Asset Pricing under Information-processing Constraints

Transcription

1 Asset Pricing under Information-processing Constraints YuleiLuo University of Hong Kong Eric.Young University of Virginia November 2007 Abstract This paper studies the implications of limited information-processing capacity (also called rational inattention ) for asset pricing in a linear-quadratic permanent income model. It is shown that rational inattention lowers asset prices and raises expected excess returns by altering the dynamic responses of consumption to endowment shocks and increasing the volatility of consumption relative to the endowment. JEL Classification Numbers: C6, D8, E2. Keywords: ational Inattention, Asset Pricing, Permanent Income. Corresponding author. School of Economics and Finance, University of Hong Kong, Hong Kong, yluo@econ.hku.hk. DepartmentofEconomics,UniversityofVirginia,Charlottesville,VA22904, ey2d@virginia.edu.

2 . Introduction The rational expectations hypothesis assumes that agents are endowed with infinite informationprocessing capacity, thereby allowing them to respond immediately and completely to changes in the economy. However, ordinary individuals in reality do not seem to have unlimited mental capacity, as mounting evidence suggests that they do not respond swiftly or thoroughly to all available information. Sims(2003) proposes rational inattention(i) to capture this fact by assuming that agents only have finite processing capacity about the information on the state of the economy. He shows that I can introduce realistic features such as sluggishness, randomness, and delays into the responses of economic variables to shocks. This paper considers a simple rational inattention version of the permanent-income model as studied in Hall (978) and examines the implications of I for the pricing of multi-period securities. The focus of the PIH model is on the relationship between aggregate consumption and aggregateincomeinanenvironmentinwhichoutputcanbestoredintheformofcapitaltosmooth consumption over time. In contrast, the focus of the intertemporal asset pricing model is on the relationship between aggregate consumption and equilibrium asset prices. Following Hansen(987) and Cochrane (chapter 2, 2005), we set up a model that is a combination of Hall s permanent incomemodelandlucas sassetpricingmodel. Themodelherecanalsobeviewedasaparametric version of the intertemporal asset-pricing model as studied by Lucas(978). We use the approach fromsims(2003)andluo(2007)tosolvethemodelwithiexplicitlyandthenexplorehowi affects asset prices within this framework. Following Cochrane(2005), we decompose the price of a risky asset into a risk-neutral perpetuity and a risk adjustment component. I alters asset pricing by increasing the size of the risk adjustment relative to the risk-neutral component, leading to a decline in the price of the asset(a riseintheriskpremiumrequiredtoholdthatasset). Thisdeclineisdrivenbytwokeyeffects I increases the volatility of the(perceived) movements in consumption and introduces persistence into consumption growth. The size of the price decline is negatively related to the channel capacity of the agents; agents with low channel capacity will require large premia to hold risky assets relative to those with unlimited capacity.

3 2.TheModel The model is a simplified version of Hansen (987) s model, in which Hall s permanent income model and Lucas s asset pricing model are combined to examine the asset pricing implications of exogenous endowment shocks. In this section, we first derive the expression of(optimal) aggregate consumption in terms of the state variables by solving an otherwise standard PIH model with I; we then price assets by treating the process of aggregate consumption that solves the I- PIH model as though it were an endowment process. Because we adopt the representative agent setup, equilibrium prices are shadow prices that leave the representative agent content with that endowment process. 2.. The Standard Permanent-Income Model A standard rational expectations(e) version of the PIH model can be formulated as follows max {c t} E 0 [ ] β t u(c t ) t0 (2.) subject to k t+ k t +e t c t, (2.2) where u(c t ) 2 (c t c) 2 is the utility function, c is the bliss point, Equation (2.2) represents a linear production technology, c t is consumption, k t is capital, e t is exogenous endowment with Gaussianwhitenoiseinnovations,β isthediscountfactor,andistherateofreturnoncapital. Let β ; then this specification implies that optimal consumption is determined by permanent income: c t ( )s t (2.3) where s t k t + j0 ( ) j E t [e t+j] (2.4) is the expected present value of lifetime resources, consisting of physical capital plus human wealth. As noted in Cochrane (chapter 2, 2005), it is not a partial equilibrium result it is a general equilibrium model with a linear production technology and an endowment process. As shown in Luo(2007),theabovePIHmodelcanbereducedtotheunivariatemodelwithiidinnovationsto permanentincomes t thatcanbesolvedinclosed-formafterintroducingi. Specifically, ifs t is 2

