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Microfoundations of Financial Economics 2004-2005 1.2 From Fisher to Arrow-Debreu

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

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  1. Microfoundations of Financial Economics2004-20051.2 From Fisher to Arrow-Debreu Professor André Farber Solvay Business School Université Libre de Bruxelles

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. Technical details PhD 01-2

  8. Using time separable utility Suppose U(c0,c1) = u(c0) + β u(c1) PhD 01-2

  9. Example As: Define: Three determinant of the interest rate: Impatience Time preference Growth rate of consumption PhD 01-2

  10. Multiperiod model – certainty Utility function: Security: price = p future cash flows = {dt} Optimum: FOC: PhD 01-2

  11. FOC: two periods version PhD 01-2

  12. 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

  13. 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

  14. Portfolio • Composition S ×1 • Payoff 1×S h’x • Price h.p =h’p inner product Example PhD 01-2

  15. Arbitrage • General definition of an arbitrage: p.h ≤ 0 and h’x≥0 (with at least one positive payoff) PhD 01-2

  16. No arbitrage • Theorem: In complete markets, NA implies that exists a unique q>> 0 such that NA PhD 01-2

  17. State price calculation PhD 01-2

  18. Geometry State 2 x(2) x q R p(x) = cst p(x) = 1 0 x(1) State 1 PhD 01-2

  19. Stochastic discount factors Define: PhD 01-2

  20. Risk neutral probabilities Define: Note: Looks like probabilities = risk neutral probabilities New pricing formula: PhD 01-2

  21. Geometry State 2 x(2) x q R p(x) = cst m p(x) = 1 0 x(1) State 1 PhD 01-2

  22. Geometry (with rescaled values) State 2 E(x) = cst m* x* 1* 0 State 1 p(x) = 0 E(x) = 0 PhD 01-2

  23. Beta pricing As: cov(m,x) = E(mx)-E(m)E(x) and E(m) = 1/Rf Define gross return: PhD 01-2

  24. Geometry State 2 p(x) = p[Projection of x on m] m* x* 0 State 1 p(x) = 0 PhD 01-2

  25. Beta representation Define : PhD 01-2

  26. Tomorrow • Where do the state prices, SDF, risk neutral proba come from? PhD 01-2

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