Macro I & II – 2011-2012 – Lecture 3

Asset Pricing in General Equilibrium

Franck Portierfportier@cict.fr

Toulouse School of Economics

Version 1.116/01/2012

Changes from version 1.0 are in red

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Disclaimer

These are the slides I am using in class. They are notself-contained, do not always constitute original material and docontain some “cut and paste” pieces from various sources that Iam not always explicitly referring to (not on purpose but because ittakes time). Therefore, they are not intended to be used outside ofthe course or to be distributed. Thank you for signalling me typosor mistakes at fportier@cict.fr.

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0.Introduction

• In this lecture, I want to

1. show how to compute asset prices in general equilibrium

2. discuss of the some quantitative properties of asset prices in(simple) GE models

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1. Asset Prices in General Equilibrium with Endowments

• I describe here the competitive equilibrium of a pure exchangeinfinite horizon economy with stochastic Markov endowments.• This is a basic setting for studying risk sharing, asset pricing,consumption.• 2 different market structures

1. Arrow-Debreu structure with complete markets in datedcontingent claims all traded at period 0

2. recursive structure with complete one-period Arrow securities.

• The 2 have different asset structures but identical consumptionallocations

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1.1. Preferences and endowments

• π(s ′|s) is a Markov chain with initial distribution π0(s)• Prob(st+1 = s ′|st = s) = π(s ′|s) and Prob(s0 = s) = πo(s)• a sequence of probability measures π(st) on historiesst = [st , st−1, ..., s0] is given by

π(st) = π(st |st−1)π(st−1|st−2)...π(s1|s0)π0(s0) (1)

and conditional probability is given by

π(st |s0) = π(st |st−1)π(st−1|st−2)...π(s1|s0) (2)

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1.1. Preferences and endowments

• Trading occurs after s0 has been observed.• the probability of state (history) st conditional on being in state(history) sτ at date τ is

π(st |sτ ) = π(st |st−1)π(st−1|st−2)...π(sτ+1|sτ ) (3)

• (because of Markov property, π(st |sτ ) does not depend onhistory sτ−1

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1.1. Preferences and endowments

• Households : i = 1, ..., I . Each owns a stochastic endowment ofone good y i

t = y i (st), and st is publicly observable• Each household purchase a history-dependant consumption planc i = c i

t(st)∞t=0

• Household objective

U(c i ) =∞∑t=0

∑st

βtu[c it(st)]π(st |s0) = E0

∞∑t=0

βtu[c it(st)] (4)

• u has all nice properties, including limc↓0 u′(c) = +∞

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1.2. Complete markets

• Household trade dated state-contingent claims to consumption• q0

t (st) = price of a claim on time-t consumption, contingent onhistory st , in terms of a numeraire not specified• the BC is

∞∑t=0

∑st

q0t (st)c i

t(st) =∞∑t=0

∑st

q0t (st)y i (st) (5)

• Hh problem : choose c i to maximize (4) s.t. (5)• Notice that one can collapse the problem into a problem with asingle budget constraint because of complete markets

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1.2. Complete markets

• let µi be the Lagrange multiplier of this constraint, FOC :

∂U(c i )

∂c it(st)

= µiq0t (st) (6)

and with the specification (4) of preferences,

∂U(c i )

∂c it(st)

= βtu′[c it(st)]π(st |s0) (7)

one getsβtu′[c i

t(st)]π(st |s0) = µiq0t (st) (8)

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1.2. Complete markets

DefinitionA price system is a sequence of functions q0

t (st)∞t=0. Anallocation is a list of sequences of functions c i

t(st)∞t=0, one foreach i . A feasible allocation satisfies∑

i

y i (st) ≥∑i

c it(st) (9)

DefinitionA competitive equilibrium is a feasible allocation and pricesystem such that the allocation solves each household problem

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1.2. Complete markets

• Notice that (8) implies

u′[c it(st)]

u′[c jt (st)]

