Consumption and Asset Pricing Yin-Chi Wang The Chinese University of Hong Kong November, 2012
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 Risk aversion Stochastic dynamic programming Brock and Mirman (1972) Real business cycle models: Prescott (1986) and Cooley (1995)
Expected Utility Theory 1 Consumer s preferences Deterministic world: ranking in consumption bundles Uncertainty: ranking in lotteries Example: A world with a single consumption good c Lottery 1: Lottery 2: { c 1 1, with prob. p 1 { c 2, with prob. 1 p 1 c 1, with prob. p 2 c2 2, with prob. 1 p 2 Expected utility from lottery i, i = 1, 2, is ( ) ( ) p i u ci 1 + (1 p i ) u ci 2 The consumer ranks lottery 1 and 2 according to ( ) ( ) p i u c1 1 + (1 p i ) u c1 2 p i u ( c 1 2 ) + (1 p i ) u ( ) c2 2
Risk aversion Expected Utility Theory 2 Many aspects of observed behavior toward risk is consistent with risk aversion If the utility function is strictly concave, then the consumer is risk averse Jensen s inequality E [u (c)] u (E [c])
Expected Utility Theory 3
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) c u (c) Example: u (c) = c 1 σ 1 1 σ
Stochastic Optimal Growth Model Brock and Mirman (1972): stochastic optimal growth model The representative consumer s preferences E 0 β t u (c t ), t=0 where 0 < β < 1 and u ( ) strictly increasing, strictly concave and twice differentiable. E 0 : expectation operator conditional on information at t = 0. Production technology y t = z t F (k t, n t ) z t : random technology disturbance {z t } t=0 : a sequence of i.i.d. random variables drawn from G (z) Law of motion for capital k t+1 = i t + (1 δ) k t, 0 < δ < 1 Resource constraint: c t + i t = y t
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 opened and the representative consumer and the representative firm trade State-contingent-commodities/claims: a promise to deliver a specified number of units of a particular object (labor or capital services) at a particular date T conditional on {z 0, z 1, z 2,..., z T } All markets clear
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 at market price The consumer makes optimal saving decision based on his/her beliefs about the prob. distribution of future prices The markets clear at every date t for every possible realization of the random shock {z 0, z 1, z 2,..., z t } All expectations are rational: beliefs of the prob. distribution=actual prob. distribution Both equilibria are Pareto Optimal (but not true in models with heterogenous agents)
Social planner s problem Social Planner s Problem max E 0 {c t,k t+1 β t u (c t ) } t=0 s.t. c t + k t+1 = z t f (k t ) + (1 δ) k t where f (k) F (K, 1) The Bellman equation: v (k t, z t ) = max [u (c t ) + βe t v (k t+1, z t+1 )] s.t. c t + k t+1 = z t f (k t ) + (1 δ) k t Note that c t is know but c t+i, i = 1, 2, 3,..., is unknown (uncertain) Goal: solve for v (k, z) and the optimal decision rule k t+1 = g (k t, z t ) and c t = z t f (k t ) + (1 δ) k t k t+1
Example u (c t ) = ln c t, f (k t ) = k α t n 1 α t, 0 < α < 1, y t = z t F (k t, n t ), δ = 1 and E [ln z t ] = µ Guess and verify Guess that the value function takes the form It can be solved that v (k t, z t ) = A + B ln k t + D ln z t k t+1 = αβz t k α t c t = (1 αβ) z t k α t The economy will NOT converge to a steady state Technolgy disturbances will cause persistent fluctuations in output, consumption and investment Stochastic steady state Problems with the model Var (ln k t+1 ) = Var (ln y t ) = Var (ln c t ) : not the case in the data
Asset Pricing Model Lucas (1978) Treat consumption and output as exogenous and asset prices as endogenous Also called as the ICAPM (intertemporal capital asset pricing model) or consumption-based capital asset pricing model
Representative agent The Economy 1 E 0 t=0 β t u (c t ) Output 0 < β < 1, u ( ) strictly increasing, strictly concave, twice differentiable Exists n productive units (fruit trees), denote the productive unit by i, i = 1,..., n y it : quantity of output produced/yielded by production unit i at time t, a random variable Equilibrium: c t = n y it i=1
