Chapter Thirteen

Chapter Thirteen

Risky Assets

Mean of a Distribution

A random variable (r.v.) w takes values w1,…,wS with probabilities 1,...,S (1 + · · · + S = 1).

The mean (expected value) of the distribution is the av. value of the r.v.;

E[ ] .w ww s ss

S

1

Variance of a Distribution The distribution’s variance is the

r.v.’s av. squared deviation from the mean;

Variance measures the r.v.’s variation.

var[ ] ( ) .w ww s w ss

S

2 2

1

Standard Deviation of a Distribution

The distribution’s standard deviation is the square root of its variance;

St. deviation also measures the r.v.’s variability.

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

S

2 2

1

Mean and Variance

Probability

Random Variable Values

Two distributions with the samevariance and different means.

Mean and Variance

Probability

Random Variable Values

Two distributions with the samemean and different variances.

Preferences over Risky Assets

Higher mean return is preferred. Less variation in return is preferred

(less risk).


Higher mean return is preferred. Less variation in return is preferred

(less risk). Preferences are represented by a

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


Preferred Higher mean return is a good.Higher risk is a bad.

Mean Return,

St. Dev. of Return,



Mean Return,

St. Dev. of Return,


How is the MRS computed?


How is the MRS computed?

dU U d U d

U d U d

dd

UU

0

//

.


Mean Return,

St. Dev. of Return,


dd

UU

//

Budget Constraints for Risky Assets

Two assets. Risk-free asset’s rate-or-return is rf . Risky stock’s rate-or-return is ms if

state s occurs, with prob. s . Risky stock’s mean rate-of-return is

r mm s ss

S

.

1


A bundle containing some of the risky stock and some of the risk-free asset is a portfolio.

x is the fraction of wealth used to buy the risky stock.

Given x, the portfolio’s av. rate-of-return is r xr x rx m f ( ) .1


r xr x rx m f ( ) .1x = 0 r rx f and x = 1 r rx m .


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


r xr x rx m f ( ) .1x = 0 r rx f and x = 1 r rx m .Since stock is risky and risk is a bad, for stockto be purchased must have r rm f .So portfolio’s expected rate-of-return rises with x(more stock in the portfolio).


Portfolio’s rate-of-return variance is x s f x s

s

Sxm x r r2 2

11

( ( ) ) .



s

Sxm x r r2 2

11

( ( ) ) .

r xr x rx m f ( ) .1



s

Sxm x r r2 2

11

( ( ) ) .


x s f m f ss

Sxm x r xr x r2 2

11 1

( ( ) ( ) )



s

Sxm x r r2 2

11

( ( ) ) .


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

( ( ) ( ) )

( )



s

Sxm x r r2 2

11

( ( ) ) .


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

( ( ) ( ) )

( ) ( )



s

Sxm x r r2 2

11

( ( ) ) .


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

( ( ) ( ) )

( ) ( ) .


x mx2 2 2Variance

x mx .so st. deviation


x mx2 2 2

x = 0 and x = 1 x 0 x m .

Variance



x mx2 2 2

x = 0 and x = 1 x 0 x m .

Variance


So risk rises with x (more stock in the portfolio).


Mean Return,

St. Dev. of Return,


r xr x rx m f ( ) .1 x mx .

x r rx f x 0 0,

0

rf

Mean Return,

St. Dev. of Return,



x r rx f x 0 0,

m0

rmx r rx m x m 1 ,

rf

Mean Return,

St. Dev. of Return,



x r rx f x 0 0,

m0

rmx r rx m x m 1 ,

Budget line

rf

Mean Return,

St. Dev. of Return,



x r rx f x 0 0,

m0

rmx r rx m x m 1 ,

Budget line, slope =

rf

r rm f

m

Mean Return,

St. Dev. of Return,

Choosing a Portfolio

m0

rmBudget line, slope =

rf

r rm f

m

Mean Return,

St. Dev. of Return,

is the price of risk relative tomean return.


m0


rf

r rm f

m

Where is the most preferredreturn/risk combination?

Mean Return,

St. Dev. of Return,


m0


rf

r rm f

m


Mean Return,

St. Dev. of Return,


m0


rf

r rm f

m


rx

x

Mean Return,

St. Dev. of Return,


m0


rf

r rMRSm f

m


rx

x

Mean Return,

St. Dev. of Return,


m0


rf

r r UU

m f

m

//


rx

x

Mean Return,

St. Dev. of Return,


Suppose a new risky asset appears, with a mean rate-of-return ry > rm and a st. dev. y > m.

Which asset is preferred?


Suppose a new risky asset appears, with a mean rate-of-return ry > rm and a st. dev. y > m.

Which asset is preferred? Suppose r r r ry f

y

m f

m

.


m0

rm

rf

rx

x

Budget line, slope = r rm f

m

Mean Return,

St. Dev. of Return,


m0


rf

r rm f

m

rx

x

ry

y

Mean Return,

St. Dev. of Return,


m0

rm

rf


m

rx

x

ry

y

Budget line, slope = r ry f

y

Mean Return,

St. Dev. of Return,


m0

rm

rf


m

rx

x

ry

y

Budget line, slope = r ry f

y

Higher mean rate-of-return andhigher risk chosen in this case.

Mean Return,

St. Dev. of Return,

Measuring Risk

Quantitatively, how risky is an asset? Depends upon how the asset’s value

depends upon other assets’ values. E.g. Asset A’s value is $60 with

chance 1/4 and $20 with chance 3/4. Pay at most $30 for asset A.

Measuring Risk

Asset A’s value is $60 with chance 1/4 and $20 with chance 3/4.

Asset B’s value is $20 when asset A’s value is $60 and is $60 when asset A’s value is $20 (perfect negative correlation of values).

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

Measuring Risk

Asset A’s risk relative to risk in the whole stock market is measured by

Arisk of asset A

risk of whole market .

Measuring Risk

Asset A’s risk relative to risk in the whole stock market is measured by

Arisk of asset A

risk of whole market .

AAcovariance(

variance(

r rr

mm

, ))

where is the market’s rate-of-returnand is asset A’s rate-of-return.rA

rm

Measuring Risk

asset A’s return is not perfectly correlated with the whole market’s return and so it can be used to build a lower risk portfolio.

1 1A .

A 1

Equilibrium in Risky Asset Markets

At equilibrium, all assets’ risk-adjusted rates-of-return must be equal.

How do we adjust for riskiness?


Riskiness of asset A relative to total market risk is A.

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


Riskiness of asset A relative to total market risk is A.

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

So cost of asset A’s risk is pAm.

pr rm f

m

.


Risk adjustment for asset A is

Risk adjusted rate-of-return for asset A is

pr r

r rmm f

mm m f

A A A

( ).

r r rm fA A ( ).


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

The risk-free asset’s = 0 so its adjusted rate-of-return is just

Hence,

for every risky asset A.

r r r r

r r r rf m f

f m f

A A

A Ai.e.

( )

( )

rf .


That at equilibrium in asset markets is the main result of the Capital Asset Pricing Model (CAPM), a model used extensively to study financial markets.

r r r rf m fA A ( )

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Chapter Thirteen

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