=µi

µj(10)

which means thats ratios of marginal utilities between pairs ofagents are constant across all sates and dates.• An equilibrium allocation solves (10), (9),and (5). Note that (10)implies

c it(st) = u′−1

u′[c1

t (st)]µi

µ1

(11)

and substituting into feasibility condition (9) at equality gives

∑i

u′−1

u′[c1

t (st)]µi

µ1

=∑i

y i (st) (12)

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1.2. Complete markets

∑i

u′−1

u′[c1

t (st)]µi

µ1

=∑i

y i (st) (12)

• the RHS of (12) does not depend on the entire history st , butonly on current state st , therefore the LHS, therefore c1

t (st). Then,from (11), it is also the case for all c i

t(st). One then has thefollowing proposition

Proposition

The competitive equilibrium allocation is not history dependent ;c it(st) = c i (st)

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1.3. Equilibrium pricing function

• Let c i , i = 1, ...I b an equilibrium allocation. Then (6) or (8)gives the price system q0

t (st) as a function of the allocation to Hhi , for any i .• The price system is a stochastic process• Because the units of the price system are arbitrary, one cannormalized one of the multipliers at any positive value. I setµ1 = u′[c1(s0)], so that q0

0(s0) = 1, i.e. the price system is in unitsof time-0 goods. (one has therefore µi = u′[c i (s0)] for all i)

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1.4. Example : Risk sharing

• suppose u(c) = (1− γ)−1c1−γ , γ > 0 (CRRA). Then (10)implies

c it = c j

t

(µi

µj

)− 1γ

(13)

• time-t elements of consumption allocations to distinct agents areconstant fractions of one another.• The individual consumption is perfectly correlated with theaggregate endowment or aggregate consumption.• The fractions assigned to each individual are independent of therealization of st .• There is extensive cross-time cross-state consumption smoothing.

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1.4. Asset pricingPricing Redundant Assets

• Let d(st)∞t=0 be a stream of claims on time t, state st

consumption, where d(st) is a measurable function of st . The priceof an asset entitling the owner to this stream must be

a00 =

∞∑t=0

∑st

q0t (st)d(st) (14)

(this can be understood as an arbitrage equation)

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1.4. Asset pricingRiskless Consol

• A riskless consol offers for sure one unit of consumption at eachperiod, i.e. dt(st) = 1 for all t and st . The price is

a00 =

∞∑t=0

∑st

q0t (st)

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1.4. Asset pricingRiskless strips

• Consider a sequence of strips of returns on the riskless consol.The time-t strip is the return process dτ = 1 if τ = t ≥ 0, and 0otherwise. The price of time-t strip at 0 is∑

st

q0t (st)

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1.4. Asset pricingTail assets

• Consider the stream of dividends d(st)t≥0

• For τ ≥ 1, suppose that we strip off the first τ − 1 periods of thedividend and want to get the time-0 value of the dividend streamd(st)t≥τ .• Let a0

τ (sτ ) be the time-0 price of an asset that entitles thedividend stream d(st)t≥τ if history sτ is realized :

a0τ (sτ ) =

∑t≥τ

∑st :sτ=sτ

q0t (st)d(st) (15)

• Let us convert this price into units of time τ , state sτ by dividingby q0

τ (sτ ) :

aττ (sτ ) =a0τ (sτ )

q0τ (sτ )

=∑t≥τ

∑st :sτ=sτ

q0t (st)

q0τ (sτ )

d(st) (16)

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1.4. Asset pricingTail assets

• Notice that for all consumers i

qτt (st) = q0t (st)

q0τ (sτ )

= βtu′[c it(st)]π(st)

βτu′[c iτ (sτ )]π(sτ )

= βt−τ u′[c it(st)]

u′[c iτ (sτ )]π(st |sτ )

(17)

• Here qτt (st) is the price of one unit of consumption delivered attime t, state st in terms of the date-τ , state-sτ consumption good.• The price at t for the tail asset is

aττ (sτ ) =∑t≥τ

∑st :sτ=sτ

qτt (st)d(st) (18)

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1.4. Asset pricingTail assets

• This tail asset formula is useful if one wants to create in a modela time series of equity prices : an equity purchased at time τentitles the owner to the dividends from time τ forward, and theprice is given by (18).• Note : The relative price is (17) is not history dependent, givenProposition 1. This is stated in the following proposition :

Proposition

The equilibrium price of date-t ≥ 0, state-st consumption goodexpressed in terms of date τ (0 ≤ τ ≤ t), state sτ consumptiongood is not history dependent : qτt (st) = qj

t(sk) for j , k ≥ 0 suchthat t − τ = k − j and [st , st−1, . . . , sτ ] = [sk , sk−1, . . . , sj ].