Asset holding The Economy 2 p it : price of tree i at time t z it : shares of tree held at time tgoal: determine the prices of the trees Shares are traded in competitive market Endowment: z i0 = 1, i =,..., n The fruits/output on each tree is proportionally distributed to their share holders according to their share holding After the distributing of fruits, the shares are traded again Budget constraint for t = 0, 1, 2,... n n p it z i,t+1 + c t = z it (p it + y it ) (1) i=1 i=1
The Bellman equation Optimization 1 v (z t, p t, y t ) = max c t,z t+1 [u (c t ) + βe t v (z t+1, p t+1, y t+1 )] s.t. c t = n i=1 z it (p it + y it ) Rewrite the Bellman equation as v (z t, p t, y t ) = max c t,z t+1 FOC and envelope theorem p it u ( n i=1 n i=1 p it z i,t+1 [ u ( n i=1 z it (p it + y it ) n i=1 p it z i,t+1 ) +βe t v (z t+1, p t+1, y t+1 ) z it (p it + y it ) v z i,t+1 = (p it + y it ) u for i = 1,..., n. ( n n i=1 i=1 p it z i,t+1 ) z it (p it + y it ) + βe t v z i,t+1 = 0 n i=1 p it z i,t+1 ) ]
Optimization 2 The basic formula for asset pricing p }{{} it = E t curret price of tree i (p i,t+1 + y i,t+1 ) }{{} future payoff of the share βu (c t+1 ) u (c t ) (2) }{{} the IMRS
Let Optimization 3 π it = p i,t+1 + y i,t+1 p i,t m t = βu (c t+1 ) u (c t ) (gross return) Equation (2) can be rewritten as E t (π it m t ) = 1 Use cov(x, Y ) = E (XY ) E (X )E (Y ) What is a good asset? cov (π it, m t ) + E t (π it ) E t (m t ) = 1 A good asset is one that pays you well when your consumption is low.
Optimization 4 Apply the law of iterated expectations E t [E t+s x t+s ] = E t x t+s, s 0, s 0 The price of tree i at time t can be rewritten as [ ] p it = E t (p i,t+1 + y i,t+1 ) βu (c t+1 ) u (c t ) βu (c t+1 ) u (c t y ) i,t+1 + β2 u (c t+2 ) = E t = E t [ s=t+1 u (c t ) + β3 u (c t+3 ) u (c t y ) i,t+3 +... ] β s t u (c s ) u y is (c t ) y i,t+2 That is, the current price of any asset is the expected discounted value of future dividends, where the discount factor is the IMRS
Example 1 Assume that y it is i.i.d., and hence p it is i.i.d. ( )] n E t [(p i,t+1 + y i,t+1 ) u y i,t+1 = A i i=1 for i = 1,..., n, A i > 0 is a constant. From the basic asset-pricing formula p it = βa ( i ) n u y i,t i=1 n If y i,t low u high p it low i =1 n If y i,t high u low p it high i =1
Example 2: Risk Neutral Agent Assume that u (c) = c From the basic asset-pricing formula p it = βe t [p i,t+1 + y i,t+1 ] = E t [ pi,t+1 + y i,t+1 p it p it ] = 1 β 1 Familiar formula in finance: only hold when people are risk neutral p it can be rewritten as p it = E t β s t y is s=t+1
Example 3 Assume { that u (c) = ln (c), n = 1 and y1 w.p. π y t = y 2 w.p. 1 π, y 1 > y 2, y t is i.i.d. Let p i denote the price of a share when y t = y i for i = 1, 2. [ p 1 = β π y 1 (p 1 + y 1 ) + (1 π) y ] 1 (p 2 + y 2 ) y 1 y 2 [ p 1t = β π y 2 (p 1 + y 1 ) + (1 π) y ] 2 (p 2 + y 2 ) y 1 y 2 Can solve for p 1 = βy 1 1 β p 2 = βy 2 1 β since y 1 > y 2, it follows that p 1 > p 2.
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 a big equity premium in Lucas asset pricing model, the implied IES of consumption must be very large, lying outside of the range of estimates for IES in empirical work 2 assets: a risk-free asset and a risky asset Risky-asset Pr [ y t+1 = y j y t = y i ] = πij where π ii = ρ, 0 < ρ < 1 Derive p i, q i, i = 1, 2 when y t = y i for i = 1, 2 and derive R 1, R 2, r 1, r 2 The average equity premium e (β, σ, ρ, y 1, y 2 ) = 1 2 (R 1 r 1 ) + 1 2 (R 2 r 2 ) 0.06
The Equity Premium Puzzle 2 Two explanations (u (c) = c 1 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 smooth consumption over time. Or, the higher the σ is, the less willing the agents are to save for future consumption. Hence, to induce the agent to save more, have to give more compensation (r t ) 2. σ also measures for risk aversion. The higher is σ the larger is the expected return on equity, as agents must be compensated more for bearing risk Fitting the model into data: not enough variability in aggregate consumption to produce a large enough risk premium, given plausible levels of risk aversion