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Microfoundations of Financial Economics 2004-2005 2 Choices under uncertainty - Equilibrium

Explore preferences under uncertainty, risk aversion measurement, demand for securities, optimum investment strategies, and consumption decisions under uncertainty. Analyze interest rates, CCAPM, mean-variance frontier, equity premium puzzle, and Hansen-Jagannathan Bounds.

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Microfoundations of Financial Economics 2004-2005 2 Choices under uncertainty - Equilibrium

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  1. Microfoundations of Financial Economics2004-20052 Choices under uncertainty - Equilibrium Professor André Farber Solvay Business School Université Libre de Bruxelles

  2. Preferences under uncertainty • Standard approach based on axioms of cardinal utility – • von Neuman Morgenstern (VNM). • Suppose Y is a random variable: {Y(s), π(s)} • u() is a cardinal utility function • u’(.) > 0 • u” attitude toward risk • risk lover u” >0 • risk neutral u” = 0 • risk averter u” <0 PhD 02

  3. Example u(c)=ln(c) 40 1/2 c =10 1/2 PhD 02

  4. Measuring risk aversion: ARA • Suppose Y = W + x E(x)=0 • What is the risk premium p such that: E[u(W+x)] = u(W – p) • Using Taylor expansion: Absolute risk aversion: PhD 02

  5. Measuring risk aversion: RRA • Suppose now that the uncertainty is proportional to wealth: x = r W • Y = W(1+r) • As: Relative risk aversion: PhD 02

  6. Quadratic utility function increasing increasing PhD 02

  7. Exponential utility constant increasing PhD 02

  8. Log utility decreasing constant PhD 02

  9. Power utility decreasing constant PhD 02

  10. What value for RRA? For an updated version see:Bliss and Panigirtzoglou, “Options-Implied Risk Aversion Estimates”, Journal of Finance, 59, 1 (Feb.2004) PhD 02

  11. Demand for securities • Let’s first ignore c0 • Initial wealth: W • 2 security: riskless bond (gross return Rf) and risky asset (gross return R) • Let a be the amount invested in the risky asset Note: f’(a) = E[u’(c)(R – Rf)] and f”(a)<0 ???????? PhD 02

  12. Optimum • a* > 0 f’(0)>0 u’(W)E(R - Rf) > 0 E(R) > Rf • FOC: PhD 02

  13. Quadratic utility PhD 02

  14. Exponential utility + normally distributed returns If z is normal: PhD 02

  15. Log utility Suppose R can take two values: Ruwith proba πRdwith proba (1-π)Ru >Rf>Rd FOC: PhD 02

  16. Power utility Suppose R can take two values: Ruwith proba πRdwith proba (1-π)Ru >Rf>Rd FOC: PhD 02

  17. How does a change when W vary? • Decreasing absolute risk aversion implies da/dW>0 • Decreasing relative risk aversion implies that the fraction invested in the risky asset increases with wealth PhD 02

  18. Two-period models of consumption decisions under uncertainty • Early models: Leland 1968, Sandmo • 1 risky asset • General representation of consumer’s preferences • U(c0,c1) = E[u(c0,c1)] • Budget constraint c0 + zp = W FOC: PhD 02

  19. Utility: time-separable Von Neuman –Morgenstern function FOC: Remember p = E(mx) PhD 02

  20. Numerical example (DD Chap 8) Endowment (Illustration using Excel file) PhD 02

  21. Using power utility Suppose consumption growth is lognormal Define PhD 02

  22. Understanding interest rates • Interest rates are high when: • People are impatient - δ high • In good times - E(Δln(c)) high - γ controls intertemporal substitution) • In safe times - σ²(Δln(c)) low -γ controls intertemporal substitution PhD 02

  23. CCAPM Start from: Power utility: Assets pay a higher expected return if: covary negatively with m covary positively with consumption growth PhD 02

  24. Looking at the data • Annual data US 1948-2002 (Source: Cochrane) percent E(R – Rf) σ(R) E(Δc) σ(Δc) corr(Δc,R) 7.21 18.0 1.31 1.93 0. 39 7.2 = γ×0.135 γ = 53!!! HUGE Equity premium puzzle (Mehra Prescott) PhD 02

  25. Suppose γ = 53. What about Rf? Risk free rate Puzzle Either δ negative (people prefer future) or real interest rate = 17% + δ PhD 02

  26. Toward a Mean Variance Frontier Start from: For ρi,m = +1: For ρi,m = -1: PhD 02

  27. Mean-Variance Frontier Slope = PhD 02

  28. Equity puzzle again Pick a frontier portfolio Rmv E(m) ≈ 1 and σ(m) = γσ(Δc) US, last 80 years: Sharpe ratio ≈ 0.50 and σ(Δc) ≈ 1%  σ(m) = 50% Is this realistic ? (remember E(m) ≈ 1)  γ≈ 50 This is the equity premium puzzle stated differently PhD 02

  29. Hansen-Jagannathan Bounds E(R) σ(m) SR 1/E(m) E(m) σ(R) PhD 02

  30. Hansen-Jagannathan bounds: recent estimates Lustig, Hanno N. and Van Nieuwerburgh, Stijn, "Housing Collateral, Consumption Insurance and Risk Premia: An Empirical Perspective" (March 15, 2004). EFA 2004 Maastricht Meetings Paper No. 1403. http://ssrn.com/abstract=556101 - also Journal of Finance June 2005 PhD 02

  31. Tomorrow • CAPM: traditional derivations PhD 02

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