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1.4. Asset pricingPricing One Period Returns

• The one-period version of equation (17) is

qττ+1(sτ+1) = βu′(c i

τ+1)

u′(c iτ )

π(sτ+1|sτ )

• The RHS is the one-period pricing kernel at time τ .• The price at time τ in state sτ of a claim to a random payoffω(sτ+1) is given, using the pricing kernel, by

pττ (sτ ) =∑

sτ+1qττ+1(sτ+1)ω(sτ+1)

= Eτ[β u′(cτ+1)

u′(cτ ) ω(sτ+1)] (19)

where superscripts i and dependence to sτ have been deleted.

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1.4. Asset pricingPricing One Period Returns

• Let denote the one-period gross return on the asset byRτ+1 = ω(sτ+1)/pττ (sτ ). Then, for any asset, equation (19) implies

1 = Eτ

[β

u′(cτ+1)

u′(cτ )Rτ+1

](20)

• The term mτ+1 = β u′(cτ+1)u′(cτ ) is a stochastic discount factor.

Equation (20) can be understood as a restriction on theconditional moments of returns and mτ+1.• Applying the law of iterated expectations to equation (20), onegets the unconditional moments restrection :

1 = E

[β

u′(cτ+1)

u′(cτ )Rτ+1

](21)

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1.5. A Recursive Formulation : Arrow Securities

• One introduce another market structure that preserves theequilibrium allocation from our competitive equilibrium. Thissetting also preserves the one-period asset-pricing formula (19).• Arrow (1964) : one-period securities are enough to implementcomplete markets, provided that new one-period markets arereopened for trading each period• See Ljundqvist and Sargent for a formal proof

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2. Quantitative Model : Mehra & Prescott2.1. Data

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2. Quantitative Model : Mehra & Prescott2.1. Data

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2. Quantitative Model : Mehra & Prescott2.1. Data

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2. Quantitative Model : Mehra & Prescott2.1. Data

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2. Quantitative Model : Mehra & Prescott2.1. Data

• The risk premium is high (6.18 %), as the s.d. of real returns is5.67% for riskless asset and 16.54% for risky asset.• Mehra and Prescott have proposed a relatively simpleendowment economy to quantitatively reproduce this fact.

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2.2. A Pure Exchange EconomyEnvironment

• representative agent, E0∑∞

t=0 βtu(ct) ; 0 < β < 1,

u(c ;α) =c1−α − 1

1− α

• One productive unit (a tree) gives yt units of a perishablegood. This tree is an equity share that is competitively traded, andyt is its dividend.• the growth rate of yt is stochastic : yt+1 = xt+1yt• x is markov : xt+1 ∈ λ1, ..., λn, Prob(xt+1 = λj |xt = λi ) = φij .Is is assumed that this markov chain is ergodic, and that λi > 0,y0 > 0.• yt is observed at the beginning of the period and securities aretraded ex-dividend.

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2.2. A Pure Exchange EconomyEquilibrium

Proposition

Define A = [aij ], aij = βφijλ1−αj and assume limm−→∞ Am = 0.

Then, a Debreu competitive equilibrium exists.

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2.2. A Pure Exchange EconomyPricing

• In this economy, the ex dividend price of a security withdividends dt is

Pt = Et

[ ∞∑s=t+1

βs−tu′(ys)

u′(yt)ds

]• For the equity, given the functional forms and d = y ,

Pet = Pe(yt , xt) = Et

[ ∞∑s=t+1

βs−tyαtyαs

ys

]• (yt , xt) is a sufficient description of the past history. It definesthe state of the economy.

