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Microfoundations of Financial Economics 2004-2005 1.2 From Fisher to Arrow-Debreu. Professor André Farber Solvay Business School Université Libre de Bruxelles. Theory of asset pricing under certainty. 1930. Fisher Theory of Interest. Williams Theory of Investment Value. 1940. 1950.

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Microfoundations of financial economics 2004 2005 1 2 from fisher to arrow debreu

Microfoundations of Financial Economics2004-20051.2 From Fisher to Arrow-Debreu

Professor André Farber

Solvay Business School

Université Libre de Bruxelles


Theory of asset pricing under certainty
Theory of asset pricing under certainty

1930

FisherTheory of Interest

WilliamsTheory of Investment Value

1940

1950

HirshleiferTheory of Optimal Investment Decisions

1960

PhD 01-2


Theory of asset pricing under uncertainty
Theory of asset pricing under uncertainty

1950

ArrowState prices

MarkowitzPortfolio theory

1960

Arrow DebreuGeneral equilibrium

Sharpe LintnerCAPM

1970

Black Scholes MertonOPM

RossAPT

LucasAsset Prices

RossRisk neutral pricing

VasiceckTerm structure

Harrison KrepsMartingales

Cox Ross RubinsteinBinomial OPM

1980

1990

Cochrane – Campbell: p = E(MX)

2000

PhD 01-2


Three views of asset pricing
Three views of asset pricing

General equilibrium

Mean variance efficiency

Beta pricing

Stochastic discount factors

Factor model+No arbitrage

Risk-neutral pricing

State priceslinear pricing rule

Complete marketsNo arbitrage (NA)Law of one price (LOOP)

Adapted from Cochrane Figure 6.1

PhD 01-2


Certainty irving fisher
Certainty: Irving Fisher

  • Setting:

    • 1 good – price at time 0 = 1 (numeraire)

    • Constant price (no inflation)

    • 1 security: zero-coupon, face value = 1, price at time t=0: m

  • Gross interest rate: Rf = 1/m

  • Consider future payoff x

  • Price at time t = 0: p(x) = m x

  • Why?

    • otherwise, ARBITRAGE

  • PhD 01-2


    Where does m come from
    Where does m come from?

    • Consumption over time:

    • Max utility function: U(c0, c1) U’i >0, concave

    • subject to budget constraint: c0 + mc1 = W

    • FOC:

    PhD 01-2



    Using time separable utility
    Using time separable utility

    Suppose U(c0,c1) = u(c0) + β u(c1)

    PhD 01-2


    Example
    Example

    As:

    Define:

    Three determinant of the interest rate:

    Impatience

    Time preference

    Growth rate of consumption

    PhD 01-2


    Multiperiod model certainty
    Multiperiod model – certainty

    Utility function:

    Security: price = p future cash flows = {dt}

    Optimum:

    FOC:

    PhD 01-2



    Introducing uncertainty
    Introducing uncertainty

    • Setting

      • 1 period

      • S states of nature :S = {s} finite

      • Probabilities: π(s)

      • S traded securities: price p (S×1 vector)

      • Future payoffs conditional on state of nature: x(s)

      • xi =[xi(1), xi(2), …, xi(S)] 1 ×S vector

      • Matrix of payoffs: S×S matrix

    PhD 01-2


    Assumptions
    Assumptions

    • Payoff space: X: set of all the payoffs that investor can purchase

    • Complete markets: X = RS

    • Portfolio formation:

    • Law of one price, linear pricing rule:

    PhD 01-2


    Portfolio
    Portfolio

    • Composition S ×1

    • Payoff 1×S h’x

    • Price h.p =h’p inner product

    Example

    PhD 01-2


    Arbitrage
    Arbitrage

    • General definition of an arbitrage:

      p.h ≤ 0 and h’x≥0 (with at least one positive payoff)

    PhD 01-2


    No arbitrage
    No arbitrage

    • Theorem: In complete markets, NA implies that exists a unique q>> 0 such that

    NA

    PhD 01-2



    Geometry
    Geometry

    State 2

    x(2)

    x

    q

    R

    p(x) = cst

    p(x) = 1

    0

    x(1)

    State 1

    PhD 01-2


    Stochastic discount factors
    Stochastic discount factors

    Define:

    PhD 01-2


    Risk neutral probabilities
    Risk neutral probabilities

    Define:

    Note:

    Looks like probabilities = risk neutral probabilities

    New pricing formula:

    PhD 01-2


    Geometry1
    Geometry

    State 2

    x(2)

    x

    q

    R

    p(x) = cst

    m

    p(x) = 1

    0

    x(1)

    State 1

    PhD 01-2


    Geometry with rescaled values
    Geometry (with rescaled values)

    State 2

    E(x) = cst

    m*

    x*

    1*

    0

    State 1

    p(x) = 0

    E(x) = 0

    PhD 01-2


    Beta pricing
    Beta pricing

    As: cov(m,x) = E(mx)-E(m)E(x) and E(m) = 1/Rf

    Define gross return:

    PhD 01-2


    Geometry2
    Geometry

    State 2

    p(x) = p[Projection of x on m]

    m*

    x*

    0

    State 1

    p(x) = 0

    PhD 01-2


    Beta representation
    Beta representation

    Define :

    PhD 01-2


    Tomorrow
    Tomorrow

    • Where do the state prices, SDF, risk neutral proba come from?

    PhD 01-2


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