4 definedasanewstatevariable,wecanrewritetheevolutionequationofs t as s t+ s t c t +ζ t+ (2.5) wherethet+innovationζ t+ is ζ t+ jt+ ( ) j (t+) (E t+ E t )[e j ]. (2.6) This reduction is critically important for the I model because multi-dimensional I problems do notremainwithinthelinear-quadratic-gaussianclassthatcanbesolvedanalytically. Under the E hypothesis, consumption growth can be written as ( c t ( ) ) j (E t E t ) e t+j ( )ζ t, j0 which relates the innovations to consumption to endowment shocks Optimal Consumption under I Following Sims (2003) and Luo (2007), the consumer s information-processing constraint can be characterized by the equation H(s t+ I t ) H(s t+ I t+ )κ, (2.7) where I t is the consumer s currently processed information, κ is the consumer s channel capacity, H(s t+ I t ) denotes the entropy of the state prior to observing the new signal at t+, and H(s t+ I t+ ) is the entropy after observing the new signal. The concept of entropy is from information theory, and it characterizes the uncertainty in a random variable. Hence,(2.7) means that the reduction in the uncertainty about the state variable gained from observing a new signal is bounded by κ. As shown in Sims (2003), D t is a normaldistribution N ( ŝ t,σ 2 t) ; as a result, One should interpret the operator (E t+ E t) as generating the difference between an expectation taken with informationattimet+andthattakenwithinformationattimet. LuoandYoung(2007)containsadiscussionof the problems multi-state I models present. 3

5 (2.7)canbereducedto log ψ 2 t log σ 2 t+ 2κ (2.8) whereσ 2 t+ var[s t+ I t+ ]andψ 2 t var[s t+ I t ]aretheposteriorvarianceandpriorvariance ofthestatevariable,respectively. 2 Itisstraightforwardtoshowthatintheunivariatecase(2.8)hasasteadystateσ 2 ω 2 ζ exp(2k) 2. In this steady state the consumer behaves as if observing a noisy measurement of permanent income s t+ s t++ξ t+, where ξ t+ is the endogenous noise with variance λ 2 t var [ ξ t+ I t ] determined by the usual updating formula of the variance of a Gaussian distribution based on a linear observation: σ 2 t+ψ 2 t ψ 2 t ( ψ 2 t +λ 2 ) ψ 2 t t. (2.9) Note that in the steady state σ 2 ψ 2 ψ 2( ψ 2 +λ 2) ψ 2, which can be solved to obtain λ 2 [ (σ 2 ) ( ψ 2 ) ]. AsshowninLuo(2007),theconsumptionfunctionunderIis c t ( )ŝ t, (2.0) wheretheconditionalmeanŝ t evolvesaccordingtothekalmanfilterequation ŝ t+ ( )ŝ t +(s t+ +ξ t+ ). (2.) σ 2 /λ 2 /exp(2κ) [0,] is the constant optimal weight on any new observation. Straightforward calculations imply that [( c t ( ) ζ t ( )L 3. Equilibrium Asset Prices under I ) ( + ξ t )] ξ t. (2.2) ( )L Notethatinthemodelisthereturnontechnologyandisnotyettheinterestrate(theequilibrium rate of return on one-period claims to consumption). As proposed in Cochrane(chapter 2, 2005), we first find optimal consumption and then price one-period claims from the equilibrium consumption 2 Note thathere we use thefact thatthe entropy of a Gaussian random variableis equal to half of its logarithm variance plus some constant term. 4

6 stream. Denotingtheriskfreerateby f,wehavethefollowingeulerequation: f E t β u (c t+ ) u (c t ) c ct+ βe t β, wheree t [ ]istheconsumer sexpectationoperatorconditionalonhisprocessedinformationattime t. Wecannowusethebasicpricingequation,pE[mx], 3 tocomputethepriceofthestreamof aggregate consumption (it is treated as the stream of endowments) as p t E t (m t,t+j c t+j ) (3.) j ( β jcc t c 2 t var ) t[c t+j ] j c t Ξ, where ( Ξ β j var t [c t+j ] ) (3.2) j andweusethefactsthat m t,t+j β ju (c t+j ) u (c t ) E t c 2 t+j var t [c t+j ]+c 2 t. β j+j, Denotingtherisk-neutralcomponentoftheassetpricebyp rn t and the risk-adjusted component by p rc t,wehave p rn t c t (3.3) and p rc t ( β j var t [c t+j ] ). (3.4) j Expressions (3.) yield the following implications. The first term in (3.) is the risk-neutral component, thevalueofaperpetuitypayingc t. Thesecondtermisarisk-adjustedcomponent, 3 As argued in Cochrane (2005), this equation tells us only what the price should be given the joint distribution of consumption (thediscountfactor)andtheasset payoff. We know E[mx]aftersolving the PIH modelgiven the statevariablesandcanusethemtodeterminetheassetpricep. 5