Pet = Et

β

u′(yt+1)

u′(yt)(Pe

t+1 + yt+1)

• Given that ys = yt × xt+1 × xt+2 · · · × xs , Pe

t is homogenous ofdegree 1 in yt , which is the current endowment of consumptiongood.

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2.2. A Pure Exchange EconomyPricing

• Given that equilibrium values of the economy are time invariantfunctions of (yt , xt), the subscript t can be dropped. The state canbe written as (c, i), where yt = c and xt = λi .• With these notations, the price of an equity satisfies

Pe(c , i) = β∑n

j=1 φij︸︷︷︸ (λjc)−α︸ ︷︷ ︸ [Pe(λjc , j) + cλj ]︸ ︷︷ ︸ cα︸︷︷︸i ii iii iv

withi : probability of state j knowing iii : marginal utility of tomorrow consumption

in state jiii : tomorrow price + dividend in state jiv : inverse of marginal utility of today consumption

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2.2. A Pure Exchange EconomyPricing

• Given that Pe is homogenous of degree 1 in c , we can write

Pe(c , i) = wic

where wi is a constant. Then the pricing equation becomes

wi = β

n∑j=1

φijλ(1−α)j (wj + 1) ∀ i = 1, ..., n

• This is a system of n linear equations in n unknowns (the wi ) this has a unique positive solution when a competitive equilibriumexists we can derive prices

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2.2. A Pure Exchange EconomyPrices

• The return of an equity if current state is c , i and next state j is

r eij =Pe(λjc, j) + λjc − Pe(c , i)

Pe(c , i)=λj(wj + 1)

wi− 1

and the equity expected return is, conditional on state i :

Rei =

n∑j=1

φij reij

• Let us also consider a riskless security that pays 1 unit of good ineach state, i.e. di = 1 ∀i . The price P f of this asset is

P fi = P f (c , i) = β

n∑j=1

φiju′(λjc)

u′(c)dj = β

n∑j=1

φijλ−αj

and R fi = 1/P f

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2.2. A Pure Exchange EconomyPrices

• Now we can compute expected returns w.r.t. the stationarydistribution• Let π ∈ Rn be the vector of stationary probabilities of themarkov chain : π is such that

π = φ′π

with∑n

i=1 πi = 1 and φ = φij• Then we define the expected returns as

Re =∑n

i=1 πiRei

R f =∑n

i=1 πiRfi

and the risk premium is given by Re − R f .

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2.3. Results

• preference parameters : α and β• Technology parameters φij, λi : it is assumed that λ takestwo values : λ1 = 1 + µ+ δ and λ2 = 1 + µ− δ, andφ11 = φ22 = φ, φ12 = φ21 = 1− φ.• For US data over 1889-1978, consumption growth = .018,consumption growth s.d. = .036, consumption growth serialcorrelation = -.14 µ = .018, δ = .036, φ = .43• Then, we search for (α, β) so that the average risk free rate andthe equity risk premium are reproduced.• α = people’s willingness to substitute consumption betweensuccessive yearly time period not greater that 10.• β ∈]0, 1[

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2.3. Results

• The model cannot reproduce a equity premium of more that.35%, while it is 6 in the data (given that the risk free rate is .8%)

• There is therefore an Equity Premium Puzzle• A large literature has been devoted to this question• See Kocherlakota 1996 for a nice survey• Here I present a quantitative solution of this (quantitative)puzzle, as proposed by Jerman 1998

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2.4. A Possible Resolution of the PuzzleThe Model

• Jerman (1998, JME) proposed a model with production, capital,habit formation and K adjustment costs that is quantitativelysatisfactory.• Firms

max Et

∞∑k=0

βkΛt+k

Λt[At+kF (Kt+k ,Xt+kNt+k)− wt+kNt+k − It+k ]

where βk Λt+k

Λtis the MRS of the owners of the firm.

