ECON 815. Uncertainty and Asset Prices

Transcription

1 ECON 815 Uncertainty and Asset Prices Winter 2015 Queen s University ECON 815 1

2 Adding Uncertainty Endowments are now stochastic. endowment in period 1 is known at y t two states s {1, 2} in period 2 with endowment y s,t+1. there is a probability distribution (π 1, π 2 ) People now maximize expected utility u(c 1 ) + βe[u(c 2 )] Key idea: They face different budget constraints depending on the state. Consumption in different states is a different good. Queen s University ECON 815 2

3 Extending the Framework With more periods, it is convenient to allow tomorrow s state to depend on today s state. Example 1: Markov chain with two states [ ] π11 π Π = 12 π 21 π 22 where π ij describes the probability of going from state i today to state j tomorrow. Example 2: AR(1) process where ɛ t N (0, σ) and ρ (0, 1) y t = ρy t 1 + ɛ t We then have that tomorrow s expected values are functions of today s state or E t [y t+1 ] = E[y t+1 y t,... ] Queen s University ECON 815 3

4 Decisions under Uncertainty With two periods, people solve max E t [u(c t ) + βu(c t+1 )] subject to c t + a t = y t c s,t+1 = y s,t+1 + (1 + r t+1 )a t for all s where a denote savings now and r t is a risk-free interest rate. Solution: or E t [u (c t )] + E t [βu (c t+1 )(1 + r t+1 )] = 0 u (c t ) E t [βu (c t+1 )] = 1 + r t+1 Queen s University ECON 815 4

5 Warning! 1) Arithmetic and harmonic means are different. u [ (c t ) u ] E t [βu (c t+1 )] E (c t ) t βu (c t+1 ) 2) Jensen s Inequality f < 0 E[f(x)] < f(e[x]) 3) Covariance matters End Warning! E t [XY ] E t [X]E t [Y ] Queen s University ECON 815 5

6 Asset Prices What is an asset? Anything that delivers a payoff in units of consumption across states tomorrow. Could be a contract, a machine, capital, anything. More generally, think of a tree. price of the tree today is p t payoff consist of dividend tomorrow d t and price of the tree tomorrow p t+1 Queen s University ECON 815 6

7 The return from buying a tree is 1 + r(s t+1 s t ) = d(s t+1 s t ) + p(s t+1 s t ) p(s t ) in state s t+1 tomorrow given today s state is s t. An asset is risk-free if it has the same return across all states tomorrow. Otherwise it is a risky asset, with an expected return of [ ] dt+1 + p t+1 E t [1 + r t+1 ] = E t. p t We use our model the intertemporal Euler equation, expectations and asset payoffs to derive a theory of asset prices {p(s t+1 s t )}. Queen s University ECON 815 7

8 Arrow-Debreu Securities To do so, we first will price elementary securities called Arrow-Debreu securities. tomorrow s states s {1, 2,..., S} today s AD security s pays exactly one unit of consumption in state s tomorrow and nothing in any other state or period its price is called the state price s think of them as one-period zero coupon bonds All assets can be thought of as portfolios of AD securities. Key Idea: If we can price all AD securities, we can price any other security through arbitrage. This is known as the consumption-based capital asset pricing model (CCAPM) and relies on the notion of complete markets. Queen s University ECON 815 8

9 Pricing Securities Suppose there are two states tomorrow and people can only choose AD securities to invest in. max u(c t ) + βe t [u(c t+1 )] subject to c t + q(1 s t )a(1 s t ) + q(2 s t )a(2 s t ) y t c(s t+1 s t ) y t+1 + a(s t+1 s t ) for s t+1 {1, 2} where a(s t+1 s t ) is the amount of AD security s t+1 {1, 2} they buy. Solution: q(s t+1 s t ) = βπ(s t+1 s t )u (c(s t+1 s t )) u (c t ) where π(s t+1 s t ) is the conditional probability for state s t+1 occurring in period t. Queen s University ECON 815 9

10 Example 1: Consider a one-period risk-free bond that pays 1 unit of consumption in each state tomorrow. Payoffs for the bond are given by: ( ) ( 1 1 = ) ( Hence, its price is equal to a portfolio consisting of one unit of each of the two AD securities. Thus, q = q(1 s t ) + q(2 s t ) = βπ(1 s t)u (c(1 s t )) u (c t ) ]. = βe t [ u (c t+1 ) u (c t ) ) + βπ(2 s t)u (c(2 s t )) u (c t ) This implies that the risk-free interest rate q = 1/(1 + r f ) rate satisfies [ βu ] (c t+1 ) 1 = E t u (1 + r f t+1 (c t ) ) Queen s University ECON

11 Example 2: Consider any asset with arbitrary payoff across states equal to (x 1, x 2 ). Payoffs for this asset are given by ( ) ( x1 1 = x 1 0 x 2 ) ( 0 + x 2 1 ) Its price must be given by q x = x 1 q(1 s t ) + x 2 q(2 s t ) = βe t [ u (c t+1 )x t+1 u (c t ) ] Interpret this as equity with payoff x t+1. We then have that the return on equity satisfies [ u ] (c t+1 ) 1 = βe t u (c t ) (1 + re t+1) Queen s University ECON

12 Consumption Insurance and Risk Premia We have E[xy] = E[x]E[y] + Cov[xy]. This implies for asset pricing that [ ] [ ] [ 1 = E t β u (c t+1 ) xt+1 u (c t+1 ) u E t + βcov (c t ) q x u (c t ), x ] t+1 q x What matters for asset prices? the average payoff and the covariance of payoffs with consumption if negative, it is a hedge which increases the price if positive, people require an additional risk premium which decreases the price Queen s University ECON