7 which lowers the asset price relative to the risk-neutral level because c t c. The following proposition states our key result. Proposition. Therisk-neutralcomponentp rn t oftheassetpriceisindependentofthedegreeof inattention,whiletherisk-adjustedcomponentintheassetpricep rc t increaseswiththedegreeof inattention. The ratio of the risk-adjusted component under I to that under E is Proof. Expression(2.2) can be rewritten as ( F prc t (<) p rc t () +2 ) 2 ( ). (3.5) c t+ ρ c c t +( ) ( ε t+ +ξ t+ ξ t ), (3.6) whereρ c ( ). Thevarianceofconsumptiongrowthis σ 2 c var[ c t] Substituting(3.6)into j( β j var t [c t+j ] ) yields Ξ ( β j var t [c t+j ] ) j ( β j var t [c t+j c t ] ) j 2 ( ) ω2 ζ. (3.7) βσ 2 c +β2( 2σ 2 c +2cov[ c t+2, c t+ ] ) +β 3 3σ 2 c+2cov[ c t+2, c t+ ] + +2cov[ c t+3, c t+ ]+2cov[ c t+2, c t+3 ] ( jβ j ) σ 2 c++2β 2[ ρ c +β ( 2ρ c +ρ 2 ) c +β 2 ( 3ρ c +2ρ 2 c+ρ 3 ) c + ] σ 2 c j ( jβ j ) +2 βρ c β βρ j c ( β) 2 ( +2 βρ ) c β βρ c ( β) 2σ2 c ( +2 ) ( β ( β) 2( )2 ( )( 2 2 ( ) σ 2 c 2 ( ) ω2 ζ ) ) ω 2 ζ, (3.8) 6

8 whereweusethefactthatβ.if,ξω 2 ζ. Hence, F 2 2 ( ). Expression (3.8) clearly shows that I reduces the asset price by two channels: First, I increasesthevolatilityofconsumptiongrowthσ 2 c;second,iintroducespersistenceintoconsumption growthρ c. 4 Furthermore,giventheexpressionofσ 2 c,itisobviousthatthehigherthevolatilityof theinnovationtopermanentincomeω 2 ζ,thelowertheprice. Notethatassetpricesarealsoaffected byboththepersistenceandvolatilityofthefundamentalendowmentshockssinceω 2 ζ dependson them. We will discuss this in subsection 5. It is immediate that we obtain the following corollary (underthemildparameterrestriction> 2 2 ). 5 Corollary 2. F is decreasing and convex in and increasing and convex in. The cross-partial is negative. Proof. By direct calculation, F 2 + ( 2 2 <0 (3.9) ( )) 2 F 2 2 ( 2 + ) 2 ( 2 3 >0 (3.0) ( )) F 2( 2)( ) ( 2 2 <0 (3.) ( )) 2 F 2 2( 2)( ) +32 ( ) ( 2 3 >0 (3.2) ( )) 2 F ( ) ( 2 ( )) 3 <0. (3.3) and 4 Bydirectcalculation σ 2 c 2 ( 2 ( )) 2 >0 ρ c 0 iff. 5 This restriction is needed in many linear-quadratic-gaussian I models to avoid anomalous results as channel capacity converges to 0; see Luo(2007). 7

9 Figure plots the relationship between the degree of attention and F; for example, when 0.67and.0weobtainF 2,meaningIdoublestherisk-adjustedcomponentofasset prices(sincef if). 6 Oneimplicationof(3.9)isthatthepriceoftheassetisincreasing with in the representative consumer context; that is, the equity premium is a decreasing function of channel capacity. Furthermore, the return on the linear technology also has effects on asset prices,asseeninfigure;theeffectofisstrongerwhenissmall. ThelowerpriceintheI economy reflects both the aggregate (macroeconomic) risk and the induced noise due to limited information-processing ability. 4. Equilibrium Asset eturns under I In this section we examine the implications of I for asset returns. Given(3.),(3.8), and(3.5), the expected asset return can be written as pt+ +c t+ E t [ t,t+ ] E t p t [ E t + c t+ + Ξ c t c c t Ξ E t [ c t+ Ξ+ Ξ + Ξ ] c t Ξ +E t c t+ c c t Ξ where we use the fact that c t E t [c t+ ]. To evaluate E t [ ] Ξ, (4.) c t+ c function c t+ c aroundtheconditionalexpectationofc t+,e t [c t+ ],toobtain ] we approximate the concave c t+ c E t [c t+ ] c (E t[c t+ ] c) 2 (c t+ E t [c t+ ])+(E t [c t+ ] c) 3 (c t+ E t [c t+ ]) 2 ; 6 Wechoosethisnumericalparametrizationbecauseitisconsistentwithapercentannualrisk-freerateandLuo and Young(2007) found that 0.67 implies a relative volatility of consumption to income roughly consistent with US data. 8

