lecture 11: multi-period equilibrium modelsmarkus/teaching/fin501/11lecture.pdf · conditional vs....
TRANSCRIPT
Fin 501:Asset Pricing I
Slide 11-1
Lecture 11: Multi-period
Equilibrium Models
Prof. Markus K. Brunnermeier
Fin 501:Asset Pricing I
Slide 11-2
Time-varying R*t (SDF)
• If one-period SDF mt is not time-varying (i.e.
distribution of mt is i.i.d., then
Expectations hypothesis holds
Investment opportunity set does not vary
Corresponding R* of single factor state-price beta
model can be easily estimate (because over time one
more and more observations about R*)
• If not, then mt (or corresponding R*t)
depends on state variable
multiple factor model
Fin 501:Asset Pricing I
Slide 11-3
R* depends on state variable
• R*t=R*(zt), with state variable zt
• Example:
zt = 1 or 2 with equal probability
Idea:
• Take all periods with zt=1 and figure out R*(1)
• Take all periods with zt=2 and figure out R*(2)
Can one do that?
• No – hedge across state variables
• Potential state-variables: predict future return
Fin 501:Asset Pricing I
Slide 11-5
Deriving ICAPM
• ICAPM allows consumption to depend on state
variable zt, which predicts future returns,
e.g. price-dividend ratio, risk-free rate
• Hence, value function V depends on both wealth
Wt and on state variable z
• Bellman equation
E[P1
t=0 ±tu(ct; zt)]
V (Wt; zt) = supct;Wtfu(ct; zt) + ±Et[V (Wt+1; zt+1]g
Fin 501:Asset Pricing I
Slide 11-6
Deriving ICAPM
• Recall Wt+1 = RWt+1(Wt-ct).
Differentiate w.r.t. ct and Wt
• Therefore u’(ct,zt) = VW (Wt,zt)
V (Wt; zt) = supct;Wtfu(ct; zt) + ±Et[V (Wt+1; zt+1]g
0 = u0(ct; zt)¡ ±Et[VW (Wt+1; zt+1)RWt+1]
VW (Wt; zt) = ±Et[VW (Wt+1; zt+1)RWt+1]
Fin 501:Asset Pricing I
Slide 11-7
Deriving ICAPM
• Hence equation
becomes
• Using a first order approximation
we obtain
Where ° is relative risk aversion coefficient of V
Second term are additional “risk factors”
E[Rit+1]¡Rf =¡Covt(u
0(ct+1);Rit+1)
Et[u0(ct+1]
E[Rit+1]¡Rf =¡Covt(VW (Wt+1;zt+1);R
it+1)
Et[VW (Wt+1;zt+1)]
VW(Wt+1; zt+1) ¼ VW(Wt; zt) +VWW(Wt; zt)¢Wt+1 +VWz(Wt; zt)¢zt+1
E[Rit+1]¡Rf = ¡°Covt(¢Wt+1; R
it+1) +
VWz
Et[VW ]Cov(¢zt+1; R
it+1)
Fin 501:Asset Pricing I
Slide 11-8
Static problem = intertemporal problem
• In general ICAPM setting
CRRA with γ≠1 and changing investment
opportunity sets
• Special cases
1. CRRA and i.i.d. returns and constant rf
• SR and LR investors have the same portfolio weights.
• Solve static problem instead of intertemporal problem
2. Log utility and non-i.i.d. returns => same result
Fin 501:Asset Pricing I
Slide 11-9
Digression: Multi-period Portfolio Choice
Theorem 4.10 (Merton, 1971): Consider the above canonical
multi-period consumption-saving-portfolio allocation problem.
Suppose U() displays CRRA, rf is constant and {r} is i.i.d.
Then a/st is time invariant.
Fin 501:Asset Pricing I
Slide 11-10
(Dynamic) Hedging Demand
• Illustration with noise trader risk:
Suppose fundamental value is constant v=1, but price is
noisy (due to noise traders)
If the asset is underpriced, e.g. p=.9 , then it might be
even more underpriced in the next period
• Myopic risk-averse investor:
buy some of the asset and push price towards 1, but not fully
• Forward-looking risk-averse investor:
yes, there can be intermediate losses if price declines in next
period, but then investment opportunity set improves even
more i.e. if returns are bad, then I have great opportunity
(dynamic hedge)
Fin 501:Asset Pricing I
Slide 11-11
Dynamic hedging demand
• Trade-off
Low return realization in next period
• Good since opportunity going forward is high
Invest more
Bad since marginal utility is high
Consume and invest less
High return realization in next period ….
• Utility
1 first (second) effect dominates
1(log-utility) both effects offset each other
Fin 501:Asset Pricing I
Slide 11-12
Conditional vs. unconditional CAPM
• If of each subperiod CAPM are time-
independent, then
conditional CAPM = unconditional CAPM
• If s are time-varying they may co-vary with Rm
and hence CAPM equation does not hold for
unconditional expectations.
Additional co-variance terms have to be considered!
Move from single-factor setting to multi-factor
setting