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

The geometry of mean variance

E

Note: returns are in excess of the risk free rate

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

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

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