10 takingtheconditionalexpectatione t [ ]ofbothsidesgives E t c t+ c E t [c t+ ] c +(E t[c t+ ] c) 3 var t [ c t+ ] c t c +(c t c) 3 var t [ c t+ ] c t c, (4.2) sincec t c. Usingthisresult,wecanestablishanupperboundfortheexpectedreturnas E t [ t,t+ ] + ( ) 2 Ξ c t ( ) ( )Ξ t,t+. (4.3) Clearly, t,t+ isincreasingwiththedegreeofinattentionbecause Ξ F ω2 ζ <0. Theexpectedreturnontheassetisgivenby E t [ t,t+ ] + E t + c t Ξ Ξ + ( )2 ( )(c t c) 2 σ 2 c Ξ (4.4) c t ( ) ( )Ξ 2 + ( ) 2 Ξ ( ) 2 Ξ, (4.5) c t ( )/( ) Ξ whereweusethefactthat σ 2 c 2 ( β) 2 Ξ. β Expression(4.5) shows that the expected excess return is increasing with the degree of inattention: E t [ t,t+ ] <0. (4.6) ( ) because Ξ < 0 and 2 < 0. Hence, I can drive up the expected excess return of the asset by reducing the asset price. Note that(4.4) implies that I raises the expected excess return, E t [ t,t+ ],holdingandσ 2 c constant. 5. Applications to Some Particular Income Processes Inthissectionweprovideexplicitformulaefortwoincomeprocesses astationarya()anda difference-stationary A() process. 9

11 5.. A() Endowment Process Suppose that the endowment follows an A() process in levels: e t+ eρ (e t e)+ε t+. (5.) (2.6) implies that In this case, ζ t ρ ε t. (5.2) ΞF ( ρ ) 2ω2 ε, (5.3) which means that Ξ increases with ρ and ω 2 ε. In other words, I can amplify the impact of endowment shocks on the risk-adjusted component in asset prices and expected returns, and thus drive down asset prices and drive up expected returns Difference-stationary Endowment Process Suppose that the endowment follows a difference-stationary process(an A() in differences): e t+ e t ρ 2 (e t e t )+ε t+. (5.4) (2.6) implies that In this case, ζ t ΞF ( ρ 2 )( ) ε t. (5.5) 3 ( ρ 2 ) 2 ( ) 2ω2 ε, (5.6) whichmeansthatξincreaseswithρ 2 andω 2 ε. 6. Conclusion In this paper we examine the implications of limited information-processing capacity rational inattention in the sense of Sims (2003) for asset prices in an otherwise standard permanentincome model. We find that I raises the risk-adjusted component of the asset price by increasing both the relative volatility and persistence of consumption growth. 0

12 eferences [] Campbell, John(2003), Consumption-Based Asset Pricing, in Constantinides, George, Milton Harris, and ené Stultz, eds., Handbook of the Economics of Finance Volume B, Amsterdam: North-Holland, pp [2] Cochrane, John(2005), Asset Pricing, Princeton University Press. [3] Hansen, Lars(987), Calculating Asset Prices in Three Example Economies, in Truman F. Bewley ed., Advances in Econometrics, Fifth World Congress, Cambridge University Press. [4] Hall, obert E.(978), Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence, Journal of Political Economy 86(6), pp [5] Lucas, obert Jr. (978), Asset Prices in an Exchange Economy, Econometrica 46(6), pp [6] Luo, Yulei(2007), Consumption Dynamics Under Information Processing Constraints, forthcoming in eview of Economic Dynamics. [7] Luo, Yulei and Eric. Young (2007), ational Inattention and Aggregate Fluctuations, working paper. [8] Mehra, ajnish and Prescott, Edward (985), The Equity Premium: A Puzzle, Journal of Monetary Economics 5, [9] Sims, Christopher A.(2003), Implications of ational Inattention, Journal of Monetary Economics 50(3), pp

13 The Effects of I for Asset Prices Figure 2