Kt+1 = (1− δ)Kt + φ

(ItKt

)Kt

• φ(·) < 1 is a positive concave function that models adjustmentcosts on capital the shadow price of one installed unit of capital,q, differs from the price of one new unit of capital (Tobin’s q)• Firms are financed by retained earnings, and dividends are givenby

Dt = AtF (Kt ,XtNt)− wtNt − It38 / 44

2.4. A Possible Resolution of the PuzzleThe Model

• Hh

max Et

∞∑k=0

βku(ct+k)

s.t. wtNt + a′t(V at + Da

t ) ≥ Ct + a′t+1V at (Λt)

• at is a vector of financial assets held at t and chosen at t − 1.this vector contains the representative firm, + possibly otherassets. V a is the vector of asset prices and Da the vector ofdividends payments.• The Hh also face a time constraint : Nt + Lt = 1, and we assumehabit persistence : u = u(ct − αct−1)• Market equilibrium : AtF (Kt ,XtNt) = Ct + It .• Shocks are to the technology A

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2.4. A Possible Resolution of the PuzzleModel Solution

• If we use a log-linear approximation to solve the model, theexpected returns will be the same for all agents no possibility toaccount for the risk premium• The model is solved by log-linearization, but asset prices arecomputed in a second round using lognormal pricing formulas (seeHansen & Singleton, 1983)• The model solution can be written

st = Mst−1 + εt (22)

where ε could be a multivariate normal iid shock (In this model, itis univariate).• Then we use basic asset pricing formula : a claim on futurepayment Dt+k(st+k) has a value

Vt(st) = βkEt

[Λt+k(st+kDt+k(st+k)

Λt(st)

](23)

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2.4. A Possible Resolution of the PuzzleModel Solution

• If Λ and D are lognormal, with distribution given by (22), thenthe risk free rate ca be computed from (23) with k = 1 andD(st+1) = 1 :

E (Rt,t+1(st)) =γβ? exp

12 (var(Etλt+1 − λt)− var(λt+1 − Etλt+1)

where λt = Λt

Xt, Xt = γXt−1, β? = βγt

• The return on equity is given by

Rdt,t+1(st , st+1) =

Vt+1(st+1) + Dt+1(st+1)

Vt(st)

• Jerman uses simulations to find the unconditional mean.

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2.4. A Possible Resolution of the PuzzleQuantitative predictions

• easy part :

1. long run restrictions : Cobb-Douglas elasticity on labor = .64,γ = 1.005 (per quarter), δ = .025

2. Productivity shocks : A follows a AR(1) process, withpersistence .95 or 1, with s.d. of the innovation which is suchthat postwar US gdp s.d. is reproduced

3. Risk aversion :

c1−τ

1− ττ = 5

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2.4. A Possible Resolution of the PuzzleCalibration

• Difficult part :

1. habit formation parameter α2. K adjustment cost elasticity ξ3. time preference β4. shock persistence ρ

• Let θ1 = [α, β, ξ, ρ]′. θ1 is chosen (estimated) to minimize

F = (θ2 − f (θ1))′Ω(θ2 − f (θ1))

where θ2 is a vector of moments to match, f (θ1) is the vector ofcorresponding moments generated by the model and Ω a weightingmatrix. (Simulated Method of Moments)• θ2 =(s.d. of c growth/s.d. of y growth, s.d. of i growth/s.d. ofy , mean risk free rate, equity premium), Ω is identity• The solution is that F = .00001 with

α ξ β? ρ.82 .23 .99 .99

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2.4. A Possible Resolution of the PuzzleQuantitative predictions

• The simulation results are

Table: Simulation results, Jerman 1998

σ∆cσ∆y

σ∆iσ∆y

E (r f ) E (r e − r f )

Data .51 2.65 .80 6.18Standard RBC .77 1.54 4.26 .02Standard RBC + τ = 10 .78 1.53 3.36 .26Habit persistence .33 3 4.2 .03K adj. costs 1.14 .68 3.91 .67Benchmark .51 2.65 .8 6.18

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