Daucus carrota

Professor André FarberSolvay Business SchoolUniversité Libre de Bruxelles

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Theory of asset pricing under certainty

1930

1940

1950

1960

FisherTheory of Interest

WilliamsTheory of Investment Value

HirshleiferTheory of Optimal

Investment Decisions

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Theory of asset pricing under uncertainty

1950

1960

1970

1980

1990

2000

MarkowitzPortfolio theory

Sharpe LintnerCAPM

Black Scholes MertonOPM

ArrowState prices

Cox Ross RubinsteinBinomial OPM

Arrow DebreuGeneral equilibrium

Cochrane – Campbell: p = E(MX)

VasiceckTerm structure

RossRisk neutral pricing

RossAPT Lucas

Asset Prices

Harrison KrepsMartingales

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Three views of asset pricing

Adapted from Cochrane Figure 6.1

General equilibrium

Stochastic discount factors

Mean variance efficiency

Complete marketsNo arbitrage (NA)

Law of one price (LOOP)

Beta pricing

State priceslinear pricing rule

Factor model+

No arbitrage Risk-neutral pricing

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Review: valuing uncertain cash flows

)(1)(

fMf rrrCEV

( ) cov( , )1

M

f

E C C rVr

2

M f

M

r r

Mean Variance Pricing

u u d dV v C v C 1

1 fu

dr

vu d

11 f

d

ur

vu d

1u u d d

f

p V p VV

r

CAPMSharpeLintner

APT Ross

Law of one price

No arbitrage opportunities

State prices

Risk neutralpricing

(1 )u F up r v

(1 )d f dp r v

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Data

Company Price Future payoff (m€)

RainProba = 0.40

SunProba = 0.60

Mkt Portf. 100 80.46 128.61

Gov. Bond 100 105 105

Type of weather

Rain Sun

Payoff 45m€ 35m€

Market

Project

Investment = 30m

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1. Market portfolio - statistics

Company Price Future payoff (m€)

RainProba = 0.40

SunProba = 0.60

Mkt Portf. 100 80.46 128.61

Returns -19.54% 28.61%

Expected return: rM = 9.35% Market risk premium rM – rf = 4.35%

Variance

Price of covariance risk

2 0.056 23.59%M M

2

.0435 0.782.056

M f

M

r r

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2. Project

Type of weather Rain (0.40) Sun (0.60)

Payoff C 45m€ 35m€

Return Mkt Portfolio -19.54% 28.61%

Expected cash flow

Covariance with mkt port.

Certainty equivalent:

( ) 39E C ( ) 4.9C

cov( , ) ( ) ( ) ( ) 1.1556M M Mr C E r C E r E C

( ) cov( , ) 39 (0.782)( 1.1556) 39.90ME C r C

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Details of previous calculations

22 2 2

( )

( ) ( ) ( ) ( )

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

i ii

E x x

x E x E x E x E x

x y E x E x y E y E xy E x E y

Sun (0.60) Rain (0.40)

Mkt Portfolio 28.61% -19.54%

Project 35 45

2 2 2 2

(0.60)(28.61%) (0.40)( 19.54%) 9.35%

(0.60)(0.2861) (0.40)( .1954) (.0935) 0.056( ) (0.60)(35) (0.40)(45) 39

cov( , ) (0.60)(.2861)(35) (0.40)( .1945)(45) (.0935)(39) 1.1556

M

M

M

r

E Cr C

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3. Net Present Value of the project.

PV (Payoff)

NPV = 38 – 30 = +8 >0

Expected return = 2.62%

Beta

39.90 381.05

0.54f

M f

r rr r

Sun 35 -7.90%

Rain 45 18.41%

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4. State prices valuation

11

0.485fu

dr

vu d

11

0.467fd

ur

vu d

0.485 45 0.467 35 30 38 30 8NPV

Price Sun / Up Rain / DownBond 1.00 1.05 1.05

Market 1.00 1.29 0.80

Bond 0.77 0.80 0.80Market 1.00 1.29 0.80

Mkt - Bond = 0.23 0.48 0.000.485 1.000 0.000

Bond 1.22 1.29 1.29Market 1.00 1.29 0.80

Bond - Mkt = 0.22 0.00 0.480.467 0.000 1.000

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5. Risk neutral pricing

( )1

u u d du u d d

f

p C p CPV Payoff v C v Cr

(1 ) 0.485 1.05 0.51

(1 ) 0.467 1.05 0.49u u f

d d f

p v r

p v r

0.51 35 0.49 45 39.9030 30 81.05 1.05

NPV

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John CampbellAsset Pricing at the MilleniumJournal of Finance 55,4 (Aug. 2000)

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Stochastic Discount Factors

( )

u u d d

u du u d d

u d

u u u d d d

V v C v C

v vC C

m C m CE mC

Stochastic discount factor