empirical financial economics asset pricing and mean variance efficiency

Empirical Financial Economics

Asset pricing and Mean Variance Efficiency

Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors satisfy

Eigenvectors diagonalize covariance matrix

1,,

0,i i i i j

i j

i j

,

0,i

i j

i jor D

i j

1

1 ,G D then G G I GG

Normal Distribution results

Basic result used in univariate tests:

2

22 2

2

( , ) ( ,1)

(1, )

rr N z N

z Noncentral

Multivariate Normal results

Direct extension to multivariate case:

2 1 2 1

1

( , ) ( , )

' ' ( , )m

ii

r MVN G r z MVN G I

z z z r GG r r r Noncentral m

Mean variance facts

1 2

1 11 2 1 2

1 11 2 1 2

1 22 21 11 2 1 2

1. . ( ) (1 )

21

. . . :

,1

xMin x x

s t x E L x x E x x

x

F O C x x

E x a b cE b a bE

ac b ac bx b c

The geometry of mean variance

a

b

a

b

E

2 1a

1 1

2

1/1/

0

bx b

22

2

2a bE cE

ac b

Note: returns are in excess of the risk free rate

fr

Tests of Mean Variance Efficiency

Mean variance efficiency implies CAPM

For Normal with mean and covariance matrix ,

is distributed as noncentral Chi Square with

degrees of freedom and noncentrality

11/

1/

1/m

x bx

x x b x x Ex x

x b

z 1z z

m 1

MacBeth T2 test

Regress excess return on market excess return

Define orthogonal return Market efficiency implies ,

estimate .

; ,f m fy w y r r w r r

z

0Ez ̂

22

212

2

1

1 1 ˆˆVar 1 1ˆ

( )

T

tt

mTw

tt

ww

T TT w w

MacBeth T2 test (continued)

The T2 test statistic is distributed as noncentral Chi Square with m degrees of freedom and noncentrality parameter

The quadratic form is interpreted as the Sharpe ratio of the optimal orthogonal portfolio

This is interpreted as a test of Mean Variance Efficiency

Gibbons Ross and Shanken adjust for unknown

12 1ˆ ˆ ˆ1 mT

12 1ˆ1 mT

1

Gibbons, M, S. Ross and J. Shanken, 1989 A test of the efficiency of a given portfolio

Econometrica 57, 1121-1152


E


fr

2 1

2 1

Multiple period consumption-investment problem

Multiperiod problem:

First order conditions:

Stochastic discount factor interpretation:

0

Max ( )jt t j

j

E U c

,( ) (1 ) ( )jt t i t j t jU c E r U c

, , ,

( )1 (1 ) ,

( )t jj

t i t j t j t jt

U cE r m m

U c

Stochastic discount factor and the asset pricing model

If there is a risk free asset:

which yields the basic pricing relationship

, , , , ,,

11 (1 ) (1 )

(1 )t f t j t j r t j t t j t t jf t j

E r m r E m E mr

1 (1 )

(1 ( )

(1 ) [ ] ( )

(1 ) ( )f f

E r m

E r m

E m E r m

r r E r m

Stochastic discount factor and mean variance efficiency

Consider the regression model

The coefficients are proportional to the negative of minimum variance portfolio weights, so

( ) ( )m E m r

1

(1 ) ( )( )

1( )

1

f f

ff

r r E r r

rr

2

2

(1 ) ( ) (1 ) ( )

(1 )(1 )

f f f MV

MV ff MV i

f MV

r r E r m b r E r r

rb r b

r

MVm a br


a

b

a

b

E

2 1a

1 1

2

1/1/

0

bx b

22

2

2a bE cE

ac b


fr

Hansen Jagannathan Bounds

Risk aversion times standard deviation of consumption is given by:

“Equity premium puzzle”: Sharpe ratio of market implies a risk aversion coefficient of about 50

Consider

2(1 ) MV fm

f MV MVm MV

rr b

[(1 ) ] 1m

MV f fm m

m m MV

m a br m E r m

r r r r

Non negative discount factors

Negative discount rates possible when market returns are high

Consider a positive discount rate constraint:

1, 2 0

2MV MV MV MVr m

,

( )

(1 ) ( )

(1 ) ( ) (1 ) ( )( )

(1 )

M

f f

f f M

f M c c

m a b r c

r r E r m

r a E r b r E r r c

b r LPM

Stochastic discount factor and the asset pricing model

If there is a risk free asset:

which yields the basic pricing relationship

, , , , ,,

11 (1 ) (1 )

(1 )t f t j t j r t j t t j t t jf t j

E r m r E m E mr

1 (1 )

(1 ( )

(1 ) [ ] ( )

(1 ) ( )f f

E r m

E r m

E m E r m

r r E r m

Where does m come from?

