Asset Pricing. Question: What is the equilibrium price of a stock? Defn: a stock is a claim to a future stream of dividends. # X E β t U(c t ) t=0

Transcription

1 Asset Pricing 1 Lucas (1978, Econometrica) Question: What is the equilibrium price of a stock? Defn: a stock is a claim to a future stream of dividends. 1.1 Environment Tastes: " # X E β t U( ) t=0 Technology: Each project or tree bears uncertain dividends (stochastic endowments of fruit): y t (= d t ) D a finite set in R ++ with Π(y t+1 y t )= prob(y t+1 = y 0 y t = y). 1.2 Equilibrium Ownership of projects or trees is determined each period in a competitive stock market. Each tree has one outstanding perfectly divisible equity share. A share (denoted s t ) entitles its owner at the beginning of period t to all of the tree s output in period t. Shares are traded after payment of dividends at price p t. Household sequence problem max {,s t+1} E " X # β t U( ) t=0 (1) subject to + p t s t+1 = s t (y t + p t ) Def. An equilibrium is a sequence t =0, 1,... of functions: a price function p t = P (y t ) and an allocation of consumption and stocks (s t,y t ) and s t+1 (s t,y t ) which satisfies household optimization (1), stock market clearing s t =1, and goods market clearing = y t. The first order conditions characterizing the optimal choice of stocks satisfies p t U 0 ( )=βe t [U 0 (+1 )(y t+1 + p t+1 )] (2) which is the standard (asset pricing) Euler equation. 1

2 Rearranging (2), substituting the market clearing condition = y t,and iterating forward one period yields U 0 (y t+2 ) p t+1 = βe t+1 U 0 (y t+1 ) (y t+2 + p t+2 ) into (2) yields p t = βe t U 0 (y t+1 ) U 0 (y t ) = E t ½ U 0 ¾ (y t+2 ) y t+1 + βe t+1 U 0 (y t+1 ) (y t+2 + p t+2 ) 2X β j U 0 (y t+j ) U 0 (y t ) y t+j + β 2 U 0 (y t+2 ) U 0 (y t ) p t+2 where we have used the law of iterated expectations", that E t [E t+1 [ex]] = E t [ex]. Successive forward iterations yields TX p t = E t β j U 0 (y t+j ) U 0 (y t ) y t+j + β T U 0 (y t+t ) U 0 (y t ) p t+t If there exists a bounded price function, then X lim p t = E t β j U 0 (y t+j ) T U 0 (y t ) y t+j which says that the stock price is the present discounted value of future dividends, ³ where the stochastic discount factor" or pricing kernel m t+j β j U 0 (y t+j) U 0 (y t ) for any date t + j is just equal to the marginal rate of substitution in consumption. Notice that if U is linear (i.e. households are risk neutral), then lim T p t = X β j E t [y t+j ] Recall that if x and z are random variables, then E [x z] =E [x] E [z]+ cov(x, z).then X lim p t = E t E t [m t+j ] E t [y t+j ]+cov t (m t+j,y t+j ) T 2

3 If U( )= c1 ψ t cov µ ³ yt y t ψ ψ,yt+1 ³, then m t+j = β j ψ y t+j y t, in which case covt (m t+j,y t+j )= < 0. That is, dividends on this asset go up just when the marginal utility of consumption is lowest and dividends go down just when you value extra utility. Thus, the asset doesn t provide a good hedge against consumption risk. In that case, its price will be lower. Since its price is lower, the return is higher to compensate for the extra risk. This is an explanation for the higher return on stocks than bonds. Now let s return to the issue of existence of a bounded price function. The function P (y t ) must satisfy the necessary condition (2) or P (y t )U 0 (y t )=βe t [U 0 (y t+1 )(y t+1 + P (y t+1 )]. Let f(y t ) P (y t )U 0 (y t ) and g(y t ) β P y t+1 D Π(y t+1 y t )U 0 (y t+1 )y t+1. Then we can define an operator T : B + (D) B + (D) where B + (D) is the space of non-negative, bounded functions such that for any element f B + (D), we can express the rhs of the Euler equation as: (Tf)(y) =g(y)+β X Π(y 0 y)f(y 0 ). y 0 D Think of taking f(y 0 ) from B + (D) to B + (D). As long as what we add is bounded, it will remain bounded. If you started even with a constant function for f(y 0 ) = 0, then it would stay in B + (D) provided g(y) is bounded. Notice that since y t D R ++,U 0 (y t ) is finite for the types of utility functions we are considering. Thus, g(y) is bounded, so that we can use a contraction mapping argument to establish the existence of a unique fixed point f = Tf. But that fixed point is just the lhs of the Euler equation (2). 2 Mehra-Prescott (1985, JME) Facts: From , the average return on equity was 7% and the standard deviation was 16.5 The average return on short term debt was less than 1% and the standard deviation was 5.7 Question: Can the 6% equity premium over riskless debt be explained using a version of Lucas asset pricing model? 3

