Consumption and Asset Pricing - Yin-Chi Wang's...

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