Stein’s lemmaIf the vector ft+1 and rt+1 are jointly

Normal

Taylor series expansionLinear term: CAPM, higher order

terms? Put option payoff

11 1

( )( )

( )t

t tt

u cm g f

u c

1 1 1 1 1 1 1

1 1

[( ) ( )] [ ( )] [( ) ]

. . ( [ ( )] )t t t t t t t

t f ft t t

E r g f E g f E r f

r i e the APT assumes E g f exists

21 1 1( ) ...t t tg f a bf cf

1( ) ( )t Mg f a b r c

Multivariate Asset Pricing

Consider

Unconditional means are given by

Model for observations is

m m m mr b f e

r Bf e

fr B

fr r B Bf e

Principal Factors

Single factor caseDefine factor in terms of

returnsWhat factor maximizes

explained variance?

Satisfied by with criterion equal to

r f e ( )f w r

2 2

1

. . . : ( ) ( ) 0

m

i fw

i

w wMax

w w

F O C w w w w w w

jw k j

Principal Factors

Multiple factor caseCovariance matrix Define and the

first columnsThen This is the “principal factor”

solutionFactor analysis seeks to

diagonalize

Satisfied by with criterion equal to

r Bf e

efB B D *B D

*

kB k* * * *

k kB B B B

Importance of the largest eigenvalue

The Economy

1 1

( ) 11, ,

( )

it it i t ki kt it

i

i b

r b f b f

E bi m

Cov b D

What does it mean to randomly select security i?

Restrictive?

Harding, M., 2008 Explaining the single factor bias of arbitrage pricing models in finitesamples Economics Letters 99, 85-88.

k Equally important factors

Each factor is priced and contributes equally (on average) to variance:

Eigenvalues are given by

2 2

22 2

1 2

22 2

2 2

21

1, ,

( 1) 1(1 )

( 1) 1(1 )

fj f

b

k b

k m

j k

Rm km

k R

Rm

k R

Important result

The larger the number of equally important factors, the more certain would a casual empirical investigator be there was only one factor!

1

22

1k b

dkdm

ddm

Numerical example

2

2

2

1

2

1

4

: .1235

.0045

.01:

0.00063456

0.00000158

0.0

b

k

k m

k

Brown and Weinstein R

d

dmd

dmd

dm

What are the factors?

Where W is the Helmert rotation:

*1

*2

*

1 1 2 1 2 3 1 ( 1)

1 1 2 1 2 3 1 ( 1)

1 0 2 2 3 1 ( 1)

1 0 0 ( 1) ( 1)

s

k

B BWD

k k k

k k k

W k k k

k k k k

b

b

The average is one and

the remaining average to zero

Implications for pricing

Regress returns on factor loadings

Suppose k factors are priced:

Only one factor will appear to be priced!

1 11 1 1

111 2 1 2

1 2

* *1 2

ˆ ˆ( )

ˆ( ) , 0

( ) ( ) ( ) 2

( ) 2 ( ) 0

k

k

k

B B B r Var B B

Var B B I where

If t t t

Then t k and t

Application of Principal Components

Yield curve factors: level, slope and curvature

1 1 11 1

2 2 2 * *2 3

3 3 33 2

4 4 4

* *,

t t t

t t tt t

t t tt t t t

t t tt t

t t t

t t

y Bf e

y e ef f

y e ef f B f e

y e ef f

y e e

B B f f where

1 0 0

0 0 1 .

0 1 0

Note I

A more interesting example

Yield curve factors: level, slope and curvature

1 1 11 1

2 2 2* * *2 2

3 3 3*3 3

4 4 4

* *

?

?

?

?

,

t t t

t t tt t

t t tt t t t

t t tt t

t t t

t t

y Bf e

y e ef f

y e ef f B f e

y e ef f

y e e

B B f f wher

1 0 0

1 10 .

2 21 1

02 2

e Note I

Application of Principal Components

Procedure:

1. Estimate B* using principal components

2. Choose an orthogonal rotation to minimize a function

that penalizes departures from

*( )

. .

Min h B

s t I

B

(.)h

Conclusion

Mean variance efficiency and asset pricing

Important role of Sharpe ratioImplicit assumption of

Multivariate NormalityLimitations of data driven

approach

empirical financial economics asset pricing and mean variance efficiency

Documents