ( ) ( ) )

( ) ( )


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Assets


x s f x s

s

S xm x r r 2 2

1

1= +

=

( ( ) ) .

r xr x r x m f = + ( ) .1

x s f m f ss

S

s m ss

Ss m s

s

Sm

xm x r xr x r

xm xr x m r x

2 2

1

2

1

2 2

1

2 2

1 1= +

= = =

=

=

=

( ( ) ( ) )

( ) ( ) .


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Assets

x mx2 2 2

=Variance

x mx= .so st. deviation


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Assets

x mx2 2 2

=

x= 0 andx= 1 x= 0 x m= .

Variance



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Assets

x mx2 2 2

=

x= 0 andx= 1 x= 0 x m= .

Variance


So risk rises withx(more stock in the portfolio).


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AssetsMean Return,

St. Dev. of Return,


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Assets

r xr x r x m f = + ( ) .1 x mx= .

x r r x f x = = =0 0,

0

rf

Mean Return,

St. Dev. of Return,


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Assets

r xr x r x m f = + ( ) .1 x mx= .

x r r x f x = = =0 0,

m0

rm

x r r x m x m= = =1 ,

rf

Mean Return,

St. Dev. of Return,


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Assets

r xr x r x m f = + ( ) .1 x mx= .

x r r x f x = = =0 0,

m0

rm

x r r x m x m= = =1 ,

Budget line

rf

Mean Return,

St. Dev. of Return,


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Assets

r xr x r x m f = + ( ) .1 x mx= .

x r r x f x = = =0 0,

m0

rm

x r r x m x m= = =1 ,

Budget line, slope =

rf

r rm f

m

Mean Return,

St. Dev. of Return,


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Choosing a Portfolio

m0

rmBudget line, slope =

rf

r rm f

m

Mean Return,

St. Dev. of Return,

is the price of riskrelative to

mean return.


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m0


rf

r rm f

m

Where is the most preferredreturn/risk combination?

Mean Return,

St. Dev. of Return,


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m0


rf

r rm f

m


Mean Return,

St. Dev. of Return,


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m0


rf

r rm f

m


rx

x

Mean Return,

St. Dev. of Return,


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m0


rf

r r MRSm f

m

=


rx

x

Mean Return,

St. Dev. of Return,


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m0


rf

r r UU

m f

m

=

/

/


rx

x

Mean Return,

St. Dev. of Return,


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x Suppose a new risky asset appears,with a mean rate-of-return ry > rmand a

st. dev. y > m.x Which asset is preferred?


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x Suppose a new risky asset appears,with a mean rate-of-return ry > rmand a

st. dev. y > m.x Which asset is preferred?

x Supposer r r r y f

y

m f

m

>

.


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m0

rm

rf

rx

x

Budget line, slope =r rm f

m

Mean Return,

St. Dev. of Return,


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m0


rf

r rm f

m

rx

x

ry

y

Mean Return,

St. Dev. of Return,


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m0

rm

rf

Budget line, slope =r rm f

m

rx

x

ry

y

Budget line, slope =r ry f

y

Mean Return,

St. Dev. of Return,


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m0

rm

rf

Budget line, slope = r rm f

m

rx

x

ry

y

Budget line, slope =r ry f

y

Higher mean rate-of-return and

higher risk chosen in this case.

Mean Return,

St. Dev. of Return,


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Measuring Risk

x Quantitatively, how risky is an asset?

x Depends upon how the assets value

depends upon other assets values.x E.g. Asset As value is $60 with

chance 1/4 and $20 with chance 3/4.

x

Pay at most $30 for asset A.


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Measuring Risk

x Asset As value is $60 with chance 1/4and $20 with chance 3/4.

x Asset Bs value is $20 when asset Asvalue is $60 and is $60 when asset Asvalue is $20 (perfect negativecorrelation of values).

x Pay up to $40 > $30 for a 50-50 mix ofassets A and B.


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Measuring Risk

x Asset As risk relative to risk in thewhole stock market is measured by

Arisk of asset A

risk of whole market= .


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Measuring Risk

x Asset As risk relative to risk in thewhole stock market is measured by

A

risk of asset A

risk of whole market= .

AAcovariance(

variance(=

r r

rm

m

, )

)

where is the markets rate-of-returnand is asset As rate-of-return.rA

rm


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Measuring Risk

x asset As return is not

perfectly correlated with the wholemarkets return and so it can be usedto build a lower risk portfolio.

+1 1A .

A < + 1

E ilib i i Ri k A t


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Equilibrium in Risky Asset

Markets

x At equilibrium, all assets risk-adjusted rates-of-return must beequal.

x How do we adjust for riskiness?

E ilib i i Ri k A t


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Markets

x Riskiness of asset A relative to totalmarket risk is

A.

x

Total market risk is m.x So total riskiness of asset A is

A m.

E ilib i i Ri k A t


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Markets

x Riskiness of asset A relative to totalmarket risk is A.

x

Total market risk is m.x So total riskiness of asset A is A m.x Price of risk is

x So cost of asset As risk is p A m.p

r rm f

m

=

.

E ilib i i Ri k A t


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Markets

x Risk adjustment for asset A is

x Risk adjusted rate-of-return for assetA is

pr r

r rmm f

mm m f

A A A=

= ( ).

r r rm fA A ( ).

E ilib i i Ri k A t


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Markets

x At equilibrium, all risk adjusted rates-of-return for all assets are equal.

x The risk-free assets = 0 so itsadjusted rate-of-return is just

x Hence,

for every risky asset A.

r r r r

r r r r

f m f

f m f

=

= +

A A

A Ai.e.

( )

( )

rf.

E ilib i i Ri k A t


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Markets

x Thatat equilibrium in asset markets is themain result of the Capital AssetPricing Model (CAPM), a model usedextensively to study financialmarkets.

r r r r f m fA A= + ( )

Risky Assets

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