Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Consumption and Asset Pricing
Yin-Chi Wang
The Chinese University of Hong Kong
November, 2012
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
References:
• Williamson’s lecture notes (2006) ch5 and ch 6• Further references:
• Stochastic dynamic programming: Acemoglu (2010) ch16,Stokey-Lucas-Prescott (1989) ch9
• Asset pricing: Cochrane (2005)
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Background Knowledge
• Expected Utility Theory• Risk aversion
• Stochastic dynamic programming• Brock and Mirman (1972)
• Real business cycle models: Prescott (1986) and Cooley(1995)
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Expected Utility Theory 1Consumer’s preferences
• Deterministic world: ranking in consumption bundles• Uncertainty: ranking in lotteries
• Example: A world with a single consumption good c
Lottery 1:{c11 ,c21 ,
with prob. p1with prob. 1− p1
Lottery 2:{c12 ,c22 ,
with prob. p2with prob. 1− p2
• Expected utility from lottery i , i = 1, 2, is
piu(c1i)+ (1− pi ) u
(c2i)
• The consumer ranks lottery 1 and 2 according to
piu(c11)+ (1− pi ) u
(c21)R piu
(c12)+ (1− pi ) u
(c22)
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Expected Utility Theory 2• Risk aversion
• Many aspects of observed behavior toward risk is consistentwith risk aversion
• If the utility function is strictly concave, then the consumer isrisk averse
• Jensen’s inequality
E [u (c)] ≤ u (E [c ])
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Expected Utility Theory 3
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Expected Utility Theory 4
• Anomalies in observed behavior towards risk• The Allais Paradox
• Two measures of risk aversion• Absolute risk aversion
ARA(c) = −u′′ (c)u′ (c)
• Example: u (c) = −e−αc .
• Relative risk aversion
RRA(c) = −u′′ (c) cu′ (c)
• Example: u (c) = c 1−σ−11−σ
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Stochastic Optimal Growth Model• Brock and Mirman (1972): stochastic optimal growth model• The representative consumer’s preferences
E0∞∑t=0
βtu (ct ) ,
• where 0 < β < 1 and u (·) strictly increasing, strictly concaveand twice differentiable.
• E0 : expectation operator conditional on information at t = 0.• Production technology
yt = ztF (kt , nt )
• zt : random technology disturbance• {zt}∞
t=0 : a sequence of i .i .d . random variables drawn fromG (z)
• Law of motion for capital
kt+1 = it + (1− δ) kt , 0 < δ < 1
• Resource constraint: ct + it = yt
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Arrow-Debreu Eqm vs Rational Expectation Eqm 1
Competitive equilibrium: Two approaches
1. Arrow and Debreu (Arrow (1983) and Debreu (1983))
• At t = 0, market for contingent claims is openedand the representative consumer and therepresentative firm trade
• State-contingent-commodities/claims: a promiseto deliver a specified number of units of aparticular object (labor or capital services) at aparticular date T conditional on{z0, z1, z2, ..., zT }
• All markets clear
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Arrow-Debreu Eqm vs Rational Expectation Eqm 2
2. Spot market trading with rational expectations(Muth (1960))
1. • At each date the consumer rents capital and sells labor atmarket price
• The consumer makes optimal saving decision based on his/herbeliefs about the prob. distribution of future prices
• The markets clear at every date t for every possible realizationof the random shock {z0, z1, z2, ..., zt}
• All expectations are rational: beliefs of the prob.distribution=actual prob. distribution
• Both equilibria are Pareto Optimal (but not true in modelswith heterogenous agents)
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Social Planner’s Problem• Social planner’s problem
max{ct ,kt+1}
E0∞
∑t=0
βtu (ct )
s.t. ct + kt+1 = zt f (kt ) + (1− δ) kt
where f (k) ≡ F (K , 1)• The Bellman equation:
v (kt , zt ) = max [u (ct ) + βEtv (kt+1, zt+1)]
s.t. ct + kt+1 = zt f (kt ) + (1− δ) kt
• Note that ct is know but ct+i , i = 1, 2, 3, ..., is unknown(uncertain)
• Goal: solve for v (k, z) and the optimal decision rulekt+1 = g (kt , zt ) and ct = zt f (kt ) + (1− δ) kt − kt+1
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Example• u (ct ) = ln ct , f (kt ) = kα
t n1−αt , 0 < α < 1, yt = ztF (kt , nt ) ,
δ = 1 and E [ln zt ] = µ• Guess and verify
• Guess that the value function takes the form
v (kt , zt ) = A+ B ln kt +D ln zt
• It can be solved that
kt+1 = αβztkαt
ct = (1− αβ) ztkαt
• The economy will NOT converge to a steady state• Technolgy disturbances will cause persistent fluctuations inoutput, consumption and investment
• Stochastic steady state• Problems with the model
• Var (ln kt+1) = Var (ln yt ) = Var (ln ct ) : not the case in thedata
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Asset Pricing Model
• Lucas (1978)• Treat consumption and output as exogenous and asset pricesas endogenous
• Also called as the ICAPM (intertemporal capital asset pricingmodel) or consumption-based capital asset pricing model
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
The Economy 1
• Representative agent
E0∞
∑t=0
βtu (ct )
• 0 < β < 1, u (·) strictly increasing, strictly concave, twicedifferentiable
• Output• Exists n productive units (fruit trees), denote the productiveunit by i , i = 1, ..., n
• yit : quantity of output produced/yielded by production unit iat time t, a random variable
• Equilibrium:
ct =n
∑i=1yit
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
The Economy 2
• Asset holding• pit : price of tree i at time t• zit : shares of tree held at time tGoal: determine the prices ofthe trees
• Shares are traded in competitive market• Endowment: zi0 = 1, i =, ..., n• The fruits/output on each tree is proportionally distributed totheir share holders according to their share holding
• After the distributing of fruits, the shares are traded again
• Budget constraint
n
∑i=1pitzi ,t+1 + ct =
n
∑i=1zit (pit + yit ) (1)
for t = 0, 1, 2, ...
