1. Microfoundations of Financial Economics 2004-2005 2 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
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3. Example
u(c)=ln(c)
40 1/2
c = 10 1/2
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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:
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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:
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6. Quadratic utility function
increasing
increasing
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7. Exponential utility
constant
increasing
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8. Log utility
decreasing
constant
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9. Power utility
decreasing
constant
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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)
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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 ????????
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12. Optimum
a* > 0  f’(0)>0  u’(W)E(R - Rf) > 0 E(R) > Rf
FOC:
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13. Quadratic utility
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14. Exponential utility + normally distributed returns
If z is normal:
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15. Log utility
Suppose R can take two values: Ru with proba π Rd with proba (1-π) Ru >Rf>Rd
FOC:
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16. Power utility
Suppose R can take two values: Ru with proba π Rd with proba (1-π) Ru >Rf>Rd
FOC:
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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
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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:
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19. Utility: time-separable Von Neuman –Morgenstern function
FOC:
Remember
p = E(mx)
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20. 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)
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21. Using power utility Suppose consumption growth is lognormal Define
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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
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23. CCAPM Start from: Power utility:
Assets pay a higher expected return if:
covary negatively with m
covary positively with consumption growth
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24. 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)
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25. Risk free rate Puzzle Suppose γ = 53. What about Rf ?
Either δ negative (people prefer future)
or real interest rate = 17% + δ
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26. Toward a Mean Variance Frontier
Start from:
For ρi,m = +1:
For ρi,m = -1:
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27. Mean-Variance Frontier
Slope =
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28. 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
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29. Hansen-Jagannathan Bounds
E(R)
σ(m) SR
1/E(m)
E(m)
σ(R)
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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 also Journal of Finance June 2005
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31. Tomorrow
CAPM: traditional derivations
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