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Microfoundations of Financial Economics 2004-2005 2 Choices under uncertainty - Equilibrium
Professor André Farber Solvay Business School Université Libre de Bruxelles
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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
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Example u(c)=ln(c) 40 1/2 c = 10 1/2 PhD 02
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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
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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
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Quadratic utility function
increasing increasing PhD 02
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Exponential utility constant increasing PhD 02
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Log utility decreasing constant PhD 02
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Power utility decreasing constant PhD 02
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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
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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
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Optimum a* > 0 f’(0)>0 u’(W)E(R - Rf) > 0 E(R) > Rf FOC: PhD 02
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Quadratic utility PhD 02
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Exponential utility + normally distributed returns
If z is normal: PhD 02
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Log utility Suppose R can take two values: Ru with proba π Rd with proba (1-π) Ru >Rf>Rd FOC: PhD 02
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Power utility Suppose R can take two values: Ru with proba π Rd with proba (1-π) Ru >Rf>Rd FOC: PhD 02
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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
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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
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Utility: time-separable Von Neuman –Morgenstern function
FOC: Remember p = E(mx) PhD 02
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Numerical example (DD Chap 8)
Endowment t = 0 t = 1 State 1 (B) Proba = 1/3 State 2 (G) Proba = 2/3 Al 10 1 2 Ben 5 4 6 (Illustration using Excel file) PhD 02
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Using power utility Suppose consumption growth is lognormal Define
PhD 02
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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
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CCAPM Start from: Power utility:
Assets pay a higher expected return if: covary negatively with m covary positively with consumption growth PhD 02
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Looking at the data Annual data US (Source: Cochrane) percent E(R – Rf) σ(R) E(Δc) σ(Δc) corr(Δc,R) 39 7.2 = γ×0.135 γ = 53!!! HUGE Equity premium puzzle (Mehra Prescott) PhD 02
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Risk free rate Puzzle Suppose γ = 53. What about Rf ?
Either δ negative (people prefer future) or real interest rate = 17% + δ PhD 02
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Toward a Mean Variance Frontier
Start from: For ρi,m = +1: For ρi,m = -1: PhD 02
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Mean-Variance Frontier
Slope = PhD 02
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Equity puzzle again Pick a frontier portfolio Rmv
E(m) ≈ 1 and σ(m) = γ σ(Δc) US, last 80 years: Sharpe ratio ≈ and σ(Δc) ≈ 1% σ(m) = 50% Is this realistic ? (remember E(m) ≈ 1) γ ≈ This is the equity premium puzzle stated differently PhD 02
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Hansen-Jagannathan Bounds
E(R) σ(m) SR 1/E(m) E(m) σ(R) PhD 02
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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 also Journal of Finance June 2005 PhD 02
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Tomorrow CAPM: traditional derivations PhD 02
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