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Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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Page 1: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Hal VarianIntermediate Microeconomics

Chapter Thirteen

Risky Assets

Page 2: Hal Varian Intermediate Microeconomics 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

Page 3: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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

Page 4: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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

Page 5: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Mean and Variance

Probability

Random Variable Values

Two distributions with the samevariance and different means.

Page 6: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Mean and Variance

Probability

Random Variable Values

Two distributions with the samemean and different variances.

Page 7: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Preferences over Risky Assets

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

(less risk).

Page 8: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Preferences over Risky Assets

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 .

Page 9: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Preferences over Risky Assets

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

Mean Return,

St. Dev. of Return,

Page 10: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Preferences over Risky Assets

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

Mean Return,

St. Dev. of Return,

Page 11: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Preferences over Risky Assets

How is the MRS computed?

Page 12: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Preferences over Risky Assets

How is the MRS computed?

dUUd

Ud

Ud

Ud

dd

UU

0

//

.

Page 13: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Preferences over Risky Assets

Mean Return,

St. Dev. of Return,

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

dd

UU

//

Page 14: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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

Page 15: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

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

Page 16: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

r xr x rx m f ( ) .1

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

Page 17: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

r xr x rx m f ( ) .1

x = 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 .

Page 18: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

r xr x rx m f ( ) .1

x = 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).

Page 19: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

Portfolio’s rate-of-return variance is

x s f x ss

Sxm x r r2 2

11

( ( ) ) .

Page 20: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

Portfolio’s rate-of-return variance is

x s f x ss

Sxm x r r2 2

11

( ( ) ) .

r xr x rx m f ( ) .1

Page 21: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

Portfolio’s rate-of-return variance is

x s f x ss

Sxm x r r2 2

11

( ( ) ) .

r xr x rx m f ( ) .1

x s f m f ss

Sxm x r xr x r2 2

11 1

( ( ) ( ) )

Page 22: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

Portfolio’s rate-of-return variance is

x s f x ss

Sxm x r r2 2

11

( ( ) ) .

r xr x rx 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

( ( ) ( ) )

( )

Page 23: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

Portfolio’s rate-of-return variance is

x s f x ss

Sxm x r r2 2

11

( ( ) ) .

r xr x rx 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

( ( ) ( ) )

( ) ( )

Page 24: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

Portfolio’s rate-of-return variance is

x s f x ss

Sxm x r r2 2

11

( ( ) ) .

r xr x rx 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

( ( ) ( ) )

( ) ( ) .

Page 25: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

x mx2 2 2Variance

x mx .so st. deviation

Page 26: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

x mx2 2 2

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

Variance

x mx .so st. deviation

Page 27: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

x mx2 2 2

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

Variance

x mx .so st. deviation

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

Page 28: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

Mean Return,

St. Dev. of Return,

Page 29: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

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

x r rx f x 0 0,

0

rf

Mean Return,

St. Dev. of Return,

Page 30: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

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

x r rx f x 0 0,

m0

rmx r rx m x m 1 ,

rf

Mean Return,

St. Dev. of Return,

Page 31: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

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

x r rx f x 0 0,

m0

rmx r rx m x m 1 ,

Budget line

rf

Mean Return,

St. Dev. of Return,

Page 32: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Budget Constraints for Risky Assets

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

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,

Page 33: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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.

Page 34: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

m0

rmBudget line, slope =

rf

r rm f

m

Where is the most preferredreturn/risk combination?

Mean Return,

St. Dev. of Return,

Page 35: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

m0

rmBudget line, slope =

rf

r rm f

m

Where is the most preferredreturn/risk combination?

Mean Return,

St. Dev. of Return,

Page 36: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

m0

rmBudget line, slope =

rf

r rm f

m

Where is the most preferredreturn/risk combination?

rx

x

Mean Return,

St. Dev. of Return,

Page 37: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

m0

rmBudget line, slope =

rf

r rMRSm f

m

Where is the most preferredreturn/risk combination?

rx

x

Mean Return,

St. Dev. of Return,

Page 38: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

m0

rmBudget line, slope =

rf

r r UU

m f

m

//

Where is the most preferredreturn/risk combination?

rx

x

Mean Return,

St. Dev. of Return,

Page 39: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

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

Which asset is preferred?

Page 40: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

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

.

Page 41: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

m0

rm

rf

rx

x

Budget line, slope = r rm f

m

Mean Return,

St. Dev. of Return,

Page 42: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

m0

rmBudget line, slope =

rf

r rm f

m

rx

x

ry

y

Mean Return,

St. Dev. of Return,

Page 43: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

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,

Page 44: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Choosing a Portfolio

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 andhigher risk chosen in this case.

Mean Return,

St. Dev. of Return,

Page 45: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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.

Page 46: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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.

Page 47: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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 .

Page 48: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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

m

m

, ))

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

rm

Page 49: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

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

Page 50: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Equilibrium in Risky Asset Markets

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

How do we adjust for riskiness?

Page 51: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Equilibrium in Risky Asset Markets

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

Total market risk is m.

So total riskiness of asset A is Am.

Page 52: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Equilibrium in Risky Asset Markets

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

.

Page 53: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Equilibrium in Risky Asset Markets

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 ( ).

Page 54: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Equilibrium in Risky Asset Markets

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 r

f m f

f m f

A A

A Ai.e.

( )

( )

rf .

Page 55: Hal Varian Intermediate Microeconomics Chapter Thirteen Risky Assets

Equilibrium in Risky Asset Markets

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 ( )