4 Answer: No. For reasonable parameter values, get the premium to only be 4/10 of one percentage point. The puzzle is just quantitative. Theoretically, the asset pricing model can explain a premium, it just can t explain the size of the premium for reasonable parameter values. Puzzle: To get a low risk free interest rate in a growing economy, you need a high elasticity of intertemporal substitution (see Phillipe Weil )). 1 To get a large equity premium, need a high coefficient of relative risk aversion. But one is the reciprocal of the other. What is a puzzle? When a model s implications are inconsistent with the data. This illustrates an excellent approach to writing good papers. 2.1 Environment Just a growth version of Lucas (1978). Preferences U(c) = c1 ψ 1 ψ Technologies (Endowments vary stochastically according to an n-state Markov process in growth rates): y t+1 = x t+1 y t where x t+1 {λ 1,..., λ n } and π ij =Pr[x t+1 = x j x t = x i ]. 1 The elasticity of intertemporal substitution is defined to be ³ ct+1 d R t ³ dr ct+1. t With CRRA preferences, the first order conditions yield +1 =(βr t ) 1/ψ Then and the elasticity is d ³ ct+1 dr t ³ ct+1 d dr t = 1 ψ 1 ψ [βrt] ψ R t 1 ³ ct+1 = ψ [βr t] 1 ψ ψ β β R t (βr t ) 1/ψ = 1 ψ. To see why we need the intertemporal elasticity to be high, recall for R t = (1+gt)ψ, to be β low, we need ψ to be small, which means 1 is high. ψ 4

5 2.1.1 Primer on Markov Processes Def. Let s t S where S = {s 1,..., s k } is a finite set. A (first order) Markov process is a stochastic process with the property that prob(a s t,s t 1,..., s t m )=prob(a s t ) for any {s t+1,..., s t+n } A. That is, a Markov process has the property that given the current realization s t, future realizations are independent of the past. A probability distribution (more formally a measure on (S, S) where S is the power set of S) attimet is represented as a vector p t in the k- dimensional simplex k = {p t R k : p t 0 and P k i=1 pi t =1}. Then we define the stationary transition function Π =[π ij ], just a k k matrix where π ij = prob(s t+1 = s j s t = s i ) which satisfies P k π ij =1. In the k =2case t\t +1 s 1 s 2 s 1 π 11 π 12 s 2 π 21 π 22 where π 12 =1 π 11 and π 21 =1 π 22. Given any current probability distribution p t, we can consider next period s probability distribution p t+1.the law of motion is just p t+1 = Π 0 p t. It is easy to show that if p t k,thensoisp t+1. In the k =2case, the probability of being in state s 1 at t +1 is just p 1 t+1 = π 11 p 1 t + π 21 p 2 t and p 2 t+1 = π 12 p 1 t + π 22 p 2 t.specifically, in the k =2 case, p t+1 = Π 0 p t is just: p 1 t+1 p 2 t+1 = π 12 π 22 p 2 t π11 π 21 p 1 t Given p t+2 = Π 0 p t+1,then p t+2 = Π 0 (Π 0 p t ) = (Π 0 ) 2 p t andingeneral p t+n =(Π 0 ) n p t. Using this we can define the stationary distribution p = lim n p t+n.a simple way to calculate this is as the solution to p = Π 0 p. In the k =2case, p 1 = π 11 p 1 + π 21 p 2 and p 2 = π 12 p 1 + π 22 p 2 where π 12 =1 π 11,π 21 =1 π 22, and p 1 + p 2 =1. These two equations are not distinct; in particular, the second is just 1 times the first. Hence we need only consider the first: (1 π 11 )p 1 = π 21 (1 p 1 ) wherewehaveused the fact that p 1 + p 2 =1. This yields p 1 π 21 =. (3) 1 π 11 + π 21 Clearly p 1 0 since 1 π 11 +π 21 > 0 and 1 p 1 since 1 π 11 +π 21 π 21. 5