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Optimization 1• The Bellman equation
v (zt , pt , yt ) = maxct ,zt+1
[u (ct ) + βEtv (zt+1, pt+1, yt+1)]
s.t. ct =n
∑i=1zit (pit + yit )−
n
∑i=1pitzi ,t+1
• Rewrite the Bellman equation as
v (zt , pt , yt ) = maxct ,zt+1
[u (∑n
i=1 zit (pit + yit )−∑ni=1 pitzi ,t+1)
+βEtv (zt+1, pt+1, yt+1)
]• FOC and envelope theorem
−pitu′(
n
∑i=1zit (pit + yit )−
n
∑i=1pitzi ,t+1
)+ βEt
∂v∂zi ,t+1
= 0
∂v∂zi ,t+1
= (pit + yit ) u′(
n
∑i=1zit (pit + yit )−
n
∑i=1pitzi ,t+1
)for i = 1, ..., n.
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Optimization 2
• The basic formula for asset pricing
pit︸︷︷︸curret price of tree i
= Et
(pi ,t+1 + yi ,t+1)︸ ︷︷ ︸future payoff of the share
βu′ (ct+1)u′ (ct )︸ ︷︷ ︸the IMRS
(2)
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Optimization 3• Let
πit =pi ,t+1 + yi ,t+1
pi ,t(gross return)
mt =βu′ (ct+1)u′ (ct )
• Equation (2) can be rewritten as
Et (πitmt ) = 1
• Use cov(X ,Y ) = E (XY )− E (X )E (Y )
cov (πit ,mt ) + Et (πit )Et (mt ) = 1
• What is a good asset?• A good asset is one that pays you well when yourconsumption is low.
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Optimization 4• Apply the law of iterated expectations
Et [Et+sxt+s ′ ] = Etxt+s ′ , s′ ≥ 0, s ≥ 0
• The price of tree i at time t can be rewritten as
pit = Et
[(pi ,t+1 + yi ,t+1)
βu′ (ct+1)u′ (ct )
]
= Et
βu ′(ct+1)u ′(ct )
yi ,t+1 +β2u ′(ct+2)u ′(ct )
yi ,t+2
+ β3u ′(ct+3)u ′(ct )
yi ,t+3 + ...
= Et
[∞
∑s=t+1
βs−tu′ (cs )u′ (ct )
yis
]
• That is, the current price of any asset is the expecteddiscounted value of future dividends, where the discount factoris the IMRS
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Example 1
• Assume that yit is i .i .d ., and hence pit is i .i .d .
Et
[(pi ,t+1 + yi ,t+1) u′
(n∑i=1yi ,t+1
)]= Ai
for i = 1, ..., n, Ai > 0 is a constant.
• From the basic asset-pricing formula
pit =βAi
u′(
n∑i=1yi ,t
)
• Ifn∑i=1
yi ,t low → u′ high → pit low
• Ifn∑i=1
yi ,t high → u′ low → pit high
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Example 2: Risk Neutral Agent
• Assume that u (c) = c• From the basic asset-pricing formula
pit = βEt [pi ,t+1 + yi ,t+1]
=⇒ Et
[pi ,t+1 + yi ,t+1 − pit
pit
]=1β− 1
• Familiar formula in finance: only hold when people are riskneutral
• pit can be rewritten as
pit = Et∞∑
s=t+1βs−tyis
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
Example 3• Assume that u (c) = ln (c), n = 1 and
yt ={y1y2
w.p. πw.p. 1− π
, y1 > y2, yt is i .i .d .
• Let pi denote the price of a share when yt = yi for i = 1, 2.
p1 = β
[πy1y1(p1 + y1) + (1− π)
y1y2(p2 + y2)
]p1t = β
[πy2y1(p1 + y1) + (1− π)
y2y2(p2 + y2)
]• Can solve for
p1 =βy11− β
p2 =βy21− β
since y1 > y2, it follows that p1 > p2.
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
The Equity Premium Puzzle 1
• The average rate of return on equity is approximately 6%higher than the average rate of return on risk-free debt
• Mehra and Prescott (1985) showed that to generate such abig equity premium in Lucas’asset pricing model, the impliedIES of consumption must be very large, lying outside of therange of estimates for IES in empirical work
• 2 assets: a risk-free asset and a risky asset• Risky-asset
Pr[yt+1 = yj |yt = yi
]= πij
where πii = ρ, 0 < ρ < 1• Derive pi , qi , i = 1, 2 when yt = yi for i = 1, 2 and deriveR1,R2, r1, r2
• The average equity premium
e (β, σ, ρ, y1, y2) =12(R1 − r1) +
12(R2 − r2) ≈ 0.06
Background Knowledge Expected Utility Theory Stochastic Dynamic Programming Asset Pricing
The Equity Premium Puzzle 2
• Two explanations (u (c) = c1−σ/(1− σ))• 1. The higher the σ is, the lower the IES is, and the greater is
the tendency of the representative consumer to smoothconsumption over time. Or, the higher the σ is, the lesswilling the agents are to save for future consumption. Hence,to induce the agent to save more, have to give morecompensation (rt ↑)
2. σ also measures for risk aversion. The higher is σ the larger isthe expected return on equity, as agents must be compensatedmore for bearing risk
• Fitting the model into data: not enough variability inaggregate consumption to produce a large enough riskpremium, given plausible levels of risk aversion