Microfoundations of Financial Economics2004-2005

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

PhD 03 |2April 18, 2023

CAPM: the real stuff

• Today we will look at various classical derivations of the CAPM.

• 1. Mossin

• Equilibrium of an exchange economy

• Based on quadratic utility functions

• 2. Mathematics of the efficient frontier

PhD 03 |3April 18, 2023

William Forsyth Sharpe

• From Wikipedia, the free encyclopedia.

• William Forsyth Sharpe (born June 16, 1934) is Professor of Finance, Emeritus at Stanford University's Graduate School of Business and the winner of the 1990 Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel.

• Dr. Sharpe taught at the University of Washington and the University of California at Irvine. In 1970 he joined the Stanford University. He was one of the originators of the Capital Asset Pricing Model, created the Sharpe ratio for risk-adjusted investment performance analysis, contributed to the development of the binomial method for the valuation of options, the gradient method for asset allocation optimization, and returns-based style analysis for evaluating the style and performance of investment funds.

• He served as a President of the American Finance Association.

• He received his Ph.D., M.A., and B.A. in Economics from the University of California at Los Angeles. He is also the recipient of a Doctor of Humane Letters, Honoris Causa from DePaul University, a Doctor Honoris Causa from the University of Alicante (Spain), a Doctor Honoris Causa from the University of Vienna and the UCLA Medal, UCLA's highest honor.

– Bibliography

Portfolio Theory and Capital Markets (McGraw-Hill, 1970 and 2000)

Asset Allocation Tools (Scientific Press, 1987)

Fundamentals of Investments (with Gordon J. Alexander and Jeffrey Bailey, Prentice-Hall, 2000)

Investments (with Gordon J. Alexander and Jeffrey Bailey, Prentice-Hall, 1999)

PhD 03 |4April 18, 2023

John Lintner

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PhD 03 |5April 18, 2023

Jan Mossin

• From Wikipedia, the free encyclopedia.

• Jan Mossin (b. 1936 in Oslo – d. 1987) was a Norwegian economist. He graduated with a siviløkonom degree from the Norwegian School of Economics and Business Administration (NHH) in 1959. After a couple of years in business, he started his PhD studies in the Spring semester of 1962 at Carnegie Mellon University (then Carnegie Institute of Technology).

• One of the papers in his doctoral dissertation was a very important contribution to the Capital Asset Pricing Model (CAPM). At Carnegie Mellon he was, among others, awarded the Alexander Henderson Award for 1968 for this contribution. If Jan Mossin had lived longer he would most likely had been a candidate for the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel in 1990 together with Professors Sharpe and Lintner.

• After he had finished his PhD he returned to NHH where he in 1968 was tenured professor.

PhD 03 |6April 18, 2023

CAPM à la Mossin

• 1 period model

• Investor i has quadratic utility function over future wealth Y

• n firms issue shares

• 1 share outstanding per firm

• pj price per share

• xj(s) payoff of firm j in state s

• σjk covariance of payoffs of firm j and k

²)( cYYEYuE

PhD 03 |7April 18, 2023

Investor’s problem

ij

iii

z

cYYEYuE 2)(Max

j

jijii pzmW

jj

fjiji

f

jjiji

fi

pRxzWR

xzmRY

)(

PhD 03 |8April 18, 2023

FOC

njpRxYuEz

Uj

fji

ij

i ,...,1 0))(('

cYYu 21)('

k

if

ij

fjk

fkj

fjjkik WR

cpRxEpRxEpRxEz )

2

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

)2

1(*

if

ijij WR

czz Note:

with zj* solution of:

k

jf

jkf

kjf

jjkk pRxEpRxEpRxEz ))(())()()((*

j=1,…,n

j=1,…,n

PhD 03 |9April 18, 2023

Market clearing conditions

i

ijz 1 j=1,…,n

0i

im

PhD 03 |10April 18, 2023

Equilibrium

12

1*

i ii

f

ij

iij WR

czz

i ii

f

i

j

WRc

z

2

11*

i

i ii

f

i

if

iij z

WRc

WRc

z

2

12

1

PhD 03 |11April 18, 2023

Equilibrium (2)

0)(2

1)(

k i kk

ij

fjjk xE

cpRxE

FOC (in equilibrium) can be written as:

i kk

i

kjk

jfj

xEc

xER

p)(

2

1)(

1

Solving for pj:

PhD 03 |12April 18, 2023

Equilibrium (3)

k

kjk

jkj xxb ),cov(

i kk

i

xEc

)(2

11

Define:

λ is a measure of the market risk aversion, the same for all companies

bj is the contribution of company j to the market’s total variance

j j k j

jjkj xVarb )(

we can write the equilibrium value of the firm as:

f

jjj R

bxEp

)(

PhD 03 |13April 18, 2023

Beta formulation

),cov()( MMMf

M RRpRRE

k

kM pp

The equilibrium price can be written as: k

kjjf

j xxxERp ),cov()(

Define:

j

jj p

xR

kkM xx

),cov(

),cov()(

MjMf

Mjf

j

RRpR

xRRRE

2

)(

M

fM

M

RREp

2

),cov()()(

M

MifM

fj

RRRRERRE

PhD 03 |14April 18, 2023

Mean-Variance Frontier Calculation: brute force

Mean variance portfolio: i j

ijjw

iw

Pn

wwwMin 2

,..,2

,1

jj

jjj

w

Eew

1

s.t.

Matrix notations:

w

VwwMin '

Eew '

1w’u=

PhD 03 |15April 18, 2023

Some math…

)'1(2)'(2'),,( 2121 uwewEVwwwL

01'

0'

021

wu

Ewe

ueVw

UEX 12

11

1''

''1

21

1

12

11

uVueVu

EuVeeVe

2111 ' ' ' ABCDuVuCeVeBuVeA

D

EABD

AEC

2

1

Ehgw

Lagrange:

FOC:

Define:

eVD

AuV

D

Bh

uVD

AeV

D

Cg

11

11

PhD 03 |16April 18, 2023

Interpretation

1g+h

0H

E

The frontier can be spanned by two frontier returns

Minimum variance portfolio MVPA/C

Ehgw

C

1

PhD 03 |17April 18, 2023

Zero covariance portfolio

CC

ARE

C

ARE

D

CRR qpqp

1)()(),cov(

The covariance between any two frontier portfolios p and q is:

For any two frontier portfolios p (except the MVP), there exists a unique frontier portfolio with which p has zero covariance:

CA

RE

CD

C

ARE

p

pzc

)()(

2

)(

PhD 03 |18April 18, 2023

0.800

0.900

1.000

1.100

1.200

1.300

1.400

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400

Zero Covariance Portfolio in the σ, E space

p

zc(p)

E[Rzc(p)]

E(Rp)

E(R)

σ(R)

PhD 03 |19April 18, 2023

Toward a Zero-Beta Capital Asset Pricing Model

0.800

0.900

1.000

1.100

1.200

1.300

1.400

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400

p q

zc(p)

E(Rp)

E[Rzc(p)]

σ(R)

PhD 03 |20April 18, 2023

Some math

)(),cov( qqp RERR

)(0 )( pzcRE

)()(),cov( )( pzcqqp RERERR

)()(),cov( )(2

pzcpppp RERERR

)()(

)()(),cov(

)(

)(

2pzcp

pzcq

p

qp

RERE

RERERR

Proof on demand – see DD Chap 7

Apply to ZC portfolio:

Apply to p:

Divide:

Rearrange: )()()()( )()( pzcpqppzcq RERERERE

PhD 03 |21April 18, 2023

Another proof (more intuitive??)

0)(

)(

aP

P

Rd

RdE

)(

)()( )(

p

pzcp

R

RERE

)(2))(),(cov(2

1)()(

)(

)(

)(

)(

)(

)(

0

2

2

0

ppqp

pq

aP

p

p

p

aP

P

RRRR

RERE

Rd

Rd

Rd

da

da

RdE

Rd

RdE

Consider a fraction a invested in stock q and (1-a) in p

The slope

is equal to the slope of tangent

As:

)()()()( )()( pzcpqppzcq RERERERE

PhD 03 |22April 18, 2023

Zero-Beta CAPM

)()()()( )()( MzcMjMMzcj RERERERE

In equilibrium, the market portfolio is on the efficient frontier

If there exist a risk free asset: E(Rzc(M)) = Rf

Empirical test: Roll critique

If proxy used for the market portfolio, linear relationship doesn’t hold

PhD 03 |23April 18, 2023

Next session

• Efficient frontier in Hilbert space (wooow..)

• Where are the SDF in the CAPM?