6 Def. If the probability distribution over the initial state is p, then it is also p in every successive period. A vector with this property is called an invariant distribution. 2.2 Equilibrium Asset Pricing Rather than the formal fixed point argument establishing existence of the bounded price function in Lucas model, here we will actually solve for the price function explicitly to establish existence under certain conditions (i.e. an invertibility condition). Just as in (2) of Lucas, with CRRA preferences U 0 (c) =c ψ : " µ # ψ yt p t = βe (y t+1 + p t+1) x t,y t y t+1 P (y, i) =β µ ψ y π ij [λ j y + P (λ j y, j)]. λ j y (4) Note that the dependence of P on i is due to π ij. Clearly P (y, x) is homogeneous of degree one in y: Toseethis, P (γy,i)=β µ ψ 1 π ij [λ j γy + P (λ j γy,j)] λ j But if P (λγy, j) =γp(λy, j), then P (γy,i) =γβ µ ψ 1 π ij [λ j y + P (λ j y,j)] = γp(y, i) λ j (i.e. we conjectured P (y, x) was homogeneous of degree 1 and then showed thepricefunctionwasconsistentwiththisconjecture). Thisislikean operator argument. If P is a vector valued function which is homogeneous of degree one, then (TP) returns a vector valued function. Given that P (y, x) is homogeneous of degree 1 in y, then conjecture a (linear) solution of the form P (y, i) =w i y, where we must determine the constants w i,i=1,..., n. This provides another application of the method of undetermined coefficients. Substituting this (and P (λ j y, j) =λ j w j y)into(4)yields w i y = β µ ψ 1 π ij [λ j y + λ j w j y] λ j 6

7 or w i = β which is just n equations in n unknowns w i. In matrix notation with w = w 1 :, Λ = w n π ij λ 1 ψ j (1 + w j ),i=1,..., n (5) w = βλw + κ π 11λ 1 ψ 1.. π 1n λ 1 ψ n : : π n1 λ 1 ψ 1.. π nn λ 1 ψ n,κ= β P n π 1jλ 1 ψ j : β P n π njλ 1 ψ j which has solution provided I βλ 6= 0. w =[I βλ] 1 κ (6) Notice that if β, π ij,and λ j are known (from the calibration below), then we can compute undetermined coefficients w from (6). Nowwemustcomputeexpectedreturnstoequityandtheriskfreeasset in the model to compare against the time averages in the data (just as we computed second moments in the RBC model). Conditional realized real return to holding equity: denoted r e ij starting in state (y,i) and moving to (λ j y, j) is the capital gain and the dividend r e ij = P (λ jy, j) P (y, i)+λ j y P (y,i) = λ jw j y w i y + λ j y w i y = λ j(w j +1) w i 1 Conditional expected real return to holding equity: denoted R i starting in state i: Ri e = π ij rij e Conditional real return to a risk free bond: The price of a bond (which pays off in every state next period) starting in state i is given by p f i = βe U 0 (λ j y) U 0 (y) Hence the return is given by R f i = 1 p f i 1 y, i = β 1 π ij λ ψ j 7

8 Unconditional real returns: To calculate the model s average return on equity or risk free bonds, we need to know the unconditional probability of being in a given state i denoted μ i is the solution to μ = Π 0 μ where μ =(μ 1,..., μ n ) 0 and Π =[π ij ]. Recall in the two state case, the π solution (3) is given yields μ 1 = 21 1 π 11+π 21 and μ 2 =1 μ 1. Then the average return on equity, for example, is just R e = P n i=1 μ iri e. 2.3 Calibration and Findings The parameters defining technology are [π ij ]and [λ i ] while the parameters defining preferences are ψ and β. To make calibration simple, M-P considered a symmetric 2 state markov process (n =2).Then where π (0, 1). λ 1 = 1+g + σ, λ 2 =1+g σ π 11 = π 22 = π, π 12 = π 21 =1 π This parameterization allowed them to independently vary average growth rate of consumption/output by varying g, its variability by varying σ, and its serial correlation by varying π. Calibration: (g, σ, π) selected to match the average (0.018), standard deviation (0.036), and autocorrelation ( 0.14) of the growth rate of per capita consumption. Result (g, σ,π) =(0.018, 0.036, 0.43). Given these values, the nature of the test is to search for parameters ψ (0, 10) and β (0, 1) such that the model s average risk free rate and equity risk premium matches the data. 8

9 Result: Set of admissable average equity premia and average risk free rate not even close to matching the 1% risk free return and the 6% premium. See Figure 4. 9